Discrete gravitational
approaches to cosmology
Rex Gerry Liu
Trinity College
This thesis is submitted on 1 December 2014 for the degree of Doctor of
Philosophy.

Abstract
Exact solutions to the Einstein field equations are notoriously difficult to find.
Most known solutions describe systems with unrealistically high degrees of sym-
metry. A notable example is the FLRW metric underlying modern cosmology:
the universe is assumed to be perfectly homogeneous and isotropic, but in the
late universe, this is only true on average and only at large scales. Where an
exact solution is not available, discrete gravitational approaches can approximate
the system instead. This thesis investigates several cosmological systems using
two distinct discrete approaches. Closed, flat, and open ‘lattice universes’ are
first considered where matter is distributed as a regular lattice of identical point
masses in constant-time hypersurfaces. Lindquist and Wheeler’s Schwarzschild–
cell method is applied where the lattice cell around each mass is approximated
by a perfectly spherical cell with Schwarzschild space–time inside. The result-
ing dynamics and cosmological redshifts closely resemble those of the dust-filled
FLRW universes, but with certain differences in redshift behaviour attributable
to the lattice universe’s lumpiness. The application of Regge calculus to cos-
mology is considered next. We focus exclusively on the closed models developed
by Collins, Williams, and Brewin. Their approach is first applied to a universe
where an exact solution is already well-established, the vacuum Λ-FLRW model.
The resulting models are found to closely reproduce the dynamics of the contin-
uum model being approximated, though certain constraints on the applicability
of the approach are also uncovered. Then using this knowledge, we next model
the closed lattice universe. The resulting evolution closely resembles that of the
closed dust-filled FLRW universe. Constraints on the placement of the masses
in the Regge skeleton are also uncovered. Finally, a ‘lattice universe’ with one
perturbed mass is modelled. The evolution is still stable and similar to that of
the unperturbed model. The thesis concludes by discussing possible extensions
of our work.

Declaration
This thesis is the result of my own work and includes nothing which is the outcome
of work done in collaboration except as declared in the Acknowledgements and
specified in the text. It is not substantially the same as any that I have submitted,
or, is being concurrently submitted for a degree or diploma or other qualification
at the University of Cambridge or any other university or similar institution
except as declared in the Acknowledgements and specified in the text. I further
state that no substantial part of my thesis has already been submitted, or, is being
concurrently submitted for any such degree, diploma or other qualification at the
University of Cambridge or any other university or similar institution except as
declared in the Acknowledgements and specified in the text. It does not exceed
the prescribed word limit for the relevant Degree Committee.
Rex Gerry Liu
1 December 2014

Acknowledgements
The person I first wish to thank is my research supervisor, Dr Ruth Williams.
The work in Chapter 5 and Chapter 6 were done in collaboration with her, and
Chapter 2 greatly benefited from discussions with her. Yet more importantly,
throughout my PhD, she has always been patient and dedicated in her mentor-
ship, highly accommodating, and generous with her time, her kind support even
extending at times to the pastoral affairs that a PhD student might encounter
from time to time outside his research; as such, she has sometimes acted as a
second Tutor to me, far exceeding the narrow remit of her roˆle as research super-
visor. In many respects, this thesis would most certainly not have been possible
without her help, and for that, I owe her an immense debt of gratitude.
I wish to thank Timothy Clifton and Pedro Ferreira for much helpful discussion
while trying to understand the Lindquist–Wheeler models of Chapter 2; I wish
to thank Leo Brewin for kindly answering some questions regarding his papers
on Regge cosmology; and I wish to thank Adrian Gentle, who, during his visit to
Cambridge, generously spent some of his time helpfully discussing the research
that led to Chapter 5 and Chapter 6.
I wish to thank my PhD thesis examiners Ulrich Sperhake and Timothy Clifton
who, during my viva, provided much-appreciated comments and suggestions that
greatly improved the quality of this thesis. These included informing me of the
relevant cosmological context for my work, making me aware of related research
in the area, and suggesting that I compare my Lindquist–Wheeler redshift results
with the integrated Sachs–Wolfe effect.
The numerical simulations of the Lindquist–Wheeler models in Chapter 2 were
performed on the COSMOS Shared Memory system at DAMTP, University of
Cambridge, operated on behalf of the STFC DiRAC HPC Facility. This equip-
ment is funded by BIS National E-infrastructure capital grant ST/J005673/1 and
STFC grants ST/H008586/1, ST/K00333X/1, and ST/J001341/1.
Additionally, I wish to thank the COSMOS team in general for support in
adapting my C++ code to the COSMOS system and James Briggs in particular
for his kind assistance in parallelising my C++ code with OpenMP and for his
helpful tips in making the code perform faster. I would also like to thank Jed
Liu for helpful advice on a wide range of computational issues, ranging from tips
on various C++ programming issues to typesetting with LaTeX to generating
graphs for the figures in this thesis.
This PhD was financed in part by a bursary from the Cambridge Common-
wealth Trust. Additionally, I was twice granted generous Rouse Ball Travel-
ling Studentships by Trinity College, Cambridge, as well as commensurate travel
funds by DAMTP to attend academic conferences in Beijing, China, and Jena,
Germany.
I wish to thank my friend Sohaib for his encouragement and confidence in me
throughout my PhD, and for reminding me, in my busy life, to always see life
itself as an art.
Finally, and most importantly, I wish to thank my family – 阿公, 阿嬤,
爸爸, 媽媽, Jed, and Richard – for their encouragement, their support, their
guidance, their love. To 阿公, 阿嬤, 爸爸, 媽媽, since my infancy they have
always encouraged the best in me, in every possible way, and they have laid the
foundations that I might, in my own way, strive to be better still. The fruits
of my labours today were, in truth, seeded by them long ago and continuously
nurtured over the many intervening years up to the present. To my brothers,
Jed and Richard, they’ve kept me sane and whole as a person, readily providing
me with much needed humour and encouragement from far away, reminding me
always of that other important part of life beyond books and physics, that part
without which a person could never be truly complete.
For 阿公, 阿嬤, 爸爸, 媽媽, Jed, and Richard: to them do I dedicate the
labours herein.
[古之]欲修其身者，先正其心
欲正其心者，先誠其意
欲誠其意者，先致其知
致知在格物
—《大學》, 孔子(前551年－前479年)
[The Ancients] wishing to cultivate their persons,
first rectified their hearts;
wishing to rectify their hearts,
first sought to be sincere in their thoughts;
wishing to be sincere in their thoughts,
first extended to the utmost their knowledge.
Such extension of knowledge lay in the investigation of things.
— The Great Learning, Confucius (551–479 BC)
ὁ γὰρ τὸ ἐpiίστασθαι δι᾿ αὑτὸ αἱρούμενος τὴν μάλιστα ἐpiιστήμην μάλιστα
αἱρήσεται, τοιαύτη δ᾿ ἐστὶν ἡ τοῦ μάλιστα ἐpiιστητοῦ, μάλιστα δ᾿ ἐpiιστητὰ
τὰ piρῶτα καὶ τὰ αἴτια, διὰ γὰρ ταῦτα καὶ ἐκ τούτων τἆλλα γνωρίζεται
ἀλλ᾿.
— τὰ μετὰ τὰ φυσικά, Α᾿ριστοτέλης (384–322 piΧ)
For he who desires knowledge for its own sake will most desire the
most perfect knowledge, and this is the knowledge of the most know-
able, and the things which are most knowable are first principles and
causes ; for it is through these and from these that other things come
to be known.
— Metaphysics, Aristotle (384–322 BC)

Contents
1 Introduction 21
2 Lattice universes and the Lindquist–Wheeler models 31
2.1 Constructing the LW approximation . . . . . . . . . . . . . . . . 34
2.2 The cosmological scale factor and the Friedmann equations . . . . 42
2.3 Redshifts in the lattice universe . . . . . . . . . . . . . . . . . . . 50
2.4 Propagating photons across cell boundaries . . . . . . . . . . . . . 51
2.5 Numerical implementation of the LW model . . . . . . . . . . . . 55
2.6 Redshifts from lattice universe simulations . . . . . . . . . . . . . 63
2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3 An introduction to Regge calculus 93
3.1 The geometry of Regge calculus . . . . . . . . . . . . . . . . . . . 94
3.2 The Regge action . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.3 Non-simplicial piecewise linear manifolds . . . . . . . . . . . . . . 100
3.4 The Cauchy problem and Regge calculus . . . . . . . . . . . . . . 101
4 Regge calculus of closed FLRW universes 105
4.1 The Collins–Williams Regge skeleton . . . . . . . . . . . . . . . . 107
4.2 The fully-triangulated CW skeleton . . . . . . . . . . . . . . . . . 113
4.3 Global and local Regge equations . . . . . . . . . . . . . . . . . . 116
4.4 Brewin’s children models . . . . . . . . . . . . . . . . . . . . . . . 117
4.4.1 Co-ordinates for a child tetrahedron’s 4-block . . . . . . . 119
4.5 Comparison with the ADM formalism . . . . . . . . . . . . . . . . 122
5 Regge calculus of the closed vacuum Λ-FLRW universe 125
5.1 The parent models . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.1.1 Embedding Cauchy surfaces into a 3-sphere . . . . . . . . 127
5.1.2 Global variation of the parent models . . . . . . . . . . . . 132
5.1.3 Local variation of the parent models . . . . . . . . . . . . 138
5.1.4 Relationship between the global and local Regge equations 144
5.1.5 Initial value equation of the parent models . . . . . . . . . 146
5.1.6 Discussion of the parent models . . . . . . . . . . . . . . . 147
5.2 The children models . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.2.1 The 3-sphere embedding of children Cauchy surfaces . . . 156
5.2.2 Geometric quantities for the children model . . . . . . . . 163
5.2.3 Varying the Regge action . . . . . . . . . . . . . . . . . . . 170
5.2.4 Initial value equation of the children models . . . . . . . . 175
5.2.5 Discussion of the children models . . . . . . . . . . . . . . 178
6 Regge models of closed lattice universes 185
6.1 Regge calculus of closed lattice universes . . . . . . . . . . . . . . 187
6.2 Perturbation of a single mass . . . . . . . . . . . . . . . . . . . . 195
6.2.1 Global and local Regge equations . . . . . . . . . . . . . . 199
6.2.2 Particle trajectories . . . . . . . . . . . . . . . . . . . . . . 202
6.2.3 Geometric quantities for the Regge equation . . . . . . . . 204
6.2.4 Solving the Regge equations . . . . . . . . . . . . . . . . . 211
6.3 Initial value equation for perturbed models . . . . . . . . . . . . . 218
6.4 Discussion of the models . . . . . . . . . . . . . . . . . . . . . . . 222
7 Conclusions and future directions 229
A Regular lattices in 3-spaces of constant curvature 235
B Radial velocities in Regge Schwarzschild space–time 239
C The Schwarzschild Regge block’s radial length 247
D Boundary conditions for Schwarzschild-cells of unequal masses 251
E The average radius of a CW Cauchy surface 255
F Constraint and evolution equations in Λ-FLRW Regge calculus 257
G Variation of the particle path-lengths 261
Bibliography 282
List of symbols 285
Index 287

List of figures
2.1 Particle at interface of two Schwarzschild-cell boundaries . . . . . 35
2.2 Constant-t hypersurfaces of neighbouring Schwarzschild-cells in-
tersecting at cell boundaries . . . . . . . . . . . . . . . . . . . . . 37
2.3 Constant-τ hypersurfaces of neighbouring Schwarzschild-cells mesh-
ing correctly at cell boundaries . . . . . . . . . . . . . . . . . . . 37
2.4 ‘Gap’ region inside Schwarzschild-cells for closed universes . . . . 39
2.5 Rescaling of a lattice . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6 Evolution of dust-filled FLRW universes . . . . . . . . . . . . . . 43
2.7 Embedding of spherical cells in a 3-sphere . . . . . . . . . . . . . 49
2.8 Embedding of spherical cells in a flat hypersuface . . . . . . . . . 49
2.9 Embedding of spherical cells in a hyperbolic hypersurface . . . . . 49
2.10 Propagation of photons from one cell to the next . . . . . . . . . 52
2.11 Propagation of photons across a lattice . . . . . . . . . . . . . . . 53
2.12 Schwarzschild Regge blocks . . . . . . . . . . . . . . . . . . . . . 56
2.13 Plot of zLW against zFLRW in the rb0 = 3× 104R flat LW universe 66
2.14 Plot of redshifts z against robs in the flat universe . . . . . . . . . 67
2.15 Plot of redshifts z against robs for a photon starting later along the
θ0 =
1
2
pi trajectory in Figure 2.14 . . . . . . . . . . . . . . . . . . 67
2.16 Plot of photon frequencies νLW against robs within single cells for
the θ0 =
1
2
pi trajectory . . . . . . . . . . . . . . . . . . . . . . . . 69
2.17 Plot of photon frequencies νLW against robs within single cells for
the θ29 =
311
3000
pi trajectory . . . . . . . . . . . . . . . . . . . . . . 71
2.18 Plot of redshifts zLW against robs within a single cell for the θ0 =
1
2
pi
trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.19 Plot of zLW against zFLRW for a photon at large initial radius in the
rb0 = 3× 104R flat universe . . . . . . . . . . . . . . . . . . . . . 73
2.20 Plot of (zLW − zFLRW ) against reference graph zFLRW in the rb0 =
3× 104R flat universe . . . . . . . . . . . . . . . . . . . . . . . . 74
2.21 Plot of zLW against zFLRW in the rb0 = 10
5R flat universe . . . . . 76
2.22 Plot of zLW against zFLRW in the rb0 = 10
6R flat universe . . . . . 76
2.23 Plot of zLW against zFLRW in the rb0 = 10
7R flat universe . . . . . 77
2.24 Plot of zLW against zFLRW in the rb0 = 10
8R flat universe . . . . . 77
2.25 Plot of zLW against zFLRW for a photon at large initial radius in the
rb0 = 10
8R flat universe . . . . . . . . . . . . . . . . . . . . . . . 78
2.26 Plot of (zLW − zFLRW ) against reference graph zFLRW in the rb0 =
108R flat universe . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.27 Plot of the gradients of zLW vs zFLRW graphs against initial direction
of travel in various flat universes . . . . . . . . . . . . . . . . . . . 81
2.28 Plot of zLW against zFLRW in the Eb = 1.1 open universe . . . . . . 82
2.29 Plot of the gradients of zLW vs zFLRW graphs against initial direction
of travel in various open universes . . . . . . . . . . . . . . . . . . 83
2.30 Plot of the zLW vs zFLRW gradients against Eb for a photon initially
in the direction of θ0 =
1
2
pi . . . . . . . . . . . . . . . . . . . . . . 84
2.31 Plot of the zLW vs zFLRW gradients against Eb for a photon initially
in the direction of θ11 =
149
1200
pi . . . . . . . . . . . . . . . . . . . . 84
2.32 Plot of zLW against zFLRW along two entire photon trajectories in
the closed universe . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.33 Plot of zLW against zFLRW along outgoing photon trajectories in
the closed universe . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.34 Plot of zLW against zFLRW along ingoing photon trajectories in the
closed universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.35 Plot of zLW vs zFLRW gradients against initial direction of travel in
a closed universe . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.1 Illustration of a hinge and its attached faces . . . . . . . . . . . . 95
3.2 Conical singularity at a hinge . . . . . . . . . . . . . . . . . . . . 96
3.3 Dihedral angle between faces at a hinge . . . . . . . . . . . . . . . 97
3.4 Two different quadrilaterals with the same edge-lengths . . . . . . 101
4.1 The 4-block of an equilateral tetrahedron . . . . . . . . . . . . . . 110
4.2 Triangulation of a 4-block’s six trapezoidal hinges. . . . . . . . . . 114
4.3 A partially subdivided parent tetrahedron . . . . . . . . . . . . . 118
5.1 Parent CW tetrahedron embedded into a 3-sphere . . . . . . . . . 128
5.2 Plots of the expansion rate of the universe’s volume against the
volume itself for the Λ-FLRW parent models . . . . . . . . . . . . 150
5.3 Plots of 3-sphere volume expansion rates versus the volume itself
for the Λ-FLRW parent models, using 3-sphere radii set equal to
a(t) at the moment of minimum expansion . . . . . . . . . . . . . 150
5.4 Plots of the radius expansion rates versus the radius itself for the
Λ-FLRW parent models, using the radius set equal to a(t) at the
moment of minimum expansion . . . . . . . . . . . . . . . . . . . 151
5.5 Plots of the radius expansion rates versus the radius itself for the Λ-
FLRW parent models, using the average radius of the CW Cauchy
surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.6 Plots of the radius expansion rates versus the radius itself for the
Λ-FLRW parent models, using the radius to vertices . . . . . . . . 152
5.7 Plots of the radius expansion rates versus the radius itself for the
Λ-FLRW parent models, using the radius to centres of edges . . . 152
5.8 Plots of the radius expansion rates versus the radius itself for the
Λ-FLRW parent models, using the radius to centres of triangles . 153
5.9 Plots of the radius expansion rates versus the radius itself for the
Λ-FLRW parent models, using the radius to tetrahedral centres . 153
5.10 4-block with space-like struts . . . . . . . . . . . . . . . . . . . . 154
5.11 Plots of the ratio of the Hubble radius to the tetrahedral edge-
length l(t) versus the edge-length . . . . . . . . . . . . . . . . . . 155
5.12 Spokes of vertices around a mid-point vertex in a subdivided CW
Cauchy surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.13 Plots of the expansion rate of the universe’s volume against the
volume itself for the Λ-FLRW parent and children models . . . . . 179
5.14 Plots of the expansion rate of the universe’s volume against the vol-
ume itself for the 600-tetrahedral parent and its 7200-tetrahedral
child model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
5.15 Plots of the radius expansion rates versus the radius itself for the
Λ-FLRW parent and children models, using the radius set equal to
a(t) at the moment of minimum expansion . . . . . . . . . . . . . 182
5.16 Plots of the expansion rates of 3-sphere volumes versus the volume
itself for the Λ-FLRW parent models, using 3-sphere radii set equal
to a(t) at the moment of minimum expansion . . . . . . . . . . . 183
6.1 Plots of the volume expansion rate against the volume itself for
the unperturbed lattice models, with masses in the Regge models
at tetrahedral centres . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.2 Plots of the volume expansion rate against the volume itself for
the unperturbed lattice models, with masses in the Regge models
at mid-points of tetrahedral edges . . . . . . . . . . . . . . . . . . 194
6.3 Plots of the volume expansion rate against the volume itself for
perturbed lattice models . . . . . . . . . . . . . . . . . . . . . . . 223
6.4 Extended plot of the volume expansion rate against the volume
itself for the δM
M
= 0.1 model . . . . . . . . . . . . . . . . . . . . . 224
6.5 Plot of δl˙(0)(t) against t for perturbed lattice models . . . . . . . . 224
6.6 Plot of δl˙(1)(t) against t for perturbed lattice models . . . . . . . . 225
6.7 Plot of δl(0)(t) against t for perturbed lattice models . . . . . . . . 226
6.8 Plot of δl(1)(t) against t for perturbed lattice models . . . . . . . . 226
A.1 Coxeter and non-Coxeter lattices in 2-dimensions . . . . . . . . . 237
B.1 Plot of relationship between particle’s initial radial velocity and
maximum radius attained in Schwarzschild Regge space–time . . . 240
B.2 Plot of relationship between simulated and expected radial veloci-
ties in Schwarzschild Regge space–time . . . . . . . . . . . . . . . 241
B.3 Graphs showing evolution of simulated radial velocity with radius
in Regge Schwarzschild space–time before correction . . . . . . . . 243
B.4 Plot of relationship between initial radial velocity and maximum
radius attained for a particle starting at large initial radius in
Schwarzschild Regge space–time . . . . . . . . . . . . . . . . . . . 244
B.5 Graph showing difference between expected and simulated velocity
for ingoing particle in Regge Schwarzschild space–time . . . . . . 244
C.1 Qualitative graph illustrating Schwarzschild Regge block’s inter-
polation of radial distances . . . . . . . . . . . . . . . . . . . . . . 248
List of tables
2.1 Regression equations for zLW versus zFLRW graphs for photon red-
shifts in the rb0 = 3× 104R flat universe . . . . . . . . . . . . . . 75
2.2 Regression equations for zLW versus zFLRW graphs for photon red-
shifts in the rb0 = 10
8R flat universe . . . . . . . . . . . . . . . . 80
4.1 Parent Collins–Williams skeletons . . . . . . . . . . . . . . . . . . 108
4.2 Sub-simplices of a subdivided parent tetrahedron . . . . . . . . . 119
5.1 3-sphere embedding co-ordinates of parent vertices . . . . . . . . . 129
5.2 3-sphere embedding co-ordinates of children vertices . . . . . . . . 159
5.3 Minimum volume and fractional difference from the FLRW mini-
mum for each CW model . . . . . . . . . . . . . . . . . . . . . . . 181
6.1 Numerical values for l˙, δl˙(0), and δl˙(1) at t = Tmax . . . . . . . . . 225
A.1 Coxeter lattices on 3-surfaces of constant curvature . . . . . . . . 235
B.1 Percentage difference between simulated and expected radial ve-
locities in Schwarzschild Regge space–time . . . . . . . . . . . . . 242

CHAPTER 1
Introduction
The start of the previous century saw one of the great revolutions of thought that
profoundly re-shaped our understanding of the universe at its most fundamen-
tal level. In a span of 25 years, the Newtonian orthodoxy that had prevailed in
physics for the previous 200 years had been overthrown. Newton’s key achieve-
ments in physics, the laws of motion and universal gravitation, had been over-
turned. In its place were erected the two radical edifices of quantum theory and
relativity theory, both of which greatly challenged common intuition. The former
replaced a deterministic view of the universe with one where all matter moves in
a probabilistic manner, where electrons can be in multiple places at once, where
‘God plays dice with the universe’. Yet this seemingly bizarre theory has success-
fully accounted for a wide range of phenomena including the nature of chemical
bonds between atoms, the periodicity of the elements according to the periodic
table, as well as the interactions between all known fundamental particles that
make up all visible matter in the universe. Relativity on the other hand has pre-
sented a view of the universe where space and time are no longer absolute, where
clocks can run at different rates and rulers can be at different lengths for different
observers, where whether one event happened before or after another can some-
times depend on whom one asks. Moreover, the intellectual significance of these
two theories extends even beyond physics, as their counter-intuitive implications
have prompted much philosophical discussion in trying to fully understand them
and the nature of science itself.1
1Indeed, physicists these days have, broadly speaking, adopted one of two philosophical
outlooks, platonism or positivism. Platonists generally believe that the entities described by
21
1. INTRODUCTION
It is generally accepted that the behaviour of all matter and even the space–
time in which it sits is governed by these two theories. Yet one lingering issue
remains unresolved even a century after the theories first appeared: that is how
to unify these two disparate theories into a single self-consistent framework. All
matter must obey the rules of quantum mechanics, yet all matter also gravitates,
some much more than others, and gravitation is the demesne of the general the-
ory of relativity. Thus there must exist some theory of quantum gravity, which
requires a further modification of relativity theory or quantum theory or both,
and the pursuit of this theory has driven much of the research programme in
fundamental physics for the past several decades.
Yet notwithstanding the pursuit of this ‘final theory’, the two theories have
in themselves remained highly fecund areas of study; even a century after they
were first proposed, they are still not completely understood, with many novel
features and implications continuing to be uncovered. Moreover, it is hoped that
the insights gleaned may help illuminate the nature of this ‘final theory’. Thus
the focus of this thesis will be on the study of general relativistic systems. General
relativity famously describes gravitation as distortions in the geometry of space–
time itself induced by the presence of matter, or as John Wheeler succinctly
puts it, ‘space–time tells matter how to move; matter tells space–time how to
curve’ [2]; and this relationship between the space–time geometry and the matter
content is governed by the Einstein field equations [3]
Gµν + Λ gµν = 8piTµν , (1.0.1)
where Tµν is the stress–energy tensor, Λ is a cosmological constant, Gµν is the
Einstein tensor, defined as
Gµν := Rµν − 1
2
Rgµν , (1.0.2)
Rµν is the Ricci tensor, R is the Ricci scalar, and gµν is the metric tensor; all
of these tensors have been defined on a pseudo-Riemannian manifold M that is
physical theories, such as particles or quantum fields, are real in some ontological sense. Pos-
itivists take a more limited view that physical theories should simply be self-consistent and
accurately predict all observable phenomena and that all other questions are beyond the realm
of physics, a viewpoint perhaps best summarised by Ludwig Wittgenstein’s statement, ‘What
can be said at all can be said clearly; and whereof one cannot speak thereof one must be
silent’ [1], or perhaps more succinctly by the slogan, ‘Shut up and calculate!’.
22
1. INTRODUCTION
the space–time itself; and geometric units, where c = G = 1, have been used.
This deceptively simple field equation actually belies 10 coupled, second-order,
non-linear partial differential equations for the metric tensor, and solutions for
such a system of equations are notoriously difficult to find. Indeed, most known
solutions of the Einstein field equations describe systems with an unrealistically
high degree of symmetry.
One of the best known examples of such a highly symmetric solution is the
Friedmann–Lemaˆıtre–Robertson–Walker metric that underlies standard cosmol-
ogy. Standard cosmology is founded upon the so-called Copernican principle,
which posits that the universe ‘looks’ on average to be the same regardless of
where one is in the universe or in which direction one looks. In other words,
every point in the universe is identical; no point is special. More formally, the
Copernican principle states that Cauchy surfaces of the universe can be admit-
ted that are homogeneous and isotropic, and this symmetry can be expressed
mathematically by writing the universe’s metric in the form [4–7]
ds2 = −dt2 + a2(t)
(
dr2
1− kr2 + r
2
(
dθ2 + sin2 θ dφ2
))
, (1.0.3)
where a(t) is a time-dependent function known as the scale factor and k is a
curvature constant. The sign of k determines whether Cauchy surfaces of constant
time t will be open, flat, or closed, with k < 0 being open, k = 0 being flat, and
k > 0 being closed; one can always re-scale a(t) and k such that k = +1, 0, or −1
as appropriate, in which case a(t) becomes the radius of curvature for the open
and closed universes. The metric (1.0.3) is known as the Friedmann–Lemaˆıtre–
Robertson–Walker (FLRW) metric.
If k > 0, Cauchy surfaces of constant t can also be embedded in 4-dimensional
Euclidean space E4 where they would form 3-spheres. The 3-sphere radius is given
by a(t) after re-scaling such that k = +1, and the embedding is given by
r = sinχ,
x1 = a(t) cosχ,
x2 = a(t) sinχ cos θ,
x3 = a(t) sinχ sin θ cosφ,
x4 = a(t) sinχ sin θ sinφ,
(1.0.4)
23
1. INTRODUCTION
for 0 ≤ χ, θ ≤ pi and 0 ≤ φ < 2pi. This 3-sphere has a volume given by
UFLRW(t) = 2pi
2a(t)3, (1.0.5)
and a rate of expansion given by
U˙FLRW(t) = 6pi
2a(t)2 a˙(t). (1.0.6)
The FLRW metric can then be written as
ds2 = −dt2 + a2(t) [dχ2 + sin2 χ (dθ2 + sin2 θ dφ2)] . (1.0.7)
Though the form of the FLRW metric is fixed by consideration of symmetries
alone, the function for the scale factor a(t) is instead determined by general
relativity. Inserting the FLRW metric into the Einstein field equations (1.0.1)
yields (
a˙
a
)2
=
1
3
(
8piρ+ Λ
)
− k
a2
, (1.0.8)
a¨
a
= −4pi
3
(
ρ+ 3p
)
+
Λ
3
, (1.0.9)
where ρ and p are the energy density and pressure of any perfect fluid filling
the space, and where Λ is the cosmological constant. This pair of differential
equations is known as the Friedmann equations, and their solution determines
a(t).
These FLRW models have had great success in explaining much of the uni-
verse’s behaviour, including most notably the Hubble expansion of the universe
[8], the cosmic microwave background (CMB) [9,10], and baryon acoustic oscilla-
tions [11,12]. Indeed, the underlying assumption of homogeneity and isotropy ap-
pears well-supported by precision measurements showing the CMB to be isotropic
to within one part in 100,000 [13,14]. Yet in spite of this, observations also clearly
show that the late, matter-dominated universe is not homogeneous and isotropic
except at the coarsest of scales. Instead, matter is distributed predominantly
in clusters and superclusters of galaxies with large voids in between, and the
physical effects of such a ‘lumpy’ universe are still not fully understood.
Indeed, there has been intense interest recently over the possible importance of
24
1. INTRODUCTION
inhomogeneities to observational cosmology. Perhaps the area of greatest interest
concerns the implications of recent redshift measurements from Type Ia super-
novae (SN1a). When fitted to the homogeneous FLRW models, this data has led
to the conclusion that the universe’s expansion is actually accelerating [15–17],
and to account for this acceleration, cosmologists have posited the existence of
some exotic matter, known generally as dark energy ; two of the most-favoured
dark energy models are an FLRW model with non-zero cosmological constant
Λ, which describes a negative-pressure fluid acting repulsively under gravity and
permeating the entire universe, and a scalar field theory called quintessence [18].
However attempts to directly observe any exotic matter and determine its nature
have so far proven unsuccessful, and some cosmologists have begun to question
the exotic matter approach. Instead, they have sought to explain the observed ac-
celeration as an apparent effect arising from fitting data from an inhomogeneous
universe onto a homogeneous model [19–23].
As an example, it has been shown that Lemaˆıtre–Tolman–Bondi (LTB) mod-
els can also account for all known cosmological observations, including SN1a
redshifts, without requiring any exotic matter [24–36]. However, these models
require the observer to be at the centre of a Hubble-scale under-dense region in a
spherically-symmetric, dust-dominated universe; thus, these models incorporate
inhomogeneity by completely breaking the Copernican principle at all scales. It
remains to be seen whether it is also possible to account for cosmological ob-
servations using an inhomogeneous model without exotic matter but which still
satisfies the Copernican principle in a coarse-grained manner.
Because of the possibly important implications of inhomogeneities to cosmol-
ogy, there has been much recent effort in quantifying their relative importance.
Ideally, one would want an exact solution to the Einstein field equations that
describes the universe’s actual structure, but as the structure is highly compli-
cated, such a solution remains out of reach. For this reason, discrete gravitational
approaches to cosmology are of great interest. Where an exact solution is not
available, discrete approaches can be used to approximate the system instead,
thereby allowing the system’s behaviour and properties to be studied. Moreover,
discrete approaches have the virtue of being non-perturbative: there has been
much debate on whether perturbed FLRW models can adequately capture the
non-linear structure of inhomogeneities [37–39], so a non-perturbative approach
is necessary to unambiguously understand inhomogeneities.
25
1. INTRODUCTION
There have been attempts to apply a discrete approach to cosmological mod-
elling as early as 1945, when Einstein and Straus first published their ‘Swiss
cheese’ models [40]. In this construction, co-moving spherical volumes are excised
from an FLRW background and replaced by Schwarzschild regions; any number
of spheres can be replaced, so long as they do not overlap. By an appropriate
fitting of these Schwarzschild regions into the FLRW surroundings, the resulting
space–time will still be an exact solution to the Einstein field equations. In light
of the inhomogeneity question, these models have received much renewed interest
recently and have even been extended to use LTB regions instead [41–46]. How-
ever, even though these models can incorporate ‘lumpy’ matter content through
Schwarzschild regions, the masses are still surrounded by a fluidic region rather
than pure vacuum, in contrast to actual universe’s matter distribution. More-
over, this fluidic region will still be governed by the FLRW metric, and therefore,
the models will still be dynamically identical to FLRW. And although there is
no a priori reason to believe the models’ optical properties should be identical to
those of FLRW as well, recent studies have shown that they are, in fact, broadly
similar, though there are still subtle differences [42–46].
In this thesis, we shall develop and explore two alternative discrete approaches
to gravitational modelling, and we shall apply these approaches to explore the
properties of several different cosmological systems. Although we shall not focus
on the exotic matter question itself, it is hoped the work in this thesis may lay
the foundations for future investigations into this question.
Unlike the inhomogeneous models discussed above, one of the cosmological
systems we shall consider will not, in any way, be based on FLRW models; it will
instead have a truly discrete matter content comprising point masses separated by
vacuum rather than fluid; this should more accurately reflect the matter content
of the late universe. For simplicity however, this universe will still possess a high
degree of symmetry, though not as great as that of FLRW universes. We shall
consider the so-called lattice universes where the matter content on each Cauchy
surface consists of identical point masses arranged into a regular lattice. We shall
focus on lattices that are constructed by tessellating a 3-space of constant cur-
vature with identical regular polyhedral cells; the possible lattices that can be
constructed from such a tessellation have been summarised in Appendix A. To
‘construct’ a lattice universe then, one of the lattices in Appendix A is selected,
and all mass in each cell is ‘concentrated’ into a point at its centre. The metric
26
1. INTRODUCTION
consistent with such a matter distribution is expected to be invariant under the
same symmetry transformations that leave the lattice invariant, symmetries which
include discrete translation symmetries, discrete rotational symmetries, and re-
flection symmetries at the cell boundaries. In other words, the lattice universe
should have a metric of the form
ds2 = −dt2 + γ(3)ab (t,x) dxadxb, (1.0.10)
where γ(3) (t,x) is the 3-dimensional metric for constant t hypersurfaces, and
Latin indices a, b = 1, 2, or 3 denote spatial co-ordinates only; the spatial met-
ric γ(3) (t,x) at constant t would possess the lattice symmetries. Effectively,
the Copernican symmetries of FLRW universes have been reduced to just these
symmetries.
There has been some progress towards determining the 3-metric γ(3) (t,x).
For the closed universe, the 3-metric γ(3) (t = 0,x) on the time-symmetric hy-
persurface was first determined by Wheeler for the 5-cell universe [47] and then
more recently by Clifton et al. for all closed lattices [48]; this work has been fur-
ther generalised by Korzyn´ski who examined closed universes with an arbitrary
number of identical masses, not necessarily arranged in a lattice [49]. The time
evolution of this initial data has been investigated numerically by Bentivegna
and Korzyn´ski [50] and analytically by Clifton et al. [51, 52]; however Clifton
et al. only considered the evolution of certain highly-symmetric curves on the
initial surface, so the complete, analytic 4-metric of the closed universe is still
lacking. The flat lattice universe has been modelled perturbatively by Bruneton
and Larena [53] and numerically by various authors [54–56]; using their per-
turbative model, Bruneton and Larena have also investigated the flat universe’s
optical properties [57]. Yet in spite of the progress in determining γ(3) (t,x),
a complete, non-perturbative solution for any lattice universe remains elusive.
Moreover, similar analyses of the open universe, analytical or numerical, have yet
to be performed; and the optical properties of the closed universe have yet to be
examined as well.
Since the complete 4-metric for any lattice universe is still unknown, con-
structing a non-perturbative approximation of it instead may help provide fur-
ther insights into the universe’s properties. Such an approximation may, for
instance, offer qualitative insights into the universe’s dynamics as well as into the
27
1. INTRODUCTION
behaviour of photons as they propagate along any arbitrary direction; indeed, it
would be difficult to propagate photons through any universe without having a
full 4-metric, so for this reason, an analytic approximation of the metric would
be especially invaluable. Thus in this thesis, we shall consider approximations
constructed using discrete gravitational approaches.
The first approximation we shall consider was actually devised by Lindquist
and Wheeler (LW) in 1957 [58,59]; they were, in fact, the first to investigate the
lattice universe by any method. In their construction, each polyhedral lattice
cell gets approximated by a spherical one instead with Schwarzschild geometry
inside. The original construction, however, was restricted to approximating closed
universes. Yet even in this limited case, Lindquist and Wheeler obtained a scale
factor that very closely resembled the scale factor of closed dust-filled FLRW
universes. Indeed, the LW scale factor would asymptotically approach the FLRW
one as the total number of masses in the universe increased while the universe’s
total mass was held constant.
More recently, Clifton and Ferreira (CF) [60] have re-visited the LW construc-
tion and extended it in several notable ways. First, they were able to generalise
the construction so that flat and open universes could be modelled as well. Sec-
ondly, while Lindquist and Wheeler had defined a ‘cosmological time’ co-ordinate
in the neighbourhood of the cell boundaries, Clifton and Ferreira were able to
extend the co-ordinate to cover the entire space–time globally. This opened the
way to a definition of co-moving observers for lattice universes, thereby allowing
quantities such as cosmological redshifts to be defined. Finally, Clifton and Fer-
reira were able to define a set of boundary conditions that photons must obey
when crossing from one cell into the next, thereby allowing photons to be prop-
agated across the entire LW model. With their extension, Clifton and Ferreira
then modelled the flat lattice universe and found that its evolution and redshifts
generally agreed well with those of the flat dust-filled FLRW universe but that
its angular diameters and luminosity distances differed [61].
Chapter 2 will re-visit models of the lattice universes using Lindquist and
Wheeler’s approximation as well as Clifton and Ferreira’s extensions. In addi-
tion to reviewing this construction, several new results will be presented. These
include scale factors for the flat and open universes, allowing us to qualitatively
describe these universes’ dynamical behaviour, and redshifts for the closed, flat,
and open universes. Presently, only redshifts in the flat lattice universe have
28
1. INTRODUCTION
been studied, by Bruneton and Larena using a perturbative method [53] and by
Clifton, Ferreira, and O’Donnell using the LW formalism [60,61]; redshifts for the
closed and open universes currently do not exist anywhere in the literature, so
this part of our results is entirely new. Furthermore, although the method used
for propagating photons is strongly influenced by Clifton and Ferreira’s approach,
several modifications have been adopted which will be justified in the chapter.
Thus our specific redshift results for the flat universe are, in a certain sense, also
original. The chapter will show that both the evolution and the cosmological
redshifts of all lattice universes generally behave rather similarly to their FLRW
analogues though with certain differences in the redshift behaviour attributable
to the lattice universe’s ‘lumpy’ matter distribution.
The thesis will next consider the discretisation scheme of Regge calculus. First
proposed by Tullio Regge in 1961 [62], this scheme discretises space–times into
piecewise linear manifolds constructed of flat blocks glued together at their faces;
as the blocks are flat, the metric inside is the Minkowski metric. The main value
of this scheme is its versatility as it can in principle approximate any space–
time, with the approximation becoming more accurate as the resolution of the
discretisation becomes finer. The formalism of Regge calculus will be presented
in Chapter 3. This will be predominantly a review chapter.
The rest of this thesis will use a Regge discretisation of FLRW space–times
formulated by Collins and Williams (CW) [63] and further developed by Brewin
[64]. Though this formalism was devised for closed FLRW universes, it can readily
be extended to flat and open universes as well. Nevertheless, the focus for the
remainder of this thesis will be on closed universes alone. The CW discretisation
will be modified according to the cosmological system being studied. Thus to
provide the necessary background, Chapter 4 will review the CW formulation
and Brewin’s extension. This will also be predominantly a review chapter.
Chapter 5 will further explore the CW formalism as well as Brewin’s extension
by modelling a universe for which the continuum solution is well-established,
that is, the closed vacuum Λ-FLRW universe; this is the second cosmological
system this thesis will consider. In particular, we shall demonstrate the success
of the formalism in reproducing the behaviour of the continuum space–time being
approximated. However, this chapter will also reveal certain constraints on the
applicability of the formalism, particularly with regards to certain modifications
considered by Brewin. The material presented in this chapter will consist almost
29
1. INTRODUCTION
entirely of original research.
With the knowledge gained from modelling the closed vacuum Λ-FLRW uni-
verse, we shall now apply the CW formalism to approximate the closed lattice
universe; this will be the focus of Chapter 6. The Regge field equations governing
the universe’s evolution will be derived, and it will be shown that the evolution
closely resembles that of a closed dust-filled FLRW universe. This chapter will
also derive constraints on where the point masses can be placed in each cell of the
CW Cauchy surfaces if the Regge model is to be stable; however these stability
constraints are an artefact of the Regge approximation being used rather than a
feature of the underlying continuum space–time. The latter half of this chapter
will then consider a perturbed lattice universe where a single mass is changed
from M to M ′. The Regge field equations governing such a universe’s evolu-
tion will also be derived, and the dynamical behaviour compared against that
of the unperturbed lattice universe. It will be shown that the evolution of the
perturbed universe closely resembles that of its unperturbed counterpart, albeit
with a slightly larger overall volume when M ′ > M . This chapter will consist
entirely of original research.
Finally, Chapter 7 will discuss possible directions in which this thesis’ re-
search may be extended, including how these different models and approaches
may complement each other.
Throughout this thesis, we shall be using geometric units where c = G = 1;
we shall also be using Greek tensor indices to denote space–time indices, which
run from 0 to 3, and Latin tensor indices to denote spatial indices only, which
run from 1 to 3.
30
CHAPTER 2
Lattice universes and the
Lindquist–Wheeler models
Lattice universes offer perhaps some of the simplest models featuring a ‘lumpy’
matter distribution. It is hoped that by studying such universes, one could gain
insight into what observable effects the actual universe’s inhomogeneous mat-
ter distribution might yield. Such universes are considered ‘simple’ by the sole
virtue of their regular matter distribution; in actuality, solving the Einstein field
equations for the metric of such universes is rather non-trivial, and a complete
non-perturbative solution remains to be obtained, although as discussed in Chap-
ter 1, there has been significant progress recently in special cases. Yet in spite of
the difficulty in obtaining a complete solution, the regularity of the matter distri-
bution has actually facilitated the construction of several approximation schemes
for the lattice universe, thereby allowing some of its properties to be studied.
In this chapter, we shall study the lattice universe using one such scheme.
Lindquist and Wheeler [58, 59] have devised a method to approximate space–
time around each mass by Schwarzschild space–time. For this approximation
to work though, the masses are required to be of such magnitude and separation
that the geometry around each can be reasonably approximated by Schwarzschild
geometry; for example, should the masses get too close together, the deviation
around each mass from Schwarzschild geometry would become too great, and the
LW approximation would consequently break down. More recently, Clifton and
Ferreira have re-visited the LW construction and extended it in several ways [60].
31
2. LINDQUIST–WHEELER MODELS
They have generalised the construction so that flat and open universes can also
be modelled – Lindquist and Wheeler’s original construction was constrained to
closed universes only. They have also extended the construction so that particles
like photons can be propagated through the universe, thus opening the way for
cosmological redshifts, in particular, to be studied.
In this chapter, we shall further explore the properties of the lattice universe
using the LW approximation with the CF extensions. We shall consider all three
types of universes, closed, flat, as well as open. First, we shall derive the scale
factors for all three universes; Lindquist and Wheeler had derived the scale factor
for the closed universe only, but we shall extend this by deriving analogous results
for the flat and open universes. These scale factors are related to the radius of
the cell boundary by a constant scaling, and in all three cases, the scale factors
have the same functional form as their FLRW counterparts. Secondly, as Clifton
and Ferreira have shown, the LW cell boundary radius satisfies a Friedmann-like
equation for all three types of lattice universes; they have also shown that a
similar equation holds for the closed universe’s scale factor. By using the LW
scale factors we derived rather than cell radii, we can make the analogy between
the CF Friedmann equation and the FLRW Friedmann equation more salient;
specifically, we shall demonstrate that the CF Friedmann equation takes a form
essentially identical to (1.0.8) for Λ = 0 and with k in the LW case also equalling
+1, 0, or −1 according to whether the universe is open, flat, or closed respectively.
Moreover, we find that the density ρ in the LW case equals a cell’s Schwarzschild
mass divided by the spherical volume 4pir3b/3 where rb is the radius of a cell; thus,
the density behaves as if the cell were a Euclidean sphere. Thirdly, as mentioned
in Chapter 1, Clifton and Ferreira managed to define a ‘cosmological time’ that
covered the entire universe globally. However, we have found their co-ordinate
system to be unsuitable for the closed universe. We shall show that in this case,
the CF co-ordinates do not actually cover the cell’s interior completely but leave
a gap region instead. For this reason, we must adopt an alternative co-ordinate
system if we wish to propagate photons through closed universes, and in fact, a
suitable system was proposed by Lindquist and Wheeler themselves at the end of
their original paper [58]. Fourthly, though our method for propagating photons
across the universe is strongly influenced by Clifton and Ferreira’s approach, we
have adopted several modifications to the set of conditions used to propagate
photons across cell boundaries; these will be justified below. Finally, we shall
32
2. LINDQUIST–WHEELER MODELS
examine the redshifts of photons travelling along a range of trajectories in closed,
flat, and open universes. So far, redshifts have only been studied in the flat
universe using both a perturbative approach [57] and the LW formalism [60,61];
redshifts for the closed and open universes currently do not exist anywhere in
the literature, so this part of our results are entirely new. We shall show that
for all three types of universes, the redshifts behave broadly in the same way as
their FLRW counterparts though with certain differences that can be attributed
to the inhomogeneities of the lattice. One of our most striking results is that
LW redshifts can differ from their FLRW counterparts by as much as 30%, which
has implications on estimating the age of the photon’s source, and that an LW
universe can give rise to a non-zero integrated Sachs–Wolfe effect without needing
any cosmological constant.
This chapter is organised as follows. The first section will explain the origins
of the LW approximation and detail its construction; the CF generalisation to flat
and open universes will also be reviewed. This section will also compare the CF
and LW global co-ordinate system and demonstrate the existence of a gap region
if the CF system is applied to closed universes. The subsequent section will derive
and examine the LW analogues of the FLRW scale factor as well as the modified
CF Friedmann equation for all three types of lattice universes. The latter half of
this chapter will examine the behaviour of redshifts in the lattice universe. The
third section will briefly explain the manner by which redshifts are computed;
the method used follows that of Clifton and Ferreira. The fourth section will
explain the boundary conditions that are used to propagate photons across cell
boundaries between two contiguous cells. The fifth section will explain the nu-
merical method used to simulate the propagation of photons through the lattice.
We shall apply Williams and Ellis’ Regge calculus formalism for Schwarzschild
space–time to each cell [65,66]; however we have derived an improvement to their
rules for tracing geodesics through their Regge Schwarzschild space–time which
significantly enhances numerical accuracy. Our improvement will be derived in
this section with supporting numerical evidence presented in Appendix B; the
improvement is not specific to the models being considered here but is applicable
to any situation where the Williams–Ellis scheme is being used to numerically
simulate geodesics through Schwarzschild space–time. The penultimate section
presents and analyses the redshift results of our simulations. The final section
discusses possible directions in which this work might be extended.
33
2.1. CONSTRUCTING THE LW APPROXIMATION
2.1 Constructing the LW approximation
The LW approximation of lattice universes was inspired by Wigner and Seitz’s
method [67] for approximating electronic wavefunctions in crystal lattices. The
Wigner–Seitz method approximates the polyhedral cell of a crystal lattice by a
sphere of the same volume. Any conditions that the wavefunction must satisfy
on the original cell boundary get imposed on the spherical boundary instead.
For example in the original lattice, reflection symmetry at the cell boundary
means that the wavefunction Ψ of the lowest energy free electron must satisfy
n · ∇Ψ = 0 at the boundary, where the vector n is orthogonal to the boundary.
In the Wigner–Seitz approximation, this same vanishing-derivative condition gets
imposed instead on the spherical boundary; that is, Ψ must now satisfy ∂Ψ/∂r =
0 at the boundary; and this effectively assumes the electron potential within
a cell to be spherically symmetric. The higher the symmetries of the original
polyhedral cell, the closer the cell resembles a sphere, and the more accurate
the results obtained from the Wigner–Seitz models. Indeed when applied to
crystals where exact solutions are known, the Wigner–Seitz construction yields
very accurate results [68,69].
In analogy with the Wigner–Seitz construction, Lindquist and Wheeler ap-
proximate each elementary cell of the lattice universe by a spherical cell and the
metric inside each cell by the Schwarzschild metric [70]
ds2 = −
(
1− 2m
r
)
dt2 +
dr2(
1− 2m
r
) + r2 (dθ2 + sin2 θdφ2) . (2.1.1)
They have called this spherical cell the Schwarzschild-cell. Without knowing
the true metric, one cannot directly assess how well is a polyhedral cell in the
lattice universe approximated by a spherical one. However Lindquist and Wheeler
have shown that polyhedra in a closed or flat 3-space of constant curvature are
reasonably approximated by spheres of equal volume [58, 59]; hence when such
polyhedra are combined into lattices in such 3-spaces, the polyhedral lattice cells
would be reasonably well-approximated by spherical cells. They therefore provide
this as partial evidence that the approximation would probably be reasonable as
well for the polyhedral lattice cells in the lattice universe.
In contrast to the Wigner–Seitz lattice, the LW lattice is itself dynamical.
34
2. LINDQUIST–WHEELER MODELS
A test particle sitting at the boundary between two Schwarzschild-cells will by
symmetry always remain at the boundary. Yet like any other test particle in a
Schwarzschild geometry, this test particle must also be radially falling towards
the centre of one of the Schwarzschild-cells. And by the same reasoning, this
particle is also radially falling towards the centre of the other Schwarzschild-
cell. Therefore this test particle, and hence the cell boundary itself, is radially
falling towards both cell centres simultaneously, as depicted in Figure 2.1. The
Figure 2.1: A test particle co-moving with the boundary between two cells must be
simultaneously falling towards both central masses. Thus the boundary itself must
expand and contract accordingly.
boundary’s motion is hence given by the equation of motion for a radial time-like
geodesic, (
dr
dτ
)2
= E − 1 + 2m
r
, (2.1.2)
where τ is the proper time of a test particle following the geodesic and E is a
positive constant of motion; it can be shown that
√
E is the particle’s energy per
unit mass at radial infinity. However this simultaneous free-fall of the bound-
ary towards the two masses is actually the result of the two masses themselves
falling towards each other under their mutual gravitational attraction. This mu-
tual attraction of all the point–masses thereby gives rise to the expansion and
contraction of the lattice itself, which manifests as the expansion and contraction
of the Schwarzschild-cell boundary.
We can see from (2.1.2) that the value of E = Eb at the boundary determines
whether the cell boundaries will expand indefinitely or eventually re-collapse, and
hence whether the underlying lattice universe is open, flat, or closed. For Eb < 1,
35
2.1. CONSTRUCTING THE LW APPROXIMATION
the boundaries will expand until reaching a maximum radius of
rmax =
2m
1− Eb (2.1.3)
before re-collapsing; this corresponds to a closed universe. For Eb > 1, the
boundaries will expand indefinitely; this corresponds to an open universe. And
for Eb = 1, the boundaries travel at the escape velocity and just reach radial
infinity; this corresponds to a flat universe. Thus it is through this constant Eb
that Clifton and Ferreira generalise Lindquist and Wheeler’s work to flat and
open universes in a natural manner.
In order to ‘glue’ the individual cells together into a lattice, we require that
the 3-space of constant time in one Schwarzschild-cell mesh at the boundary with
the corresponding 3-space of the neighbouring cell. As Lindquist and Wheeler
have pointed out, two 3-spaces will mesh together if and only if they intersect
their common boundary orthogonally. Surfaces of constant Schwarzschild time
t do not satisfy this meshing condition; instead, they intersect each other when
they meet, as illustrated in Figure 2.2, and some other time co-ordinate must be
found.
Both Lindquist and Wheeler as well as Clifton and Ferreira devised new time
co-ordinates that satisfied the meshing condition. The LW time co-ordinate is
defined for closed universes only while the CF time co-ordinate can only be ap-
plied to flat and open universes. Both time co-ordinates are defined to equal the
proper times of a congruence of radial time-like geodesics, which must include
the geodesics of test particles co-moving with the boundary. Test particles fol-
lowing these geodesics will travel at the same velocity if at the same radius, thus
forming freely falling shells. All geodesics’ clocks are calibrated to read identical
proper time on some initial space-like hypersurface that intersects the geodesics
orthogonally. Then if the congruence was well-chosen, all other constant proper
time hypersurfaces will intersect the congruence orthogonally as well. In partic-
ular, such hypersurfaces would always intersect the boundary orthogonally, thus
satisfying the meshing condition, as illustrated in Figure 2.3. Such a definition
of time is well-defined globally and can therefore be used as a ‘cosmological time’
parametrising the lattice universe. Such a construction of cosmological time is
natural because we expect co-moving cosmological observers to be stationary with
respect to constant-time hypersurfaces and also to follow freely falling geodesics.
36
2. LINDQUIST–WHEELER MODELS
Coordinate patch of cell 1
r
t
Ω
r
Ω
t
Coordinate patch of cell 2
Figure 2.2: Hypersurfaces of constant Schwarzschild time t intercept each other rather
than mesh together at the cell boundaries. The normals from different cells point in
different directions at the boundary. Therefore, Schwarzschild t cannot serve as a global
cosmological time co-ordinate for our model.
r
τr
Ω
Coordinate patch of cell 1 Coordinate patch of cell 2
Ω
τ
Figure 2.3: Hypersurfaces of constant τCF and τLW mesh at the boundaries, and the
normals from the two cells coincide. Therefore either can serve as cosmological time.
37
2.1. CONSTRUCTING THE LW APPROXIMATION
However there is no unique choice of congruence; any choice that satisfies our
constraints above is acceptable. Thus herein lies the difference between LW time
and CF time. Note that (2.1.2) actually describes any radial time-like geodesic in
Schwarzschild space–time, including boundary geodesics for which r = rb. Clifton
and Ferreira’s congruence consists of geodesics all having the same constant E,
namely that of the boundary Eb. Lindquist and Wheeler’s congruence uses differ-
ent E < 1 for different geodesics. Indeed, it may be possible to find an alternative
choice of congruence based on some stronger physical motivation, but we shall
not consider this problem here. Instead, we shall use LW time for the closed
universe and CF time for the flat and open universes.
For boundaries following out-going trajectories, the CF time co-ordinate τCF
is given by
dτCF =
√
Eb dt−
√
Eb − 1 + 2mr(
1− 2m
r
) dr, (2.1.4)
where Eb is the same positive constant as E in (2.1.2) for the boundary geodesic.
The Schwarzschild metric (2.1.1) now becomes
ds2 = − 1
Eb
(
1− 2m
r
)
dτ 2CF −
2
Eb
√
Eb − 1 + 2m
r
dτCF dr +
dr2
Eb
+ r2dΩ2. (2.1.5)
The boundary’s equation of motion is still given by (2.1.2) but with the proper
time τ replaced by τCF , as it can be shown that τCF is identical to the boundary’s
proper time. In CF co-ordinates, the 4-vector tangent to any trajectory satisfying
(2.1.2) for E = Eb is
ua =
(
1,
√
Eb − 1 + 2m
r
, 0, 0
)
. (2.1.6)
For an arbitrary vector n tangent to a constant τCF surface,
na =
(
0, nr, nθ, nφ
)
, (2.1.7)
it can be shown that u · n = 0 and hence that surfaces of constant τCF are
orthogonal to all geodesics satisfying (2.1.2) for E = Eb, including particularly
the boundary geodesics. Thus the meshing condition is satisfied, and thus τCF
can serve as a cosmological time.
38
2. LINDQUIST–WHEELER MODELS
If we attempt to apply the CF co-ordinate system to closed universes though,
we encounter problems. When the boundary is contracting, its radial velocity is
given by the negative root of (2.1.2) instead, and as a result, the square-roots in
(2.1.4)–(2.1.6) must change sign. However the combined co-ordinate patches do
not correctly cover the interior of the Schwarzschild-cell. Rather, they leave an
uncovered region centred on the boundary’s moment of maximum expansion, as
illustrated in Figure 2.4.
t
r
r = 2M
Figure 2.4: A qualitative plot of various co-moving test-particle trajectories (dashed
line) inside a closed Schwarzschild-cell and of various constant τCF surfaces (dotted
line); the plot is of Schwarzschild co-ordinates r versus t. The cell boundary’s trajectory
has been drawn with a solid line. The r axis passes through the boundary’s moment
of maximum expansion, which happens when τCF = τmax. Since the radial velocity
must change sign abruptly when the boundary starts closing, there is a noticeable
discontinuity in the gradients of the test particle graphs at the r axis. The constant
τCF surfaces for τCF just greater than and less than τmax are not identical. Between
the two surfaces is a ‘gap’ region, shaded in the figure, where space–time points do not
have a well-defined cosmological time τCF .
We shall denote by τmax the boundary’s proper time at maximum expansion.
For τCF < τmax, surfaces of constant τCF are generated by the integral curves of
(2.1.4) for dτCF = 0, that is, by
dr
dt
=
√
Eb
(
1− 2m
r
)√
Eb − 1 + 2mr
> 0 ∀ rb ≥ r > 2m. (2.1.8)
This has a solution of the form t = f(r) + t0 where t0 is the Schwarzschild
time co-ordinate of the surface when it passes through r = r0 and r0 is the
Schwarzschild radial co-ordinate satisfying f(r0) = 0. Thus on the r–t plane,
like the one shown in Figure 2.4, all of these surfaces are identical apart from a
39
2.1. CONSTRUCTING THE LW APPROXIMATION
horizontal shift corresponding to different t0 constants. Increasing t0 shifts the
surface to the right. This implies the surface would intercept the boundary later
in the boundary’s trajectory, and therefore τCF would increase as well. As we
keep increasing t0, we shall eventually reach a limiting surface τCF → τmax from
the right, and this surface we denote by τ−max.
For τCF > τmax, surfaces of constant τCF are generated instead by the negative
of (2.1.8). This has the solution t = −f(r)+t0, and again, t0 is the Schwarzschild
time co-ordinate of the surface when it passes through r = r0. On the r–t plane,
all τCF > τmax surfaces are also identical to each other apart from a horizontal
shift. Decreasing t0 shifts the surface to the left and decreases τCF . As we decrease
t0, we shall eventually reach a limiting surface τCF → τmax from the left, and this
surface we denote by τ+max.
The complete set of constant τCF surfaces will completely cover the interior
of the cell without any overlaps or gaps if and only if surfaces τ−max and τ
+
max are
identical. This implies that
f(r) + t−0 = −f(r) + t+0
=⇒ f(r) = constant,
where t−0 and t
+
0 are the t0 constants for τ
−
max and τ
+
max respectively. However we
know that function f(r) is not constant, so we have a contradiction.
Instead, we actually have a ‘gap’ between τ−max and τ
+
max. Both surfaces meet
at (τmax, rb(τmax)) = (τmax, rmax). If we move along τ
−
max into the cell, then
decreasing r would decrease t because dr
dt
is always positive along this surface. If
we move along τ+max into the cell, then decreasing r would increase t because
dr
dt
is always negative along this surface. On the r–t plane, τ−max moves to the left
and τ+max to the right as r decreases from rmax. Since τ
−
max is the rightmost of the
τCF < τmax surfaces and τ
+
max the leftmost of the τCF > τmax surfaces, the region
between τ−max and τ
+
max is not covered by any τCF surface and is hence a gap.
We shall therefore use the LW co-ordinate system, which covers closed cells
correctly. This system is constructed from a congruence of closed geodesics that
attain maximum radii at the same Schwarzschild time. Being a congruence, the
geodesics have maximum radii spanning the entire range of 2m < r ≤ rb, and
from a generalised form of (2.1.3), the geodesics must therefore have different
constants E. The geodesics’ clocks are calibrated to have identical proper time
40
2. LINDQUIST–WHEELER MODELS
at maximum radii. The LW time co-ordinate τLW at any point is then defined
to be the proper time of the geodesic passing through it. Lindquist and Wheeler
also defined a new radial co-ordinate r˜LW that is constant along each geodesic
and equals the geodesic’s maximum radius in Schwarzschild co-ordinates.
The co-ordinate transformation to (τLW , r˜LW ) is then given implicitly by
τLW = τ0 ±
(
r˜LW
2m
)1/2 [
r
1/2 (r˜LW − r)1/2 + r˜LW cos−1
(
r
r˜LW
)1/2]
, (2.1.9)
t = t0 ±
[[(
r˜LW
2m
− 1
)
(r˜LW − r)r
]1/2
+ 2m
(
r˜LW
2m
− 1
)1/2(
r˜LW
2m
+ 2
)
cos−1
(
r
r˜LW
)1/2
+ 2m ln
r1/2 (r˜LW/2m− 1)1/2 + (r˜LW − r)1/2
r1/2 (r˜LW/2m− 1)1/2 − (r˜LW − r)1/2
]
,
(2.1.10)
where t0 and τ0 are the time co-ordinates at which the boundary attains its
maximum expansion; the positive signs are taken when the boundary is expanding
and the negative signs when contracting. The Schwarzschild metric now becomes
ds2 = − dτ 2LW +
[
r˜LW − r
4 r r˜LW (r˜LW/2m− 1)
][
(3 r˜LW − r)
[
r
r˜LW − r
]1/2
+ 3 r˜LW cos
−1
[
r
r˜LW
]1/2]2
dr˜2LW + r
2 dΩ2,
(2.1.11)
where r is now a function of τLW and r˜LW . From this metric, it is clear that surfaces
of constant τLW are always orthogonal to Lindquist and Wheeler’s congruence.
Thus the meshing condition is also satisfied, and τLW is also suitable for use as a
cosmological time.
We shall henceforth drop any subscript labels from the cosmological time τ .
In the case of the closed universe, τ will denote τLW , while in the flat or open
universe, it will denote τCF .
41
2.2. THE COSMOLOGICAL SCALE FACTOR
2.2 The cosmological scale factor and the Fried-
mann equations
There is a clear analogy between the size of the Schwarzschild-cell rb(τ) and the
FLRW scale factor a(t), as both provide a scale of their respective universe’s size.
Indeed we expect the lattice universe’s scale factor a(τ) should be a function of
rb alone, that is a(τ) = a(rb(τ)).
1 If we expand rb, then each cell of the lattice
will expand by some scale ξ, and therefore the lattice as a whole will expand by
ξ. We shall take a(τ) to be related linearly to rb(τ), that is, a
(
rb(τ)
)
= α rb(τ)
for some constant α > 0. As we shall see, a(τ) then depends on τ in a manner
analogous to how a(t) depends on t for FLRW universes.
ξrb
ξ`
`
rb
Figure 2.5: Rescaling the cell size by ξ rescales the entire lattice universe by ξ as well.
Perhaps the closest FLRW analogue to the lattice universe would be the
dust-filled Λ = 0 universes, because as the number of masses increases, the lat-
tice universe’s matter content should asymptotically approach homogeneous and
isotropic dust. For these dust-filled FLRW universes, the Friedmann equations
(1.0.8) and (1.0.9) can be solved to obtain a set of relations between a and t given
1We shall use a(t) when we refer to the FLRW scale factor and a(τ) when referring to the
lattice universe scale factor. The two functions are completely independent.
42
2. LINDQUIST–WHEELER MODELS
by
a =
a0
2
(1− cos η)
t =
a0
2
(η − sin η)
for k > 0, (2.2.1)
a =
(
9
4
a0
)1/3
t2/3 for k = 0, (2.2.2)
a =
a0
2
(cosh η − 1)
t =
a0
2
(sinh η − η)
for k < 0, (2.2.3)
where
a0 =
8piρ0
3
, (2.2.4)
and where ρ0 is the energy density when a = 1. When k > 0, a0 also corresponds
to the maximum value of a. These relations have been plotted in Figure 2.6.
a
t
closed
flat
open
Figure 2.6: Plots of a versus t for open, flat, and closed dust-filled FLRW universes.
Indeed Lindquist and Wheeler were able to derive a parametric relationship
between a(τ) and τ closely resembling that of (2.2.1), thus showing that a(τ)
behaves identically to its FLRW counterpart a(t); the only difference was the
factor of a0. However, their results were limited to the closed lattice universe.
We shall generalise Lindquist and Wheeler’s results, showing that this identical
43
2.2. THE COSMOLOGICAL SCALE FACTOR
behaviour between a(τ) and a(t) holds for lattices of all curvatures. Equation
(2.1.2) can be integrated to obtain
τ =
√
rE
2m
(√
r (rE − r) + rE cos−1
√
r
rE
)
for 0 < E < 1, (2.2.5)
τ =
(
4
9 rE
)1/2
r3/2 for E = 1, (2.2.6)
τ =
√
rE
2m
(√
r (rE + r)− rE sinh−1
√
r
rE
)
for E > 1, (2.2.7)
where
rE =

2m
|1− E| for E 6= 1 and E > 0,
2m for E = 1.
(2.2.8)
In the case of E < 1, rE corresponds to the maximum radius attained by the
geodesic, and for the boundary geodesic, where E = Eb, this maximum radius is
identical to (2.1.3). For boundary geodesics, where r = rb, equations (2.2.5) to
(2.2.7) can be re-cast into the form
rb =
rEb
2
(1− cos η)
τ =
rEb
2
√
rEb
2m
(η − sin η)
for 0 < Eb < 1, (2.2.9)
rb =
(
9
4
rEb
)1/3
τ 2/3 for Eb = 1, (2.2.10)
rb =
rEb
2
(cosh η − 1)
τ =
rEb
2
√
rEb
2m
(sinh η − η)
for Eb > 1, (2.2.11)
where rEb simply denotes rE when E = Eb and η is simply a parametrisation. If
we choose α to be
α =
√
rEb
2m
, (2.2.12)
44
2. LINDQUIST–WHEELER MODELS
then the scale factor a(τ) is given by
a =
rEb
2
√
rEb
2m
(1− cos η) for 0 < Eb < 1, (2.2.13)
a =
(
9
4
rEb
√
rEb
2m
)1/3
τ 2/3 for Eb < 1, (2.2.14)
a =
rEb
2
√
rEb
2m
(cosh η − 1) for Eb > 1. (2.2.15)
Comparing the relations just obtained for a(τ) and τ with their counterparts in
(2.2.1)–(2.2.3), we find that a(τ) and τ have the same functional form as their
FLRW counterparts for all background curvatures. Moreover, we have deduced an
expression for the factor α that is the same for all universes. The only difference
between the lattice and FLRW relations is that a0 for the lattice universe is
a0 = rEb
√
rEb
2m
. (2.2.16)
Equivalently, by equating (2.2.16) with (2.2.4), we can say that the density ρ˜0 for
the lattice universe is
ρ˜0 =
m
4
3
piα−3
. (2.2.17)
The denominator is simply the ‘Euclidean volume’ of a Schwarzschild-cell with
radius rb corresponding to a = 1.
As Clifton and Ferreira have noted, the radius of the cell boundary rb(τ)
satisfies an equation strongly resembling the Friedmann equation (1.0.8) for Λ =
0; they also found a similar relation for the closed LW universe’s scale factor.
By consistently using the LW scale factors just derived rather than rb(τ), we can
re-express the CF Friedmann equation in a form that makes its resemblance to
(1.0.8) much more salient; the CF Friedmann equation then becomes(
a˙(τ)
a(τ)
)2
=
8piρ˜
3
− k
a(τ)2
, (2.2.18)
where k = α2(1 − Eb) plays the roˆle of the curvature constant for the lattice
universe, and density ρ˜ is given by
ρ˜ =
m(
4
3
pia(τ)3
)
/α3
=
m
4
3
pir3b
. (2.2.19)
45
2.2. THE COSMOLOGICAL SCALE FACTOR
As noted earlier, Eb determines whether the universe will be open, flat, or closed,
and we see that k will take on the correct sign accordingly. In fact, if we make use
of (2.2.8) and (2.2.12), k simplifies to −1, 0,+1 for open, flat, and closed lattice
universes respectively, much like its FLRW analogue.
To complete our model, all that remains is to specify α or Eb. However,
Lindquist and Wheeler have actually provided an argument to specify α and an
independent argument to effectively2 specify Eb. What is intriguing is that their
choices of α and Eb are consistent with the relationship between the two quantities
required by (2.2.12). We shall summarise Lindquist and Wheeler’s arguments
leading to their choices. Although they dealt only with closed universes, we shall,
in this chapter, generalise their arguments to flat and open universes as well.
We begin with Lindquist and Wheeler’s choice of α. Note that hypersurfaces
of constant t in the FLRW metric (1.0.3) correspond to 3-spaces of constant
curvature, and through an appropriate choice of scaling, the radius of curvature
is given by the scale factor a(t). We embed the same type of lattice as the
lattice universe on such a hypersurface with the appropriate curvature.3 We shall
call this hypersurface the comparison hypersurface. We then approximate each
polyhedral cell by a sphere of the same volume.4 For closed hyperspheres, we
define ψ to be the angle between the centre of the spherical cell and its boundary
as measured from the centre of the hypersphere. Then ψ is given implicitly by
1
N
=
2ψ − sin 2ψ
2pi
, (2.2.20)
where N is the total number of cells in the lattice. For hyperbolic spaces, we
define ψ analogously in terms of hyperbolic angles. If r0 is the radius of one such
spherical cell, then we can relate r0 to the comparison hypersurface’s radius of
curvature aLW by
aLW =
r0
χ(ψ)
, (2.2.21)
2Recall that the quantity Eb was introduced to the LW construction afterwards by
Clifton and Ferreira. However, Lindquist and Wheeler imposed a tangency condition on the
Schwarzschild-cells, which we shall soon describe, that effectively specifies Eb.
3That is, for closed lattice universes, we use hyperspheres; for flat lattice universes, Euclidean
space; and for open lattice universes, hyperbolic space.
4This is actually Clifton and Ferreira’s generalisation. Lindquist and Wheeler’s original
condition was that the spheres occupy 1N of the comparison hypersphere’s total volume, where
N is the total number of cells. Such a condition clearly needs to be modified for flat and open
universes, as both N and the hypersurface volume are infinite in these cases.
46
2. LINDQUIST–WHEELER MODELS
where the function χ(ψ) is given by
χ(ψ) =

sinψ for closed universes,
1 for flat universes, 5
sinhψ for open universes.
(2.2.22)
Lindquist and Wheeler identified the Schwarzschild-cell radius rb of the lattice
universe with r0 and used the corresponding aLW as given by (2.2.21) to be the
lattice universe’s scale factor; that is,
aLW (τ) =
rb(τ)
χ(ψ)
. (2.2.23)
Hence they have chosen α = 1/χ(ψ). According to (2.2.12), Eb would therefore
correspond to
Eb =

cos2 ψ for closed universes,
1 for flat universes,
cosh2 ψ for open universes.
(2.2.24)
However, Lindquist and Wheeler prescribed Eb independently of (2.2.12).
They instead embedded the Schwarzschild-cell in the comparison hypersurface
and required it to be in some sense tangent to the hypersurface. Their prescrip-
tion for the embedding is as follows. Suppose we replaced each of the original
spherical cells in the hypersurface with a Schwarzschild-cell from the lattice uni-
verse. Then we want to choose an Eb to make the Schwarzschild-cell tangent to
the hypersurface. Lindquist and Wheeler formulate this tangency condition as
follows. Take any great circle on the boundary of the original sphere and compare
its circumference with that of the corresponding great circle on an infinitesimally
smaller sphere. Depending on which hypersurface the cell is embedded in, the
circumferences will obey the relation
1
2pi
d(circumference)
d(radial distance)
5Note that when k = 0 in the FLRW metric (1.0.3), we have complete freedom to make
a re-scaling of the form r → ξr and a → a/ξ. Hence we can always choose a comparison
hypersurface such that aLW = r0.
47
2.2. THE COSMOLOGICAL SCALE FACTOR
=

1
2pi
d(2piaLW sinψ)
aLWdψ
for closed hyperspheres,
1 for flat hypersurfaces,
1
2pi
d(2piaLW sinhψ)
aLWdψ
for open hypersurfaces,
=

cosψ,
1,
coshψ,
as depicted by Figure 2.7 to Figure 2.9. Then for the Schwarzschild-cell to be
tangent to the hypersphere, we also require that at its boundary,
1
2pi
d(circumference)
d(radial distance)
=

cosψ for closed universes,
1 for flat universes,
coshψ for open universes.
(2.2.25)
From the Schwarzschild metric expressed in CF co-ordinates (2.1.5), we can see
that d(circumference) = 2pidrb and that d(radial distance) =
drb√
Eb
. Thus,
1
2pi
d(circumference)
d(radial distance)
=
√
Eb.
From the Schwarzschild metric in LW co-ordinates (2.1.11), it can also be shown,
by making use of (2.1.9), that 1
2pi
d(circumference)
d(radial distance)
is the same. Solving for Eb, we
then obtain the same result as in (2.2.24).
We close this section with a few remarks about the large-N behaviour of the
closed LW model. By taking N → ∞, we can deduce what limiting behaviour
the model would approach as the number of masses increases while the universe’s
total mass is held constant. It can be shown from (2.2.20) that as N → ∞, the
angle ψ → ( 3pi
2N
)1/3
. If M = Nm is the total mass of the universe, then
lim
N→∞
α3m = lim
N→∞
M
N sin3 ψ
=
2M
3pi
,
48
2. LINDQUIST–WHEELER MODELS
a L
W
ψ dψ
aLW sinψ
a
LW dψ
Figure 2.7: A spherical cell of radius aLW sinψ and an infinitesimally smaller shell,
indicated by the dashed line, have been embedded in a 3-dimensional hypersphere of
radius aLW , with one of the angular dimensions suppressed. The radial distance between
the two shells, as measured along the 3-sphere, is aLWdψ. Because of the curvature of
the underlying 3-sphere, we have that d(circumference)/d(radial distance) = 2pi cosψ.
draLW
Figure 2.8: An analogous embedding in a flat hypersurface. In this case, the circum-
ference of the boundary is 2piaLW ; the radial distance between the two shells is dr; and
d(circumference)/d(radial distance) = 2pi.
a
LW dψ
ψ
dψ
aLW sinhψ
Figure 2.9: An analogous embedding in a hyperbolic hypersurface. The circumference
of the boundary is 2piaLW sinhψ; the radial distance between the two shells is aLWdψ;
and d(circumference)/d(radial distance) = 2pi coshψ.
49
2.2. THE COSMOLOGICAL SCALE FACTOR
and therefore the density ρ˜0 for the lattice universe, as given by (2.2.17), becomes
lim
N→∞
ρ˜0 =
M
2pi2
. (2.2.26)
In closed FLRW space–time, a hypersphere of constant t has a volume of 2pi2a(t)3.
Since the FLRW ρ0 is defined to be the density when a(t) = 1, then ρ0 also equals
M
2pi2
. Therefore as N → ∞, the lattice universe density ρ˜0 approaches its FLRW
equivalent ρ0, and hence the lattice universe factor a0 approaches its FLRW
equivalent as well. Consequently, a(τ) in (2.2.13) approaches a(t) in (2.2.1), and
τ in (2.2.9) becomes identical to t in (2.2.1). We also note that ρ˜ in (2.2.19)
becomes
lim
N→∞
ρ˜ =
M
2pi2a3
, (2.2.27)
which is identical to the FLRW energy density ρ. This implies that the CF
Friedmann equation (2.2.18) would be identical to the FLRW Friedmann equation
(1.0.8) for k = 1 and Λ = 0. Therefore, as N increases and the closed lattice
universe approaches the continuum limit, it becomes increasingly similar to the
closed dust-filled FLRW universe, which is consistent with the matter content
itself becoming increasingly similar to homogeneous and isotropic dust. In fact,
Korzyn´ski has proven that this large-N correspondence between the two closed
universes is indeed true for the exact solution of the Einstein field equations, at
least on the hypersurface of time-symmetry [49]. He actually considered universes
where there could be an arbitrary number N of identical Schwarzschild-like black
holes and demonstrated that as N →∞, the hypersurface approaches its FLRW
counterpart exactly, both in terms of the scale factor and in terms of the energy
density. Therefore in this regard, the LW approximation agrees with the exact
solution.
2.3 Redshifts in the lattice universe
Cosmological redshifts, 1 + zFLRW , in FLRW universes are defined with respect
to sources and observers that are co-moving with the universe’s expansion. The
redshifts are given by
1 + zFLRW =
ao
ae
, (2.3.1)
50
2. LINDQUIST–WHEELER MODELS
where ao and ae are the cosmological scale factors at the moments of observation
and emission respectively.
We should like to find the lattice universe analogue to zFLRW so that we
can compare redshifts in the two types of universes. In general, redshifts 1 +
z are defined as the ratio of the photon frequency measured at the source to
the frequency measured by the end observer. The closest analogy to co-moving
sources and observers in the lattice universe would be sources and observers that
are co-moving with respect to a constant τ surface. All such observers would
be following radial geodesics that obey (2.1.2), with E fixed to be Eb in the flat
and open universes but geodesic-dependent in the closed universe. If u is the
4-velocity of an observer and k the 4-momentum of a photon as it passes the
observer, then the photon frequency measured by the observer would be −u · k.
For the lattice universe, the cosmological redshift is therefore given by
1 + zLW =
us ·ks
uo ·ko , (2.3.2)
where the subscripts s and o denote ‘source’ and ‘observer’ respectively. Clifton
and Ferreira have constrained their consideration to observers that are co-moving
with the photon’s source: that is, at any time τ , the observer would be at the
same radius as the source in their respective cells’ Schwarzschild co-ordinates;
this has been illustrated in Figure 2.10. To facilitate comparison with Clifton
and Ferreira’s results, we shall compute redshifts for the same set of observers.
Following Clifton and Ferreira’s example, we shall also use the lattice uni-
verse’s scale factor a(τ) at the moments of emission and observation to compute
zFLRW for comparison; we thus re-express (2.3.1) as
1 + zFLRW =
rb(τo)
rb(τe)
. (2.3.3)
2.4 Propagating photons across cell boundaries
In order to propagate photons through the lattice universe, we must first specify
what boundary conditions trajectories must satisfy whenever they pass from one
cell into the next. Before discussing the boundary conditions though, we first note
a difference between Clifton and Ferreira’s choice of boundary geometry and ours.
51
2.4. PROPAGATING PHOTONS ACROSS CELL BOUNDARIES
(τf , rf )
(τi, ri)
(τf , rf )
(τi, ri)
Figure 2.10: A photon travels from radius ri at cosmological time τi in one cell to
radius rf at time τf in the next, as indicated by the long solid arrow. We assume that
the photon always passes through the boundary in the manner illustrated here: that is,
the point of crossing is a point of tangency between the boundaries of the two cells. A
photon is emitted at (τi, ri) by a source travelling along a geodesic given by (2.1.2) and
observed by an observer in another cell. Although in a different cell from the source,
the observer is still ‘co-moving’ with the source: that is, for any τ , the observer is
always at the same radius as the source in their respective cells, and this requires the
observer to travel along a geodesic with the same E as the source.
Clifton and Ferreira converted from spherical cell boundaries back to polyhedral
ones, deducing the polyhedral boundary velocity from the requirement that the
spherical cell always have the same ‘Euclidean volume’ as the polyhedral cell. We
shall instead continue to use spherical boundaries and propagate photons across
boundaries in the manner illustrated in Figure 2.10; that is, wherever a photon
crosses, we always regard the point of crossing as a point of tangency between the
two neighbouring cell boundaries. We assume that the boundaries are tangent not
just in (3+1)-dimensional space–time as a whole, but also in each 3-dimensional
constant-τ hypersurface, which would be the case had we been using the true
polyhedral lattice cells. As Lindquist and Wheeler have argued, spherical bound-
aries should be a good approximation to the shape of polyhedral boundaries, with
the approximation improving as the number of symmetries increases. Therefore
any errors due to this approximation would be small to begin with. We further
argue that although spherical cells may tile the lattice universe with gaps and
overlaps, an arbitrary photon would on average travel through an equal number
of gaps and overlaps such that the overall error approximately cancels out, as
illustrated in Figure 2.11. Additionally, we note that in Clifton and Ferreira’s
52
2. LINDQUIST–WHEELER MODELS
A
B
Figure 2.11: A flat lattice universe with cubic cells approximated by spheres. The
central masses are depicted by dots. A photon will on average travel through an equal
number of gap and overlap regions. Because we are using spherical boundaries, at
boundary crossing A, the photon will jump over a gap region, but at crossing B, it
will pass through the overlap twice. The cubic boundaries are tangent to each other
everywhere along the boundaries while the spherical boundaries are nowhere tangent
to each other except where they intersect; though even there, the tangency is only in
the entire (3+1)-dimensional space–time, not in any specific 3-dimensional constant-τ
hypersurface. However in the LW approximation, we shall assume that the spherical
boundaries are everywhere tangent to each other, both in the entire (3+1)-dimensional
space–time and in each 3-dimensional constant-τ hypersurface; this is because we are
treating the spherical boundaries as if they were approximately identical to the cubic
ones.
53
2.4. PROPAGATING PHOTONS ACROSS CELL BOUNDARIES
polyhedral cells, the meshing of constant-τ hypersurfaces at cell boundaries is
lost, with the hypersurfaces now meshing in an average manner only. In contrast,
the hypersurfaces in the original spherical cells, by construction, mesh exactly.
Finally, the idea of replacing polyhedral boundaries completely with a spherical
approximation seems closer in spirit to the original idea of Wigner and Seitz.
Therefore according to our choice of boundary geometry, if a photon exits its
current cell at a radius r1 = rb, we require it to enter the next cell at radius
r2 = rb. Because of cell 2’s spherical symmetry, we are free to choose any θ and
φ co-ordinate for the entry point.
We now present the conditions that photon trajectories must satisfy when
propagated across boundaries. These conditions are applied locally at any pair
of exit and entry points. Our conditions are founded upon Clifton and Ferreira’s
principle that any physical quantity should be independent of which cell’s co-
ordinate system an observer co-moving with the boundary may choose to use. In
the context of photon trajectories, we require that
1) the photon frequency match across the boundary,
u1 ·k1 = u2 ·k2, (2.4.1)
2) and the projection of the photon’s 4-momentum onto the vector nr orthog-
onal to the boundary match across the boundary,6
nr1 ·k1 = −nr2 ·k2; (2.4.2)
it can be shown that (nr)a = (0,
√
Eb, 0, 0) for both the LW and CF co-
ordinate systems. There is a negative sign in the above equation because nr
always points radially out of its respective cell, whereas the radial direction
of k would be out of one cell and into the next.
These conditions along with the normalisation k ·k = 0 are sufficient to deduce
the components of k2 in terms of u1, u2, and k1.
The vectors u and nr imply a decomposition of k into the form
k = −(k·u)u+ (k·nr)nr + kΩ, (2.4.3)
6Because of the normalisation condition, this second requirement is equivalent to requiring
that the space-like projection of k onto the boundary be identical in both cells.
54
2. LINDQUIST–WHEELER MODELS
where it can be shown that kΩ·u = kΩ·nr = 0. We note that because of the cell’s
spherical symmetry, we can always choose polar co-ordinates such that k lies in
the θ = pi
2
plane, thereby allowing us to suppress the θ co-ordinate. In this case,
kΩ would take the form (kΩ)a = (0, 0, 0, kΩ). The above conditions imply that
(kΩ1 )
2 = (kΩ2 )
2, and by a suitable choice of the φ co-ordinates, we can always make
kΩ1 = k
Ω
2 . We also note that there is no physical reason for the photon trajectory
to refract when passing through a boundary, and therefore the θ = pi
2
planes of
the two cells should be aligned.
Using the above decomposition and the two boundary conditions, we can
therefore express the photon’s 4-momentum k2 in the new cell as
k2 = −(k1 ·u1)u2 − (k1 ·nr1)nr2 + kΩ1 . (2.4.4)
2.5 Numerical implementation of the LW model
To numerically simulate photons propagating through the lattice universe, we
have chosen to implement Williams and Ellis’ scheme [65] for propagating photons
through a discretised Schwarzschild space–time. Their discretisation actually
comes from Regge calculus, and although a full exposition of the formalism of
Regge calculus will not come until the subsequent chapter, it is not required here;
we shall simply present and implement Williams and Ellis’ scheme rather than
re-derive it, although we shall make some modifications to the geodesic-tracing
rules to improve numerical accuracy.
Under Willliams and Ellis’ scheme, Schwarzschild space–time is discretised as
follows. A grid is constructed in the Schwarzschild space–time such that along any
particular gridline, only one Schwarzschild co-ordinate (t, r, θ, φ) changes while
the other three are held constant. The lines intersect at constant ∆t, ∆r, ∆θ, or
∆φ intervals, thus forming the edges of curved rectangular blocks. Each of these
blocks gets mapped to flat rectangular Regge blocks such that the straight edges
of the Regge blocks have the same lengths as the curved edges of the original
blocks; Figure 2.12 illustrates an example of a curved Schwarzschild block with
its corresponding Regge block. The Schwarzschild co-ordinates of the original
blocks’ vertices (ti, rj, θk, φl) are now taken over as labels for the Regge blocks’
vertices. As the Regge blocks are flat, the metric inside is simply the Minkowski
metric.
55
2.5. NUMERICAL IMPLEMENTATION OF THE LW MODEL
(ri, φi+1, ti)
t φ
r
(ri+1, φi+1, ti)
(ri, φi, ti+1)
(ri, φi+1, ti+1)
(ri+1, φi+1, ti+1)
(ri+1, φi, ti)
(ri, φi, ti)
(ri+1, φi, ti+1)
τ ψ
ρ
(ρi, ψi+1, τi+1)
(ρi, ψi+1,−τi+1)
(−ρi,−ψi, τi)
(−ρi,−ψi,−τi)
(−ρi, ψi, τi)
2di (ρi,−ψi+1,−τi+1)
(ρi,−ψi+1, τi+1)
(−ρi, ψi,−τi)
Figure 2.12: On the left is an example of the original Schwarzschild block with one
angular co-ordinate suppressed, and on the right is the Regge block to which it is
mapped. (τ, ρ, ψ) is a Minkowski co-ordinate system within the Regge block.
Suppose a Regge block had vertices (ti, rj, θk, φl) and (ti+1, rj+1, θk+1, φl+1);
then we shall use the label (ti, rj, θk, φl) to refer to this block as a whole. The
edge-lengths for this block are
d[(ti, rj, θk, φl), (ti+1, rj, θk, φl)] =
√
1− 2m
rj
(
ti+1 − ti
)
,
d[(ti, rj, θk, φl), (ti, rj+1, θk, φl)] =
∫
rj+1
rj
dr√
1− 2m
r
,
d[(ti, rj, θk, φl), (ti, rj, θk+1, φl)] = rj(θk+1 − θk),
d[(ti, rj, θk, φl), (ti, rj, θk, φl+1)] = rj sin θk(φl+1 − φl).
Since photons always move in planes, we can always suppress one of the angu-
lar co-ordinates by setting θ to be pi/2. We then set-up a Minkowski co-ordinate
system (τ, ρ, ψ) in the block, with block vertices located at (± τi,−ρi,±ψi) and
56
2. LINDQUIST–WHEELER MODELS
(± τi+1, ρi,±ψi+1), where τi, ρi, and ψi are defined by7
2τi =
√
1− 2m
ri
(
tj+1 − tj
)
,
2ψi = ri(φj+1 − φj),
ρi =
[
d2i +
(τi+1 − τi)2
4
− (ψi+1 − ψi)
2
4
]1/2
,
(2.5.1)
and where8
2di =
∫
ri+1
ri
dr√
1− 2m
r
=
{
r
√
1− 2m
r
− 2m ln
[√
r
2m
−
√
r
2m
− 1
]}ri+1
ri
.
(2.5.2)
Williams and Ellis propagate geodesics through Regge blocks on the principle
that geodesics should follow straight lines both within a block and on crossing
from one block into the next. There is an apparent refraction of the geodesic
in crossing into a new block because the (τ, ρ, ψ) co-ordinate systems of the two
blocks are not aligned; therefore the same tangent vector of a geodesic would
be represented differently in different blocks’ co-ordinate systems. Williams and
Ellis have demonstrated that their scheme successfully reproduces the orbits and
redshifts of particles travelling in Schwarzschild space–time [65,66].
Referring to Figure 2.12, we now summarise the rules for propagating geo-
desics from one block into the next. Suppose the particle is at position (τ0, ρ0, ψ0)
and travelling in direction ka = (k1, k2, k3). It exits the block at (τ
′, ρ′, ψ′) =
(τ0, ρ0, ψ0) + λk, where the value of λ depends on the face exited.
1) If the particle exits the block by the top face, then
λ =
τ˜ − τ0 + ρ0 tanh βi
k1 − k2 tanh βi , (2.5.3)
7Note that an arbitrary index j appears in the expressions for τi and ψi. However, because
both (tj+1 − tj) and (φj+1 − φj) are actually independent of j, it does not matter what value
of j is used to calculate τi and ψi.
8In [65], there was a missing factor of 2m in front of the logarithm for this equation; this
factor has been restored here.
57
2.5. NUMERICAL IMPLEMENTATION OF THE LW MODEL
where τ˜ = (τi+1 + τi)/2 and tanh βi = (τi+1− τi)/2ρi. In the new block, the
particle’s new trajectory is given by applying to k the matrix cosh 2βi − sinh 2βi 0− sinh 2βi cosh 2βi 0
0 0 1
 , (2.5.4)
and the particle’s starting position is given by (−τ ′, ρ′, ψ′).
2) If the particle exits by the back face, then
λ =
ψ˜ − ψ0 + ρ0 tanαi
k3 − k2 tanαi , (2.5.5)
where ψ˜ = (ψi+1 + ψi)/2 and tanαi = (ψi+1 − ψi)/2ρi. In the new block,
the particle’s trajectory k gets transformed by1 0 00 cos 2αi sin 2αi
0 − sin 2αi cos 2αi
 , (2.5.6)
and the particle’s starting position is (τ ′, ρ′,−ψ′).
3) If the particle exits by the front face, then9
λ =
−ψ˜ − ψ0 − ρ0 tanαi
k3 + k2 tanαi
, (2.5.7)
where ψ˜ and tanαi are the same as above. In the new block, the particle’s
trajectory k gets transformed by the same rotation matrix as (2.5.6) but
with angle −αi instead. The particle’s starting position in the new block is
again (τ ′, ρ′,−ψ′), since ψ′ < 0 at the front face of the original block.
4) If the particle exits by the right/left face, then
λ =
±ρi − ρ0
k2
, (2.5.8)
and ρ′ is simply ρ′ = ±ρi. There is no refraction as the particle enters the
9In [65], there were a few sign errors for this equation which have been corrected here.
58
2. LINDQUIST–WHEELER MODELS
next block. Its new starting position is (τ ′,−ρi+1, ψ′) if entering the block
to the right and (τ ′, ρi−1, ψ′) if entering the block to the left.
However from numerical simulations, we have discovered an empirical relation
for the tangent 4-vectors of radially out-going time-like geodesics. This has led
us to introduce a correction to the above propagation rules. We have found that
as a function of ri, these 4-vectors obey
uaRegge =
(
τ˙ , ρ˙, ψ˙
)
=
(1− 2mr˜max
1− 2m
ri
)1/2
,
(
2m
ri
− 2m
r˜max
1− 2m
ri
)1/2
, 0
 , (2.5.9)
where r˜max is a constant of motion. Our numerical results supporting this have
been presented in Appendix B.
This relation can be derived analytically by comparing uRegge with its coun-
terpart 4-vector in continuum Schwarzschild space–time. The tangent 4-vector of
a radially out-going time-like geodesic in Schwarzschild space–time is
uaSchwarz =
(
t˙, r˙, Ω˙
)
=

√
1− 2m
rmax
1− 2m
r
,
(
2m
r
− 2m
rmax
)1/2
, 0
 , (2.5.10)
where rmax is also a constant of motion and is related to E in (2.1.2) by rmax =
2m/(1 − E). For in-going geodesics, the radial components of both uRegge and
uSchwarz would have an additional negative sign. If we take the scalar product
of uRegge and uSchwarz with the unit vector in the time direction, that is, with
τˆ a = (1, 0, 0) for Regge space–time and tˆ a =
((
1− 2m
r
)−1/2
, 0, 0
)
for continuum
Schwarzschild space–time, we have that
τˆ ·uRegge =
(
1− 2m
r˜max
1− 2m
ri
)1/2
,
and that
tˆ·uSchwarz =
(
1− 2m
rmax
1− 2m
r
)1/2
.
If we identify rmax with r˜max, then these two expressions are identical whenever
r = ri in the continuum space–time. Similarly, if we take the scalar product
59
2.5. NUMERICAL IMPLEMENTATION OF THE LW MODEL
with the unit vector in the radial direction, that is, with ρˆ a = (0, 1, 0) for Regge
space–time and r˜ a =
(
0,
(
1− 2m
r
)1/2
, 0
)
for continuum space–time, we have that
ρˆ·uRegge = −
(
2m
ri
− 2m
r˜max
1− 2m
ri
)1/2
,
and that
rˆ ·uSchwarz = −
(
2m
ri
− 2m
rmax
1− 2m
r
)1/2
.
Again if we identify rmax with r˜max, then the two expressions are also identical
whenever r = ri in the continuum space–time. Therefore provided r˜max = rmax,
we see that (2.5.9) is indeed the Regge analogue of (2.5.10). Furthermore, the
choice of rmax and r˜max determines whether the resulting particle orbit will be
closed or open in the corresponding space–time. If 2m/rmax > 0, then the orbit in
the continuum space–time will be closed and rmax would indeed be the maximum
radius of the orbit. Similarly, we found that if 2m/r˜max > 0, then the orbit in
Regge space–time will also be closed, and the maximum radius would be r˜max. If
2m/rmax = 0, then the geodesic will just reach spatial infinity in the continuum
space–time, and (2.5.10) gives the particle’s escape velocity as a function of its
radial position r > 2m. Similarly, we show in Appendix B that when 2m/r˜max =
0, then (2.5.9) gives the escape velocity for a test particle in the Regge space–
time. Finally if 2m/rmax < 0, then the orbit in continuum space–time will be
open. And if 2m/r˜max < 0, then the orbit in Regge space–time will also be
open, as we show in Appendix B. We shall henceforth make the identification of
r˜max = rmax.
Inspired by this, we can generalise the expression for uRegge to any geodesic in
Regge Schwarzschild space–time. The Lagrangian for particles moving in contin-
uum Schwarzschild space–time can be written as
L = −
(
1− 2m
r
)
t˙2 +
r˙2(
1− 2m
r
) + r2Ω˙2,
where the dot denotes differentiation with respect to some parameter λ. Since
60
2. LINDQUIST–WHEELER MODELS
0 = ∂L
∂t
, we have a constant of motion E, which we define by the relation
E :=
(
1− 2m
r
)2
t˙2. (2.5.11)
Similarly, since 0 = ∂L
∂Ω
, we have another constant of motion J defined by
J := r2Ω˙. (2.5.12)
These two constants correspond to the square of the particle’s energy per unit
mass at radial infinity and to the angular momentum. For radial time-like
geodesics, E here is the same as E in (2.1.2). In terms of these constants, the
tangent 4-vector vSchwarz to the particle’s geodesic can be expressed as
vaSchwarz =
(
t˙, r˙, Ω˙
)
=
( √
E(
1− 2m
r
) , r˙, J
r2
)
, (2.5.13)
which is clearly a function of r alone, since r˙ can be deduced from t˙ and Ω˙ through
the normalisation of vSchwarz.
Let vRegge denote the Regge analogue of vSchwarz. If we assume that vRegge
and vSchwarz are related in a manner analogous to how uRegge and uSchwarz are
related, then we can deduce the components of vRegge from vSchwarz. Specifically,
by equating τˆ ·vRegge to tˆ·uSchwarz, we deduce τ˙ to be
τ˙ =
√
E(
1− 2m
r
) 1
2
.
Similarly by equating ψˆ ·vRegge to Ωˆ ·uSchwarz, where ψˆa = (0, 0, 1) and Ωˆa =
(0, 0, r−1), we deduce ψ˙ to be
ψ˙ =
J
r
.
Thus the components of vRegge are given by
vaRegge =
 √E(
1− 2m
r
) 1
2
, ρ˙,
J
r
 , (2.5.14)
with ρ˙ deducible from the normalisation of vRegge. Finally, it is easy to verify that
61
2.5. NUMERICAL IMPLEMENTATION OF THE LW MODEL
ρˆ·vRegge and rˆ ·vSchwarz are then consistent.
Our generalised expression for tangent 4-vectors can be used for both null and
time-like geodesics following both radial and non-radial trajectories. For time-
like radial geodesics, where J = 0, it can be shown that vRegge reduces to uRegge.
As we wish to simulate geodesics in continuum rather than Regge Schwarzschild-
cells, we have introduced a correction to the Williams–Ellis scheme: whenever
a geodesic crosses a right or left face, its tangent 4-vector is changed back to
(2.5.14).
Having established the modified Williams–Ellis scheme, we now discuss its
application to the Schwarzschild-cell. The boundary is simulated by propagating
a test particle that is co-moving with the boundary. Whenever the photon reaches
the same radial and time co-ordinates as the boundary particle, then at that point
the photon exits one cell and enters the next, and we need to apply conditions
(2.4.1) and (2.4.2) to determine the photon’s new trajectory. However vector
components will differ from previously as we are now working in Regge space–
time.
Let ua = (uτ , uρ, 0) be the boundary particle velocity and ka = (kτ , kρ, kψ)
be the photon’s 4-momentum at the boundary. We shall use subscripts 1 and 2
to indicate the cell being exited and the cell being entered respectively. As in
(2.4.3), we decompose the Regge vector k into
k = νu+ nρ + nψ, (2.5.15)
where ν = −u·k, (nψ)a = (0, 0, nψ), and nρ satisfies nρ ·u = nρ ·nψ = 0. If nˆρ is
the normalised vector of nρ, that is nρ = n nˆρ, then the components of nˆρ are
(nˆρ)a = (uρ, uτ , 0),
as this satisfies all orthogonality relations required of nρ. We then deduce n to
be
n = nˆρ ·k = (uτkρ − uρkτ ) . (2.5.16)
Condition (2.4.1) implies that
ν1 = ν2. (2.5.17)
62
2. LINDQUIST–WHEELER MODELS
Condition (2.4.2) in this context is equivalent to
k1 ·nˆρ1 = −k2 ·nˆρ2,
which implies that
n1 = −n2. (2.5.18)
As at the end of Section 2.4, these conditions imply that (nψ1 )
2 = (nψ2 )
2, and we
again have freedom to choose polar co-ordinates such that
nψ1 = n
ψ
2 , (2.5.19)
which we shall henceforth assume to be the case. Using relation (2.5.16), condi-
tions (2.5.17) and (2.5.18), relation (2.5.19), and decomposition (2.5.15), we can
express the components of k2 as
ka2 = (νu
τ
2 + (u
ρ
1k
τ
1 − uτ1kρ1)uρ2, νuρ2 + (uρ1kτ1 − uτ1kρ1)uτ2, kψ1 ). (2.5.20)
We note that constant E in (2.5.14) will differ between k1 and k2, but J will
remain the same since kψ = J
r
is identical on both sides of the boundary.
2.6 Redshifts from lattice universe simulations
We have simulated the propagation of photons through multiple Schwarzschild-
cells for closed, flat, and open LW universes. Each time, a photon was propa-
gated outwards in various directions from an initial radius of 10R, R being the
Schwarzschild radius. The initial direction of travel was given, in terms of block
co-ordinates, by
ka = (1, cos θn, sin θn),
with the range of θn starting from the purely tangential direction of θ0 = pi/2 and
decreasing until the direction was almost completely radial. Both LW and FLRW
redshift factors were computed whenever the photon, while travelling outwards
again, passed an observer co-moving with the source.
We can in principle re-scale the initial k by a constant factor λ, but this should
not affect the redshifts: such a re-scaling would not alter the corresponding null
geodesic, so the space–time points at which the observer and photon meet would
63
2.6. REDSHIFT RESULTS
remain unchanged; at these points, the only change in the photon’s frequencies
would be a re-scaling by λ, but this factor would then cancel out of the redshifts.
We verified this invariance of the redshifts by simulating a photon with different
scalings of k, and the results, not shown, were indeed invariant for at least 11
significant figures; any discrepancies can be attributed to numerical error.
All length-scales in our simulations have been specified in terms of R. By
simply re-scaling R in one set of results, we can readily obtain the results for
an equivalent simulation where the only difference is the magnitude of R. In
particular, because redshifts are dimensionless quantities, they would not depend
on the choice of R, so we therefore made the arbitrary choice of setting R = 1
for all our simulations.
We have chosen the dimensions of our Regge block as follows. The angular
length 2ψi was chosen so that ∆φ =
2pi
3×1010 . For the closed universe, the radial
length 2di was chosen to represent a fixed interval of ∆r = 10
−7R. For the flat
and open universes, it was instead chosen to lengthen for blocks further away from
the cell centre. This lengthening was implemented so as to increase computation
speed with only a marginal expense to the accuracy: the underlying continuum
Schwarzschild space–time becomes flatter as one moves further away from the
centre, so a flat Regge block would approximate the region more accurately. Our
exact method for determining the block’s length has been described in Appendix
C. Although ∆r is no longer constant, it is constrained to be an integral multiple
of a minimum interval ∆r0, a parameter we can freely specify, and the Regge
grid is fixed so that its first set of blocks covers the region between R+ ∆r0 and
R + 2∆r0.
10 As we shall see below, ∆r0 was chosen according to which universe
was being simulated. Finally, the temporal interval was chosen to be ∆t = 10 ∆r0
for the flat and open universes and simply ∆t = 10 ∆r for the closed universe.
In this section we shall present the redshift results of our simulations. We
begin with the flat universe, for which E = 1. To investigate the effect of the
10As an informal check on the accuracy, we simulated photons propagating across 100 cells in
the flat, E = 1 universe using both fixed and increasing block-lengths. The specific simulation
parameters were an initial Schwarzschild-cell radius of rb0 = 3 × 104R with an initial photon
radius of 10R; the Regge blocks’ non-radial dimensions were ∆t = 10 ∆r and ∆φ = 2pi/(3×107);
the radial dimension ∆r was fixed at 10−4R for the fixed-length simulation and was set to be
at least that length for the increasing-length simulation. The two simulations’ redshift results,
not shown, agreed for at least five significant figures, even at the largest radii; however the fixed
block-length simulation required about 6.5 hours of computation time, while the increasing
block-length simulation required only 20 minutes.
64
2. LINDQUIST–WHEELER MODELS
initial cell size rb0 on the redshifts, we have simulated flat universes for a range of
initial sizes from rb0 = 3× 104R to rb0 = 108R. Following the example of Clifton
and Ferreira, we have chosen the largest initial size to approximate cells with
Milky Way–like masses at their centres, as this is thought to best represent our
actual universe. Depending on the initial cell size, ∆r0 ranged from ∆r0 = 10
−5R
for rb0 = 3× 104R to ∆r0 = 10−3R for rb0 = 108R.
Figure 2.13 plots zLW against zFLRW for the universe where rb0 = 3 × 104R.
Each trajectory was traced across 50 cells; only results for 15 trajectories are
shown, although we simulated trajectories for 30 angles θn ranging from θ0 =
1
2
pi to θ29 =
311
3000
pi in decrements of 41
3000
pi. Apart from a brief curve at the
start, each graph clearly demonstrates a strong linear relationship between zLW
and zFLRW , and the gradient of the line is different for each angle. There are
several differences, however, between our results and those of Clifton and Ferreira.
Clifton and Ferreira’s graphs showed more initial scatter, which varied depending
on the trajectory’s initial angle, but their graphs would eventually converge upon
a common mean graph as the photon passed through an increasing number of
cells. Our graphs do not display such scatter nor any common mean. However,
their common mean agreed rather closely with FLRW redshifts, as it was given
by 1 + zLW ≈ (1 + zFLRW )0.98 [61]; thus in their model, zLW and zFLRW are almost
completely linear. Our graphs, apart from the initial curve, are completely linear,
but the gradients depend on the trajectory followed.
However Clifton and Ferreira’s relation has the desirable feature of passing
through the origin, since there must be zero redshift at the start of any trajectory;
that is, both zLW and zFLRW must be zero. All our graphs instead show a jump
between the origin and the next data point, and this was true of our simulations
of all other universes as well. When we plotted the redshifts against the radius
of observation for the rb0 = 3 × 104R flat universe, as shown in Figure 2.14,
we found that the zFLRW graph would extend naturally outwards from the zero-
redshift point at the starting radius of 10R, but that the zLW graph would jump
suddenly from the zero-redshift point to the next data point. Suppose we ignored
our current zero-redshift data point and assumed the photon actually began its
trajectory at the next data point; then when we re-calculated all subsequent
redshifts based on our new initial frequency for the photon, we found that the
resulting zLW graph progresses naturally from the new zero-redshift point to the
65
2.6. REDSHIFT RESULTS
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70
z L
W
zFLRW
zLW = 1.0994 zFLRW + 0.1225, RMS of residuals = 7.493× 10−4
zLW = 1.0696 zFLRW + 0.0920, RMS of residuals = 7.290× 10−4
zLW = 1.0400 zFLRW + 0.0618, RMS of residuals = 7.088× 10−4
zLW = 1.0108 zFLRW + 0.0321, RMS of residuals = 6.890× 10−4
zLW = 0.9823 zFLRW + 0.0030, RMS of residuals = 6.695× 10−4
zLW = 0.9547 zFLRW − 0.0253, RMS of residuals = 6.507× 10−4
zLW = 0.9281 zFLRW − 0.0524, RMS of residuals = 6.326× 10−4
zLW = 0.9028 zFLRW − 0.0783, RMS of residuals = 6.153× 10−4
zLW = 0.8789 zFLRW − 0.1026, RMS of residuals = 5.991× 10−4
zLW = 0.8567 zFLRW − 0.1253, RMS of residuals = 5.839× 10−4
zLW = 0.8362 zFLRW − 0.1462, RMS of residuals = 5.700× 10−4
zLW = 0.8177 zFLRW − 0.1651, RMS of residuals = 5.574× 10−4
zLW = 0.8012 zFLRW − 0.1819, RMS of residuals = 5.462× 10−4
zLW = 0.7870 zFLRW − 0.1964, RMS of residuals = 5.365× 10−4
zLW = 0.7750 zFLRW − 0.2086, RMS of residuals = 5.285× 10−4
1.5708
1.4849
1.3991
1.3132
1.2273
1.1414
1.0556
0.9697
0.8838
0.7980
0.7121
0.6262
0.5404
0.4545
0.3686
Figure 2.13: A plot of zLW against zFLRW for 15 trajectories in a flat universe with
an initial cell size of rb0 = 3 × 104R. The initial angle θn of the trajectory is given in
the legend. Each trajectory was traced across 50 cells. A linear regression has been
performed for each graph, with the first five data points excluded so as to focus only
on the linear regime. The regression equations and corresponding root-mean-squares
of the residuals are listed above in order of decreasing θn.
66
2. LINDQUIST–WHEELER MODELS
−0.20
−0.10
0.00
0.10
0.20
0.30
0.40
0 2000 4000 6000 8000 10000 12000 14000
z
robs/R
zFLRW for θ0 =
1
2
pi
zLW for θ0 =
1
2
pi
zFLRW for θ29 =
311
3000
pi
zLW for θ29 =
311
3000
pi
Figure 2.14: A plot of redshifts z against the radius of observation robs for photons
travelling in the flat universe with an initial cell size of rb0 = 3 × 104R. The photons
were travelling initially in the directions of θ0 =
1
2pi and θ29 =
311
3000pi. Plots for both
LW and FLRW redshifts are shown. All four series start at the same data point, and
there is a clear jump in the LW graphs between the first and second data points.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
2000 4000 6000 8000 10000 12000 14000
z
robs/R
zFLRW
zLW
Figure 2.15: A plot of redshifts z against the radius of observation robs for a photon
starting instead with a position and trajectory corresponding to the second data point
of the θ0 =
1
2pi trajectory in Figure 2.14. Plots for both LW and FLRW redshifts are
shown. This time, both graphs extend smoothly out from the zero-redshift data point.
67
2.6. REDSHIFT RESULTS
next data point without any jumps, as shown in Figure 2.15.11
To investigate this initial jump further, we note that in universes using CF co-
ordinates, the frequency νLW measured by a co-moving observer can be expressed
as a function of the radius at which the measurement is made, that is, the radius
robs at which the photon and observer intercept. This implies the redshifts zLW
can be expressed as a function of robs. Recall that the 4-velocity of a co-moving
observer is given by (2.5.14) in Regge Schwarzschild space–time, with E = Eb
and J = 0; this vector is a function of the observer’s radial position alone. The 4-
momentum of the photon is given by the same relation as well but with E = Eph
and J an arbitrary constant; as long as the photon does not cross into the next
cell, Eph will be constant, so this vector is also a function of the photon’s radial
position alone. When the observer and photon intercept, they will have the same
radial position, that is, the same value for r, and we have denoted above this
common r by robs. By taking the scalar product of these two vectors, we obtain
the measured frequency as a function depending only on the value of robs,
νLW =
√
EbEph
1− 2m
robs
−
[(
Eb
1− 2m
robs
− 1
)(
Eph
1− 2m
robs
− J
2
r2obs
)] 1
2
. (2.6.1)
We note that if the photon were to cross into the next cell, this relation would
still hold, but from the boundary conditions, Eph only would change.
12 In Figure
2.16 and Figure 2.17, we have plotted this function for photons travelling at ini-
tial angles of 1
2
pi and 311
3000
pi respectively in the rb0 = 3 × 104R flat universe. We
have included plots for different Eph corresponding to the first few cells traversed.
We have also simulated a photon’s propagation across the first cell for both tra-
jectories, and for a selection of radii, we have computed the frequencies that a
co-moving observer would measure if at those radii. These numerical results are
also included in the figures, and we see that they agree very closely with their
analytic counterparts. In Figure 2.18, we show redshifts zLW instead against robs
11We did check this result by simulating a photon starting at the same radius and direction
as that of the original photon when it generated the second data point; the resulting graph was
indeed identical to that of Figure 2.15.
12Although we have an analytic function for zLW based on robs, we must rely on the simulation
to tell us where robs is, that is, where the photon intercepts the observer. And although we can
also determine Eph analytically from the boundary condition, to do this, we still need to know
the radius at which the photon and cell boundary intercept, and we must rely on the simulation
to tell us this radius as well.
68
2. LINDQUIST–WHEELER MODELS
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
10 100 1000 10000 100000
ν L
W
robs/R
cell 1 (Eph = 0.9), numerical
cell 1 (Eph = 0.9), analytic
cell 2 (Eph = 0.8795), analytic
cell 3 (Eph = 0.8596), analytic
cell 4 (Eph = 0.8403), analytic
cell 5 (Eph = 0.8214), analytic
cell 6 (Eph = 0.8032), analytic
cell 7 (Eph = 0.7854), analytic
Figure 2.16: A plot of photon frequencies νLW against the radius of observation robs.
Each graph represents the photon’s frequencies within a single cell. The photon’s initial
direction is θ0 =
1
2pi, and the initial cell size is rb0 = 3×104R. These are the frequencies
that would be seen by a co-moving observer if the photon intercepted the observer at
robs. The analytic frequencies are given by (2.6.1), and graphs for the first seven cells
traversed are shown. Frequencies were also computed numerically by simulating the
propagation of a photon across the first cell only. The corresponding graph is shown
and completely overlaps with its analytic counterpart.
69
2.6. REDSHIFT RESULTS
for the θ0 =
1
2
pi photon travelling through cell 1 only.
Based on these graphs, we make some speculations on the origin of the jump
as well as the initial curve in the zLW vs zFLRW graphs. We note that for large
robs, the frequency asymptotically approaches
νasym =
√
Eph
(√
Eb −
√
Eb − 1
)
=
√
Eph for Eb = 1.
Equivalently, the redshift asymptotically approaches
zasym =
νe√
Eph
(√
Eb −
√
Eb − 1
) − 1
=
νe√
Eph
− 1 for Eb = 1.
As the photon traversed ever more cells in our simulations, we found that it would
intercept the co-moving observer at ever larger robs. So after enough cells, zLW
should be very close to the asymptotic value zasym. Also from our simulations,
we found that Eph would decrease with each subsequent cell implying that zasym
would increase, which is consistent with the zLW vs zFLRW graphs’ positive gradi-
ents. Thus the asymptotic, large-radius behaviour of the redshifts is responsible
for the linear behaviour of the zLW vs zFLRW graphs. The graphs’ initial behaviour
however must be explained by the behaviour of the redshifts at low robs, where the
photon is still sufficiently close to the central mass to feel its influence strongly.
We note that the frequencies always approach νasym from below, implying that
the redshifts always approach zasym from above. This is consistent with the initial
curve in the zLW vs zFLRW graphs, as this curve converges into the linear regime
from above as well. At much smaller robs however, the redshift may deviate much
more significantly from the asymptotic value. In particular, the deviation of the
zero-redshift point, where robs is smallest, may be much greater than the devi-
ation of the next data point, where robs may have increased sufficiently for the
redshift to be much closer to the asymptotic value. This would account for the
sudden initial jump seen in the zLW vs zFLRW graphs. Therefore, the initial jump
and the initial curve in the zLW vs zFLRW graphs can be attributed to the photon
being closer to the central mass and thus feeling its effects more strongly.
To test our conjecture, we simulated photons in the same universe but starting
70
2. LINDQUIST–WHEELER MODELS
0.65
0.70
0.75
0.80
0.85
0.90
0.95
10 100 1000 10000 100000
ν L
W
robs/R
cell 1 (Eph = 0.9), numerical
cell 1 (Eph = 0.9), analytic
cell 2 (Eph = 0.8795), analytic
cell 3 (Eph = 0.8596), analytic
cell 4 (Eph = 0.8403), analytic
cell 5 (Eph = 0.8214), analytic
cell 6 (Eph = 0.8032), analytic
cell 7 (Eph = 0.7854), analytic
Figure 2.17: The equivalent plot to Figure 2.16 for frequencies of a photon initially
in the direction of θ29 =
311
3000pi instead. Once again, the numerical and analytic graphs
for cell 1 overlap completely.
71
2.6. REDSHIFT RESULTS
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
10 100 1000 10000 100000
z L
W
robs/R
cell 1 (Eph = 0.9), numerical cell 1 (Eph = 0.9), analytic
Figure 2.18: The equivalent plot to Figure 2.16 but showing the redshifts zLW of cell
1 only against the radius of observation robs.
from a large radius, specifically r = 104R, such that the central mass’ influence
would be much reduced. Figure 2.19 shows the resulting plot of zLW against
zFLRW for the most radial and the most tangential trajectories simulated. The
graphs now follow completely straight lines passing very close to the origin, with
the initial jump completely gone and the initial curve nearly absent. All other
trajectories at intermediate angles displayed similar behaviour, with the graphs’
gradients increasing as the initial trajectory became more tangential. We also
note that the gradients are nearly unity, indicating that zLW is nearly identical
to zFLRW . In Figure 2.20, we have plotted the difference between the zLW graphs
and the reference graph of z˜LW = zFLRW . The deviation from linearity at low robs
becomes clearer now, although we also see that it is short-lived. From this figure,
we see that linearity is relatively well established by zFLRW ≈ 0.2; thus for each
trajectory, we have regressed the corresponding zLW vs zFLRW graph for the region
zFLRW > 0.2. All regression equations and corresponding root-mean-squares of the
residuals have been listed in Table 2.1, and two of these regressions are shown in
Figure 2.19. These equations show that our observations for the graphs in Figure
2.19 holds generally for all trajectories, specifically that the gradients are nearly
72
2. LINDQUIST–WHEELER MODELS
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
zLW = 0.9973 zFLRW + 0.0048, RMS of residuals = 1.84× 10−5
zLW = 0.9884 zFLRW − 0.0017, RMS of residuals = 7.94× 10−5
z L
W
zFLRW
1.5708 0.3686
Figure 2.19: A plot of zLW against zFLRW for a tangential and a nearly radial trajec-
tory in the flat universe with initial cell size of rb0 = 3× 104R. This is the equivalent
plot to that of Figure 2.13 but with the photon starting at 104R instead. The regres-
sion lines now pass very close to the origin, and the initial curve is nearly absent; we
also note that the gradients are nearly unity. Other trajectories, not shown, showed
similar behaviour.
73
2.6. REDSHIFT RESULTS
−0.010
−0.008
−0.006
−0.004
−0.002
0.000
0.002
0.004
0.006
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70
z L
W
−
z F
L
R
W
zFLRW
1.5708
1.4849
1.3991
1.3132
1.2273
1.1414
1.0556
0.9697
0.8838
0.7980
0.7121
0.6262
0.5404
0.4545
0.3686
Figure 2.20: A plot of zLW − zFLRW against zFLRW for 15 trajectories in the flat
universe with initial cell size of rb0 = 3× 104R. Each graph still shows an initial curve,
but it is small and short-lived, and linearity is soon established.
74
2. LINDQUIST–WHEELER MODELS
Initial angle Regression equation RMS of residuals
1.5708 zLW = 0.9973 zFLRW + 4.75× 10−3 1.84× 10−5
1.4849 zLW = 0.9966 zFLRW + 4.07× 10−3 1.88× 10−5
1.3991 zLW = 0.9959 zFLRW + 3.41× 10−3 2.03× 10−5
1.3132 zLW = 0.9952 zFLRW + 2.78× 10−3 2.29× 10−5
1.2273 zLW = 0.9944 zFLRW + 2.18× 10−3 2.63× 10−5
1.1414 zLW = 0.9937 zFLRW + 1.62× 10−3 3.05× 10−5
1.0556 zLW = 0.9929 zFLRW + 1.09× 10−3 3.53× 10−5
0.9697 zLW = 0.9922 zFLRW + 6.06× 10−4 4.04× 10−5
0.8838 zLW = 0.9915 zFLRW + 1.75× 10−4 4.89× 10−5
0.7980 zLW = 0.9908 zFLRW − 2.35× 10−4 5.49× 10−5
0.7121 zLW = 0.9902 zFLRW − 6.06× 10−4 6.07× 10−5
0.6262 zLW = 0.9897 zFLRW − 9.37× 10−4 6.62× 10−5
0.5404 zLW = 0.9892 zFLRW − 1.23× 10−3 7.12× 10−5
0.4545 zLW = 0.9888 zFLRW − 1.48× 10−3 7.57× 10−5
0.3686 zLW = 0.9884 zFLRW − 1.69× 10−3 7.94× 10−5
Table 2.1: Linear regression equations and the root-mean-squares of the corresponding
residuals for zLW versus zFLRW graphs from simulations of photons in the flat universe.
The initial cell size was rb0 = 3× 104R. This is the same simulation as for Figure 2.19
and Figure 2.20. The constants in the regression equations here are much smaller than
those in the regression equations in Figure 2.13.
unity and that the graphs pass very close to the origin. Thus, the influence of
the masses on redshifts has been significantly suppressed, and the redshifts now
behave nearly identically to FLRW, much like Clifton and Ferreira’s mean redshift
result.
For flat universes with larger initial cell sizes, our simulations yielded results
similar to those of the rb0 = 3 × 104R universe. Figure 2.21 to Figure 2.24
depict plots of zLW against zFLRW for the θ0 =
1
2
pi trajectory for universes with rb0
ranging from 105 to 108. All graphs again converged quickly to a straight line from
above, including graphs, not shown, for the other trajectories. One noticeable
development though was that the curve at the start of the graphs became more
pronounced as the initial size increased. However when we simulated photons
starting from a large radius again, the initial curve was again suppressed, and the
initial jump was again absent, as shown in Figure 2.25, Figure 2.26, and Table 2.2
for photons starting from r = 107R in the rb0 = 10
8R universe; the initial curve
is even more short-lived, as seen in Figure 2.26, and the regression lines for all
trajectories pass even closer to the origin, as demonstrated by the even smaller
75
2.6. REDSHIFT RESULTS
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
z L
W
zFLRW
Figure 2.21: A plot of zLW against zFLRW for the θ0 =
1
2pi photon trajectory in the
rb0 = 10
5R universe. ∆r0 was 10
−5R for this simulation.
0.120
0.125
0.130
0.135
0.140
0.145
0.150
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035
z L
W
zFLRW
Figure 2.22: A plot of zLW against zFLRW for the θ0 =
1
2pi photon trajectory in the
rb0 = 10
6R universe. ∆r0 was 10
−5R for this simulation.
76
2. LINDQUIST–WHEELER MODELS
0.115
0.116
0.117
0.118
0.119
0.120
0.121
0.122
0.123
0.124
0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
z L
W
zFLRW
Figure 2.23: A plot of zLW against zFLRW for the θ0 =
1
2pi photon trajectory in the
rb0 = 10
7R universe. ∆r0 was 10
−4R for this simulation.
0.1125
0.1130
0.1135
0.1140
0.1145
0.1150
0.1155
0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035
z L
W
zFLRW
Figure 2.24: A plot of zLW against zFLRW for the θ0 =
1
2pi photon trajectory in the
rb0 = 10
8R universe. ∆r0 was 10
−3R for this simulation.
77
2.6. REDSHIFT RESULTS
−0.0005
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035
zLW = 1.0215 zFLRW + 5.4× 10−5, RMS of residuals = 1.66× 10−6
zLW = 0.9967 zFLRW − 1.6× 10−7, RMS of residuals = 1.24× 10−8
z L
W
zFLRW
1.5708 0.3901
Figure 2.25: A plot of zLW against zFLRW for a tangential and a nearly radial tra-
jectory starting at radius 107R in the rb0 = 10
8R flat universe. Each trajectory was
traced across 15 cells. Compared to Figure 2.19, the regression lines here pass even
closer to the origin; the initial curve is even more suppressed; and the gradients are
even closer to unity. Other trajectories, not shown, showed similar behaviour.
78
2. LINDQUIST–WHEELER MODELS
−2.0× 10−5
0.0× 10−5
2.0× 10−5
4.0× 10−5
6.0× 10−5
8.0× 10−5
10.0× 10−5
12.0× 10−5
0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035
z L
W
−
z F
L
R
W
zFLRW
1.5708
1.4635
1.3561
1.2488
1.1414
1.0341
0.9268
0.8194
0.7121
0.6048
0.4974
0.3901
Figure 2.26: A plot of zLW−zFLRW against zFLRW for 15 trajectories in the rb0 = 108R
flat universe. Compared to Figure 2.20, the initial curve here is even smaller and more
short-lived.
79
2.6. REDSHIFT RESULTS
Initial angle Regression equation RMS of residuals
1.5708 zLW = 1.0215 zFLRW + 5.4× 10−5 1.66× 10−6
1.4635 zLW = 1.0200 zFLRW + 2.9× 10−5 1.45× 10−6
1.3561 zLW = 1.0167 zFLRW + 1.5× 10−5 1.04× 10−6
1.2488 zLW = 1.0130 zFLRW + 8.8× 10−6 6.97× 10−7
1.1414 zLW = 1.0095 zFLRW + 5.4× 10−6 4.62× 10−7
1.0341 zLW = 1.0066 zFLRW + 3.4× 10−6 3.08× 10−7
0.9268 zLW = 1.0041 zFLRW + 2.2× 10−6 2.04× 10−7
0.8194 zLW = 1.0020 zFLRW + 1.4× 10−6 1.31× 10−7
0.7121 zLW = 1.0002 zFLRW + 9.2× 10−7 6.34× 10−8
0.6048 zLW = 0.9988 zFLRW + 4.5× 10−7 3.10× 10−8
0.4974 zLW = 0.9976 zFLRW + 1.0× 10−7 7.08× 10−9
0.3901 zLW = 0.9967 zFLRW − 1.6× 10−7 1.24× 10−8
Table 2.2: Linear regression equations and the root-mean-squares of the corresponding
residuals for zLW versus zFLRW graphs from simulations of photons in the rb0 = 10
8R
flat universe. This is the same simulation as for Figure 2.25 and Figure 2.26. Regression
was performed on the zFLRW > 0.001 region. Both the regression equation constants
and the residuals are even smaller here than in Table 2.1.
constants in the regression equations in Table 2.2.
We also notice, in Figure 2.13, that the gradients of zLW vs zFLRW graphs de-
crease with θn. In fact, if we plot the gradients against cos(θn), we find a strongly
linear relationship between the two quantities, as shown in Figure 2.27. As the
figure shows, this is also true of the gradients for all the other flat universes we
simulated. Since cos(θn) determines the radial component ρ˙init of the photon’s
initial velocity, this implies that the gradients are linearly related to ρ˙init/τ˙init. As
the photon’s initial velocity becomes increasingly radial, the photon’s frequency
becomes increasingly blueshifted relative to the FLRW redshift. We believe this
is due to the central mass’ stronger influence on the photon, as a more radial
trajectory would bring the photon closer to the central mass. Quite remarkably,
this influence can alter the lattice universe redshifts zLW from their FLRW coun-
terparts zFLRW by as much as 30%, as shown by the right end of the rb0 = 10
8
graph. Since the lattice universe can generate lower redshift values, an observer
fitting these redshifts to an FLRW model could significantly underestimate the
age of the photons’ source.
Everything we noticed about redshifts in the flat universe also applied to
open universes. We simulated open universes for values of Eb in the range of
80
2. LINDQUIST–WHEELER MODELS
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
gr
ad
ie
n
t
of
z L
W
v
s
z F
L
R
W
cos(θ)
gradient = −0.3477 cos(θ) + 1.0994
RMS of residuals = 7.01× 10−6
gradient = −0.3380 cos(θ) + 1.0689
RMS of residuals = 2.11× 10−6
gradient = −0.3322 cos(θ) + 1.0504
RMS of residuals = 2.16× 10−7
gradient = −0.3234 cos(θ) + 1.0226
RMS of residuals = 2.11× 10−8
gradient = −0.3104 cos(θ) + 0.9814
RMS of residuals = 1.99× 10−9
rb0 = 3× 104
rb0 = 10
5
rb0 = 10
6
rb0 = 10
7
rb0 = 10
8
Figure 2.27: A plot of the gradients of zLW vs zFLRW graphs against cos(θn) for flat
universes of different initial cell size rb0 . A linear regression has been performed on
each graph, and the regression equations are displayed in order of increasing rb0 .
1.1 ≥ Eb > 1. In each simulation, the initial cell size was fixed to be rb0 = 3×104,
and photons were propagated along 12 trajectories with initial angles ranging
from θ0 =
1
2
pi to θ11 =
149
1200
pi in decrements of 41
1200
pi. Depending on Eb, ∆r0
ranged from 10−5R for Eb = 1 to 10−3R for Eb = 1.1. Figure 2.28 shows a clear
linear relationship again between zLW and zFLRW for the Eb = 1.1 universe. The
regression equations’ non-zero intercepts indicate the presence again of a jump
between the zero-redshift point and the first non-zero redshift point. Similar
behaviour was seen for universes corresponding to other values of Eb. Figure
2.29 shows that for all Eb, there is again a clear negative linear relation between
the gradients of zLW vs zFLRW graphs and cos(θn), or equivalently between the
gradients and ρ˙init/τ˙init.
To see how the gradients of zLW vs zFLRW change with Eb, we have plotted the
gradients against Eb in Figure 2.30 and Figure 2.31 for the trajectories of θ0 =
1
2
pi
and θ11 =
149
1200
pi respectively. Both graphs indicate that the gradients approach
some asymptotic value as Eb increases, and we have found similar asymptotic
81
2.6. REDSHIFT RESULTS
0
2000
4000
6000
8000
10000
12000
14000
0 2000 4000 6000 8000 10000 12000
z L
W
zFLRW
zLW = 1.1109 zFLRW + 0.0908, RMS of residuals = 0.0353
zLW = 1.0732 zFLRW + 0.0539, RMS of residuals = 0.0341
zLW = 1.0361 zFLRW + 0.0174, RMS of residuals = 0.0328
zLW = 0.9997 zFLRW − 0.0183, RMS of residuals = 0.0317
zLW = 0.9647 zFLRW − 0.0526, RMS of residuals = 0.0305
zLW = 0.9313 zFLRW − 0.0854, RMS of residuals = 0.0294
zLW = 0.9000 zFLRW − 0.1161, RMS of residuals = 0.0283
zLW = 0.8711 zFLRW − 0.1444, RMS of residuals = 0.0274
zLW = 0.8450 zFLRW − 0.1700, RMS of residuals = 0.0265
zLW = 0.8220 zFLRW − 0.1926, RMS of residuals = 0.0257
zLW = 0.8022 zFLRW − 0.2120, RMS of residuals = 0.0251
zLW = 0.7860 zFLRW − 0.2279, RMS of residuals = 0.0245
1.5708
1.4635
1.3561
1.2488
1.1414
1.0341
0.9268
0.8194
0.7121
0.6048
0.4974
0.3901
Figure 2.28: A plot of zLW against zFLRW for 12 trajectories in the Eb = 1.1 universe.
For this simulation, ∆r0 was 10
−3R. The initial angle θn of the trajectory is given in
the legend. Each trajectory was traced across 15 cells. A linear regression has been
performed for each graph. The regression equations and corresponding root-mean-
squares of the residuals are listed above in order of decreasing θn.
82
2. LINDQUIST–WHEELER MODELS
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
gradient = −0.3291 cos(θ) + 1.0407
RMS of residuals = 6.57× 10−6
gradient = −0.3321 cos(θ) + 1.0499
RMS of residuals = 6.58× 10−6
gradient = −0.3377 cos(θ) + 1.0678
RMS of residuals = 6.53× 10−6
gradient = −0.3407 cos(θ) + 1.0773
RMS of residuals = 6.44× 10−6
gradient = −0.3463 cos(θ) + 1.0950
RMS of residuals = 6.03× 10−6
gradient = −0.3478 cos(θ) + 1.0999
RMS of residuals = 5.82× 10−6
gradient = −0.3512 cos(θ) + 1.1109
RMS of residuals = 7.13× 10−6
gr
ad
ie
n
t
of
z L
W
v
s
z F
L
R
W
cos(θ)
Eb = 1.00000
Eb = 1.00001
Eb = 1.00005
Eb = 1.00010
Eb = 1.00050
Eb = 1.00100
Eb = 1.10000
Figure 2.29: Plot of the gradients of zLW vs zFLRW graphs against cos(θn) for open
universes of different values of Eb. Also shown is a plot for the flat universe, corre-
sponding to Eb = 1. A linear regression has been performed on each graph, and both
the regression equation and the root-mean-square of the residuals are displayed in order
of increasing Eb.
83
2.6. REDSHIFT RESULTS
1.04
1.05
1.06
1.07
1.08
1.09
1.10
1.11
1.12
0.98 1.00 1.02 1.04 1.06 1.08 1.10
gr
ad
ie
n
t
of
z L
W
v
s
z F
L
R
W
Eb
Figure 2.30: A plot of the zLW vs zFLRW gradients against Eb for a photon initially in
the direction of θ0 =
1
2pi. The graph approaches an asymptotic value as Eb increases.
0.73
0.74
0.75
0.76
0.77
0.78
0.79
0.98 1.00 1.02 1.04 1.06 1.08 1.10
gr
ad
ie
n
t
of
z L
W
v
s
z F
L
R
W
Eb
Figure 2.31: The analogous graph to Figure 2.30 for a photon travelling in the direc-
tion of θ11 =
149
1200pi.
84
2. LINDQUIST–WHEELER MODELS
behaviour, not shown, along the other trajectories simulated.
Finally, we shall now discuss our simulation of the closed universe. We have
chosen to focus on an LW universe built from the 600-cell Coxeter lattice de-
scribed in Appendix A, as it is the most finely subdivided closed Coxeter lattice
possible. Using (2.2.20) and (2.2.24), we find that Eb ≈ 0.96080152145 for this
universe, and the maximum cell size is therefore approximately 25.5R. If we
embed a Schwarzschild-cell into the comparison hypersphere, as described in Sec-
tion 2.2, then the ratio between the angular distance, ψh, from the cell centre
to the horizon, as measured from the hypersphere’s centre, and the correspond-
ing distance, ψb, to the cell boundary is approximately 0.0392;
13 for comparison,
Clifton et al. [48] have shown that on the time-symmetric hypersurface, the exact
3-metric is conformally equivalent to that of a 3-sphere, and on this 3-sphere, the
equivalent ratio is approximately 0.0147; our value is about 2.6 times greater than
theirs, although their ratio uses the shortest angular distance to the cell bound-
ary, which is not spherical in the exact lattice. In any case, the 600-cell universe
is much smaller than the universes we have heretofore been considering, and the
central mass’ influence will therefore be much stronger. We note that in the LW
co-ordinates for the closed universe, we do not have freedom to choose the initial
size of the cell; once the photon’s initial co-ordinates are chosen, then the cell
boundary’s position is determined, since both the boundary and the source must
reach their respective maximum radii at the same time τ . For our simulation of
this universe, we chose to propagate photons along 30 trajectories ranging from
θ0 =
1
2
pi to θ29 =
17
150
pi going in decrements of 2
150
pi.
The simulation results were again similar to those for the flat and open uni-
verses but with several slight differences this time. The graphs of zLW against
zFLRW again followed straight lines but the gradient was slightly different for when
the photon was outgoing and when it was ingoing, as shown in Figure 2.32. As
the figure also shows, the graphs did not necessarily intercept the origin either.
Several of the graphs for ingoing photons also had a subtle bend in their low z
end. We believe this bend to be analogous to the initial curve we saw in the flat
universe graphs previously: it corresponds to redshifts measured at low robs and
is more pronounced in the more radial trajectories, that is, the smaller θn tra-
jectories, which pass closer to the central mass; thus, like the initial curve in the
13This value has been calculated using (2.2.23), (2.2.22), (2.2.8), and (2.2.24); from these
equations, it follows that sinψh = sin
3 ψb, where ψb is given by (2.2.20) for N = 600.
85
2.6. REDSHIFT RESULTS
−0.50
−0.25
0.00
0.25
0.50
0.75
1.00
1.25
−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
z L
W
zFLRW
1.5708 (outgoing)
1.5708 (ingoing)
0.3560 (outgoing)
0.3560 (ingoing)
Figure 2.32: A plot of zLW against zFLRW for two trajectories of photons, θ0 =
1
2pi and
θ29 =
17
150pi. Each trajectory’s graph is divided into a subgraph for when the trajectory
is outgoing and a separate subgraph for when the same trajectory becomes ingoing;
the two subgraphs meet at the redshift data point closest to maximum expansion of
the universe. There is a slight bend in the blueshift regime of the ingoing θ29 =
17
150pi
graph.
flat universe graphs, we believe the bend is caused by the central mass’ stronger
influence at low robs.
14 In Figure 2.33, we show graphs of zLW against zFLRW for
when the photons are outgoing, and in Figure 2.34, we show the corresponding
graphs for when the same photons are ingoing.
In Figure 2.35, we have plotted the gradients of zLW vs zFLRW against cos(θn).
Two graphs are shown, one for the gradients when the photons are outgoing, and
the other for when they are ingoing. To exclude the initial bend in the graphs for
ingoing photons, we excluded data points corresponding to robs less than 10R,
14We note that the earlier analysis of the dependence of νLW on robs, given by function
(2.6.1), would require some modification in this context. In particular, as we are now using
LW co-ordinates, function (2.6.1) would change, with parameter Eb replaced by a parameter
Eo corresponding to the constant E for the co-moving observer’s geodesic. The new function
though would still have the same functional form as the original and therefore the same general
behaviour. However since the radius of the observer’s geodesic is bounded, the new function’s
domain is bounded from above by robs ≤ 2m/(1−Eo), otherwise the square-root in the function
would turn imaginary; therefore there cannot be any large-robs asymptotic regime in the new
function, and hence this aspect of our previous analysis is not transferable to the closed universe.
Nevertheless, we believe that the general conclusion still applies, that the initial jump and the
bend in zLW vs zFLRW graphs is caused by the central mass’ stronger influence at low robs.
86
2. LINDQUIST–WHEELER MODELS
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
z L
W
zFLRW
zLW = 0.9990 zFLRW + 0.0342, RMS of residuals = 8.461× 10−4
zLW = 0.9935 zFLRW + 0.0251, RMS of residuals = 1.141× 10−3
zLW = 0.9866 zFLRW + 0.0170, RMS of residuals = 1.116× 10−3
zLW = 0.9781 zFLRW + 0.0105, RMS of residuals = 7.763× 10−4
zLW = 0.9665 zFLRW + 0.0063, RMS of residuals = 9.523× 10−4
zLW = 0.9547 zFLRW + 0.0024, RMS of residuals = 7.152× 10−4
zLW = 0.9404 zFLRW + 0.0000, RMS of residuals = 1.422× 10−3
zLW = 0.9267 zFLRW − 0.0032, RMS of residuals = 9.969× 10−4
zLW = 0.9085 zFLRW − 0.0049, RMS of residuals = 1.946× 10−3
zLW = 0.8892 zFLRW − 0.0085, RMS of residuals = 8.650× 10−4
1.5708
1.4451
1.3195
1.1938
1.0681
0.9425
0.8168
0.6912
0.5655
0.4398
Figure 2.33: A plot of zLW against zFLRW while photons are outgoing. Graphs for
a selection of photon trajectories θn are shown. A linear regression was performed on
each graph, and the regression equation is listed in order of decreasing θn.
87
2.6. REDSHIFT RESULTS
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
z L
W
zFLRW
zLW = 1.0271 zFLRW − 0.0028, RMS of residuals = 1.552× 10−3
zLW = 1.0201 zFLRW − 0.0098, RMS of residuals = 6.617× 10−4
zLW = 1.0128 zFLRW − 0.0173, RMS of residuals = 6.439× 10−4
zLW = 1.0060 zFLRW − 0.0260, RMS of residuals = 1.085× 10−3
zLW = 1.0008 zFLRW − 0.0375, RMS of residuals = 6.022× 10−4
zLW = 0.9955 zFLRW − 0.0491, RMS of residuals = 9.149× 10−4
zLW = 0.9891 zFLRW − 0.0605, RMS of residuals = 9.778× 10−4
zLW = 0.9821 zFLRW − 0.0718, RMS of residuals = 1.498× 10−3
zLW = 0.9700 zFLRW − 0.0805, RMS of residuals = 1.517× 10−3
zLW = 0.9525 zFLRW − 0.0870, RMS of residuals = 1.710× 10−3
1.5708
1.4451
1.3195
1.1938
1.0681
0.9425
0.8168
0.6912
0.5655
0.4398
Figure 2.34: A plot of zLW against zFLRW while photons are ingoing. Graphs for the
same trajectories as in Figure 2.33 are shown. A linear regression was performed on
each graph, and the regression equation is listed in order of decreasing θn. Though in
this case, the regression only includes redshifts measured at robs equal to at least 10R,
the photon’s starting radius, so as to exclude points in the initial bend.
88
2. LINDQUIST–WHEELER MODELS
0.86
0.89
0.92
0.95
0.98
1.01
1.04
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
gr
ad
ie
n
t
of
z L
W
v
s
z F
L
R
W
cos(θ)
outgoing
ingoing
Figure 2.35: A plot of zLW vs zFLRW gradients against cos(θn), where θn is the initial
direction of the photon. The gradients for when the photons are outgoing and for when
they are ingoing are plotted separately, and all gradients are obtained by performing a
linear regression on zLW vs zFLRW data. To avoid the initial bend in the zLW vs zFLRW
graphs for ingoing photons, the ingoing gradients are obtained by regressing over only
data points for which robs is at least equal to the photon’s starting radius of 10R.
the photons’ starting radius.15 The figure shows that, as with the other universes,
the gradient decreases as the photon’s trajectory becomes more radial, but now,
the relationship between the gradients and cos(θn) is no longer linear.
Thus to summarise, our simulations have demonstrated several features com-
mon to redshifts in all LW universes. The LW redshifts generally increase linearly
with their FLRW counterparts in any universe and for any photon trajectory.
When robs is small such that the central mass has a stronger influence, there is
some deviation away from this linear behaviour. Any influence from the cen-
tral mass can be suppressed by starting the photon at very large radii; zLW then
becomes completely proportional to zFLRW . The LW redshifts also generally de-
crease relative to their FLRW counterparts as the photon takes a more radial
trajectory; this can also be attributed to the stronger influence of the central
mass since a more radial trajectory would pass closer to it. The mass’ influence
on redshifts can sometimes be quite significant, causing LW redshifts to deviate
from their FLRW counterparts by as much as 30%. This would have significant
15In fact, including all data points in the regression introduced some ‘jaggedness’ in the graph,
especially towards the low cos(θn) regime. By excluding these points, the ‘jaggedness’ smoothed
away, and we were left with a curve much more similar to that for the outgoing gradients.
89
2.6. REDSHIFT RESULTS
implications if we attempted to fit redshift data from a lattice universe onto an
FLRW model, as quantities such as the age of the universe may be incorrectly
estimated. Thus we see certain significant effects arising from the ‘lumpiness’ of
the LW universe which an FLRW-based model could not adequately capture.
This influence of the masses on redshifts can be understood as an integrated
Sachs–Wolfe effect (ISW). The ISW describes the net redshift induced on a photon
as it passes through a fluctuating gravitational potential caused by fluctuations
in the energy density [71–74]. In a flat FLRW universe, a time-varying potential
is necessary to generate a non-zero ISW; however when the universe is matter-
dominated, the potential is static, and therefore the ISW would be zero. If there
is a non-zero cosmological constant though, the potential in a matter-dominated
universe does become time-varying, and hence a non-zero ISW can result [75].
Indeed, recent precision observations of the cosmos have not only established that
our universe is very nearly flat but that there is definitely a non-zero ISW [76,77];
naturally, this has been interpreted as evidence of a non-zero cosmological con-
stant in our universe. However, our results suggest that this might be explainable
within a matter-dominated universe without needing a cosmological constant; we
may simply require a model that better reflects the inhomogeneous matter dis-
tribution than FLRW space–times allow. Because of the significant implications
a non-zero ISW may have on the dark energy question, this modelling problem
deserves further investigation.
2.7 Discussion
We have investigated the properties of the Lindquist–Wheeler universes, as we
hope they might provide some insight into what observable effects the ‘lumpy’
matter distribution of the actual universe might yield. And although the LW
universes are only approximations rather than exact solutions to the Einstein
field equations, we believe they model enough of the underlying physics to yield
at least meaningful qualitative insights into the behaviour of the actual universe.
Much of the LW universes’ dynamics bear strong resemblance to those of the
matter-dominated FLRW universes. Additionally, photon redshifts in LW uni-
verses behaved roughly similarly to their FLRW counterparts. Yet there were
also certain direction-dependent effects in the redshifts due to the ‘lumpiness’ of
90
2. LINDQUIST–WHEELER MODELS
the universe’s matter distribution.
Our investigation can be extended in several ways. It would be interesting to
examine the optical properties of the LW universe, as this may have important
implications for actual astrophysical observations. Clifton and Ferreira have al-
ready done this for their implementation of the LW universe, but we have used
a different implementation from theirs, and this has led to certain differences in
the behaviour of redshifts. However, Clifton and Ferreira have suggested that
the optical properties, unlike the redshifts, may actually be insensitive to the
choice of boundary conditions [61]. Nevertheless, it would be very interesting to
see whether that is indeed the case for our boundary conditions. It would also
be interesting to extend our study to LW universes with a non-zero cosmological
constant and attempt to evaluate by how much inhomogeneities reduce the need
for such a constant. Clifton and Ferreira have already constructed an appropriate
extension of the LW universe based on the Schwarzschild de-Sitter metric [60,78],
and they have shown that the corresponding Friedmann-like equation strongly
resembles its FLRW counterpart as well. Moreover, using their boundary con-
ditions, they have also shown that the cosmological constant density ratio ΩΛ
can be reduced by about 10% [61, 78], and again this was rather insensitive to
which boundary condition they used. It should be possible to include Clifton and
Ferreira’s Λ-Schwarzschild-cells in our implementation and to investigate the re-
sulting model. Finally, our model has a still very idealised distribution of matter.
Each mass is identical and distributed on a perfect lattice which is clearly not
the case in the actual universe. We should like to extend our model to allow for
different sized masses and cells. To this end, we have derived in Appendix D a set
of conditions that must be satisfied at the boundary between two neighbouring
cells of different mass m. We leave the numerical simulation of such universes
and the detailed investigation of their properties to future work.
91

CHAPTER 3
An introduction to Regge
calculus
Regge calculus has proven to be one of the most versatile discrete formulations of
gravity. First proposed by Tullio Regge in 1961 [62], it was originally developed
as a means to approximate any solution to Einstein’s field equations, particularly
in situations where analytic solutions had so far proven elusive; thus, its first
great success was in numerical general relativity [79]. Yet it soon became a topic
of wider interest with quantum gravity research, as it was adopted to formulate a
wide range of path-integral approaches to quantum gravity; these included most
notably quantum Regge calculus [79,80], dynamical triangulations [80,81], causal
dynamical triangulations [82,83], and the spin-foam formulation of loop quantum
gravity [84–88]. Indeed, the use of a discrete formulation of gravity is particularly
apposite for these different theories as they share the common proposition that
space–time is actually discrete at the smallest of scales, some theories taking this
as a fundamental assumption, others deducing it from their formulation.
Our interest in Regge calculus, however, will be in approximating cosmological
space–times; the rest of this thesis will focus solely on this discretisation scheme.
To that end, we shall provide a brief introduction in this chapter to the formalism.
The central idea of Regge calculus is to discretise space–time using a manifold
built out of flat simplices ‘glued’ together at shared faces. The first section
introduces and describes the geometry of these simplicial manifolds. The next
section introduces the Regge action, the Regge analogue of the Einstein–Hilbert
93
3. AN INTRODUCTION TO REGGE CALCULUS
action, and the Regge field equations are then derived from it. Several forms of
the action, describing different physical scenarios, are also presented. The third
section briefly considers the application of Regge calculus to discrete manifolds
built out of other types of polytopes. In the final section, the Cauchy problem in
the context of Regge calculus is explored: first the problem in continuum general
relativity is quickly reviewed; then the initial value equations are derived; and
lastly, the application of these equations to Regge Cauchy surfaces is discussed.
To the best of our knowledge, one of our methods for applying the initial value
equations to Regge calculus is novel; otherwise, this chapter is a review of pre-
existing results.
3.1 The geometry of Regge calculus
The key idea behind Regge calculus is to formulate general relativity on a piece-
wise linear simplicial manifold rather than a smooth differential manifold. The
simplicial manifolds of Regge calculus are composed of 4-simplices ‘glued’ to-
gether such that each simplex shares an entire face with a neighbouring simplex,
unless the face lies on the manifold’s boundary. The interior of each simplex is
Minkowksi space–time. In Regge calculus, the simplicial manifold itself is usually
referred to as a skeleton, and sub-simplices of co-dimension 2 as the hinges of the
skeleton. Note that all geometry of a single simplex, including angles between
edges or faces, can be determined solely by specifying the lengths of all edges of
the simplex. Therefore, to specify the entire geometry of a skeleton, it is sufficient
to specify just the lengths of all its edges.
These lengths serve as the analogue of the metric of standard general relativity.
In general, an n-simplex is composed of n+ 1 vertices, and each possible pair of
vertices is joined together by an edge; hence, there are(
n+ 1
2
)
=
(n+ 1)n
2
edges in total, and this is identical to the number of independent components in
an n-dimensional metric. Indeed for any arbitrary set of vertex co-ordinates, the
lengths of the simplex’s edges and the components of the metric in this co-ordinate
94
3. AN INTRODUCTION TO REGGE CALCULUS
system are related by
lij =
√
gµν ∆x
µ
ij ∆x
ν
ij, (3.1.1)
where lij is the distance between vertices i and j and ∆x
µ
ij is the displacement
vector between them. Therefore, specifying the components of the metric would
determine the lengths of all edges of the simplex; conversely, specifying the lengths
of all edges would determine all components of the metric in the given co-ordinate
system.
Curvature in a skeleton is concentrated at the hinges in the form of coni-
cal singularities, unlike differential manifolds where it is distributed continuously
throughout. Consider the hinge depicted in Figure 3.1. If the geometry surround-
θ(j)
Figure 3.1: Illustration of a hinge and its attached faces. Only the space orthogonal
to the hinge is represented, so the hinge appears as a vertex and the faces as lines. One
dihedral angle θ(j) has been labelled between two of the faces.
ing the hinge were flat, then the dihedral angles θ(j) between adjacent pairs of
faces should sum to 2pi. The difference  from 2pi, given by
 = 2pi −
∑
j
θ(j), (3.1.2)
is known as the deficit angle and provides a measure of curvature at the hinge.
If the hinge were space-like, then the deficit angle would be complex. A non-
zero deficit angle corresponds to a conical singularity at the hinge: using polar
co-ordinates (ρ, φ, z1, . . . , zn−2) centred on the hinge, we can express the metric
around it as
ds2 = dρ2 + ρ2dφ2 + dz21 + · · ·+ dz2n−2, (3.1.3)
but with φ constrained to lie in the interval [0, 2pi − ); points corresponding to
95
3.1. THE GEOMETRY OF REGGE CALCULUS
φ = 2pi−  would get identified with points at φ = 0, as illustrated in Figure 3.2,
where the dashed face gets identified with the solid face bounding . Moreover, if
θ(j)
q
p
p
q

Figure 3.2: The conical singularity at a hinge, as represented in the plane orthogonal
to the hinge. One dihedral angle θ(j) has been labelled between two of the faces. The
space–time in the neighbourhood of the hinge is Minkowski, but the dihedral angles
around the hinge do not sum to 2pi, leaving a deficit angle of , with the two faces
bounding  identified as a single face. A pair of orthogonal vectors, p and q, are
parallel-transported around the dotted path enclosing the hinge; the path is actually
closed as it starts and terminates at the same face. Owing to the deficit angle, the
vectors will not have a completed a 2pi rotation, stopping short instead by .
we parallel-transport a vector around any loop enclosing the hinge, then by the
end of the loop, the vector component orthogonal to the hinge will have rotated
by  from its original orientation, as illustrated by p and q in Figure 3.2, while
the component parallel to the hinge will remain unchanged.
There are two ways to calculate the dihedral angle θ(j) between a pair of faces
at a hinge. The first method is to take the scalar product of the unit normals to
the two faces, illustrated by n1 and n2 in Figure 3.3, as this yields cos θ
(j). The
second method instead takes the scalar product between the two unit vectors
parallel to the faces but orthogonal to the hinge, u1 and u2 in Figure 3.3, as
this also yields cos θ(j). In practice, there are situations where the first method is
easier to use, but care is needed with the relative sign between the two normals, as
using the incorrect unit normal, such as −n2, would lead to an incorrect dihedral
angle; with the second method, this ambiguity is more easily avoided.
96
3. AN INTRODUCTION TO REGGE CALCULUS
n2
Face 2
u2
−n2
θ(j)
Hinge
Face 1u1
n1
Figure 3.3: Illustration of the dihedral angle θ(j) between two faces meeting at a
hinge. The plane orthogonal to the hinge has been represented. The unit vectors
normal to the two faces, n1 and n2, are shown, as are unit vectors parallel to the faces
but orthogonal to the hinge, u1 and u2.
3.2 The Regge action
Once a simplicial manifold has been chosen, then as noted earlier, to completely
specify the skeleton, all that remains is to specify the lengths of all simplicial
edges. The analogue of these lengths in general relativity would be the metric
gµν , and this is determined by the Einstein field equations. These equations can be
obtained by varying the Einstein–Hilbert action with respect to gµν . Analogously,
in Regge calculus, the skeletal edge-lengths are determined by the Regge field
equations, and these are obtained by varying the Regge action with respect to the
edge-lengths.
The Regge action is obtained by applying the Einstein–Hilbert action to sim-
plicial manifolds; this is the manner by which Regge calculus applies general
relativity to simplicial manifolds. For (3+1)-dimensional vacuum space–time,
the Einstein–Hilbert action is
SEH = 1
16pi
∫
R
√−g d4x, (3.2.1)
where g is the determinant of the metric and R the Ricci scalar. Simplicial man-
ifolds however are flat everywhere except at the hinges; therefore R has support
at the hinges only and would be a function of the deficit angle. Indeed, it has
97
3.2. THE REGGE ACTION
been shown that for simplicial manifolds, the integral in (3.2.1) yields [62,89]∫
R
√−g d4x =
∑
i∈{hinges}
2Ai i, (3.2.2)
where the summation is over all hinges, Ai is the area of a hinge, and i is its
deficit angle. Hence, the Einstein–Hilbert action reduces to
SRegge = 1
8pi
∑
i∈{hinges}
Ai i. (3.2.3)
This is a Regge action. This action has also been derived from continuum general
relativity by regarding the skeleton as a limit of a sequence of continuum surfaces
[90]. It has also been demonstrated that the Einstein–Hilbert action can be
recovered from the Regge action when the continuum limit is taken [91].
To obtain the Regge field equations, the Regge action is then varied with
respect to the skeletal edge-lengths `m, yielding
0 =
δSR
δ`m
=
∑
i∈{hinges}
1
8pi
(
∂Ai
∂`m
i + Ai
∂i
∂`m
)
. (3.2.4)
However, the variation of the deficit angles cancels out owing to the well-known
Schla¨fli identity1 ∑
j
Aj
∂θ(j)
∂`m
= 0,
where the summation is over all hinges in a single 4-simplex and θ(j) is the dihedral
1Schla¨fli had actually proven a form of this identity for the case of spherical simplices [92].
Kneser subsequently proved a generalised form that applied equally to both hyperbolic and
spherical simplices [93,94]. Regge provided a proof specifically for flat simplices [62], and this is
the form that appears above. More generally, for an n-simplex in a space of constant curvature
κ, the Schla¨fli identity is
(n− 1)κ∂V
(n)
∂`m
=
∑
j
Aj
∂θ(j)
∂`m
,
where V (n) is the volume of the n-simplex; Alekseevskij, Vinberg, and Solodovnikov [95] as
well as Milnor [96] have provided proofs. Mathematical literature generally refers to this as the
Schla¨fli differential formula or the Schla¨fli differential volume formula. We immediately see that
for flat simplices, where κ = 0, this generalised identity reduces to the form that appears above.
The generalised identity has been used to formulate Regge calculus using curved simplices [97],
where the volume term effectively acts like a cosmological constant term in the Regge equations.
For further generalisations of the Schla¨fli identity, see [98–103].
98
3. AN INTRODUCTION TO REGGE CALCULUS
angle between faces of the 4-simplex at hinge j. As a result, the second term in
(3.2.4) vanishes, and the Regge field equations are therefore
0 =
∑
i∈{hinges}
∂Ai
∂`m
i. (3.2.5)
The Regge action can easily be modified to describe non-vacuum space–times.
In the presence of massive particles, the Einstein–Hilbert action becomes
SEH = 1
16pi
∫
R
√−g d4x −
∑
i∈{particles}
Mi
∫
dsi, (3.2.6)
where the summation is over all particles in the space–time, Mi is the mass
of particle i, and dsi is its line element. The corresponding Regge action then
becomes
SRegge = 1
8pi
∑
i∈{hinges}
Ai i −
∑
i∈{particles}
j ∈{4-simplices}
Mi sij, (3.2.7)
where sij denotes the length of particle i’s path through 4-simplex j and the
second summation is over all particles and all 4-simplices of the skeleton; note
that if particle i never passes through 4-simplex j, then sij will accordingly be
zero. In the presence of a cosmological constant Λ, the Einstein–Hilbert action
is given by
SEH = 1
16pi
∫
(R− 2 Λ)√−g d4x, (3.2.8)
and the Regge action becomes
SRegge = 1
8pi
 ∑
i∈{hinges}
Ai i −
∑
i∈{4-simplices}
ΛV
(4)
i
, (3.2.9)
where V
(4)
i denotes the 4-volume of a 4-simplex and the second summation is over
all 4-simplices.
In general, the Regge equations should completely determine the lengths of the
skeletal edges. However, there is one notable case where this is not entirely true.
It has been shown that the edge-lengths of flat skeletons will not be completely
determined by the Regge equations, thus leaving residual degrees of freedom
[104, 105]; vertices can be translated in any direction and still give a solution
99
3.2. THE REGGE ACTION
of the Regge equations, and this translational degree of freedom is understood
as a form of diffeomorphism symmetry. For curved skeletons, there may also
be some residual symmetries in the Regge equations such that not all skeletal
edge-lengths would be completely determined [106]. Whether curved skeletons
also possess diffeomorphism symmetries is still a subject of debate: some authors
have argued that diffeomorphism symmetries are still present in general [107]
while others have argued that they are generically broken instead [108,109].
As approximations to continuum space–times, the solutions of the Regge equa-
tions are generally believed to converge at second order in the edge-lengths to
solutions of the Einstein field equations. Interestingly though, the Regge equa-
tions themselves do not appear to converge to the Einstein equations; rather, the
residuals of the Regge equations, when evaluated on continuum solutions, scale
as O(1) as the lattice is refined [110–112]: that is, if one constructs a simplicial
lattice in a continuum vacuum space–time, evaluates the geodesic lengths of the
simplicial edges, and then assigns those lengths to the corresponding edges in a
flat simplicial lattice with the same lattice structure, one finds that the Regge
equation, when evaluated on the flat lattice, will differ from zero by an amount
that scales as O(1) with the lattice spacing. Recall that since the space–time is
vacuum, the Regge equation should equal zero exactly.
3.3 Non-simplicial piecewise linear manifolds
Regge calculus can also be performed on non-simplicial piecewise linear mani-
folds, that is, where the fundamental building blocks are polytopes more complex
than just simplices. In such cases, extra constraints would have to be imposed on
the polytopes since the edge-lengths would not be sufficient to completely spec-
ify the geometry. Consider for instance the quadrilateral, where without extra
constraints, it would be impossible to tell whether it was a rectangle or a paral-
lelogram from just its four external edge-lengths, as Figure 3.4 shows. In general,
the extra constraints are equivalent to constraining the internal diagonals of the
polytope to be specific functions of the polytope’s external edge-lengths. In the
case of the quadrilateral, one of the internal diagonals would be specified as a
function of the external edges; for instance, if the quadrilateral is to be a rectan-
gle, then an internal diagonal would be constrained to have length d1 =
√
`21 + `
2
2.
100
3. AN INTRODUCTION TO REGGE CALCULUS
d2
`1
`2 d1
`1
`2
Figure 3.4: Two different quadrilaterals with the same edge-lengths. The internal
diagonals and angles differ.
Thus a non-simplicial manifold is actually equivalent to a simplicial one but with
certain edges, the diagonals of the polytopal cells, constrained to be functions
of others, the external edges of the polytopes. When the Regge action is var-
ied, it can only be varied directly with respect to the unconstrained edges, but
constrained edges appearing in the action would get varied implicitly.
3.4 The Cauchy problem and Regge calculus
It is a well-known fact of differential calculus that partial differential equations
will not yield a unique solution without the specification of appropriate boundary
conditions; the Einstein field equations, being a set of second-order differential
equations, are no exception. Even in Regge calculus, where the Regge equations
are no longer differential equations, appropriate boundary conditions are still
required to determine a unique skeleton.
It is quite common to study general relativity from the framework of a (3+1)-
splitting of space–time whereby space–time is foliated into a one-parameter family
of space-like Cauchy surfaces. In such contexts, one would specify an appropriate
set of data on some initial Cauchy surface and then evolve the set forwards in
time to determine the rest of the space–time. Naturally, the evolution equation
is derived from the Einstein field equations.
To determine a unique solution to a second-order partial differential equation,
one generally requires two sets of constraints. For the initial value problem, these
constraints would usually be the values of the function and of the first-order
partial derivatives of the function on the initial Cauchy surface. Thus in general
relativity, where the function in question is the metric tensor g, one might na¨ıvely
101
3.4. THE CAUCHY PROBLEM AND REGGE CALCULUS
believe that one must specify the values of the metric gµν |Σ0 and its co-variant
derivatives gµν;σ|Σ0 on the initial surface Σ0. However, not all the solutions yielded
would be physically distinct, with some space–times differing from others by only
a diffeomorphism. Thus, the required set of constraints is actually much smaller:
it has been shown that to specify a unique space–time, one need only specify the
first and second fundamental forms on Σ0 [113]. The first fundamental form h
corresponds to the projection of the metric g into Σ0 and is given by
hµν = gµν |Σ0 + nµ nν |Σ0 , (3.4.1)
where n is a field of normalised one-forms everywhere orthogonal to Σ0; it effec-
tively determines the intrinsic curvature of Σ0 because it implies a 3-dimensional
metric connection in Σ0 with which one can calculate the 3-dimensional Riemann
curvature tensor (3)Rµνσρ of Σ0. The second fundamental form χ is given by
χµν = h
σ
µ h
ρ
ν nσ;ρ
∣∣
Σ0
, (3.4.2)
and effectively specifies the extrinsic curvature of Σ0 within the overall space–
time. Indeed, these two measures of curvature are related to the 4-dimensional
intrinsic curvature of Σ0 by the Gauss equation [114]
(3)Rµνσρ = Rαβγδ h
α
µ h
β
ν h
γ
σ h
δ
ρ − χµσ χνρ + χµρ χνσ. (3.4.3)
Alternatively, the 4-dimensional intrinsic curvature can also be related to the
second fundamental form by the Gauss–Codazzi equation [115–117]
Rσρ n
σhρµ = χ
σ
µ|σ − χσσ|µ (3.4.4)
where Rσρ is the Ricci tensor and | denotes covariant differentiation with respect
to the metric connection implied by h.
Using the Gauss and Gauss–Codazzi equations, we can express the Einstein
field equations (1.0.1) on Σ0 in terms of the initial data h and χ. In so doing,
we can obtain four further constraint equations between h and χ, indicating
that not all components of h and χ are completely independent. By taking
102
3. AN INTRODUCTION TO REGGE CALCULUS
G (n,n) = 8pi T (n,n) and applying the Gauss equation (3.4.3), we obtain
(3)R + (χµν hµν)
2 − χµν χρσ hµρ hνσ = 16piρ, (3.4.5)
where (3)R is the 3-dimensional Ricci scalar of Σ0 and ρ is the energy density of
the matter source as measured by an observer co-moving with respect to Σ0. By
taking G (n,ui) = 8pi T (n,ui), where {ui} for i = 1, 2, 3 is a set of normalised
bases tangent to Σ0, and by applying the Gauss–Codazzi equation (3.4.4), we
obtain (
χσµ|µ hσν − χσµ|ν hσµ
)
uνi = 8pi Tµν n
µ uνi, (3.4.6)
which is actually a set of three equations, one for each i. In the Arnowitt–Deser–
Misner (ADM) formalism [118], the first constraint is known as the Hamiltonian
constraint and comes about by extremising the ADM action with respect to
the lapse function, while the latter three constraints are known as momentum
constraints and come about by extremising with respect to the three shift func-
tions [119]. More generally, this set of equations are known as the initial value
equations and must be satisfied for the initial data on Σ0 to be consistent with
the Einstein field equations.
A Cauchy surface is defined to be at a moment of time-symmetry when its
extrinsic curvature χ vanishes. In such a situation, the initial value equations
simplify greatly, with (3.4.5) becoming
(3)R = 16piρ (3.4.7)
and (3.4.6) vanishing.
The initial value equations can also be applied to Regge skeletons to help
determine the geometry of an initial Cauchy surface. However, there are certain
caveats to be aware of. On a skeleton, the Einstein field equations and hence
the initial value equations will only be satisfied in an average manner. Curvature
in the skeleton is concentrated at the hinges only, yet matter can be distributed
away from the hinges where the skeleton is flat; thus the two sides of (3.4.7),
for example, will not agree in a point-wise manner. This contradiction arises
because the Einstein field equations actually apply to smooth manifolds rather
than Regge skeletons; they come about by varying the Einstein–Hilbert action
when the underlying manifold is smooth rather than discrete. Thus by using the
103
3.4. THE CAUCHY PROBLEM AND REGGE CALCULUS
Einstein equations in this manner, we are effectively varying the Einstein–Hilbert
action on a smooth manifold first and then applying the resulting field equations
on a discrete manifold afterwards. The standard approach in Regge calculus is to
use a discrete manifold from the very beginning, with the field equations obtained
being different as a result. Clearly, the two approaches are not equivalent.
When applied to Regge Cauchy surfaces, the time symmetric initial value
equation (3.4.7) can be further simplified. By integrating both sides of (3.4.7)
over the entire Cauchy surface and applying the 3-dimensional analogue of (3.2.2),
we obtain ∑
i∈{hinges}
`i i = 8pi
∫
Σ0
ρ d3x, (3.4.8)
where we have continued to denote the initial time Cauchy surface by Σ0 for
Regge space–time, and where the integral measure is unity because the Regge
blocks in Σ0 are flat; since the Cauchy surface is a 3-dimensional skeleton, its
hinges are the edges, the ‘areas’ of the hinges are the edge-lengths `i, and i
are the corresponding deficit angles. To the best of our knowledge, this form of
the Regge initial value equation is novel. Alternatively, if the Cauchy surface is
sufficiently uniform such that there is a well-defined volume per vertex, then the
Ricci scalar (3)R can be expressed as [59]
(3)R =
∑
i `i i
‘volume per vertex’
, (3.4.9)
where the summation is over all edges radiating from a single vertex2, and the
initial value equation (3.4.7) becomes∑
i `ii
‘volume per vertex’
= 16piρ. (3.4.10)
2One can recover the 3-dimensional analogue of (3.2.2) from (3.4.9) by first multiplying
(3.4.9) by the ‘volume per vertex’ and then summing over all vertices; under this double sum-
mation, each edge will appear twice because each edge is connected to two vertices; hence this
double summation is equivalent to the single summation in the analogue of(3.2.2), which would
just sum over all edges once.
104
CHAPTER 4
Regge calculus of closed
FLRW universes
In a 1973 paper [63], Collins and Williams presented a Regge calculus formula-
tion of the closed FLRW universe that strongly mirrored the formulation of the
continuum model in standard general relativity. The Regge skeleton was con-
structed from a one-parameter family of space-like Cauchy surfaces foliating the
entire skeleton. Each surface consisted of a 3-skeleton constructed out of equi-
lateral tetrahedra such that all vertices, edges, and triangles were identical to
each other. In this way, the Cauchy surfaces captured as closely as possible the
Copernican principle, which required each Cauchy surface to be homogeneous and
isotropic. As with the FLRW universes, all CW Cauchy surfaces were required to
be identical to each other apart from an overall scaling, represented by the length
l(ti) of the tetrahedral edge, ti being a discrete time parameter labelling the fo-
liation. Thus, l(ti) became the Regge analogue of the FLRW scale factor a(t),
and whereas a(t) was determined by the Einstein field equations via the Fried-
mann equations, l(ti) was determined by the Regge field equations. To complete
the construction of the 4-skeleton, the CW Cauchy surfaces were glued together
by a series of time-like edges connecting vertices in one surface to their time-
evolved counterparts in the next. By then taking the limit where the separation
between surfaces goes to zero, Collins and Williams could generate a continuum
time Regge model of the FLRW universe. Their formalism was first applied to
study closed dust-filled FLRW universes, and the continuum time function l(t)
105
4. REGGE CALCULUS OF CLOSED FLRW UNIVERSES
for the edge-lengths behaved very similarly to the equivalent FLRW scale-factor
a(t), with models with a greater number of tetrahedra yielding higher accuracy.
The CW formulation was further explored and extended by Brewin [64]. One
feature of the CW skeleton was that the fundamental building blocks of the 4-
skeleton were not 4-simplices but rather 4-blocks corresponding to the truncated
world-tubes of the tetrahedra as they evolved from one Cauchy surface to the
next. Thus, the CW skeleton is actually an example of a non-simplicial skeleton
as discussed in Chapter 3.3. The exact geometry of this 4-block will be presented
below. However, such a skeleton allowed Collins and Williams to constrain all
identical edges to have the same lengths prior to varying the Regge action. As a
result, when the length of a tetrahedral edge was varied in one Cauchy surface, the
lengths of all edges in that surface had to be varied; the same applied to the time-
like edges between pairs of Cauchy surfaces. This is different from standard Regge
calculus where each edge is varied individually before any constraints are imposed
on their lengths. In standard general relativity, Collins and Williams’ approach
would be analogous to requiring the metric in the Einstein–Hilbert action to
be of FLRW form (1.0.3), and then varying the action with respect to a(t);
this imposes the Copernican symmetries prior to varying the Einstein–Hilbert
action. The more standard approach would be to vary the Einstein–Hilbert action
first, yielding the Einstein field equations (1.0.1), and then setting the metric to
be of FLRW form. We shall refer to Collins and Williams’ approach as global
variation and the more standard approach as local variation. The relationship
between solutions of the global Regge equations and the local Regge equations
was explored in depth by Brewin.
Brewin also proposed a scheme for subdividing the CW Cauchy surfaces to
generate finer approximations to FLRW Cauchy surfaces. However under this
scheme, Cauchy surfaces of these new secondary models would no longer consist
of identical nor necessarily equilateral tetrahedra. The virtue of Brewin’s scheme
is that it could in principle be repeated indefinitely, thereby yielding even finer
approximations to the underlying FLRW surfaces, although it sacrifices some of
the symmetries inherent in the original CW surfaces. We shall refer to the un-
subdivided models as parent models and the subdivided ones as children models.
Brewin’s children models were also applied to closed dust-filled FLRW universes,
and they were found to mimic the continuum universe even more closely than
their parent counterparts.
106
4. REGGE CALCULUS OF CLOSED FLRW UNIVERSES
The CW formalism will serve as the template upon which we shall base our
cosmological Regge models in the chapters to come. To introduce the necessary
background, this chapter will present the CW formalism as well as Brewin’s sub-
sequent extensions. The first section reviews Collins and Williams’ construction
of the CW Regge skeleton and presents several geometric quantities which will
be used in the Regge actions of subsequent chapters. The second section presents
a triangulation of the CW skeleton as well as several of the new geometric quan-
tities resulting from this triangulation. In the third section, the relationship
between global and local Regge solutions, as explored by Brewin, is discussed.
In the penultimate section, Brewin’s subdivision scheme is reviewed, and a gen-
eral co-ordinate system for the child tetrahedron’s 4-block is introduced. This
co-ordinate system, which will be used to describe the child 4-block’s geometry,
is an entirely novel construction specific to the work presented in this thesis: it is
a modification of Collins and Williams’ co-ordinate system for the parent 4-block
and differs from Brewin’s co-ordinate system, which is based on non-Cartesian
co-ordinates. The final section compares Collins and Williams’ (3+1)-formulation
of Regge calculus with the ADM formalism.
4.1 The Collins–Williams Regge skeleton
As mentioned above, the CW skeleton consists of a one-parameter family of
Cauchy surfaces, each identical to the other apart from an overall scaling of
the edge-lengths and each labelled by a discrete time parameter ti; we shall de-
note the CW Cauchy surface at ti by Σi. In the closed FLRW universe, each
constant-time Cauchy surface, if embedded in a 4-dimensional Euclidean space,
would correspond to a 3-sphere. Collins and Williams therefore constructed their
Regge Cauchy surfaces by triangulating such 3-spheres with equilateral tetrahe-
dra; as mentioned above, all vertices, edges, and faces in the Cauchy surface were
required to be identical to each other. According to Coxeter [120], such a trian-
gulation of the 3-sphere is only possible with 5, 16, and 600 tetrahedra. Thus
there can only be three possible parent models; Table 4.1 tabulates the numbers
of vertices, edges, triangles, and tetrahedra in each of these models.
To glue the Cauchy surfaces together, each vertex in one surface was con-
nected to its time-evolved image in the next by a set of time-like edges called
107
4.1. THE COLLINS–WILLIAMS REGGE SKELETON
Tetrahedra
(N3)
Triangles
(N2)
Edges
(N1)
Vertices
(N0)
Triangles
per edge
5 10 10 5 3
16 32 24 8 4
600 1200 720 120 5
Table 4.1: The numbers of simplices in Cauchy surfaces of the three parent models
as well as the numbers of triangles meeting at any edge. We introduce N3, N2, N1,
and N0 to denote the numbers of parent tetrahedra, triangles, edges, and vertices in a
surface.
struts. Because all vertices in a surface were identical, all struts between pairs
of consecutive surfaces had to be identical as well. With this construction, each
Cauchy surface Σi could be characterised by just two distinct lengths, the tetra-
hedral edge-length l(ti) = li and the strut-length m(ti) = mi.
The geometry of the 4-block was determined by two requirements: (i) that
all struts have the same length; and (ii) that there be no twist or shear along
the 4-tube. Brewin has likened these two requirements to a choice of lapse and
shift function in the ADM formalism; indeed, the standard form of the FLRW
metric (1.0.3) also implies a certain foliation of FLRW space–time, and Collins
and Williams’ choice seems closest to the lapse and shift implicit in this foliation.
The geometry of the 4-block is easiest described by introducing a Cartesian
co-ordinate system into the 4-block. The vertices of the tetrahedron in surface
Σi will be denoted A, B, C, D, and their time-evolved images in the next surface
Σi+1 will be denoted A
′, B′, C ′, D′, respectively. In our co-ordinate system, the
108
4. REGGE CALCULUS OF CLOSED FLRW UNIVERSES
vertices in Σi are located at
1
A =
(
− li
2
,− li
2
√
3
,− li
2
√
6
, ιti
)
,
B =
(
li
2
,− li
2
√
3
,− li
2
√
6
, ιti
)
,
C =
(
0,
li√
3
,− li
2
√
6
, ιti
)
,
D =
(
0, 0,
√
3 li
2
√
2
, ιti
)
,
(4.1.1)
A Euclidean metric is being used to help simplify calculations, but the imaginary
unit ι has been introduced to the time co-ordinate so that inner products would
effectively yield a signature of (+,+,+,−). An analogous expression to (4.1.1)
gives the co-ordinates for the time-evolved image in Σi+1, with each vertex re-
placed by its primed counterpart and each subscript i replaced by i + 1. Thus
in this co-ordinate system, Brewin’s two requirements lead to the tetrahedron
simply expanding or contracting uniformly about its centre in the spatial dimen-
sions. Figure 4.1 illustrates this 4-block. For simplicity, we shall sometimes refer
to the tetrahedron in Σi+1 as the upper tetrahedron and the tetrahedron in Σi as
the lower tetrahedron, as that is how they appear in Figure 4.1.
The 4-block co-ordinates just introduced greatly facilitate the calculation of
any geometric quantity relevant to the CW skeleton. However by using these co-
ordinates, one may potentially introduce the time difference δti into the quantity
being calculated. Since in Regge calculus, it is the skeletal edge-lengths that are
to be varied, we must be able to convert δti into edge-lengths. Only the length mi
of the struts depends on δti, since increasing the time separation between a pair
of consecutive Cauchy surfaces lengthens mi but leaves li and li+1 unchanged.
Therefore by using (4.1.1) to calculate the length of a strut, such as AA′, we
obtain a relation between mi and δti given by
m2i =
(
3
8
l˙ 2i − 1
)
δt2i , (4.1.2)
1We could more generally have used a time co-ordinate ιTi := ιT (ti) instead for the co-
ordinates in (4.1.1), but this would effectively re-parametrise the Cauchy surfaces and would
still in general lead to the same set of final equations. So for simplicity, we choose to use ti as
the vertices’ time co-ordinate.
109
4.1. THE COLLINS–WILLIAMS REGGE SKELETON
ti+1
D′
C ′
B′
mi
B
C
D
A
A′
li
li+1
ti
Figure 4.1: An equilateral tetrahedron of edge-length li in surface Σi evolves to a
tetrahedron of edge-length li+1 in surface Σi+1, tracing out a 4-dimensional world-tube.
The struts are all of length mi.
where we have introduced the notation
l˙i :=
li+1 − li
ti+1 − ti .
The CW skeleton has only two distinct types of hinges, the time-like trape-
zoidal hinges and the space-like triangular hinges. Each trapezoidal hinge is
generated by the world-sheet of a tetrahedral edge as it evolves from one Cauchy
surface to the next; an example would be hinge ABA′B′, which is generated by
edge AB. The triangular hinges are simply the triangular faces of the tetrahedra;
an example would be triangle ABC. The area of any trapezoidal hinge between
Σi and Σi+1 is
Atrapi =
ι
2
(li+1 + li)
[
1
4
(li+1 − li)2 −m2i
] 1
2
, (4.1.3)
110
4. REGGE CALCULUS OF CLOSED FLRW UNIVERSES
while the area of any triangular hinge in Σi is
Atrii =
√
3
4
l 2i . (4.1.4)
As we shall discuss briefly at the end of this chapter and also explore more
fully with the Λ-FLRW models in Chapter 5, it is often sufficient to determine
the evolution of the entire universe from just the Regge equation obtained from
varying with respect to the struts. Therefore, if the two hinge areas are varied
with respect to mj, only the variation of A
trap
i will be non-zero, yielding
∂Atrapi
∂mj
= − ι
2
mi(li+1 + li)
[
1
4
(li+1 − li)2 −m2i
]− 1
2
δij. (4.1.5)
Then in such cases, we would only need the trapezoidal hinges’ deficit angle to
determine the Regge equation. In the context of the CW formalism, we shall
denote all deficit angles by δi rather than i.
Because all simplices are identical, the deficit angle on a trapezoidal hinge
simplifies to
δ trapi = 2pi − nθi, (4.1.6)
where n is the number of faces meeting at the hinge and θi is the dihedral angle
between any two adjacent faces. Since each trapezoidal hinge corresponds to the
world-sheet of a tetrahedral edge and each face on this hinge to the world-tube of
a triangle at this edge, n is equal to the number of triangles meeting at an edge;
this number is listed in the last column of Table 4.1.
Faces ABCA′B′C ′ and ABDA′B′D′ meeting at hinge ABA′B′ will be sep-
arated by a dihedral angle of θi; hence, θi can be determined from the scalar
product of the two faces’ unit normals. Let nˆ1 denote the unit normal pointing
into ABCA′B′C ′ and nˆ2 the unit normal out of ABDA′B′D′; then in co-ordinate
system (4.1.1), they have components
nˆµ1 =
(
0, 0, 1,−ι 1
2
√
6
l˙i
)
(
1− 1
24
l˙ 2i
) 1
2
(4.1.7)
111
4.1. THE COLLINS–WILLIAMS REGGE SKELETON
and
nˆµ2 =
(
0,−2√2, 1, ι
√
3
2
√
2
l˙i
)
3
(
1− 1
24
l˙ 2i
) 1
2
; (4.1.8)
and therefore θi is given by
cos θi =
1 + 1
8
l˙ 2i
3− 1
8
l˙ 2i
. (4.1.9)
The continuum time limit of the Regge equations can be obtained by taking
δti → 0, because in this limit, mi → 0, and hence the separation between Cauchy
surfaces goes to zero. We shall denote the CW Cauchy surface at continuum
time parameter t by Σt . To determine the continuum time form of the Regge
equations, it is sufficient to substitute in the continuum time form of the geometric
quantities it depends on. In the continuum time limit, we have that
li → l(t),
li+1 → l(t) + l˙ dt+O
(
dt2
)
,
θi → θ(t) +O(dt) ,
where the overdot now denotes a continuum time–derivative. Therefore, the
geometric quantities above now become
l˙i → l˙ +O(dt) , (4.1.10)
m2i →
(
3
8
l˙ 2 − 1
)
dt2 +O
(
dt3
)
, (4.1.11)
∂Atrapi
∂mi
→ l m˙
[
1
8
l˙ 2 − 1
]− 1
2
+O(dt) , (4.1.12)
cos θi → cos θ ≈
1 + 1
8
l˙ 2
3− 1
8
l˙ 2
+O(dt) , (4.1.13)
where m˙ denotes the quantity
m˙ =
[
3
8
l˙ 2 − 1
] 1
2
. (4.1.14)
We note that in the continuum time limit of the strut-length, where mi → m as
112
4. REGGE CALCULUS OF CLOSED FLRW UNIVERSES
δti → 0, the quantity m˙ is actually related to m through the Taylor expansion
m ≈ m˙ dt +O(dt2) ,
which is obtained by first expanding mi, as given by (4.1.2), in terms of δti and
then taking δti → 0; since the leading order of mi is O(δti), the zeroth order term
of m vanishes as δti → 0. Relation (4.1.13) can be inverted to parametrise l˙ in
terms of θ, thus yielding
l˙ 2 = 8
[
1− 2 tan2
(
1
2
θ
)]
. (4.1.15)
Finally, it is often easiest to study a parent model’s evolution by considering
the evolution of its Cauchy surface’s volume. This volume is given by
UN3(t) =
N3
6
√
2
l(t)3, (4.1.16)
where N3 is the number of tetrahedra in the Cauchy surface and is given by the
first column of Table 4.1. This expression applies equally to both the continuum
time model and the discrete time model; in the latter case, we would simply set
t = ti. In the continuum time model, we can also define the volume’s rate of
expansion; this is given by
U˙N3(t) =
N3
2
√
2
l(t)2 l˙(t). (4.1.17)
4.2 The fully-triangulated CW skeleton
In order to vary the Regge action locally, whereby every edge would be varied
independently of all the others, the CW skeleton must first be fully triangulated.
Every trapezoidal hinge must be divided into a pair of time-like triangular hinges,
as illustrated by the first hinge in Figure 4.2. If appropriately chosen, these
diagonals would fully triangulate the skeleton, dividing each 4-block into four
4-simplices.
To triangulate the region between two Cauchy surfaces, we first label all ver-
tices in Cauchy surface Σi with distinct letters in any arbitrary order; we then
113
4.2. THE FULLY-TRIANGULATED CW SKELETON
AAi
C ′
A
A′
C
B′ C ′
CB
B′
DB
D′
DC
C ′ D′A′
DA
D′
B′
B
A′
A
mAim
B
i di
li+1
li
ABi
Figure 4.2: The six trapezoidal time-like hinges of a 4-block. A diagonal di divides
each hinge into a pair of triangular hinges, and the two struts on the sides of the hinge
are now considered independent quantities. The two triangular areas have been labelled
by AAi and A
B
i on the first hinge, with the superscripts A and B denoting the lower and
upper triangles respectively. Also labelled are the diagonal di and the two independent
struts mAi and m
B
i , with the strut’s superscript matching that of the triangular area to
which the strut is attached.
label all vertices in Cauchy surface Σi+1 with the same letter as their counterparts
in Σi but with an additional prime.
2 For example, in the 5-tetrahedra model, a
Cauchy surface has only five vertices, and we would label the set in Σi by A, B,
C, D, E, and their counterparts in Σi+1 by A
′, B′, C ′, D′, E ′. The diagonal in
each trapezoidal hinge is chosen so that the letter of the diagonal’s vertex in Σi
alphabetically precedes the letter of the diagonal’s vertex in Σi+1. Thus in the
5-tetrahedra model, where there are 10 trapezoidal hinges between Σi and Σi+1,
the 10 diagonals would be AB′, AC ′, AD′, AE ′, BC ′, BD′, BE ′, CD′, CE ′, and
DE ′. This algorithm for choosing trapezoidal diagonals ensures that each 4-block
gets properly triangulated into 4-simplices. We also note that in this algorithm,
it is the ordering of the labels and not the labels themselves that matter.
2If the Cauchy surface has more than 26 vertices, we could label with numbers instead; the
principle would still be the same.
114
4. REGGE CALCULUS OF CLOSED FLRW UNIVERSES
Let us now consider one typical 4-block. The labels of the lower tetrahedron’s
vertices will be ordered in some way, and without loss of generality, let us re-
label them A, B, C, D while preserving the relative ordering. Then the upper
tetrahedron’s vertices will be re-labelled A′, B′, C ′, D′, and with the new labelling,
the 4-block would resemble the one in Figure 4.1. Then under the algorithm
above, the six diagonals would be chosen to be AD′, BD′, CD′, AC ′, BC ′, and
AB′, as Figure 4.2 shows, and the four 4-simplices that result are ABCDD′,
ABCC ′D′, ABB′C ′D′, and AA′B′C ′D′. Since all 4-blocks are the same, all
geometric quantities relevant to the triangulated skeleton can be determined by
just considering this typical 4-block. In particular, we can continue using (4.1.1)
to calculate any geometric quantity.
Using (4.1.1), we therefore deduce that the diagonal has length
d 2i =
1
3
l 2i +
1
24
(3 li+1 + li)
2 − δt 2i . (4.2.1)
There is actually a new type of hinge in the triangulated skeleton, in ad-
dition to the space-like triangular hinge from before and the pair of time-like
triangular hinges. We shall call these new hinges the diagonal hinges, as they
consist of isosceles triangles formed from two diagonal edges and one tetrahedral
edge, an example being ABC ′. However, we have found that the deficit angles of
these hinges are actually zero, so by virtue of the Schla¨fli identity, they will not
contribute to the Regge equations. Therefore they can henceforth be ignored.
Using Heron of Alexandria’s formula of classical antiquity, we can express the
areas AAi and A
B
i of the triangular time-like hinges in terms of the edge-lengths,
giving
AAi =
1
4
[
−l 4i −
(
mAi
)4 − d 4i + 2(l 2i d 2i + l 2i (mAi )2 + d 2i (mAi )2)] 12 , (4.2.2)
ABi =
1
4
[
−l 4i+1 −
(
mBi
)4 − d 4i + 2(l 2i+1d 2i + l 2i+1 (mBi )2 + d 2i (mBi )2)] 12 . (4.2.3)
Differentiating with respect to their respective strut-lengths, we obtain
∂AAi
∂mAj
=
mAi
8AAi
(
l 2i + d
2
i −
(
mAi
)2)
δij, (4.2.4)
∂ABi
∂mBj
=
mBi
8ABi
(
l 2i+1 + d
2
i −
(
mBi
)2)
δij. (4.2.5)
115
4.2. THE FULLY-TRIANGULATED CW SKELETON
If we now substitute in the strut and diagonal lengths given by (4.1.2) and (4.2.1),
and then take the continuum time limit, these derivatives simplify to
∂AAi
∂mAj
=
∂ABi
∂mBj
=
l
2
m˙[
1
8
l˙ 2 − 1
] 1
2
δij +O(dt) . (4.2.6)
After edge-lengths of the same type have been set equal, the deficit angles
δAi and δ
B
i of any triangular time-like hinge become identical to the deficit angle
δ trapi of the original trapezoidal hinge; that is, δ
A
i = δ
B
i = δ
trap
i . Setting the
edge-lengths equal makes the two triangular hinges be co-planar both with each
other and with the original trapezoidal hinge; thus the 4-blocks meeting at the
triangular hinges would be flat; the unit normals of the triangulated faces would
be identical to the unit normals of the original faces; and the dihedral angles
between the triangulated faces would be identical to the dihedral angles between
the original faces. Since the number of faces meeting at the triangulated hinge is
the same as the number of faces at the original hinge, the deficit angles for the
triangulated hinge and the original hinge are identical. Additionally, since the
dihedral angles are unchanged, then in the continuum time limit, l˙ would still be
given by (4.1.15).
Finally, the deficit angle of the space-like triangular hinges should be identical
in both the original and triangulated skeletons, simply because the unit normals
to the faces meeting at the hinge will not change because of the triangulation.
4.3 Global and local Regge equations
As Brewin has pointed out, the global Regge equations can usually be related to
the local Regge equations through the chain rule. A necessary condition is that
the global and local Regge actions be identical. Any action will always include a
component given by (3.2.3), and this component will be identical in the global and
local actions for the following reasons: the diagonal hinges do not contribute to
the local action; the area of each trapezoidal hinge would always equal the areas
of its two constituent triangular hinges while the deficit angles would always be
identical; and both the areas and the corresponding deficit angles of the triangular
space-like hinges would always be identical in both the original and triangulated
skeletons. When there is a non-zero cosmological constant, the action is given by
116
4. REGGE CALCULUS OF CLOSED FLRW UNIVERSES
(3.2.9), which has an extra component involving 4-volumes; since the volume of
a 4-block should equal the sum of the volumes of its four constituent 4-simplices,
the global and local form of (3.2.9) would be identical. When massive particles
are present, the action is given by (3.2.7), which has an extra component involving
the path-length of the particles; clearly, the path through the 4-block should not
depend on whether the 4-block has been triangulated or not, and therefore the
global and local form of (3.2.7) would also be identical.
By the chain rule, the variation of the action S with respect to an arbitrary
global edge-length q can be expressed as
∂S
∂q
=
∑
i
∂S
∂li
∂li
∂q
+
∑
i
∂S
∂mi
∂mi
∂q
+
∑
i
∂S
∂di
∂di
∂q
, (4.3.1)
where li denotes the length of a local tetrahedral edge, di the length of a local
diagonal, and mi the length of a local strut between Σi and Σi+1. This chain rule
clearly shows that the global Regge equation is a linear combination of the local
Regge equations. Since the local Regge equations are given by
0 =
∂S
∂li
, 0 =
∂S
∂mi
, 0 =
∂S
∂di
,
it is clear then that any solution of the local equations will automatically be
a solution of the global Regge equation 0 = ∂S
∂q
. However, the converse is not
necessarily true; the local solutions would in general form only a subset of the
global solutions.
4.4 Brewin’s children models
From any parent model, one can always generate a secondary model by trian-
gulating each parent tetrahedron into a set of smaller tetrahedra. Brewin has
devised a method that can subdivide any tetrahedra such that not only can it
be applied to parent tetrahedra to generate secondary models, but it can also
be applied to each child tetrahedron afterwards to generate even finer-grained
models. Thus, Brewin’s scheme can in principle be applied indefinitely. However,
we shall only consider the first generation of children models.
Under Brewin’s scheme, each parent tetrahedron gets divided into 12 children
117
4.4. BREWIN’S CHILDREN MODELS
tetrahedra. Seven new vertices are introduced, six at the parent edges’ mid-
points, the seventh at the tetrahedral centre. If the parent vertices are labelled A,
B, C, D, then (XY ) will denote the mid-point vertex between any pair of parent
vertices X, Y ∈ {A,B,C,D}, while (ABCD) will denote the central vertex. A
partially subdivided parent tetrahedron is shown in Figure 4.3.
(ABCD)
C
A (AB)
(BD)(AD) vi
vivi
uiui
D
B
Figure 4.3: A partially subdivided parent tetrahedron. Three of the mid-point ver-
tices, (AB), (AD), and (BD), as well as new edges of length vi connecting them have
been shown. The parent edges AB, AD, and BD, have now been subdivided into two
edges of length ui each: A(AB) and (AB)B, A(AD) and (AD)D, and B(BD) and
(BD)D. Also depicted is the central vertex (ABCD).
The scheme also introduces three distinct types of edges. Edges connecting
parent vertices to mid-point vertices, such as A(AB), have length ui. As there are
six parent edges in a parent tetrahedron, so there are 12 length-ui edges. Edges
connecting mid-points to mid-points, such as (AB)(AC), have length vi. Each
face has three of these edges, and as there are four faces per parent tetrahedron,
so there are 12 of these edges per parent tetrahedron as well. Finally, edges
connecting mid-points to (ABCD), such as (AB)(ABCD) have length pi. Each
mid-point contributes one such edge and each parent edge contributes one mid-
point, so there are six length-pi edges per parent tetrahedron. No edges connect
parent vertices to (ABCD).
As a result, there will also be three distinct types of subdivided tetrahedra,
each with four members per parent tetrahedron, thus giving the total of 12 chil-
dren tetrahedra. Each Type I tetrahedron consists of an equilateral base formed
by three mid-point vertices and an apex at a parent vertex; an example would be
A(AB)(AC)(AD). There are four of these tetrahedra in a parent tetrahedron,
118
4. REGGE CALCULUS OF CLOSED FLRW UNIVERSES
one per parent vertex. Type II tetrahedra share the same equilateral base as
Type I but have their apices at (ABCD) instead. Because they share the same
base, there is a 1–1 correspondence between Type I and Type II tetrahedra, so
there are four Type II tetrahedra per parent as well. Finally, a Type III tetra-
hedron consists of an apex at (ABCD) as well and an equilateral base formed
by the three mid-point vertices on a single parent face; an example would be
(AB)(AC)(BC)(ABCD). Since each parent face is associated with one Type
III tetrahedron, there are four Type III tetrahedra per parent tetrahedron. We
note that in terms of their edge-lengths, Type II and Type III tetrahedra are
identical. The different vertices, edges, and tetrahedra in the subdivided parent
tetrahedron have been summarised in Table 4.2.
Simplex type Example
Number per parent
tetrahedron
Parent vertices A, B, C, D 4
Mid-point vertices
(AB), (AC), (AD),
(BC), (BD), (CD)
6
Central vertex (ABCD) 1
Edge ui A(AB) 12
Edge vi (AB)(AC) 12
Edge pi (AB)(ABCD) 6
Type I tetrahedra A(AB)(AC)(AD) 4
Type II tetrahedra (AB)(AC)(AD)(ABCD) 4
Type III tetrahedra (AB)(AC)(BC)(ABCD) 4
Table 4.2: The vertices, edges, and tetrahedra of a subdivided parent tetrahedron.
Finally, each vertex’s world-line will generate a strut connecting one Cauchy
surface to the next. As there are three sets of vertices, there will also be three
sets of struts, with all struts in the same set sharing the same length.
4.4.1 Co-ordinates for a child tetrahedron’s 4-block
To facilitate the calculation of geometric quantities in the children models, we
shall introduce a co-ordinate system similar to (4.1.1). Our approach is a modi-
119
4.4. BREWIN’S CHILDREN MODELS
fication of Collins and Williams’ original approach and differs from that followed
by Brewin, who uses non-Cartesian co-ordinates. Let us consider a typical tetra-
hedron of the model in Cauchy surface Σi. It will always have an equilateral base
of edge-length vi regardless of the tetrahedron’s type. We label the base’s vertices
by A, B, C and the apex by D. The three edges meeting at the apex will all be
of identical length, either ui or pi, and without loss of generality, we shall work
with ui. Then we have the freedom to set the co-ordinates for A, B, C, D to be
A =
(
−vi
2
,− vi
2
√
3
, 0, ιti
)
,
B =
(
vi
2
,− vi
2
√
3
, 0, ιti
)
,
C =
(
0,
vi√
3
, 0, ιti
)
,
D = (0, 0, hi, ιti),
(4.4.1)
where hi is the height of the tetrahedron and is given by
hi =
√
u2i −
1
3
v2i . (4.4.2)
This tetrahedron evolves to another in surface Σi+1 with vertices labelled A
′,
B′, C ′, D′. The vertices’ co-ordinates are determined by the constraints on the
edge-lengths and the requirement that the 4-block have no twist or shear. The
edge-length constraints are that the lengths of struts AA′, BB′, CC ′ be equal,
the lengths of edges A′B′, A′C ′, B′C ′ be vi+1, and the lengths of edges A′D′,
B′D′, C ′D′ be ui+1. Brewin has commented that, as with the parent 4-block,
the constraint on the strut-lengths and the requirement of no twist or shear are
analogous to a choice of shift and lapse functions in the ADM formalism.
Evolution would preserve the equilateral symmetry of the tetrahedron’s base,
and hence, the base would simply expand or contract uniformly about its centre.
However, it might undergo a vertical displacement δzi; this displacement would
be fixed by the length of the struts.
The requirement that apex D′ be located at distance ui from each base vertex
is equivalent to the two constraints that D′ lie at distance hi+1 from the base’s
centre and that the central axis connecting D′ to the base’s centre lie orthogonally
to the base. As base A′B′C ′ defines a 2-dimensional plane in a (3+1)-dimensional
120
4. REGGE CALCULUS OF CLOSED FLRW UNIVERSES
Minkowski space–time, the subspace orthogonal to this base would be a (1+1)-
dimensional plane, and the tetrahedron’s central axis can be oriented along any
direction in this plane. Combined with the first constraint, the second constraint
implies that D′ will lie on a hyperbola in this (1+1)-dimensional plane; exactly
where on the hyperbola it lies depends on the length of strut DD′, which need
not be identical to the length of AA′. Thus the axis of the upper tetrahedron
may in general be Lorentz-boosted relative to the axis of the lower tetrahedron.
Therefore, the upper vertices’ co-ordinates are given most generally by
A′ =
(
−vi+1
2
,− vi+1
2
√
3
, δzi, ιti+1
)
,
B′ =
(
vi+1
2
,− vi+1
2
√
3
, δzi, ιti+1
)
,
C ′ =
(
0,
vi+1√
3
, δzi, ιti+1
)
,
D′ = (0, 0, hi+1 coshψi + δzi, ιti+1 + ιhi+1 sinhψi),
(4.4.3)
where ψi is the relative boost between the upper and lower tetrahedral axes.
However if the 4-block is to have no twist or shear, then we also require that
the world-sheet generated by each tetrahedral edge between Σi and Σi+1 be flat.
In other words, the four vectors parallel to the four sides of this world-sheet must
be co-planar. When this requirement is imposed on any world-sheet involving D′,
such as ADA′D′, we obtain the further constraint that
0 = Aψ −Bψ coshψi + Cψ sinhψi, (4.4.4)
where
Aψ = δti hi vi+1,
Bψ = δti hi+1 vi,
Cψ = hi+1 (δzi vi + hi+1 δvi) .
This constraint effectively determines ψi, and solving it leads to a quadratic
equation with two solutions. However, if we take the limit where ui → vi and
δzi → − 12√6 δvi, we expect to recover the 4-block of an equilateral tetrahedron;
in this limit, ψi should vanish, and this only happens for one of the solutions.
121
4.4. BREWIN’S CHILDREN MODELS
Therefore, we require that
ψi = ln
−Aψ +
√
A2ψ + C
2
ψ −B2ψ
Cψ −Bψ
 . (4.4.5)
Such a choice of ψi would generally result in strut DD
′ having a different
length from those of AA′, BB′, and CC ′. Through a suitable choice of δzi, it
may still be possible to set the length of DD′ equal to the length of the other
struts, but this generally depends on the model. If we do not have the freedom
to choose δzi, then we would only have freedom to choose the length of one strut;
that choice would determine the lengths of all other struts. As mentioned earlier,
Brewin has likened a choice of strut-lengths in the CW formalism to a choice of
lapse function in the ADM formalism and the choice of having no twist or shear
in the 4-blocks to a choice of shift function; we see here that the choice of shift
function has constrained the freedom to choose the lapse function to just the
freedom to choose a single strut-length; this is different from the ADM formalism
where both the lapse and shift functions can be freely chosen independently of
each other throughout the Cauchy surface. This conclusion applies as well to the
parent 4-block since its geometry can be recovered from that of the child 4-block
by setting all tetrahedral edge-lengths to be equal.
Since the co-ordinates of the upper tetrahedron depend on δzi and ψi and since
these are model-dependent quantities, we shall leave the calculation of specific
geometric quantities to later chapters when a specific model is being considered.
4.5 Comparison with the ADM formalism
There appears to be a strong analogy between this (3+1)-formulation of Regge
calculus and the ADM formalism, though this analogy can only be taken so far.
Brewin [89] has suggested that the struts and diagonals of the CW skeleton could
be considered as Regge analogues of the ADM lapse and shift functions, and that
the tetrahedral edge-lengths, being carriers of the 3-geometry, as analogues of
the 3-metric h of a foliation. In the ADM formalism, varying the ADM action
with respect to h leads to an evolution equation for h, while varying with respect
to the lapse and shift functions leads to a set of constraint equations, known
respectively as the Hamiltonian and momentum constraints. These constraint
122
4. REGGE CALCULUS OF CLOSED FLRW UNIVERSES
equations are first integrals of the evolution equation, so if they are satisfied on
an initial Cauchy surface, the evolution equation ensures they remain satisfied
on all subsequent Cauchy surfaces. Additionally, after the initial Cauchy surface,
one also has complete freedom to specify all lapse and shift functions, and this
freedom can be understood as a manifestation of diffeomorphism symmetries in
the ADM formalism. However as Brewin has pointed out, an analogous situation
is not present in Regge calculus: none of the Regge equations can be considered
trivial; rather, all would generally be needed to evolve Cauchy surfaces’ edges,
including not only the tetrahedral edges but also the struts and diagonals. This
lack of freedom to specify the struts and diagonals would seem to imply that
diffeomorphism symmetries are broken in Regge calculus, though as we noted
at the end of Chapter 3.2, the question of whether diffeomorphism symmetries
are broken or not in Regge calculus is still a matter of debate. Nevertheless, in
analogy with the ADM formalism, the Regge equations obtained by varying the
action with respect to the tetrahedral edge-lengths have sometimes been called the
Regge ‘evolution equations’, and the equations obtained by varying with respect
to the struts have sometimes been called the Regge ‘Hamiltonian constraints’.
Brewin [121] has also shown though that in taking the continuum time limit,
the Regge Hamiltonian constraints may reduce to first integrals of the Regge evo-
lution equations. If this happens, one would recover the freedom to specify the
lapse of the Regge skeleton. Moreover, if the Hamiltonian constraints are first
integrals of the evolution equations and if there are sufficient constraint equa-
tions, then one could use them in place of the evolution equations to determine
the geometry of all subsequent Cauchy surfaces. This will be our approach in
subsequent chapters.
123

CHAPTER 5
Regge calculus of the closed
vacuum Λ-FLRW universe
Before proceeding to model any new space–times, it seems most appropriate to
first apply the Collins–Williams formalism to a space–time where the continuum
solution is well-established. This would help us understand the various versions
of this new formalism better, especially its properties and effectiveness. Thus in
this chapter, we shall model the closed vacuum FLRW universe with non-zero
cosmological constant Λ, that is, universes where k > 0, ρ = p = 0, and Λ 6= 0.
From solving the Friedmann equations (1.0.8) and (1.0.9), we find that the
scale factor a(t) for the continuum space–time is
a(t) =
1
2
√
3
Λ
(
e−
√
Λ
3
t + e
√
Λ
3
t
)
, (5.0.1)
where the integration constant has been chosen so that a˙(t) = 0 when t = 0, and
where we have chosen a scaling of a(t) such that it corresponds to the radius of
curvature of constant-t Cauchy surfaces, in which case k = 1.
In this chapter, we shall examine all three of CW’s parent skeletons as well as
Brewin’s first generation of children skeletons; for each model, we shall consider
both global and local variation of the associated Regge action as well as both
the Hamiltonian constraint and the evolution equation. It will be shown that for
the parent models, only global variation leads to a viable model; the diagonals
introduced for local variation seem to break the residual Copernican symmetries
125
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
in the CW Cauchy surfaces, thereby rendering invalid the assumption that all
tetrahedral edge-lengths should be equal. For the children models, we shall focus
on the case where the lengths of all struts have been set equal, regardless of the
strut-type. As will be discussed below, this does not lead to a viable model with
local variation. However, the model obtained from global variation is viable if the
tetrahedra are all equilateral, otherwise the Hamiltonian constraint would not in
general be a first integral of the evolution equation.
In this chapter, we shall also examine the embedding of CW Cauchy surfaces
into 4-dimensional Euclidean space E4. Recall that the constant-time Cauchy
surfaces of closed continuum FLRW universes can be embedded into 3-spheres in
E4, with the embedding given by (1.0.4); we can similarly embed CW Cauchy
surfaces of parent and children models into E4. Such an embedding was briefly
noted by Collins and Williams [63] and much developed by Brewin [64]. Since
the CW Cauchy surfaces are approximations of 3-spheres, this embedding would
map all vertices from a parent Cauchy surface into a single 3-sphere, and it would
in general map each of the three sets of vertices in a child Cauchy surface into
its own 3-sphere. However in Brewin’s embedding of the child Cauchy surface,
he considered only the case where the three sets of vertices had been constrained
to lie on the same 3-sphere. He then proposed that the corresponding 3-sphere
radius R(t) could provide a more accurate analogue to the FLRW scale factor
a(t) than the tetrahedral edge-length. We shall generalise Brewin’s investigation
in two ways: we shall look at alternative radii that can also serve as analogues to
a(t), and we shall also consider models where the three sets of children vertices
do not necessarily lie on the same 3-sphere.
The first half of this chapter will focus on the parent models. We start by
presenting the embedding of these models into 3-spheres and then consider the
different radii of the embedded Cauchy surface that can serve as analogues to
a(t). We then proceed to vary the Regge action globally with respect to the
struts and the tetrahedral edge-lengths. We find that the Hamiltonian constraint
equations, obtained from varying the struts, are actually first integrals of the
evolution equation, obtained from varying the tetrahedral edge-lengths; thus, the
Hamiltonian constraint equations would be sufficient to determine the evolution
of the Regge model. We next vary the Regge action locally with respect to each
of the struts, the diagonals, and the tetrahedral edge-lengths; in this case, we find
that the Hamiltonian constraint equations are again first integrals of the evolu-
126
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
tion equations provided we also satisfy the constraint equations obtained from
varying the diagonals; however these diagonal equations impose rather unphysical
constraints, and we therefore dismiss the local model as unviable. By relating the
global and local Regge equations through the chain rule, we arrive at the same
conclusion again. We next demonstrate that the global models do satisfy the
initial value equation at the moment of time-symmetry, as discussed in Chapter
3.4. We conclude our investigation of the parent models with a brief specula-
tion on the reasons for the local models’ breakdown before finally examining the
evolution of the global models.
The second half of this chapter will focus on the children models. We again
start by presenting the embedding of children Cauchy surfaces into 3-spheres.
We next present the geometric quantities needed to compute the varied Regge
action. After briefly discussing the local variation of the Regge action, we de-
termine the global Regge equations obtained by varying with respect to both
the struts and the tetrahedral edge-lengths, and we then briefly discuss the con-
ditions under which the Hamiltonian constraint would be a first integral of the
evolution equation. We continue on to the initial value equation for the children
models, showing the equation is satisfied by these models. We finally examine
the evolution of the children models, comparing their performance against that
of the parent models, and we speculate briefly on how the children models might
be extended.
5.1 The parent models
5.1.1 Embedding Cauchy surfaces into a 3-sphere
As Collins and Williams first noted and Brewin fully explored, a CW Cauchy
surface Σi can be embedded into E
4 such that all vertices lie on a 3-sphere of
radius Ri. It is most natural to parametrise this 3-sphere using a set of polar
co-ordinates (χ, θ, φ). If (x1, x2, x3, x4) is a set of Cartesian co-ordinates in E4,
127
5.1. THE PARENT MODELS
then points on the 3-sphere can be parametrised by
x1 = Ri cosχ,
x2 = Ri sinχ cos θ,
x3 = Ri sinχ sin θ cosφ,
x4 = Ri sinχ sin θ sinφ,
(5.1.1)
with χ, θ ∈ [0, pi] and φ ∈ [0, 2pi). This embedding provides a natural framework
within which to study and elucidate the underlying geometry of the CW Cauchy
surface. Most importantly, it makes clearer the relationship between the CW
Cauchy surfaces and the FLRW 3-spheres they approximate.
θ
li
φ
li
li
Ri
θ
Figure 5.1: A schematic diagram of an equilateral tetrahedron of edge-length li em-
bedded into a 3-sphere of radius Ri. One dimension has been projected out.
We begin by embedding a subset of the vertices into the 3-sphere. A schematic
diagram illustrating our embedding is given in Figure 5.1. By symmetry of the
3-sphere, we can always choose polar co-ordinates such that one vertex is located
at (χ, θ, φ) = (0, 0, 0). Then, we also have freedom to choose θ and φ co-ordinates
such that one of the neighbouring vertices is at (χ, θ, φ) = (χ0, 0, 0) for some χ0.
If n is the number of triangles meeting at an edge, then there will be n vertices
surrounding the edge formed by the first two vertices. To see this, we can consider
the first two vertices as forming a common base for the n triangles and each of
the n vertices as forming the apex of each of the n triangles. The values for
n corresponding to the different models are listed in the final column of Table
128
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
4.1. We again have freedom to choose the φ co-ordinate such that one of these n
vertices is located at (χ, θ, φ) = (χ1, θ0, 0) for some χ1 and θ0. Once this choice is
made, the remaining n−1 vertices would be located at (χ1, θ0, 2pin ), (χ1, θ0, 4pin ),
. . . ,
(
χ1, θ0,
2(n−1)pi
n
)
. The co-ordinates of these vertices have been summarised
in Table 5.1. By requiring all distances to be li between any pair of neighbouring
Vertex χ θ φ
1 0 0 0
2 χ0 0 0
3 χ1 θ0 0
4 χ1 θ0
2pi
n
...
...
...
...
n+ 2 χ1 θ0
2(n−1)pi
n
Table 5.1: The polar co-ordinates of n+ 2 neighbouring vertices.
vertices, we obtain the equations
(L12)
2 = l2i = 2R
2
i (1− cosχ0),
(L13)
2 = l2i = 2R
2
i (1− cosχ1),
(L23)
2 = l2i = 2R
2
i (1− cosχ0 cosχ1 − sinχ0 sinχ1 cos θ0),
(L34)
2 = l2i = 2R
2
i sin
2 χ1 sin
2 θ0
(
1− cos 2pi
n
)
,
(5.1.2)
where Lij denotes the distance between vertices i and j. We then solve these
equations to obtain
χ0 = χ1, (5.1.3)
cosχ0 =
cos 2pi
n
1− 2 cos 2pi
n
, (5.1.4)
cos θ0 =
cos 2pi
n
1− cos 2pi
n
, (5.1.5)
Z0 :=
li
Ri
=
√
2
[
1− 3 cos 2pi
n
1− 2 cos 2pi
n
] 1
2
. (5.1.6)
These are the relations Brewin [64] obtained for the parent models’ embedding,
although we have derived them here independently of him.
129
5.1. THE PARENT MODELS
Two interesting features of the Cauchy surface geometry come to light from
this embedding. First, we can now see that li and Ri are related by the constant
ratio Z0. Most notably, this ratio is independent of the label i and hence of time
ti. Thus, we can define a radius R(ti) = l(ti)/Z0 for our CW Cauchy surfaces,
and this serves as a natural analogue to the FLRW scale factor a(t). Secondly,
we see that as the edge-lengths l(ti) expand and contract, the 3-sphere simply
expands and contracts about its centre, and the vertices simply move radially
inwards or outwards accordingly; their angular positions remain constant.
The embedding above has yielded one possible definition of radius for the CW
Cauchy surface, namely the vertices’ embedding radius. This was the definition
Brewin chose as his Regge analogue to the FLRW scale factor of a(t). However,
there are other equally plausible definitions of radius for the Cauchy surface: one
could just as well have chosen the radius of any other point in the tetrahedra,
as there will always be a set of points in the Cauchy surface sharing that same
radius. Some possibilities include the radius R1 to the centres of edges
R1(ti) =
1√
2
[
1− cos 2pi
n
1− 2 cos 2pi
n
] 1
2
R(ti), (5.1.7)
Z1 :=
l(ti)
R1(ti)
= 2
[
1− 3 cos 2pi
n
1− cos 2pi
n
] 1
2
, (5.1.8)
the radius R2 to the centres of triangles
R2(ti) =
1√
3
[
1
1− 2 cos 2pi
n
] 1
2
R(ti), (5.1.9)
Z2 :=
l(ti)
R2(ti)
=
√
6
[
1− 3 cos 2pi
n
] 1
2
, (5.1.10)
or the radius R3 to the centres of tetrahedra
R3(ti) =
1
2
[
1 + cos 2pi
n
1− 2 cos 2pi
n
] 1
2
R(ti), (5.1.11)
Z3 :=
l(ti)
R3(ti)
= 2
√
2
[
1− 3 cos 2pi
n
1 + cos 2pi
n
] 1
2
. (5.1.12)
The main point to notice is that regardless of which radius we choose, the ratio
130
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
between that radius and l(ti) is always a constant independent of ti.
Since no particular choice of radius seems preferred, one natural choice would
be to average over all radii across the entire Cauchy surface. We have numerically
computed this average radius R¯ in terms of R(ti) to be
R¯(ti) =

0.484066R(ti) for the 5 tetrahedra model,
0.627392R(ti) for the 16 tetrahedra model,
0.940901R(ti) for the 600 tetrahedra model.
(5.1.13)
The derivation of these numbers has been explained in Appendix E.
We can also consider the effective radius R˜(ti) obtained by treating the volume
of the Cauchy surface as if it were the volume of a 3-sphere. The volume of a
3-sphere of radius R˜(t) is
U(t) = 2pi2R˜(t)3, (5.1.14)
while the volume of an N3-tetrahedra CW universe is given by (4.1.16). If we
equate the two expressions, we find that the effective radius is
R˜(ti) =
(
N3
12
√
2pi2
)1/3
l(ti). (5.1.15)
For comparison with the average radii in (5.1.13), the numerical values for
R˜(t) in terms of the vertex radius R(ti) are
R˜(ti) =

0.490488R(ti) for the 5 tetrahedra model,
0.646482R(ti) for the 16 tetrahedra model,
0.945651R(ti) for the 600 tetrahedra model,
(5.1.16)
and the fractional difference between these numerical factors and those in (5.1.13)
are
R¯(ti)− R˜(ti)
R¯(ti)
=

−0.0132668 for the 5 tetrahedra model,
−0.0304271 for the 16 tetrahedra model,
−0.00504837 for the 600 tetrahedra model.
(5.1.17)
The two radii are very close to each other, but with R˜(ti) consistently greater
than R¯(ti) by a slight amount.
131
5.1. THE PARENT MODELS
Finally, we shall consider one more possible definition of the radius when we
have obtained the equations for l(t) and l˙(t) in the continuum time limit. Like
all other radii, this radius Rˆ(t) is related to l(t) by a constant Zˆ,
Rˆ(t) = Zˆ l(t), (5.1.18)
and we define Zˆ by requiring Rˆ = a(t) when both dRˆ
dt
= 0 and a˙ = 0.
5.1.2 Global variation of the parent models
In general, the Regge action for a skeleton with non-zero cosmological constant
Λ is given by (3.2.9). However, as discussed in Chapter 4.1, the original CW
skeletons have only two distinct sets of hinges associated with any Cauchy surface,
the time-like trapezoidal hinges and the space-like triangular hinges. Thus when
the Regge action (3.2.9) is applied to such a skeleton, we obtain the action
8piSglobal =
∑
i∈
{
trapezoidal
hinges
}A
trap
i δ
trap
i +
∑
i∈
{
triangular
hinges
}Atrii δ trii − Λ
∑
i∈
{
4-blocks
}V
(4)
i . (5.1.19)
We shall now derive the Regge equations by global variation. As mentioned
in Chapter 4.1, there are only two distinct types of edges characterising a global
skeleton, the struts and the tetrahedral edges. If we vary the action with respect
to a strut of length mi, we obtain the Regge equation
0 = N1
∂Atrapi
∂mi
δ trapi −N3 Λ
∂V
(4)
i
∂mi
, (5.1.20)
where N1 and N3 are the numbers of edges and tetrahedra in a Cauchy surface
and equal the numbers of trapezoidal hinges and 4-blocks respectively between
any two consecutive Cauchy surfaces Σi and Σi+1. If we vary with respect to a
tetrahedral edge of length li, we obtain the Regge equation
N1
[
∂Atrapi
∂li
δ trapi +
∂Atrapi−1
∂li
δ trapi−1
]
+N2
∂Atrii
∂li
δ trii = N3 Λ
[
∂V
(4)
i
∂li
+
∂V
(4)
i−1
∂li
]
,
(5.1.21)
where N2 is the number of triangles in a Cauchy surface; all three numbers N1,
N2, and N3 are given in Table 4.1. From these equations, we see that there are
132
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
only three types of geometric quantities relevant to the global Regge equations:
the varied hinge areas, the corresponding deficit angles, and the varied 4-volumes.
We shall now derive each in turn. Once again, we shall work with the 4-block
co-ordinates given by (4.1.1) and their counterparts for A′, B′, C ′, D′; in these
co-ordinates, the relationship between the strut-length mi and the time difference
δti is again given by (4.1.2).
We shall first consider the trapezoidal hinges, where the area Atrapi is once
again given by (4.1.3). Varying Atrapi with respect to mj again yields (4.1.5).
When the tetrahedral edges in the skeleton are varied, there will actually be two
sets of trapezoidal hinges that get affected because each edge li is attached to two
trapezoidal hinges, one between surfaces Σi and Σi+1 and the other between Σi
and Σi−1. Varying the area of a ‘future’ hinge with respect to li yields
∂Atrapi
∂li
= − ι
2
1
2
li(li+1 − li) +m2i√
1
4
(li+1 − li)2 −m2i
, (5.1.22)
and varying the area of the ‘past’ hinge yields
∂Atrapi−1
∂li
=
ι
2
1
2
li(li − li−1)−m2i−1√
1
4
(li − li−1)2 −m2i−1
. (5.1.23)
Finally, the deficit angle δ trapi of this hinge is again given by (4.1.6), with the
dihedral angle θi between faces at the hinge given by (4.1.9).
We now turn to the triangular hinges. The area for the hinges on Σi is given
by (4.1.4). Varying this with respect to li yields
∂Atrii
∂li
=
√
3
2
li. (5.1.24)
We can again express the hinge’s deficit angle in terms of the dihedral angles
between the faces meeting at the hinge. Four faces meet at a triangular hinge in
Σi. For hinge ABC in Figure 4.1, three of these faces are ABCD, ABCA
BC,
and ABCABC, where we use superscripts  and  to denote the counterparts
to vertices A, B, C in Σi+1 and Σi−1 respectively. The fourth face corresponds
to the neighbouring tetrahedron, which we denote by ABCE. By symmetry,
ABCD and ABCE will form the same dihedral angle with ABCABC and
133
5.1. THE PARENT MODELS
with ABCABC; hence, there will only be two distinct dihedral angles sur-
rounding this hinge. We take φi to be the angle ABCD and ABCE form with
ABCABC, and φi to be the angle they form with ABCA
BC. Thus the
deficit angle of ABC is
δ trii = 2pi − 2φi − 2φi. (5.1.25)
To determine the two dihedral angles, we shall take the scalar product of the
unit vectors tangent to the two faces but orthogonal to the hinge; this is the
second method described towards the end of Chapter 3.1. We shall deduce φi
first. In the 4-block co-ordinates of (4.1.1), the unit vector tangent to ABCD
but orthogonal to ABC is simply
uˆµABCD = (0, 0, 1, 0),
while the equivalent vector tangent to ABCABC is
uˆµ
ABCABC =
(
0, 0,− 1
2
√
6
l˙i, ι
)
√
1
24
l˙2i − 1
.
Thus φi is given by
cosφi = −
1
2
√
6
l˙i√
1
24
l˙2i − 1
. (5.1.26)
By swapping li+1 for li−1, which appears implicitly in l˙i, we immediately obtain
the corresponding expression for φi, that is,
cosφi =
1
2
√
6
l˙i−1√
1
24
l˙2i−1 − 1
. (5.1.27)
The final geometric quantities required for the Regge equations are the 4-block
volumes. The volume V
(4)
i of ABCDA
′B′C ′D′ is given by
V
(4)
i =
ι
24
√
2
(l2i+1 + l
2
i )(li+1 + li)
[
3
8
(li+1 − li)2 −m2i
] 1
2
. (5.1.28)
134
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
Varying this with respect to mj yields
∂V
(4)
i
∂mj
= − ι
24
√
2
mi (l
2
i+1 + l
2
i )(li+1 + li)
[
3
8
(li+1 − li)2 −m2i
]− 1
2
δij. (5.1.29)
When entire 4-blocks are varied with respect to the tetrahedral edge-lengths, the
situation is similar to the trapezoidal hinges: each edge lj is associated with a
‘past’ 4-block between Σj and Σj−1 and a ‘future’ 4-block between Σj and Σj+1.
Varying a ‘future’ 4-block with respect to lj yields
∂V
(4)
i
∂lj
= − ι
24
√
2
 32 l3i (li+1 − li) +m2i (l2i+1 + 2 li+1li + 3 l2i )√
3
8
(li+1 − li)2 −m2i
 δij, (5.1.30)
and varying a ‘past’ 4-block yields
∂V
(4)
i−1
∂lj
=
ι
24
√
2
 32 l3i (li − li−1)−m2i−1 (l2i−1 + 2 li−1li + 3 l2i )√
3
8
(li − li−1)2 −m2i−1
 δij. (5.1.31)
We can now substitute these geometric quantities into the Regge equations
above. For the moment, we shall only do this for (5.1.20), which yields
l2i+1 + l
2
i = 12
√
2
N1
N3 Λ
[ 3
8
(li+1 − li)2 −m2i
1
4
(li+1 − li)2 −m2i
] 1
2
(2pi − nθi) . (5.1.32)
The other equation simplifies greatly in the continuum time limit, so we shall
only present its continuum time form later on.
We now take the continuum time limit, where δti → 0, to obtain a differential
equation for l(t). In this limit, the tetrahedral edge-lengths and dihedral angles
take the form
li → l(t),
li+1 → l(t) + l˙ dt+ 1
2
l¨ dt2 +O
(
dt3
)
,
li−1 → l(t)− l˙ dt+ 1
2
l¨ dt2 +O
(
dt3
)
,
θi → θ(t) + θ˙ dt+O
(
dt2
)
,
θi−1 → θ(t) + θ˙ dt+O
(
dt2
)
,
135
5.1. THE PARENT MODELS
φi → φ(t) + φ˙ dt+O
(
dt2
)
,
φi → φ(t) + φ˙ dt+O
(
dt2
)
,
where in the context of continuum time, the overdot denotes a time-derivative.
Some of these quantities also appear in Chapter 4.1, but in a slightly different
form; here, we are expanding to a higher order in dt, and we have introduced
superscripts  and  to distinguish ‘future’ dihedral angles from ‘past’ dihedral
angles. The various geometric quantities now become
l˙i → l˙ dt+ 1
2
l¨ dt2 +O
(
dt3
)
,
m2i →
(
3
8
l˙2 − 1
)
dt2 +O
(
dt3
)
,
∂Atrapi
∂mi
→ l m˙
[
1
8
l˙ 2 − 1
]− 1
2
+O(dt) ,
∂Atrapi
∂li
→ − ι
2
1[
1− 1
8
l˙2
] 1
2
[
1
2
ll˙
+
dt
1− 1
8
l˙2
[
1
4
ll¨ +
1
2
l˙2 − 3
64
l˙4 − 1
]]
+O
(
dt2
)
,
∂Atrapi−1
∂li
→ ι
2
1[
1− 1
8
l˙2
] 1
2
[
1
2
ll˙
− dt
1− 1
8
l˙2
[
1
4
ll¨ +
1
2
l˙2 − 3
64
l˙4 − 1
]]
+O
(
dt2
)
,
∂Atrii
∂li
→
√
3
2
l,
cos θi →
1 + 1
8
l˙2
3− 1
8
l˙2
+
1
2
l˙ l¨[
3− 1
8
l˙2
]2 dt+O(dt2) ,
cos θi−1 →
1 + 1
8
l˙2
3− 1
8
l˙2
−
1
2
l˙ l¨[
3− 1
8
l˙2
]2 dt+O(dt2) ,
θ = θ = arccos
(
1 + 1
8
l˙2
3− 1
8
l˙2
)
+O(dt) ,
θ˙ = −θ˙ = − 1
4
√
2
1[
1− 1
8
l˙2
] 1
2
l˙ l¨
3− 1
8
l˙2
dt+O
(
dt2
)
,
136
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
cosφi →
ι
2
√
6
1[
1− 1
24
l˙2
] 1
2
[
l˙ +
1
2
l¨dt
1− 1
24
l˙2
]
+O
(
dt2
)
,
cosφi → −
ι
2
√
6
1[
1− 1
24
l˙2
] 1
2
[
l˙ −
1
2
l¨dt
1− 1
24
l˙2
]
+O
(
dt2
)
,
φ = pi − φ = arccos
 ι2√6 l˙[
1− 1
24
l˙2
] 1
2
+O(dt) ,
φ˙ = φ˙ = − ι
4
√
6
l¨
1− 1
24
l˙2
dt+O
(
dt2
)
,
∂V
(4)
i
∂mi
→ − ι
6
√
2
l3
[
3
8
l˙2 − 1
] 1
2
+O(dt) ,
∂V
(4)
i
∂li
→ − ι
24
√
2
[
3
2
l3l˙ + dt
(
3
4
l3l¨ + 6l2
(
3
8
l˙2 − 1
))]
+O
(
dt2
)
,
∂V
(4)
i−1
∂li
→ ι
24
√
2
[
3
2
l3l˙ − dt
(
3
4
l3l¨ + 6l2
(
3
8
l˙2 − 1
))]
+O
(
dt2
)
.
The variation above of Atrapi with respect to the strut mi is exactly identical to
(4.1.12), but we have included it here for ease of reference. Since θ = θ, we shall
henceforth denote this quantity simply by θ.
After substituting these results into the Regge equations (5.1.32) and (5.1.21),
we obtain, to leading order in dt,
l2 = 6
√
2
N1
N3 Λ
(2pi − nθ)[
1− 1
8
l˙2
] 1
2
, (5.1.33)
0 =
N1
1− 1
8
l˙2
 (2pi − nθ)[
1− 1
8
l˙2
] 1
2
(
1
4
ll¨ +
1
2
l˙2 − 3
64
l˙4 − 1
)
+
n
8
√
2
ll˙2l¨
3− 1
8
l˙2
− N2
2
√
2
ll¨
1− 1
24
l˙2
− N3 Λ
12
√
2
l2
[
3
4
ll¨ + 6
(
3
8
l˙2 − 1
)]
.
(5.1.34)
The first equation is the Regge Hamiltonian constraint, and the second equa-
137
5.1. THE PARENT MODELS
tion is the Regge evolution equation. The constraint equation shows that on some
initial Cauchy surface, we need only specify l(t = 0) = l0 as initial data, and then
l˙(t = 0) will follow from the constraint.
It can be shown that the constraint equation is actually a first integral of the
evolution equation; a proof has been provided in Appendix F. This implies that
the constraint equation is sufficient to determine the evolution of l(t), so we shall
henceforth work with (5.1.33) only.
In the continuum time limit, l˙(t) is again given parametrically by (4.1.15).
We can then use this relation to express l(t) parametrically in terms of θ as well,
thus obtaining
l2 = 6
N1
N3 Λ
(2pi − nθ)
tan
(
1
2
θ
) . (5.1.35)
For the strut-length to be time-like, that is, for m(t)2 < 0, we require θ > pi
3
, and
for l˙ to be real, we require θ ≤ 2 arctan
(
1√
2
)
. Hence, θ must lie in the range
pi
3
< θ ≤ 2 arctan
(
1√
2
)
. (5.1.36)
Now that we have l(t) and l˙(t), we can determine Rˆ(t). Recall that we chose
the constants in a(t) so that a˙ = 0 when t = 0. At this point, a2 = 3
Λ
. On the
other hand, dRˆ
dt
will be zero when l˙ = 0, which happens when θ0 = 2 arctan
(
1√
2
)
.
Inserting θ0 into (5.1.35), we have that
l20 =
3
Λ
2
√
2N1
N3
[
2pi − 2n arctan
(
1√
2
)]
.
Therefore, we find that
Rˆ(t) =
 N3
2
√
2N1
1(
2pi − 2n arctan
(
1√
2
))
 12 l(t). (5.1.37)
5.1.3 Local variation of the parent models
For comparison, we shall now derive the Regge equations by locally varying the
action. As discussed in Chapter 4.2, the triangulated parent skeleton has three
types of hinges; these are the time-like triangular hinges, the space-like triangular
138
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
hinges, and the diagonal hinges, though as we mentioned in Chapter 4.2, the
diagonal hinges have zero deficit angle and can therefore be ignored. Thus the
Regge action for the triangulated skeleton can be written as
8pi Slocal =
∑
i∈
{
trapezoidal
hinges
}
[
AAi δ
A
i + A
B
i δ
B
i
]
+
∑
i∈
{
triangular
hinges
}Atrii δ trii − Λ
∑
i∈
{
4-blocks
}V
(4)
i , (5.1.38)
where, again, the superscripts A and B denote the lower and upper triangular
subdivisions of the trapezoidal hinge as depicted in Figure 4.2. Here, we continue
using V
(4)
i as a short-hand to denote the sum of the entire 4-block volume even
though the 4-block is actually triangulated.
We shall focus on obtaining the constraint equation first, that is, on varying
with respect to the strut-lengths mi. Locally varying with respect to an arbitrary
strut-length mj, we obtain an equation of the form
0 = NA
∂AAi
∂mj
δAi +NB
∂ABi
∂mj
δBi − Λ
∂V¯
(4)
i
∂mj
, (5.1.39)
where
∂V¯
(4)
i
∂mj
= NAA′B′C′D′
∂V olAA
′B′C′D′
∂mj
+NABB′C′D′
∂V olABB
′C′D′
∂mj
+NABCC′D′
∂V olABCC
′D′
∂mj
+NABCDD′
∂V olABCDD
′
∂mj
,
and where V olX denotes the 4-volume of 4-simplex X situated between Σi and
Σi+1. NA and NB are the numbers of lower and upper triangular time-like hinges
meeting at mj, while NAA′B′C′D′ , NABB′C′D′ , NABCC′D′ , and NABCDD′ are the
numbers of 4-simplices corresponding respectively to AA′B′C ′D′, ABB′C ′D′,
ABCC ′D′, and ABCDD′ meeting at mj as well. This equation only depends
on geometric quantities involving the time-like hinges and the volumes of the
triangulated 4-block.
We first consider the pair of triangular time-like hinges. Their areas are once
again given by (4.2.2) for AAi and (4.2.3) for A
B
i ; the variation of these areas with
respect to their respective strut-lengths is given by (4.2.4) and (4.2.5), and the
continuum time limit of these expressions is given by (4.2.6). As explained in
Chapter 4.2, the deficit angles δAi and δ
B
i corresponding to these hinges become
139
5.1. THE PARENT MODELS
identical to the deficit angle δ trapi of the original trapezoidal hinge when corre-
sponding edge-lengths have been set equal; thus δAi and δ
B
i would be given by
(4.1.6), with θi given by (4.1.9), and in the continuum time limit, l˙ would be
given by (4.1.15).
We next consider the volumes of the 4-simplices. To compute these volumes,
we shall use the Cayley–Menger determinant [122–124], which generalises Heron’s
formula from areas of triangles to volumes of n-simplices. Suppose we label the
vertices of an n-simplex with numbers 0 to n; then the volume of this simplex
can be computed by the formula
V ol(n) =
[
(−1)n+1
2n(n!)2
det(B)
] 1
2
, (5.1.40)
where B is a symmetric matrix given by
Bij =

l2i−2 j−2 for i, j > 1,
0 for i = j = 1,
1 otherwise,
(5.1.41)
and where lij = lji is the distance between vertices i and j. So for the volume of
a 4-simplex, we have
V ol(4) =
1
96
√
− det(B), (5.1.42)
and
B =

0 1 1 1 1 1
1 0 l201 l
2
02 l
2
03 l
2
04
1 l201 0 l
2
12 l
2
13 l
2
14
1 l202 l
2
12 0 l
2
23 l
2
24
1 l203 l
2
13 l
2
23 0 l
2
34
1 l204 l
2
14 l
2
24 l
2
34 0

. (5.1.43)
We now differentiate each 4-simplex volume with respect to its strut-length.
In our 4-block, each strut will be an edge of exactly one of the four 4-simplices,
and each of the four 4-simplices will be attached to exactly one of the four struts.
When we differentiate each volume with respect to its associated strut-length, the
resulting expression simplifies greatly if we then take the continuum time limit;
therefore we shall present only these continuum-time expressions here. In the
140
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
continuum time limit, we find that
∂V olAA
′B′C′D′
∂mAA
′
i
=
∂V olABB
′C′D′
∂mBB
′
i
=
∂V olABCC
′D′
∂mCC
′
i
=
∂V olABCDD
′
∂mDD
′
i
=
l3
24
√
2
[
1− 3
8
l˙2
] 1
2
+O(dt) .
(5.1.44)
With the continuum time form of the relevant geometric quantities, we can
now take the continuum time limit of the constraint equation (5.1.39) to obtain a
differential equation for l(t). Before doing this though, we can make some further
simplifications to the continuum time form of (5.1.39). We saw in the continuum
time limit that all derivatives of the time-like hinges’ areas became identical, as
did all derivatives of the 4-simplices’ volumes. Using this knowledge, we can
significantly simplify the continuum time form of (5.1.39) to
Nedges/vertex
∂AA
∂m
δA = ΛNtetrahedra/vertex
∂V AA
′B′C′D′
∂m
.
The constants Nedges/vertex and Ntetrahedra/vertex are the numbers of edges and tetra-
hedra respectively meeting at any single vertex; they follow from the fact that
each strut corresponds to the world-line of a vertex, each time-like hinge at that
strut to an edge at the vertex, and each 4-simplex at that strut to a tetrahedron
at the vertex; therefore, the total number of triangular hinges meeting at the
strut is identical to the number of edges at a vertex, Nedges/vertex, and the total
number of 4-simplices at the strut is identical to the number of tetrahedra at a
vertex, Ntetrahedra/vertex.
Substituting all geometric quantities into the above equation, we obtain
l2 =
12
√
2
Λ
Nedges/vertex
Ntetrahedra/vertex
(2pi − nθ)[
1− 1
8
l˙2
] 1
2
.
However, Nedges/vertex and Ntetrahedra/vertex are related to N1 and N3 by the relations
Nedges/vertex =
2N1
N0
and
Ntetrahedra/vertex =
4N3
N0
,
141
5.1. THE PARENT MODELS
where N0 is the number of vertices in a Cauchy surface and is also given by Table
4.1. We therefore recover equation (5.1.33); thus in this case, local variation
yields the same constraint equation as global variation.
By locally varying action (5.1.38) with respect to li, we can also obtain an
evolution equation that is identical to (5.1.34). To do this however, it turns out
we must also make use of the diagonal Regge equations, that is, the equations
obtained by locally varying the action with respect to each di. It can be shown
that such equations are of the form
0 = Λ
∑
j
∂V
(4)
j
∂di
, (5.1.45)
where the summation is over all 4-blocks containing the diagonal being varied.
The area terms vanish from this equation, as it can be shown that
∂AAi
∂di
= −∂ABi
∂di
for any triangulated trapezoidal hinge.
To understand the relationship between the diagonal Regge equations and the
evolution equations, let us first consider the variation of the 4-simplex volumes
with respect to the diagonals and the tetrahedral edges. These derivatives actu-
ally depend on which tetrahedral edge or diagonal is being varied. Consider the
triangulated 4-blocks lying between Σi−1 and Σi+1; if we vary the volumes with
respect to the diagonals and then take the continuum time limit, we obtain
∂V
(4)
i
∂dAB
′
i
→ − ιδt
24
√
2
l2
(
1
8
l˙2 + 1
)
+O
(
dt2
)
,
∂V
(4)
i
∂dAC
′
i
→ − ιδt
24
√
2
l2
(
1
4
l˙2
)
+O
(
dt2
)
,
∂V
(4)
i
∂dAD
′
i
→ − ιδt
24
√
2
l2
(
3
8
l˙2 − 1
)
+O
(
dt2
)
,
∂V
(4)
i
∂dBC
′
i
→ − ιδt
24
√
2
l2
(
1
8
l˙2 + 1
)
+O
(
dt2
)
,
∂V
(4)
i
∂dBD
′
i
→ − ιδt
24
√
2
l2
(
1
4
l˙2
)
+O
(
dt2
)
,
∂V
(4)
i
∂dCD
′
i
→ − ιδt
24
√
2
l2
(
1
8
l˙2 + 1
)
+O
(
dt2
)
.
Clearly the diagonals give different derivatives. If we do a similar variation with
142
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
respect to the tetrahedral edge-lengths, it can be shown that all six derivatives
can be expressed in the form
∂
(
V
(4)
i + V
(4)
i−1
)
∂l xyi
→ ιδt
24
√
2
l2
[
2
(
1− 3
8
l˙2
)
− 1
4
ll¨
]
− ∂V
(4)
i
∂dxy′
,
where l xyi is the length of the edge between vertices x and y and d
xy′ is the length
of the diagonal triangulating the world-sheet between Σi and Σi+1 of edge l
xy
i .
Thus each tetrahedral edge also gives a different derivative.
Now if we vary the action with respect to an arbitrary edge l xyi , we obtain an
evolution equation that can be expressed in the form
0 =
(
RHS of (5.1.34)
)
+ Λ
∑ ∂V (4)
∂dxy′
, (5.1.46)
where the summation is over all 4-blocks containing diagonal dxy
′
; and by the
diagonal Regge equation (5.1.45), this summation is equal to zero. We thus
recover the same evolution equation (5.1.34) as we obtained from globally varying
the action.
Since the diagonal equation is just a sum of different volume derivatives, we
deduce from the form of the volume derivatives above that the diagonal equation
will necessarily have the form
0 = −Λ ιδt
24
√
2
l2
(
1
8
P l˙2 +Q
)
,
for some constants P and Q. Unfortunately, this equation does not give any
physically meaningful solutions as it implies that either l = 0 or l˙2 = −8Q/P .
If P and Q have the same signs, then l˙ would be imaginary. Even if P and Q
have opposite signs, l˙2 would still be a constant, and relation (4.1.15) for l˙ would
only equal this constant at a single value of θ; this implies that there is only
one moment in the evolution of the universe, as given parametrically by relations
(5.1.35) and (4.1.15), where such a diagonal equation could be satisfied.
Furthermore, without knowing how the triangulated 4-blocks fit together in
the skeleton globally, we cannot determine the exact values for P and Q. The
reason is as follows: a single diagonal will be shared by n 4-blocks; the diagonal
might behave like a AB′-type diagonal in one 4-block but like a AC ′-type diag-
143
5.1. THE PARENT MODELS
onal in a neighbouring 4-block, and these two 4-blocks will clearly have different
contributions to the sum in (5.1.45); therefore it is even conceivable that different
diagonals will give different Regge equations.
We can however obtain one constraint equation by summing the diagonal
Regge equations over all diagonals in the Cauchy surface Σt; it then follows that
0 =
∑
d xy
′
in Σt
∑
V (4)
at d xy
′
∂V (4)
∂dxy′
=
∑
V (4)
in Σt
∑
d xy
′
in V (4)
∂V (4)
∂dxy′
= −N3 ιδt
24
√
2
l2
(
5
4
l˙2 + 2
)
.
If l 6= 0, then this implies that l˙2 = −8
5
, which, as we have just remarked, is
clearly non-physical.
5.1.4 Relationship between the global and local Regge
equations
In Chapter 4.3, it was argued that the global and local Regge actions for a CW
skeleton with cosmological constant would be identical. It was also shown that
the corresponding Regge equations can be related through a chain rule of the
form (4.3.1), and this implied that any solution of the local equations would also
be a solution of the global equation, though the converse was not necessarily true.
We shall now further explore the relationship between the global and local Regge
equations specific to the CW model of the Λ-FLRW universe. However, we sus-
pect several features of this relationship may apply generally to the relationship
between any global and local Regge equations obtained from the CW formalism.
Using the chain rule (4.3.1), we can express the global variation of S with
respect to a global strut-length m as
0 =
∂S
∂m
=
∑
i
∂S
∂m`i
∂m`i
∂m
+
∑
i
∂S
∂di
∂di
∂m
, (5.1.47)
where the summations are constrained to the region between a single pair of
144
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
Cauchy surfaces Σj and Σj+1, and where m
`
i denotes the length of a local strut
and di the length of a local diagonal. Since we shall be setting all strut-lengths
to be equal, then
∂m`i
∂m
= 1 for all i. Additionally, as we saw above, ∂S
∂m`i
is O(1)
to leading order in dt. Thus the first summation has an overall leading order of
O(1). On the other hand, it can be shown that
∂di
∂m
=
m
d
,
which becomes m
l
in the continuum time limit. However m has a leading order of
O(dt) in this limit, and as we saw above, ∂S
∂di
has a leading order of O(dt). Thus
the second summation has an overall leading order of O(dt2), which is higher
than that of the first summation. It therefore does not contribute to the Regge
equation at leading order, and we can consequently simplify (5.1.47) to just
0 =
∂S
∂m
=
∑
i
∂S
∂m`i
; (5.1.48)
this result is consistent with what we saw above, where the global constraint
equation and the local constraint equation were in fact identical.
Using the chain rule, we can also express the global variation with respect to
the tetrahedral edge-lengths l as
0 =
∂S
∂l
=
∑
i
∂S
∂l`i
∂l`i
∂l
+
∑
i
∂S
∂di
∂di
∂l
+
∑
i
∂S
∂di−1
∂di−1
∂l
, (5.1.49)
where the summations are constrained to a single Cauchy surface Σj as well as the
regions between Σj and its neighbours Σj−1 and Σj+1, and where l`i denotes the
length of a local tetrahedral edge. Because the tetrahedral edge-lengths will all be
set equal, we have that
∂l`i
∂l
= 1. Additionally, it can be shown that ∂di
∂l
= ∂di−1
∂l
= 1
2
in the continuum time limit. Thus, we can simplify the above equation to
0 =
∂S
∂l
=
∑
i
∂S
∂l`i
+
∑
i
∂S
∂di
, (5.1.50)
where we have made use of the fact that
∑
i
∂S
∂di
=
∑
i
∂S
∂di−1
in the continuum
time limit. However, it can be shown that in this case, both ∂S
∂l`i
and ∂S
∂di
are O(dt)
at leading order. Thus in contrast to the situation with the struts, a solution to
145
5.1. THE PARENT MODELS
0 = ∂S
∂l`i
by itself is not sufficient to be a solution to 0 = ∂S
∂l
; we must also satisfy
0 = ∂S
∂di
. This is what we saw above, where we were able to recover the global
evolution equation (5.1.34) from its local counterpart (5.1.46) only when we also
made use of the diagonal Regge equation (5.1.45). We shall discuss possible
reasons for the local model’s unviability in Section 5.1.6.
5.1.5 Initial value equation of the parent models
The parent models have a moment of time-symmetry at the point of minimum
expansion. This happens when l˙ = 0, corresponding to a dihedral angle of
θ0 = 2 arctan
(
1√
2
)
= arccos
(
1
3
)
,
and from the Regge equation (5.1.35), the edge-lengths would then be
l20 = 6
√
2
N1
N3 Λ
(2pi − nθ0). (5.1.51)
We shall now demonstrate that this result is consistent with the initial value
equation at the moment of time-symmetry, specifically in the form of (3.4.10).
In the Einstein equation with cosmological constant, as given by (1.0.1), we can
always absorb the cosmological constant term Λgµν into Tµν ; it effectively acts as
a perfect fluid source where ρΛ = −pΛ = Λ8pi . Therefore, the initial value equation
for the vacuum Λ-universe can be expressed as∑
i liδi
‘volume per vertex’
= 2Λ, (5.1.52)
where, once again, the summation is over all edges attached to a single vertex and
δi is the 3-dimensional deficit angle of an edge li in the Cauchy surface at time-
symmetry. Since all vertices and tetrahedra are identical, there is a well-defined
‘volume per vertex’ given by
‘volume per vertex’ =
N3
N0
(volume of one tetrahedron)
=
N3
6
√
2N0
l30.
Also, as there are 2N1
N0
edges radiating from any single vertex, the summation in
146
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
(5.1.52) can be expressed as
∑
i liδi =
2N1
N0
l0 δ, where δ is the common deficit
angle of all edges. Substituting these quantities into (5.1.52), we obtain
l20 = 6
√
2
N1
N3 Λ
δ, (5.1.53)
which is exactly identical to (5.1.51) provided δ = (2pi − nθ0).
Since n triangular faces do meet at any single edge, then the deficit angle
δ will have the form (2pi − nθ¯0), where θ¯0 is the 3-dimensional dihedral angle
between triangles in the Cauchy surface. Thus to complete our proof, we must
show that θ¯0 = θ0. Consider a typical tetrahedron in E
3 with vertices A,B,C,D,
and assign the vertices to have co-ordinates identical to the spatial co-ordinates
of their counterparts in (4.1.1). Edge AB has faces ABC and ABD meeting at
it; the unit normal to ABC is
nˆaABC = (0, 0, 1),
and the unit normal to ABD is
nˆaABD =
(
0,−2
√
2
3
,
1
3
)
;
thus the dihedral angle between the two faces is given by
cos θ¯0 = nˆABC · nˆABD
=
1
3
.
We therefore see that θ¯0 = θ0 = arccos
(
1
3
)
, and hence our models do satisfy the
initial value equation at the moment of time-symmetry.
5.1.6 Discussion of the parent models
Before examining the behaviour of our global Regge models, we shall first pos-
tulate on the reasons for the local models’ failure. After we have obtained the
Regge equations from local variation, the standard approach in Regge calculus
would be to specify a set of initial data on a single Cauchy surface and then
use the Regge equations alone to determine the edge-lengths on all subsequent
147
5.1. THE PARENT MODELS
surfaces. Rather than doing just this, we have additionally constrained edges on
each subsequent surface to be identical. As a result, the only variable left for the
Regge equations to determine is the overall scaling of the edge-length on each
surface. Our extra constraints were motivated by the Copernican principle, as we
expected each Cauchy surface to be homogeneous and isotropic like the FLRW
Cauchy surfaces they are intended to approximate. Our constraints implicitly as-
sume that if we evolve from a Cauchy surface with identical edges, our subsequent
surfaces will continue having identical edges because of the Copernican principle;
we had assumed this would be the outcome even if we did not explicitly impose
the constraints, so our constraints were really expected to only simplify calcula-
tions. The Regge equations presented in (5.1.33) and (5.1.46) were obtained after
the constraints were imposed.
In the global model, all Cauchy surfaces did possess Copernican symmetries
as we could readily swap any pair of vertices, tetrahedral edges, or struts on a
surface without really changing the surface itself. However this was no longer
the case in the local model when we triangulated the Cauchy surfaces, as the
diagonals seemed to disrupt homogeneity. Vertices, for example, were no longer
identical and thus could not be swapped without non-trivially altering the surface
itself; in a 4-block, like the one depicted in Figure 4.1 and Figure 4.2, vertices
D and A′ would not be assigned any diagonals while A and D′ would each be
assigned three. Because we no longer have perfect homogeneity, our expectation
for edges to remain identical under evolution was no longer well founded. Rather,
we should allow the initial surface to evolve according to the unconstrained Regge
equations; without the additional constraints, we should expect to have a different
equation for each edge on the Cauchy surface.
There may perhaps be a third method of varying the Regge skeleton, lying
somewhere between a completely global variation and a completely local one, such
that each edge of the original CW skeleton could be varied individually without
having to break the symmetries inherent in the Cauchy surfaces. We shall refer
to this third approach as semi-local variation. In Chapter 3.3, we remarked
that Regge calculus with non-simplicial manifolds was possible but that the non-
simplicial blocks’ internal geometry had to first be specified; for instance, one
could constrain the internal diagonals to be functions of the external edges. The
Regge action would then be varied with the constraints on the internal geometry
in place. For the CW skeleton however, the constraints must be chosen in a way
148
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
that keeps all tetrahedral edges identical, otherwise the model may still not be
viable. We leave to future consideration whether such a choice of constraints is
possible.
We now turn to comparing the global Regge models against the continuum
model. We shall consider both the 3-sphere radii of the different models as well
as the total volume of the universe. In Figure 5.2, we have plotted the rate of
expansion of the universe’s volume against the volume itself for each of the models.
The volumes of Regge universes were given by (4.1.16), while the volume of the
FLRW universe was given by (1.0.5). We see that as the number of tetrahedra
increases, the model’s accuracy improves, with the 600 tetrahedra model matching
the FLRW model especially well when the universe is small.
For comparison, we have also plotted analogous graphs in Figure 5.3 where the
Regge universes’ volumes were taken to be volumes of 3-spheres of radius Rˆ(t), as
given by (5.1.37). In these graphs, the Regge models also very closely approximate
the FLRW universe at low volumes before diverging as the universe gets larger;
the approximation again improves as the number of tetrahedra increases.
In Figure 5.4 to Figure 5.9, we have plotted the expansion rate of the universe’s
radius against the radius itself, with each figure using a different measure of 3-
sphere radius for the Regge models. In all cases, the approximation to FLRW
again improves as the number of tetrahedra increases but gradually diverges as
the universe expands. The figures also reveal that radius Rˆ(t) gives the best
approximation to the FLRW model. This was somewhat expected given that
Rˆ(t) was deliberately defined so that it would match a(t) exactly at the point
of minimum expansion, the point which corresponds to the beginning of all the
graphs. However, the radius R2(t) to the centres of triangles also gives a very
good measure, as Figure 5.8 reveals; this clearly indicates that centres of the
triangles lie very close to 3-spheres of radius Rˆ, but it is not clear if there exists
any underlying reason for this.
All of our graphs terminated at an end-point on the right; this corresponds to
the moment when the time-like struts turn null. Brewin remarked on a similar
feature in his dust-filled FLRW models [64]: he obtained analogous graphs that
also terminated at an end-point when the struts turned null. However, Brewin
was considering a closed universe, and the end-points appeared while the universe
was contracting, whereas we are considering an open universe, and the end-points
appear while it is expanding. Brewin noted, in his case, that at the point when
149
5.1. THE PARENT MODELS
0
100
200
300
400
500
600
700
800
100 200 300 400 500 600
d
U d
t
U(t)
FLRW
5 tetrahedra
16 tetrahedra
600 tetrahedra
Figure 5.2: The rate of expansion of the universe’s volume dUdt versus the volume
U itself for the FLRW universe and the three different Regge models. The Regge
universe volume is given by the sum of the volumes of the Cauchy surface’s constituent
tetrahedra.
0
100
200
300
400
500
600
700
800
100 200 300 400 500 600
d
U d
t
U(t)
FLRW
5 tetrahedra
16 tetrahedra
600 tetrahedra
Figure 5.3: The rate of expansion of 3-sphere volumes dUdt versus the volume U itself
for the FLRW universe and the three different Regge models. We have used Rˆ(t) as
defined in (5.1.37) to be the Regge universes’ 3-sphere radii. As Rˆ(t) was defined for
all models to equal the FLRW scale factor a(t) when dRˆdt = a˙ = 0, then the volumes for
all Regge models should equal the volume for the FLRW universe when dUdt = 0; hence
all graphs above coincide at dUdt = 0.
150
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
0.0
0.5
1.0
1.5
2.0
2.5
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
d
(R
a
d
iu
s)
d
t
Radius(t)
FLRW
5 tetrahedra
16 tetrahedra
600 tetrahedra
Figure 5.4: The expansion rate of the universe’s radius versus the radius itself for the
FLRW model and the three different Regge models, where we are using radius Rˆ(t), as
given by (5.1.37), for the Regge models.
0.0
0.5
1.0
1.5
2.0
2.5
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
d
(R
a
d
iu
s)
d
t
Radius(t)
FLRW
5 tetrahedra
16 tetrahedra
600 tetrahedra
Figure 5.5: The expansion rate of the universe’s radius versus the radius itself for the
FLRW model and the three different Regge models, where we are using the average
radius R¯(t) as given by (5.1.13) for the Regge models.
151
5.1. THE PARENT MODELS
0.0
0.5
1.0
1.5
2.0
2.5
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
d
(R
a
d
iu
s)
d
t
Radius(t)
FLRW
5 tetrahedra
16 tetrahedra
600 tetrahedra
Figure 5.6: The expansion rate of the universe’s radius versus the radius itself, using
the radius R(t) to vertices, as given by (5.1.6), for the Regge models.
0.0
0.5
1.0
1.5
2.0
2.5
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
d
(R
a
d
iu
s)
d
t
Radius(t)
FLRW
5 tetrahedra
16 tetrahedra
600 tetrahedra
Figure 5.7: The expansion rate of the universe’s radius versus the radius itself, using
the radius R1(t) to the centres of edges, as given by (5.1.7), for the Regge models.
152
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
0.0
0.5
1.0
1.5
2.0
2.5
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
d
(R
a
d
iu
s)
d
t
Radius(t)
FLRW
5 tetrahedra
16 tetrahedra
600 tetrahedra
Figure 5.8: The expansion rate of the universe’s radius versus the radius itself, using
the radius R2(t) to the centres of triangles, as given by (5.1.9), for the Regge models.
0.0
0.5
1.0
1.5
2.0
2.5
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
d
(R
a
d
iu
s)
d
t
Radius(t)
FLRW
5 tetrahedra
16 tetrahedra
600 tetrahedra
Figure 5.9: The expansion rate of the universe’s radius versus the radius itself, using
the radius R3(t) to tetrahedral centres, as given by (5.1.11), for the Regge models.
153
5.1. THE PARENT MODELS
mt
t+ dt
mt
Σt
Σt+dt
t
Figure 5.10: A schematic diagram, projected onto a 1+1 plane, of a 4-block with
space-like struts. The dashed lines correspond to null curves that originate from the
4-block vertices in Σt. These lines separate out the region of the 4-block interior that
can be reached by causal curves from Σt, the white region, from those that cannot, the
shaded region. Because of this causal structure, no past-directed causal curve from any
vertex in Σt+dt will intercept Σt, and therefore, Σt+dt lies outside the future domain
of dependence of Σt. For Brewin’s case, the above figure should be vertically inverted,
since for a contracting universe, Σt+dt would be smaller than Σt.
the struts turn null, the surface Σt would no longer lie in the past domain of
dependence of the surface Σt+dt; there would be points in Σt+dt which cannot be
reached by any causal curves from any point in Σt. Since our Cauchy surfaces
are expanding instead, it is the reverse that happens; that is, Σt+dt no longer lies
in the future domain of dependence of Σt, as illustrated by Figure 5.10. Brewin
suspected, in his case, that the end-point signalled the local curvature had become
too large for the approximation to handle. Nevertheless, in both Brewin’s models
and ours, increasing the number of tetrahedra does postpone the appearance of
the end-point. In his case, he was able to reach smaller volumes with a larger
number of tetrahedra, while we are able to reach larger volumes, as Figure 5.3
shows.
To investigate whether there is any relationship between the struts’ turning
null and the ratio of the Hubble radius 1/H0 to the tetrahedral edge-length l(t), we
have plotted this ratio against l(t) in Figure 5.11. The Regge Hubble parameter
H0 is defined to be l˙(t)/l(t).
1 The figure shows that the Hubble radius always
1In the FLRW universe, Hubble parameter H0 is defined to be a˙/a. We can define the
154
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
0
1
2
3
4
5
1 2 3 4 5 6 7 8 9 10
H
−1 0
l(
t)
−1
l(t)
5 tetrahedra
16 tetrahedra
600 tetrahedra
Figure 5.11: The ratio of the Hubble radius 1/H0 to the tetrahedral edge-length l(t)
versus the edge-length for each of the three Regge models; the graphs terminate when
the struts turn null. The Regge Hubble parameter H0 is given by l˙(t)/l(t).
falls below the edge-length before the struts turn null; thus there does not appear
to be any relationship between the Hubble radius and the edge-length at the
moment the struts turn null.
In all of our graphs, we noticed that the models diverge increasingly from
FLRW as the universe gets larger. We believe this is due to the finite resolution
of our models trying to approximate an ever-expanding universe. Regardless of
the universe’s size, the number of tetrahedra in any given approximation is always
kept fixed; therefore the resolution will degrade as the universe gets larger. This is
consistent with our observation that the graph diverges more slowly from FLRW
as the number of tetrahedra is increased, as the resolution of the models with a
higher number of tetrahedra are able to ‘keep up’ longer with FLRW.
Regge parameter analogously, using 3-sphere embedding radii rather than the FLRW scale
factor a(t); however, regardless of which embedding radius we use, the quantity will always
reduce to l˙(t)/l(t).
155
5.2. THE CHILDREN MODELS
5.2 The children models
In this section, we shall model the Λ-FLRW universe using children skeletons
obtained by subdividing the parent skeletons according to Brewin’s scheme [64].
Since all Cauchy surfaces in continuum FLRW space–time are identical to each
other apart from a time-dependent scale factor a(t), we shall analogously require
our subdivided Cauchy surfaces to be identical to each other as well apart from
an overall time-dependent scale factor. However, Brewin’s subdivision scheme
will generate three distinct sets of edges in any Cauchy surface, as discussed in
Chapter 4.4, with the lengths denoted by vi, ui, and pi. Therefore our requirement
that the subdivided Cauchy surfaces be identical implies that
ui
vi
= α ∀i
pi
vi
= β ∀i,
(5.2.1)
where α and β are constants independent of the Cauchy surface. This require-
ment is natural because our CW Cauchy surfaces are intended to approximate
FLRW Cauchy surfaces; so if the only difference between two FLRW surfaces
is an overall scaling a0 → λ a0, then the only difference between the two CW
surfaces approximating them should be a re-scaling of all lengths by λ as well.
This is assuming that the two CW surfaces triangulate their respective FLRW
surfaces in the same way; that is, no extra vertices, edges, or tetrahedra appear
in one Regge surface but not the other.
If we now apply (5.2.1) to relation (4.4.5) for the Lorentz boost parameter ψi,
it follows that ψi has to be zero. That is, for the 4-blocks to have no twist or
shear in the child model, the upper tetrahedron cannot be Lorentz boosted with
respect to the lower tetrahedron.
5.2.1 The 3-sphere embedding of children Cauchy sur-
faces
The subdivided Cauchy surface will have a slightly different embedding from that
of the parent Cauchy surfaces. The main difference is that each of the three sets
of vertices can lie on its own 3-sphere, as the three sets are independent of each
other. Nonetheless, the three 3-spheres will share a common centre in E4. As the
156
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
Cauchy surface expands or contracts, the radii of the three 3-spheres will increase
or decrease correspondingly. Each 3-sphere can be parametrised using polar co-
ordinates as given by (5.1.1) though with different radii. We shall denote the
three radii by R
(1)
i for the parent vertices, R
(2)
i for the mid-points, and R
(3)
i for
the central vertices. As mentioned at the start of this chapter, our approach to
this embedding will be completely different from that of Brewin [64], who instead
constrained all three sets of vertices to lie on the same 3-sphere.
We shall first embed a representative subset of the mid-point vertices. As with
the parent model, we can always choose our co-ordinates such that one of these
vertices lies at (0, 0, 0). This vertex will have n+n nearest mid-point neighbours
to it, where n is given by the final column of Table 4.1. To help understand the
positions of these nearest neighbours, we shall refer to Figure 4.3. Suppose our
vertex at (0, 0, 0) corresponds to the mid-point vertex (AB) in Figure 4.3; then on
each parent triangle containing this vertex, there will be two nearest mid-point
neighbours, for instance (AD) and (BD) for triangle ABD in the figure. Since n
parent triangles share a parent edge, there will be n parent triangles containing
vertex (AB); thus we shall have a total of n nearest neighbours from (AD) and
its analogues and another n nearest neighbours from (BD) and its analogues.
This gives a combined set of n+ n nearest mid-point neighbours. We shall refer
to one of the two sets as the ‘upper’ set and the other as the ‘lower’ set. As these
neighbours are all located at a common distance of vi from the first vertex, they
are situated on a common 2-sphere centred on this vertex. Hence, we can choose
the χ co-ordinate of these neighbours to be determined by just the radius of this
2-sphere. We denote this co-ordinate by χ = χ0.
The upper and lower sets each form a spoke around the first vertex. An
example of such spokes is illustrated in Figure 5.12 for the case of n = 4. We
shall label the upper vertices by 1 . . . n and the lower ones by 1′ . . . n′ such that
if m ∈ [1, n] is an (AD)-type vertex then m′ ∈ [1′, n′] would be the (BD)-type
counterpart located on the same parent triangle. With this choice of labelling,
we note that each m and m′ pair are nearest neighbours as well. We can choose
the θ co-ordinate to be such that the upper vertices are all located at θ = θ0.
Then if we choose vertex 1 to be at φ = 0, upper vertex m will be located
at φ = 2(m − 1)pi/n. Having fixed the upper vertices, the lower vertices are
constrained to be at θ = pi − θ0 with vertex m′ having the same φ co-ordinate as
its upper counterpart m. These results are summarised in Table 5.2.
157
5.2. THE CHILDREN MODELS
θ0
vi
vi
Parent vertex
1′4′
vi
pi pi
vi
3′
θ0
Parent vertex
pi
ui
pi
vi
ui
vi
0
ui
ui
vi
2
4
2′
Central vertex
3
Central vertex
1
Figure 5.12: The upper and lower spokes of mid-point vertices, shown in solid lines,
surrounding mid-point vertex 0 for the case of n = 4. One dimension has been sup-
pressed. All vertices in the two spokes are situated on a sphere of radius vi. Each m,
m′ pair of vertices are also nearest neighbours to each other, being located on the same
parent triangle; they are hence separated by vi. Each vertex is also separated from
its two neighbours in the same spoke by a distance of vi. Also depicted are the two
nearest parent vertices and two of the nearest central vertices. The upper parent vertex
is equidistant to each vertex of the upper spoke and to vertex 0, while the lower parent
vertex is equidistant to each vertex of the lower spoke and to vertex 0. The left central
vertex is equidistant to vertices 0, 3, 4, 3′, 4′, and the right central vertex to 0, 1, 2, 1′, 2′.
158
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
Vertex χ θ φ
0 0 0 0
1 χ0 θ0 0
2 χ0 θ0
2pi
n
...
...
...
...
n χ0 θ0
2(n−1)pi
n
1′ χ0 pi − θ0 0
2′ χ0 pi − θ0 2pin
...
...
...
...
n′ χ0 pi − θ0 2(n−1)pin
Table 5.2: The polar co-ordinates of a mid-point vertex and its 2n nearest mid-point
neighbours.
The parameters χ0, θ0, R
(2)
i can be related to the edge-length vi. As mentioned
above, the χ0 co-ordinate is fixed by the distance between vertex 0 and its nearest
neighbours; this yields the relation
v2i = 2
(
R
(2)
i
)2
(1− cosχ0). (5.2.2)
From the distance between vertices 1 and 2, we have the relation
v2i = 2
(
R
(2)
i
)2
sin2 χ0 sin
2 θ0
(
1− cos 2pi
n
)
, (5.2.3)
and from the distance between 1 and 1′, we have the relation
v2i = 4
(
R
(2)
i
)2
sin2 θ0 cos
2 χ0. (5.2.4)
Solving these equations yields
cosχ0 =
1 + cos 2pi
n
2 (1− cos 2pi
n
)
, (5.2.5)
cos θ0 =
[
1− cos 2pi
n
3− cos 2pi
n
] 1
2
, (5.2.6)
Z(2) :=
vi
R
(2)
i
=
[
1− 3 cos 2pi
n
1− cos 2pi
n
] 1
2
. (5.2.7)
159
5.2. THE CHILDREN MODELS
We see that once again, the ratio between the edge-length vi and the 3-sphere
radius R
(2)
i is independent of the Cauchy surface label i and hence of time. We
see as well that the angular co-ordinates of the vertices are also independent of
i, and therefore, as the edge-length vi expands or contracts, the 3-sphere simply
expands and contracts about its centre.
We now consider the embedding of the nearest parent vertices. There will be
two nearest parent neighbours to vertex 0. One of them will also be a nearest
neighbour of all n upper vertices and the other of all n lower vertices, as illustrated
in Figure 5.12. By symmetry, the upper parent vertex will be located on the 2-
dimensional plane containing vertex 0 but orthogonal to the plane containing
the upper n vertices. We can obtain one vector p(1) in this plane by taking the
average of the n vertices and subtracting the position of vertex 0. This gives the
vector (
p(1)
)µ
=
(−Z(2), 1, 0, 0) . (5.2.8)
A second basis q(1) for the plane can be obtained by requiring orthogonality to
both p(1) and any vector joining any two of the upper n vertices. This gives
(
q(1)
)µ
=
(
1, Z(2), 0, 0
)
. (5.2.9)
In terms of these two vectors, the location r(1) of the parent vertex is then
(
r(1)
)µ
= R
(2)
i (1, 0, 0, 0) + λ
(1)
(
p(1)
)µ
+ µ(1)
(
q(1)
)µ
,
since both vectors p(1) and q(1) are given with respect to the position of vertex
0. We also require r(1) be equidistant to vertex 0 and all upper n vertices; this
constraint fixes λ(1) to be
λ(1) =
vi
1 + (Z(2))2
, (5.2.10)
while µ(1) remains a free parameter.
It can be shown that q(1) is actually parallel to the radial vector connecting
the parent vertex to the centre of the 3-sphere, that is,
r(1) = R
(1)
i qˆ
(1), (5.2.11)
where qˆ(1) is simply q(1) normalised. We can therefore express r(1) more simply
160
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
as (
r(1)
)µ
=
R
(1)
i√
1 + (Z(2))2
(1, Z(2), 0, 0), (5.2.12)
and radius R
(1)
i is related to parameter µ
(1) by
(
R
(1)
i
)2
=
(
Z(2) µ
(1)
λ(1)
+ 1
)2
1 + (Z(2))2
(
R
(2)
i
)2
. (5.2.13)
Since the free parameter µ(1) only changes r(1) along the direction of q(1), chang-
ing µ(1) will only change the radius of the 3-sphere.
We can deduce the embedding of the nearest central vertices in a similar
manner. As there are n parent tetrahedra hinging on a parent edge, there will be
a total of n central vertices that are nearest neighbours to vertex 0. Each of these
n vertices will be located along the central axis to each vertex quadruplet of the
form m, m′, (m+ 1), (m+ 1)′ for all m ∈ [1, n], with vertices (n+ 1) and (n+ 1)′
being identified with vertices 1 and 1′. Two examples of central vertices are shown
in Figure 5.12. We shall focus on the central vertex equidistant to vertices 0, 1,
1′, 2, 2′. Similar to the situation with the parent vertex, the central vertex will
be located on the 2-dimensional plane containing vertex 0 but orthogonal to the
plane containing vertices 1, 1′, 2, 2′. A vector p(2) in this plane can be found
by taking the average of vertices 1, 1′, 2, 2′ and then subtracting the position of
vertex 0; this gives
(
p(2)
)µ
=
−Z(2), 0, 2− (Z(2))2√
3− (Z(2))2
,
[
2− (Z(2))2
3− (Z(2))2
] 1
2
 . (5.2.14)
The second basis q(2) can be obtained by requiring orthogonality with p(2) and
with any vector connecting any of vertices 1, 1′, 2, 2′; this yields
(
q(2)
)µ
=
√2− (Z(2))2, 0, Z(2) [2− (Z(2))2
3− (Z(2))2
] 1
2
,
Z(2)√
3− (Z(2))2
 . (5.2.15)
Then in terms of these two vectors, the position r(2) of the vertex is
(
r(2)
)µ
= R
(2)
i (1, 0, 0, 0) + λ
(2)
(
p(2)
)µ
+ µ(2)
(
q(2)
)µ
.
161
5.2. THE CHILDREN MODELS
Requiring r(2) to be equidistant to vertices 0, 1, 1′, 2, 2′ yields
λ(2) =
vi
2
, (5.2.16)
while µ(2) remains a free parameter as well.
As with the parent vertex, it can be shown that q(2) is actually parallel to
the radial vector pointing from the centre of the 3-sphere to the central vertex,
that is,
r(3) = R
(3)
i qˆ
(2), (5.2.17)
where qˆ(2) is simply q(2) normalised. Therefore, we can also express r(2) as
(
r(2)
)µ
=
R
(3)
i√
2
√2− (Z(2))2, 0, Z(2) [2− (Z(2))2
3− (Z(2))2
] 1
2
,
Z(2)√
3− (Z(2))2
 ,
(5.2.18)
where R
(3)
i is related to µ
(2) by
(
R
(3)
i
)2
=
1
2
(
R
(2)
i
)2(vi
2
Z(2)µ(2) +
√
2− (Z(2))2
)2
. (5.2.19)
Since the free parameter µ(2) only changes r(2) along the direction of q(2), chang-
ing µ(2) will only change the radius of the 3-sphere.
We have previously related vi to R
(2)
i in (5.2.7). We can also relate edge-
lengths ui and pi to the 3-sphere radii. The relationship between ui and R
(1)
i
is
u2i =
(
R
(1)
i
)2
− 2R
(1)
i R
(2)
i√
1 + (Z(2))2
+
(
R
(2)
i
)2
, (5.2.20)
while the relationship between pi and R
(3)
i is
p2i =
(
R
(3)
i
)2
− 2R(3)i R(2)i
√
2− (Z(2))2 +
(
R
(2)
i
)2
. (5.2.21)
Since µ(1) and µ(2) are free parameters that determine the 3-sphere radii R
(1)
i
and R
(3)
i respectively, we can re-express (5.2.13) and (5.2.19) in the form
R
(1)
i = α¯ R
(2)
i ,
R
(3)
i = β¯ R
(2)
i ,
(5.2.22)
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5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
where the scaling factors α¯ > 0 and β¯ > 0 now become the free parameters. This
effectively means that R
(1)
i and R
(3)
i can be freely chosen for some initial Cauchy
surface, and from (5.2.20) and (5.2.21), this choice would effectively determine ui
and pi. Therefore, the freedom to choose α¯ and β¯ is equivalent to a freedom to
choose α and β in (5.2.1) for some initial Cauchy surface.
By altering α¯ or β¯, we would expand or contract one of the 3-spheres relative
to the others. The vertices on the altered 3-sphere would simply shift radially
inwards or outwards, but not angularly. Because the shift is purely radial, a
vertex on this 3-sphere would still remain equidistant to its nearest neighbours
on the same 3-sphere, although that distance would change. Similarly, the vertex
would remain equidistant to its nearest neighbours on the R
(2)
i 3-sphere but with
the distance altered as well.
As mentioned at the start of this section, all FLRW 3-spheres are identical to
each other apart from an overall scaling a(t), and we approximate this symmetry
by requiring ratios α and β as defined in (5.2.1) to be constant; then the evolution
of two sets of the Cauchy surface edge-lengths can be determined by the third
set alone, which we shall take to be vi. Since α and β are equivalent to α¯ and
β¯ respectively, we can equivalently require that all of our Regge 3-spheres be
identical to each other apart from an overall scaling; we would freely specify α¯
and β¯ for some initial Cauchy surface, but our requirement would constrain α¯
and β¯ to be the same for all subsequent surfaces. As a result, the evolution of
two of our 3-sphere radii can be determined by the third radius alone, and we
shall choose R
(2)
i to be that sole dynamical radius. The analogy between CW
Cauchy surfaces and FLRW Cauchy surfaces is much clearer when working with
these embedding 3-spheres and 3-sphere radii rather than with tetrahedral edge-
lengths. However regardless of whether we work with radii or edge-lengths, there
is only one dynamical length parametrising the entire system.
5.2.2 Geometric quantities for the children model
To compute the geometric quantities for the children model, we shall make use
of the 4-block co-ordinates given by (4.4.1) and (4.4.3). However, as we noted
at the start of this section, the scaling relations (5.2.1) requires that the boost
parameter ψi be zero; therefore, the upper tetrahedron’s co-ordinates (4.4.3) can
163
5.2. THE CHILDREN MODELS
be simplified to
A′ =
(
−vi+1
2
,− vi+1
2
√
3
, δzi, ιti+1
)
,
B′ =
(
vi+1
2
,− vi+1
2
√
3
, δzi, ιti+1
)
,
C ′ =
(
0,
vi+1√
3
, δzi, ιti+1
)
,
D′ = (0, 0, hi+1 + δzi, ιti+1).
(5.2.23)
The parameter δzi can be deduced from the requirement that all struts have
equal length. Equating the lengths of struts AA′ and DD′ gives the equation
1
3
δv2i + δz
2
i =
[(√
α2 − 1
3
)
δvi + δzi
]2
,
where 4-block co-ordinates (4.4.1) and (5.2.23) have been used, and where once
again, δvi denotes δvi := vi+1 − vi. By solving this, we obtain
δzi =
1
2
√
α2 − 1
3
(
2
3
− α2
)
δvi. (5.2.24)
Requiring all struts to have equal length also places certain constraints on the
relationship between α and β. Using co-ordinates (4.4.1) and (5.2.23), we can
express the length of a parent vertex’s strut as
m2i =
α4
4
(
α2 − 1
3
) δv2i − δt2i , (5.2.25)
where again, δti denotes δti := ti+1− ti. By simply swapping α for β in the above
expression, we obtain an equivalent expression for the length of a central vertex’s
strut, that is,
m2i =
β4
4
(
β2 − 1
3
) δv2i − δt2i .
As we require all strut-lengths to be equal between any pair of consecutive ver-
tices, it follows that
α4(
α2 − 1
3
) = β4(
β2 − 1
3
) , (5.2.26)
164
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
which implies that
β = α or β =
α√
3
(
α2 − 1
3
) . (5.2.27)
This relationship between α and β implies that one of the three edge-lengths, and
hence one of the three 3-sphere radii, can no longer be independent of the other
two edge-lengths and radii.
We shall now derive the geometric quantities relevant to the Regge equations.
However, we shall only present those quantities relevant to the Hamiltonian con-
straint equation, that is, to varying the action with respect to the strut-lengths,
as our plan is to use the Hamiltonian constraint to study the behaviour of the
children models, much like how we studied the behaviour of the parent models.
Moreover, we shall only present the local variation of those quantities, since global
variation can always be related to local variation by the chain rule, as discussed
in Chapter 4.3. As with the parent models, the only geometric quantities relevant
to the children models’ constraint equations are the varied time-like hinges gen-
erated by the tetrahedral edges’ world-sheets, the corresponding deficit angles,
and the 4-blocks’ varied volumes.
We begin with the variation of the time-like hinges. As with the parent models,
the time-like hinges can again be triangulated in the same manner as depicted
in Figure 4.2; and we can again use equations (4.2.2) to (4.2.5) to obtain the
variation of the resulting triangular hinges with respect to their strut-lengths mi.
For the hinges generated by an edge of length ui, we find that
∂AA1i
∂mAi
=
∂AB1i
∂mBi
=
αmi
2δti
vi+1 + vi√
1
3
α2
α2− 1
3
v˙2i − 4
, (5.2.28)
where v˙i denotes v˙i :=
δvi
δti
. For the hinges generated by an edge of length pi, we
can swap α for β to obtain
∂AA3i
∂mAi
=
∂AB3i
∂mBi
=
β mi
2δti
vi+1 + vi√
1
3
β2
β2− 1
3
v˙2i − 4
. (5.2.29)
165
5.2. THE CHILDREN MODELS
Finally for the hinges generated by vi, we find that
∂AA2i
∂mAi
=
∂AB2i
∂mBi
=
mi (vi+1 + vi)
4
√
m2i − 14 δv2i
. (5.2.30)
As we shall ultimately take the continuum time limit of the Regge equations,
we shall at this stage present the continuum time limit of the quantities above
for later use. As δti → 0, equation (5.2.28) becomes
∂AA1i
∂mAi
=
∂AB1i
∂mBi
→ αv
 14 α4α2− 13 v˙2 − 1
1
3
α2
α2− 1
3
v˙2 − 4
 12 ; (5.2.31)
equation (5.2.29) becomes
∂AA3i
∂mAi
=
∂AB3i
∂mBi
→ βv
 14 β4β2− 13 v˙2 − 1
1
3
β2
β2− 1
3
v˙2 − 4
 12 ; (5.2.32)
and equation (5.2.30) becomes
∂AA2i
∂mAi
=
∂AB2i
∂mBi
→ v
 14 α4α2− 13 v˙2 − 1(
α4
α2− 1
3
− 1
)
v˙2 − 4
 12 . (5.2.33)
We next turn to determining the deficit angles. In the 4-block described by
(4.4.1) and (5.2.23), there will be two different dihedral angles, one at hinges
generated by the edges of length vi and the other at hinges generated by the
edges of length ui.
Let us first consider the dihedral angle at trapezoidal hinge ABA′B′, which is
generated by a length-vi edge. Because this hinge is co-planar, the two triangular
hinges that subdivide it will have the same dihedral angle as that of the original
hinge. The two faces in the 4-block meeting at ABA′B′ are ABCA′B′C ′ and
ABDA′B′D′, and their unit normals are respectively
nˆµ1 =
[
1−
(
α2 − 2
3
)2
4
(
α2 − 1
3
) v˙2i
]− 1
2
0, 0, 1,−ι α2 − 23
2
√
α2 − 1
3
v˙i
 (5.2.34)
166
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
and
nˆµ2 =
[
3
(
4α2 − 1)− α4 v˙2i
4
(
α2 − 1
3
)]− 12
0,−2√3(α2 − 1
3
)
, 1, ι
α2v˙i
2
√
α2 − 1
3
 .
(5.2.35)
Therefore the dihedral angle between the two faces is given by
cos θ
(1)
i = nˆ1 · nˆ2
=
1 +
α2− 2
3
4(α2− 13)
α2v˙2i[
1− (α2−
2
3)
2
4(α2− 13)
v˙2i
] 1
2
[
3 (4α2 − 1)− α4 v˙2i
4(α2− 13)
] 1
2
.
(5.2.36)
Let us next consider the dihedral angle at hinge ADA′D′, which is generated
by length-ui edges. This hinge is also co-planar, and so the two triangular hinges
that subdivide it will also have the same dihedral angle. The faces meeting at
this hinge are ABDA′B′D′ and ACDA′C ′D′. Face ABDA′B′D′ repeats from
before and so has unit normal nˆ2, while face ACDA
′C ′D′ has unit normal
nˆµ3 =
 3 (α2 − 13)
3 (4α2 − 1)− α4 v˙2i
4(α2− 13)

1
2 −√3, 1, 1√
3(α2 − 1
3
)
, ι
α2v˙i
2
√
3
(
α2 − 1
3
)
 .
(5.2.37)
Thus the dihedral angle is given by
cos θ
(2)
i = nˆ2 · nˆ3
=
3 (2α2 − 1) + α4 v˙2i
4(α2− 13)
3 (4α2 − 1)− α4 v˙2i
4(α2− 13)
.
(5.2.38)
There will be another pair of dihedral angles, analogous to the two above, in
the 4-block with edges of length pi instead of ui. These angles can immediately
be obtained by swapping α for β in the two expressions above, thus giving
cos θ
(3)
i =
1 +
β2− 2
3
4(β2− 13)
β2v˙2i[
1− (β2−
2
3)
2
4(β2− 13)
v˙2i
] 1
2
[
3 (4β2 − 1)− β4 v˙2i
4(β2− 13)
] 1
2
(5.2.39)
167
5.2. THE CHILDREN MODELS
and
cos θ
(4)
i =
3 (2β2 − 1) + β4 v˙2i
4(β2− 13)
3 (4β2 − 1)− β4 v˙2i
4(β2− 13)
. (5.2.40)
Therefore the child model has a total of four distinct dihedral angles.
With these dihedral angles, we can now deduce the deficit angles at each hinge.
We begin with hinges generated by length-ui edges. There will be n space-like
triangular faces meeting at such an edge, with n again given by the last column
of Table 4.1, and hence there will be n faces meeting at the hinge generated by
that edge. Between each adjacent pair of faces is a dihedral angle of θ
(2)
i , and
hence the hinge’s deficit angle is
δ
(1)
i = 2pi − nθ(2)i . (5.2.41)
We next consider the deficit angle at hinges generated by length-vi edges. As
discussed in Chapter 4.4, the children models will be comprised of three types
of tetrahedra, which we have called Type I, Type II, and Type III tetrahedra.
Length-vi edges are attached to all three types, and each type contributes a
different dihedral angle to the corresponding hinges’ overall deficit angle; each
Type I tetrahedron contributes θ
(1)
i , while each Type II and Type III tetrahedron
contributes θ
(3)
i . The number of Type I tetrahedra at a length-vi edge is given by
NType I =
(No. of Type I)(3 vi-edges per Type I)
(No. of vi-edges)
=
(4 Type I per parent)(No. of parents)(3 vi-edges per Type I)
(3 vi-edges per parent triangle)(No. of parent triangles)
= 2,
where ‘parent’ refers to a parent tetrahedron, and where we have used the fact
that, regardless of the model,
(No. of parent triangles)
(No. of parents)
= 2; (5.2.42)
this last identity follows from the fact that there are always four triangles per
tetrahedron, but every triangle is always shared by two tetrahedra, so there are
only half as many triangles per tetrahedron in a skeleton. By similar combina-
168
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
torics, we can show that the number of Type II tetrahedra is given by
NType II =
(No. of Type II)(3 vi-edges per Type II)
(No. of vi-edges)
= 2,
and the number of Type III by
NType III =
(No. of Type III)(3 vi-edges per Type III)
(No. of vi-edges)
= 2.
Therefore the deficit angle is
δ
(2)
i = 2pi − 2θ(1)i − 4θ(3)i . (5.2.43)
Finally, we consider the deficit angle at hinges generated by length-pi edges.
Only Type II and III tetrahedra have such edges, and each will contribute a
dihedral angle of θ
(4)
i . The number of Type II and III tetrahedra at an edge is
NType II & III =
(No. of Type II & III per parent)(3 pi-edges per Type II or III)
(No. of pi-edges per parent)
= 4,
and therefore the deficit angle is
δ
(3)
i = 2pi − 4θ(4)i . (5.2.44)
The final geometric quantities to derive are the 4-simplices’ varied volumes. A
4-block generated by a tetrahedron can again be triangulated in the same manner
as shown in Figure 4.2, and the volumes of the four 4-simplices generated can
again be calculated in the same manner as with the parent models, using equa-
tions (5.1.40) to (5.1.43). Again, we vary each of these volumes with respect to
their associated strut-length. The resulting expressions also simplify greatly after
the continuum time limit is taken, so we therefore present only the continuum
time expressions again. For the 4-block generated by a Type I tetrahedron, the
169
5.2. THE CHILDREN MODELS
derivatives of all four 4-simplices’ volumes simplify to
∂V
(4)
I
∂mi
→ v
3
16
√
3
√√√√(α2 − 1
3
)(
1− α
4
4
(
α2 − 1
3
) v˙2)+O(dt) . (5.2.45)
The 4-block generated by Type II and Type III tetrahedra are identical, and the
derivatives of all 4-simplices’ volumes simplify to
∂V
(4)
II & III
∂mi
→ v
3
16
√
3
√√√√(β2 − 1
3
)(
1− β
4
4
(
β2 − 1
3
) v˙2)+O(dt) . (5.2.46)
5.2.3 Varying the Regge action
We shall now derive the Regge equations for the children models. We initially
attempted to locally vary the Regge action with respect to the strut lengths,
however this resulted in a mutually inconsistent set of equations. Because each
child model has three distinct sets of struts, local variation would yield a total of
three distinct constraint equations, one for each set. Any of the constraint equa-
tions could then be used to determine the behaviour of our dynamical variable vi,
but with three constraint equations determining a single vi, the system of equa-
tions risked being over-determined. This indeed happened: we obtained three
constraint equations that gave mutually inconsistent relations for vi regardless of
which choice we made for β in (5.2.27). Therefore, we can perform only a global
variation of the action, and this is what we shall now present.
The Regge action for a child model can be expressed as
S = 1
8pi
∑
i
[
NArea 1
(
AA1i + A
B1
i
)
δ
(1)
i +N
Area 2
(
AA2i + A
B2
i
)
δ
(2)
i
+NArea 3
(
AA3i + A
B3
i
)
δ
(3)
i
− 4N3 Λ
(
V
(4)
i, I + 2V
(4)
i, II & III
)]
,
(5.2.47)
where the summation i is over the Cauchy surfaces, N3 denotes the number of
parent tetrahedra, and V
(4)
i,X is the volume of the entire 4-block of a Type X
tetrahedron, that is the combined volume of all four constituent 4-simplices of
the 4-block. Each coefficient NArea X, where X = 1, 2, 3, denotes the number of
170
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
AX, BX triangular hinge pairs in a Cauchy surface and would equal the number
of tetrahedral edges that generate the pairs. So NArea 1 is given by twice the
number of parent edges N1, that is,
NArea 1 = 2N1. (5.2.48)
As there are three vi-edges per parent face, then N
Area 2 is
NArea 2 = 3N2 = 6N3, (5.2.49)
where N2 is the number of parent faces, and the second equality follows from
(5.2.42). Finally, NArea 3 is determined by the fact that each parent tetrahe-
dron has six pi-edges, none of which is shared with any other parent tetrahedra.
Therefore,
NArea 3 = 6N3. (5.2.50)
We can use the chain rule to express any globally varied quantity as a sum of
locally varied quantities. For a common strut-length mi, the chain rule takes the
form
∂
∂mi
=
∂m
(1)
j
∂mi
∂
∂m
(1)
j
+
∂m
(2)
j
∂mi
∂
∂m
(2)
j
+
∂m
(3)
j
∂mi
∂
∂m
(3)
j
+
∂d
(1)
j
∂mi
∂
∂d
(1)
j
+
∂d
(2)
j
∂mi
∂
∂d
(2)
j
+
∂d
(3)
j
∂mi
∂
∂d
(3)
j
.
Since all struts have equal length, then
∂m
(k)
j
∂mi
= δij ∀k.
We also have that
∂d
(k)
j
∂mi
=
mi
d
(k)
i
δij = O(dt) ∀k.
However we found that the leading order of the Regge equation 0 = ∂S
∂m
(k)
i
was
O(1) for all k, so once again, the diagonal derivatives do not contribute. Hence
171
5.2. THE CHILDREN MODELS
to leading order, the chain rule can be simplified to the form
∂
∂mi
=
∂
∂m
(1)
i
+
∂
∂m
(2)
i
+
∂
∂m
(3)
i
.
We have used the chain rule to globally vary the Regge action with respect
to mi to obtain the Hamiltonian constraint for the children models; and we have
then taken the continuum time limit. The resulting equation is
v2 =
4
√
3
Λ
α2
β2
[√
α2 − 1
3
(
α2
β2
+ 2
)]−1 α N1N3 2pi − nθ
(2)√
4− 1
3
α2
α2− 1
3
v˙2
+ 3
2pi − 2θ(1) − 4θ(3)√
4−
(
α4
α2− 1
3
− 1
)
v˙2
+ 3 β
2pi − 4θ(4)√
4− 1
3
α2
β2
α2
α2− 1
3
v˙2
 ,
(5.2.51)
where relation (5.2.26) has been used to simplify the expression. From (5.2.38),
we can parametrise v˙ in terms of cos θ(2) by the expression
v˙2 =
12
(
α2 − 1
3
) [
(4α2 − 1) cos θ(2) − (2α2 − 1)]
α4 (1 + cos θ(2))
. (5.2.52)
As with the parent model, the range of θ(2) is bounded from above by the
requirement that v˙2 ≥ 0 and from below by the requirement that the strut-
lengths be time-like, that is, (mi)
2 < 0; this leads to the range of
pi
3
< θ(2) ≤ arccos
(
2α2 − 1
4α2 − 1
)
. (5.2.53)
We have also varied the action with respect to vi to obtain the evolution
equation for the children models. The calculation, which we do not show, is
similar to that of the parent models. In the continuum time limit, we obtained
0 =
N1
N3
1[
4− 1
3
α2v˙2
α2− 1
3
] 1
2
 3nα6vv˙2v¨
2
√
α2 − 1
3
[
3(4α2 − 1)− α4v˙2
4(α2− 13)
] [
3− α2v˙2
4(α2− 13)
] 1
2
172
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
+
16α
(
2pi − nθ(2))
4− 1
3
α2v˙2
α2− 1
3
[
α4
4
vv¨ − α
6v˙4
48
(
α2 − 1
3
)2 + α4v˙22 (α2 − 1
3
) − α2v˙2
4
− 1
]
+
1[
4−
(
α4
α2− 1
3
− 1
)
v˙2
] 1
2
3√3vv˙2v¨
×
 4α2
(
α2 − 2
3
)
+ 2
(
α2 − 1
3
) (
α4
α2− 1
3
− 1
)
− 1
2
α2
(
α2 − 2
3
) (
α4
α2− 1
3
− 1
)
v˙2√
α2 − 1
3
[
4−
(
α4
α2− 1
3
− 1
)
v˙2
] 1
2
[
1− (α2−
2
3)
2
4(α2− 13)
v˙2
] [
3 (4α2 − 1)− α4v˙2
4(α2− 13)
]
+
8β2
(
β2 − 2
3
)
+ 4
(
β2 − 1
3
) (
β4
β2− 1
3
− 1
)
− β2 (β2 − 2
3
) (
β4
β2− 1
3
− 1
)
v˙2√
β2 − 1
3
[
4−
(
β4
β2− 1
3
− 1
)
v˙2
] 1
2
[
1− (β2−
2
3)
2
4(β2− 13)
v˙2
] [
3 (4β2 − 1)− β4v˙2
4(β2− 13)
]

+
48
(
2pi − 2θ(1) − 4θ(3))
4−
(
α4
α2− 1
3
− 1
)
v˙2
1
4
vv¨ −
(
α4
α2− 1
3
− 1
)
α4v˙4
16
(
α2 − 1
3
) + α4v˙2
2
(
α2 − 1
3
) − v˙2
4
− 1


+
6[
4− 1
3
β2v˙2
β2− 1
3
] 1
2
 3β6vv˙2v¨√
β2 − 1
3
[
3(4β2 − 1)− β4v˙2
4(β2− 13)
] [
3− β2v˙2
4(β2− 13)
] 1
2
+
8β
(
2pi − 4θ(4))
4− 1
3
β2v˙2
β2− 1
3
[
β4
4
vv¨ − β
6v˙4
48
(
β2 − 1
3
)2 + β4v˙22 (β2 − 1
3
) − β2v˙2
4
− 1
]
+
12
√
3
(
α2 − 1
4
)
α2vv¨√
α2 − 1
3
[
3 (4α2 − 1)− α4v˙2
4(α2− 13)
]
+
3
√
3
2
vv¨
 α2 − 23√
α2 − 1
3
[
1− (α2−
2
3)
2
4(α2− 13)
v˙2
] + β2 − 23√
β2 − 1
3
[
1− (β2−
2
3)
2
4(β2− 13)
v˙2
]

173
5.2. THE CHILDREN MODELS
+
24
√
3
(
β2 − 1
4
)
β2vv¨√
β2 − 1
3
[
3 (4β2 − 1)− β4v˙2
4(β2− 13)
]
− Λ
4
√
3
v2
 α4√
α2 − 1
3
+ 2
β4√
β2 − 1
3
(vv¨ + 3v˙2)
− 12
(√
α2 − 1
3
+ 2
√
β2 − 1
3
) . (5.2.54)
If this evolution equation is to be consistent with the Hamiltonian constraint
(5.2.51) and with relation (5.2.52), then we require the Hamiltonian constraint
to be its first integral. In Appendix F, we have proven that this is possible if
α = β = 1, that is, if the children tetrahedra are all equilateral, and we have
found indications that this requirement fails for any other values of α and β.
By taking α = β = 1, we can further simplify our model. The dihedral angles
now become
cos θ(1) = cos θ(2) = cos θ(3) = cos θ(4)
=
1 + 1
8
v˙2
3− 1
8
v˙2
,
(5.2.55)
which is identical to its counterpart (4.1.9) for the parent models. Since all
dihedral angles are identical, we shall henceforth drop the superscript. Then, the
Hamiltonian constraint (5.2.51) simplifies to
v2 =
√
2
Λ
1√
1− 1
8
v˙2
[
N1
N3
(2pi − nθ) + 6 (2pi − 5θ)
]
, (5.2.56)
and relation (5.2.52) for v˙ simplifies to
v˙2 = 8
[
1− 2 tan2
(
1
2
θ
)]
, (5.2.57)
which is identical to its parent model counterpart as given by (4.1.15). We can
now simply use (5.2.56) and (5.2.57) rather than the evolution equation to deter-
mine the behaviour of our models.
Finally, we present the volume of the subdivided Regge universe, which is
again simply the sum of the volumes of all constituent tetrahedra in a Cauchy
174
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
surface. This volume is
U˜N3(ti) =
1√
3
N3
(
1 + 2
β2
α2
)√
α2 − 1
3
v(ti)
3, (5.2.58)
which reduces to
U˜N3(ti) =
√
2N3 v(ti)
3, (5.2.59)
when α = β = 1.
5.2.4 Initial value equation of the children models
Like the parent models, the children models also admit a moment of time-
symmetry at the moment of minimum expansion, when v˙ = 0. At this moment,
it follows from (5.2.51) that
v20 =
2
√
3
Λ
α2
β2
[√
α2 − 1
3
(
α2
β2
+ 2
)]−1 [
α
N1
N3
(
2pi − nθ(2)0
)
+ 3
(
2pi − 2θ(1)0 − 4θ(3)0
)
+ 3 β
(
2pi − 4θ(4)0
)]
,
(5.2.60)
where
cos θ
(1)
0 =
1√
3(4α2 − 1) ,
cos θ
(2)
0 =
2α2 − 1
4α2 − 1 ,
cos θ
(3)
0 =
1√
3(4β2 − 1) ,
cos θ
(4)
0 =
2β2 − 1
4β2 − 1 .
The three deficit angles δ
(1)
i , δ
(2)
i , δ
(3)
i , given respectively by (5.2.41), (5.2.43), and
(5.2.44), now become
δ
(1)
0 = 2pi − nθ(2)0
δ
(2)
0 = 2pi − 2θ(1)0 − 4θ(3)0
δ
(3)
0 = 2pi − 4θ(4)0 .
(5.2.61)
We shall now demonstrate that (5.2.60) is also consistent with the initial value
175
5.2. THE CHILDREN MODELS
equation, as discussed in Chapter 3.4. Unlike the parent models however, vertices
in children Cauchy surfaces are no longer identical, so the notion of a ‘volume
per vertex’ becomes harder to define. Therefore, we shall use the form of the
initial value equation given by (3.4.8) instead. As with the parent models, we
shall consider the cosmological constant as a perfect fluid where ρΛ = −pΛ = Λ8pi ;
therefore, the integral appearing in (3.4.8) evaluates to
8pi
∫
Σ0
ρΛ d
3x = Λ U˜N3(t0),
where U˜N3(t0) is the volume of Σ0 and is given by (5.2.58) when v(ti) = v0, and
therefore the initial value equation can be expressed as
∑
i∈{edges}
liδi =
1√
3
N3 Λ
(
1 + 2
β2
α2
)√
α2 − 1
3
v30, (5.2.62)
where the summation is over all edges in Σ0.
Recall that a child Cauchy surface has only three distinct edge-lengths, which
for surface Σ0 are u0, v0, and p0, and that all edges with the same length are
identical, meaning that they would be associated with the same deficit angle
in the 3-dimensional skeleton of Σ0. We shall denote the 3-dimensional deficit
angle associated with u0 by δ¯
(1)
0 , with v0 by δ¯
(2)
0 , and with p0 by δ¯
(3)
0 . Then the
summation on the left-hand side of (5.2.62) can be expressed as∑
i∈{edges}
liδi =
(
NEdge u0
)
u0 δ¯
(1)
0 +
(
NEdge v0
)
v0 δ¯
(2)
0 +
(
NEdge p0
)
p0 δ¯
(3)
0 ,
where NEdge u0 , NEdge p0 , and NEdge v0 denote the numbers of edges with lengths
u0, p0, and v0 respectively in Σ0. Making use of the scaling relations (5.2.1) and
the fact that
NEdge u0 = NArea 1, NEdge v0 = NArea 2, NEdge p0 = NArea 3,
we further simplify the summation to∑
i∈{edges}
liδi = 2
[
N1 α δ¯
(1)
0 + 3N3 δ¯
(2)
0 + 3N3 β δ¯
(3)
0
]
v0,
176
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
where we have used relations (5.2.48) to (5.2.50) to substitute for NArea 1, NArea 2,
and NArea 3.
Substituting back into (5.2.62), we solve for v0 to get
v20 =
2
√
3
Λ
α2
β2
[√
α2 − 1
3
(
α2
β2
+ 2
)]−1 [
α
N1
N3
δ¯
(1)
0 + 3 δ¯
(2)
0 + 3 β δ¯
(3)
0
]
. (5.2.63)
This expression is identical to (5.2.60) provided the 3-dimensional deficit angles
δ¯
(i)
0 equal their 4-dimensional counterparts δ
(i)
0 , as given by (5.2.61), for i = 1, 2, 3.
The deficit angles δ
(i)
0 and δ¯
(i)
0 do have identical forms though. Recall that
each edge in the hypersurface generates a trapezoidal hinge, and each triangle
meeting at the edge generates a 3-dimensional face meeting at the hinge. Thus
from each of the deficit angles δ
(i)
0 , we can immediately deduce the form of its
counterpart δ¯
(i)
0 ; it follows that
δ¯
(1)
i = 2pi − nθ¯(1)0 ,
δ¯
(2)
i = 2pi − 2θ¯(1)0 − 4θ¯(3)0 ,
δ¯
(3)
i = 2pi − 4θ¯(4)0 ,
where θ¯
(i)
0 is the dihedral angle between the two triangles that generate the two
3-dimensional faces separated by dihedral angle θ
(i)
0 . To complete our proof then,
we need only demonstrate θ¯
(i)
0 = θ
(i)
0 for all i.
Consider a typical tetrahedron in E3 with vertices A,B,C,D. Let the ver-
tices’ co-ordinates be identical to the spatial co-ordinates of their counterparts in
(4.4.1). This tetrahedron has two distinct hinges, AB and AD. We first consider
the dihedral angle at AB. Triangles ABC and ABD meet at this hinge and are
separated by the dihedral angle θ¯
(1)
0 . The unit normal to ABC is (0, 0, 1), and
the unit normal to ABD is
1√
u20 − 14v20
(
0, −
√
u20 −
1
3
v20,
v0
2
√
3
)
=
1√
α2 − 1
4
(
0, −
√
α2 − 1
3
,
1
2
√
3
)
.
177
5.2. THE CHILDREN MODELS
From their scalar product, we find that
cos θ¯
(1)
0 =
1√
3(4α2 − 1) ,
and thus we see that θ¯
(1)
0 = θ
(1)
0 .
Next, we consider the dihedral angle at hinge AD. Triangles ABD and ACD
meet at this hinge and are separated by the dihedral angle θ¯
(2)
0 . The unit normal
to ABD is the same as before, while the unit normal to ACD is
1√
3 (4u20 − v20)
(
3
√
u20 −
1
3
v20, −
√
3u20 − v20, −v0
)
=
1√
3 (4α2 − 1)
(
3
√
α2 − 1
3
, −
√
3α2 − 1, −1
)
.
The scalar product gives
cos θ¯
(2)
0 =
2α2 − 1
4α2 − 1 ,
and we can similarly conclude that θ¯
(2)
0 = θ
(2)
0 .
Finally, we can readily obtain analogous expressions for θ¯
(3)
0 and θ¯
(4)
0 by swap-
ping u0 for p0 or equivalently α for β in all of the above expressions. From θ¯
(1)
0 ,
we immediately obtain
cos θ¯
(3)
0 =
1√
3(4β2 − 1) ,
and from θ¯
(4)
0 , we obtain
cos θ¯
(4)
0 =
2β2 − 1
4β2 − 1 .
Therefore θ¯
(3)
0 = θ
(3)
0 and θ¯
(4)
0 = θ
(4)
0 , and we can now conclude that the Regge
equation (5.2.51) for the children models does indeed satisfy the initial value
equation (3.4.8).
5.2.5 Discussion of the children models
We now compare the behaviour of the children models against their corresponding
parent models as well as against the FLRW universe. We shall only consider the
models where the children tetrahedra are all equilateral, as these are the only
178
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
models for which the evolution and constraint equations are consistent.
0
500
1000
1500
2000
0 200 400 600 800 1000 1200 1400 1600
d
U d
t
U(t)
FLRW
5 tetrahedra
16 tetrahedra
600 tetrahedra
60 tetrahedra
192 tetrahedra
7200 tetrahedra
Figure 5.13: The expansion rate of the universe’s volume against the volume itself
for all parent models and their children models.
Figure 5.13 and Figure 5.14 compare the graphs of dU
dt
against U for each
of our models, with U given by (5.2.59) for the subdivided Regge models and
by (4.1.16) for the parent models. The graphs in Figure 5.14 focus on the 600-
tetrahedral parent and its 7200-tetrahedral child, with the bottom plot extended
so as to reveal the graphs’ endpoints. At low volumes, subdividing the tetrahedra
actually made the Regge approximation worse. We can see this more concretely in
Table 5.3, where we list the minimum volumes for each model and their fractional
difference from the FLRW minimum. While increasing the number of tetrahedra
in the parent models brought the minimum volume closer to the FLRW value,
increasing the number in the children models actually brought it further away. In
fact, the worst parent model was still more accurate than the best child model.
We also see that all models again diverge from the FLRW model as the uni-
verse expanded; however increasing the number of tetrahedra reduces the rate of
divergence, and in this sense, increasing the number of tetrahedra improves the
Regge approximation. We again believe this lower rate of divergence is the result
of more tetrahedra providing a higher resolution approximation that can ‘keep
up’ longer with the FLRW model. We also note that each model terminates at
large volumes whenever the strut becomes null and that this endpoint gets in-
creasingly delayed as the number of tetrahedra is increased. Thus these figures
179
5.2. THE CHILDREN MODELS
0
1000
2000
3000
4000
5000
6000
7000
0 500 1000 1500 2000 2500 3000 3500 4000
d
U d
t
U(t)
FLRW
600 tetrahedra
7200 tetrahedra
0
10000
20000
30000
40000
50000
0 5000 10000 15000 20000 25000 30000 35000
d
U d
t
U(t)
FLRW
600 tetrahedra
7200 tetrahedra
Figure 5.14: The expansion rate of the universe’s volume against the volume itself for
the 600-tetrahedral parent and its 7200-tetrahedral child model. The top graph focuses
on the region around the origin while the bottom graph shows both Regge graphs in
their entirety.
180
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
Model Minimum volume
Fractional difference
from FLRW
FLRW 102.567937639753 0
5-tetrahedral parent 171.741398309775 0.67442
16-tetrahedral parent 135.70186007972 0.32304
600-tetrahedral parent 105.692461545881 0.03046
60-tetrahedral child 172.637934789289 0.68316
192-tetrahedral child 179.191098180344 0.74705
7200-tetrahedral child 1268.30953855058 11.36556
Table 5.3: The minimum volume and the fractional difference from the FLRW mini-
mum for each Regge model and for the FLRW model.
reveal that each child model provides an approximation that starts off worse than
its parent but is later much better by virtue of its more robust resolution. Indeed,
if one were to extrapolate all graphs past their end-points to very large volumes,
the 7200-tetrahedral model would ultimately provide the best performance.
As with the parent models, we can define a 3-sphere radius Rˆ(t) analogous to
(5.1.18) for Cauchy surfaces of the children models such that Rˆ(0) = a(0); that
is, Rˆ(t) and a(t) match at the moment of minimum expansion; thus, we define
Rˆ(t) to be
Rˆ(t) =
a(0)
vmin
v(t), (5.2.64)
where vmin is the minimum value of v and is given by (5.2.51) when θ
(2) =
arccos
(
2α2−1
4α2−1
)
= arccos
(
1
3
)
. Such a definition is possible because, as discussed
at the end of Section 5.2.1, all dynamical length-scales in the system are related
to each other by time-independent scalings such that there is really only one
independent dynamical length-scale describing the entire model; therefore any
dynamical length-scale, when re-scaled by an appropriate constant, will yield the
same Rˆ(t). Note that when n = 5, then l(t) as given by (5.1.33) and v(t) as given
by (5.2.56) will be identical apart from an overall constant factor. When this
happens, the two corresponding models will have the same Rˆ(t). This happens
for the 600-tetrahedral parent model and its 7200-tetrahedral child.
We now examine the behaviours of dRˆ
dt
versus Rˆ(t) and of the correspond-
ing 3-sphere volumes. Figure 5.15 shows the relationship between dRˆ
dt
and Rˆ(t)
181
5.2. THE CHILDREN MODELS
0.0
0.5
1.0
1.5
2.0
2.5
3.0
1 2 3 4 5 6
d
(R
a
d
iu
s)
d
t
Radius(t)
FLRW
5 tetrahedra
16 tetrahedra
600 tetrahedra
60 tetrahedra
192 tetrahedra
7200 tetrahedra
Figure 5.15: A combined graph of the radius’ expansion rate against the radius itself,
using Rˆ(t) as the Regge models’ radius.
for each of the Regge models and the relationship between da
dt
and a(t) for the
FLRW model. Figure 5.16 shows the relationship between the corresponding 3-
sphere volumes and their rates of expansion. We note that in both figures, the
graphs for the 600-tetrahedral model and its 7200-tetrahedral child coincide, as
expected. The definition of Rˆ(t) has clearly removed any variability in the initial
performance of the Regge models; we now see that the Regge models with more
tetrahedra consistently outperform those with fewer. The rate of divergence from
FLRW is again reduced and the graphs’ end-points further delayed as the num-
ber of tetrahedra is increased. Thus in these graphs, increasing the number of
tetrahedra clearly improves the Regge approximation.
As mentioned at the start of Section 5.2.3, we were unable to obtain a consis-
tent set of Hamiltonian constraints when we varied the Regge action for the child
model locally. We suspect this may be due to an inappropriate specification of
strut-lengths, thereby causing the model to be over-constrained. As we remarked
at the end of Chapter 4.4.1, we do not in general have complete freedom to spec-
ify all struts independently of each other, unlike lapse functions in the ADM
formalism; therefore, our choice of having all strut-lengths be equal may not be
appropriate. There may instead be some other choice of strut-lengths that would
lead to a consistent set of constraint equations. We note though that so long as
the strut-lengths remain interdependent, which we expect to be the case, there
182
5. REGGE CALCULUS OF THE Λ-FLRW UNIVERSE
0
200
400
600
800
1000
1200
0 200 400 600 800 1000
d
U d
t
U(t)
FLRW
5 tetrahedra
16 tetrahedra
600 tetrahedra
60 tetrahedra
192 tetrahedra
7200 tetrahedra
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0 500 1000 1500 2000 2500 3000 3500
d
U d
t
U(t)
FLRW
5 tetrahedra
16 tetrahedra
600 tetrahedra
60 tetrahedra
192 tetrahedra
7200 tetrahedra
Figure 5.16: A combined graph of the 3-sphere volume’s expansion rate against the
volume itself, using Rˆ(t) as the Regge models’ 3-sphere radius. The top graph focuses
on the region around the origin while the bottom graph shows all Regge graphs in their
entirety.
183
5.2. THE CHILDREN MODELS
will also be some constraint between the tetrahedral edge-length ratios α and β
similar to (5.2.27); this would follow from a similar reasoning to that which led
to (5.2.27). Indeed, one could perhaps view the set of three constraint equations
as follows: one equation determines the evolution of the tetrahedral edge-lengths,
while the other two determines the evolution of the two non-independent sets of
struts, one equation for each set. After determining the lengths of these two sets,
one could then deduce the constraint between α and β.
Nevertheless, we may still run into the same problem encountered when lo-
cally varying the parent model. There, we discovered that for the evolution
equation (5.1.46) to be a first integral of the Hamiltonian constraint (5.1.35),
the constraint equation (5.1.45), obtained from varying the diagonals, must also
be satisfied. However these latter constraints required the model to behave in
an unphysical manner, and thus, the model broke down. We suspected the un-
derlying cause to be the breaking of Copernican symmetries from introducing
the diagonals. With the children models, we should also expect the diagonals
to break Copernican symmetries, possibly rendering for example the mid-point
vertices inequivalent to each other. However, even if the local model were not vi-
able, it would, through the chain-rule relationship of (5.1.47), still point towards
an alternative but also viable global model, and we believe the properties of this
new model to be worthy of further investigation. Indeed, it may possess some
desirable advantages over our current child model, such as, for instance, having
a Hamiltonian constraint that is unconditionally a first integral of the evolution
equation. At this point though, we shall leave a more thorough examination into
the viability and properties of such models, global and local, to future study.
184
CHAPTER 6
Regge models of closed lattice
universes
With a better understanding of the CW formalism obtained from modelling the
Λ-FLRW universe, we shall now apply it to the lattice universe; our focus however
will be on modelling closed lattice universes. The CW formalism naturally lends
itself to modelling such universes since the Cauchy surfaces of the three parent
models themselves form three of the closed Coxeter lattices listed in Table A.1 of
Appendix A. Our ultimate interest will be to perturb one of the lattice masses
and study the resulting universe’s behaviour, as this would help bring us closer
to approximating the actual universe’s matter distribution; but before doing this,
it is necessary to first test the formalism on the standard lattice universe itself.
From our investigations in the previous chapter, we can draw on some lessons
to guide us in our application of the formalism here. First, we saw that locally
varying the skeleton in the previous chapter led to an unviable model, and we
believed this arose from a contradiction between assuming that all edge-lengths
in the Cauchy surface should remain identical and the actuality that the diag-
onals introduced broke symmetries between the edges in the surface. We shall
therefore only consider global models in this chapter, though we shall again take
the continuum time limit of all models, where δti → 0. Secondly, we saw that
in the continuum time limit, the Hamiltonian constraints, obtained by varying
the Regge action with respect to the struts, were first integrals of the evolution
equations, obtained by varying with respect to the tetrahedral edges. As a re-
185
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
sult, we could determine the evolution of the entire universe by examining the
constraint equations alone. Similarly, it can be shown that for the models of the
unperturbed lattice universe, which will be considered below, the Hamiltonian
constraints are also first integrals of the evolution equations, though we shall not
provide the proof. We shall assume a similar conclusion holds for models of the
perturbed lattice universe. This conclusion means that we can determine the
models’ evolution from their constraint equations alone.
This chapter is organised as follows. In the first section, we shall examine the
behaviour of the unperturbed lattice universe as a prelude to our investigation
of the perturbed universe itself. In particular, we shall show that the universe’s
behaviour depends on where the masses are located: the universe becomes uncon-
ditionally divergent if the masses are placed at the vertices of the Cauchy surface
but is unconditionally convergent if the masses are placed anywhere in a spherical
region around the centres of the tetrahedra. We thereafter work only with models
where the masses are at the tetrahedral centres. We also make this choice for
the following reason: in FLRW universes, a test particle that is co-moving with
the universe would be following a geodesic as well; if we also want particles in
our model to be both co-moving and following geodesics across the entire Regge
space–time, then we must place the particles at the centres of the tetrahedra; at
any other location, co-moving particles will not follow global geodesics. In the
second section, we shall perturb one of these masses, construct the correspond-
ing perturbed Regge model, and derive the relevant Regge equations. We shall
focus exclusively on the 5-tetrahedra model: this model would involve only two
sets of perturbed edges whereas the other two parent models would require many
more, with each set having its own independent length. In the third section,
we shall consider the application of the initial value equation at the moment of
time-symmetry to this model; we shall derive certain conditions that the initial
conditions of the Regge equations must satisfy in order to be consistent with this
equation. In the final section, we shall examine the behaviour of the model for
various perturbations and compare it against that of the unperturbed model; we
shall then close with a brief discussion of certain assumptions inherent in our
model.
186
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
6.1 Regge calculus of closed lattice universes
For a space–time whose only matter content are massive point particles, the
Regge action is given by (3.2.7). However, since all masses are identical in the
unperturbed lattice universe, the Regge action can be simplified to
8piS =
∑
i∈
{
trapezoidal
hinges
}A
trap
i δ
trap
i +
∑
i∈
{
triangular
hinges
}Atrii δ trii − 8piNpM
∑
i∈{ti}
si, (6.1.1)
where M is the common mass of each particle, Np the total number of particles
in the universe, and si the length of one particle’s trajectory between Cauchy
surfaces Σi and Σi+1. As we shall be varying with respect to the struts alone, the
space-like triangular hinges can be ignored.
When this action is varied globally with respect to the struts mj, we obtain
the Regge equation
0 =
∑
i
∂Atrapi
∂mj
δ trapi − 8piNpM
∑
i∈{ti}
∂si
∂mj
, (6.1.2)
where the first summation is still over all trapezoidal hinges.
There are several possible ways to arrange the masses into a regular lattice on
a Cauchy surface; examples include placing masses at the centres of the tetrahe-
dra, the centres of the triangles, the mid-points of the tetrahedral edges, or the
tetrahedral vertices. Each of these configurations will yield a regular lattice; the
new cell boundaries would lie along planes equidistant to pairs of masses that are
nearest neighbours to each other, and the masses would consequently lie at the
centres of the new cells. A 2-dimensional example has been illustrated in Fig-
ure A.1 of Appendix A, where the original lattice, drawn in solid lines, consists
of equilateral triangles tessellating flat 2-dimensional space; masses placed at the
mid-points of the triangular edges result in a new lattice consisting of rhomboidal
cells, drawn in dashed lines. However as discussed in Appendix A, not all new
lattices would correspond to Coxeter lattices; those that do not would have non-
regular polytopes as lattice cells and would consequently have reduced lattice
symmetries. However, we see that in this way, the CW formalism can allow us to
go beyond Coxeter lattices and model other lattice universes, something which
would not be possible with the LW formalism of Chapter 2 because spherical
187
6.1. REGGE CALCULUS OF CLOSED LATTICE UNIVERSES
cells, which that formalism uses, can only well-approximate lattice cells that are
regular polytopes. Once a particular arrangement of masses has been chosen,
then by symmetry, the masses will maintain that arrangement on all subsequent
Cauchy surfaces. The masses are therefore co-moving with respect to the Cauchy
surface.
Let us consider masses located in each tetrahedron at the general location of
vi = αA + βB + γC + δD,
with vectors A, B, C, D denoting the position vectors of the tetrahedron’s four
vertices relative to the tetrahedral centre, and with constants α, β, γ, δ satisfying
0 ≤ α, β, γ, δ ≤ 1 and α + β + γ + δ = 1. Each mass is located at a distance |vi|
from the tetrahedral centre given by
vi · vi = 1
8
[
3(α2 + β2 + γ2 + δ2)− 2(αβ + αγ + αδ + βγ + βδ + γδ)] l2i , (6.1.3)
with li being the length of a tetrahedral edge; as
|vi|
li
is a constant of time, we
shall denote this constant by v.
We shall once again work with 4-block co-ordinates given by (4.1.1) and their
Σi+1 counterparts to determine all geometric quantities; we can consider position
vectors A, B, C, D as being identical to the position of vertices A, B, C, D in
(4.1.1). In these co-ordinates, the length si of each mass’ line element between
Σi and Σi+1 is given by
s2i = v
2δl2i − δt2i ,
where δli denotes the difference δli := li+1 − li. In terms of the strut-length mi,
s2i can be expressed as
s2i =
(
v2 − 3
8
)
δl2i +m
2
i , (6.1.4)
where we have made use of (4.1.2). Then varying si with respect to mj yields
∂si
∂mj
=
mi
si
δij. (6.1.5)
All other geometric quantities appearing in the Regge equation (6.1.2) have
already been calculated in Chapter 4.1, so we can proceed now to determine
the constraint equation. By substituting (4.1.5), (4.1.6), (4.1.2), and (6.1.5) into
188
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
(6.1.2), we are led to the constraint equation
li = 8piM
Np
N1
[
1
8
l˙2i − 1
v2 l˙2i − 1
] 1
2
1
2pi − nθi , (6.1.6)
where N1 is the total number of tetrahedral edges on a Cauchy surface. By using
(4.1.9), we can express l˙i as a function of the dihedral angle θi, and then (6.1.6)
can be expressed as
li = 8piM
Np
N1
tan θi
2[
8v2 tan2 θi
2
− 1
2
(8v2 − 1)] 12 12pi − nθi . (6.1.7)
If we next take the continuum time limit, such that as δti → 0,
θi → θ +O(dt) ,
li → l,
li+1 → l + l˙dt+O
(
dt2
)
,
then (4.1.9) again yields the expression for l˙ given by (4.1.15), while (6.1.7)
becomes
l = 8piM
Np
N1
tan θ
2[
8v2 tan2 θ
2
− 1
2
(8v2 − 1)] 12 12pi − nθ . (6.1.8)
Equations (4.1.15) and (6.1.8) provide a parametric description of the universe’s
evolution in terms of the parameter θ.
Both l˙ and the lengths of the struts place constraints on the range of θ. In
the continuum time limit, the strut-length is given by relation (4.1.11), and for
the strut to remain time-like, we require that θ > pi
3
. On the other hand, for l˙2
to be positive, we require that θ ≤ 2 arctan 1√
2
. Thus, θ is constrained to lie in
the range
pi
3
< θ ≤ 2 arctan 1√
2
. (6.1.9)
For the square root in (6.1.8) to be real, we also require that
θ > 2 arctan
[
v2 − 1
8
2v2
] 1
2
; (6.1.10)
189
6.1. REGGE CALCULUS OF CLOSED LATTICE UNIVERSES
thus l diverges if θ = 2 arctan
[
v2− 1
8
2v2
] 1
2
. We can compare this with the range
of θ, as given by (6.1.9), to see under what conditions will divergence occur. If
2 arctan
[
v2− 1
8
2v2
] 1
2 ≥ pi
3
, then l will definitely diverge; this happens if
v2 ≥ 3
8
.
Equality would correspond to placing the masses at the tetrahedral vertices. How-
ever the lower bound of θ > pi
3
came from requiring that the struts be time-like,
not from any direct constraints on l itself. Thus, even if the model satisfies
2 arctan
[
v2− 1
8
2v2
] 1
2
< pi
3
, we may still see the beginnings of a divergence in l that
is abruptly cut off by the θ = pi
3
bound. Therefore, for l to be unconditionally
convergent, we require the more stringent constraint that 0 < −1
2
(8v2−1), which
comes from requiring the square-root in (6.1.8) to be real and positive, even when
θ = 0; thus v2 must satisfy
v2 <
1
8
.
When v2 = 1
8
, the masses are located at the mid-points of the tetrahedral edges.
Therefore, the universe will only be unconditionally convergent if the masses are
placed within a spherical region in the centre of the tetrahedron with a boundary
that just touches the mid-points of the tetrahedral edges. Such a region would
include the centres of the triangles and of the tetrahedra itself.
Interestingly, Collins and Williams [63] found a similar result in their study
of closed dust-filled universes. They were considering different ways to measure
the ‘time’ of the universe by using the proper time τ of test particles located at
different positions in the tetrahedron. We note that τ is actually identical to the
continuum time limit of si, that is, the continuum time limit of the square-root of
(6.1.4). They found that
dUN3
dτ
, where UN3 is the volume of the universe as given by
(4.1.16), would diverge if the test particle was outside the same spherical region of
convergence as the one we obtained. We suspect that this region of convergence
may be a generic feature of any model based on parent CW skeletons.
Although masses situated at or near vertices will cause the resulting model
to diverge, there may be a way around this problem. Each closed Coxeter lattice
admits a dual Coxeter lattice centred on the original lattice’s vertices, as noted in
Appendix A, although the dual lattice may not necessarily be a tetrahedral lattice.
190
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
Each parent model thus admits a dual model using Cauchy surfaces based on the
dual lattice, and we can always extend the CW formalism for global parent models
to perform Regge calculus with these non-tetrahedral models. Where masses
would have been at the vertices in the original parent model, when translated to
the dual model, they would now be at the centres of the dual cells. As a result,
the particles would now be both co-moving with respect to the Cauchy surface
and following geodesics globally across the entire Regge space–time; this follows
because even with non-tetrahedral cells, each cell would simply expand or contract
uniformly about its centre as the Cauchy surface evolves, so we can establish an
analogous 4-block co-ordinate system to that of (4.1.1), and in such a system,
one sees that a particle co-moving with the cell centre always moves along the
temporal direction only regardless of the cell’s shape. We additionally conjecture
that in non-tetrahedral models, there would also be an analogous spherical region
of convergence regarding the placement of massive particles within the cell, with
the region of convergence being centred on the cell centres. However, it remains to
be seen whether the spherical regions of both the original and dual models would
completely cover the entire Cauchy surface such that for any configuration of the
particles, there is always a model for which this configuration is well-behaved.
As noted in Appendix A, the 5-tetrahedra lattice is dual to itself, so in this case,
the region of convergence in the dual model is exactly identical to the region of
convergence in the original model, and if masses are positioned at the vertices of
the original model, then when they are translated to the dual model, the resulting
model will behave in exactly the same way as the original model with masses at
its cell centres.
In the foregoing discussion, we have only considered universes consisting of a
single set of regularly distributed masses. We can also consider the more general
case where we have not one but several different sets of masses, each set being
regularly distributed and having equal magnitude, though the magnitude can dif-
fer between sets. If we work in the original lattice, then each set of masses would
be arranged in a way that preserves the lattice symmetries, since, as mentioned
above, the sets would lie at such sites as the lattice vertices or the mid-points
of the edges, for instance. Therefore, even if the magnitudes of the masses dif-
fer between sets, because the lattice symmetries remain preserved, the edges in
particular would all remain identical to each other, and hence, the lattice would
still have only one length-scale l. A similar argument applies when the masses
191
6.1. REGGE CALCULUS OF CLOSED LATTICE UNIVERSES
are translated to the dual model. Therefore for such models with multiple sets of
masses, it can be shown that the continuum time equation for l becomes
l = 8pi
∑
i
Mi
Ni
N1
tan θ
2[
8v2i tan
2 θ
2
− 1
2
(8v2i − 1)
] 1
2
1
2pi − nθ , (6.1.11)
where i labels the set, the summation is over all sets, Ni is the number of particles
in set i, Mi is the magnitude of the mass in set i, and vi denotes the parameter
v for set i. It is clear that this equation has the same convergence conditions
as those for the single-set universe. We note though that there are certain set
combinations that would not lead to well-behaved models, even when translated
to dual models. One example would be to have particles at both the tetrahedral
centres and the vertices of the 5-tetrahedra model; one of the two sets would
always lie outside the region of convergence, regardless of whether the original
model or the dual model was being considered.
Returning to the single-set universes, to illustrate the behaviour of our various
Regge models, we have plotted dU
dt
against U in Figure 6.1 and Figure 6.2, where
U is the universe’s volume as given by (4.1.16) for the Regge models. For com-
parison, we have also plotted the corresponding graph for the dust-filled closed
FLRW model, with the universe’s volume given by (1.0.5) and the FLRW scale
factor a(t) given parametrically by (2.2.1) and (2.2.4); and we have plotted the
corresponding graphs for the equivalent LW models in Chapter 2, where the LW
universe’s ‘volume’ is given by the volume of a 3-sphere with the radius set equal
to the lattice universe scale factor a(τ) as given parametrically by (2.2.9) and
(2.2.13)1. Two classes of Regge models are shown: on the one extreme, Fig-
ure 6.1 shows the behaviour when particles are positioned at the centres of the
tetrahedra, while on the other, Figure 6.2 shows the behaviour when they are at
the mid-points of the tetrahedral edges, right on the boundary of the region of
unconditional convergence.
There are several features to note from these graphs. The evolution of all
lattice models are well-behaved and closed. Moreover, the behaviour of the ap-
1We do not imply anything physical by this LW 3-sphere; it has merely been defined to
facilitate comparison of length-scales between LW and Regge models. We could instead have
chosen to graph a˙(τ) against a(τ) for the LW universes and l˙/Z against l/Z, where Z is some
suitably chosen constant that re-scales the Regge edge-lengths; the comparison of the models
would effectively be equivalent.
192
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
0.0
2.0
4.0
6.0
8.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
d
U d
t
U(t)
dust-filled FLRW
5 tetrahedra Regge model
16 tetrahedra Regge model
600 tetrahedra Regge model
5 tetrahedra LW model
16 tetrahedra LW model
600 tetrahedra LW model
Figure 6.1: The expansion rate of the universe’s volume dUdt versus the volume U
itself for the dust-filled FLRW universe, for the three different Regge models where
a mass is positioned at the centre of each tetrahedron, and for the three equivalent
LW models. The Regge universe volume is given by the sum of the volumes of the
constituent tetrahedra of the Cauchy surface, while the LW universe volume is given
by the volume of a 3-sphere with a radius equal to the lattice universe scale factor a(τ).
The universe’s total mass is the same across all universes.
proximations converge to that of the FLRW universe as the number of particles
is increased, which is to be expected because the matter content then becomes
more dust-like. However, the Regge models and the LW models converge from
opposite directions. It is difficult to say a priori whether the LW approximation
or the Regge approximation is more representative of the actual lattice universe:
on the one hand, the average matter density is reduced as the number of par-
ticles is reduced, and this would weaken the gravitational interaction between
particles, but on the other, the mass of each particle gets increased, and this
would strengthen the interaction. However, comparison with exact initial value
data on the time-symmetric hypersurface seems to favour the LW models [48]:
using various measures for the lattice scale factor alatt on such a hypersurface,
Clifton et al. have shown that the ratio alatt
aFLRW
would always approach unity from
above as the number of lattice masses increases.
193
6.1. REGGE CALCULUS OF CLOSED LATTICE UNIVERSES
0.0
2.0
4.0
6.0
8.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
d
U d
t
U(t)
dust-filled FLRW
5 tetrahedra Regge model
16 tetrahedra Regge model
600 tetrahedra Regge model
5 tetrahedra LW model
16 tetrahedra LW model
600 tetrahedra LW model
Figure 6.2: An equivalent graph to Figure 6.1 but for the case where, in the Regge
models, the masses are positioned at the mid-points of the tetrahedral edges. The
universe’s total mass is again the same across all universes.
We note that there is a degree of ambiguity in choosing a common measure
with which to compare the two approximations. As we noted in the previous
chapter, CW Cauchy surfaces can be embedded into 3-spheres in E4, and there
are multiple ways to define a radius for the 3-sphere, although each radius would
simply be a constant re-scaling of the tetrahedral edge-length l(t). Each radius
implies a different 3-sphere volume, which results in volume graphs of different
magnitudes, although regardless of which scaling is chosen, the graphs would still
remain closed and well-behaved. Thus, it is within the realm of possibility that
the models are actually equivalent and that the apparent difference is due to an
inappropriate choice of scaling.
However, we suspect that the principal factor behind the two approximations’
discrepancy is the finite resolution of the lattice cell’s geometry in the Regge
models. We have constructed Regge models where the entire geometry of each
cell has been reduced to one quantity, that of the tetrahedral edge-length; the
cell’s interior geometry consists uniformly of Minkowski space–time throughout,
with no hinges or extra edges to provide additional geometric information, and
194
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
such a model is clearly very coarse-grained. In contrast, LW models approximate
each cell’s interior geometry with Schwarzschild space–time, and, although not
perfect, this approximation should still be more accurate than completely uni-
form Minkowski space–time. One could perhaps fine-grain the Regge models by
subdividing each lattice cell into smaller tetrahedra; this would certainly intro-
duce more edges and hence more geometric information into each cell, although
the smaller tetrahedra would no longer be identical nor equilateral. Alternatively,
one might be able to construct a finer-grained 5-cell universe for instance, by dis-
tributing five massive particles in a regular manner in a 600-tetrahedra Cauchy
surface; it remains to be seen whether such a regular distribution of the masses
can be obtained. Assuming it is possible, then each cell would now be associ-
ated with several edge-lengths rather than just one, and this would also provide
greater geometric information about each lattice cell. Nevertheless, we see that
even these coarse-grained models can still reveal much qualitative information
about the lattice universe’s behaviour, most notably its stability and its closed
evolution; thus we shall continue to use them to study the lattice universe.
We shall henceforth work with universes where the masses are located at the
centres of the tetrahedra. This would place the masses right in the middle of the
region of convergence. Furthermore, as mentioned above, this is the only position
in the tetrahedra where co-moving masses would also be following geodesics of
the Regge space–time, which would be closest to what happens in an FLRW
universe.
6.2 Perturbation of a single mass
We shall now construct a universe where the magnitude of a single central mass
gets perturbed fromM toM ′. This would induce a commensurate perturbation in
the surrounding geometry, specifically in the lengths of the edges surrounding the
perturbed mass. In general, there will be several sets of edges, each with its own
independent length, and we must determine what those sets are. We begin with
the edges in the tetrahedron enclosing the perturbed mass. The Cauchy surface
will remain symmetric about this single perturbed mass; therefore the enclosing
tetrahedron will still remain equilateral but with perturbed edge-lengths l
(1)
i . We
shall refer to this tetrahedron as a Type I tetrahedron. Next, we consider the
195
6.2. PERTURBATION OF A SINGLE MASS
four tetrahedra that share a face with the Type I tetrahedron. We shall refer
to these as Type II tetrahedra. These tetrahedra will no longer be equilateral:
they will each have an equilateral base corresponding to the face shared with the
Type I tetrahedron, but they will also have three other edges identical to each
other but different from the l
(1)
i edges. In the 5-tetrahedra model, these two sets
of edges are the only ones present, and we denote the length of the second set
by l
(0)
i . In the 16 and 600-tetrahredra models, we must keep going; we must
next consider the new sets of tetrahedra neighbouring the ones we have already
considered; each additional set of tetrahedra may introduce a new set of edges;
and each additional set of edges will generally have a length different from the
lengths of all other edges so far considered. There will be far more than just two
sets of edges in these models. For example, the 16-tetrahedra model has four
distinct edge-lengths. One might consider simplifying the model by constraining
all edges to have a common length l
(0)
i apart from the edges l
(1)
i of the equilateral
tetrahedron enclosing the perturbed mass; that is, all perturbation in geometry
gets localised to just this equilateral tetrahedron. We attempted this but found
that the resulting model was self-inconsistent. Therefore, we cannot study all
three models in a general manner but must construct and examine each one
individually. We have chosen to focus only on the 5-tetrahedra model as it is the
most tractable, involving only two distinct edge-lengths.
We shall once again work with a co-ordinate system for the tetrahedra’s 4-
blocks, though each tetrahedral type requires its own system. Since the Type
I tetrahedron is equilateral, we can use the same co-ordinates for its 4-block as
those in (4.1.1) and its Σi+1 counterpart, although we must replace li and li+1
with l
(1)
i and l
(1)
i+1 respectively; we shall continue to label the lower tetrahedron’s
vertices by A, B, C, D and their upper counterparts by A′, B′, C ′, D′.
For a Type II tetrahedron’s 4-block, we shall label the vertices of the lower
tetrahedron by A, B, C, E, with E denoting the tetrahedron’s apex, and we shall
denote their counterparts in the upper tetrahedron by A′, B′, C ′, E ′; vertices A,
B, C are shared with the Type I tetrahedron in Σi and A
′, B′, C ′ with the
Type I tetrahedron in Σi+1. Geometrically, the Type II tetrahedron is essentially
identical to Brewin’s child tetrahedron described in Chapter 4.4.1; this can be
seen if we identify vertices D and D′ there with E and E ′ respectively here and
lengths vi and ui there with l
(1)
i and l
(0)
i respectively here. Thus after making the
appropriate substitutions, we can assign co-ordinates (4.4.1) to the vertices of the
196
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
lower tetrahedron, with hi again given by (4.4.2). By symmetry, the base A
′B′C ′
of the upper tetrahedron should simply be a uniform expansion or contraction of
the lower tetrahedron’s base; there may also be a shift in the spatial direction
orthogonal to the base, but that is all. This is again similar to what happens
with Brewin’s child tetrahedron, so we can also assign the co-ordinates of A′,
B′, C ′ in (4.4.3) to vertices A′, B′, C ′ here; however, for the sake of generality,
we shall for the moment denote the temporal co-ordinates of these vertices by
ιt
(1)
i+1; this co-ordinate may or may not be identical to the temporal co-ordinate
in the Type I tetrahedron. Both δzi and t
(1)
i+1 can be deduced from the fact
that (i) struts AA′, BB′, CC ′ are shared with the Type I tetrahedron and must
therefore have the same length in both co-ordinate systems; and (ii) diagonals
such as AB′, AC ′, BC ′ are shared with the Type I tetrahedron and must therefore
have the same length in both co-ordinate systems. From these two constraints,
it can be deduced that ti+1 = t
(1)
i+1 and that δzi = ± 12√6δl
(1)
i ; but in the limit
where M ′ → M , we must recover the unperturbed model, so we take δzi to be
δzi = − 12√6 δl
(1)
i . The only constraint on apex E
′ is that it lie at distance l(0)i
from each of A′, B′, C ′ and that there be no twist or shear in the 4-block; these
are the same two constraints imposed on D′ in Brewin’s child tetrahedron, and
therefore, the co-ordinates of E ′ can be given by those of D′ in (4.4.3) as well,
though again with the appropriate substitutions made, and the Lorentz boost
parameter ψi in (4.4.3) can similarly be given by (4.4.5). In the end, this means
that a 4-block of the Type II tetrahedron is geometrically equivalent to that of
Brewin’s child tetrahedron. For ease of reference, we shall summarise the 4-block
co-ordinate system specific to the Type II tetrahedron here, with the appropriate
substitutions made; in this system, the co-ordinates of the lower vertices are
A =
(
− l
(1)
i
2
,− l
(1)
i
2
√
3
, 0, ιti
)
,
B =
(
l
(1)
i
2
,− l
(1)
i
2
√
3
, 0, ιti
)
,
C =
(
0,
l
(1)
i√
3
, 0, ιti
)
,
E = (0, 0, hi, ιti),
(6.2.1)
197
6.2. PERTURBATION OF A SINGLE MASS
and those of the upper vertices are
A′ =
(
− l
(1)
i+1
2
,− l
(1)
i+1
2
√
3
,− δl
(1)
i
2
√
6
, ιti+1
)
,
B′ =
(
l
(1)
i+1
2
,− l
(1)
i+1
2
√
3
,− δl
(1)
i
2
√
6
, ιti+1
)
,
C ′ =
(
0,
l
(1)
i+1√
3
,− δl
(1)
i
2
√
6
, ιti+1
)
,
E ′ =
(
0, 0, hi+1 coshψi − δl
(1)
i
2
√
6
, ιti+1 + ιhi+1 sinhψi
)
,
(6.2.2)
with hi given by
hi =
√(
l
(0)
i
)2
− 1
3
(
l
(1)
i
)2
, (6.2.3)
and the parameters Aψ, Bψ, Cψ in (4.4.5) by
Aψ = hi l
(1)
i+1,
Bψ = hi+1 l
(1)
i ,
Cψ = hi+1l˙
(1)
i
[
hi+1 − l
(1)
i
2
√
6
]
.
(6.2.4)
We note that because of the non-zero boost parameter ψi, the time-like hinges
of this skeleton will be quadrilateral but not necessarily trapezoidal, which is
different from all other CW skeletons considered so far. Therefore, we shall
denote the areas and deficit angles of quadrilateral hinges by Aquadi and δ
quad
i
instead. Moreover, we note that such a ψi implies the lengths of the struts will
in general not be identical, with the length of EE ′ being different. Finally, we
note that in the continuum time limit where δti → 0, ψi must become ψ˙ dt to
leading order in dt; this is because the relative boost between the lower and upper
tetrahedra must become infinitesimally small as the separation between Cauchy
surfaces tends to zero, and therefore, the zeroth order term of ψi must be zero.
We can now express the global Regge action for this model as
8piS =
∑
i∈
{
quadrilateral
hinges
}A
quad
i δ
quad
i +
∑
i∈
{
triangular
hinges
}Atrii δ trii − 8pi
∑
i∈{particles}
j ∈{4-blocks}
Mi sij, (6.2.5)
198
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
where sij is the path-length for the mass of magnitude Mi in the 4-block labelled
j; when varied with respect to the struts mk, it yields the Regge equation
0 =
∑
i
∂Aquadi
∂mk
δ quadi − 8pi
∑
i∈{particles}
j ∈{4-blocks}
Mi
∂sij
∂mk
, (6.2.6)
where the first summation is still over all quadrilateral hinges.
6.2.1 Global and local Regge equations
Before continuing with our derivation of the Regge equations, we shall consider
again the relationship between the global and local Regge equations though this
time specific to the perturbed model. We mentioned earlier that we would focus
exclusively on the global Regge equation. Our actual approach however will be
to vary the Regge action locally first and then use the chain rule to obtain the
global equation. As discussed in Chapter 4.3, solutions of local variation will also
be solutions of global variation, though the converse is not necessarily true.
To vary the action locally, we must again triangulate the skeleton using the
algorithm described in Chapter 4.2. Indeed, the triangulation of a 5-tetrahedra
skeleton and its 10 diagonals was given explicitly there, so we can readily apply it
to our perturbed skeleton here. We shall work with the triangulation where the
four vertices of the equilateral tetrahedron corresponds to vertices A, B, C, D in
the triangulation of Chapter 4.2 and where the fifth vertex of the skeleton, the one
that is the apex for all non-equilateral tetrahedra, corresponds to vertex E. Then
the diagonals of all hinges involving E and E ′ would terminate at E ′ and not at E.
In terms of a Type II tetrahedron’s 4-block as described above, this means that
the diagonals of hinges involving 4-block vertices E and E ′ would terminate at E ′
and not at E as well; this is consistent with the 4-block triangulation algorithm
also described in Chapter 4.2.
Like the time-like trapezoidal hinges we considered previously, the diagonals
divide each time-like quadrilateral hinge into a pair of triangular hinges, and
by the same argument as in Chapter 4.2, the deficit angles δAi and δ
B
i of the
two triangular hinges would become identical to the deficit angle δ quadi of the
original quadrilateral hinge after the constraints on the edge-lengths have been
applied; that is, δAi = δ
B
i = δ
quad
i . The argument still applies to the triangulated
199
6.2. PERTURBATION OF A SINGLE MASS
quadrilateral hinges because they become entirely planar as well when the relevant
edge-length constraints are applied.
Since the deficit angles are identical, then so far as the struts are concerned,
the Regge action for the triangulated skeleton effectively equals the Regge action
for the original skeleton as given by (6.2.5); this is because the only difference
between the two actions, quantities corresponding to the diagonal hinges, does
not depend on the struts. Then using the chain rule (4.3.1), we can again express
the global Regge equation as a linear combination of local Regge equations to
obtain
0 =
∂S
∂mi
=
∑
j
∂S
∂m`i
∂m`i
∂mi
+
∑
j
∂S
∂dj
∂dj
∂mi
, (6.2.7)
where ∂S
∂m`i
is a local variation with respect to a single strut m`i , with superscript `
indicating that these struts are to be considered as local struts rather than global
struts, and where ∂S
∂dj
is a local variation with respect to a single diagonal dj; the
first summation is over all local struts and the second over all diagonals.
As we shall perform a local variation first, we note that our skeleton has two
distinct sets of struts which we must independently vary, one corresponding to
the struts of the Type I tetrahedron, and the other to the EE ′ strut of the Type
II tetrahedron. The first set of struts all have the same length, which we shall
denote by mAA
′
i since the set includes strut AA
′. The length is given by (4.1.2)
as well, but with l˙i replaced by l˙
(1)
i , that is, by(
mAA
′
i
)2
=
[
3
8
(
l˙
(1)
i
)2
− 1
]
δt2i . (6.2.8)
The length of strut EE ′ will be denoted by mEE
′
i and is given by(
mEE
′
i
)2
=
[
hi+1 coshψi − hi − 1
2
√
6
δl
(1)
i
]2
−
[
δti + hi+1 sinhψi
]2
. (6.2.9)
We shall first examine the contribution of the first summation in (6.2.7) to
the global Regge equation. We shall always choose the global strut-length mi to
equal either mEE
′
i or m
AA′
i ; without loss of generality, let us assume then that
mi = m
AA′
i . Then it trivially follows that
∂mAA
′
j
∂mi
= δij, which is clearly O(1) to
leading order in dt when the continuum time limit is taken. If we substitute for
δti in (6.2.9) using (6.2.8), it can also be shown that the leading order of
∂mEE
′
j
∂mi
200
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
will be at least O(1) as well. We shall be explicitly calculating ∂S
∂m`i
later on,
and at the end of the calculation, we shall see that the leading order of ∂S
∂m`i
is
also O(1). Therefore unless the second summation in (6.2.7) has negative leading
order, the first summation will definitely contribute to the leading order of the
global Regge equation.
We now turn to the second summation in (6.2.7). Our skeleton has two
diagonals, one subdividing quadrilateral hinges generated by l
(0)
i edges, and the
other subdividing quadrilateral hinges generated by l
(1)
i edges. The first set of
hinges are the ones that involve 4-block vertices E and E ′, so an example of a
diagonal on such a hinge would be AE ′; these diagonals have length dAE
′
i given
by(
dAE
′
i
)2
=
2
3
(
l
(1)
i
)2
−
(
l
(0)
i
)2
+
(
mEE
′
i
)2
+hi
[
2hi+1 coshψi − 1√
6
δl
(1)
i
]
. (6.2.10)
An example of a diagonal on the second set of hinges would be AB′, and these
diagonals have length dAB
′
i given by(
dAB
′
i
)2
= l
(1)
i+1 l
(1)
i +
(
mAA
′
i
)2
. (6.2.11)
It can be shown that
∂dAB
′
i
∂mi
=
∂dAB
′
i
∂mAA
′
i
=
mAA
′
i
dAB
′
i
.
In the continuum time limit, where l
(1)
i → l(1)(t), we have that m
AA′
i
dAB
′
i
→ mAA
′
i
l(1)
,
which is an O(dt) term. Thus,
∂dAB
′
i
∂mi
raises the leading order of the second sum-
mation in (6.2.7) by one. It can also be shown that in the continuum time limit,
the leading order term of
∂dAE
′
j
∂mi
is at least O(dt) as well. Therefore
∂dj
∂mi
will be at
least O(dt) to leading order for all diagonals.
So unless the leading order of ∂S
∂dj
is negative, the leading order of the second
summation in (6.2.7) will be at least O(dt). Naturally, verifying that the leading
order of ∂S
∂dj
is not negative requires a direct calculation of ∂S
∂dj
. However, such
a calculation is beyond the scope of this thesis. Instead, we shall assume the
conclusion. First, if ∂S
∂dj
had negative leading order, then the corresponding Regge
equation would diverge in the continuum time limit; thus our model would break
down. We are assuming this is not the case. Secondly, we have found many
201
6.2. PERTURBATION OF A SINGLE MASS
similarities between the Regge model we are studying here and the parent Regge
models of the Λ-FLRW universe in Chapter 5. In Chapter 5.1.4, we similarly
considered the relationship between global and local Regge equations through an
essentially identical chain rule (5.1.47). We found the leading order of
∂dj
∂mi
to be
O(dt) as well for all diagonals. Furthermore,
∂m`i
∂mi
was also O(1) to leading order;
this followed trivially because all struts between pairs of consecutive Cauchy
surfaces had equal length, so
∂m`i
∂mi
would be unity for struts between the same
pair of surfaces. Finally, the leading order of ∂S
∂m`i
was O(1) as well. Since the two
models have identical leading orders for three of the partial derivatives appearing
in (5.1.47) and (6.2.7), we suspect they would have identical leading order for the
final partial derivative, ∂S
∂dj
. For the Λ-FLRW model, the order of this term was
O(dt), so we suspect it would be the same for this model.
If this is true, then the second summation in (6.2.7) would have a higher
leading order than the first summation and would therefore not contribute. As
a result, the solutions to 0 = ∂S
∂m`i
would by themselves satisfy the global Regge
equation 0 = ∂S
∂mi
, much like the situation with the Λ-FLRW Regge models.
6.2.2 Particle trajectories
The final term of the Regge action (6.2.5) determines the effect of the masses
on the behaviour of the universe. This term depends on sij, the length of the
trajectory followed by the mass Mi through the 4-block labelled j; thus to fully
specify our Regge model, we must specify what trajectory the masses will follow.
Ideally, we should like our masses to follow geodesics throughout the entire
universe and also be co-moving with respect to the Cauchy surfaces, as we expect
this to be the situation in the continuum universe. As mentioned at the start of
this chapter, this is the situation in the perfectly smooth FLRW universe, where
test particles co-moving with respect to constant-t Cauchy surfaces are also follow-
ing geodesics of the space–time. In the lattice universe, the point-masses should
similarly be co-moving with respect to the universe’s Cauchy surfaces so as to
preserve the lattice symmetries – we do not expect the gravitational interactions
between a symmetric distribution of masses to give rise to an asymmetric mo-
tion of the masses. Yet these particles should also be following geodesics as well.
However we have seen that in the unperturbed Regge model, it is not possible for
the particle to be simultaneously co-moving and following geodesics across the
202
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
entire space–time unless the particles are positioned at the centres of the equilat-
eral tetrahedra. Even in CW approximations of the perfectly homogeneous and
isotropic FLRW space–times, co-moving test particles will not follow geodesics
globally either unless the particles are at the centres of the tetrahedra. Consider,
for example, test particles co-moving with respect to the centres of the triangles.
The centres themselves trace out piecewise linear trajectories as the underlying
CW skeleton is piecewise linear. Therefore, test particles co-moving with these
points will follow the same piecewise linear trajectories, deflecting each time they
cross from one Cauchy surface into the next. Only trajectories traced out by the
centres of the tetrahedra will have no deflection, since the tetrahedra expand or
contract uniformly about their centres as they evolve, and this thereby leaves the
tetrahedral centres spatially fixed with time. We note though that each linear
segment of a piecewise linear trajectory will still be a local geodesic within the
4-block it traverses because straight-line segments are always geodesics according
to the Minkowski metric. If we take the continuum space–time limit of a CW ap-
proximation to an FLRW space–time, we expect to recover the continuum FLRW
space–time itself in which any co-moving particle will indeed follow geodesics as
well. The reason only the tetrahedral centres follow global geodesics is because
Cauchy surfaces of the CW skeleton are not perfectly isotropic and homogeneous;
thus, not all points on the Cauchy surface have been ‘created equal’. If co-moving
particles do not follow geodesics in CW approximations of the perfectly homo-
geneous and isotropic FLRW space–times, there is even less reason for them to
follow geodesics in approximations of the lattice universe, both perturbed and un-
perturbed. Thus we shall only require the point-masses of the perturbed Regge
lattice universe to be co-moving with respect to the Cauchy surfaces. But in
the continuum space–time limit, we do hope that these co-moving particles will
indeed follow geodesics as well.
As mentioned previously, we have chosen to work with a lattice universe where
the masses would be co-moving with the centres of the tetrahedra when the
universe is unperturbed. For the particle in the equilateral Type I tetrahedron,
the trajectory is straightforward: by symmetry, the particle should remain at
the centre of the tetrahedron for its entire trajectory. For particles in Type II
tetrahedra, it is less clear where the particles should be positioned because the
tetrahedra are no longer equilateral. The only clear symmetry here is in the
equilateral base. We can therefore say that a particle should remain above the
203
6.2. PERTURBATION OF A SINGLE MASS
centre of the equilateral base for the entirety of its trajectory. The issue lies in
fixing the particle’s position above the base. We know that the vertices themselves
should be co-moving with respect to the Cauchy surface, so we shall use them as
reference points to express the trajectory of the co-moving particle. The particle’s
position pi on Cauchy surface Σi can be expressed as
pi = α (A + B + C) + β E, (6.2.12)
where A,B,C,E are the position vectors of vertices A,B,C,E respectively, and
where α and β are yet to be determined constants. For the particle to be inside the
tetrahedron, α and β must be non-negative and satisfy the constraint 3α+β = 1.
For the particle to be co-moving, we require that its position pi+1 on Cauchy
surface Σi+1 be given by (6.2.12) as well, but with vectors A,B,C,E replaced by
A′,B′,C′,E′ respectively. In the 4-block between Σi and Σi+1, the particle then
propagates in a straight line from pi to pi+1, and such a trajectory is considered
co-moving with respect to the Cauchy surfaces.
There is one situation where there is clearly a unique choice for α and β.
Should l
(1)
i become equal to l
(0)
i at any moment, then the corresponding tetrahe-
dron will be equilateral; we would then require the particle to lie at the tetrahe-
dron’s centre, which means α and β must be 1
4
at this moment. Yet based on
our definition of co-moving trajectories, α and β must be constant over the par-
ticle’s entire trajectory. Therefore, α and β must be 1
4
over the particle’s entire
trajectory. However we shall take α and β to be 1
4
for all particles, regardless of
whether their tetrahedra become equilateral or not. Such a choice would place
the particles at the tetrahedra’s centroids, which would in some sense generalise
our requirement that the particles be at the tetrahedra’s centres.
6.2.3 Geometric quantities for the Regge equation
We now turn to deriving the geometric quantities relevant for the Regge equation.
From (6.2.6), it is clear that we need to derive three types of quantities: the
varied areas of the time-like hinges,
∂Ai
∂mk
; the corresponding deficit angles δi, or
equivalently, the dihedral angle θi between neighbouring faces; and the varied
lengths of the particles’ trajectories across 4-blocks,
∂sij
∂mk
.
We shall be taking the continuum time limit of these quantities so that we
204
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
can express the Regge equation in its continuum time form. Thus, we shall be
needing the continuum time form of the lengths and the boost parameters; these
are given by
l
(0)
i → l(0)(t),
l
(0)
i+1 → l(0) + l˙(0) dt+O
(
dt2
)
,
l
(1)
i → l(1)(t),
l
(1)
i+1 → l(1) + l˙(1) dt+O
(
dt2
)
,
ψi → ψ˙ dt+O
(
dt2
)
,
dAE
′
i → l(0) +O(dt) ,
dAB
′
i → l(1) +O(dt) ,
mEE
′
i →
[(
1
2
√
6
l˙(1) − h˙
)2
−
(
hψ˙ + 1
)2] 12
dt+O
(
dt2
)
,
mAA
′
i →
[
3
8
(
l˙(1)
)2
− 1
] 1
2
dt+O
(
dt2
)
,
where
h˙ =
l(0)l˙(0) − 1
3
l(1)l˙(1)
h
, (6.2.13)
ψ˙ =
h˙l(1) − hl˙(1)
hl˙(1)
[
h − l(1)
2
√
6
] , (6.2.14)
and where h =
√
(l(0))
2 − 1
3
(l(1))
2
denotes the continuum time limit of hi.
As mentioned previously, our skeleton has two types of time-like hinges cor-
responding to the world-sheets of l
(0)
i and l
(1)
i edges. We shall refer to the quadri-
lateral hinge generated by l
(1)
i as a Type I hinge and its triangular components
as hinges A1 and B1, counterparts to triangles A and B respectively in Figure
4.2. We can again use Heron’s formula to compute these triangular hinges’ areas.
Hinge A1 has area
AA1i =
1
4
[
2
[(
l
(1)
i
)2 (
dAB
′
i
)2
+
(
l
(1)
i
)2 (
mAA
′
i
)2
+
(
dAB
′
i
)2 (
mAA
′
i
)2]
−
[(
l
(1)
i
)4
+
(
dAB
′
i
)4
+
(
mAA
′
i
)4]] 12
,
(6.2.15)
205
6.2. PERTURBATION OF A SINGLE MASS
and, when varied with respect to mAA
′
i , yields
∂AA1i
∂mAA
′
i
=
mAA
′
i
8AA1i
[(
l
(1)
i
)2
+
(
dAB
′
i
)2
−
(
mAA
′
i
)2] 12
. (6.2.16)
Hinge B1 has area
AB1i =
1
4
[
2
[(
l
(1)
i+1
)2 (
dAB
′
i
)2
+
(
l
(1)
i+1
)2 (
mAA
′
i
)2
+
(
dAB
′
i
)2 (
mAA
′
i
)2]
−
[(
l
(1)
i+1
)4
+
(
dAB
′
i
)4
+
(
mAA
′
i
)4]] 12
,
(6.2.17)
and, when varied with respect to mAA
′
i , yields
∂AB1i
∂mAA
′
i
=
mAA
′
i
8AB1i
[(
l
(1)
i+1
)2
+
(
dAB
′
i
)2
−
(
mAA
′
i
)2] 12
. (6.2.18)
It can be shown that in the continuum time limit, the varied areas become
∂AA1
∂mAA′
=
∂AB1
∂mAA′
=
l(1)
2
m˙AA
′
[
1
8
(
l˙(1)
)2
− 1
]− 1
2
; (6.2.19)
that is, ∂A
A1
∂mAA′ and
∂AB1
∂mAA′ are identical to at least leading order.
We shall refer to the quadrilateral hinge generated by l
(0)
i as a Type II hinge
and its triangular components as hinges A2 and B2, also counterparts to triangles
A and B respectively in Figure 4.2. Hinge A2 has area
AA2i =
1
4
[
2
[(
l
(0)
i
)2 (
dAE
′
i
)2
+
(
l
(0)
i
)2 (
mEE
′
i
)2
+
(
dAE
′
i
)2 (
mEE
′
i
)2]
−
[(
l
(0)
i
)4
+
(
dAE
′
i
)4
+
(
mEE
′
i
)4]] 12
,
(6.2.20)
and, when varied with respect to mEE
′
i , yields
∂AA2i
∂mEE
′
i
=
mEE
′
i
8AA2i
[(
l
(0)
i
)2
+
(
dAE
′
i
)2
−
(
mEE
′
i
)2] 12
. (6.2.21)
206
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
Hinge B2 has area
AB2i =
1
4
[
2
[(
dAE
′
i
)2 (
l
(0)
i+1
)2
+
(
mAA
′
i
)2 (
l
(0)
i+1
)2
+
(
dAE
′
i
)2 (
mAA
′
i
)2]
−
[(
l
(0)
i+1
)4
+
(
dAE
′
i
)4
+
(
mAA
′
i
)4]] 12
,
(6.2.22)
and, when varied with respect to mAA
′
i , yields
∂AB2i
∂mAA
′
i
=
mAA
′
i+1
8AB2i
[(
l
(0)
i+1
)2
+
(
dAE
′
i
)2
−
(
mAA
′
i
)2] 12
. (6.2.23)
It can be shown that in the continuum time limit, the varied areas become
∂AA2i
∂mEE
′
i
→ ∂A
A2
∂mEE′
=
l(0)
2
m˙EE
′
1
3
(
l(1)
l(0)
)2(
l˙(1)
2
√
6
− h˙
)2
−
(
hψ˙ + 1
)2− 12+O(dt) ,
(6.2.24)
∂AB2i
∂mAA
′
i
→ ∂A
B2
∂mAA′
=
l(0)
2
m˙AA
′
1
3
(
l(1)
l(0)
)2 ( l˙(1)
2
√
6
− h˙
)2
−
(
l˙(0)
)2
−
(
l˙(1)
2
√
6
− h˙ + h
l(0)
l˙(0)
)2
+
(
m˙AA
′
)2− 12 +O(dt) .
(6.2.25)
Since the Cauchy surface is no longer composed solely of equilateral tetra-
hedra, we must use (3.1.2) to calculate the hinges’ deficit angles. Each 4-block
will contribute one dihedral angle to each of its hinges. The 4-block of the Type
I tetrahedron will contribute the same dihedral angle to all of its hinges: the
tetrahedron is equilateral, so all of its associated A1 hinges are geometrically
identical, as are all of its B1 hinges; moreover, when all edges are constrained to
be identical, each pair of A1 and B1 hinges become co-planar, as mentioned pre-
viously, and the unit normals to the two faces meeting at A1 become identical to
the unit normals to the two faces meeting at B1; thus all dihedral angles become
207
6.2. PERTURBATION OF A SINGLE MASS
identical, and this angle θ
(0)
i is given by (4.1.9) but with l˙i replaced by l˙
(1)
i . In the
continuum time limit, we have that θ
(0)
i → θ(0), and θ(0) is then given by (4.1.13)
but with l˙ replaced by l˙(1); that is, θ(0) is given by
cos θ(0) =
1 + 1
8
(
l˙(1)
)2
3− 1
8
(
l˙(1)
)2 +O(dt) . (6.2.26)
The 4-block of a Type II tetrahedron has four distinct types of hinges: a pair
of A1 and B1 hinges for each l
(1)
i edge and a pair of A2 and B2 hinges for each
l
(0)
i edge. We found that all hinges of the same type had the same dihedral angle,
even though they are not entirely identical because of the way the Regge skeleton
is triangulated. Additionally, we found that A1 hinges and B1 hinges had the
same dihedral angle to leading order in the continuum time limit. Therefore in
the continuum time limit, the 4-block of a Type II tetrahedron will contribute
three distinct dihedral angles, which we shall denote θ(1), θ(2), and θ(3) for A1,
A2, and B2 respectively. Each dihedral angle has been calculated by taking the
scalar product of the unit normals to the two faces meeting at the corresponding
hinge, that is, by the first method described at the end of Chapter 3.1.
One example of an A1 hinge is ABB′, which is shared with the Type I tetra-
hedron’s 4-block. The two faces meeting at this hinge, ABB′C ′ and ABB′E ′,
are separated by a dihedral angle of θ
(1)
i . To leading order in the continuum time
limit, this angle is given by
cos θ(1) =
[(
1
8
(
l˙(1)
)2
− 3
)
l(1) − 1
2
√
3
2
(
l˙(1)
)2
h
]
×
[(
1
8
(
l˙(1)
)2
− 3
)[(
9− 7
8
(
l˙(1)
)2)(
l(1)
)2
+ 3
((
l˙(1)
)2
− 12
)(
l(0)
)2 −√3
2
h
(
l˙(1)
)2
l(1)
]]− 1
2
+O(dt) ,
(6.2.27)
The B1 counterpart hinge is AA′B′, and as mentioned previously, we found the
two faces meeting on this hinge, AA′B′C ′ and AA′B′E ′, to be separated by the
same dihedral angle at lowest order in dt as well.
An example of an A2 hinge is AEE ′. The two faces meeting at this hinge,
208
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
ABEE ′ and ACEE ′, are separated by a dihedral angle of θ(2)i . This angle is given
by
cos θ
(2)
i =
1
2
((
l
(0)
i
)2
− 1
2
(
l
(1)
i
)2)(
mEE
′
i
)2 − h2i [hi+1 coshψi − hi − δl(1)i2√6]2((
l
(0)
i
)2
− 1
4
(
l
(1)
i
)2)(
mEE
′
i
)2 − h2i [hi+1 coshψi − hi − δl(1)i2√6]2
,
and in the continuum time limit, this expression becomes
cos θ(2) =
1
2
((
l(0)
)2 − 1
2
(
l(1)
)2) (
m˙EE
′)2 − h2 [h˙ − l˙(1)
2
√
6
]2
(
(l(0))
2 − 1
4
(l(1))
2
)
(m˙EE′)2 − h2
[
h˙ − l˙(1)
2
√
6
]2 +O(dt) . (6.2.28)
Finally, the B2 counterpart to the above A2 hinge is AA′E ′. The two faces
meeting at this hinge, AA′B′E ′ and AA′C ′E ′, are separated by a dihedral angle
of θ
(3)
i . This angle is given by
cos θ
(3)
i =
[(l(1)i+1)2 − 4(l(0)i+1)2] [18 (l˙(1)i )2 − 1
]
+ 4
[
1
12
l
(1)
i+1 l˙
(1)
i + hi+1
(
l˙
(1)
i
2
√
6
coshψi + sinhψi
)]2−1
×
2[1
6
l
(1)
i+1 l˙
(1)
i − hi+1
(
l˙
(1)
i
2
√
6
coshψi + sinhψi
)]2
−
[(
l
(1)
i+1
)2
− 2
(
l
(0)
i+1
)2] [1
8
(
l˙
(1)
i
)2
+ 1
]
209
6.2. PERTURBATION OF A SINGLE MASS
and in the continuum time limit, this expression becomes
cos θ(3) =
[[(
l(1)
)2 − 4 (l(0))2] [1
8
(
l˙(1)
)2
− 1
]
+ 4
[
1
12
l(1) +
1
2
√
6
h
]2 (
l˙(1)
)2]−1
×
[
2
[
1
6
l(1) − 1
2
√
6
h
]2 (
l˙(1)
)2
−
[(
l(1)
)2 − 2 (l(0))2] [1
8
(
l˙(1)
)2
+ 1
]+O(dt) .
(6.2.29)
The last geometric quantity we require is the variation
∂sij
∂mk
of the particle’s
path-length sij through a 4-block with respect to each strut-length mk. Between
a pair of Cauchy surfaces Σi and Σi+1, there are only two distinct types of path-
lengths: one corresponds to the path of the unperturbed masses, and we shall
denote this path-length by si; the other corresponds to the path of the perturbed
mass, and we shall denote this path-length by s′i. It can be shown that varying
si with respect to each of the relevant struts and then taking the continuum time
limit yields
∂s
∂mA
=
∂s
∂mB
=
∂s
∂mC
=
1
4
m˙AA
′
s˙0
(
1
4
hψ˙ + 1
)
+O(dt) , (6.2.30)
∂s
∂mE
=
1
4
m˙EE
′
s˙0
1
4
hψ˙ + 1
hψ˙ + 1
+O(dt) , (6.2.31)
where mX denotes the length, in the continuum time limit, of the strut attached
to lower vertex X, and where s˙0 denotes the quantity
s˙0 =
(1
4
h˙ − l˙
(1)
2
√
6
)2
−
(
1
4
hψ˙ + 1
)2 12 (6.2.32)
and is related to the continuum time limit of si, where si → s as δti → 0, through
the Taylor expansion
s ≈ s˙0 dt+O
(
dt2
)
; (6.2.33)
this expansion has no zeroth order term because as δti → 0, the separation
between Cauchy surfaces becomes infinitesimal, and therefore the particle’s path-
length from one surface to the next would become infinitesimal as well. It can
210
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
also be shown that varying s′i with respect to each of the struts and then taking
the continuum time limit yields
∂s′
∂mA
=
∂s′
∂mB
=
∂s′
∂mC
=
∂s′
∂mD
= − ι
4
m˙AA
′
. (6.2.34)
The derivation of these results has been explained at length in Appendix G.
6.2.4 Solving the Regge equations
Having determined all relevant geometric quantities, we can now substitute them
into (6.2.6) to obtain the corresponding Regge equations. As we have two distinct
types of struts, locally varying the Regge action will lead to two distinct equations,∑
i
∂Ai
∂mEE
′
k
δi =
∑
i,j
8piMi
∂sij
∂mEE
′
k
, (6.2.35)
∑
i
∂Ai
∂mAA
′
k
δi =
∑
i,j
8piMi
∂sij
∂mAA
′
k
. (6.2.36)
We can directly obtain the continuum time limit of these equations by substituting
in the continuum-time form of the geometric quantities, and this is how we shall
proceed.
We begin with the first equation. Between each pair of consecutive Cauchy
surfaces, there is only one strut with length mEE
′
k , namely the strut EE
′ in the
Type II tetrahedron’s 4-block. Thus the only relevant geometric quantities are
those involving strut EE ′ and vertex E. We begin by working out the left-hand
side of (6.2.35), starting with the quantity
∂Ai
∂mEE
′
k
. The only edges meeting at
vertex E are AE-type edges, all of which have length l
(0)
i . Thus the only hinges
meeting at strut EE ′ are hinges like AEE ′, what we have called A2 hinges above,
and for each of these hinges,
∂Ai
∂mEE
′
k
is given by (6.2.24) in the continuum time limit.
Next, we consider the corresponding deficit angle. In the 5-tetrahedra model,
three faces meet at each hinge, and hence three dihedral angles contribute to the
hinge’s deficit angle. The only dihedral angle at A2 hinges is θ(2), which is given
by (6.2.28) in the continuum time limit; thus the deficit angle is δi = 2pi − 3 θ(2).
We finally perform the summation on the left-hand side of (6.2.35). Because
∂Ai
∂mEE
′
k
δi is identical for all hinges meeting at EE
′, performing the summation is
equivalent to multiplying this term by the number of hinges at EE ′. As there
211
6.2. PERTURBATION OF A SINGLE MASS
are four edges meeting at vertex E, there can only be four hinges.
Next, we consider the right-hand side of (6.2.35). There are four tetrahedra
meeting at vertex E, and each carry an unperturbed mass M ; thus Mi = M
for all i. Since all four masses and all four tetrahedra are identical, the quantity
∂sij
∂mEE
′
k
will be identical for all i as well; when index j = k, its continuum time
form is given by (6.2.31); otherwise it is zero. Because Mi
∂sij
∂mEE
′
k
is identical for
all i, performing the summation on the right-hand side of (6.2.35) is equivalent
to multiplying M
∂sij
∂mEE
′
k
by four.
Finally we substitute everything into (6.2.35) to obtain the constraint equation
l(0)
(
2pi − 3 θ(2)) = 4piM 14hψ˙ + 1
s˙0
1
3
(
l(1)
l(0)
)2 l˙(1)2√6 − h˙
hψ˙ + 1
2 − 1

1
2
. (6.2.37)
We next consider the second Regge equation (6.2.36). Struts of length mAA
′
are connected to the four vertices labelled A, B, C, D in the Type I tetrahedron.
As we shall see, varying any of the four associated struts will lead to the same
Regge equation. We begin with the left-hand side of (6.2.36) as well. Each of the
four vertices has four tetrahedral edges attached to it: three edges are attached
to the three other vertices in the tetrahedron and have length l(1); the fourth
is attached to vertex E of a Type II tetrahedron and has length l(0). Each of
the four edges generates a pair of time-like triangular hinges, one of which is
attached to the vertex’s strut. The length-l(0) edge contributes hinges like AA′E ′
in a Type II tetrahedron’s 4-block, what we have called B2 hinges above, and
for such hinges,
∂Ai
∂mAA
′
k
is given by (6.2.25). The other three hinges correspond to
either A1 or B1 hinges, and in either case,
∂Ai
∂mAA
′
k
is given by (6.2.19). Once again,
the deficit angles at each hinge involve three dihedral angles. At a B2 hinge, the
only relevant dihedral angle is θ(3), which is given by (6.2.29). So a B2 hinge has a
deficit angle of δi = 2pi−3 θ(3). At A1 and B1 hinges, the relevant dihedral angles
are θ(0) and θ(1). In general, each dihedral angle at a hinge comes from a 4-block
meeting at the hinge: each 4-block has two faces meeting at the hinge, and the
dihedral angle contributed by the 4-block would be the dihedral angle between
these two faces. Additionally, each 4-block is generated by a tetrahedron attached
to the edge that generates the hinge. So in the case of A1 and B1 hinges, each
length-l(1) edge is always attached to the Type I tetrahedron and to two Type II
212
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
tetrahedra; the 4-block of the Type I tetrahedron contributes a single θ(0) to the
hinge’s deficit angle; the two Type II tetrahedra each contribute a θ(1) angle; so
the deficit angle of both A1 and B1 hinges is δi = 2pi−θ(0)−2 θ(1). Combining the
contributions from all four hinges, we can express the left-hand side of (6.2.36)
as
∑
i
∂Ai
∂mAA
′
k
δi =
3
2
l(1) m˙AA
′
[
1
8
(
l˙(1)
)2
− 1
]− 1
2 (
2pi − θ(0) − 2 θ(1))
+
l(0)
2
m˙AA
′
1
3
(
l(1)
l(0)
)2 ( l˙(1)
2
√
6
− h˙
)2
−
(
l˙(0)
)2
−
(
l˙(1)
2
√
6
− h˙ + h
l(0)
l˙(0)
)2
+
(
m˙AA
′
)2− 12 (2pi − 3 θ(3)) .
We now move on to the right-hand side of (6.2.36). Varying the strut-length
will affect the trajectory length sij of four neighbouring masses. One of these will
be the perturbed mass of magnitude M ′ while the other three will be masses of
magnitude M . Thus the right-hand side will be
∑
ij
8piMi
∂sij
∂mAA
′
k
= 8pi
[
M ′
∂s′
∂mA
+ 3M
∂s
∂mA
]
,
with the quantity ∂s
′
∂mA
given by (6.2.34) and ∂s
∂mA
by (6.2.30).
Finally substituting everything into (6.2.36), we obtain
4pi
[
M ′ + ι 3M
1
4
hψ˙ + 1
s˙0
]
= 3 l(1)
[
1− 1
8
(
l˙(1)
)2]− 12 (
2pi − θ(0) − 2 θ(1))
+ l(0)
( l˙(1)
2
√
6
− h˙ + h
l(0)
l˙(0)
)2
−
(
m˙AA
′
)2
− 1
3
(
l(1)
l(0)
)2 ( l˙(1)
2
√
6
− h˙
)2
−
(
l˙(0)
)2− 12 (2pi − 3 θ(3)) ,
(6.2.38)
where the quantity ι
s˙0
appearing on the left-hand side would actually be real;
213
6.2. PERTURBATION OF A SINGLE MASS
this follows because if the path-length s of the unperturbed particles is time-
like, then s would have to be imaginary, and by virtue of its relationship to s˙0
through Taylor expansion (6.2.33), s˙0 would have to be imaginary as well. We
note that both equations (6.2.37) and (6.2.38) came from the O(1) term of the
Regge equations 0 = ∂S
∂mEE′ and 0 =
∂S
∂mAA′ ; thus, the Regge equations are O(1)
to leading order, as claimed in Section 6.2.1.
These Regge equations however involve both l(0) and l(1) in a non-linear man-
ner, which makes solving for them difficult. We shall therefore linearise these and
all subsequent equations by performing a perturbative expansion up to first order
in δM = M ′ −M . Under this expansion, we must have that
l(0) ≈ l + δl(0),
l˙(0) ≈ l˙ + δl˙(0),
l(1) ≈ l + δl(1),
l˙(1) ≈ l˙ + δl˙(1),
ψ ≈ δψ,
ψ˙ ≈ δψ˙,
θ(i) ≈ θ + δθi for i = 0, 1, 2, 3,
as the zeroth order terms must match the corresponding quantities for the un-
perturbed model. The zeroth order angle θ is given by relation (4.1.13).
It can then be shown that the zeroth order terms for both Regge equations
yield
l =
4piM
2pi − 3 θ
[
1− 1
8
l˙2
] 1
2
, (6.2.39)
which is equivalent to the unperturbed Regge equation (6.1.8) for the 5-tetrahedra
model with the masses at the tetrahedral centres: for the 5-tetrahedra model, we
would have Np = 5, N = 10, and n = 3; and since the masses are at the
tetrahedral centres, we would have v2 = 0, because recall that v is the ratio
between |vi|, the distance of a mass to its tetrahedron’s centre as given by (6.1.3),
and li, the tetrahedral edge-length. By using (4.1.15) to substitute for l˙, we can
parametrise l entirely in terms of θ, yielding
l =
4
√
2piM
2pi − 3 θ tan
(
1
2
θ
)
. (6.2.40)
214
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
An expression for δψ˙ can be deduced by taking the perturbative expansion of
ψ˙ as given by (6.2.14); we thus obtain
δψ˙ =
√
6
l2 l˙
[
l
(
δl˙(0) − δl˙(1)
)
− l˙ (δl(0) − δl(1))] . (6.2.41)
The quantities δθi can be deduced from the perturbative expansions of re-
lations (6.2.26) to (6.2.29). The zeroth order terms of these relations are all
identical to (4.1.13), as expected. The first order terms yield
δθ0 = − δl˙
(1)
4
√
2
√
3 cos θ − 1
1− cos θ (1 + cos θ) , (6.2.42)
δθ1 = − δl˙
(1)
4
√
2
√
3 cos θ − 1
1− cos θ (1 + cos θ)
+
1
8
√
2
(2pi − 3 θ)
piM
(1 + cos θ)
(
δl(0) − δl(1)) , (6.2.43)
δθ2 =
(2pi − 3 θ)
4
√
2 piM
(
1 + cos θ
1− cos θ
)
(2 cos θ − 1) (δl(0) − δl(1))
− 1
4
√
2
√
3 cos θ − 1
1− cos θ (1 + cos θ)
(
2 δl˙(0) − δl˙(1)
)
+
4√
3
piM
(2pi − 3 θ) (3 cos θ − 1) δψ˙, (6.2.44)
δθ3 = −(2pi − 3 θ)
4
√
2piM
(
1 + cos θ
1− cos θ
)
cos θ
(
δl(0) − δl(1))
− δl˙
(1)
4
√
2
√
3 cos θ − 1
1− cos θ (1 + cos θ) , (6.2.45)
where we have made use of (4.1.15) and (6.2.40) to express these as functions of
θ. We note that only δθ2 depends upon the boost parameter ψ; if we substitute
in relation (6.2.41) for δψ˙, then δθ2 becomes
δθ2 = − 1
4
√
2
[
(2pi − 3 θ)
piM
(
1 + cos θ
1− cos θ
)
cos θ
(
δl(0) − δl(1))
+
√
3 cos θ − 1
1− cos θ (1 + cos θ) δl˙
(1)
]
,
(6.2.46)
and we note that δl˙(0) has now dropped out of this expression; in fact, now none
of the angle perturbations depends on δl˙(0).
215
6.2. PERTURBATION OF A SINGLE MASS
From the perturbative expansion of (6.2.37), the first order term yields
δl(0) (2pi − 3 θ)− 3 l δθ2 = − piM√
1− 1
8
l˙2
1
2
l˙
[
l˙
l
(
δl(0) − δl(1))+ δl˙(1)] , (6.2.47)
where we have substituted for δψ˙ using (6.2.41), and from the perturbative ex-
pansion of (6.2.38), the first order term yields
4piδM
√
1− 1
8
l˙2 =
(
δl(0) + 3 δl(1)
)
(2pi − 3 θ)− 3 l (δθ0 + 2δθ1 + δθ3)
+
(2pi − 3 θ)
1− 1
8
l˙2
[
1
8
l˙2
(
δl(0) − δl(1))+ 1
2
l l˙ δl˙(1)
]
.
(6.2.48)
We note that none of these relations depends on δl˙(0) either.
In the unperturbed model, we used the dihedral angle θ to parametrise l˙
through relation (4.1.15). We shall do something similar here and parametrise
δl˙(1) with respect to one of the angle perturbations. It is easiest to do this with
relation (6.2.42), which then yields
δl˙(1) = −4
√
2
√
1− cos θ
3 cos θ − 1
δθ0
(1 + cos θ)
. (6.2.49)
Since none of the angle perturbations depends on δl˙(0), a similar parametrisation
is not possible for δl˙(0). However, the first order terms (6.2.47) and (6.2.48) of
the two Regge equations (6.2.37) and (6.2.38) do not depend on δl˙(0) anyway, so
such a parametrisation of δl˙(0) is not necessary.
Using relations (6.2.42), (6.2.43), (6.2.46), (6.2.45) to substitute for δθ0, δθ1,
δθ2, δθ3, relation (6.2.49) to substitute for δl˙
(1), and relations (6.2.40) and (4.1.15)
to substitute for l and l˙, we can now solve (6.2.47) and (6.2.48) for δl(0) and δl(1)
and express them exclusively in terms of the parameters θ and δθ0. We find that
216
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
δl(0) =
4
√
2piM
3 (2pi − 3 θ)
[
2 sin θ
(
1 + 2 cos θ
1 + cos θ
)
+ (2pi − 3 θ)
]−1
×
[
δM
M
[
1− cos θ
1 + cos θ
] 1
2
[
6 sin θ
[
cos θ
1 + cos θ
]
+ (2pi − 3 θ)
[
3 cos θ − 1
1 + cos θ
]]
+
3 δθ0
(2pi − 3 θ)
1
1 + cos θ
[
3 sin θ + (2pi − 3 θ)
]
×
[
2 sin θ
(
1 + 2 cos θ
1 + cos θ
)
+ (2pi − 3 θ)
]]
(6.2.50)
and that
δl(1) =
4
√
2piM
3 (2pi − 3 θ)
[
2 sin θ
(
1 + 2 cos θ
1 + cos θ
)
+ (2pi − 3 θ)
]−1
×
[
δM
M
[
1− cos θ
1 + cos θ
] 1
2
[
6 sin θ
[
cos θ
1 + cos θ
]
+ (2pi − 3 θ)
]
+
3 δθ0
(2pi − 3 θ)
1
1 + cos θ
[
3 sin θ + (2pi − 3 θ)
]
×
[
2 sin θ
(
1 + 2 cos θ
1 + cos θ
)
+ (2pi − 3 θ)
]]
.
(6.2.51)
Finally, we note that our two parameters θ and δθ0 are not independent of each
other; rather both are functions of the underlying time parameter t; therefore if
one parameter evolves, so must the other. We can relate the two parameters to
t through the system of differential equations
l˙ =
d
dt
l, (6.2.52)
δl˙(1) =
d
dt
δl(1), (6.2.53)
where the left-hand side denotes the quantities given by (4.1.15) and (6.2.49),
while the right-hand side denotes the explicit differentiation of (6.2.40) and
(6.2.51) with respect to t. The first equation involves only θ; it can be solved on
its own to yield θ(t). This can then be substituted into the second equation to
give a differential equation for δθ0. We shall solve these equations numerically.
217
6.2. PERTURBATION OF A SINGLE MASS
To determine a unique solution though, we must also specify a set of initial con-
ditions: we shall require the perturbed model to obey the initial value equation
at its moment of time-symmetry; this implies a condition on θ and δθ0 which we
shall derive in the next section.
The range of the parameter t will be constrained by the requirement that all
struts remain time-like. As this constraint depends on δM , the resulting range
of t will also depend on δM .
Before leaving this section, we wish to remark on an advantage that local
variation has afforded over global variation. For each model, local variation has
yielded a pair of Regge equations that, when expanded perturbatively, gave three
distinct equations, an identical equation from their zeroth order terms and two
distinct equations from their first order terms. Had we directly varied the action
globally instead, we would only have obtained one Regge equation corresponding
to a linear combination of the two local Regge equations, and the perturbative
expansion of this global equation would give just two independent equations.
Thus local variation has provided us with an extra independent equation, allowing
us to specify one more of the five quantities, l, l˙, δl(0), δl(1), δl˙(1), that we needed
to solve for. This has allowed us to pick one unique solution out of many in the
solution space of the global Regge equations.
6.3 Initial value equation for perturbed models
We shall now determine the set of initial conditions on θ and δθ0 necessary for our
models to satisfy the initial value equation at their moment of time-symmetry;
for perturbed models, the moment of time-symmetry would correspond to the
moment when all lengths cease expanding or contracting, that is, when l˙(0) =
l˙(1) = 0. We shall use the initial value equation as given by (3.4.8). For a Cauchy
surface Σ0 of the perturbed lattice universe, the integral in (3.4.8) evaluates to∫
Σ0
ρ d3x = 5M + δM.
Therefore, the initial value equation for the perturbed model can be expressed as∑
i∈{edges}
liδi = 8pi (5M + δM) , (6.3.1)
218
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
with the summation being over the edges of Σ0 because these would be the hinges
of a 3-dimensional skeleton.
The deficit angle at an edge must be determined from the dihedral angles
between the triangular faces meeting at that edge. A Cauchy surface of the
perturbed universe has only two distinct types of hinges, the edges of length l
(0)
i
and the edges of length l
(1)
i . Each edge is connected to three faces separating
three tetrahedra, so each tetrahedron at the edge will contribute one dihedral
angle to the edge’s deficit angle. A Cauchy surface of the perturbed universe also
has only two distinct types of tetrahedra, the Type I and Type II tetrahedra. As
the Type I tetrahedron is equilateral, it will contribute the same dihedral angle to
each of its six edges, and we denote this angle by φ
(0)
i . In the Type II tetrahedron,
all edges of l
(0)
i are identical to each other, as are all edges of length l
(1)
i . Thus
this tetrahedron will contribute the same dihedral angle φ
(1)
i to each of its l
(0)
i
edges and the same dihedral angle φ
(2)
i to each of its l
(1)
i edges. To determine
the dihedral angle φ
(i)
i between any pair of faces, we shall again take the scalar
product of the unit normals to the two faces, as this product will yield cosφ
(i)
i .
Let us first consider the dihedral angles in the Type I tetrahedron. We can use
co-ordinate system (4.1.1) for this tetrahedron, dropping the time co-ordinate so
that we work in a purely 3-dimensional spatial co-ordinate system and replacing
lengths li with l
(1)
i , as this tetrahedron has edges of length l
(1)
i ; we can then use
this co-ordinate system to assign co-ordinates to any of the normal vectors. We
can calculate φ
(0)
i using the faces meeting at edge AB; these faces are ABC and
ABD, and the scalar product of their unit normals yields
cosφ
(0)
i =
1
3
. (6.3.2)
For the Type II tetrahedron, we can work with co-ordinate system (6.2.1),
again dropping the time co-ordinate to obtain a purely 3-dimensional spatial
system. We can use edge AE to calculate the dihedral angle φ
(1)
i at an edge of
length l
(0)
i ; the faces meeting at AE are ABE and ACE, and the scalar product
of their unit normals yields
cosφ
(1)
i =
1
2
(
l
(0)
i
)2
− 1
4
(
l
(1)
i
)2
(
l
(0)
i
)2
− 1
4
(
l
(1)
i
)2 . (6.3.3)
219
6.3. INITIAL VALUE EQUATION FOR PERTURBED MODELS
Similarly, we can use edge AB to calculate the dihedral angle φ
(2)
i of an l
(1)
i edge;
the faces meeting at AB are ABC and ABE, and the scalar product of their unit
normals yields
cosφ
(2)
i =
l
(1)
i
2
√
3
1√(
l
(0)
i
)2
− 1
4
(
l
(1)
i
)2 . (6.3.4)
To obtain the continuum time limit of these expressions to leading order in dt, we
simply need to drop all subscripts i. If we next take the perturbative expansion
of these continuum time expressions, such that
φ(1) ≈ φ+ δφ(1),
φ(2) ≈ φ+ δφ(2),
we find that
φ =
1
3
, (6.3.5)
and that
δφ(1) = −δφ(2) = −
√
2
3 l
(
δl(0) − δl(1)) . (6.3.6)
To calculate the deficit angles at an edge, we simply subtract the three relevant
dihedral angles from 2pi. Since only Type II tetrahedra have l(0) edges, an l(0)
edge must be connected exclusively to Type II tetrahedra, and its deficit angle
δ(0) must therefore be
δ(0) = 2pi − 3φ(1)
≈ 2pi − 3 arccos 1
3
+
√
2
l
(
δl(0) − δl(1)) . (6.3.7)
As there is only one Type I tetrahedron on the entire Cauchy surface, an l(1) edge
can only be connected to one Type I tetrahedron, and its other two tetrahedra
must be Type II; its deficit angle δ(1) must therefore be
δ(0) = 2pi − φ(0) − 2φ(2)
≈ 2pi − 3 arccos 1
3
− 2
√
2
3 l
(
δl(0) − δl(1)) . (6.3.8)
220
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
We can now substitute these deficit angles into the left-hand side of (6.3.1).
As mentioned previously, a 5-tetrahedra Cauchy surface will have a total of 10
edges; six must come from the Type I tetrahedron and must therefore be of
length l(1); the remaining four must therefore be of length l(0). Thus (6.3.1) can
be expressed as
8pi (5M + δM) = 4 l(0)
[
2pi − 3 arccos 1
3
+
√
2
l
(
δl(0) − δl(1))]
+ 6 l(1)
[
2pi − 3 arccos 1
3
− 2
√
2
3 l
(
δl(0) − δl(1))]
≈ [10 l + 2 (2 δl(0) + 3 δl(1))](2pi − 3 arccos 1
3
)
. (6.3.9)
The zeroth order term corresponds to the initial value equation for the un-
perturbed model. At the moment of time-symmetry in the unperturbed model,
we have that l˙ = 0, and from (4.1.13), it follows that θ is
θ = arccos
1
3
. (6.3.10)
Substituting this into the unperturbed model’s Regge equation (6.2.39), we de-
duce that
4piM = l
(
2pi − 3 arccos 1
3
)
, (6.3.11)
which is identical to the initial value equation for the unperturbed model. Thus
the Regge equation of the unperturbed model satisfies its initial value equation
at the moment of time-symmetry. Therefore, for the zeroth order component of
the perturbed models’ initial value equation to be satisfied, we must also require
that θ satisfy condition (6.3.10).
We can next deduce the condition on δθ0 by solving for it from the first order
term of (6.3.9). After using (6.3.10) to substitute for θ as well as (6.2.50) and
(6.2.51) to substitute for δl(0) and δl(1), we find that δθ0 must satisfy
δθ0 = 0. (6.3.12)
We note that the behaviour of δl˙(1) near time-symmetry cannot be determined
from (6.2.49). On the one hand, condition (6.3.12) suggests it may approach
221
6.3. INITIAL VALUE EQUATION FOR PERTURBED MODELS
zero, while on the other, condition (6.3.10) suggests it may diverge. As we shall
see below however, δl˙(1) is indeed well-behaved and tends towards zero as time-
symmetry is approached.
Finally, even after imposing conditions (6.3.10) and (6.3.12) at some moment
t = Tmax, we must still ensure that δl˙
(0) = δl˙(1) = 0 there; otherwise, t = Tmax
would not be a moment of time-symmetry. We have just mentioned that δl˙(1)
does tend toward zero as time-symmetry is approached. Given this information,
it can also be shown that δl˙(0) will be zero as well. First, by comparing (6.2.50)
and (6.2.51), we note that δl(0) can be expressed in terms of δl(1) to give
δl(0) = δl(1) − 8
√
2piδM
3
[
2 sin θ
(
1 + 2 cos θ
1 + cos θ
)
+ (2pi − 3 θ)
]−1 [
1− cos θ
1 + cos θ
] 3
2
.
(6.3.13)
If we differentiate this with respect to t, we obtain an expression of the form
δl˙(0) = δl˙(1) − 8
√
2piδM
3
F1 (θ) θ˙.
At the moment of time-symmetry, we have said that δl˙(1) will vanish and that
condition (6.3.10) must be satisfied; then it can be shown that F1
(
θ = arccos 1
3
)
will not vanish, and therefore δl˙(0) will vanish if and only if θ˙ vanishes. To see that
θ˙ does indeed vanish, we next differentiate l, as given by (6.2.40), with respect
to t to obtain an expression of the form
l˙ = F2 (θ) θ˙.
The left-hand side is given by (4.1.15) and is zero at time-symmetry. It can also
be shown that F2
(
θ = arccos 1
3
)
will not vanish; thus for the two sides of the
expression to equal, it follows that θ˙ must vanish at time-symmetry. Therefore,
we deduce that δl˙(0) vanishes at time-symmetry provided δl˙(1) vanishes as well.
6.4 Discussion of the models
We shall now examine the behaviour of the Regge models just obtained, compar-
ing the behaviour for various mass perturbations against each other and against
the behaviour of the unperturbed 5-tetrahedra model. We begin by examining
222
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
0.0 0.1 0.2 0.3 0.4 0.5 0.6
0.00
0.25
0.50
0.75
1.00
1.25
1.50
U(t)
dU
/dt

M
M
= 0

M
M
= 0.001

M
M
= 0.01

M
M
= 0.1
Figure 6.3: The expansion rate of the universe’s volume dUdt versus the volume U
itself. Graphs corresponding to four different values of δMM are shown, with
δM
M = 0
corresponding to the unperturbed model. Owing to δMM = 0.001 being a very small
perturbation, its red graph has completely covered the black graph of δMM = 0. The
left of the graphs have been truncated at the moment the struts turn null. In all four
models, the mass M has been fixed to be M = 15 .
the expansion rate of the universe’s volume dU
dt
against the volume U itself, a
relation which has been plotted in Figure 6.3. The volume of the unperturbed
universe is given by (4.1.16) and its expansion rate by (4.1.17). To first order in
the perturbative expansion, the volume of a perturbed universe is given by
U =
5
6
√
2
l3 +
1
2
√
2
l2
(
2 δl(0) + 3 δl(1)
)
, (6.4.1)
and the expansion rate by
dU
dt
=
5
2
√
2
l2 l˙ +
1√
2
ll˙
(
2 δl(0) + 3 δl(1)
)
+
1
2
√
2
l2
(
2 δl˙(0) + 3 δl˙(1)
)
, (6.4.2)
where δl˙(0) would be given by the explicit time-derivative of δl(0). Across all
perturbations in mass, the evolution of the universe’s volume is very stable, in-
deed closely resembling the evolution of the unperturbed universe. The effect of
increasing the perturbation δM
M
is for the universe to attain larger volumes and
faster expansion rates. All graphs have been truncated on the left at the moment
the struts turn null. However, we note that these models actually remain well-
behaved past this point all the way back to t = 0, as Figure 6.4 shows for the
δM
M
= 0.1 model.
Figure 6.5 and Figure 6.6 show the behaviour of δl˙(0) and δl˙(1) as functions
223
6.4. DISCUSSION OF THE MODELS
0.0 0.1 0.2 0.3 0.4 0.5 0.6
0.00
0.25
0.50
0.75
1.00
1.25
1.50
U(t)
dU
/dt

M
M
= 0.1
Figure 6.4: The same graph as in Figure 6.3 for the δMM = 0.1 model, but with the
graph extended all the way back to t = 0.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
t

l
(0)

M
M
= 0.001

M
M
= 0.01

M
M
= 0.1
Figure 6.5: Plot of δl˙(0)(t) against t. The graphs have been extended all the way back
to t = 0.
of time; these graphs have also been extended all the way back to t = 0. They
reveal that δl˙(0) and δl˙(1) are indeed well-behaved near t = Tmax and approach
zero as t → Tmax, which is required for a moment of time-symmetry. They also
reveal that δl˙(0) and δl˙(1) start from zero as well at t = 0. Finally, they show that
the absolute magnitudes of the graphs increase with δM
M
.
We have computed the numerical values for l˙, δl˙(0), and δl˙(1) at t = Tmax to
verify that they are indeed zero; the results are given in Table 6.1. As far as the
computer is concerned, δl˙(1) is exactly zero for all δM
M
. All other quantities are
infinitesimally small, and the difference from zero can be attributed to numerical
error. Thus, we can conclude that l˙, δl˙(0), and δl˙(1) are indeed all zero at t = Tmax,
and hence t = Tmax is a moment of time-symmetry, as required for our choice of
initial conditions to be valid.
224
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-0.07-0.06
-0.05-0.04
-0.03-0.02
-0.010.00
t

l
(1)

M
M
= 0.001

M
M
= 0.01

M
M
= 0.1
Figure 6.6: Plot of δl˙(1)(t) against t. The graphs have been extended all the way back
to t = 0.
δM
M l˙ δl˙
(0) δl˙(1)
0.001 5.16191× 10−8 −1.5465× 10−11 0
0.01 5.16191× 10−8 −1.5465× 10−10 0
0.1 5.16191× 10−8 −1.5465× 10−9 0
Table 6.1: The numerical values for l˙, δl˙(0), and δl˙(1) at t = Tmax.
Figure 6.7 and Figure 6.8 show the behaviour of the length perturbations
δl(0) and δl(1) as functions of time; these graphs have also been extended back
to t = 0. We see that the magnitudes of these graphs also increase with δM
M
,
which is consistent with the models’ attaining larger volumes in Figure 6.3 as the
perturbation is increased. We also see that δl(0) and δl(1) are always well-behaved
and non-zero. The latter fact implies that there is never a moment when the
tetrahedra have edges of zero length. Rather, there is actually a moment when
the tetrahedra are equilateral; this happens at t = 0, because θ is then zero, and
from (6.3.13), it therefore follows δl(0) and δl(1) are equal. This suggests that
our choice of α = β = 1
4
for co-moving particle trajectories in Section 6.2.2 was
appropriate. Finally, we see that the length perturbations decrease with time,
consistent with the fact that both δl˙(0) and δl˙(1) are always negative, as Figure
6.5 and Figure 6.6 show.
Before closing, we wish to comment on our choice of approach to the per-
turbed Regge models. These models were based on a specific triangulation of the
CW skeleton where all diagonals on Type II quadrilateral hinges terminated at
E ′. This triangulation had the virtue of simplicity, but there were other equally
225
6.4. DISCUSSION OF THE MODELS
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.00
0.01
0.02
0.03
0.04
0.05
t

l
(0)

M
M
= 0.001

M
M
= 0.01

M
M
= 0.1
Figure 6.7: Plot of δl(0)(t) against t. The graphs have been extended all the way back
to t = 0.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.00
0.01
0.02
0.03
0.04
0.05
t

l
(1)

M
M
= 0.001

M
M
= 0.01

M
M
= 0.1
Figure 6.8: Plot of δl(1)(t) against t. The graphs have been extended all the way back
to t = 0.
valid but inequivalent triangulations we could have worked with. In terms of
the triangulation algorithm of Chapter 4.2, we used a labelling of the Cauchy
surface’s five vertices such that the common apex to the Type II tetrahedra was
ordered last. By shifting the order of that vertex’s label, one could generate an
alternative triangulation that would certainly not be equivalent to the one used
above. For instance, if the orders of vertices A and E were swapped, then the
diagonals on all hinges attached to strut AA′ would now terminate on A′ instead;
none would terminate on A. Thus, compared to the original triangulation, some
of the diagonals on Type II quadrilateral hinges would get swapped, and the new
pair of triangular hinges that result would be geometrically different from the
original pair. It remains to be seen how many distinct triangulations there are,
226
6. REGGE MODELS OF CLOSED LATTICE UNIVERSES
although we know there must be at least five, as there are five possible orderings
of the label for the Type II tetrahedra’s common apex; given a specific ordering
for this vertex, it remains to be seen whether permutations of the other vertices’
ordering would generate inequivalent triangulations or not. More importantly,
it remains to be seen whether these alternative triangulations would lead to the
same Regge equations or to something new. For example, it is possible that in
the continuum time limit, the alternative triangulations would still reduce to the
same Regge equations at lowest order in dt. Even if we had a new set of Regge
equations, the solutions of both this new set and the original set would be equally
valid as both of them would satisfy the global Regge equation, as discussed in
Section 6.2.1. We shall leave a more thorough investigation of these alternative
models to future study.
227

CHAPTER 7
Conclusions and future
directions
This thesis has demonstrated that discrete approaches to gravity provide a vi-
able means to model universes with an inhomogeneous matter distribution. We
have shown that both the LW approximation and the CW Regge models appear
to reasonably capture the qualitative dynamics of lattice universes and that the
behaviour of redshifts in LW models do appear to reflect the underlying inho-
mogeneous matter distribution. Of course, without knowing the behaviour of
an actual lattice universe, we cannot say with certainty that our approximations
are accurate, but comparisons with dust-filled FLRW universes and with exact
results, where available, have so far been broadly favourable. Nevertheless, much
work remains before one can begin to reasonably model the actual universe with
its inhomogeneous matter distribution, and indeed, there are several immediate
directions in which our work as a whole may be extended.
So far, the CW formalism has been restricted to modelling closed universes
only, using Cauchy surfaces based on just three of the Coxeter lattices. The
immediate next step would perhaps be to expand this formalism to include all
Coxeter lattices, so that we can model both a wider range of lattice universes as
well as universes of other curvatures. We again suspect that only models obtained
by globally varying the Regge action would be viable and that triangulating
any global skeleton for the purposes of local variation would again break the
symmetries inherent in these lattices; we would consequently be forced to consider
229
7. CONCLUSIONS AND FUTURE DIRECTIONS
a much larger set of independent edges than would otherwise be the case in the
global models. It is nevertheless possible that some of these lattices admit a
symmetric triangulation, even if the symmetries are reduced, so the triangulation
of these skeletons should still be investigated. Indeed, we speculate that if there
are any lattices that could admit a symmetric triangulation, it would be the
infinite ones. Consider, for instance, the flat cubic lattice. Suppose we had
a triangulation for one lattice 4-block. We might be able to repeat this same
triangulation in the 4-block’s nearest neighbours without needing to rotate the
triangulation so that shared faces would fit properly. If this were possible, we
could repeat this step and continue triangulating nearest neighbouring 4-blocks
in a similar manner ad infinitum, thereby triangulating the entire lattice. With
such a triangulation, all 4-blocks would be identical to each other modulo a
lattice translation but not a rotation. Thus, all edges that are separated by a
lattice translation would be identical, and the number of independent edges in the
entire lattice would be reduced to some finite number dependent on the number of
distinct edges in a single 4-block. However unlike the global skeletons, the number
of independent edges in a Cauchy surface may still not reduce to just one. In
the case of closed lattices, the main difficulty is that as one keeps triangulating
nearest neighbours, the triangulation will close up on itself; when this happens,
one would have to make sure that shared faces still match up properly, which may
not necessarily be the case. However, it may still be possible that some closed
lattices admit an analogous triangulation wherein all 4-blocks are identical to
each other modulo a lattice translation only. In any case, even if models obtained
by local-variation were not viable, they may still prove useful in obtaining global
models through a similar chain-rule relationship to that of (4.3.1).
There are several additional issues that we must also address, however, before
we can begin using this expanded CW formalism. First, if we are constructing
further models of lattice universes, then for each new Coxeter-based Cauchy sur-
face, we must also determine whether there exists a region of convergence for the
placement of particles analogous to the one in Chapter 6.1 for closed tetrahedral
surfaces. Secondly, if we are modelling open FLRW universes instead, then since
all discretisations of such universes involve an infinite number of blocks, it is less
transparent which discretisation would provide the best approximation, so this
question also requires further investigation; with closed universes, one would sim-
ply expect the discretisation with the highest number of blocks to be the best,
230
7. CONCLUSIONS AND FUTURE DIRECTIONS
and this is consistent with what we saw in Chapter 5 for both parent and children
models. Finally, in Chapter 5, we remarked that when modelling infinitely ex-
panding universes such as vacuum Λ-FLRW universes, the Regge approximations
would eventually break down because their fixed resolutions would ultimately be
unable to keep up with the universe’s expansion. This resolution problem should
apply to flat and open Regge models as well, though because the associated Cox-
eter lattices are themselves infinite, there might possibly be more latitude with
these models in getting around the problem; this question also requires further
investigation.
Expanding the CW formalism to all Coxeter lattices would open up several
further avenues of investigation. In Chapter 6.1, we have remarked on the poten-
tial for the CW formalism to model lattice universes that are not based on any
Coxeter lattice. In fact, there are two approaches one could take to modelling
such universes; the first is to continue using the original Coxeter lattice for the
Cauchy surface, that is assuming the resulting model is stable; the second is to
construct a new lattice from the original Coxeter lattice such that the particles
now lie at the centres of the new lattice cells. We noted in Appendix A that some
of these new lattices are themselves Coxeter lattices, particularly closed lattices
where the new cells are centred on the vertices of the original cells. It remains to
be determined what other Coxeter lattices can generate new lattices that are also
Coxeter lattices. For example, we note that the flat cubic lattice can give rise to
several other possible flat cubic lattices, including for instance a lattice centred
on the original lattice’s vertices, a lattice centred on the original edge mid-points,
and a lattice centred on the original square centres; in these cases, the new lat-
tices would be identical to the original, just translated or rotated. If the new
lattices are indeed Coxeter lattices, then we can readily apply the expanded CW
formalism to the new models; if they are not, however, then we must expand the
CW formalism even further, though that should hopefully be easily achievable
using the knowledge gained by this point.
In Chapter 6.1, we have also remarked on one limitation of using the CW
formalism in its current form to model lattice universes; that is the fact that
all geometric information for a lattice cell has been reduced to just one single
quantity, the length of the cell’s edge. Hence, the CW formalism offers a very
coarse-grained approximation of the lattice universe. However, as we have noted
in Chapter 6.1, there are two ways by which the resolution can be improved. The
231
7. CONCLUSIONS AND FUTURE DIRECTIONS
first would be to space out the particles such that not every Coxeter cell in the
Cauchy surface receives a mass; in this way, a single lattice universe cell would
actually consist of several Coxeter cells, thereby introducing multiple length-
scales into the lattice universe cell. Such a spacing-out of masses might actually
be easier to achieve with flat or open lattices, where the relevant Coxeter lattice
would extend outwards indefinitely in all directions. For instance, in the flat
cubic lattice, one could easily place a particle at the centre of every nth cube for
some suitably large n, and each lattice universe cell would therefore consist of n3
Coxeter cubes; one would then develop the Regge model by applying the expanded
CW formalism. The second way to improve the lattice cell resolution would be
to subdivide each lattice cell in a manner analogous to Brewin’s subdivision of
parent CW tetrahedra in Chapter 4.4; the extra edges introduced into the cell
would thereby increase the number of length-scales describing its geometry.
Finally and perhaps most importantly, it would be very interesting to compare
the results obtained from the two approaches of the LW approximation and Regge
calculus to modelling lattice universes. Such a comparison serves a twofold pur-
pose. First, consistency in results between the two approaches would strengthen
our confidence in the reliability of both approaches. Secondly, such a comparison
may better elucidate the relative merits of the two approaches, better informing
us as to which approach we should follow for which situations. Therefore, once
we have developed a reliable finer-grained Regge model of the lattice universe,
the first logical step would be to compare its evolution again with that of the
LW models in Chapter 2. The second would be to determine redshifts in our
Regge model and compare them with those from our LW models, as presented in
Chapter 2. The third would be to determine and compare the optical properties
of both LW and Regge models. Lastly, we should like to further expand our lat-
tice universe models, for instance, by introducing a cosmological constant or by
perturbing multiple masses. As mentioned at the end of Chapter 2, LW models
can be extended to include a cosmological constant by using Clifton and Fer-
reira’s Λ-Schwarzschild-cells [60,78], and they can be extended to allow for mass
perturbations by using our unequal Schwarzschild-cells developed in Appendix
D. Regge models of lattice universes can be extended to include a cosmological
constant by combining the Regge actions of (3.2.7) and (3.2.9), and they can be
extended to allow for mass perturbations by following an approach similar to that
in Chapter 6.2.
232
7. CONCLUSIONS AND FUTURE DIRECTIONS
One of the main interests in investigating these lattice universes, particularly
their perturbed variants, is that they may point the way towards building mod-
els of the actual universe that more accurately reflect its inhomogeneous matter
distribution; it is hoped that such models would elucidate what types of obser-
vational effects the actual matter distribution might give rise to that could not
be determined from models based on the perfectly homogeneous and isotropic
FLRW space–times. Therefore an even longer term goal would be to perturb a
lattice universe model in such a way as to provide a reasonable approximation to
the actual universe’s matter distribution. Though given that such a model would
probably involve a large number of independent dynamical variables, it would
probably have to be implemented computationally.
Indeed, one of the most important questions driving theoretical cosmologi-
cal research today is the origin and nature of the observed acceleration of the
universe’s expansion. As elaborated in Chapter 1, this is presently a topic of
intense debate. The currently most-favoured explanation is to introduce some
form of exotic matter into the standard models. Some of the postulated matter
would be completely different in nature to any other matter currently known to
exist, possibly having, for instance, negative pressure, and its existence would
have important implications for other branches of theoretical physics, such as
the standard model of particle physics and possibly even quantum field theory.
Yet attempts to directly observe any exotic matter and determine its origin and
exact nature have so far proven unsuccessful. Some cosmologists have instead
proposed that the acceleration is an apparent effect arising from the actual inho-
mogeneous structure of the universe, an effect that is not be deducible from the
averaged FLRW-based models, even if perturbations are introduced. Thus it is
hoped that our discrete models may help shed light on this question of the uni-
verse’s observed acceleration. We have already constructed models of Λ-FLRW
universes in Chapter 5, and by comparing the redshifts and optical properties of
such models against those obtained from modelling the actual universe’s inhomo-
geneous matter distribution, we could perhaps derive certain observational tests
that could determine which of the two scenarios is correct, that there is indeed
exotic matter in the universe or that the universe’s acceleration is an apparent
effect and general relativity alone is sufficient to explain it. Answering such a
fundamental question of Nature may perhaps be the most important motivation
for applying discrete gravitational approaches to cosmology.
233

APPENDIX A
Regular lattices in 3-spaces of
constant curvature
In this appendix, we shall list all possible lattices that cover 3-spaces of constant
curvature with a single regular polyhedral cell. The cell is tiled to completely
cover the 3-space without any gaps or overlaps. This tessellation problem has been
thoroughly studied by Coxeter [120]. Clifton and Ferreira [60] have succinctly
summarised Coxeter’s results relevant to our discussion, and we have presented
their summary in Table A.1.
Elementary
cell shape
Number of cells
at a lattice edge
Background
curvature
Total cells
in lattice
tetrahedron 3 + 5
cube 3 + 8
tetrahedron 4 + 16
octahedron 3 + 24
dodecahedron 3 + 120
tetrahedron 5 + 600
cube 4 0 ∞
cube 5 - ∞
dodecahedron 4 - ∞
dodecahedron 5 - ∞
icosahedron 3 - ∞
Table A.1: All possible lattices obtained by tessellating 3-spaces of constant curvature
with a regular polyhedron. We note that the second column, which indicates how many
cells meet at any lattice edge, effectively determines the lattice’s structure.
235
A. REGULAR LATTICES IN 3-SPACES OF CONSTANT CURVATURE
The Coxeter lattices are the only possible lattices that use a single regular
polytope as its elementary cell. However, if we allow for elementary cells that are
not regular polytopes, then further regular lattices are possible. For instance, one
can obtain a new lattice from the closed 600-tetrahedra lattice by using the centres
of the original lattice’s triangles as the new cell centres; one would then partition
out new cells by erecting new boundaries in between pairs of nearest-neighbouring
triangular centres. Each tetrahedra has four triangles, but each triangle is shared
between two tetrahedra, so the 600-tetrahedra lattice has a total of 1200 triangles;
thus this new lattice has a total of 1200 cells. Clearly, this cannot be a Coxeter
lattice, and its cell therefore cannot be a regular polytope; yet this new lattice
is regular because the triangles are distributed in a regular manner. For an
analogous situation, consider a lattice tessellating flat 2-dimensional space with
equilateral triangles, as shown in Figure A.1, and a new lattice with cells centred
on the mid-points of the original edges. This new lattice is still regular, but its
cells do not correspond to regular polytopes: their internal angles are not equal,
even though their edges are; thus the cells are actually rhombi. Since the internal
angles are not equal, the vertices themselves are not identical either, with three
dual cells meeting at vertices like the one marked A and six at vertices like the one
marked B. Therefore, such cells would not have as high a rotational symmetry as
its regular counterpart, the square; the rhombus has only an order 2 rotational
symmetry, while the square has order 4. For such reasons, the Coxeter lattices
are the lattices with the highest symmetry.
For the closed lattices, if one constructs new lattices using cells centred on
the original lattice vertices, it appears these form Coxeter lattices as well. We
shall refer to these new lattices as dual lattices. The 5-tetrahedra lattice is dual
to itself, as is the 24-octahedra lattice. The 8-cube and 16-tetrahedra lattices
are duals of each other, as are the 120-dodecahedra and 600-tetrahedra lattices.
However, new lattices using cells centred on the mid-points of edges or the centres
of faces are not Coxeter lattices in general.
236
A. REGULAR LATTICES IN 3-SPACES OF CONSTANT CURVATURE
C
B
A
D
Figure A.1: A lattice of equilateral triangles tessellating flat 2-dimensional space
with a derived lattice centred on the mid-points of the triangular edges; the triangular
lattice has been drawn in solid lines and the derived lattice in dashed lines. The new
lattice cells do not correspond to regular polytopes but are instead rhombi; although
the edge-lengths are equal, the interior angles are not. For the new cell marked ABCD,
the angles at the centres of triangles, marked A and C, are 120°, while the angles at
vertices of triangles, marked B and D, are 60°. These angles imply that three dual cells
would meet at A and C, and six at B and D; thus the cell vertices are not identical
either.
237

APPENDIX B
Radial velocities in Regge
Schwarzschild space–time
In this appendix, we shall present our numerical results supporting (2.5.9) to be
the 4-velocity tangent to radial time-like geodesics in Regge Schwarzschild space–
time. These geodesics include, most importantly, those of test particles co-moving
with a Schwarzschild-cell boundary. We began by determining numerically the
escape velocity for a test particle. We propagated a test particle radially outwards
from a series of initial radii rinit and with a series of initial velocities ρ˙init. We
started ρ˙init from 0 and increased it until the maximum radius rmax attained by
the particle was very large. For each rinit, we then plotted R/rmax against its
corresponding ρ˙init, where R is the Schwarzschild radius. Both rinit and rmax
refer to the lower Schwarzschild label of the block in which the particle is found.
Our plots are shown in Figure B.1. Each graph is very well-fitted by a quadratic
curve of the form
R
rmax
= −A ρ˙2init +B.
Moreover, the co-efficients A and B always satisfy the relations A + B = 1 and
B = R/rinit.
1 From these relations, we can therefore infer a relation for ρ˙ corre-
sponding to that given in (2.5.9), and the τ˙ component follows from normalisation.
We have just provided numerical support for (2.5.9) for test particles following
geodesics where rmax ≥ 0, corresponding to closed orbits or orbits at the escape
1We have also looked at graphs for different R and rinit, not shown, and they also conform
to this pattern.
239
B. RADIAL VELOCITIES IN REGGE SCHWARZSCHILD SPACE–TIME
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
R
/r
m
a
x
ρ˙init
R/rmax = −0.8750(ρ˙init)2 + 0.1250
RMS of residuals = 2.51× 10−7
R/rmax = −0.9688(ρ˙init)2 + 0.0313
RMS of residuals = 5.04× 10−8
R/rmax = −0.9922(ρ˙init)2 + 0.0078
RMS of residuals = 8.13× 10−9
R/rmax = −0.9981(ρ˙init)2 + 0.0020
RMS of residuals = 1.94× 10−8
rinit/R = 8 rinit/R = 32 rinit/R = 128 rinit/R = 512
Figure B.1: A plot of R/rmax versus ρ˙init for various rinit. Each graph has 100 data
points. A quadratic regression has been performed on each graph, and the regression
equations are ordered from rinit = 8R to 512R. The regressions were performed
without a linear term; regression was also attempted with a linear term present, but
it was found to be many orders of magnitude smaller than the other two terms. The
simulation’s block parameters were R = 31, ∆t = 10∆r, ∆r = R/105, and ∆φ = 2pi
3×107 .
velocity. At this point, we conjectured that for open orbits, (2.5.9) would still
apply but with rmax < 0. To test this, we propagated a test particle outwards with
an initial velocity given by (2.5.9) but for rmax < 0, and we examined how the
velocity evolved with r. Let us denote the velocity of our simulated particle by ˙˜ρ.
Along the particle’s trajectory, we compared ˙˜ρ 2 against (uρ)2 for the same radius,
where uρ is given by the radial component of (2.5.9). Our comparison is presented
graphically in Figure B.2. In all cases, the graphs’ gradients were effectively unity
and the constants effectively zero, thus indicating that the particle’s velocity does
indeed obey (2.5.9). In Table B.1, we provide a partial list of the values of ˙˜ρ 2
and (uρ)2 as well as their percentage difference. The percentage difference is
very small in all rows, although there appears to be an increasing trend with
increasing r. We find that our conjecture is supported by numerical results, and
hence (2.5.9) appears valid for all types of radial time-like geodesics.2
2Again, we have also simulated the cases of rmax = −R,−21R,−41R,−61R, and −81R,
240
B. RADIAL VELOCITIES IN REGGE SCHWARZSCHILD SPACE–TIME
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.2 0.4 0.6 0.8 1.0 1.2
uρ = ρ˙ + 1.0× 10−11
RMS of residuals = 2.86× 10−6
uρ = ρ˙ + 1.0× 10−11
RMS of residuals = 2.77× 10−6
uρ = ρ˙ + 1.0× 10−11
RMS of residuals = 2.54× 10−6
uρ = ρ˙ + 1.0× 10−11
RMS of residuals = 2.49× 10−6
uρ = ρ˙ + 1.0× 10−11
RMS of residuals = 2.56× 10−6
(u
ρ
)2
(ρ˙)2
R/rmax = −1/11
R/rmax = −1/31
R/rmax = −1/51
R/rmax = −1/71
R/rmax = −1/91
Figure B.2: A plot of (uρ)2 against ˙˜ρ 2 for various rmax < 0, where u
ρ is given by
(2.5.9). Each plot consists of velocities computed at 100 different radii along the course
of the test particle’s trajectory. A linear regression was performed on each graph, and
the corresponding equations are ordered from R/rmax = −1/11 to −1/91. The graphs
of the regressions completely overlap each other. The gradients only begin deviating
from unity at the 10−5 order of magnitude. The simulation’s block parameters were
rinit = 2R, R = 21, ∆t = ∆r = R/10
5, and ∆φ = 2pi
3×107 .
241
B. RADIAL VELOCITIES IN REGGE SCHWARZSCHILD SPACE–TIME
r/R ˙˜ρ 2 (uρ)2 % difference
2 1.0219794649 1.02197802197802 0.00014118894047312
102 0.021001226724 0.0209988015595972 0.0115477273528382
202 0.016021230625 0.0160188121184031 0.0150956356191602
302 0.014350362849 0.0143477802197414 0.0179969613713785
402 0.013512667536 0.0135101783683129 0.0184209940820187
502 0.013009455481 0.0130069512260618 0.0192494985037111
602 0.012673806084 0.0126711894819952 0.0206457475160902
702 0.012433811049 0.0124312192608481 0.0208446802168433
802 0.012253604416 0.0122511667237755 0.0198936748870944
...
...
...
...
9202 0.011101572496 0.0110988891274903 0.0241710668530401
9302 0.011100308164 0.0110977080370158 0.0234239171181774
9402 0.011099254609 0.0110965512181986 0.02435650768163
9502 0.011097990409 0.0110954196820738 0.023163895728025
9602 0.011096936964 0.0110943114060724 0.0236602040377857
9702 0.011095883569 0.0110932263900415 0.0239474300718204
9802 0.011094830224 0.0110921626114406 0.0240437438476478
9902 0.011093776929 0.011091121081329 0.0239399772323361
Table B.1: A list of ˙˜ρ 2 and (uρ)2 at various radii, and the percentage difference
between them. This data is for R/rmax = −91.
We have just found that the initial velocity ρ˙init of a simulated particle must
equal at least that of (2.5.9) in order for the particle to ‘escape’. We next exam-
ined the long-term behaviour of the particle’s velocity ˙˜ρ as the particle propagated
outwards from an initial velocity equalling at least the escape velocity. We found
that ˙˜ρ would not stay equal to uρ as given by (2.5.9) but would instead decrease
at a slower rate as a function of the particle’s radius. For example if we started
a particle at the escape velocity at rinit = 2R and propagated it outwards, then
by the time it reached r = 3 × 104R, there was a very significant discrepancy
between ˙˜ρ and uρ, as shown in Figure B.3, with ˙˜ρ being consistently larger than
not shown here, and they all display the same behaviour as in Figure B.2. The percentage
difference seen between ˙˜ρ 2 and (uρ)2 is also small but slowly increasing in all rmax that we
looked at.
242
B. RADIAL VELOCITIES IN REGGE SCHWARZSCHILD SPACE–TIME
0
20
40
60
80
100
120
140
160
180
0 5000 10000 15000 20000 25000 30000
1/
ρ˙
r/R
∆r = R/101
∆r = R/102
∆r = R/103
∆r = R/104
∆r = R/105
uρ
Figure B.3: A plot of 1/ρ˙ versus r/R comparing simulated particle velocities with
uρ. For the simulated particles’ graphs ρ˙ = ˙˜ρ, while for the uρ graph, ρ˙ = uρ. The
simulation was run for various sizes ∆r of the Regge block. For all simulations, the
particle was propagated from r = 2R to 3 × 104R. The simulation’s other block
parameters were R = 1, ∆t = ∆r, and ∆φ = 2pi
3×107 .
uρ. This discrepancy reduced if we improved the radial resolution ∆r of our
Regge blocks, but it was still present even for resolutions as high as ∆r = R/105.
If instead, using the same method as previously, we numerically determined the
escape velocity for a particle starting at rinit = 3× 104R, we again obtained the
same quadratic behaviour between R/rmax and ˙˜ρ as that in Figure B.1, thus in-
dicating the escape velocity should still correspond to (2.5.9). Indeed, this result
was obtained even when the radial resolution was as low as ∆r = R/10, as shown
in Figure B.4, even though there is a huge discrepancy at this resolution between
˙˜ρ and uρ in Figure B.3. Therefore even if a particle began at the correct escape
velocity for its initial radius, it would gradually travel faster than the correct
escape velocity for its subsequent radii as it propagated along its trajectory.
The discrepancy between ˙˜ρ and uρ can be understood by considering the
manner by which particles are deflected in the Regge model. Suppose a particle
enters a Regge block of radius ri by the −ρi face and travels at the escape velocity
as given by (2.5.9). If the particle then enters the next block by the −ρi+1 face
as well, it will have undergone no deflection and hence continue having the same
velocity as at ri. However, (2.5.9) indicates that the escape velocity at ri+1 should
be smaller than that at ri, and hence we see the beginning of a discrepancy. If
243
B. RADIAL VELOCITIES IN REGGE SCHWARZSCHILD SPACE–TIME
0
0.5× 10−5
1.0× 10−5
1.5× 10−5
2.0× 10−5
2.5× 10−5
3.0× 10−5
3.5× 10−5
0 0.001 0.002 0.003 0.004 0.005 0.006
R
/r
m
a
x
ρ˙init
R/rmax = −0.9999765 (ρ˙init)2 + 3.333× 10−5
RMS of residuals = 8.367× 10−11
Figure B.4: A plot of R/rmax versus ρ˙init for rinit = 3×104R. A quadratic regression
has been performed and the corresponding equation shown. The simulation’s block
parameters were R = 1, ∆t = 10∆r, ∆r = R/10, and ∆φ = 2pi
3×107 .
−1.1
−0.9
−0.7
−0.5
−0.3
−0.1
0 10 20 30 40 50 60 70 80 90 100
˙˜ ρ
r/R
simulated velocity
uρ
Figure B.5: A plot of ˙˜ρ versus r/R comparing a simulated particle velocity with
uρ. The simulation began at rinit = 3 × 104R, but only data points from r = 2R to
100R are shown. The simulated velocity is always greater than or equal to uρ. The
simulation’s block parameters were R = 1, ∆t = ∆r = R, and ∆φ = 2pi
3×107 .
244
B. RADIAL VELOCITIES IN REGGE SCHWARZSCHILD SPACE–TIME
the particle passes through a sizeable number of ρ faces in succession, it would
accumulate a significant discrepancy such that the particle’s velocity would be
noticeably larger than the escape velocity for its radius. This explains why the
graphs for simulated particles in Figure B.3 are always smaller than the graph
for uρ. Only when the particle crosses a τ surface does it undergo a change in
velocity, but this change may not be enough to overcome the discrepancy already
accumulated. By a similar argument, we expect the radial velocity of a radially
in-going particle to decrease less quickly than uρ in (2.5.9), because if the particle
crosses a +ρ face with the correct escape velocity, it will cross the neighbouring
+ρ face with the same unchanged velocity, which would be less negative than uρ
for the same radius; and indeed, this is what we see in Figure B.5. By improving
the resolution of our Regge model, we slow the growth rate of the discrepancy,
as we saw in Figure B.3, because our model becomes a better approximation to
the underlying continuum Schwarzschild space–time and its geodesics therefore
become more similar to those in the continuum space–time.
Thus combined with our analytic arguments provided at the end of Chapter
2.5, we conclude that (2.5.9) gives the correct velocity as a function of r for
particles following radial time-like geodesics.
245

APPENDIX C
The Schwarzschild Regge
block’s radial length
In our simulations of the flat and open universes, our Regge blocks’ radial lengths
were not constant but were increased as the blocks were further away from the
Schwarschild-cell centre. This decision was motivated by the fact that the un-
derlying space–time being approximated becomes increasingly flat. We therefore
attempted to maintain higher resolution while the curvature was high but then
decrease the resolution with minimal impact on accuracy as the curvature de-
creased. This technique allowed us to perform larger-scale simulations within
more attainable computation times. We now describe our method for specifying
the block’s radial length.
In Schwarzschild space–time, the radial distance between radius ri > 2m and
some arbitrary r > ri is given by
d(r) =
{
x
√
1− 2m
x
− 2m ln
[√
x
2m
−
√
x
2m
− 1
]}r
x= ri
. (C.0.1)
When r = ri+1, this is identical to (2.5.2). Distances along ‘radial edges’ of Regge
blocks approximate radial distances in Schwarzschild space–time by linearly in-
terpolating d(r) between ri and ri+1, as shown in Figure C.1. Since d(r) is convex,
the interpolation will always underestimate the true distance, and the error ε of
the approximation is given by ε = max [d(r)− lin(r)], where lin(r) denotes the
interpolating function.
247
C. THE SCHWARZSCHILD REGGE BLOCK’S RADIAL LENGTH
ra
d
ia
l
d
is
ta
n
ce
r
Schwarzschild space-time
Regge block interpolation
Figure C.1: The Regge block interpolation intersects the Schwarzschild graph at r =
ri and r = ri+1, and it interpolates the radial distance for any r in between. Since the
Schwarzschild distance function is convex, the interpolation will always underestimate
the distance.
Our goal is as follows. Given some error-tolerance εtol, we want our blocks to
be as long as possible in the radial direction while still satisfying ε ≤ εtol. If we
enter a new block from its left or right face, then we would need to re-calculate
the block’s radial length h = ri+1− ri in accordance with our goal. It will always
be the case that we know one of ri or ri+1 depending on which face we entered
by, and we would need to deduce the other quantity. However our computation
must give consistent results for h regardless of which quantity we used. In our
particular implementation, we have also chosen to impose the constraint that h
must be some integer units of a minimal interval ∆r0, a parameter which we can
freely specify.
The most natural approach would be to solve d′(r) = lin′(r), which maximises
d(r) − lin(r), to obtain r = rmax, the point at which the error is greatest. Also
through its dependence on lin(r), rmax would be a function of h. We would then
set ε = εtol and solve d(rmax) − lin(rmax) = εtol for h. However, this second
equation can only be solved numerically, and from a programming point of view,
it was easier to implement a different approach instead.
We note that for r > ri, the functions d(r) and lin(r) are bounded from above
by
d+(r) =
√
ri
ri − 1(r − ri) + d(ri), (C.0.2)
248
C. THE SCHWARZSCHILD REGGE BLOCK’S RADIAL LENGTH
since this is just the equation of the tangent to d(r) at ri. And also, for r > ri,
the two functions are bounded from below by
d−(r) = (r − ri) + d(ri); (C.0.3)
this follows because the gradient of d(r) goes asymptotically to unity from above
as r → ∞, which means d(r) will always increase more quickly than d−(r).
Therefore, after the two functions have intercepted at r = ri, d(r) will always
be strictly greater than d−(r). The function lin(r) also intercepts both d(r)
and d−(r) at r = ri. But it intercepts d(r) again at ri+1 > ri, a point where
d(r) > d−(r). Because lin(r) is a linear function, this interception at ri+1 implies
that lin(r) must have a greater gradient than d−(r), and therefore lin(r) must
also be strictly greater than d−(r) in the region r > ri.
Thus we can bound our error by ε ≤ max [d+(r)− d−(r)] = d+(ri+1) −
d−(ri+1). We shall therefore require that
εtol ≥ d+(ri+1)− d−(ri+1),
and solve for ri+1 from this equation; this yields the solution
h =
εtol√
ri
ri−1 − 1
.
To get h as a function of ri+1 instead, we set ri to be ri = ri+1 − h and solve
for h in the preceding equation; this yields
h =
εtol
2(2εtol − 1)
[
2ri+1 + εtol − 2 +
√
4ri+1(ri+1 − 1− εtol) + εtol(4 + εtol)
]
.
As mentioned above, we want h to satisfy h = n∆r0 for some integer n.
Therefore our actual h is obtained by taking n to be
n =
 εtol
∆r0
(√
ri
ri−1 − 1
)
 (C.0.4)
249
C. THE SCHWARZSCHILD REGGE BLOCK’S RADIAL LENGTH
or
n =
⌊
εtol
2∆r0(2εtol − 1)
[
2ri+1 + εtol − 2
+
√
4ri+1(ri+1 − 1− εtol) + εtol(4 + εtol)
] ⌋ (C.0.5)
as appropriate, where bxc gives the greatest integer less than or equal to x, and
then substituting this n into h = n∆r0.
Finally, we want to ensure that our algorithm would give the same h regardless
of whether it used ri or ri+1. To satisfy this requirement, whenever the program
calculated ri+1 from ri, it would check to see if it could recover the same h using
the new value for ri+1. If not, then it would decrement h by one unit of ∆r0
and check again, and it would continue decrementing until obtaining agreement.
The program can only decrement if the error is to remain within tolerance. We
had also required that h be at least ∆r0; thus the decrementing would stop if h
reached this length.
250
APPENDIX D
Boundary conditions for
Schwarzschild-cells of unequal
masses
An observer sitting at the interface of two cell boundaries must observe the same
physics regardless of which co-ordinate system the observer uses. In particular,
the following two conditions must be satisfied:
1) the observer must measure the same proper time regardless of which cell’s
metric is used;
2) and the observer must measure the same spatial distances locally along the
boundary regardless of which cell’s metric is used.
However if the cells in the lattice are no longer identical, we would no longer
have the same lattice symmetries as before. In particular, the exact location of
the boundary between two unequal cells becomes less transparent, as it is no
longer equidistant to the two cell centres. Recall that in the perfectly symmetric
lattice, a test particle sitting on the boundary between two cells would by sym-
metry always remain at the boundary and yet fall simultaneously towards both
cell centres. Although the symmetry is no longer present when the two cells are
no longer identical, we shall still assume that test particles at the boundary fall
simultaneously to both centres. This condition defines the location of the cell
boundary for us: it is effectively defined to be where the two cells’ gravitational
251
D. SCHWARZSCHILD-CELLS OF UNEQUAL MASSES
influences are equal. On one side of the boundary, the gravitational influence
of one mass dominates, and we approximate the space–time by a Schwarzschild
space–time centred on that mass. On the other side, the other mass dominates,
and we approximate the space–time with a Schwarzschild space–time centred
on that mass. Once again, we can understand this simultaneous free-fall of the
boundary towards the two centres as actually the motion of the two masses to-
wards each other under their mutual attraction. This mutual attraction gives
rise to the expansion and contraction of the lattice itself, which manifests as the
expansion and contraction of the cell boundary. Given this definition and our
two boundary conditions, we can now derive a set of constraints that Eb and the
cell radius rb must satisfy.
Local spatial distances along the boundary are given by
rb dΩ. (D.0.1)
Since by assumption, particles co-moving with the boundary follow radial geo-
desics, then this distance evolves as
drb
dτ
dΩ. (D.0.2)
By condition (2), we require that
rb1 dΩ1 = rb2 dΩ2, (D.0.3)
where the numerical subscripts refer to cells 1 and 2; combined with condition
(1), we additionally require that
drb1
dτ
dΩ1 =
drb2
dτ
dΩ2, (D.0.4)
since dτ1 = dτ2 = dτ . Combining these two conditions, we obtain
1
rb1
drb1
dτ
=
1
rb2
drb2
dτ
, (D.0.5)
252
D. SCHWARZSCHILD-CELLS OF UNEQUAL MASSES
and using (2.1.2), we can express this equivalently as
1
rb1
√
Eb1 − 1 +
2m1
rb1
=
1
rb2
√
Eb2 − 1 +
2m2
rb2
. (D.0.6)
By condition (1), we require that ∆τ1 = ∆τ2. Recall that the proper time
of a freely falling particle is given by equations (2.2.5) to (2.2.8). However, from
the form of equations (2.2.5) to (2.2.7), it is clear that we cannot satisfy this
condition unless both cells are of the same type, that is, both cells are open, flat,
or closed. This constrains the cases we need to consider to just three, and we
shall consider each in turn.
If Eb = 1 for both cells, then (2.2.6) and (2.2.8) imply that
∆τ1 = ∆τ2
2
3
1√
2m1
r
3
2
b1
− τ0 = 2
3
1√
2m2
r
3
2
b2
− τ0,
where τ0 is a constant of integration. From this, we obtain the relation
rb2 =
(
m2
m1
) 1
3
rb1 . (D.0.7)
It can be checked that this relation also satisfies (D.0.6) and hence condition (2)
as well.
If Eb < 1 for both cells, then (2.2.5) and (2.2.8) imply that
2m1
(1− Eb1)
3
2
[√
1− Eb1
2m1
rb1
√
1− 1− Eb1
2m1
rb1 + cos
−1
√
1− Eb1
2m1
rb1
]
− τ0
=
2m2
(1− Eb2)
3
2
[√
1− Eb2
2m2
rb2
√
1− 1− Eb2
2m2
rb2 + cos
−1
√
1− Eb2
2m2
rb2
]
− τ0.
The two sides of this equation can be made equal if we simultaneously equated
1− Eb1
2m1
rb1 =
1− Eb2
2m2
rb2
and
2m1
(1− Eb1)
3
2
=
2m2
(1− Eb2)
3
2
.
253
D. SCHWARZSCHILD-CELLS OF UNEQUAL MASSES
The second equation leads to the constraint
(Eb2 − 1) =
(
m2
m1
) 2
3
(Eb1 − 1) , (D.0.8)
and if we substitute this into the first equation, we recover (D.0.7). Again, it can
be checked that (D.0.7) and (D.0.8) combined satisfy (D.0.6) and hence condition
(2) as well.
Finally if Eb > 1, then (2.2.7) and (2.2.8) imply that
2m1
(Eb1 − 1)
3
2
[√
Eb1 − 1
2m1
rb1
√
1 +
Eb1 − 1
2m1
rb1 − sinh−1
√
Eb1 − 1
2m1
rb1
]
− τ0
=
2m2
(Eb2 − 1)
3
2
[√
Eb2 − 1
2m2
rb2
√
1 +
Eb2 − 1
2m2
rb2 − sinh−1
√
Eb2 − 1
2m2
rb2
]
− τ0.
As with the Eb < 1 case, the two sides of this equation can be made equal if we
simultaneously equated
Eb1 − 1
2m1
rb1 =
Eb2 − 1
2m2
rb2
and
2m1
(Eb1 − 1)
3
2
=
2m2
(Eb2 − 1)
3
2
.
This clearly leads to the same constraints as in the Eb < 1 case.
We therefore find that for all cases, two neighbouring cells must satisfy con-
straints (D.0.7) and (D.0.8) at the boundary.1 And when m1 = m2, we recover
Eb1 = Eb2 and rb1 = rb2 .
1The author wishes to acknowledge that constraints (D.0.7) and (D.0.8) were actually first
derived by Ruth Williams in an unpublished calculation, where she derived the boundary con-
ditions using the Israeli junction conditions instead. Equating the two metrics on the boundary
gave the condition
drb1
dτ1
=
drb2
dτ2
, and equating Trκ, where κ is the extrinsic curvature, gave
condition (D.0.5).
254
APPENDIX E
The average radius of a CW
Cauchy surface
In this appendix, we shall present the computation of the numerical factors in
(5.1.13) for the average radius of a CW Cauchy surface. To compute these factors,
we must first determine the radius to any arbitrary point v in the Cauchy surface.
In terms of the position of vertices A, B, C, D of our representative tetrahedron,
the position of v in the tetrahedron can be expressed as
v = αA + βB + γC + δD, (E.0.1)
where A, B, C, D are the position vectors of vertices A, B, C, D relative to
the embedding 3-sphere’s centre, and where constants α, β, γ, δ satisfy 0 ≤
α, β, γ, δ ≤ 1 and α + β + γ + δ = 1. Working in the E4 co-ordinate system of
(5.1.1), we can take vertices A, B, C, D to have co-ordinates given by vertices 1
to 4 respectively in Table 5.1. Then the radius of v is given by
Rv(ti) =
R(ti)√
1− 2 cos 2pi
n
[(
1− 2 cos 2pi
n
)
(α2 + β2 + γ2 + δ2)
+ 2 cos
2pi
n
(αβ + αγ + αδ + βγ + βδ + γδ)
] 1
2
,
(E.0.2)
where R(ti) is the radius to any of the vertices and is determined from the edge-
lengths l(ti) using (5.1.6).
255
E. THE AVERAGE RADIUS OF A CW CAUCHY SURFACE
To get the average radius, we therefore need to compute the multiple integral
R¯(ti) =
1
N
∫ 1
α=0
∫ 1−α
β=0
∫ 1−α−β
γ=0
Rv dα dβ dγ, (E.0.3)
where the normalisation N is given by
N =
∫ 1
α=0
∫ 1−α
β=0
∫ 1−α−β
γ=0
dα dβ dγ =
1
6
. (E.0.4)
For each of the three models, the integral (E.0.3) was evaluated numerically to
obtain the factors in (5.1.13).
256
APPENDIX F
Constraint and evolution
equations in Λ-FLRW Regge
calculus
In this appendix, we shall first prove that the constraint equation (5.1.33) is a first
integral of the evolution equation (5.1.34) for the parent CW models in Λ-FLRW
Regge calculus. We shall next provide a partial proof that constraint equation
(5.2.51) is a first integral of the evolution equation (5.2.54) for the children models.
To facilitate comparison with the constraint equation, we shall first simplify
the evolution equation of the parent model. Note that the number of edges N1
in a Cauchy surface is related to the number of triangles N2 by the relation
N1 =
3N2
n
, where n is the number of triangles meeting at an edge; this relation
follows because a triangle has three edges, but an edge joins n triangles together.
By using this relation to substitute for N2 and by substituting (5.1.33) into the
l2 factor in the last term of (5.1.34), we can simplify (5.1.34) to get
0 =
1
1− 1
8
l˙2
1
8
(2pi − nθ)[
1− 1
8
l˙2
] 1
2
(
3
8
l˙2 − 1
)(
ll¨ + 2l˙2 − 16
)
+
n
2
√
2
ll¨
 . (F.0.1)
Now we consider the constraint equation (5.1.33) itself. To demonstrate that
257
F. CONSTRAINT EQUATIONS OF Λ-FLRW REGGE MODELS
this is a first integral of (5.1.34), we first differentiate it; this gives
ll˙ = 3
√
2
N1
N3 Λ
1[
1− 1
8
l˙2
] 1
2
[
−nθ˙ + 1
8
l˙ l¨
(2pi − nθ)
1− 1
8
l˙2
]
. (F.0.2)
The derivative θ˙ can be determined simply by differentiating θ = arccos
(
1+ 1
8
l˙2
3− 1
8
l˙2
)
,
yielding
θ˙ = − 1
2
√
2
1[
1− 1
8
l˙2
] 1
2
l˙ l¨
3− 1
8
l˙2
,
and (F.0.2) now becomes
l = 3
√
2
N1
N3 Λ
1[
1− 1
8
l˙2
] 1
2
 n
2
√
2
1[
1− 1
8
l˙2
] 1
2
l¨
3− 1
8
l˙2
+
1
8
l¨
(2pi − nθ)
1− 1
8
l˙2
 . (F.0.3)
We next multiply through by l and replace the l2 on the left-hand side by (5.1.33).
After further simplification, we arrive at
0 =
1
8
(2pi − nθ)[
1− 1
8
l˙2
] 1
2
(
ll¨ + 2l˙2 − 16
)
+
n
2
√
2
ll¨(
3
8
l˙2 − 1
) ,
which is clearly identical to (F.0.1) apart from an irrelevant overall factor of(
3
8
l˙2−1
1− 1
8
l˙2
)
. The numerator of this factor is simply the square of the strut-length and
is therefore constrained to be negative. This in turn implies that the denominator
must be strictly positive. Hence, the factor is never singular nor zero. Therefore,
we can conclude that the constraint equation is indeed a first integral of the
evolution equation for the parent models.
We now turn to the children models. Following the example of the parent
model, we similarly differentiate the constraint equation (5.2.51) with respect to
time and simplify the resulting expression to get
0 =
N1
N3
1[
4− 1
3
α2v˙2
α2− 1
3
] 1
2
 3nα4vv¨
2
√
α2 − 1
3
[
3(4α2 − 1)− α4v˙2
4(α2− 13)
] [
3− α2v˙2
4(α2− 13)
] 1
2
258
F. CONSTRAINT EQUATIONS OF Λ-FLRW REGGE MODELS
+
α
(
2pi − nθ(2))
4− 1
3
α2v˙2
α2− 1
3
[
1
3
α2
α2 − 1
3
vv¨ +
2
3
α2v˙2
α2 − 1
3
− 8
]
+
1[
4−
(
α4
α2− 1
3
− 1
)
v˙2
] 1
2
6√3vv¨
×
 3
(
α2 − 1
3
)2(
1− α2
9(α2− 13)
2
)
− 1
4
(
α2 − 2
3
) (
α4
α2− 1
3
− 1
)
α2v˙2√
α2 − 1
3
[
4−
(
α4
α2− 1
3
− 1
)
v˙2
] 1
2
[
1− (α2−
2
3)
2
4(α2− 13)
v˙2
] [
3 (4α2 − 1)− α4v˙2
4(α2− 13)
]
+
6
(
β2 − 1
3
)2(
1− β2
9(β2− 13)
2
)
− 1
2
(
β2 − 2
3
) (
β4
β2− 1
3
− 1
)
β2v˙2√
β2 − 1
3
[
4−
(
β4
β2− 1
3
− 1
)
v˙2
] 1
2
[
1− (β2−
2
3)
2
4(β2− 13)
v˙2
] [
3 (4β2 − 1)− β4v˙2
4(β2− 13)
]

+
(
2pi − 2θ(1) − 4θ(3))
4−
(
α4
α2− 1
3
− 1
)
v˙2
[
3
(
α4
α2 − 1
3
− 1
)
vv¨ + 6
(
α4
α2 − 1
3
− 1
)
v˙2 − 24
]
+
1[
4− 1
3
β2v˙2
β2− 1
3
] 1
2
 18β4vv¨√
β2 − 1
3
[
3(4β2 − 1)− β4v˙2
4(β2− 13)
] [
3− β2v˙2
4(β2− 13)
] 1
2
+
β
(
2pi − 4θ(4))
4− 1
3
β2v˙2
β2− 1
3
[
β2
β2 − 1
3
vv¨ +
2β2v˙2
β2 − 1
3
− 24
] . (F.0.4)
If we use (5.2.51) to expand out the v2 term at the end of (5.2.54), we hope to
find that this expanded form of (5.2.54) will be identical to (F.0.4) apart from
some overall factor. If this is the case, then we shall have demonstrated that
(5.2.51) is a first integral of (5.2.54).
Note that (5.2.54) and (F.0.4) can be separated into four distinct compo-
nents corresponding to a N1
N3
(
2pi − nθ(2)) term, a (2pi − 2θ(1) − 4θ(3)) term, a(
2pi − 4θ(4)) term, and everything else. To identify the factor, our approach
259
F. CONSTRAINT EQUATIONS OF Λ-FLRW REGGE MODELS
will be to match each component independently of the rest. We begin with the
N1
N3
(
2pi − nθ(2)) component which requires the following equality to hold:
16
[
α4
4
α2 − 4
3
α2 − 1
3
vv¨ +
1
24
α6v˙4(
α2 − 1
3
)2 − α2
(
α2 + 1
3
)
v˙2
2
(
α2 − 1
3
) + α6vv˙2v¨
48
(
α2 − 1
3
)2 + 2
]
= factor ×
[
1
3
α2
α2 − 1
3
vv¨ +
2
3
α2v˙2
α2 − 1
3
− 8
]
.
The highest order terms on the left-hand side are the v˙4 and vv˙2v¨ terms, which
are of order v˙2 higher than the right-hand side. Thus we make the ansatz that
the factor is of the form
factor = A+B v˙2
for some constants A and B to be determined. By matching the numerical con-
stants on the two sides, we find that A = −4. However if we now match the vv¨
terms, we obtain the expression
4α4
α2 − 4
3
α2 − 1
3
= −4
3
α2
α2 − 1
3
.
This equality can only be true if α = ±1 or ± 1√
3
. However since α is the ratio
between the lengths of two edges, we require α > 0, and for the strut-length to be
finite, we also require α > 1√
3
. Therefore the only acceptable solution is α = 1.
After making this choice, we can continue matching the remaining terms to find
that B = 3
2
. Thus the factor is given by
factor = −4
(
1− 3
8
v˙2
)
.
By matching the
(
2pi − 4θ(4)) component in a similar manner, we also find
that β = 1 and that the terms from the two equations differ by a factor of
−4 (1− 3
8
v˙2
)
.
If we now make the substitution α = β = 1, we find that the evolution
equation (5.2.54) is indeed identical to the time-derivative of the constraint equa-
tion (F.0.4) multiplied by the overall factor of −4 (1− 3
8
v˙2
)
. Therefore when
α = β = 1, the constraint equation is a first integral of the evolution equation.
260
APPENDIX G
Variation of particle
path-lengths with respect to
the struts
In this appendix, we shall explain the derivation of results (6.2.30), (6.2.31), and
(6.2.34), which give the local variation of the particle path-lengths with respect to
each strut-length. Let us consider the length si of an arbitrary particle through
a 4-block; we shall work with the 4-block of a Type II tetrahedron because it
is more general; it can easily be reduced to the Type I case by setting ψi = 0,
l
(1)
i = l
(0)
i , and l
(1)
i+1 = l
(0)
i+1. In order to vary si with respect to each strut locally,
we need to express it first in terms of the lengths of all four struts. We denote the
lengths of the four struts by mAi , m
B
i , m
C
i , and m
E
i , with the superscript labelling
the lower vertex to which the strut is attached. Our approach will be to use a
new co-ordinate system for the 4-block such that the co-ordinates of the vertices
are given in terms of the lengths of the edges, including the tetrahedral edges,
the diagonals, and the struts. As we are only interested in varying the struts, we
can greatly simplify our co-ordinate system if we first impose the symmetries on
all other edges, constraining them to be l
(0)
i , l
(0)
i+1, l
(1)
i , l
(1)
i+1, d
AE′
i , d
AE′
i+1 , d
AB′
i , d
AB′
i+1
accordingly; this is permissible because when we locally vary with respect to one
edge, all other edges must be held constant. We then calculate si in this new
co-ordinate system, differentiate it with respect to each of mAi , m
B
i , m
C
i , m
E
i , and
then impose the relevant strut-length constraints on mAi , m
B
i , m
C
i , m
E
i .
261
G. VARIATION OF THE PARTICLE PATH-LENGTHS
Once again, the 4-block has been triangulated in the manner described in
Chapter 4.2, and this introduces the diagonals AE ′, BE ′, CE ′, which have length
dAE
′
i , and the diagonals AB
′, AC ′, and BC ′, which have length dAB
′
i .
We shall now construct our new co-ordinate system. We can freely fix the
co-ordinates of the upper tetrahedron’s vertices to be
A′ =
(
− l
(1)
i+1
2
,− l
(1)
i+1
2
√
3
, 0, 0
)
,
B′ =
(
l
(1)
i+1
2
,− l
(1)
i+1
2
√
3
, 0, 0
)
,
C ′ =
(
0,
l
(1)
i+1√
3
, 0, 0
)
,
E ′ = (0, 0, hi+1, 0).
(G.0.1)
Since the upper tetrahedron’s co-ordinates are fixed, the dependence on the strut-
lengths must appear in the lower tetrahedron’s co-ordinates.
Vertex A is constrained by the lengths∣∣∣−−→AA′∣∣∣ = mAi , ∣∣∣−−→AB′∣∣∣ = ∣∣∣−−→AC ′∣∣∣ = dAB′i , ∣∣∣−−→AE ′∣∣∣ = dAE′i .
As A is equidistant to B′ and C ′, its co-ordinates will have the form
A =
(
−1
2
aA, − 1
2
√
3
aA, cA, −ιdA
)
.
Using our new co-ordinates for the vertices, we can calculate the edge-lengths
above in terms of aA, cA, and dA. This leads to the equations(
mAi
)2
=
1
4
(
l
(1)
i+1 − aA
)2
+
1
12
(
l
(1)
i+1 − aA
)2
+ c2A − d2A,(
dAB
′
i
)2
=
1
4
(
l
(1)
i+1 + aA
)2
+
1
12
(
l
(1)
i+1 − aA
)2
+ c2A − d2A,(
dAE
′
i
)2
=
1
3
a2A + (hi+1 − cA)2 − d2A.
Since we are only interested in the first derivative of si with respect to the strut-
lengths, we need only determine aA, cA, and dA and similar quantities to first
order in δmAi , δm
B
i , δm
C
i , and δm
E
i . So by expressing m
A
i as m
A
i ≈ mAA′i + δmAi
262
G. VARIATION OF THE PARTICLE PATH-LENGTHS
and making use of (6.2.8) to (6.2.11), we can solve the above system of equations
to first order in δmAi , obtaining
aA ≈ l(1)i − 2
mAA
′
i
l
(1)
i+1
δmAi ,
cA ≈ δl
(1)
i
2
√
6
coshψi + sinhψi δti +
1
3
mAA
′
i
hi+1
δmAi ,
dA ≈ δl
(1)
i
2
√
6
sinhψi + coshψi δti
+
1
3
mAA
′
i
hi+1
δl
(1)
i
2
√
6
coshψi + sinhψi δti −
[
2
l
(1)
i
l
(1)
i+1
+ 1
]
hi+1
δl
(1)
i
2
√
6
sinhψi + coshψi δti
δmAi .
Vertex B is constrained by the lengths∣∣∣−−→BB′∣∣∣ = mBi , ∣∣∣−−→BC ′∣∣∣ = dAB′i , ∣∣∣−−→BE ′∣∣∣ = dAE′i , ∣∣∣−→AB∣∣∣ = l(1)i ,
and we shall express its co-ordinates in the form
B =
(
1
2
aB, − 1
2
√
3
bB, cB, −ιdB
)
.
Furthermore, based on symmetries and the co-ordinates just obtained for A, we
know what the co-ordinates of B should be when mBi = m
AA′
i , so we can express
aB, bB, cB, and dB as
aB ≈ l(1)i + δaB,
bB ≈ l(1)i + δbB,
cB ≈ δl
(1)
i
2
√
6
coshψi + sinhψi δti + δcB,
dB ≈ δl
(1)
i
2
√
6
sinhψi + coshψi δti + δdB,
where δaB, δbB, δcB, δdB are linear in δm
A
i and δm
B
i . We can use the new
co-ordinates to express the lengths above in terms of aB, bB, cB, and dB, yielding(
mBi
)2
=
1
4
(
l
(1)
i+1 − aB
)2
+
1
12
(
l
(1)
i+1 − bB
)2
+ c2B − d2B,
263
G. VARIATION OF THE PARTICLE PATH-LENGTHS
(
dAB
′
i
)2
=
1
4
a2B +
1
3
(
l
(1)
i+1 +
1
2
bB
)2
+ c2B − d2B,(
dAE
′
i
)2
=
1
4
a2B +
1
12
b2B + (hi+1 − cB)2 − d2B,(
l
(1)
i
)2
=
1
4
(aA + aB)
2 +
1
12
(bB − aA)2 + (cB − cA)2 − (dB − dA)2 .
Then by taking mBi ≈ mAA′i + δmBi and making use of (6.2.8) to (6.2.11), we
match the first order terms and solve to obtain
δaB ≈ 2 m
AA′
i
l
(1)
i+1
δmAi ,
δbB ≈ −2 m
AA′
i
l
(1)
i+1
(
δmAi + 2 δm
B
i
)
,
δcB ≈ 1
3
mAA
′
i
hi+1
(
δmAi + 2 δm
B
i
)
,
δdB ≈ −1
3
mAA
′
i
[
δl
(1)
i
2
√
6
sinhψi + coshψi δti
]−1
×
1− δl(1)i2√6 coshψi + sinhψi δti
hi+1
(δmAi + 2 δmBi )
− l
(1)
i
l
(1)
i+1
(
δmAi − δmBi
) .
Vertex C is constrained by the lengths∣∣∣−−→CC ′∣∣∣ = mCi , ∣∣∣−→AC∣∣∣ = ∣∣∣−−→BC∣∣∣ = l(1)i , ∣∣∣−−→CE ′∣∣∣ = dAE′i ,
and we shall express its co-ordinates in the form
C =
(
aC ,
1√
3
bC , cC , −ιdC
)
.
We also know what the co-ordinates of C should be when mCi = m
AA′
i , so we can
express aC , bC , cC , and dC as
aC ≈ δaC ,
bC ≈ l(1)i + δbC ,
264
G. VARIATION OF THE PARTICLE PATH-LENGTHS
cC ≈ δl
(1)
i
2
√
6
coshψi + sinhψi δti + δcC ,
dC ≈ δl
(1)
i
2
√
6
sinhψi + coshψi δti + δdC ,
where δaC , δbC , δcC , δdC are linear in δm
A
i , δm
B
i , and δm
C
i . Following a similar
method to that of the previous two vertices, we find that
δaC ≈ m
AA′
i
l
(1)
i+1
(
δmAi − δmBi
)
,
δbC ≈ m
AA′
i
l
(1)
i+1
(
δmAi + δm
B
i
)
,
δcC ≈ 1
3
mAA
′
i
hi+1
(
δmAi + δm
B
i + 3 δm
C
i
)
,
δdC ≈ 1
3
mAA
′
i
[
δl
(1)
i
2
√
6
sinhψi + coshψi δti
]−1
×
 δl(1)i2√6 coshψi + sinhψi δti
hi+1
− 1
(δmAi + δmBi + 3 δmCi )
+
l
(1)
i
l
(1)
i+1
(
δmAi + δm
B
i
) .
Finally, vertex E is constrained by the lengths∣∣∣−−→EE ′∣∣∣ = mEi , ∣∣∣−→AE∣∣∣ = ∣∣∣−−→BE∣∣∣ = ∣∣∣−−→CE∣∣∣ = l(0)i ,
and we shall express its co-ordinates in the form
E = (aE, bE, cE, −ιdE) .
We also know what the co-ordinates of E should be when mEi = m
EE′
i , so we can
express aE, bE, cE, and dE as
aE ≈ δaE,
bE ≈ δbE,
cE ≈ δl
(1)
i
2
√
6
coshψi + sinhψi δti + hi coshαi + δcE,
265
G. VARIATION OF THE PARTICLE PATH-LENGTHS
dE ≈ δl
(1)
i
2
√
6
sinhψi + coshψi δti + hi sinhαi + δdE,
where αi is a yet to be determined boost parameter and δaE, δbE, δcE, δdE
are linear in δmAi , δm
B
i , δm
C
i , and δm
E
i . Unlike the other struts, m
E
i has the
perturbative expansion mEi ≈ mEE′i + δmEi . We can follow a similar method to
that of the previous vertices to solve the equations above for αi, δaE, δbE, δcE,
δdE. By matching the zeroth order terms, we deduce that αi is
αi = ψi. (G.0.2)
Next, by matching the first order terms and then solving, we find that
δaE ≈ m
AA′
i
l
(1)
i+1
δmAi −
1
3
mAA
′
i
l
(1)
i
[
l
(1)
i
l
(1)
i+1
+ 2
hi
hi+1
coshψi
]
δmBi
+
1
3
mAA
′
i
l
(1)
i
hi sinhψi
δl
(1)
i
2
√
6
sinhψi + coshψi δti
 l(1)i
l
(1)
i+1
(
3 δmAi − δmBi
)
− 2
1− δl(1)i2√6 coshψi + sinhψi δti
hi+1
 δmBi
 ,
δbE ≈ 1√
3
mAA
′
i
l
(1)
i+1
1 + hi sinhψi
δl
(1)
i
2
√
6
sinhψi + coshψi δti
(δmAi + δmBi )
− 2√
3
mAA
′
i
l
(1)
i
hi
hi+1
δti + hi+1 sinhψi
δl
(1)
i
2
√
6
sinhψi + coshψi δti
δmCi ,
δcE ≈ − m
EE′
i sinhψi
δti + hi+1 sinhψi
δmEi
+
1
3
mAA
′
i
hi+1
1 + hi sinhψi
δl
(1)
i
2
√
6
sinhψi + coshψi δti
(δmAi + δmBi + δmCi ) ,
δdE ≈ − m
EE′
i coshψi
δti + hi+1 sinhψi
δmEi
+
1
3
mAA
′
i
hi+1
(
δmAi + δm
B
i + δm
C
i
)
×
δl
(1)
i
2
√
6
coshψi + sinhψi δti + hi coshψi − hi+1
δl
(1)
i
2
√
6
sinhψi + coshψi δti
.
266
G. VARIATION OF THE PARTICLE PATH-LENGTHS
Using this new co-ordinate system, we shall now vary the particle’s path-
length with respect to each of the struts. Based on the particle’s position given
by (6.2.12) and its counterpart on Cauchy surface Σi+1, the particle should follow
a trajectory si given by
si =
1
4
(−−→
AA′ +
−−→
BB′ +
−−→
CC ′ +
−−→
EE ′
)
.
We note that since si is just a linear combination of the four strut vectors, si
will always be time-like if all four strut vectors are time-like. For each vertex
X = A,B,C,E, let us express the perturbative expansion of the corresponding
strut vector
−−→
XX ′ as −−→
XX ′ ≈mX + δmX + · · · ,
where mX denotes a vector corresponding to the zeroth order component of−−→
XX ′ and δmX denotes a vector corresponding to the component of
−−→
XX ′ that is
first order in δmAi , δm
B
i , δm
C
i , and δm
E
i . Then the trajectory length si can be
expressed as the perturbative expansion
si = |s0|+ s0|s0| · δs+ · · ·
= |s0|+ ∂si
∂mAi
δmAi +
∂si
∂mBi
δmBi +
∂si
∂mCi
δmCi +
∂si
∂mEi
δmEi + · · · ,
where s0 denotes the vector
s0 =
1
4
(mA +mB +mC +mE) ,
and δs the vector
δs =
1
4
(δmA + δmB + δmC + δmE) .
In the new co-ordinate system, it can be shown that s0 has co-ordinates
s0 =
(
0, 0,
1
4
(hi+1 − hi coshψi)−
(
δl
(1)
i
2
√
6
coshψi + sinhψi δti
)
,
ι
4
hi sinhψi + ι
(
δl
(1)
i
2
√
6
sinhψi + coshψi δti
))
.
267
G. VARIATION OF THE PARTICLE PATH-LENGTHS
Since the first two co-ordinates are zero, then to calculate s0|s0| ·δs and hence ∂si∂mXi ,
we need only the third and fourth co-ordinates of δs. The third co-ordinate is
−1
3
mAA
′
i
hi+1
1 + 14 hi sinhψi
δl
(1)
i
2
√
6
sinhψi + coshψi δti
(δmAi + δmBi + δmCi )
+
1
4
mEE
′
i sinhψi
δti + hi+1 sinhψi
δmEi ,
and the fourth is
ι
3
mAA
′
i
hi+1
δl
(1)
i
2
√
6
coshψi + sinhψi δti − hi+1 + 14 hi coshψi
δl
(1)
i
2
√
6
sinhψi + coshψi δti
(
δmAi + δm
B
i + δm
C
i
)
−
ι
4
mEE
′
i coshψi
δti + hi+1 sinhψi
δmEi .
We can then obtain ∂si
∂mXi
for each X by reading off the factor multiplying δmXi
in s0|s0| · δs, and we find that
∂si
∂mAi
=
∂si
∂mBi
=
∂si
∂mCi
=
1
4
mAA
′
i
|s0|
1 + 14 hi sinhψi
δl
(1)
i
2
√
6
sinhψi + coshψi δti
 , (G.0.3)
∂si
∂mEi
=
1
4
mEE
′
i
|s0|
1
4
hi+1 sinhψi + δti
hi+1 sinhψi + δti
, (G.0.4)
where
|s0| =
[1
4
(hi+1 − hi coshψi)−
(
δl
(1)
i
2
√
6
coshψi + sinhψi δti
)]2
−
[
1
4
hi sinhψi +
δl
(1)
i
2
√
6
sinhψi + coshψi δti
]2 12 .
(G.0.5)
268
G. VARIATION OF THE PARTICLE PATH-LENGTHS
Taking the continuum time limit, we have that
|s0| → s˙0 dt+O
(
dt2
)
=
(1
4
h˙ − l˙
(1)
2
√
6
)2
−
(
1
4
hψ˙ + 1
)2 12 dt+O(dt2) , (G.0.6)
and in this limit, it follows that
∂s
∂mA
=
∂s
∂mB
=
∂s
∂mC
=
1
4
m˙AA
′
s˙0
(
1
4
hψ˙ + 1
)
+O(dt) ,
∂s
∂mE
=
1
4
m˙EE
′
s˙0
1
4
hψ˙ + 1
hψ˙ + 1
+O(dt) ,
which are relations (6.2.30) and (6.2.31) as required.
Finally, to obtain the equivalent results for the particle in the Type I tetra-
hedron, we simply set ψi to be zero, replace vertex E with vertex D, and replace
the lengths l
(0)
i , l
(0)
i+1, and m
EE′
i with l
(1)
i , l
(1)
i+1, and m
AA′
i respectively. Since the
mass in this tetrahedron is the perturbed mass, the path-length is now s′i. We
then find that
∂s′i
∂mAi
=
∂s′i
∂mBi
=
∂s′i
∂mCi
=
∂s′i
∂mDi
= − ι
4
mAA
′
i
δti
, (G.0.7)
and in the continuum time limit, this becomes
∂s′
∂mA
=
∂s′
∂mB
=
∂s′
∂mC
=
∂s′
∂mD
= − ι
4
m˙AA
′
,
which is relation (6.2.34) as required.
269

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List of symbols
(Pseudo)-Riemannian manifolds
M n-dimensional manifold
En n-dimensional Euclidean space
Σ Cauchy surface of M
Σ0 initial-time Cauchy surface
g metric tensor
Curvature on (pseudo)-Riemannian manifolds
R Ricci curvature scalar
Rµν Ricci curvature tensor
Rµνσρ Riemann curvature tensor
h first fundamental form
χ second fundamental form
Relativity
G Einstein tensor
T stress–energy tensor
η Minkowski metric tensor
283
LIST OF SYMBOLS
Cosmology
a (t) FLRW scale factor
Λ cosmological constant
LW lattice universes
τ cosmological time for the lattice universe
a (τ) lattice universe scale factor
rb radius of Schwarzschild-cell’s boundary
rmax maximum Schwarzschild-cell radius for the closed universe
Collins–Williams skeletons
Σt a CW Cauchy surface at continuum time parameter t
Σi a CW Cauchy surface at discrete time parameter ti
di length of a diagonal
mi length of a strut
Atrapi area of a time-like trapezoidal hinge
AAi , A
B
i areas of the lower and upper triangular subdivisions respec-
tively of a time-like trapezoidal hinge
Atrii area of a space-like equilateral triangular hinge
δi deficit angle at a hinge in the CW formalism
δ trapi deficit angle at a time-like trapezoidal hinge
δAi , δ
A
i deficit angles of the lower and upper triangular subhinges re-
spectively of a time-like trapezoidal hinge
δ trii deficit angle at a space-like triangular hinge
ψi relative Lorentz boost between the central axes of the upper and
lower tetrahedra in the 4-block of a non-equilateral tetrahedron
284
LIST OF SYMBOLS
sij path-length of particle i through 4-simplex or 4-block j
V
(4)
i 4-volume of a 4-block
N0, N1, N2, N3 numbers of parent vertices, edges, triangles, and tetrahedra re-
spectively in a skeleton
Λ-FLRW Regge models
(XY ) child vertex at mid-point between parental vertices X and Y
(ABCD) child vertex at centre of a tetrahedron with parental vertices A,
B, C, D
li length of a tetrahedral edge in the parent models
ui, vi, pi lengths of tetrahedral edges in the children models
α, β ratios of ui to vi and of pi to vi
R 3-sphere embedding radii of Cauchy surfaces
Z ratio of embedding radii to lengths of tetrahedral edges
Rˆi, Zˆ 3-sphere embedding radius scaled to match a(t) at minimum
expansion and the ratio of the edge-length to the radius
Lattice universe Regge models
li unperturbed tetrahedral edge-length
l
(0)
i , l
(1)
i perturbed tetrahedral edge-lengths
δl
(0)
i , δl
(1)
i perturbations in tetrahedral edge-lengths
dAE
′
i , d
AB′
i lengths of diagonals in the perturbed model
mEE
′
i ,m
AA′
i lengths of struts in the perturbed model
Aquadi , δ
quad
i area and dihedral angle of the quadrilateral time-like hinges in
the perturbed model
285

Index
Λ-FLRW Regge calculus
children models, performance of,
179–182
children models, ratios of length
scales, 156, 162–165
evolution equation, 132, 137–138,
142–146, 172–174
Hamiltonian constraint, 132, 137–
139, 141–142, 144–145, 170,
172, 174
parent models, performance of, 149,
155
Regge action, 132, 139, 170
ADM formalism
Collins–Williams formalism, 108,
122–123
Hamiltonian constraint, 103
momentum constraint, 103
Collins–Williams formalism
3-sphere embedding of children
Cauchy surfaces, 156–162
3-sphere embedding of parent Cau-
chy surfaces, 127–129
4-block of equilateral tetrahedron,
108–109
4-block of non-equilateral tetrahe-
dron, 119–122, 163–164
ADM formalism, see ADM formal-
ism
children skeleton, 106, 117–119
global and local variation, 106,
116–117, 144–146, 148–149
parent skeleton, 107–109
radii of children Cauchy surfaces,
156–157, 159–163, 181, 182
radii of parent Cauchy surfaces,
129–132, 138, 149
triangulated parent skeleton, 113–
115
triangulated parent skeleton, diag-
onal hinges in, 115
triangulated skeletons and Coper-
nican symmetries, 147–148
volume of Cauchy surfaces, 113,
174–175
Einstein field equations, 22
Einstein–Hilbert action, 97
FLRW cosmology, 23–24
dust-solutions, 42–43
Friedmann equations, 24
287
INDEX
vacuum Λ-FLRW, 125
Gauss equation, 102
initial value equations, 102–104
at time-symmetry, 103–104
in Regge calculus, 103–104
lattice universe, 26–28
lattice universe Regge calculus
dual models, 190–192
global and local variation, 218
initial conditions, 221
initial value equation, 221
lattice configurations, 187–188, 190–
191
particle trajectories, 202–204
perturbed constraint equation, 212–
214, 216–217
perturbed model skeleton, 195–196
perturbed model, behaviour of, 223,
225
perturbed model, time-symmetry
in, 221–222, 224
perturbed model, triangulation of,
199, 225–227
perturbed model, volume of, 223
region of convergence, 189–190
unperturbed constraint equation,
189, 192
unperturbed model, performance
of, 192–193
Lindquist–Wheeler models
rmax, 36
boundary conditions for photons,
54–55, 62–63
boundary conditions for unequal
cells, 251–254
CF co-ordinates, 36–40
CF Friedmann equation, 45–46
comparison hypersurface, tangency
with, 47–48
infinity limit of closed lattice uni-
verses, 48–50
integrated Sachs–Wolfe effect, 90
LW co-ordinates, 36–38, 40–41
meshing condition, 36–38, 41
redshifts in, 50–51, 65, 70, 80,
85–86, 89–90
scale factor a(τ), 43–45, 47, 192
Schwarzschild-cell, 34
Schwarzschild-cell boundary, 34–36
Regge action, 97–101
for vacuum space–time, 97–98
with cosmological constant, 99
with mass particles, 99, 187, 198
Regge equation
convergence of, 100
diffeomorphisms, 99–100, 123
for vacuum space–time, 98–99
multiple solutions of, 99–100
Regge skeleton, 94–96
deficit angle, 95
dihedral angle, 95–96
hinge, 94
Ricci scalar of, 98, 104
Schwarzschild Regge calculus
particle geodesics, 57–62, 243–245
Regge blocks, 55–57
288