Stability of charged rotating black
holes for linear scalar
perturbations
Damon Jay Civin
Sidney Sussex College
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
July 2014
Abstract
In this thesis, the stability of the family of subextremal Kerr–Newman space-
times is studied in the case of linear scalar perturbations.
That is, nondegenerate energy bounds (NEB) and integrated local energy decay
(ILED) results are proved for solutions of the wave equation on the domain of
outer communications. The main obstacles to the proof of these results are su-
perradiance, trapping and their interaction. These difficulties are surmounted
by localising solutions of the wave equation in phase space and applying the
vector field method. Miraculously, as in the Kerr case, superradiance and trap-
ping occur in disjoint regions of phase space and can be dealt with individually.
Trapping is a high frequency obstruction to the proof whereas superradiance
occurs at both high and low frequencies. The construction of energy currents
for superradiant frequencies gives rise to an unfavourable boundary term. In
the high frequency regime, this boundary term is controlled by exploiting the
presence of a large parameter. For low superradiant frequencies, no such pa-
rameter is available. This difficulty is overcome by proving quantitative ver-
sions of mode stability type results. The mode stability result on the real axis
is then applied to prove integrated local energy decay for solutions of the wave
equation restricted to a bounded frequency regime.
The (ILED) statement is necessarily degenerate due to the trapping effect.
This implies that a nondegenerate (ILED) statement must lose differentiabil-
ity. If one uses an (ILED) result that loses differentiability to prove (NEB),
this loss is passed onto the (NEB) statement as well. Here, the geometry of
the subextremal Kerr–Newman background is exploited to obtain the (NEB)
statement directly from the degenerate (ILED) with no loss of differentiability.
Acknowledgements
I would like to express my gratitude to my supervisor, Prof. M. Dafermos, for
suggesting the problem and his advice and guidance over the last four years.
I am also grateful to M. Joaris for her caring and patience.
During the last few years, I have enjoyed fruitful discussions with many people,
of whom I would like to mention Stefanos Aretakis, Volker Schlue and Yakov
Shlapentokh-Rothman. I apologise to those unnamed.
My colleagues and friends, especially those in the CCA and Sidney Sussex
College, have made Cambridge an interesting, supportive and fun place to
work and live. My thanks go to Sara Merino-Aceituno for her positivity and
enthusiasm, Ed Mottram for his wit and encouraging manner, Kolyan Ray for
adopting the role of captain, Marc Briant for his ideals, Bati Sengul for his
comedic timing, the Mill Road contingent of Tim Cannings, Will Matthews and
Alan Sola for their excellent company and musical taste and Graeme Ward and
Andreas Stegmu¨ller for making the transition to Cambridge an adventure.
I thank Emma Hacking for her kind assistance in administrative matters.
Finally, I am eternally grateful to my family and friends from home whose
constant encouragement and unwavering support have meant so much to me.
I acknowledge essential financial support from the Engineering and Physical
Sciences Research Council grant EP/H023348/1, the Cambridge Common-
wealth Trust, the Cambridge Centre for Analysis and Sidney Sussex College.
Statement of Originality
I hereby declare that my thesis entitled “Stability of charged rotating black
holes for linear scalar perturbations” is not substantially the same as any that
I have submitted for a degree or diploma or other qualification at any other
University. I further state that no part of my dissertation has already been or
is concurrently submitted for any such degree of diploma or other qualification.
The work contained in this thesis is original and was entirely performed by
myself at the Cambridge Centre for Analysis, University of Cambridge in the
period between October 2010 and July 2014.
Contents
1 Introduction 1
1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The black hole stability problem . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Carter’s separation & mode stability . . . . . . . . . . . . . . . . . . 5
1.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Linear stability results . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Mode stability results . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 The main difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1 Future trapped null geodesics . . . . . . . . . . . . . . . . . . . . . . 8
1.4.2 Superradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.3 Interaction of trapping and superradiance . . . . . . . . . . . . . . . 8
1.4.4 Low frequency obstructions . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5.1 Classical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5.2 Limitations of the classical analysis . . . . . . . . . . . . . . . . . . . 9
1.5.3 Modern analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 The Kerr–Newman family of spacetimes 15
2.1 The Kerr–Newman family . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.1 The underlying manifold . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.2 The Kerr–Newman metric . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.3 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.4 The maximal globally hyperbolic extension . . . . . . . . . . . . . . 20
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 Foliation and well-posedness . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 The sign of a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.3 Energy currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
CONTENTS
3 Stability of subextremal Kerr–Newman spacetimes for linear scalar per-
turbations 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.1 Key elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 The main results: (NEB) and (ILED) . . . . . . . . . . . . . . . . . 32
3.2.2 Decay results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Frequency localisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.2 The conditional version of (ILED) . . . . . . . . . . . . . . . . . . . 37
3.3.3 Carter’s separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.4 Frequency-localised energy current templates . . . . . . . . . . . . . 42
3.4 Properties of the potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.1 Critical points of the potential . . . . . . . . . . . . . . . . . . . . . 46
3.4.2 Superradiant frequencies are not trapped . . . . . . . . . . . . . . . 48
3.4.3 Trapping for fixed azimuthal mode solutions . . . . . . . . . . . . . . 52
3.5 Frequency localised estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5.1 Partitioning the frequency ranges . . . . . . . . . . . . . . . . . . . . 54
3.5.2 High superradiant frequencies F☼ . . . . . . . . . . . . . . . . . . . 55
3.5.3 Trapped frequencies F\ . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.4 Time dominated frequencies F . . . . . . . . . . . . . . . . . . . . 62
3.5.5 Angular dominated frequencies F] . . . . . . . . . . . . . . . . . . . 63
3.5.6 The bounded frequency range F[ . . . . . . . . . . . . . . . . . . . . 63
3.5.7 The general frequency-localised estimate . . . . . . . . . . . . . . . . 69
3.6 Proof of the conditional (ILED) . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.6.1 The physical space estimate . . . . . . . . . . . . . . . . . . . . . . . 71
3.6.2 Error terms in B = {r+ ≤ r ≤ R∗∞} . . . . . . . . . . . . . . . . . . . 76
3.6.3 Error terms in U = {r ≥ R∗∞} . . . . . . . . . . . . . . . . . . . . . . 77
3.6.4 Controlling the error from the conserved energy current . . . . . . . 80
3.6.5 Concluding the proof of the conditional (ILED) . . . . . . . . . . . . 81
3.6.6 Integrated decay up to null infinity . . . . . . . . . . . . . . . . . . . 82
3.7 The continuity argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.7.1 The reduction to fixed azimuthal frequency . . . . . . . . . . . . . . 85
3.7.2 The setting and non-emptiness . . . . . . . . . . . . . . . . . . . . . 87
3.7.3 Openness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.7.4 Closedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.7.5 Proof of (ILED) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.8 Proof of (NEB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
CONTENTS
3.8.1 (NEB) outside the trapping region . . . . . . . . . . . . . . . . . . . 96
3.8.2 The set up for the proof of (NEB) in the trapping region . . . . . . 98
3.8.3 Construction of wave packets . . . . . . . . . . . . . . . . . . . . . . 101
3.8.4 (ILED) for wave packets . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.8.5 (NEB) for wave packets . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.8.6 (NEB) for the full solution . . . . . . . . . . . . . . . . . . . . . . . 107
4 Quantitative mode stability for the wave equation on the Kerr–Newman
spacetime 109
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2 Mode solutions of the wave equation . . . . . . . . . . . . . . . . . . . . . . 111
4.3 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.4 The inhomogeneous equation . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.5 Statement of mode stability results . . . . . . . . . . . . . . . . . . . . . . . 114
4.6 The Whiting transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.6.1 The confluent Heun equation . . . . . . . . . . . . . . . . . . . . . . 115
4.6.2 The transformed equation . . . . . . . . . . . . . . . . . . . . . . . . 116
4.6.3 Asymptotics of the transformed solution . . . . . . . . . . . . . . . . 117
4.7 Proofs of mode stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.7.1 Qualitative results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.7.2 Quantitative results . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.8 Application: Integrated local energy decay . . . . . . . . . . . . . . . . . . . 122
Bibliography 127
CONTENTS
Chapter 1
Introduction
1
Introduction
1.1 Context
In 1915, after a seven year struggle to incorporate gravity in his theory of relativity, Ein-
stein published [Ein15]. In this pioneering work, he formulated the fundamental equations
of the general theory of relativity:
Gµν = Tµν . (Einstein field equations)
The unknown in the theory is a spacetime (M, g), where M is a Lorentzian manifold with
metric g. The Einstein tensor Gµν describes the curvature of spacetime and the energy-
momentum tensor Tµν models the energy and matter within that spacetime. For this
system to be fully determined, Tµν must be specified and equations must be specified for
the matter fields.
The Einstein field equations model the gravitational interaction of space, matter and
energy. Therefore, the geometry of the spacetime and the matter and energy present are
interconnected. In this, the Einstein equations are similar to the Maxwell equations, where
charges and currents determine an electromagnetic field.
Einstein’s geometrisation of the Newtonian theory of gravity has the immediately re-
markable consequence that the theory is nontrivial even in the case of vacuum (Tµν = 0).
In this case the Einstein field equations reduce to
Rµν = 0, (Einstein vacuum equations)
where Rµν is the Ricci curvature of (M, g).
The Einstein field equations can be viewed as a system of ten nonlinear partial differen-
tial equations for the unknown metric g. Their analysis is therefore very difficult. In fact,
the formulation and proof of well-posedness of the initial value problem for the Einstein
vacuum equations was achieved more than thirty years after [Ein15] was published. This
was done by Foure`s-Bruhat [FB52] and Choquet-Bruhat and Geroch [CBG69] after the
ground-breaking works of Friedrichs, Schauder, Sobolev, Petrovsky, Leray and others in
the interim.
The identification of explicit solutions (those that can be written in closed form) is
a useful first step in understanding a theory in which the fundamental equations are
nonlinear. The simplest solution of the vacuum Einstein equations is Minkowski space
(R4, diag(−1, 1, 1, 1)). This is the space in which the special theory of relativity is formu-
lated.
In the early years of the study of general relativity there was considerable interest in
deriving and interpreting explicit solutions of the Einstein equations under simplifying
assumptions.
2
1.1. Context
In 1916, Schwarzschild discovered a solution of the vacuum Einstein equations which
contains a region of spacetime which cannot communicate with the rest of the spacetime
[Sch03]. Such regions were later named black holes by Wheeler.1 The discovery of a
black hole solution of the Einstein–Maxwell electrovacuum equations (where Tµν is defined
through the Maxwell equations, see (2.1.5)) followed shortly after in [Rei16] and [Nor18].
This charged black hole solution is known as the Reissner–Nordstro¨m spacetime.
Both the Schwarzschild and Reissner–Nordstro¨m solutions are spherically symmetric.
It was only much later that explicit metrics for spacetimes containing rotating black holes
were discovered. In 1963, Kerr derived an explicit solution of the vacuum Einstein equa-
tions that models a rotating black hole in [Ker63]. In [NCC+65], Newman et al. derived
charged rotating black hole solutions of the Einstein–Maxwell electrovacuum equations.
These solutions are known as the Kerr–Newman family of spacetimes. The family is
parametrised by three physical parameters: the mass M , angular momentum density a
and charge Q.2 Subextremal means that 0 ≤ a2 +Q2 < M2. It is the subextremal family
(and the extremal case a2 + Q2 = M2) of Kerr–Newman spacetimes in which a charged
rotating black hole is present. The fast Kerr–Newman spacetimes (where a2 +Q2 > M2)
have profoundly different structure, see [Car73].
Table 1.1 illustrates the relationships between each of the solutions mentioned above
as subfamilies of the Kerr–Newman family.
Uncharged Q = 0 Charged Q 6= 0
Non-rotating a = 0 Schwarzschild gM Reissner–Nordstro¨m gQ,M
Rotating a 6= 0 Kerr ga,M Kerr–Newman ga,Q,M
Table 1.1: The subfamilies of the Kerr–Newman family
The Kerr–Newman solutions are of particular significance in light of the
“No-Hair” Conjecture. The domain of outer communications of a smooth, stationary,
four dimensional, electrovacuum, connected black hole solution is isometrically diffeomor-
phic to that of a member of the Kerr–Newman family of black holes.
Conditional versions of this conjecture were proved under the additional assumption
of either axisymmetry or analyticity in the work of Bunting, Carter, Hawking, Mazur
and Robinson in the 1970s and 80s (see [Heu96] for a detailed account). More recently,
conditional versions of the conjecture have been proved under much weaker assumptions
in [IK09] and [AIK10] (for the Kerr case) and [Won09] (for the Kerr–Newman case).
The existence of black holes is perhaps the most striking prediction of general relativity.
1The interested reader is referred to [DR13] for an excellent account of the intriguing history of the
Schwarzschild solution.
2The parameter Q represents the total electric charge of the spacetime, see [Wal84, §12.3]
3
Introduction
This prediction arises from the study of explicit solutions in the hope that they may be
suggestive of the behaviour of general solutions. However, if one wishes to draw any such
conclusions from an explicit solution it is imperative to prove that the solution in question
is stable in an appropriate sense.
This motivates the focus of this thesis: the study of the stability of the subextremal
Kerr–Newman family of explicit solutions of the Einstein electrovacuum equations.
1.2 The black hole stability problem
Since the Kerr–Newman solution is thought to be the unique stationary electrovacuum
black hole spacetime, the question of its stability is closely related to the plausibility of
the concept of a black hole.
The ultimate goal is to understand the dynamical stability of the Kerr–Newman solu-
tions as a family of solutions to the Cauchy problem for the Einstein–Maxwell Equations,
affirming the following:
Conjecture (Global stability of Kerr–Newman). Any small perturbation of the ini-
tial data set of a Kerr–Newman spacetime has a global future development with a complete
future null infinity which, within its domain of outer communication, behaves asymptoti-
cally like a (another) Kerr–Newman solution.
This is one of the most important unresolved issues in the theory of relativity (see
[Kla07] for an insightful discussion of this and other open problems).
1.2.1 Linear stability
The only asymptotically flat spacetime which is known to be globally stable with respect
to nonlinear perturbations is Minkowski space. This was first proved by Christodoulou
and Klainerman in the monumental [CK93]. Following their philosophy, the first step
toward the proof of the nonlinear stability of the Kerr–Newman solution is to understand
the behaviour of scalar perturbations, i.e. solutions of the linear wave equation
gM,a,Qψ = 0. (1.2.1)
This stability problem may be thought of as a “poor-man’s version” of the problem of
gravitational perturbations, obtained by linearising the Einstein equations with respect to
a fixed subextremal Kerr–Newman metric gM,a,Q.
The particular understanding of (1.2.1) required is a proof that ψ is uniformly bounded
and decays (sufficiently rapidly) in time. This stability with respect to linear scalar per-
4
1.3. Main results
turbations is proved in Chapter 3 of this thesis, see Theorem 3.2.1.3
1.2.2 Carter’s separation & mode stability
A general subextremal Kerr–Newman metric possesses two Killing fields T and Φ so the
wave equation (1.2.1) admits solutions of the form
ψ(t, r, θ, φ) = e−iωteimφψ˜(r, θ), where ω ∈ C,m ∈ Z.
Carter discovered in [Car68] that (1.2.1) can be formally separated. The wave equation
therefore admits mode solutions of the form
ψ(t, r, θ, φ) = R
(aω)
m` (r)S
(aω)
m` (θ)e
imφe−iωt,where ` ∈ Z, ` ≥ |m|. (1.2.2)
The function S
(aω)
m` (θ) solves a Sturm-Liouville problem and R
(aω)
m` (r) satisfies the Carter
ODE :
d2
dr2
R
(aω)
m` (r) +
(
ω2 − V (aω)m` (r)
)
R
(aω)
m` (r) = 0, (Carter ODE)
where V
(aω)
m` (r) is a smooth potential.
A priori, (1.2.1) may admit mode solutions that have finite energy but grow expo-
nentially in time, i.e. solutions of the form above with ω in the upper half-plane. The
(qualitative) statement that such modes do not exist is known as mode stability.
The proof of Theorem 3.2.1 (quantitative boundedness and decay for solutions of
(1.2.1)) given in Chapter 3 depends on a quantitative refinement of the qualitative state-
ment of mode stability.
The necessary refinement is proved by first extending the mode stability statement
to exclude resonances on the real axis and then refining this qualitative statement to a
quantitative estimate.
In Chapter 4 both the qualitative mode stability results (in the upper half-plane and on
the real axis) as well as the quantitative estimate are proved for the family of subextremal
Kerr–Newman spacetimes. See §4.5 for the precise statements of the mode stability results.
The application of the quantitative mode stability results required in the proof of Theorem
3.2.1 is stated as Theorem 4.8.2.
1.3 Main results
In this thesis the stability of the subextremal Kerr–Newman exterior spacetime for linear
scalar perturbations is proved. The proof appeals to quantitative mode stability results.
3In the Kerr case, the linear stability problem has been resolved in [DR11a, DRSR14], see §1.5.3.
5
Introduction
The main results are summarised below.
1.3.1 Linear stability results
The primary goal here is to provide a proof of the energy estimates (NEB) and (ILED)
below. Higher order and pointwise decay results are then derived from these key estimates.
Theorem 1.3.1. Solutions of the wave equation (1.2.1) on a subextremal Kerr–Newman
exterior spacetime satisfy the following energy estimates:
• nondegenerate energy bounds ∫
Στ
E[ψ] ≤ C
∫
Σ0
E[ψ], (NEB)
• integrated local energy decay∫ τ
0
∫
Σs
E\[ψ] + r
−3−δψ2ds ≤ C
∫
Σ0
E[ψ], (ILED)
where Σ0 and Στ are spacelike hypersurfaces and E[ψ] and E\[ψ] are appropriately weighted
square sums of the first derivatives of ψ.
The energy E\[ψ] in (ILED) is necessarily degenerate due to the presence of trapped
null geodesics (see §1.4). However, trapping is an obstacle to decay but not boundedness.
Therefore, (NEB) should not ‘see’ trapping. That is, E[ψ] in (NEB) is nondegenerate and
both sides of the inequality (NEB) contain only first order derivatives of ψ. In this thesis,
we extract the fully nondegenerate energy bound (NEB) from a degenerate integrated
local energy statement (ILED) with no loss of differentiability. To achieve this, the precise
degeneration of (ILED) due to trapping must be understood in phase space.
The precise version of the theorem above is Theorem 3.2.1. Its proof is the content of
Chapter 3. An overview of the proof can be found in §3.1.
1.3.2 Mode stability results
In the proof of estimate (ILED), a quantitative energy estimate is required for mode
solutions of the form (1.2.2) supported in the bounded frequency range
F ⊂
{
(ω,m, `) ∈ R× {Z× Z | ` ≥ |m|} |
(
|ω|+ |ω|−1 + |m|+ |`|
)
<∞
}
.
6
1.4. The main difficulties
Theorem 1.3.2. There exists a constant CF such that∫
ω∈F
∑
m,`∈F
(∣∣∣R(aω)m` (r+)∣∣∣2 + ∫ r1
r0
∣∣∣∂rR(aω)m` ∣∣∣2 + ∣∣∣R(aω)m` ∣∣∣2 dr∗) dω ≤ CF ∫
Σ0
E[ψ], (1.3.1)
where each R
(aω)
m` solves (Carter ODE) for (ω,m, `) ∈ F .
This estimate is an application of the following quantitative mode stability result
Theorem 1.3.3 (Quantitative mode stability on the real axis). Let (ω,m, `) ∈ F . The
Wronskian W (given by (4.3.1)) satisfies
sup
(ω,m,`)∈F
∣∣W−1∣∣ ≤ G(F).
where the function G can, in principle, be given explicitly.
Theorem 1.3.3 provides a quantitative upper bound for
∣∣W−1∣∣. This bound implies
that any solution of (Carter ODE) can be expressed as a superposition of solutions of
(Carter ODE) defined by the asymptotics of R
(aω)
m` (see §4.3). This rules out the existence
of resonances on the real axis.
Note that it is essential that this estimate on the Wronskian is quantitative in order
to prove Theorem 1.3.2.
In proving Theorem 1.3.3, we will also obtain the following qualitative results.
Theorem 1.3.4 (Mode Stability on the real axis). There exist no non-trivial mode solu-
tions corresponding to ω ∈ R \ {0}.
Theorem 1.3.5 (Mode Stability). There exist no non-trivial mode solutions corresponding
to Im(ω) > 0.
Chapter 4 contains the proofs of the precise versions of these theorems. An overview
of the proof can be found in §4.1.
1.4 The main difficulties
The understanding of superradiance, trapping and their interaction is crucial in the proof
of linear stability given in this thesis. Let us briefly discuss these issues before reviewing
the relevant literature.
7
Introduction
1.4.1 Future trapped null geodesics
There exist trapped null geodesics on a subextremal Kerr–Newman spacetime. These are
geodesics that remain for all affine time on, or asymptote in the future to, a constant r
value. Physically, this means that photons following such geodesics are neither scattered
to infinity, nor do they fall into the black hole. This forces any useful energy current
to degenerate in view of a general result of Sbierski [Sbi13], in the spirit of the classical
[Ral69]. Here degeneration means that any integrated local energy decay estimate (such as
(ILED)) will either have a region in which we do not control all derivatives, or the estimate
must lose differentiability. This degeneration is simple on a Schwarzschild spacetime, but
considerably more complicated on a subextremal Kerr–Newman spacetime. In fact, the
structure of this set can only be completely understood in phase space.
1.4.2 Superradiance
One of the key features of the Kerr–Newman geometry is the existence of an ergoregion.
This is a region in which the Killing vector field T is spacelike. The presence of the
ergoregion complicates the analysis of wave equations as it corresponds to a region in
which the conserved energy associated to T is not positive. Thus the conservation law
for this energy does not yield control of the solution ψ. This leads to the possibility
that the energy flux to null infinity may be larger than the initial energy, hence the term
superradiance. This phenomenon was first discussed by Zel’Dovich in [Zel71].
1.4.3 Interaction of trapping and superradiance
The main difficulty in proving the required energy estimates in the full subextremal range
of Kerr–Newman spacetimes is the interaction of trapping and superradiance. In physical
space, it appears that these phenomena must be dealt with simultaneously, as there exist
future trapped null geodesics inside the ergoregion. Dealing with the possibility of this
interaction thus involves overcoming a potentially serious obstacle. Miraculously, it turns
out that superradiance and trapping can be dealt with separately in phase space, see §1.5
for more on this.
1.4.4 Low frequency obstructions
Trapping is a high frequency phenomenon whereas superradiance occurs at low frequencies
as well. For superradiant frequencies, the frequency localised energy currents available
generate a boundary term with an unfavourable sign. For the superradiant frequencies
in the high frequency regime, a large parameter is exploited to deal with this boundary
term. In the low frequency regime no large parameter is available, making it difficult to
8
1.5. Historical overview
obtain quantitative estimates directly. This low frequency obstruction is overcome through
appeal to the estimate of Theorem 1.3.3, see §3.6.
1.5 Historical overview
1.5.1 Classical analysis
In the classical analysis of the stability of black holes, only mode solutions were studied.
This work was initiated in the Schwarzschild case in [RW57] and the literature has become
vast, see [DR10a] for an overview and references. The mode stability of the Schwarzschild
family – the statement that there are no mode solutions of gMψ = 0 with finite energy
at t = 0 and Im(ω) > 0 – follows immediately from the observation that the potential V
is non-negative in the Schwarzschild case. It is a remarkable fact that this result carries
over to gravitational perturbations of Schwarzschild [Vis70] .
The existence of mode solutions on a Schwarzschild background is a consequence of the
dimension of the Lie algebra of symmetries of that spacetime (the stationary Killing field
T and the rotations). As mentioned in §1.2.2, Carter discovered in [Car68] that, despite
the Kerr(–Newman) family possessing only two Killing vector fields, the wave equation
can be formally separated on these backgrounds. This is related to the integrability of
the geodesic equations on Kerr–Newman spacetimes. This separation was later found to
originate from the existence of a “hidden symmetry” in the Kerr–Newman metric, see
[WP70] for more on this.
Mode analysis of the Kerr–Newman spacetime reveals that superradiance is frequency
specific in the sense that the energy flux through the horizon is negative precisely in the
frequency range
0 ≤ mω < am
2
2M(M +
√
M2 − a2 −Q2)−Q2 .
A priori, superradiance may allow for the existence of mode solutions of gM,a,Qψ = 0
with finite initial energy and Im(ω) > 0. As mentioned before, such solutions grow
exponentially in time. In the celebrated [Whi89], Whiting showed that no such solutions
exist in the Kerr case. Whiting’s proof of the mode stability of Kerr is seen today as the
culmination of the classical mode analysis. However, for Kerr–Newman spacetimes, the
analogue of Whiting’s mode stability is absent in the literature.
1.5.2 Limitations of the classical analysis
Classical analysis of mode solutions alone is not enough to resolve the question of linear
stability of a spacetime. Recall from §1.2.1 that linear stability refers to solutions of
(1.2.1) being uniformly bounded and decaying (sufficiently rapidly) in time. However,
9
Introduction
mode stability is still completely consistent with general solutions of (1.2.1) with finite
initial energy growing in time without bound. Indeed, it is not a priori apparent that
general solutions of (1.2.1) can be represented as a superposition of modes with ω ∈ R.
Even with this established, statements about individual modes do not carry over to the
superposition of infinitely many modes without some additional knowledge.
It is only through the estimates of energy-type quantities that one can bound solutions
of (1.2.1) and rule out exponentially growing solutions. The modern PDE theory provides
powerful tools for the derivation of such energy estimates.
1.5.3 Modern analysis
Boundedness on Schwarzschild
The study of black hole stability from the point of view of modern PDE theory was
effectively initiated by the celebrated results of Wald [Wal79] and Kay–Wald [KW87]:
Theorem 1.5.1 (Kay–Wald). Solutions of the wave equation on a Schwarzschild back-
ground arising from sufficiently regular initial data are pointwise uniformly bounded in the
exterior region up to and including the horizon.
The proof of the Kay–Wald Theorem makes no appeal to mode analysis. Instead,
the fundamental statements are L2 based estimates of derivatives of the solution of the
wave equation. The pointwise statement follows from commuting the wave equation with
certain vector fields and applying Sobolev inequalities. These modern arguments are
essential when working with nonlinear equations, such as the Einstein equations. Indeed,
it was by these methods that Christodoulou and Klainerman proved the nonlinear stability
of Minkowski space in [CK93] (though boundedness alone does not suffice, see below).
From the perspective of modern PDE theory, the proof of the Kay–Wald Theorem
away from the horizon follows from a standard application of the energy method (since
the vector field T is timelike there). At the horizon T becomes null and the associated
conserved energy degenerates. This obstacle was overcome in [Wal79] and [KW87] to
obtain estimates up to and including the horizon. However, the geometric arguments
employed in this proof are extremely particular to the Schwarzschild solution and as such
are very delicate with respect to metric perturbations, see [DR13, §3] for further discussion.
The work of Dafermos–Rodnianski in the slowly rotating Kerr case [DR11b] yields a
simpler proof of the Kay–Wald Theorem that does not appeal to fragile geometric proper-
ties of Schwarzschild. The Dafermos–Rodnianski approach also highlights the celebrated
red-shift effect as the physical origin of the boundedness of the horizon energy flux [DR09].
The red-shift effect is discussed further below and in §2.2.3.
10
1.5. Historical overview
Decay on Schwarzschild
The boundedness statement of the Kay–Wald Theorem alone is not sufficient as a linear
stability result. Quantitative decay bounds on the solution of the wave equation are also
required, where the rate depends only on the size of the initial data. Decay provides a
physical interpretation of the linear stability result but moreover, it is the only known
mechanism for nonlinear stability (see [CK93, LR10]). It is therefore essential to prove
decay estimates in the context of the nonlinear stability problem.
Decay results on Schwarzschild have been proved by many authors [BS03, DR05,
BS06a, BS06b, DR07, DR09, Luk10, MMTT10]. These results have led to a better un-
derstanding of how geometric aspects of black hole spacetimes interact in the analysis of
the wave equation. In particular, the obstruction to decay due to trapping is captured by
use of virial-type energy currents that degenerate precisely on the trapped geodesics. The
use of virial-type estimates originates in the work of Morawetz (c.f. [Mor68]).
The final proof of boundedness and decay in the Schwarzschild case, found in [DR11b]
and [DR07] respectively, both take place entirely in physical space and do not require any
mode analysis.
Slowly rotating Kerr
In the Kerr case, the linear stability problem for scalar perturbations has been resolved
as a result of the work by several authors. The early work in this direction was restricted
to perturbations of Schwarzschild and the slowly rotating Kerr case |a| M .
In [DR11b], a proof of (NEB) was given for solutions of (1.2.1) on a class of metrics
which includes very slowly rotating and small charge Kerr–Newman, a2 +Q2 M2, as
a special case. The decay result (ILED) in the very slowly rotating Kerr case was then
proved in [DR10a]. These results make essential use of the smallness of the parameters a
and Q to control the strength of superradiance. In particular, by taking the parameters
small enough, the ergoregion can be contained in a region arbitrarily close to the horizon.
Energy estimates in the ergoregion can then by obtained from the red-shift estimate (see
§2.2.3). Furthermore, the trapping region is bounded away from the horizon, so the
difficulties of superradiance and trapping decouple in this case.
Energy decay results were also proved independently in [TT11] and [AB09]. These
results also exploit the smallness of the parameters a and Q in a crucial way.
The full subextremal range
The proof of stability for linear scalar perturbations for the full subextremal Kerr family,
|a| < M , has been achieved recently in [DR11a] and [DRSR14].
11
Introduction
The main difficulties of superradiance, trapping and their interaction were overcome
by employing a hybrid of the vector field method and classical mode analysis. These two
approaches were united by revisiting Carter’s separation of the wave equation [Car68] and
constructing frequency-localised energy currents. The success of this approach hinges on
the miraculous fact brought to light in [DR11a] – superradiance and trapping occur in
disjoint regions of phase space for the full subextremal range of Kerr spacetimes. The
deeper origin of this decoupling (if there is one) remains mysterious and may have bearing
on the Kerr(–Newman) uniqueness problem [DR11a, §7.3].
It is possible to partition phase space in a way that allows for bespoke energy estimates
to be derived by exploiting the character of each phase space regime. In this way, the
degeneration of the estimates due to trapping can be precisely captured.
To overcome superradiance, a frequency localised energy estimate can be derived in
the high frequency regime. As mentioned in §1.4.4, this approach is difficult in the low
frequency regime. This low frequency obstruction is overcome by appeal to a quantitative
refinement of Whiting’s celebrated [Whi89]. The necessary refinement was proved very
recently by Shlapentokh-Rothman in [SR13] by first extending [Whi89] to exclude reso-
nances on the real axis and then upgrading this qualitative statement to a quantitative
estimate.
Once frequency localised estimates are proved in each phase space regime, they are
summed up and inverse Fourier transformed to obtain (ILED).
In order to apply Carter’s separation, it is necessary to Fourier transform in time.
However, there is no a priori guarantee that solutions of the wave equation on a Kerr
background are sufficiently integrable to allow this. This final hurdle is leapt over by
employing a continuity argument. The separation is carried out on a class of solutions
of the wave equation that are assumed to be sufficiently integrable. It is then proved
that all solutions of the wave equation on a subextremal Kerr background lie in this class
of solutions. This continuity argument is due to [DRSR14]. The argument is simplified
by the discovery that for solutions of the wave equation supported only on a single fixed
azimuthal frequency, trapping occurs outside the ergoregion.
The (NEB) statement is then extracted from the (ILED) statement to complete the
proof of stability of the subextremal family of Kerr spacetimes for linear scalar perturba-
tions.
Turning now to the Kerr–Newman spacetimes, it turns out that all the structure
necessary to carry out the strategy of [DR11a] and [DRSR14] carries over from the Kerr
to the Kerr–Newman case. A thorough discussion of this structure is given in §3.1 and
§4.1.
In [Civ14b] (Chapter 3 of this thesis) the linear stability problem for scalar perturba-
tions for the Kerr–Newman family of spacetimes is resolved.
12
1.5. Historical overview
The argument in [Civ14b] requires the quantitative mode stability result analogous to
[SR13]. However, even the analogue of Whiting’s mode stability result is absent in the
literature for the Kerr–Newman spacetimes. In [Civ14a] (Chapter 4 of this thesis), both
the qualitative mode stability results (in the upper half-plane and on the real axis) as well
as the quantitative estimate in the spirit of [SR13] are proved.
The extremal and cosmological cases
In the extremal case, the stabilising mechanism of the red-shift effect degenerates and one
expects blow-up rather than decay results. Great progress in this direction has been made
by Aretakis, see [Are12a, Are12b, Are13a, Are13b]. See [Sch13] for an overview of the
Λ > 0 case. For the Λ < 0 case, see for example [HS13a], [HS13b] and [HS13c].
13
Introduction
14
Chapter 2
The Kerr–Newman family of
spacetimes
15
The Kerr–Newman family of spacetimes
2.1 The Kerr–Newman family
We begin with a brief review of the relevant geometric and physical features of the Kerr–
Newman spacetimes. For an in-depth treatment, the reader is referred to [Wal84].
The Kerr–Newman metric depends on three physical parameters: the mass M , angular
momentum density a and charge density Q. We express these parameters in “natural
units” , setting both the gravitational constant G and speed of light c to unity.
Here we consider the subextremal family of Kerr–Newman spacetimes in which a
charged, rotating black hole is present. Recall that subextremal means that 0 ≤ a2 +Q2 <
M2.
We first fix the underlying manifold we wish to consider, then discuss the Kerr–
Newman metric in suitable local coordinates in §2.1.2.
2.1.1 The underlying manifold
We first make a precise definition of the manifold we wish to consider. Let
M = {t∗ > −∞, y∗ ≥ 0, θ∗ ∈ [0, pi], φ∗ ∈ [0, 2pi]} .
Here θ∗ and φ∗ are standard spherical coordinates on S2. This is a manifold with boundary
∂M = H+ = {y∗ = 0}. This boundary is the event horizon. We also define the vector
fields T = ∂t∗ and Φ = ∂φ∗ and denote the one parameter family of transformations
generated by T by ϕτ .
We will define a family of metrics on M, parametrised by M , a and Q. In §3.7.4 we
will be concerned with the smooth dependence of this family on the parameters. The
precise dependence we require is given in Lemma 2.2.1.
Before defining this family of metrics, it is convenient to define coordinate systems
that depend on the parameters M , a and Q.
Kerr–Newman star coordinates
We introduce the Kerr–Newman star coordinate chart (t∗, r, θ, φ∗), which depends on the
parameters a2 +Q2 < M2. For each triple a2 +Q2 < M2, set r± = M ±
√
M2 − a2 −Q2.
Then define a new coordinate r, depending smoothly on y∗ and the parameters and such
that r = r+ on H+. We denote by Z∗ the smooth extension of the Kerr–Newman star
coordinate vector field ∂r to M.
It is often convenient to work with a rescaled version of r, denoted by r∗ and defined
16
2.1. The Kerr–Newman family
only in the interior of M by
dr∗
dr
=
r2 + a2
∆
, r∗(3M) = 0, (2.1.1)
where
∆ := (r − r+)(r − r−) = r2 − 2Mr + a2 +Q2. (2.1.2)
Note that ∆ vanishes on H+ and that the range {r > r+} corresponds to {r∗ > −∞}. In
star coordinates T = ∂t∗ and Φ = ∂φ∗ .
Remark For subextremal Kerr–Newman metrics, 0 < a2 +Q2 < M2, hence
0 < r± = M ±
√
M2 − a2 −Q2 < 2M.
This humble pair of inequalities is crucial to many of the arguments of this thesis.
Boyer–Lindquist coordinates
We define the Boyer-Lindquist coordinates (t, r, θ, φ) by applying the coordinate transfor-
mations {
t = t∗ − t¯(r), dt¯(r) = r2+a2
∆2
,
and φ = φ∗ − φ¯(r), dφ¯(r) = a∆ .
(2.1.3)
See [DR10a] for the details and explicit definitions of t¯(r) and φ¯. We will denote the
Boyer–Lindquist ∂r by ZBL. It will turn out that ZBL defines the directional derivative
that does not degenerate in the integrated decay estimate due to trapping (see §1.4.1 and
(3.5.11)) but it is Z∗ which is regular at the horizon. We therefore define the following
combination of Z∗ and ZBL.
Definition 2.1.1. For each a2 +Q2 < M2, let χ(r) be a cut-off function such that χ = 1
for r ≥ r\ and χ = 0 for r ≤ (r+ + r\)/2, where r\ is sufficiently close to r+. Finally, for
each a2 +Q2 < M2, we define the vector field
Z = χZBL + (1− χ)Z∗.
17
The Kerr–Newman family of spacetimes
2.1.2 The Kerr–Newman metric
With M , a and Q as above, we finally define a Kerr–Newman metric on the interior ofM
in Boyer–Lindquist coordinates by
gM,a,Q := −∆
ρ2
(
dt− a sin2 θdφ)2 + sin2 θ
ρ2
(
adt− (r2 + a2)dφ
)2
+
ρ2
∆
dr2 + ρ2dθ2, (2.1.4)
where ρ2 = r2 + a2 cos2 θ
and ∆ is defined by (2.1.2). Applying (2.1.3) in reverse, we see indeed that the metric
extends regularly to H+. Note that H+ is a null hypersurface.
The metric gM,a,Q together with appropriate Fµν satisfies the following system of par-
tial differential equations, known as the electrovacuum Einstein–Maxwell equations:
Rµν = 2
(
FµβFν
β − 14gµνFρσF ρσ
)
,
∇ · F = 0
and dF = ∇αFβγ +∇γFαβ +∇βFγα = 0,
(2.1.5)
where Rµν is the Ricci curvature ofM. We call the tensor Fµν the electromagnetic tensor.
The interested reader is referred to [Wal84, p. 313] for more details.
Remark The Kerr–Newman family has three well-known subfamilies. When Q = 0,
(2.1.4) is the Kerr metric. Letting a = 0, we obtain the Reissner-Nordstro¨m metric.
Finally, if we set a = Q = 0 we simply have the Schwarzschild metric. See Table 1.1.
The vector fields T and Φ are Killing and if a 6= 0 they span the Lie algebra of Killing
fields. In Boyer–Lindquist coordinates we have T = ∂t and Φ = ∂φ. Thus it is clear from
(2.1.4) that LT g = LΦg = 0. T is referred to as the stationary Killing field and Φ is called
the axisymmetric Killing field. As r → ∞, T is asymptotically future pointing timelike
and Φ is asymptotically orthogonal to T .
The determinant of the metric is simply det(gµν) = −ρ4 sin2 θ. Hence the volume form
on a (3 + 1) dimensional Kerr–Newman manifold in Boyer–Lindquist coordinates is
dV = ρ2 dt dr dVS2 = ρ
2 sin θ dt dr dθ dφ. (2.1.6)
2.1.3 The wave equation
As mentioned in the introduction, the first step in the journey toward resolution of the
nonlinear stability problem is the analysis of the linear stability problem for scalar per-
turbations, using sufficiently robust techniques. The simplest such linear problem is the
18
2.1. The Kerr–Newman family
scalar wave equation
gψ = 0,
which may be thought of as a “poor-man’s version” of the linearised Einstein equations
(taken around a subextremal Kerr–Newman metric). Thus the boundedness and energy
decay of such ψ on a Kerr–Newman background may be thought of as stability of this
spacetime for linear scalar perturbations.
The wave equation on a Lorentzian manifold can be written in coordinates as
gψ =
1√−g∂α
(√−ggαβ∂βψ) = 0. (2.1.7)
For the Kerr–Newman metric in Boyer–Lindquist coordinates, this is
1
ρ2 sin θ
[(
a2 sin2 θ − (a
2 + r2)2
∆
)
∂2t ψ −
a2
∆
∂2φψ
−2a(2Mr −Q
2)
∆
∂t∂φψ + ∂r(∆∂rψ) + ∆/ S2ψ
]
= 0, (2.1.8)
where ∆/ S2 denotes the (unit) spherical Laplacian:
∆/ S2ψ =
1
sin θ
∂θ(sin θ∂θψ) +
1
sin2 θ
∂2φψ.
Similarly, we denote the covariant derivative on the unit sphere by ∇/ S2 and the gradient
∇/ S2ψ =
∂ψ
∂θ
∂θ +
1
sin θ
∂ψ
∂φ
∂φ.
We also denote
|∇/ S2ψ|2 := (∂θψ)2 +
1
sin2 θ
(∂φψ)
2.
We introduce the related operators
∆/ψ =
1
ρ2
∆/ S2ψ and ∇/ψ =
1
ρ
∇/ S2ψ.
Note that |∇/ψ|2 = 1
ρ2
|∇/ S2ψ|2.
Carter discovered in [Car68] that (2.1.8) can be formally separated. The separation
introduces frequencies ω, m, and λ. This provides us with the means to frequency localise
and thus capture frequency specific phenomena. This separation is carried out in §3.3.3.
19
The Kerr–Newman family of spacetimes
Me
M
Σe0 i0
H+
H−
I+
I−
Figure 2.1.1: The maximal globally hyperbolic extension
2.1.4 The maximal globally hyperbolic extension
In the physical application, the asymptotically flat spacetime (M, gMa,Q) is meant to
represent the gravitational field in the vicinity of an isolated charged rotating black hole.
From a purely mathematical perspective, there is a larger manifold that we could
consider. By appropriate coordinate changes and combining coordinate patches one can
construct the maximally extended Kerr–Newman manifold ME , see [Car73] and [HE73].
However,ME is not compatible with the dynamical formulation of general relativity. If we
consider the spacetime as the solution of an initial value problem with data prescribed on a
Cauchy hypersurface,ME will necessarily contain inextendable causal geodesics which do
not intersect that Cauchy hypersurface. Indeed, the maximally extended Kerr–Newman
solutions are quit bizarre – in particular, they contain closed timelike curves.
In the dynamical formulation of the Einstein equations, the correct manifold to con-
sider is the Cauchy development of initial data prescribed on a suitable hypersurface. In
the maximally extended Kerr–Newman spacetime ME , there are two regions which are
isometric to the original exterior region M. This suggests that the Cauchy hypersurface
used in the initial value problem should have have topology S2×R with two asymptotically
flat ends. Let Σe0 be such a hypersurface. Viewing the Einstein–Maxwell equations as a
hyperbolic system of PDE, the Kerr–Newman manifold is then the solution of (2.1.5) with
appropriate initial data prescribed on Σe0, see [Wal84]. We will denote this solution by
(Me, geM,a,Q).
The manifold M that we have considered thus far is a submanifold of the Cauchy
development Me of Σe0 with geM,a,Q = gM,a,Q on M.
The Penrose diagram of Me, along the axis of symmetry, is depicted in Figure 2.1.1.
20
2.2. Preliminaries
The boundary component I+ is known as future null infinity and comprises the limit
points of future directed null rays in M along which r →∞. Similarly, I− comprises the
limit points of past directed null rays for which r →∞. We call I− past null infinity. The
remaining boundary components i0 and i± are called spacelike infinity and future (past)
timelike infinity, respectively. In the physical application, I+ is an idealisation of far away
astrophysical observers receiving radiation from the system.
The maximal globally hyperbolic developmentMe will be useful in the proof of (NEB),
see §3.8.2.
2.2 Preliminaries
The main results of this thesis are energy estimates for solutions ψ of the wave equation
(2.1.8). Here energy refers to an integral of square sum of derivatives of ψ of the form∫
Σ
∑
1≤i1+i2+i3≤j
∣∣∣∇/ i1T i2Zi3ψ∣∣∣2 dgΣ,
where Σ is a spacelike hypersurface. We typically prove estimates in the case j = 1 and
then extend the results to the higher order case j ≥ 1 as corollaries.
2.2.1 Foliation and well-posedness
In order to formulate the initial value problem for the wave equation, we must prescribe
data on a suitable hypersurface. The submanifold {t∗ ≥ 0} of M is the future Cauchy
development of Σ0 = {t∗ = 0}. We are interested in the behaviour of solutions of the
wave equation (2.1.8) in the future of Σ0. To prove energy estimates, we use a folia-
tion of the type described in [DR11a, §4]. Letting ϕτ denote the 1−parameter family of
diffeomorphisms generated by the vector field T , we define the hypersurfaces
Στ = ϕτ (Σ0) = {t∗ = τ} .
Note that each leaf of this foliation is terminates on H+ and spatial infinity i0. (More
details can be found in [DR11a, §2].) Let τ2 > τ1 so that Στ2 lies in the future of Στ1 .
Denote the region bounded by Στ1 , Στ2 and H+ by
R(τ1, τ2) =
⋃
τ1≤τ≤τ2
Στ
21
The Kerr–Newman family of spacetimes
so that the future Cauchy development of Σ0 is precisely
D+(Σ0) =
⋃
τ ′≥τ
R(0, τ ′).
The lapse function associated with the foliation is uniformly bounded above and away
from zero. That is, there exist positive uniform constants b < B such that for all τ ≥ 0,
B ≥ (−gM,a,Q(∇τ,∇τ))− 12 ≥ b > 0. (2.2.1)
In particular this means that each Στ is spacelike.
By the smooth coarea formula, for a smooth, integrable function F , there exist con-
stants c and C such that
c
∫ τ2
τ1
(∫
Στ
F
)
dτ ≤
∫
R(τ1,τ2)
F ≤ C
∫ τ2
τ1
(∫
Στ
F
)
dτ. (2.2.2)
This is used many times without further comment in what follows.
We denote the causal future and past (restricted toM) of a set A ⊂M by J+(A) and
J−(A) respectively.
Proposition 2.2.1. [Ho¨r07, Theorem 23.2.4] Let Σ0 be as above and let nΣ be the (future
directed) unit normal to Σ0. For any
ψ|Σ0 = ψ0 ∈ Hkloc(Σ0) and nΣ0ψ = ψ1 ∈ Hk−1loc (Σ0), k ≥ 1,
there exists a unique solution to the initial value problem
gψ = 0,
ψ|Σ0 = ψ0,
nΣ0ψ = ψ1,
(2.2.3)
such that
ψ(τ, ·) ∈ C([0,∞);Hkloc(Στ )) ∩ C1([0,∞);Hk−1loc (Στ )).
Furthermore, the solution depends smoothly on the parameters a and Q. The precise
dependence on Q is as follows.
Lemma 2.2.1. Let Q2 < M2 − a2 and {Qk}∞k=1 have the limit Qk → Q. Define the
22
2.2. Preliminaries
sequence {ψk}∞k=1 as the solutions of
gM,a,Qkψk = 0,
ψk|Σ0 = ψ0,
nΣ0ψ = ψ1,
where ψ0 and ψ1 are as in Proposition 2.2.1. Then, for every j ≥ 1 and τ ≥ 0,
lim
k→∞
∫
Στ
∑
1≤i1+i2+i3≤j
∣∣∣∇/ i1T i2Zi3ψk∣∣∣2 dg(k)Στ = ∫
Στ
∑
1≤i1+i2+i3≤j
∣∣∣∇/ i1T i2Zi3ψ∣∣∣2 dgΣτ .
Note that in the expression above, the geometric objects ∇/ , Z, and the volume form
dg
(k)
Στ
depend on the metric gM,a,Qk .
The analogous statement holds for a dependence, see [DRSR14, Lemma 4.1.1].
In §3.7.4, we will make explicit use of the smooth dependence of the solution on Q.
2.2.2 The sign of a
Let ψ be a solution of 2gM,a,Qψ = 0, for some M , a, Q. Then defining
ψ˜(y∗, t∗, θ∗, φ∗) = ψ(y∗, t∗, θ∗, 2pi − φ∗),
we have that ψ˜ satisfies 2gM,−a,Qψ˜ = 0.
Taking all objects and quantities defined with respect to the metric gM,−a,Q, the results
of §3.2 for ψ˜ are equivalent to those for ψ with respect to gM,a,Q. Therefore, it suffices to
consider a ≥ 0.
This reduction is of no conceptual significance and is made only to simplify the notation
when discussing the superradiant frequency range, see (3.3.21) and (3.3.22).
The reader can assume that a ≥ 0 everywhere in this thesis, though it is only strictly
necessary for statements that refer explicitly to frequency-dependent functions.
2.2.3 Energy currents
The vector field method
The vector field method is a robust technique for deriving L2-based identities with the
help of geometrically natural vector fields. These are in turn used to link the geometry
of the spacetime (M, g) to the behaviour of solutions of gψ = 0 on M. There are two
aspects to this method. Firstly, that of vector field multipliers which are used extensively
in this thesis. Secondly, by commuting the wave equation with certain vector fields one
23
The Kerr–Newman family of spacetimes
can derive higher order and pointwise estimates. For example, commutation vector fields
are used extensively in proving Corollary 3.2.4 below. A brief history of the vector field
method can be found in [Kla10].1
The vector field multiplier method is based on applying the Divergence Theorem to
a particular class of 1-forms, known as energy currents. Energy currents are constructed
from the energy momentum tensor. The energy momentum tensor associated with the
wave equation (2.1.7) on a Lorentzian manifold is given by
Tµν [ψ] = ∂µψ∂νψ − 1
2
(gαβ∂αψ∂βψ)gµν .
Given a vector field V and a (sufficiently regular) function ψ, we define the following
energy currents.
JVν [ψ] = Tµν [ψ]V ν ,
KV [ψ] = Tµν [ψ]∇µV ν
and EV [ψ] = div(T)V = (ψ)dψ(V ).
Note that
∇µ JVµ [ψ] = KV [ψ] + EV [ψ].
The wave equation is satisfied if and only if the divergence of the energy momentum
tensor vanishes. If V is a Killing vector field then KV [ψ] = 0. These facts are vital in the
construction which follows.
We will be working with an inhomogeneous wave equation gψ = F due to the neces-
sity of cutting off in time (see §3.3.3). As such, the divergence of the associated energy
momentum tensor will not vanish. Thus
EV [ψ] = FV ν∂νψ.
It will be useful to augment JVν [ψ] with a (sufficiently regular) function w. We define
JV,wν [ψ] = JVν [ψ] +
1
8
w∂µ(ψ
2)− 1
8
(∂µw)(ψ
2).
Hence KV,w[ψ] = KV [ψ]− 1
8
(w)(ψ2) + 1
4
w∇αψ∇αψ
and EV,w[ψ] = EV [ψ] + 1
4
(wψ)(ψ).
The vector field method essentially refers to applying the Divergence Theorem to JV,wν [ψ]
within a region such as R(τ1, τ2) for carefully chosen V and w, to obtain the associated
1Available online at http://web.math.princeton.edu/~seri/homepage/papers/John2010.pdf.
24
2.2. Preliminaries
energy identity.∫
Στ2
JV,wµ [ψ]n
µ
Στ
+
∫
H+∩R(τ1,τ2)
JV,wµ [ψ]n
µ
H+ +
∫
R(τ1,τ2)
KV,w[ψ]
+
∫
R(τ1,τ2)
EV,w[ψ] =
∫
Στ1
JV,wµ [ψ]n
µ
Σ0
.
In implementing the vector field method, it is often useful to arrange for the boundary
terms to have a “good” sign and treat the bulk terms as error terms. For example, if V is
a Killing field and ψ satisfies the wave equation, then the bulk terms vanish and we obtain
the following conservation law∫
Στ2
JVµ [ψ]n
µ
Στ
+
∫
H+∩R(τ1,τ2)
JVµ [ψ]n
µ
H+ =
∫
Στ1
JVµ [ψ]n
µ
Σ0
,
which is a version of Noether’s Theorem. There are other situations in which it is desirable
to have bulk terms with a sign (see for example Proposition 3.6.4).
The following proposition and its corollary give the essential definiteness properties
that make the energy currents compatible with Sobolev estimates.
Proposition 2.2.2. Let V and W be two future directed timelike vector fields. Then for
any function ψ, T[ψ](V,W ) is positive definite. By continuity, if V or W is null then
T[ψ](V,W ) is non-negative definite.
Proof. The proof is an immediate application of the Cauchy–Schwarz inequality.
The redshift estimate
A subextremal Kerr–Newman spacetime possesses the following Killing field
K := T +
a
(r2+ + a
2)2
Φ = T +
a
Mr+ −Q2 Φ (2.2.4)
known as the Hawking vector field. It is a null generator of H+. Therefore, the event
horizon is a Killing horizon. Note the identity
∇KK = κK where κ = r+ − r−
2(r2+ + a
2)
> 0.
The quantity κ is the surface gravity. The positivity of κ allows for the construction of
a nondegenerate energy on (and near) the horizon which has the divergence properties
needed to prove energy estimates [DR09] (see Theorem 2.2.3).
Remark In the extremal case, κ = 0, so this stabilising mechanism breaks down. In fact,
25
The Kerr–Newman family of spacetimes
our results do not hold in the extremal case! Aretakis has recently made great progress in
classifying the stability and instabilities of extremal black holes, see for example [Are11],
[Are12a] and [Are12b].
The following vector field will be useful:
Ke := T +
a(r2 + a2 −∆)
(r2 + a2)2
Φ = T +
a(2Mr −Q2)
(r2 + a2)2
Φ. (2.2.5)
This vector field has the following important properties.
Lemma 2.2.2. The vector field Ke defined by (2.2.5) is null on the horizon H+ and
timelike in M\H+.
Proof. We compute g(Ke,Ke). First, note that ∆ = (r − r+)(r − r−) so ∆ = 0 on the
horizon. Therefore, the vector field defined by (2.2.5) reduces to (2.2.4) on the horizon. A
simple computation shows that g(Ke,Ke)|r=r+ = g(K,K)|r=r+ = 0.
Off the horizon, we need to show that
ρ2g
(
T +
a(r2 + a2 −∆)
(r2 + a2)2
Φ, T +
a(r2 + a2 −∆)
(r2 + a2)2
Φ
)
= −∆ + sin2 θ
(
a2 − a
2(r2 + a2 −∆)2
(r2 + a2)2
− ∆a
4(r2 + a2 −∆)2 sin2 θ
(r2 + a2)4
)
< 0.
We need only consider the case that the term in parentheses is positive. It then suffices
to show that
−∆ + a2 − a
2(r2 + a2 −∆)2
(r2 + a2)2
< 0.
Multiplying through by −(r2 + a2)2, we would like to have
(∆− a2)(r2 + a2)2 + a2(r2 + a2 −∆)2 > 0.
Now
(∆− a2)(r2 + a2)2 + a2(r2 + a2 −∆)2 = ∆(r2 + a2)2 + ∆2(r2 + a2)2 − 2a2∆(r2 + a2)
= (∆(r2 + a2)[r2 − a2 + ∆(r2 + a2)] > 0,
since r > M > a.
Lemma 2.2.3. There exists an 0 such that K (defined by (2.2.4)) is timelike for r ∈
(r+, r+ + 0).
26
2.2. Preliminaries
Proof. We consider re = r+ + for < 0 and note that ∆ > 0 for r > r+ and ∆ = 0 at
r = r+, so ∆(re) = O(). So
ρ2g
(
T +
a(
r2+ + a
2
)Φ, T + a(
r2+ + a
2
)Φ)
= −∆ + sin2 θ
(
a2 − 2a
2(r2 + a2)
(r2+ + a
2)
− a
2(r2 + a2)
(r2+ + a
2)2
)
+O()
= −∆ + sin2 θ
(
a2[(r2+ − r2)− a2 − r2]
(r2+ + a
2)
− a
2(r2 + a2)
(r2+ + a
2)2
)
+O() < 0,
taking 0 small enough and noting that r > r+.
Dafermos and Rodnianski showed in [DR13] that for all stationary black hole space-
times with Killing horizons of positive surface gravity, there exists a timelike vector field
N whose multiplier current JNµ [ψ] captures the red-shift effect. In the Kerr–Newman case,
we have
Theorem 2.2.3. Let a2 +Q2 < M2, g = gM,a,Q be a Kerr–Newman metric and D
+(Σ0)
be as defined in §2.2.1. There exist positive constants b = b(a,Q,M) and B = B(a,Q,M),
parameters r1(a,Q,M) > re(a,Q,M) > r+ and a ϕt-invariant timelike vector field N =
N(a,Q,M) on D such that
1. KN [ψ] ≥ b JNµ [ψ]Nµ for r ≤ re,
2. −KN [ψ] ≤ B JNµ [ψ]Nµ for r ≥ re,
3. N = T for r ≥ r1,
where the currents are defined with respect to g.
One of the most important uses of this current is that it yields the following estimate
for nondegenerate energy near the horizon, provided we can control the last term on the
right hand side of (2.2.6).
Proposition 2.2.4. [DR10a] Let a2 +Q2 < M2 and Στ = ϕτ (Σ0). For all r+ < r0 ≤ re
and δ > 0, there exists a constant B0 = B(Σ0, r0, δ) such that for all solutions of (2.2.3)
we have ∫
R(0,τ)∩{r+≤r≤r0}
[
JN [ψ] ·N + (|r − r+|−1 log |r − r+|−2)ψ2
]
+
∫
H+(0,τ)
JN [ψ] · nH+ +
∫
Στ∩{r+≤r≤r0}
JN [ψ] · nΣτ
≤ B0
∫
Σ0
JN [ψ]nµΣ0 +B0
∫
R(0,τ)∩{R0≤r≤r0+δ}
[
JN [ψ] · nΣ0 + ψ2
]
. (2.2.6)
27
The Kerr–Newman family of spacetimes
In what follows, we sometimes use the shorthand |∂ψ|2 to denote the quantity con-
trolled by JNµ [ψ]n
µ
Στ
, where nΣτ is the normal to Στ = {t∗ = τ}. More explicitly, let us
set
|∂ψ|2 = (∂t∗ψ)2 + (∂rψ)2 + |∇/ψ|2. (2.2.7)
In what follows, we often need to estimate a weighted L2 norm of a function by
energy quantities (which contain only derivatives). This is accomplished by using Hardy
inequalities of the following form:
For a square-integrable, differentiable function f vanishing as x → ∞ with square
integrable derivative: ∫ ∞
0
f2(x)dx ≤ C
∫ ∞
0
x2
(
df
dx
)2
dx. (2.2.8)
To prove a Hardy inequality, one integrates the right hand sign by parts, eliminates
the boundary terms and identifies a factor on the right hand side as the square root of
the left hand side. The bound then follows by an application of the L2 Cauchy–Schwarz
inequality.
28
Chapter 3
Stability of subextremal
Kerr–Newman spacetimes for
linear scalar perturbations
29
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
3.1 Introduction
Our primary goal here is the proof of the following energy estimates:
• nondegenerate energy bounds ∫
Στ
E[ψ] ≤ C
∫
Σ0
E[ψ], (NEB)
• integrated local energy decay∫ τ
0
∫
Σs
E\[ψ] + r
−αψ2ds ≤ C
∫
Σ0
E[ψ], (ILED)
where α > 3 and E[ψ] and E\[ψ] are appropriately weighted square sums of the first
derivatives of ψ and Σ0 and Στ are spacelike hypersurfaces. See Theorem 3.2.1 for the
precise statement of these results. Higher order and pointwise decay results are then
derived from these key estimates.
The energy E\[ψ] in (ILED) is necessarily degenerate due to the presence of trapped
null geodesics. However, trapping is an obstacle to decay but not boundedness. Therefore,
(NEB) should not ‘see’ trapping. That is, E[ψ] in (NEB) is nondegenerate and both sides
of the inequality (NEB) contain only first order derivatives of ψ. In this chapter, we
extract the fully nondegenerate energy bound (NEB) from a degenerate integrated local
energy statement (ILED) with no loss of differentiability. To achieve this, the precise
degeneration of (ILED) due to trapping must be understood in phase space.
In this thesis we resolve the linear stability problem for scalar perturbations for the
Kerr–Newman family of spacetimes by adapting the strategy of [DR11a] and [DRSR14].
3.1.1 Key elements
The following properties of subextremal Kerr spacetimes are crucial to the proof of (NEB)
and (ILED) given in [DR11a] and [DRSR14]:
1. The span of the Killing vector fields T and Φ is timelike away from the horizon.
2. The wave equation can be separated as discovered by Carter in [Car68]. This sepa-
ration can be carried out rigorously by restricting attention to solutions of the wave
equation that are sufficiently integrable in time.
3. Superradiance and trapping are frequency specific phenomena and can be captured
in disjoint regions of phase space. This means they can be dealt with separately,
circumventing the need to deal with their possibly nontrivial interaction.
30
3.1. Introduction
4. The separated wave equation possesses an algebraic structure that allows for mode
stability results to be proved for solutions of the wave equation supported only on a
compact range of frequencies. This was first discovered by Whiting in the celebrated
[Whi89]. Whiting’s result has recently been extended and quantified in [SR13].
It is the quantitative estimates of [SR13] which are necessary in the argument of
[DRSR14].
5. For solutions of the wave equation supported only on a single fixed azimuthal fre-
quency, superradiance and trapping are disjoint in physical space.
6. The metric depends smoothly on the parameter a. This allows for the restriction
to sufficiently integrable solutions of the wave equation by a continuity argument in
that parameter.
Miraculously, all the properties listed above have analogues in the Kerr–Newman case.
Property (4) is proved in Chapter 4. The other properties of the subextremal Kerr–
Newman spacetimes are proved here and used to prove that these spacetimes are stable
with respect to linear scalar perturbations, i.e. (NEB) and (ILED).
3.1.2 Overview
The main results of this chapter are the precise versions of (NEB) and (ILED) for the
subextremal Kerr–Newman spacetimes, stated in Theorem 3.2.1.
The proof of Theorem 3.2.1 begins in §3.3 with the restriction of attention to solu-
tions ψ of the wave equation that are sufficiently integrable. Under this assumption, we
prove (ILED) by first Fourier transforming ψ. This in turn allows for the use of Carter’s
separation of the wave equation.
This separation reduces the problem to analysing a second order linear ODE with
potential V . The behaviour of this potential captures superradiance and trapping. The
analysis of this potential in §3.4 leads to the miraculous confinement of superradiance and
trapping in disjoint regions of phase space. The significance of this fact is that it allows us
to construct bespoke energy multipliers to prove the phase space analogue of (ILED) in
each frequency regime, see §3.5. Furthermore, this phase space version of (ILED) provides
the precise understanding of trapping required to prove (NEB).
There is one further obstruction in that the frequency localised energy current used
to deal with superradiance exploits a large frequency parameter. In the low frequency
superradiant regime, no frequency localised energy current is available, making it difficult
to obtain quantitative estimates directly. In the Kerr case, the mode stability result of
[Whi89] has been recently refined in [SR13]. This refinement allows for quantitative es-
timates for the bounded superradiant frequencies in the Kerr case. The analogous mode
31
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
stability results and quantitative estimates for the Kerr–Newman case are proved in Chap-
ter 4. The low frequency estimates are presented here in §3.5.6.
The (ILED) statement then follows from summing up and inverse Fourier transforming
the estimates of §3.5 and the quantitative mode stability result of Chapter 4. This is
presented in §3.6.
We then remove the integrability assumption by a continuity argument in the param-
eter Q in §3.7. This is analogous to the continuity argument in a for the Kerr case in
[DRSR14]. The proof of the (ILED) estimates under the integrability assumption is es-
sentially the closedness part of the argument. Non-emptiness follows from the analogous
(ILED) and (NEB) results for the Kerr case.
To prove openness we first note in §3.7.1 that it suffices to work with modes of fixed
azimuthal frequency. We then observe that for modes of fixed azimuthal frequency, trap-
ping is disjoint from superradiance in physical space (see Lemma 3.4.5). That is, the
degeneracy in the (ILED) statement is supported strictly outside the ergoregion. We take
advantage of this fact in §3.7.3 to prove a derivative-gaining (ILED)-type estimate. Fi-
nally, we define an interpolating metric and use the derivative-gaining estimate to prove
that the set of subextremal Kerr–Newman spacetimes for which solutions of the wave
equation are sufficiently integrable is indeed open.
In §3.8, the (NEB) statement is extracted from the (ILED) statement by constructing
bespoke physical space energy currents for solutions of the wave equation localised around
fixed degeneracies of the (ILED) estimate.
3.2 Main results
Our main result is a quantitative energy bound and energy decay result for solutions of
(2.2.3) on the full range of subextremal Kerr–Newman spacetimes.
3.2.1 The main results: (NEB) and (ILED)
Theorem 3.2.1. Let a2 + Q2 ≤ K20 < M2. Let g = gM,a,Q be a subextremal Kerr–
Newman metric and Σ0 be the Cauchy hypersurface described in 2.2.1. Let ψ be a solution
of (2.2.3). For any δ > 0 and any r+ < Re <∞, there exist constants CRe = CRe(K0,M)
and Cδ = Cδ(K0,M), such that we have the following: For τ ≥ 0 (including the limit
τ →∞), the following estimates hold:
• Nondegenerate energy bound
∫
Στ
JN [ψ] · nΣτ ≤ C
∫
Σ0
JN [ψ] · nΣ0 (NEB)
32
3.2. Main results
• Integrated local energy decay for arbitrary r+ < Re <∞
∫ τ
0
∫
Στ∩{r+≤r≤Re}
(
χ\ JN [ψ] · nΣτ + |ψ − ψ∞|2
)
dt∗ +
∫
H+
JNµ [ψ]n
µ
H+
≤ CRe
∫
Σ0
JN [ψ] · nΣ0 (ILED)
• Integrated local energy decay up to null infinity∫ τ
0
∫
Στ
(
r−1χ\|∇/ψ|2 + r−1−δχ\(Tψ)2 + r−1−δ(Zψ)2 + r−3−δ|ψ − ψ∞|2
)
dt∗
+
∫
H+
JNµ [ψ]n
µ
H+ +
∫
I+
JTµ [ψ]n
µ
I+
≤ Cδ
∫
Σ0
JN [ψ] · nΣ0 , (3.2.1)
where 4piψ∞ = limR→∞
∫
Σ0∩{r=R} r
−2ψ2.
Here χ\ is a cut-off function that vanishes in a neighbourhood of the physical space pro-
jection of the trapped set, see (3.6.2). In fact, a stronger version of (ILED) is proved in
phase space, namely (3.5.11). The constants C and CRe blow up as K0 →M .
We first prove the estimates stated in the theorem under the assumption that ψ is
sufficiently integrable to allow Fourier transform, see §3.3.1 and Theorem 3.3.2. The
integrability assumption allows us to rigorously apply Carter’s separation to the wave
equation (2.1.8) in §3.3.3 and derive the phase space estimate (3.5.11) in §3.5. This
follows the approach taken by Dafermos–Rodnianski for the Kerr case in [DR11a] and
Dafermos–Rodnianski–Shlapentokh-Rothman in [DRSR14].
In proving (3.5.11), we come across a low-frequency obstruction which is overcome
by appealing to the quantitative mode stability result Theorem 4.5.1, which itself is a
generalisation of the analogous Kerr result of [SR13].
The physical space result (ILED) is retrieved from the phase space result in §3.6.
We then remove the integrability assumption by a continuity argument in the param-
eter Q in §3.7. This is analogous to the continuity argument in a for the Kerr case in
[DRSR14]. The continuity argument makes use of the fortuitous fact that trapping occurs
outside the ergoregion for solutions of (2.1.8) supported on fixed azimuthal frequencies
(see Lemma 3.4.5).
Finally, we extract (NEB) from (3.5.11) in §3.8. Extensive use is made of the precise
understanding of trapping in phase space and the feature of the Kerr–Newman geometry
that the span of T and Φ is timelike off the horizon.
33
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
The following higher order statement is useful for various applications, e.g., proving
pointwise estimates.
Theorem 3.2.2. Under the hypotheses of Theorem 3.2.1, for all δ > 0 and all integers
j ≥ 1, there exists a constant C = C(j, δ,M) such that for all τ ≥ 0 (including the limit
τ →∞), the following inequalities hold:
∫ τ
0
∫
Σs
r−1−δ
∑
1≤i1+i2+i3≤j−1
(∣∣∣∇/ i1T i2(Z)i3+1ψ∣∣∣2 + ∣∣∣∇/ i1T i2Zi3ψ∣∣∣2)
+χ\
∑
1≤i1+i2+i3≤j
∣∣∣∇/ i1T i2(Z)i3ψ∣∣∣2
ds
+
∫
H+(0,τ)
∑
1≤i≤j−1
JNµ [N iψ]n
µ
H+ +
∫
I+
∑
1≤i≤j−1
JNµ [N iψ]I+
≤ C
∫
Σ0
∑
0≤i≤j−1
JN [N iψ] · nΣ0 , (3.2.2)∫
Στ
∑
0≤i≤j−1
JN [N iψ] · nΣτ ≤ C
∫
Σ0
∑
0≤i≤j−1
JN [N iψ] · nΣ0 . (3.2.3)
Proof. Once Theorem 3.2.1 has been established, this higher order result follows by com-
muting the wave equation with the vector fields T , Φ and Y and applying elliptic estimates,
see [DRSR14, §10] for the details.
3.2.2 Decay results
Once (NEB) and (ILED) have been proved, one can invoke the Dafermos–Rodnianski
method of [DR10b] to obtain more explicit quantitative decay results. The Dafermos–
Rodnianski method makes use of a different foliation to that described in §2.2.1. Let ς0
be a spacelike hypersurface terminating on H+ and asymptoting to null infinity (rather
than i0). The explicit form of such hypersurfaces is not important in the analysis, though
examples may be found in [DR10b] and [DR10a]). A detailed treatment of the Dafermos–
Rodnianski method allowing for a large class of spacetimes (including Kerr–Newman) and
very general asymptotics may be found in [Mos].
The foliation is then defined as before: let ϕτ denote the 1-parameter family of diffeo-
morphisms generated by the vector field T , and define the hypersurfaces
ςτ = ϕτ (ς0).
Corollary 3.2.3. With the foliation described above and under the hypotheses of Theorem
3.2.1, we have, in addition to (NEB) and (ILED),
34
3.3. Frequency localisation
• Explicit integrated energy decay∫ 2τ
τ
∫
ςτ∩{r+≤r≤Re}
|∂ψ|2dτ ≤ CReDτ−2 (3.2.4)
• Explicit decay of energy flux∫
ςτ
f(r)(∂t∗ψ)
2 + (∂rψ)
2 + |∇/ψ|2 ≤ CDτ−2, (3.2.5)
where f(r) is a positive, bounded function, such that f(r) → 0 as r → ∞. The explicit
form of f depends on the choice of ς0 (see [SR13, Lemma D.4] for more details). The
constant D denotes the square of a weighted higher-order Sobolev norm of the initial data
and τ is the time function of the foliation ∪τ≥0ςτ .
The Dafermos–Rodnianski method is a “black box” result. It requires (NEB) and the
nondegenerate form of (ILED) (obtained by commuting with the Killing fields) as input
and outputs (3.2.4) and (3.2.5).
One may obtain the following higher order boundedness and decay results through
commutation arguments (for the details of such arguments, see [Sch12] and [DRSR14,
§10])
Corollary 3.2.4. With the foliation by hypersurfaces ∪τ≥0ςτ and under the hypotheses of
Theorem 3.2.1, for any δ > 0 there exists a constant C = C(K0,M, δ,Re) > 0 such that
sup
ςτ
r|ψ − ψ∞| ≤ C
√
Dτ−
1
2
sup
ςτ∩{r≤Re}
|ψ − ψ∞| ≤ C
√
Dτ−3/2+δ
and sup
ςτ∩{r≤Re}
(|nςτψ|+ |∇ςτψ|) ≤ C
√
Dτ−2+δ
Here D denotes the square of a weighted higher-order Sobolev norm of the initial data and
τ is the time function of the foliation ∪τ≥0ςτ .
3.3 Frequency localisation
3.3.1 Assumptions
Before we begin the proof, we discuss an integrability assumption that allows us to perform
phase space analysis and a criterion that ensures we have good asymptotics near the
horizon and spacelike infinity.
35
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
Integrability assumption
Carter’s separation of the wave equation requires Fourier transform in the t variable but,
a priori, solutions of the wave equation (2.2.3) could grow exponentially in time. We
therefore restrict attention to a class of functions which are assumed to be sufficiently
integrable in the following sense:
Definition 3.3.1. Let a2 + Q2 ≤ K20 < M and let g = ga,Q,M . A smooth function
Ψ :M→ R is said to be sufficiently integrable if for every j ≥ 1 and R > r+, we have
sup
r∈[r+,R]
∫ ∞
−∞
∫
S2
∑
0≤i1+i2+i3≤j
∣∣∣∇/ i1T i2 (Z)i3 Ψ∣∣∣2 + ∣∣∣∇/ i1T i2 (Z∗)i3 gΨ∣∣∣2 sin θ dt dθ dφ <∞.
(3.3.1)
Under this assumption, Ψ and its derivatives may be Fourier transformed.
The outgoing condition
We introduce a condition that will imply that solutions of the radial ODE (3.3.12) have
outgoing boundary conditions.
Definition 3.3.2. Let K0 < M and a
2 + Q2 ≤ K20 . A smooth function Ψ is said to be
outgoing if there exists an > 0 such that for all τ ≤ −−1,
Ψ = 0 in Στ ∩ {r ≤ r+ + },
Ψ = 0 Στ ∩ {r ≥ −1}
and gM,a,QΨ = 0 for sufficiently large r.
(3.3.2)
The outgoing condition ensures that Ψ is supported away from the past event horizon
and away from past null infinity. Heuristically, this means that there is no energy entering
M. We instantiate the class of outgoing, sufficiently integrable functions by applying an
appropriate cut-off ψQ = γψ for ψ satisfying (2.2.3).
Cutting off in time
Let ψ be a solution of (2.2.3). Define
ψQ = γψ, (3.3.3)
where γ is a smooth cut-off function such that
γ(t∗) =
{
0 for t∗ ≤ 0
1 for t∗ ≥ 1.
36
3.3. Frequency localisation
This ensures that ψQ will satisfy the outgoing condition (3.3.2) and upon inverse Fourier
transform, we will be able to control ψ in terms of initial data on Σ0. The cost of this is
that ψQ satisfies the inhomogeneous wave equation
gψQ = F, where F = (γ)ψ + 2∇µγ∇µψ. (3.3.4)
Note that F is supported in
Sγ := {0 ≤ t∗ ≤ 1} ⊂
⋃
0 0 and any r+ < Re < ∞, there exist constants
CRe = CRe(K0,M) and Cδ = Cδ(K0,M) such that the following estimates hold for all
τ ≥ 0 (including the limit τ →∞):
37
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
• Integrated local energy decay for arbitrary r+ < Re <∞
∫ ∞
0
∫
Στ∩{r+≤r≤Re}
(
χ\|∂ψ|2 + |ψ − ψ∞|2
)
dt∗ ≤ CRe
∫
Σ0
JN [ψ] · nΣ0 . (3.3.6)
• Integrated local energy decay up to null infinity∫ τ
0
∫
Στ
(
r−1χ\|∇/ψ|2 + r−1−δχ\(Tψ)2 + r−1−δ(Zψ)2 + r−3−δ|ψ − ψ∞|2
)
dt∗
+
∫
H+
JNµ [ψ]n
µ
H+ +
∫
I+
JTµ [ψ]n
µ
I+
≤ Cδ
∫
Σ0
JN [ψ] · nΣ0 , (3.3.7)
Here χ\ is a cut-off function that vanishes in a neighbourhood of the physical space pro-
jection of the trapped set, see (3.6.2). In fact, a stronger version of (ILED) is proved in
phase space, namely (3.5.11).
As stated, Theorem 3.3.2 may be read as the improvement of a soft nonquantitative
statement to a uniform quantitative statement. In the case of axisymmetry, the assumption
that ψ is sufficiently integrable function can easily be removed in light of [DR10a] and
[DR11b]. In the general case, it can be seen in the context of a continuity argument.
From this point of view, Theorem 3.3.2 corresponds to the closedness part (see §3.7.4) and
the removal of the restriction (3.3.1) is then openness (see §3.7.3).
3.3.3 Carter’s separation
Carter discovered in [Car68] that the wave equation on a Kerr–Newman background can
be formally separated.
In the Kerr case, the authors of [DR10a], [DR11a] and [DRSR14] used Carter’s sepa-
ration as a geometric framework to derive frequency localised energy estimates. In partic-
ular, the frequency localisation captures trapping and superradiance in disjoint regions of
phase space. This is the key observation in proving (ILED). This new approach highlights
salient properties of individual modes and unites classical mode analysis and the vector
field method, both of which have long histories in the literature. Here we generalise this
approach to the Kerr–Newman spacetimes.
38
3.3. Frequency localisation
Oblate spheroidal harmonics
The separation of the wave equation will require the decomposition of ψ into oblate
spheroidal harmonics. Let ξ ∈ R and consider the following elliptic operator acting on the
dense subset of L2(S2) formed by the smooth functions on S2 :
Pξf = −∆/ S2f − (ξ2 cos2 θ)f.
We gather some useful facts from elliptic PDE theory in the following proposition.
Proposition 3.3.3. Let ξ ∈ R. The eigenvalues λ(ξ)m` of Pξ are real with corresponding
eigenfunctions of the form S
(ξ)
m`(cos θ)e
imφ . These eigenfunctions constitute a complete
orthonormal basis for L2(S2) and satisfy:(
Pξ − λ(ξ)m`
)
S
(ξ)
m`(cos θ)e
imφ = 0 with m, ` ∈ Z, ` ≥ |m|.
The functions S
(ξ)
m`(cos θ) are smooth in ξ and θ and λ
(ξ)
m` are smooth in ξ. Further
λ
(ξ)
m` + ξ
2 ≥ |m|(|m|+ 1)
and λ
(ξ)
m` + ξ
2 ≥ 2|mξ|.
For ξ = 0, these simplify to the standard spherical harmonics S
(0)
m` = Ym` and λ
(0)
m` =
`(`+ 1).
Proof. See [Are12a, §8.2].
The functions S
(ξ)
m` are known as oblate spheroidal harmonics. Turning to the Kerr–
Newman geometry, we will apply Proposition 3.3.3 with ξ = aω, where ω is the phase
space variable associated to Fourier transforming ψQ in time.
Performing the separation
Let ψ be and ψQ be as in Theorem 3.3.2. Since ψQ satisfies the integrability assumption
(3.3.1), it may be Fourier transformed in t. This allows us to separate the wave equation
(2.1.8) by first Fourier transforming gψQ = F in t and expanding in terms of the oblate
spheroidal harmonics. This leads to the following decomposition:
ψQ(t, r, θ, φ) =
Fourier expansion︷ ︸︸ ︷
1√
2pi
∫ ∞
−∞
∑
m,`>|m|
R
(aω)
m` (r) · S(aω)m` (cos θ)eimφ︸ ︷︷ ︸
Oblate spheroidal expansion
e−iωt dω (3.3.8)
39
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
where S
(aω)
m` (cos θ) and λ
(aω)
m` are the eigenfunctions and eigenvalues of Paω respectively
and
R
(aω)
m` (r) =
∫
S2
ψ̂Q(ω, r, θ,m) · S(aω)m` (cos θ)eimφdVS2 . (3.3.9)
Moreover, R
(aω)
m` (r) satisfies the radial equation[
∂r(∆∂r)− ω2
(
a2 − (a
2 + r2)2
∆
)
+
a2m2
∆
−2amω(2Mr −Q
2)
∆
− λ(aω)m`
]
R
(aω)
m` = F
(aω)
m` (3.3.10)
in the sense of L2(dω)`2(m, `). It is convenient to work with the coordinate r∗. Note that
in this coordinate system
{t, r∗, θ, φ} = R× R× [0, pi]× [0, 2pi),
whereas Boyer–Lindquist coordinate patch was only valid (modulo degeneration of angular
coordinates) in the range r ∈ (r+,∞).
We now define
u
(aω)
m` (r
∗) =
√
r2 + a2R
(aω)
m` (r). (3.3.11)
Writing (3.3.10) in the new coordinates, we have the radial Carter ODE :
d2
(dr∗)2
u
(aω)
m` (r
∗) +
(
ω2 − V (aω)m` (r)
)
u
(aω)
m` = H
(aω)
m` , (3.3.12)
where equality is meant in the sense of L2(dω)`2(m, `) and
V
(aω)
m` (r) =
2amω(2Mr −Q2)− a2m2 + ∆ · Λ(aω)m`
(r2 + a2)2
+
∆(3r2 + a2 +Q2 − 4Mr)
(r2 + a2)3
− 3∆
2r2
(r2 + a2)4
,
H
(aω)
m` (r) =
∆F
(aω)
m` (r)
(r2 + a2)1/2
and Λ
(aω)
m` = λ
(aω)
m` + a
2ω2,
which obeys Λ
(aω)
m` ≥ |m|(|m|+ 1) (3.3.13)
and Λ
(aω)
m` ≥ 2|amω|. (3.3.14)
Note that even though R
(aω)
m` is complex-valued, the potential V
(aω)
m` is real.
40
3.3. Frequency localisation
Physical space–Fourier space identities
The following identities are immediate consequences of Parseval’s identity and Plancherel’s
identity. They form the bridge between frequency-localised estimates and physical space
estimates.
For any fixed r > r+:∫ +∞
−∞
∑
m,`
∣∣∣u(aω)m` ∣∣∣2dω = ∫ +∞−∞
∫
S2
(ψQ)2 · (r2 + a2) dt dgS2 ,∫ +∞
−∞
∑
m,`
ω2
∣∣∣u(aω)m` ∣∣∣2dω = ∫ +∞−∞
∫
S2
(TψQ)2 · (r2 + a2) dt dgS2 ,∫ +∞
−∞
∑
m,`
∣∣∣u(aω)m` ′∣∣∣2dω = ∫ +∞−∞
∫
S2
(
∂r∗
(√
r2 + a2 · ψQ
))2
dt dgS2 ,
and∫ +∞
−∞
∑
m,`
Λ
(aω)
m`
∣∣∣u(aω)m` ∣∣∣2dω = ∫ +∞−∞ ∑m,`(λ(aω)m` + a2ω2)
∣∣∣u(aω)m` ∣∣∣2dω
=
∫ +∞
−∞
∫
S2
(−∆/ S2 − a2ω2 cos2 θ + a2ω2)|u|2dω dgS2
=
∫ +∞
−∞
∫
S2
(|∇/ S2u|2 + a2ω2 sin2 θ|u|2)dω dgS2
=
∫ +∞
−∞
∫
S2
(|∇/ S2ψQ|2 + a2 sin2 θ(TψQ)2) · (r2 + a2) dt dgS2 ,
The identities above immediately imply that for any fixed r > r+:∫ +∞
−∞
∫
S2
(ψQ)2 · r2 dt dgS2 ≤
∫ +∞
−∞
∑
m,`
∣∣∣u(aω)m` ∣∣∣2dω,∫ +∞
−∞
∫
S2
(TψQ)2 · r2 dt dgS2 ≤
∫ +∞
−∞
∑
m,`
ω2
∣∣∣u(aω)m` ∣∣∣2dω,∫ +∞
−∞
∫
S2
|∇/ψQ|2r2 dt dgS2 ≤
∫ +∞
−∞
∑
m,`
Λ
(aω)
m`
∣∣∣u(aω)m` ∣∣∣2dω
and
∫ +∞
−∞
∫
S2
(∂r∗ψQ)2 · r2 dt dgS2 ≤
∫ +∞
−∞
∑
m,`
2
∣∣∣u(aω)m` ′∣∣∣2 + 8∣∣∣u(aω)m` ∣∣∣2dω.
Boundary conditions
In view of the cut-off γ, the solution ψQ of (3.3.4) satisfies (3.3.2).
Recall that K = T + a
2Mr+−Q2 Φ is the null generator of the future horizon H+. Since
41
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
∂r∗ → K as r∗ → −∞ and ψQ is smooth at r = r+, Plancherel implies the following
asymptotic condition on u
(aω)
m` (r
∗) near the horizon:
∫ ∞
−∞
∑
m,`
∣∣∣∣(u(aω)m` )′(r∗) + i(ω − am2Mr+ −Q2
)
u
(aω)
m` (r
∗)
∣∣∣∣2dω (3.3.15)
is smooth in r and tends to 0 as r → r+.
A similar argument shows that (3.3.2) implies the following asymptotic condition on
u
(aω)
m` (r
∗) for large r (see [DRSR14, §5.3-5.4] for the details): There exists a sequence
{rn}∞n=1 such that rn →∞ and
lim
n→∞
∣∣∣(u(aω)m` )′(rn)− iωu(aω)m` (rn)∣∣∣ = 0 for almost every ω. (3.3.16)
Here, and in all that follows, u′ denotes a derivative with respect to r∗.
Almost everywhere regularity
The analysis that follows is focused on (3.3.12), which holds for u
(aω)
m` (r
∗) ∈ L2(dω)`2(m, `).
It is much more convenient to consider smooth solutions u
(aω)
m` of (3.3.12) satisfying the
boundary conditions (3.3.15) and (3.3.16).
Definition 3.3.3. Let ψQ satisfy (3.3.1) and (3.3.2) and define u(aω)m` (r∗) by (3.3.11) and
(3.3.9). Define Ω ⊂ R to be the set of frequencies ω such that for all m and `, H(aω)m`
is smooth and u
(aω)
m` is a smooth solution of (3.3.12) satisfying the boundary conditions
(3.3.15) and (3.3.16).
The following lemma allows for the reduction to classical solutions.
Lemma 3.3.4. The set {ω ∈ R} \ Ω has measure zero.
Proof. See [DRSR14, Lemma 5.4.1].
Since we will integrate over ω ∈ R to prove the physical space (ILED) estimate, it
suffices to prove frequency-localised energy estimates hold for almost every ω. Therefore,
we may restrict attention to smooth solutions u
(aω)
m` of (3.3.12) by considering ω ∈ Ω.
3.3.4 Frequency-localised energy current templates
We now turn our attention to the task of generating frequency-localised estimates. That
is, we will prove energy estimates for each u
(aω)
m` with ω ∈ Ω. To do this, we need frequency-
localised analogues of the energy currents of §2.2.3. In this section, templates for such
42
3.3. Frequency localisation
currents are described. Note that the dependence of u, H and V on aω,m and ` is
suppressed in this section. Let us also reiterate the warning that we use the notation
f ′ := dfdr∗ .
The frequency-localised conserved energy currents
The frequency-localised analogue of the conserved energy current JTµ [ψ] is
QT [u] = ωIm[u
′u¯]. (3.3.17)
From this we compute
Q′T [u] = ωIm[u
′′u¯+
∣∣u′∣∣2]
= ωIm[Hu¯− (ω2 − V )|u|2]
= ωIm[Hu¯].
The conservation identity for the QT current is∫ ∞
r+
Q′T [u](r) dr = QT [u](∞)−QT [u](r+). (3.3.18)
The boundary conditions (3.3.15) and (3.3.16) imply that
QT [u](r+) = −ω
(
ω − am
2Mr+ −Q2
)
|u(r+)|2 and QT [u](∞) = ω2|u(∞)|2, (3.3.19)
where
ω+ :=
am
2Mr+ −Q2 . (3.3.20)
Clearly, QT [u](∞) ≥ 0. Denoting
G☼ := {(ω,m) : ω(ω − ω+) < 0} , (3.3.21)
we have non-negativity of −QT [u](r+) for (ω,m) /∈ G☼. We refer to G☼ as the superradiant
regime. Thus the bulk term on the left hand side of (3.3.18) is positive in the non-
superradiant regime.
Since we assume a ≥ 0, (3.3.21) is equivalent to
G☼ = {(ω,m) : mω ∈ (0,mω+)} . (3.3.22)
We will use this simpler condition when discussing superradiance. Recall that the assump-
43
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
tion a ≥ 0 is made with no loss of generality, see 2.2.2.
The frequency-localised analogue of the conserved energy current JKµ [ψ] is
QK [u] = (ω − ω+)Im[u′u¯]. (3.3.23)
From this we compute
Q′K [u] = (ω − ω+)Im[Hu¯].
The boundary conditions (3.3.15) and (3.3.16) imply that
QK [u](r+) = −
(
ω − am
2Mr+ −Q2
)2
|u(r+)|2 and QK [u](∞) = ω(ω − ω+)|u(∞)|2.
(3.3.24)
The frequency-localised virial currents
The frequency-localised analogues of virial currents JX,wµ [ψ] (where X is in the ∂r∗ direc-
tion and w is some function) are naturally constructed from combinations of the following
templates: for arbitrary piecewise differentiable f(r∗), h(r∗) and y(r∗), define
Qf0 [u] = f [
∣∣u′∣∣2 + (ω2 − V )|u|2] + f ′Re(u′u¯)− 1
2
f ′′|u|2, (3.3.25)
Qh1 [u] = hRe(u
′u¯)− 1
2
h′|u|2, (3.3.26)
Qy2[u] = y[
∣∣u′∣∣2 + (ω2 − V )|u|2]. (3.3.27)
Note that Qf0 = Q
f ′
1 +Q
f
2 . We compute
(Qf0)
′ = 2f ′
∣∣u′∣∣2 − fV ′|u|2 − 1
2
f ′′′|u|2 + 2fRe(u′H¯) + f ′Re(uH¯),
(Qh1)
′ = h[
∣∣u′∣∣2 + (V − ω2)|u|2]− 1
2
h′′|u|2 + hRe(uH¯),
(Qy2)
′ = y′[
∣∣u′∣∣2 + (ω2 − V )|u|2]− yV ′|u|2 + 2yRe(u′H¯),
where we have made repeated use of the Carter ODE (3.3.12) and the simple identity
2Re(wz¯) = wz¯ + w¯z.
44
3.4. Properties of the potential
The frequency-localised red-shift current
The frequency-localised analogue of the red-shift current JNµ is
Qzred = z
[∣∣u′ + i (ω − ω+)u∣∣2 + (ω2 − V − |ω − ω+|2) |u|2] . (3.3.28)
Recall that ∆ = 0 at r = r+. This allows us to characterise ω+ by
ω2 − V (r+) = |ω − ω+|2. (3.3.29)
Coupling this to the boundary conditions (3.3.15) and (3.3.16) we have
∣∣u′(r+)∣∣2 = (ω − ω+)2|u(r+)|2 (3.3.30)
and
∣∣u′(∞)∣∣2 = ω2|u(∞)|2. (3.3.31)
In §3.5.2, the function z is chosen in such a way that z → ∞ as r → r+ to produce a
finite, non-zero boundary term for Qzred.
Let V˜ = V + |ω − ω+|2 − ω2. Then V˜ (r+) = 0 and V˜ ′(r) = V ′(r). Note that we are
referring to the value of r∗ for which r = r+. We compute
(Qzred)
′ = z′
∣∣u′ + i (ω − ω+)u∣∣2 − (zV˜ )′|u|2 + 2zRe ((u′ + i(ω − ω+)u)H¯) .
3.4 Properties of the potential
In §3.5, we will use the templates of §3.3.4 to derive frequency localised energy estimates
analogous to (ILED). The proof of these estimates hinges on the properties of the potential
V in the Carter ODE (3.3.12).
This argument in the Kerr case is due to Dafermos and Rodnianski and was first given
in the survey paper [DR11a]. The analysis of the potential V is extended to the Kerr–
Newman case here. In particular, the miraculous fact that trapping and superradiance
occur in disjoint regions of phase space in the Kerr case carries over to the Kerr–Newman
case. Furthermore, for solutions of (2.2.3) supported only on a single azimuthal frequency,
superradiance and trapping are disjoint in physical space as well. These properties of the
subextremal Kerr–Newman spacetimes are extremely fortunate as there is no a priori
reason to expect them to follow from the Kerr case.
Consider the potential V (r) for r ≥ r+. We begin by decomposing V into its frequency
45
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
dependent and independent parts:
V = V0 + V1,
where V0 =
2amω(2Mr −Q2)− a2m2 + ∆Λ
(r2 + a2)2
(3.4.1)
and V1 =
∆
(r2 + a2)4
((3r2 − 4Mr + a2 +Q2)(r2 + a2)− 3∆r2)
=
∆
(r2 + a2)4
(a2∆ + 2Mr3 − 2Q2r2)
>
∆
(r2 + a2)4
(a2∆ + 2r2(M2 −Q2)). (3.4.2)
We immediately see that V1 ≥ 0. We will focus mainly on proving properties of V0 and
show that these properties carry over (with slight perturbation) to V .
3.4.1 Critical points of the potential
Lemma 3.4.1. For ω ∈ Ω and any m, Λ:
1. The potential function V0 has at most one maximum, r
0
max, and one minimum, r
0
min,
on the interval (r+,∞).
2. If these extrema are achieved, r0min < r
0
max.
3. For all sufficiently large Λ, the value r0max is bounded uniformly from above provided
that either mω ≥ 0 or a2ω2 ≤ CΛ for some constant C. In the latter case the bound
for r0max may depend on C.
Proof. Compute
d
dr
V0 =
4amωM
(r2 + a2)2
+
(4r)
(
a2m2 − 2amω(2Mr −Q2))
(r2 + a2)3
+
Λ
(r2 + a2)3
[
2(r −M)(r2 + a2)− 4∆r] .
So
(r2 + a2)3
d
dr
V0 = −12amωMr2 + 4a2m2r + 4amω[Q2r +Ma2]
−2Λ [r3 − 3Mr2 + (2Q2 + a2)r + a2M] . (3.4.3)
46
3.4. Properties of the potential
Compute
d
dr
[
(r2 + a2)3
d
dr
V0
]
= −24amωMr + 4a2m2 + 4amωQ2
−2Λ [3r2 − 6Mr + (2Q2 + a2)] .
Letting σ =
amω
Λ
,
d
dr
[
(r2 + a2)3
d
dr
V0
]
= −6Λ
[
r2 − 2Mr + 4Mrσ − 2
3
(
Q2σ +
a2m2
Λ
)
+
1
3
(a2 + 2Q2)
]
.
The zeros of this function lie at
r1,2 = M(1− 2σ)±
√
M2(1− 2σ)2 − 1
3
[
a2(1− 2m
2
Λ
) + 2Q2(1− σ)
]
.
We now consider the cases σ ≥ 0 and σ < 0 separately.
Case σ ≥ 0 : In this case r2 < M < r+ so the only point where ddr
[
(r2 + a2)3 ddrV0
]
can
vanish on (r+,∞) is
r1 = M(1− 2σ) +
√
M2(1− 2σ)2 − 1
3
[
a2(1− 2m
2
Λ
) + 2Q2(1− σ)
]
.
Observe that Λ > 0 by Proposition 3.3.3, so (3.4.3) implies that
(r2 + a2)3
d
dr
V0 → −∞ as r →∞.
Since r1 is the only root of
d
dr
[
(r2 + a2)3 ddrV0
]
and (r2 + a2)3 and its derivative are pos-
itive on (r+,∞), ddrV0 has at most two zeros r0min and r0max, which must be the extrema
corresponding to their subscripts since
d2
dr2
V0(r
0
max) < 0 and
d2
dr2
V0(r
0
min) > 0.
Further, r0min < r
0
max.
If it turns out that r1 /∈ R, then ddr
[
(r2 + a2)3 ddrV0
]
is negative for all r ≥ r+, so ddrV0
can vanish at only one point, where V0 must attain a maximum.
47
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
Case σ < 0: In this case we need to do some work to show that r2 < M . We factorise
r2 = M(1− 2σ)
1−
√
1− a
2(1− 2m2Λ ) + 2Q2(1− σ)
3M2(1− 2σ)2
.
Since σ < 0, a2 +Q2 < M2 and Λ > m2:∣∣∣∣∣ a2(1− 2m
2
Λ )
3M2(1− 2σ)2
∣∣∣∣∣ < 13 and
∣∣∣∣ 2Q2(1− σ)3M2(1− 2σ)2
∣∣∣∣ < 23 .
Also, for 0 ≤ x < 1, √1− x > 1− x. Thus 1−√1− x < x. This implies
r2 < M(1− 2σ)
[
a2(1− 2m2Λ ) + 2Q2(1− σ)
3M2(1− 2σ)2
]
=
[
a2(1− 2m2Λ ) + 2Q2(1− σ)
3M(1− 2σ)
]
<
a2 + 2Q2
3M
<
3M2
M
.
So r2 < M and we may argue as in the previous case.
For the last statement of the lemma, we observe that as r → ∞, the behaviour of
(r2 + a2)3 ddrV0 is governed by (6Λ−12amωM)r2−2Λr3. For (6Λ−12amωM)r2−2Λr3 = 0,
r = 3M
(
1− 6amω
Λ
)
.
So if mω ≥ 0 or a2ω2 ≤ CΛ this quantity is bounded, consequently bounding r0max.
3.4.2 Superradiant frequencies are not trapped
Lemma 3.4.2. For ω ∈ Ω and any m, Λ:
V (r+) ≤ ω2
with equality only for ω = ω+. In particular, this implies
V0(rmin) ≤ ω2.
48
3.4. Properties of the potential
Proof. Recall that ∆ = r2+ − 2Mr+ + a2 +Q2 = 0 on the horizon and compute
ω2 − V (r+) = ω2 − 2amω(2Mr+ −Q
2)− a2m2
(r2+ + a
2)2
= ω2 − 2amω(2Mr+ −Q
2)− a2m2
(2Mr+ −Q2)2
=
ω2(2Mr+ −Q2)2 − 2amω(2Mr+ −Q2) + a2m2
(2Mr+ −Q2)2
=
(
ω(2Mr+ −Q2)− am
)2
(2Mr+ −Q2)2 ,
which is manifestly non-negative.
This result means that if rmin exists, it can only be ‘trapped’ for the threshold value
of the superradiant regime:
ω = ω+ =
am
2Mr+ −Q2 .
By Lemma 3.4.1, For rmin to exist, it is necessary that
d
drV (r+) < 0. The next lemma
shows that this is not the case for superradiant frequencies. Therefore, in the superradiant
regime, V0 has only a maximum (rmin is absent).
Lemma 3.4.3. Let ω ∈ Ω. If mω ≤ am
2
2Mr+ −Q2 , then there exists a c > 0 such that
d
dr
V (r+) ≥ d
dr
V0(r+) ≥ cΛ > 0.
Proof. To show the first inequality it suffices that
d
dr
V1(r+) =
2r2+(r+ −M)(Mr+ −Q2)
(r2+ + a
2)
> 0. (3.4.4)
We now show that ddrV0(r+) is positive. Noting that r
2
+ + a
2 = 2Mr+ −Q2, we have
(r2+ + a
2)3
d
dr
V0(r+) = (4amωM)(2Mr+ −Q2) + 4a2m2r+ − 8amrω(2Mr+ −Q2)
+2Λ(r+ −M)(2Mr+ −Q2). (3.4.5)
49
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
Using the superradiant condition we have
(2Mr+ −Q2)2 d
dr
V0(r+) = 4amωM + 4amωr+
−8amωr+ + 2Λ(r+ −M)
= 4amω(M − r+) + 2Λ(r+ −M)
= 2(r+ −M)(Λ− 2amω)
≥ 2(r+ −M)
(
Λ− 2a
2m2
2Mr+ −Q2
)
≥ 2(r+ −M)
(
(2Mr+ −Q2)Λ− 2a2m2
2Mr+ −Q2
)
> 2m2(r+ −M)
(
2Mr+ −Q2 − 2a2
2Mr+ −Q2
)
where we have used Proposition 3.3.3. Positivity holds since r+ > M .
Corollary 3.4.1. The conclusion of Lemma 3.4.3 can be extended to the range
amω ≤ a
2m2
2Mr+ −Q2 + αΛ, ω ∈ Ω, (3.4.6)
for sufficiently small constant α.
Proof. A small negative term just appears in our estimates:
(r2+ + a
2)2
d
dr
V0(r+) = 4amωM + 4amωr+ − 4αΛr+ − 8amωr+ + 2Λ(r+ −M)
= 4amω(M − r+) + 2Λ(r+ −M)− 4αΛr+
= 2(r+ −M)(Λ− 2amω)− 4αΛr+.
Using (3.4.6) we have
d
dr
V0(r+) ≥ 2(r+ −M)
(r2+ + a
2)2
(
Λ− 4αΛr+ − 2a
2m2
2Mr+ −Q2
)
− (4αΛr+)
(r2+ + a
2)
≥ 2(r+ −M)
(r2+ + a
2)2
(
(2Mr+ −Q2)(1− 4αr+)Λ− 2a2m2
2Mr+ −Q2
)
− (4αΛr+)
(r2+ + a
2)
> 2m2(r+ −M)
(
(2Mr+ −Q2)(1− 4αr+)− 2a2
2Mr+ −Q2
)
− (4αΛr+)
(r2+ + a
2)
,
so choosing α small enough we retain positivity.
The next result mathematically embodies the miraculous disunion of the superradiant
and trapped frequencies.
50
3.4. Properties of the potential
Lemma 3.4.4. For all a2 + Q2 < M2, ω ∈ Ω and 0 ≤ mω ≤ mω+ + αΛ there exists a
k > 1 such that
ω2 − V (rmax) < ω2 − V0(r0max) <
∆
2(r2+ + a
2)2
[
m2 − kΛ] < 0. (3.4.7)
Proof. It suffices to prove the lemma with α = 0.
We first consider the case when m
(
am
2Mr+−Q2 − ω
)
≤ |m|√Λ. In this case we have
ω2 − V0(r+) =
(
ω − am
2Mr+ −Q2
)2
≤ 2Λ.
Combining this with Corollary 3.4.1, we have
V0(r+ + δ)− ω2 ≥ bΛ
for some sufficiently small δ > 0 and even smaller .
In the case where ω2 ≤ Λ, we have
V0(r)− ω2 ≥ Λ
r2
+O
(
Λ
r3
)
− Λ as r →∞.
So taking r˜ sufficiently large and letting be sufficiently small,
V0(r˜)− ω2 ≥ bΛ.
Finally, consider the case where m
(
am
2Mr+
− ω
)
> |m| √Λ and ω2 > Λ.
Pick r0 such that
mω =
am2
2Mr0 −Q2 .
Then ω(2Mr0 −Q2) = am.
In this case, r0 will satisfy r0 ∈ [r+ + δ,R] for some δ > 0 and R <∞.
51
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
Now compute
ω2 − V0(r0) =
[
(r20 + a
2)2ω2 − 2amω(2Mr0 −Q2) + a2m2 −∆Λ
]
(r20 + a
2)2
=
[
(r20 + a
2)2ω2 − ω2(2Mr0 −Q2)2 −∆Λ
]
(r20 + a
2)2
=
[
ω2[(r20 + a
2)2 − (2Mr0 −Q2)2]−∆Λ
]
(r20 + a
2)2
=
∆
(r20 + a
2)2
[
ω2(r20 + a
2 + 2Mr0 −Q2)− Λ
]
=
∆
(r20 + a
2)2
[
a2m2(r20 + a
2 + 2Mr0 −Q2)
(2Mr0 −Q2)2 − Λ
]
<
∆
(r20 + a
2)2
[
a2m2r20
(2Mr0 −Q2)2 (1 +
a2
r20
+
2M
r0
)− Λ
]
.
But
(2Mr0 −Q2)2 = 4M2r20 − 4Q2Mr0 +Q4
> 4M2r20 − 4Q2r20.
Since a2 < r+ − δ < r0,
ω2 − V0(r0) < ∆
(r20 + a
2)2
[
a2m2
4(M2 −Q2)(1 +
a2
r20
+
2M
r0
)− Λ
]
<
∆
(r20 + a
2)2
[
m2(1− δ)− Λ]
which is negative by Proposition 3.3.3.
It is immediate that
ω2 < V0(r
0
max) ≤ V (r0max) ≤ V (rmax)
so that the characterisation above of the disunion of the superradiant and trapped fre-
quencies holds for the full potential.
3.4.3 Trapping for fixed azimuthal mode solutions
The following lemma shows that if we fix the azimuthal frequency m, then trapping occurs
outside the ergoregion.
Lemma 3.4.5. Let λ2 be a potentially small parameter and let λ1 and ω1 be potentially
large parameters, all of which are to be determined in §3.5. Recall that σ = amωΛ . Let m
52
3.4. Properties of the potential
be fixed, ω ∈ Ω and let (ω,Λ) lie in the trapped frequency regime
Fm\ =
{
(ω,Λ) : |ω| > ω1, λ2Λ ≤ ω2 ≤ λ−12 Λ
}
.
There exists a constant c > 0 such that the conditions |σ| ≤ c, m2 ≤ cΛ, c−1 ≤ Λ and
a2 +Q2 ≤ K20 < M2 imply that r0max > (1 +
√
2)M .
Proof. We showed in the proof of Lemma 3.4.1 that
(r2 + a2)3
d
dr
V0 = −12amωMr2 + 4a2m2r + 4amω[Q2r +Ma2]
−2Λ [r3 − 3Mr2 + (2Q2 + a2)r + a2M]
By the same lemma, r0max is the largest critical point of V0, so it suffices to show that
d
drV0(r = (1 +
√
2)M) > 0.
We compute
Λ−1 (r2 + a2)3
d
dr
V0|(r=(1+√2)M)
= −12σM2(1 +
√
2)2 + 4
a2m2
Λ
(1 +
√
2)M + 4σM [Q2(1 +
√
2) + a2]
−2
[
(1 +
√
2)3M3 − 3M2(1 +
√
2)2 + (2Q2 + a2)(1 +
√
2)M + a2M
]
= 4σ
[
−3M2(1 +
√
2)2 +MQ2(1 +
√
2) +Ma2
]
+ 4
a2m2
Λ
(1 +
√
2)M
−2
[
(7 + 5
√
2− 3(3 + 2
√
2))M3 + (Q2 + a2)(2 +
√
2)M
]
= 2(2 +
√
2)M
[
M2 − (Q2 + a2)]+ 4a2m2
Λ
(1 +
√
2)M
+4σM
[
−3M(3 + 2
√
2) +Q2(1 +
√
2) + a2
]
.
It is only the σ term which may be non-positive, so choosing c small enough completes
the proof. Since m is fixed and ω2 ∼ Λ, σ ∼ 1√
Λ
. Hence the choice of c can be made by
ensuring that Λ is large enough.
Remark The ergoregion is confined to
{
r < 2M < (1 +
√
2)M
}
. Thus the lemma above
implies that trapping occurs outside the ergoregion for modes of fixed azimuthal frequency.
This will be important in the continuity argument, see Lemma 3.7.4.
53
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
3.5 Frequency localised estimates
In this section we will construct frequency localised energy estimates that, upon summa-
tion and inverse Fourier transform, will yield the required physical space energy estimate
(ILED). To do this, it is useful to exploit the frequency specific behaviour of the potential
V obtained in §3.4. We begin by partitioning phase space into disjoint regimes in which the
potential displays certain distinctive properties. In particular, we deal with superradiance
and trapping separately in phase space.
3.5.1 Partitioning the frequency ranges
Let λ2 be a potentially small parameter and let λ1 and ω1 be potentially large parameters,
all of which are to be determined but are subject to the constraint
λ2λ1 = ω
2
1. (3.5.1)
This constraint will be enforced by choosing λ1, and ω1 as large as required and λ2 as
small as required, then either enlarging λ1 or shrinking λ2 (by a finite amount) to satisfy
(3.5.1), see §3.5.7.
We decompose phase space parametrised by the frequencies ω, m and Λ as follows :
Unbounded frequencies
F] = {(ω,m,Λ) : |ω| > ω1 or Λ > λ1}
• High superradiant frequencies
F☼ =
{
(ω,m,Λ) ∈ F] : Λ ≥
(
a
2Mr+ −Q2 + α
)−2
ω21, mω ∈ [0,mω+ + αΛ]
}
• Trapped frequencies
F\ =
{
(ω,m,Λ) ∈ F] : |ω| > ω1, λ2Λ ≤ ω2 ≤ λ−12 Λ, mω /∈ [0,mω+ + αΛ]
}
• Time dominated frequencies
F = {(ω,m,Λ) ∈ F] : |ω| > ω1, Λ < λ2ω2, mω /∈ [0,mω+ + αΛ]}
• Angular dominated frequencies
F] =
{
(ω,m,Λ) ∈ F] : Λ > λ−12 ω21, ω2 < λ2Λ, mω /∈ [0,mω+ + αΛ]
}
54
3.5. Frequency localised estimates
Bounded frequencies
F[,full = {(ω,m,Λ) : |ω| ≤ ω1, 0 ≤ Λ ≤ λ1}
• Near-stationary frequencies
F[,1 =
{
(ω,m,Λ) ∈ F[,full : |ω| ≤ ω0 1, 0 ≤ Λ ≤ λ1
}
• Non-stationary frequencies
F[,2 =
{
(ω,m,Λ) ∈ F[,full : ω0 < |ω| ≤ ω1, Λ ≤ λ1
}
The following lemma shows that the partitioning above indeed covers the whole of
phase space.
Lemma 3.5.1. For all a2 + Q2 < M2 and all (ω,m,Λ) satisfying (3.3.13) and (3.3.14),
for all choices of parameters ω1, λ2, the triple (ω,m,Λ) lies in exactly one of the frequency
ranges F☼, F\, F, F], or F[,full.
Proof. Observe that
|ω| ≥ ω1 and mω ∈
(
0,
am2
2Mr+ −Q2 + αΛ
]
⇒ Λ ≥
(
a
2Mr+ −Q2 + α
)−2
ω21.
Also note that the constraint (3.5.1) ensures that Λ ≤ λ−12 ω21 ⇒ Λ ≤ λ1. This in turn
implies that ω2 ≤ λ2Λ⇒ ω2 < ω21.
As discussed in §3.3.3, we restrict attention to frequencies ω ∈ Ω. For these frequencies,
the solutions u
(aω)
m` of (3.3.12) are smooth and satisfy the boundary conditions (3.3.15) and
(3.3.16).
The energy estimates for each frequency regime are presented below. The derivations
of the estimates are based on those for the Kerr case. Wherever details are omitted they
may be found in [DR11a, §11] for the high frequency range F] and [DRSR14, §8.7] for the
low frequency range F[.
3.5.2 High superradiant frequencies F☼
This is a large frequency regime in which superradiance occurs.
Proposition 3.5.1. Let ω ∈ Ω and (ω,m,Λ) ∈ F☼ and suppose 0 ≤ a2 +Q2 ≤ K20 < M2.
Let u be a smooth solution of (3.3.12) with boundary conditions (3.3.15) and (3.3.16).
55
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
Taking λ1, R∞ and E all sufficiently large, there exist functions f , h, ζ satisfying the
uniform bounds
|f |+ ∆−1r2∣∣f ′∣∣+ |h|+ |ζ| ≤ B,
f = 1, h = 0, ζ = 0 for r∗ ≥ R∗∞,
and positive frequency independent constants A and Γ such that, for sufficiently small
b > 0,
b
(∫ Re
r+
(
∣∣u′∣∣2 + (ω2 + Λ)|u|2dr∗) + ∫ rmax
r+
|u′ + i(ω − ω+)u|2
r − r+ dr
∗
)
+b(ω2 + Λ)
[
|u|2|r=r+ + |u|2|r=∞
]
≤ −
∫ ∞
r+
(
f ′ +Ah
)
Re(uH¯) + 2fRe(u′H¯)− 2Γ ζ
V˜
Re(u′ + i(ω − ω+)H¯)dr∗
+
∫ ∞
r+
EωIm(uH¯)dr∗. (3.5.2)
The key to the proof of this estimate is that trapping does not occur in this frequency
range, see Lemma 3.4.4. Before we proceed, we must take into account the frequency
independent part of the potential V1.
Lemma 3.5.2. In the unbounded frequency regime F], if the potential V0 attains a maxi-
mum at r0max, then the full potential V attains a maximum at rmax,∣∣rmax − r0max∣∣ ≤ cλ−11
Proof. Note that r0max is uniformly bounded above and away from r+ by Lemma 3.4.1.
(The lower bound follows from the fact that the maximum must lie beyond the value r1
defined in the proof of Lemma 3.4.1). That is,
c ≤ r0max − r+ ≤ C
where c and C do not depend on Λ. We first show that V has a critical point:
We know that ddrV0 = 0 at r
0
max > r1 + c1 and (r
2 + a2)3 ddrV0 → −∞ as r → ∞. In
light of the bound: ∣∣∣∣ ddrV1
∣∣∣∣ ≤ Cr−4,
we see that ddrV0 must eventually dominate
d
drV1, so that
d
drV = 0 at some rmax. Further-
more, this rmax is also bounded above.
We now locate the maximum of V :
56
3.5. Frequency localised estimates
Since r0max > r1 + c1, we can bound
d
dr
V0 > cλ1 in [rmin + δ, r1]
So for λ1 large enough,
d
dr
V0 > cλ1 − Cr−4 > 0 in [rmin + δ, r1].
Thus
d
dr
V > cλ1 > 0 in [rmin + δ, r1].
We can find rc ∈ (r1, r0max) such that
d
dr
V0 >
cλ1
2
in [rmin + δ, rc]
and
d
dr
(
(r2 + a2)3
d
dr
V0(r)
)
≤ −cΛr2 in [rc,∞).
Applying the mean value theorem, there exists an rd such that∣∣∣∣ ddr
(
(r2d + a
2)3
d
dr
V0(rd)
)∣∣∣∣∣∣rmax − r0max∣∣ = (r2d + a2)3∣∣∣∣ ddrV (rmax)− ddrV (r0max)
∣∣∣∣
Now since rmax and r
0
max are both greater than r1, we have∣∣rmax − r0max∣∣ ≤ cλ1 .
This bound on the location of rmax also tells us that a maximum occurs there.
Proof of Proposition 3.5.1. Recall from Proposition 3.3.3 that Λ ≥ |m|(|m|+ 1). Combin-
ing this with the superradiant condition
mω ≤ am
2
2Mr+ −Q2 ,
we have
ω2 ≤
(
am
2Mr+ −Q2
)2
<
(
a
2Mr+ −Q2
)2
Λ.
Also, ω2 + Λ > λ1 in F☼, so we conclude that Λ must be very large in this regime. Thus
a bound on |u′|+ Λ|u|2 will suffice.
We know from Lemmas 3.4.3 and 3.5.2, that in F☼, the potential V has only one critical
point, a maximum at rmax, uniformly bounded above and away from r+ by Lemma 3.4.1.
57
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
Now that we understand the behaviour of the potential in this regime we may construct
our frequency-localised current from the templates (3.3.25), (3.3.26), (3.3.28) and (3.3.17):
Q = Qf0 +AQ
h
1 + ΓQ
z
red − EQT ,
where A, E and Γ are positive frequency independent constants and f , h and z are real-
valued functions depending on the frequency triple (ω,m,Λ). The purpose of each term
in the above current is as follows:
• Applying Qf0 with suitable f gives us control over a non-negative definite expression
in |u|2 + Λ|u|2. However, the estimate we obtain will degenerate at r = rmax due to
the presence of V ′. We choose f such that
f =
−1 at r = r+,
0 at r = rmax,
1 for r ≥ R∞
and − fV ′ − 1
2
f ′′′ ≥ 0.
We further require that f ′ > 0 and f ′′′ < 0 in (r+, R∞). Such an f can be constructed
in [r+, R∞] by choosing f ′′′ = (r − rmax)3 and choosing the constants of integration
appropriately. Then we have the required control
(Qf0)
′ ≥ b(∣∣u′∣∣2 + Λ|u|2).
in (r+, R∞) \ rmax. (This degeneracy is due to the vanishing of f at rmax.)
• By (3.4.7), the degeneracy in the estimate for (Qf0) may be removed by adding a
large multiple of the current Qh1 , where h is a non-negative function supported in
[rmax − δ, rmax + δ] and
h(r) = 1 ∀ r ∈ (rmax − δ/2, rmax + δ/2).
• For non-superradiant frequencies, the boundary terms in the estimate for Qf0 +AQh1
have a favourable sign and can be controlled by simply subtracting a large multiple
of QT . In the superradiant regime, there is a lack of control on these boundary terms
as they have unfavourable sign.
We apply the current ΓQzred to ensure that the sum of all boundary terms on the
horizon has a favourable sign. We exploit the presence of the large parameter Λ by
58
3.5. Frequency localised estimates
rmax − δ rmax R∗∞
1
f
−1
r∗ = −∞ r∗ =∞
hζ
rmax − δ/2 rmax + δ/2 rmax + δ
Figure 3.5.1: The functions f , h and ζ.
taking z = −ΛV˜ −1(r)ζ(r) where
ζ(r) = 1 for r+ ≤ r ≤ rmax,
0 ≤ ζ(r) ≤ 1 for rmax < r < rmax + δ/2,
ζ(r) = 0 for rmax + δ/2 < r.
By choosing choosing A Γ and λ1 large enough,
Ah(V − ω2)−Ah′′ ≥ A(chΛ− h′′)− ΓΛ ≥ 0.
So the integrand on the left hand side of the estimate is positive and controls the
necessary terms. Note that we have again made crucial use of (3.4.7).
• Since we have the large parameter Γ, we can control the boundary terms by adding
the current −EQT with 2 < E Γ. Then the boundary terms have the ‘right’
sign. This will finalise the choice of Γ and A once we have chosen E, see the remark
below.
Applying the current constructed above yields the frequency-localised estimate for F☼.
Figure 3.5.1 illustrates the functions f , h and ζ in this construction.
Remark Let us emphasise that f , h and ζ are real-valued functions that can be uniformly
controlled so that the right hand side of (3.5.2) can be dominated by initial data (see §3.6).
The constants A and Γ are frequency independent parameters, however they depend on
the constant E. This constant will be used in applying the QT current in each frequency
range. The required size of E varies in each regime but it is always a large parameter. We
59
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
will finalise the choice of E in §3.5.7. This in turn finalises the choice of the constants A
and Γ. The constant b > 0 can always be replaced with a smaller positive value without
affecting the validity of (3.5.2).
3.5.3 Trapped frequencies F\
Consider the large frequency regime
F\ =
{
(ω,m,Λ) : |ω| > ω1, λ2Λ ≤ ω2 ≤ λ−12 Λ, mω /∈ [0,mω+ + αΛ]
}
,
This is the range in which trapping occurs. This means that any positive definite current
controlling all derivatives must necessarily degenerate at r = rmax.
Proposition 3.5.2. Let ω ∈ Ω and (ω,m,Λ) ∈ F\ and suppose 0 ≤ a2 +Q2 ≤ K20 < M2.
Let u be a smooth solution of (3.3.12) with boundary terms (3.3.15) and (3.3.16). Taking
ω1, R∞ and E all sufficiently large, there exist functions f and y satisfying the uniform
bounds
|f |+ ∆−1r2∣∣f ′∣∣+ |y| ≤ B,
f = 1, y = 0 for r∗ ≥ R∗∞
such that, for sufficiently small b > 0,
b
∫ Re
r+
[∣∣u′∣∣2 + |u|2 + (r − rmax)2(ω2 + Λ)|u|2] dr∗
+b(ω2 + Λ)
[
|u|2|r=r+ + |u|2|r=∞
]
≤ −
∫ ∞
r+
2fRe(u′H¯) + f ′Re(uH¯)− EωIm(uH¯)dr∗ −
∫ r3
r+
2yRe(u′H¯)dr∗. (3.5.3)
Proof. For trapped frequencies, the potential V0 may have at most two critical points
r0min < r
0
max. From Lemma 3.4.2, we know that ω
2 − V (r+) ≥ 0. To construct a current,
it is necessary to identify the region where (ω2 − V ) may be negative. Following the
argument in [DR11a, §11.5], we find that there exists r3 > r+,
V (r) ≤ ω2 − c
4
Λ ∀ r ∈ [r+, r3].
Either r3 is bounded above, or r3 = ∞. If r3 is finite, the potential V has a unique
nondegenerate maximum at some rmax in [r3,∞) which is Λ−1−close to r0max.
Since λ2Λ ≤ ω2 ≤ λ−12 Λ, it suffices to bound∣∣u′∣∣2 + Λ|u|2 or ∣∣u′∣∣2 + ω2|u|2.
60
3.5. Frequency localised estimates
However, we cannot control these quantities everywhere, the best we can hope for is an
estimate that degenerates precisely at rmax.
We use the templates (3.3.25), (3.3.26), (3.3.27) and (3.3.17) to construct a frequency-
localised current as follows:
Q = Qf0 −Qy2 − EQT ,
where E is a positive frequency independent constant and f and y are real-valued functions
depending on the frequency triple (ω,m,Λ). Of f , we require that f(r+) = 0, f
′ > 0 on
[r3,∞), f ′′′ < 0 on [r3, R∞), f changes sign from negative to positive at r = rmax (That
is, f(rmax) = 0, f
′(rmax) > 0) and f = 1 for r ≥ R∞.
We will take y supported in [r+, r3), with y
′ < 0 on [r+, r3). It remains to construct y
so that the left hand side of the estimate is non-negative and vanishes only at r = rmax.
In summary, it suffices that y be positive, monotonically decreasing and
− d
dr
y ≥ −Cy + C.
The function
y = CeCr
∫ r3
r
eCrdr = 1− eC(r−r3)
satisfies this differential inequality. For large enough E, the boundary conditions will have
the right sign due to the non-superradiant condition.
Figure 3.5.2 illustrates the functions f and y in this construction.
Noting that y and f vanish precisely at r = rmax, we have the estimate for the trapping
regime.
Remark The estimate for the trapped regime (3.5.3) reveals the nature of the trapped
set – the estimate for each trapped frequency must degenerate at exactly one point (where
V attains its maximum). The degeneration of these estimates carries over into physical
space: the physical space estimate must degenerate in a neighbourhood of the physical
space projection of the ‘trapped set’r : r ∈
∞⋂
L=1
⋃
l≥L
r
(aω)
m`
where r
(aω)
m` are the points rmax where the potential V attains its maximum in the F\
range.
61
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
r3 rmax R
∗
∞
1
f
−1
r∗ = −∞ r∗ =∞
y
Figure 3.5.2: The functions f and y.
3.5.4 Time dominated frequencies F
We have already dealt with the superradiant and trapped frequencies, so these obstructions
are not present here.
Proposition 3.5.3. Let ω ∈ Ω and (ω,m,Λ) ∈ F and suppose 0 ≤ a2 +Q2 ≤ K20 < M2.
Let u be a smooth solution of (3.3.12) with boundary conditions (3.3.15) and (3.3.16).
Taking ω1, λ1, λ
−1
2 , R∞ and E all sufficiently large, there exists a function f satisfying
the uniform bounds
|f | ≤ B and f = 1 for r∗ ≥ R∗∞,
such that, for sufficiently small b > 0,
b
∫ Re
r+
∆
r5
[∣∣u′∣∣2 + |u|2 + (ω2 + Λ)|u|2] dr∗ + b(ω2 + Λ)[|u|2|r=+ + |u|2|r=∞]
≤
∫ ∞
r+
EωIm(uH¯)− 2fRe(u′H¯)dr∗. (3.5.4)
Proof. Here ω2 dominates Λ so it suffices to estimate |u′|2 + ω2|u|2. We construct the
following current from the templates (3.3.27) and (3.3.17):
Q = Qf2 − EQT ,
where f is monotonically increasing with 12 ≤ f ≤ 1 in (r+, R∞), f = 1 for r ≥ R∞ and
E is a positive frequency independent constant. Again, f depends on the frequency triple
(ω,m,Λ). The subtraction of a large multiple of the conserved energy current EQT yields
62
3.5. Frequency localised estimates
boundary terms with favourable sign. We obtain
b
∫ Re
r+
[
f ′
∣∣u′∣∣2 + (f ′(ω2 − V )− fV ′) |u|2] dr∗ + b(ω2 + Λ)[|u|2|r=+ + |u|2|r=∞]
≤
∫ ∞
r+
EωIm(uH¯)− Re(u′H¯)dr∗
It just remains to check that the integrand of the left hand side of the estimate above
is positive and controls the desired quantity. This is done by choosing ω1, λ1 and λ2
appropriately.
3.5.5 Angular dominated frequencies F]
Proposition 3.5.4. Let ω ∈ Ω and (ω,m,Λ) ∈ F] and suppose 0 ≤ a2 +Q2 ≤ K20 < M2.
Let u be a smooth solution of (3.3.12) with boundary conditions (3.3.15) and (3.3.16).
Taking ω1, λ
−1
2 , R∞ and E all sufficiently large, there exist functions f and y satisfying
the uniform bounds
|f |+ ∆−1r2∣∣f ′∣∣+ |h| ≤ B,
f = 1, h = 0 for r∗ ≥ R∗∞
and a positive frequency independent constant A such that, for sufficiently small b > 0,
b
∫ Re
r+
[∣∣u′∣∣2 + |u|2 + (ω2 + Λ)|u|2] dr∗ + b(ω2 + Λ)[|u|2|r=+ + |u|2|r=∞]
≤ −
∫ ∞
r+
2fRe(u′H¯) + (f ′ +Ah)Re(uH¯)− EωIm(uH¯)dr∗. (3.5.5)
Proof. We just repeat the construction of the current used in the proof of Proposition
3.5.1, letting
Q = Qf0 +AQ
h
1 − EQT
with the same f , h and A and E. The argument is simpler than in the superradiant regime
as we may control the boundary terms directly.
3.5.6 The bounded frequency range F[
This bounded low-frequency regime depends on ω1 and λ1 but unlike the high frequency
regimes, the estimates in this section will hold for arbitrarily chosen (but finite) ω1 and
λ1.
63
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
We will consider four subcases. This requires the introduction of a small parameter
K˜0 which will be chosen later in this section.
Note that we do not need to distinguish between superradiant and non-superradiant
frequencies. In the near-stationary frequency range, we follow the approach of [DRSR14]
with by further decomposing F[,1 to take advantage of small parameters, axisymmetry
and non-vanishing of parameters respectively.
In light of Theorem 4.5.1, the desired estimate for the non-stationary frequencies may
be obtained directly.
In view of the boundedness of the frequency parameters in F[, we have∣∣u′∣∣2 + (ω2 + Λ)|u|2 ≤ max {1, λ1, ω1} (∣∣u′∣∣2 + |u|2),
so it suffices to estimate the quantity |u′|2 + |u|2.
The near-stationary subrange (small parameters case): |ω| ≤ ω0 and 0 ≤ a2 +
Q2 ≤ K˜20
Proposition 3.5.5. Let ω ∈ Ω and (ω,m,Λ) ∈ F[,1 and suppose 0 ≤ a2 +Q2 ≤ K˜20 . Let
u be a smooth solution of (3.3.12) with boundary conditions (3.3.15) and (3.3.16). Then
for all ω1 > 0, λ2 > 0, sufficiently small ω0 > 0 and K˜0 > 0 (depending on ω1 > 0 and
λ2 > 0), sufficiently large R∞ > Re and E > 2, there exist functions y, yˆ, χ2 and h,
satisfying the uniform bounds
|y|+ |yˆ|+ |h|+ |χ2| ≤ B,
y = 1, yˆ = 0, h = 0 for r∗ ≥ R∗∞,
such that
b
∫ Re
re
(∣∣u′∣∣2 + |u|2) dr∗ + b(∣∣u′∣∣2 + ω2|u|2)|r=∞
≤
∫ ∞
−∞
(
2(y + yˆ) Re(u′H) + hRe(uH) + EωIm(Hu) + χ2 (ω − ω+) Im(Hu)
)
. (3.5.6)
Proof. We construct a current analogous to that given in [DRSR14, §8.7.1]. All that is
required is that the following hold.
1. It is clear from (3.4.1) and (3.4.2) that for every −∞ < α < β < ∞, we may take
K˜0 and ω0 small enough that r ∈ [α, β]⇒ V − ω2 > 0.
2. By Lemma 3.4.1, we have V ′ < 0 for sufficiently large positive r∗.
64
3.5. Frequency localised estimates
ep
−1
R∗1 R
∗
1 R∗1 + 1 R
∗
2 − 1 R∗2 ep
−1
R∗2
1
2
1
1
2
+
hχ2
y
yˆ
−1
−1
2
− ′
−1
2
r∗ = −∞ r∗ =∞
R∗∞
Figure 3.5.3: The functions χ2, h, y and yˆ used in the proof of Proposition 3.5.5.
3. It follows from (3.4.4) and (3.4.5) and the smoothness of V that for sufficiently small
K˜0, we have V
′ > 0 for r near r+.
Now following the argument given in [DRSR14, §8.7.1], we may construct a current
from the templates (3.3.26), (3.3.27), (3.3.17) and (3.3.23):
Q[,1,small = Q
h
1 +Q
y
2 +Q
yˆ
2 − Ey(∞)QT + χ2QK .
Here, Qh1 is used to obtain a coercive estimate in a bounded interval [R1, R2], with R1
bounded away from the horizon and R2 bounded. This achieved by cutting off an indicator
function and introduces negative terms in the regions [R0, R1] and [R2, R3]. The currents
Qy2 and Q
yˆ
2 are constructed to remedy this.
It then remains to deal with the boundary terms. Choosing K˜0 small enough, we see
from (3.3.19) that this frequency regime is non-superradiant. Thus the subtraction of the
current Ey(∞)QT controls the boundary term at r∗ =∞. The χ2QK current is introduced
to absorb the boundary term at r∗ = −∞. The function χ2 is a smooth bounded cut-off
that is identically 1 in (−∞, R1] and identically 0 in [R2,∞). Applying the χ2QK current
gives control over the boundary term. Moreover, the bulk term
∫∞
−∞(χ2QK)
′ is supported
only in [R1, R2] and comes with a ω-weight. Since we already have a coercive estimate in
[R1, R2], we may absorb this term into the left hand side. See Figure 3.5.3 for the form of
the functions used in constructing the currents.
65
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
The near-stationary subrange (axisymmetric case): |ω| ≤ ω0 and m = 0
Proposition 3.5.6. Let ω ∈ Ω and (ω,m,Λ) ∈ F[,1 and suppose m = 0. Let u be a
smooth solution of (3.3.12) with boundary conditions (3.3.15) and (3.3.16). Then for all
ω1 > 0, λ2 > 0, sufficiently small ω0 > 0 sufficiently large R∞ > Re and E > 2, there
exist functions y, yˆ and h, satisfying the uniform bounds
|y|+ |yˆ|+ |h| ≤ B,
y = 1, yˆ = 0, h = 0 for r∗ ≥ R∗∞,
such that
b
∫ R∗+
R∗−
(∣∣u′∣∣2 + |u|2) dr∗ + b(∣∣u′∣∣2 + ω2|u|2)|r=∞
≤ −
∫ ∞
−∞
(
2(y + yˆ) Re(u′H) + hRe(uH) + EωIm(Hu)
)
. (3.5.7)
Proof. The properties of the potential V used to obtain (3.5.6) also hold here:
1. It is clear from (3.4.1) and (3.4.2) that if m = 0, for every −∞ < α < β < ∞, we
may take ω0 small enough that r ∈ [α, β]⇒ V − ω2 > 0.
2. By Lemma 3.4.1, we have V ′ < 0 for sufficiently large positive r∗.
3. There is no superradiance in the axisymmetric case, so it follows directly from Lemma
3.4.3 and the smoothness of V that for sufficiently small K˜0, we have V
′ > 0 for r
near r+.
The arguments from the proof of the estimate (3.5.6) may now be applied. The situa-
tion is simpler here: since there is no superradiance we do not need χ2. The result follows
from applying the following current, contructed from the templates (3.3.26), (3.3.27),
(3.3.17):
Q[,1,m=0 = Q
h
1 +Q
y
2 +Q
yˆ
2 − Ey(∞)QT ,
where E, h, y and yˆ are as before.
The near-stationary subrange (non-vanishing parameters case): |ω| ≤ ω0, m 6= 0
and a2 +Q2 ≥ K˜20
In this frequency regime, we exploit the non-vanishing of the parameters m 6= 0 and
0 < K˜20 ≤ a2 +Q2.
66
3.5. Frequency localised estimates
Proposition 3.5.7. Let ω ∈ Ω and (ω,m,Λ) ∈ F[,1 and suppose m 6= 0 and 0 < K˜20 ≤
a2 + Q2. Let u be a smooth solution of (3.3.12) with boundary conditions (3.3.15) and
(3.3.16),. Then for all ω1 > 0, λ2 > 0, sufficiently small ω0 > 0 (depending on K˜0),
sufficiently large R∞ > Re and E > 2, there exist functions y, yˆ and h, satisfying the
uniform bounds
|y˜|+ |y|+ |h|+ |χ1|+ |χ2| ≤ B
(
K˜0
)
,
and |y˜| ≤ B exp (−br) , y = 1, h = 0 for r∗ ≥ R∗∞,
such that
b
(
K˜0
)∫ R∗e
r∗e
(|u′|2 + |u|2) dr∗ + b(∣∣u′∣∣2 + ω2|u|2)|r=∞
≤
∫ ∞
−∞
(−2y˜Re(u′H)− hRe(uH)− 2yRe(u′H)) dr∗
+
∫ ∞
−∞
(−Eχ2ω Im(Hu)− 2χ1 (ω − ω+) Im(Hu)) dr∗. (3.5.8)
Proof. It suffices to show that we can adapt the argument given in [DRSR14, §8.7.2].
This amounts to verifying that
1. Since m 6= 0, there exists a constant b = b(K˜0) > 0 such that
(ω − ω+)2 =
(
ω − am
2Mr+ −Q2
)2
> b,
provided we choose ω0 small enough.
2. Since m 6= 0, Λ ≥ 2. So
V = Λr−2 +O(r−3) as r →∞.
This in turn implies that for every 1 α β < ∞, we may take ω0 small enough
that r ∈ [α, β]⇒ V − ω2 > br−2. This positivity allows for the use of a Qh1 current.
Armed with these properties of the potential, we can construct the following current
from the templates (3.3.26), (3.3.27), (3.3.23) and (3.3.17):
Q[,1,non−vanishing = Qh1 +Q
y˜
2 +Q
y
2 − χ1QK − Eχ2QT
for suitable functions h, y˜, y, χ1 and χ2 as given in [DRSR14, §8.7.2]. Refer to Figure
3.5.4 for the forms of these functions.
67
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
R∗1 R
∗
2 R
∗
∞
1 χ2
y
y˜
−1
r∗ = −∞ r∗ =∞
R∗3 e
p−1R∗3
h
χ1
Figure 3.5.4: The functions χ1, χ2, h, y and y˜ used in the proof of Proposition 3.5.7.
Application of the Qy˜2 current yields the coercive term in the estimate. A large param-
eter is needed to deal with the boundary term at r = r+. The Q
h
1 current is deployed to
provide this large parameter. The boundary term can then be dealt with by application
of χ1QK .
The cost of this construction is that it produces error terms that must then be absorbed
by application of the Qy2 current.
Finally, the boundary term at r =∞ is handled by subtracting Eχ2QT with E ≥ 2.
Remark The function y˜ is defined by
y˜(r∗) := − exp
(
−C
∫ r∗
−∞
υdr∗
)
, (3.5.9)
where υ(r) is a positive function such that
υ = ∆ near r+, υ = 1 when r
∗ ≥ R∗∞, |υ| ≤ B.
Note that y˜ (−∞) = −1 and y˜ (∞) = 0 . In particular y˜ < 0 and y˜′ 6= 0 in the r ≥ R∞
range.
This differs from all our other seed functions so the error term
∫∞
−∞ 2y˜Re(u
′H) gener-
ated by Qy˜2 must be handled separately from the other error terms, see §3.6.3.
68
3.5. Frequency localised estimates
The nonstationary subrange: |ω| ≥ ω0
The estimate in this final frequency range is relatively simple aside from the presence of
the horizon term
(|ω(ω − ω+)||u|2)r=r+ in (3.5.10). This term arises due to superradiant
frequencies in the nonstationary bounded frequency regime F[,2. There are no known
localised energy currents for dealing with these superradiant frequencies. As such, this
horizon term gives rise to a term 1{ω0≤|ω|≤ω1}∩{Λ≤λ−12 ω21}
∣∣∣u(aω)m` (−∞)∣∣∣2 on the right hand
side of (3.5.11) in the statement of Proposition 3.5.9. This troublesome term is controlled
by applying the quantitative mode stability result of Chapter 4 after summation, see §3.6.
Proposition 3.5.8. Let ω ∈ Ω and (ω,m,Λ) ∈ F[,2 and let u be a smooth solution of
(3.3.12) with boundary conditions (3.3.15) and (3.3.16). Then for all ω1 > 0, λ2 > 0,
sufficiently small ω0 > 0 (depending on K˜0), sufficiently large R∞ > Re and E > 2, there
exists a function y satisfying the uniform bounds
|y| ≤ B, and y = 1 for r∗ ≥ R∗∞,
such that
b (ω0, ω1)
∫ R∗e
r∗e
(|u′|2 + |u|2) dr∗ + b(∣∣u′∣∣2 + ω2|u|2)|r=∞
≤ B (|ω(ω − ω+)||u|2)r=r+ − ∫ ∞−∞ (2yRe(u′H)− EωIm(Hu)) . (3.5.10)
Proof. The argument of [DRSR14, §8.7.4] applies directly. We apply a current of the form
Q[,2 = Q
y
2 − EQT
where Q2 and QT are defined by (3.3.27), (3.3.17) repectively and
y(r∗) := exp
(
−C
∫ ∞
r∗
χR∗∞r
−2dr
)
.
Here C = C(ω0, ω1, λ2) is sufficiently large and the function χR∗∞ is smooth, which iden-
tically 1 on [r+, R∞ − 1) and identically 0 on [R∞,∞). Note that y|r∗≥R∗∞ = 1 and
y (−∞) = 0.
3.5.7 The general frequency-localised estimate
Note that the partitioning parameters λ1, λ2, ω0 and ω1 have been fixed in the proofs
contained in §3.5.2 to §3.5.6. Observe that these proofs all hold for all sufficiently large
but finite λ1 and all sufficiently small postive λ2. Let ω1 be fixed and let λ
∗
1 = λ1 and
69
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
λ∗2 = λ2. If it is already the case that we require λ1 so large that λ1 > (λ∗2)−1ω1, we make
a new choice of λ2 < λ
∗
2 to fulfil (3.5.1). In the case that we require λ2 small enough that
λ∗1 < λ
−1
2 ω1, we finalise the choice of λ1 by enlarging it so that (3.5.1) is satisfied.
Proposition 3.5.9. Under the hypotheses of Theorem 3.3.2, there exist frequency inde-
pendent positive constants b and E and bounded positive functions C
(aω)
m` (r
∗), D(aω)m` (r
∗)
and J
(aω)
m` (r
∗) such that for all frequency triples (ω,m,Λ) where ω ∈ Ω, the following holds
for each u
(aω)
m` satisfying (3.3.12) and the boundary conditions (3.3.15) and (3.3.16):
b
∫ R∗e
r∗e
(∣∣∣∣ ddr∗u(aω)m`
∣∣∣∣2 + ∣∣∣u(aω)m` ∣∣∣2 + χ(aω)m` (r)(ω2 + Λ)∣∣∣u(aω)m` ∣∣∣2
)
dr∗ + bω2|u|2|r=∞
≤ χ[,☼
∣∣∣u(aω)m` (−∞)∣∣∣2 + ∣∣∣∣∫ ∞
r+
[C
(aω)
m` (r
∗)− 2y˜]Re(u′H¯) +D(aω)m` (r∗)Re(uH¯) dr∗
∣∣∣∣
+
∣∣∣∣∫ ∞
r+
J
(aω)
m` (r
∗)Re(u′ + i(ω − ω+)uH¯) + EωIm
[
uH¯
]
dr∗
∣∣∣∣, (3.5.11)
where
χ[,☼(ω,m,Λ) = 1{ω0≤|ω|≤ω1}∩{Λ≤λ−12 ω21},
χ
(aω)
m` (r) =
{
(r − r(ω,m,`)max )2 for each (ω,m,Λ) ∈ F\,
1 for (ω,m,Λ) /∈ F\
,
C
(aω)
m` (r
∗) = B (|y|+ |f |+ 1) ,
D
(aω)
m` (r
∗) = B
(∣∣f ′ +Ah∣∣+ ∣∣f ′∣∣+ |h|)
and J
(aω)
m` (r
∗) = BΓ
∣∣∣∣ ζV˜
∣∣∣∣,
where B is a large positive constant.
Proof. We choose E and R∞ large enough and b > 0 small enough that (3.5.2), (3.5.3),
(3.5.4), (3.5.5), (3.5.6), (3.5.7), (3.5.8) and (3.5.10) all hold. Since every (ω,m,Λ) lies in
one of the frequency ranges for which we have a frequency localised estimate for ω ∈ Ω,
this establishes (3.5.11). The choice of E finalises the choice of the constants A and Γ in
(3.5.2) and (3.5.5).
All of the frequency dependent functions f , h, y and ζ are bounded by the functions
C
(aω)
m` , D
(aω)
m` and J
(aω)
m` . The precise degeneration of the current for trapped frequencies
was used to obtain χ
(aω)
m` . The horizon term χ[,☼
∣∣∣u(aω)m` (−∞)∣∣∣2 arises from the superradiant
frequencies in the bounded regime F[,2, see (3.5.10).
70
3.6. Proof of the conditional (ILED)
3.6 Proof of the conditional (ILED)
The time has come to return to physical space. We turn our frequency-localised estimate
(3.5.11) into a physical space estimate. We do this by summing the frequency localised
estimate (3.5.11) over m and `, integrating over ω and appealing to the Parseval-type
identities of §3.3.3. Since these identities hold in L2(dw) it suffices that we proved the
frequency localised estimates for almost every ω (see §3.3.3).
3.6.1 The physical space estimate
Proposition 3.6.1. Let ψ be a solution of (2.2.3) arising from smooth, compactly sup-
ported data on Σ0. Assume that ψQ defined by (3.3.3) satisfies (3.3.1) and (3.3.2). Then
there exist frequency independent constants b > 0 and B > E such that for any time τ > 0,
(including the limit τ →∞),
b
∫ τ
0
∫
Σt∩[r+,R∗e ]
[
(∂r∗ψQ)2 + ψ2Q + χ\
(
(TψQ)2 + (∇/ψQ)2
)]
dt∗ + b
∫
I+
JT [ψQ] · nI+
≤
∣∣∣∣∣∣∣∣
∫ ∞
−∞
∫ ∞
re
∑
m∈Z
`≥|m|
[C
(aω)
m` (r
∗)− 2y˜]Re(u′H¯) +D(aω)m` (r∗)Re(uH¯) dr∗dω
∣∣∣∣∣∣∣∣
+
∣∣∣∣∣∣∣∣
∫ ∞
−∞
∫ ∞
re
∑
m∈Z
`≥|m|
J
(aω)
m` (r
∗)Re(u′ + i(ω − ω+)uH¯) dr∗dω
∣∣∣∣∣∣∣∣
+B
∣∣∣∣∣∣∣∣
∫ ∞
−∞
∫ ∞
r+
∑
m∈Z
`≥|m|
ωIm
[
u
(aω)
m` H¯
(aω)
m`
]
dr∗dω
∣∣∣∣∣∣∣∣+B
∫
Σ0
JNµ [ψ]n
µ
Σ0
. (3.6.1)
The functions C
(aω)
m` (r
∗), D(aω)m` (r
∗) and J (aω)m` (r
∗) are as in Proposition 3.5.9 and
χ\(r) =
1√
2pi
∫ ∞
−∞
∑
m∈Z
`≥|m|
χ
(aω)
m` (r) · S(aω)m` (cos θ)eimφe−iωt dω (3.6.2)
in the L2(dω)`2(m, `) sense.
71
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
Proof. First note that∫ τ
0
∫
Σt∩[re,R∗e ]
[
(∂r∗ψQ)2 + ψ2Q + χ\
(
(TψQ)2 + (∇/ψQ)2
)]
dt∗
≤
∫ ∞
−∞
∫
Σt∩[re,R∗e ]
[
(∂r∗ψQ)2 + ψ2Q + χ\
(
(TψQ)2 + (∇/ψQ)2
)]
dt∗.
Applying the physical space–Fourier space identities in §3.3.3, we obtain for any τ > 0∫ ∞
−∞
∫
Σt∩[re,R∗e ]
[
(∂r∗ψQ)2 + ψ2Q + χ\
(
(TψQ)2 + (∇/ψQ)2
)]
dt∗
=
∫ ∞
−∞
∫
S2
∫ Re
re
[
(∂r∗ψQ)2 + ψ2Q + χ\
(
(TψQ)2 + (∇/ψQ)2
)]
ρ2 sin θ dθ dφ dr dt∗
=
∫ ∞
−∞
∑
m∈Z
`≥|m|
∫ R∗e
r∗e
∣∣∣∣ ddr∗u(aω)m`
∣∣∣∣2 + ∣∣∣u(aω)m` ∣∣∣2 + χ(aω)m` (r)(ω2 + Λ)∣∣∣u(aω)m` ∣∣∣2dr∗dω (3.6.3)
and since ∂tγ is not supported on I+,∫
I+
JT [ψ] · nI+ =
∫
I+
JT [ψQ] · nI+ =
∫ ∞
−∞
∑
m∈Z
`≥|m|
ω2
∣∣∣u(aω)m` (r =∞)∣∣∣2.
Applying (3.5.11) for ω ∈ Ω, summing over m and ` and integrating with respect to ω, we
72
3.6. Proof of the conditional (ILED)
have ∫ ∞
−∞
∑
m∈Z
`≥|m|
∫ R∗e
r∗e
∣∣∣∣ ddr∗u(aω)m`
∣∣∣∣2 + ∣∣∣u(aω)m` ∣∣∣2 + χ(aω)m` (r)(ω2 + Λ)∣∣∣u(aω)m` ∣∣∣2dr∗dω
+
∫ ∞
−∞
∑
m∈Z
`≥|m|
ω2|u|2r=∞dω
≤ b−1
∣∣∣∣∣∣∣∣
∫ ∞
−∞
∫ ∞
r+
∑
m∈Z
`≥|m|
[C
(aω)
m` − 2y˜]Re(u′H¯) +D(aω)m` (r∗)Re(uH¯) dr∗dω
∣∣∣∣∣∣∣∣
+ b−1
∣∣∣∣∣∣∣∣
∫ ∞
−∞
∫ ∞
r+
∑
m∈Z
`≥|m|
J
(aω)
m` Re(u
′ + i(ω − ω+)uH¯) dr∗dω
∣∣∣∣∣∣∣∣
+ b−1B
∣∣∣∣∣∣∣∣
∫ ∞
−∞
∫ ∞
r+
∑
m∈Z
`≥|m|
ωIm
[
u
(aω)
m` H¯
(aω)
m`
]
dr∗dω
∣∣∣∣∣∣∣∣
+B
∫
{ω0≤|ω|≤ω1}
∑
|m|(|m|+1)≤Λ
Λ≤λ−12 ω21
∣∣∣u(aω)m` (−∞)∣∣∣2dω.
Recall that we take B > E, where E is as in Proposition 3.5.9. The last term on the right
hand side is controlled by application of the quantitative mode stability result (Theorem
4.5.1). That is, by Theorem 4.8.2,∫
{ω0≤|ω|≤ω1}
∑
|m|(|m|+1)≤Λ
Λ≤λ−12 ω21
∣∣∣u(aω)m` (−∞)∣∣∣2dω ≤ BF[ ∫
Σ0
JNµ [ψ]n
µ
Σ0
.
73
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
Absorbing BF[ into B, we have thus obtained
b
∫ τ
0
∫
Σt∩[re,R∗e ]
[
(∂r∗ψQ)2 + ψ2Q + χ\
(
(TψQ)2 + (∇/ψQ)2
)]
dt∗ + b
∫
I+
JT [ψ] · nI+
≤
∣∣∣∣∣∣∣∣
∫ ∞
−∞
∫ ∞
re
∑
m∈Z
`≥|m|
[C
(aω)
m` − 2y˜]Re(u′H¯) +D(aω)m` Re(uH¯) dr∗dω
∣∣∣∣∣∣∣∣
+
∣∣∣∣∣∣∣∣
∫ ∞
−∞
∫ ∞
re
∑
m∈Z
`≥|m|
J
(aω)
m` Re(u
′ + i(ω − ω+)uH¯) dr∗dω
∣∣∣∣∣∣∣∣
+B
∣∣∣∣∣∣∣∣
∫ ∞
−∞
∫ ∞
r+
∑
m∈Z
`≥|m|
ωIm
[
u
(aω)
m` H¯
(aω)
m`
]
dr∗dω
∣∣∣∣∣∣∣∣+B
∫
Σ0
JNµ [ψ]n
µ
Σ0
. (3.6.4)
The low frequency estimates do not give control all the way up to the horizon. We therefore
couple the above estimate to an −multiple of the red-shift estimate (2.2.6) with R0 = re
and taking δ > 0 small enough that re+ δ < infω,m,` rmax(ω,m,Λ) allows us to extend the
radial region of integration on the left hand side to [r+, R
∗
e] and absorb last term on the
right hand side of (2.2.6):
bB−10
∫ τ
0
∫
Σt∩[r+,R0]
[
(∂r∗ψQ)2 + ψ2Q + χ\
(
(TψQ)2 + (∇/ψQ)2
)]
ρ2dt∗
+bB−10 (1− )
∫ τ
0
∫
Σt∩[R0,R∗e ]
[
(∂r∗ψQ)2 + ψ2Q + χ\
(
(TψQ)2 + (∇/ψQ)2
)]
ρ2dt∗
+b
∫
I+
JT [ψ] · nI+ +
∫
H+(0,τ)
JNµ [ψ]n
µ
H+
≤
∣∣∣∣∣∣∣∣
∫ ∞
−∞
∫ ∞
r+
∑
m∈Z
`≥|m|
[C
(aω)
m` − 2y˜]Re(u′H¯) +D(aω)m` Re(uH¯) dr∗dω
∣∣∣∣∣∣∣∣
+
∣∣∣∣∣∣∣∣
∫ ∞
−∞
∫ ∞
r+
∑
m∈Z
`≥|m|
J
(aω)
m` Re(u
′ + i(ω − ω+)uH¯) dr∗dω
∣∣∣∣∣∣∣∣
+ B
∣∣∣∣∣∣∣∣
∫ ∞
−∞
∫ ∞
r+
∑
m∈Z
`≥|m|
ωIm
[
u
(aω)
m` H¯
(aω)
m`
]
dr∗dω
∣∣∣∣∣∣∣∣+B
∫
Σ0
JNµ [ψ]n
µ
Σ0
.
Taking small enough and absorbing B−10 , and (1− ) into the constant b, we arrive at
the the result.
74
3.6. Proof of the conditional (ILED)
Remark In proving the estimate above, we appealed to the quantitative mode stability
results of Chapter 4. Note that appeal to mode stability is not necessary if we restrict
to a2 + Q2 M2 or require that m = 0 or m 1. In the former case superradiance
is absent. In the latter case, the unfavourably signed boundary terms that arise due to
superradiance may be dominated using the redshift current directly, see §1.5.
Proposition 3.6.2. Let ψ be a solution of (2.2.3) arising from smooth, compactly sup-
ported data on Σ0. Assume that ψQ defined by (3.3.3) satisfies (3.3.1) and (3.3.2). There
exists a uniform constant CRe > 0 such that for any time τ > 0, (including the limit
τ →∞), ∫ τ
0
∫
Σt∩[r+,R∗e ]
[
(∂r∗ψQ)2 + ψ2Q + χ\
(
(TψQ)2 + (∇/ψQ)2
)]
ρ2dt∗
+
∫
I+
JT [ψ] · nI+ +
∫
H+(0,τ)
JN [ψ] · nH+ ≤ CRe
∫
Σ0
JNµ [ψ]n
µ
Σ0
. (3.6.5)
Proof. It remains to control the right hand side of (3.6.1) by data. In order to control the
terms containing C
(aω)
m` , D
(aω)
m` and J
(aω)
m` , we take R∞ Re and split the integral in r∗
into two regions
B = {r+ ≤ r ≤ R∞} and U = {r > R∞} .
which we deal with separately. The integral over the compact region∫ ∞
−∞
∫
B
∑
m∈Z
`≥|m|
[C
(aω)
m` (r
∗)−2y˜]Re(u′H¯)+D(aω)m` (r∗)Re(uH¯)+J (aω)m` Re(u′+i(ω−ω+)uH¯) dr∗dω
is controlled §3.6.2 and the integral over the unbounded region∫ ∞
−∞
∫
U
∑
m∈Z
`≥|m|
[C
(aω)
m` (r
∗)−2y˜]Re(u′H¯)+D(aω)m` (r∗)Re(uH¯)+J (aω)m` Re(u′+i(ω−ω+)uH¯) dr∗dω
is controlled in §3.6.3.
The term ∫ ∞
−∞
∫ ∞
r+
∑
m∈Z
`≥|m|
ωIm
[
u
(aω)
m` H¯
(aω)
m`
]
dr∗dω
is dealt with in §3.6.4.
75
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
3.6.2 Error terms in B = {r+ ≤ r ≤ R∗∞}
In this section we control the following error term by data∫ ∞
−∞
∫
B
∑
m∈Z
`≥|m|
[C
(aω)
m` +J
(aω)
m` − 2y˜]Re(u′H¯) + Re
[(
D
(aω)
m` (r
∗) + i(ω − ω+)J (aω)m`
)
uH¯
]
dr∗dω.
In this region we want to absorb error terms into the left hand side of (3.6.1). For r∗ ∈ B:∣∣∣∣∣∣∣∣
∫ ∞
−∞
∑
m∈Z
`≥|m|
[C
(aω)
m` + J
(aω)
m` − 2y˜]Re(u′H¯) + Re
[(
D
(aω)
m` + i(ω − ω+)J (aω)m`
)
uH¯
]
dω
∣∣∣∣∣∣∣∣
≤ ε−1
∫ ∞
−∞
∑
m∈Z
`≥|m|
[
(C
(aω)
m` (r
∗))2 + (D(aω)m` (r
∗))2 + (J (aω)m` (r
∗))2 + 4y˜(r∗)2
]
(H
(aω)
m` )
2 dω
+ε
∫ ∞
−∞
∑
m∈Z
`≥|m|
(
[(u
(aω)
m` )
′]2 + (u(aω)m` )
2 + 1F☼(ω − ω+)2(u(aω)m` )2
)
dω.
Taking the physical/Fourier space identities in §3.3.3 into account, we have
ε
∫ ∞
−∞
∑
m∈Z
`≥|m|
(
[(u
(aω)
m` )
′]2 + (u(aω)m` )
2 + 1F☼(ω − ω+)2(u(aω)m` )2
)
dω
≤ εC
∫ ∞
0
∫
S2
∆
r2 + a2
[(r2 + a2)(∂r∗ψQ)2 + 1ˇF☼∂tψ2Q + ψ2Q] dt dgS2 .
Integrating this term over r∗ in the region B and taking ε small enough, this term can be
absorbed into the left hand side of (3.6.1). Note that this works for the ∂tψQ term as it
is only supported in the superradiant regime, where (3.6.1) does not degenerate due to
trapping.
Recall from Proposition 3.5.9 that the functions C
(aω)
m` (r
∗), D(aω)m` (r
∗), J (aω)m` (r
∗) and
y˜(r∗) are bounded (uniformly w.r.t (r, ω,m, `)) in the compact region B, so∫ ∞
−∞
∑
m∈Z
`≥|m|
ε−1
[
(C
(aω)
m` (r
∗))2 + (D(aω)m` (r
∗))2 + (J (aω)m` (r
∗))2 + (y˜(r∗))2
]
(H
(aω)
m` )
2 dω
≤ C
∫ ∞
−∞
∑
m∈Z
`≥|m|
ε−1
∆2
r2 + a2
(F
(aω)
m` )
2 dω
≤ C
∫ 1
0
∫
S2
ε−1
∆2
r2 + a2
F 2 dt dgS2 .
76
3.6. Proof of the conditional (ILED)
Now integrating over r∗ in the region B and recalling the definition of F we have∫ ∞
−∞
∫ R∞
r+
∑
m∈Z
`≥|m|
ε−1
[
(C
(aω)
m` )
2 + (D
(aω)
m` )
2 + (J
(aω)
m` )
2 + (y˜(r∗))2
] ∣∣∣H(aω)m` ∣∣∣2 dωdr∗
≤ Cε−1
∫ 1
0
∫ R∞
r+
∫
S2
∆2
r2 + a2
((γ)ψ + 2∇µγ∇µψ)2 dt∗ dgS2dr∗
≤ Cε−1
∫ 1
0
∫ R∞
r+
∫
S2
∆2
r2 + a2
[(γ)2ψ2 + |∇γ|2|∇ψ|2] dt∗ dgS2dr∗
≤ Cε−1
∫ 1
0
∫
Σt
JNµ [ψ]n
µ
Σt
dt∗.
Here we have used the Hardy inequality (2.2.8) in r to control the term containing ψ2. It
now follows from Proposition 3.3.1 that∫ 1
0
∫
Σ0
JNµ [ψ]n
µ
Σt
dt∗ ≤ eP
∫
Σ0
JNµ [ψ]n
µ
Σ0
.
We have thus controlled all error terms in the bounded region B.
3.6.3 Error terms in U = {r ≥ R∗∞}
We now control the term∫ ∞
−∞
∫
U
∑
m∈Z
`≥|m|
[C
(aω)
m` (r
∗)−2y˜]Re(u′H¯)+D(aω)m` (r∗)Re(uH¯)+J (aω)m` Re(u′+i(ω−ω+)uH¯) dr∗dω.
Recall that
C
(aω)
m` (r
∗) = B (|y|+ |f |+ 1) ,
D
(aω)
m` (r
∗) = B
(∣∣f ′ +Ah∣∣+ ∣∣f ′∣∣+ |h|)
and J
(aω)
m` (r
∗) = BΓ
∣∣∣∣ ζV˜
∣∣∣∣,
Looking back to §3.5, we see that ζ = h = y = f ′ = 0 and f = 1 in U . Therefore, C(aω)m` (r∗)
is constant and D
(aω)
m` (r
∗) = J (aω)m` (r
∗) = 0 in U . It thus remains only to estimate the term
(B − 2y˜)Re(u′H¯) in U .
Recall the definition of the inhomogeneity
F = (γ)ψ + 2∇µγ∇µψ.
77
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
For the term containing the function y˜, we note that y is not constant in U (see (3.5.9))
but since |y˜| ≤ exp (−br∗) as r∗ → ∞, we may apply Plancherel and a Hardy inequality
in r∗ to obtain∣∣∣∣∣∣∣∣∣
∫
{ω0≤|ω|≤ω1}
∑
|m|(|m|+1)≤Λ
Λ≤λ−12 ω21
∫ ∞
R∗∞
−2y˜Re(u′H¯)dr dω
∣∣∣∣∣∣∣∣∣
=
∣∣∣∣∣
∫ 1
0
∫ ∞
R∗∞
∫
S2
−2y˜ · Re
(
∂r∗
((
r2 + a2
)1/2
ψQ
)
∆ (r2 + a2)−1/2 F
)
sin θ dt dr∗ dθ dφ
∣∣∣∣∣
≤ B
∫ 1
0
∫
Σt∩[R∗∞,∞)
∆ exp (−br∗) ∣∣(γ∂r∗ψ + r−1(γ + ∂rγ)ψ) ((γ)ψ + 2∇µγ∇µψ)∣∣dt
≤ BC
∫ 1
0
∫
Σt
JN [ψ] · nΣt dt
≤ BCeP
∫
Σ0
JN [ψ] · nΣ0 ,
by Proposition 3.3.1.
Let us now deal with the term not containing y˜. By the physical/Fourier space iden-
tities in §3.3.3, and the support of γ,∫ ∞
−∞
∫ ∞
R∗∞
∑
m,`
Re(u′H¯)dωdr∗
=
∫ ∞
−∞
∫ ∞
R∞
∫
S2
Re
(
∂r∗
(
(r2 + a2)1/2ψQ
)
∆(r2 + a2)−1/2F
)
sin θdθ dφ dr∗ dt∗.
Observe that for R∞ large enough, γ depends only on t, so
F =
(
r2 + a2
)−1
ρ2
(
2gtt∂tγ∂tψ + 2g
tφ∂tγ∂φψ + g
tt∂2t γψ
)
for r ≥ R∞.
So (suppressing the factor sin θ dt dr dθ dφ),∫ ∞
0
∫ ∞
R∗∞
∫
S2
Re
(
∂r∗
((
r2 + a2
)1/2
ψQ
)
∆ (r2 + a2)−1/2 F
)
=
∫ ∞
0
∫ ∞
R∗∞
∫
S2
Re
(
∂r∗
((
r2 + a2
)1/2
ψQ
)
∆ (r2 + a2)−3/2 ρ2 (2gtt∂tγ∂tψ + 2gtφ∂tγ∂φψ)
)
+
∫ ∞
0
∫ ∞
R∗∞
∫
S2
Re
(
∂r∗
((
r2 + a2
)1/2
ψQ
)
∆ (r2 + a2)−3/2 ρ2gtt∂2t γψ
)
.
78
3.6. Proof of the conditional (ILED)
For the gtt∂tγ∂tψ term:∣∣∣∣∣
∫ ∞
0
∫ ∞
R∗∞
∫
S2
Re
(
∂r∗
((
r2 + a2
)1/2
ψQ
)
∆ (r2 + a2)−3/2 ρ2 (2gtt∂tγ∂tψ)
)∣∣∣∣∣
≤ B
∣∣∣∣∣
∫ ∞
0
∫ ∞
R∗∞
∫
S2
Re
(
(∂r∗ψQ) ∆ (r2 + a2)−1 ρ2 (2gtt∂tγ∂tψ)
)∣∣∣∣∣
+B
∣∣∣∣∣
∫ ∞
0
∫ ∞
R∗∞
∫
S2
r
(r2 + a2)1/2
Re
(
(ψQ) ∆ (r2 + a2)−3/2 ρ2 (2gtt∂tγ∂tψ)
)∣∣∣∣∣
≤ BeP
∫
Σ0
JNµ [ψ]n
µ
Στ
,
where the last inequality follows by applying the Hardy inequality (2.2.8) and Proposition
3.3.1.
Since gtφ = −a(2Mr −Q2)∆−1ρ−2 = O (r−3), the gtφ∂tγ∂φψ term can dealt with in
the same way.
Let χB be a smooth cut-off which is identically 1 for r ≤ R∞ − 1 and identically 0 for
r ≥ R∞. Then since ∂r∗γ = 0 for r ≥ Re,∣∣∣∣∣
∫ ∞
0
∫ ∞
R∗∞
∫
S2
Re
(
∂r∗
((
r2 + a2
)1/2
ψQ
)
∆ (r2 + a2)−3/2 ρ2gtt∂2t γψ
)∣∣∣∣∣
=
∣∣∣∣∣
∫ ∞
0
∫ ∞
R∗∞
∫
S2
χB∆
(
r2 + a2
)−2
ρ2gtt∂2t γγRe
(
∂r∗
((
r2 + a2
)1/2
ψ
)
(r2 + a2)1/2 ψ
)∣∣∣∣∣
=
1
2
∣∣∣∣∣
∫ ∞
0
∫ ∞
R∗∞
∫
S2
χB∂r∗
(
∆
(
r2 + a2
)−2
ρ2gtt∂2t γγ
) (
r2 + a2
) |ψ|2∣∣∣∣∣
≤ B
∫ 1
0
∫
Στ∩[Re∗ ,∞)
|ψ|2
r2
≤ BeP
∫
Σ0
JNµ [ψ]n
µ
Σ0
.
Again, the last inequality follows by applying the Hardy inequality (2.2.8) and Proposition
3.3.1. Combining everything implies∣∣∣∣∣∣
∫ ∞
R∗∞
∑
m,`
(∫ ∞
R∗∞
2Re
(
u′H
))
dω
∣∣∣∣∣∣ ≤ BeP
∫
Σ0
JNµ [ψ]n
µ
Σ0
.
79
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
3.6.4 Controlling the error from the conserved energy current
We first note that the constant B is frequency independent and that this term arises from
the energy identity for JTµ [ψQ]. By Parseval’s identity and expanding ET ,∣∣∣∣∣∣∣∣
∫ ∞
−∞
∫ ∞
r+
∑
m∈Z
`≥|m|
ωIm
[
u
(aω)
m` H¯
(aω)
m`
]
drdω
∣∣∣∣∣∣∣∣
=
∣∣∣∣∫ ∞−∞
∫ ∞
r+
∫
S2
Re[T (γψ)]F∆ dgS2drdt
∣∣∣∣
=
∣∣∣∣∫ 1
0
∫ ∞
r+
∫
S2
GRe (∂tγψ + γ∂tψ) dgS2drdt.
∣∣∣∣
≤
∣∣∣∣∫ 1
0
∫ R∞
r+
∫
S2
χBGRe (∂tγψ + γ∂tψ) dgS2drdt
∣∣∣∣ (3.6.6)
+
∣∣∣∣∫ 1
0
∫ ∞
R∞−1
∫
S2
(1− χB)GRe (∂tγψ + γ∂tψ) dgS2drdt
∣∣∣∣, (3.6.7)
were χB is a smooth cut-off which is identically 1 for r ≤ R∞ − 1 and identically 0 for
r ≥ R∞ and
G = ∆ρ
2
r2 + a2
(
2gtt∂tγ∂tψ + 2g
tφ∂tγ∂φψ + g
tt∂2t γψ
)
.
The integral over the bounded region, (3.6.6), can be controlled by data as in §3.6.2. The
other term requires more care.
Most of the terms in (3.6.7) can be dealt with painlessly:∣∣∣∣∫ 1
0
∫ ∞
−∞
∫
S2
∆ρ2(1− χB)
r2 + a2
gtφ∂tγRe
(
(∂tγψ) ∂φψ
)∣∣∣∣
=
∣∣∣∣∫ 1
0
∫ ∞
−∞
∫
S2
∆ρ2(1− χB)
r2 + a2
gtφ (∂tγ)
2 ∂φ |ψ|2
∣∣∣∣ = 0.
∣∣∣∣∫ 1
0
∫ ∞
−∞
∫
S2
∆ρ2(1− χB)
r2 + a2
gtφγ∂tγRe
(
(∂tψ) ∂φψ
)∣∣∣∣ ≤ C ∫ 1
0
∫
Στ
JNµ [ψ]n
µ
Στ
≤ C
∫
Σ0
JNµ [ψ]n
µ
Σ0
.
2
∣∣∣∣∫ 1
0
∫ ∞
−∞
∫
S2
∆ρ2(1− χB)
r2 + a2
gttγ∂tγ |∂tψ|2
∣∣∣∣ ≤ C ∫ 1
0
∫
Στ
JNµ [ψ]n
µ
Στ
≤ C
∫
Σ0
JNµ [ψ]n
µ
Σ0
.
80
3.6. Proof of the conditional (ILED)
It remains to deal with the following term:∣∣∣∣∫ 1
0
∫ ∞
−∞
∫
S2
∆ρ2(1− χB)
r2 + a2
(
Re
(
∂tγψ
(
2gtt∂tγ∂tψ + gtt∂2t γψ
))
+ Re
(
γ∂tψgtt∂2t γψ
))∣∣∣∣
≤
∫
Σ0
JNµ [ψ]n
µ
Σ0
+
∣∣∣∣∫ 1
0
∫ ∞
−∞
∫
S2
gtt
∆ρ2(1− χB)
r2 + a2
(
γ∂tγRe
(
∂2t ψψ
))∣∣∣∣ .
See [DRSR14, §9.6] for the derivation of this inequality. Instead of additional integration
by parts on the last term, we use that ψ solves the wave equation:
gtt∂2t ψ =
2a(2Mr −Q2)
ρ2∆
∂φ∂tψ − ∆− a
2 sin2 θ
∆ρ2 sin2 θ
∂2φψ
− r
2 + a2
∆ρ2
∂r∗
((
r2 + a2
)
∂r∗ψ
)− 1
ρ2 sin θ
∂θ (sin θ∂θψ) .
Substituting the right hand side of the wave equation above for gtt∂2t ψ and integrating by
parts in φ, r and θ, we have∣∣∣∣∫ 1
0
∫ ∞
−∞
∫
S2
gtt
∆ρ2(1− χB)
r2 + a2
(
γ∂tγRe
(
∂2t ψψ
))∣∣∣∣ ≤ B ∫ 1
0
∫
Σ0
JNµ [ψ]n
µ
Σ0
.
Now by Proposition 3.3.1,∣∣∣∣∫ 1
0
∫ ∞
−∞
∫
S2
gtt
∆ρ2(1− χB)
r2 + a2
(
γ∂tγRe
(
∂2t ψψ
))∣∣∣∣ ≤ B ∫
Σ0
JNµ [ψ]n
µ
Σ0
.
This concludes the proof of Proposition 3.6.2, our integrated local energy decay result
for ψQ.
3.6.5 Concluding the proof of the conditional (ILED)
We now show that (ILED) holds for ψ from the result for ψQ.
Proposition 3.6.3. For a solution ψ of (2.2.3) satisfying the hypotheses of Theorem 3.3.2
and for any time τ > 0, (including the limit τ →∞),∫ τ
0
∫
Σt∩[r∗+,R∗e ]
[
(∂r∗ψ)
2 + ψ2 + χ\
(
(Tψ)2 + |∇/ψ|2
)]
dt∗ (3.6.8)
+
∫
I+
JT [ψ] · nI+ +
∫
H+(0,τ)
JN [ψ] · nH+
≤ C
∫
Σ0
JNµ [ψ]n
µ
Σ0
, (3.6.9)
where C depends only on M , re, Re and P . Moreover, χ\ > 0 for r /∈ {re ≤ r ≤ R\ < Re}.
81
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
Proof. Let I[ψ] = (∂r∗ψ)
2 + ψ2 + χ\
(
(Tψ)2 + (∇/ψ)2).
Recall that Sγ := {0 ≤ t∗ ≤ 1} so ψ = ψQ in R(0, τ) \ Sγ ,∫
R(0,τ)∩[r∗+,R∗e ]\Sγ
I[ψ] =
∫
R(0,τ)∩[r∗+,R∗e ]\Sγ
I[ψQ].
By (3.6.5),∫
R(0,τ)∩[r∗+,R∗e ]
I[ψ] =
∫
R(0,τ)∩[r∗+,R∗e ]\Sγ
I[ψ] +
∫
R(0,τ)∩[r∗+,R∗e ]∩Sγ
I[ψ]
=
∫
R(0,τ)∩[r∗+,R∗e ]\Sγ
I[ψQ] +
∫
R(0,τ)∩[r∗+,R∗e ]∩Sγ
I[ψ]
≤
∫
R(0,τ)∩[r∗+,R∗e ]
I[ψQ] +
∫ 1
0
∫
Σt
JNµ [ψ]n
µ
Σt
dt∗
≤ C
∫
Σ0
JNµ [ψ]n
µ
Σ0
+ eP
∫
Σ0
JNµ [ψ]n
µ
Σ0
,
where we have applied Proposition 3.3.1.
We have thus proved (3.3.6).
3.6.6 Integrated decay up to null infinity
We now extend (3.3.6) up to null infinity, consequently proving (3.3.7). We make use of
the following energy estimate for large r.
Proposition 3.6.4. Fix M > 0 and a2 +Q2 ≤ K20 < M2. Let ψ be a solution of (2.2.3)
satisfying the hypotheses of Theorem 3.3.2 and ψ∞ = 0. For any δ > 0, there exist
positive constants B(δ), 2M < R0 < Re, such that for any time τ > 0, (including the
limit τ →∞),∫ τ
0
∫
Σs∩{r≥Re}
r−1(r−δ|∂rψ|2 + r−δ|∂tψ|2 + |∇/ψ|2 + r−2−δψ2) ds
≤ B(δ)
(∫
Σ0
JNµ [ψ]n
µ
Σ0
+
∫
Στ
JNµ [ψ]n
µ
Στ
)
.
Proof. Following [DR11a, §6], we let δ > 0 and apply the current JX,w[ψ] with
w = 2f ′(r∗) + 4
r − 2M
r2
f(r∗)− 2δ r − 2M
r2+δ
f(r∗),
X = f(r∗)∂r,
f(r∗) = χ(1− r−δ),
82
3.6. Proof of the conditional (ILED)
where χ is a smooth cut-off such that χ = 1 for r ≥ Re, χ = 0 for r ≤ Re − 1. Then
KX,w[ψ] =
(
r
r − 2Mf
′(r∗)− f(r
∗)δ
2r1+δ
)
(∂r∗ψ)
2 +
f(r∗)δ
2r1+δ
(∂tψ)
2
+
(
r − 3M
r2
− δ(r − 2M)
r2+δ
)
f(r∗)|∇/ψ|2 − 1
2
(w)(ψ2)
Taking Re large enough
KX,w[ψ] ≥ b(δ)
(
r−1−δ(∂r∗ψ)2 + r−1−δ(∂tψ)2 + r−1|∇/ψ|2 + r−3−δψ2
)
for r ≥ Re.
We now apply the energy identity between Σ0 and Στ . Since ∂rχ is compactly supported
and Re− 1 > R\, we can use (3.3.6) to control the spacetime error terms it generates. We
must control the error term∫ τ
0
∫
Σs
EV,w[ψ]ds
=
∫ τ
0
∫
Σs∩{r≥Re−1}
(χ(1− r−δ)∂rψ)F + 1
4
(wψF )
=
∫ 1
0
∫
Σs∩{r≥Re−1}
(χ(1− r−δ)∂rψ + 1
4
wψ)
(
2gtt∂tγ∂tψ + 2g
tφ∂tγ∂φψ + g
tt∂2t γψ
)
.
The terms involving only first order derivatives of ψ can immediately be controlled using
Proposition 3.3.1.
For zeroth order terms we first use a Hardy inequality in r and then Proposition 3.3.1.
This can immediately be done for terms containing w as they come with the required
weights in r−1.
The remaining term is dealt with by integrating by parts:∫ 1
0
∫
Σs∩{r≥Re−1}
gttχ(1− r−δ)(∂2t γ)(ψ∂rψ)ds
=
∫ 1
0
∫
Σs∩{r≥Re−1}
[
gtt(∂rχ)(1− r−δ)(∂2t γ)ψ + ∂rgttχ(1− r−δ)(∂2t γ)ψ
+gttχ(1− r−δ)(∂2t γ)∂rψ + gttχ(δr−1−δ)(∂2t γ)ψ
]
(ψ).
The first term is compactly supported so it can be controlled by (3.3.6). The other terms,
apart from the last one all have sufficient weights in r−1 to apply Hardy inequalities and
use Proposition 3.3.1. We integrate the final term by parts to obtain∫ ∞
∞
∫
Σs∩{r≥Re−1}
gttχ(δr−1−δ)(∂2t γ).(ψ
2) =
∫ 1
0
∫
Σs∩{r≥Re−1}
gttχ(δr−1−δ)(∂tγ)(ψ∂tψ).
83
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
After Cauchy-Schwarz, this term has the required r-weight to apply Hardy inequalities and
use Proposition 3.3.1. Finally, we use Hardy inequalities in r to control
∫
Σ J
X,w[ψ] · nΣ ≤
C
∫
Σ J
N [ψ] · nΣ on the spacelike hypersurfaces.
Remark Recall from §3.3.1 that the assumption ψ∞ = 0 can be made without loss of
generality.
We must take care in applying Proposition 3.6.4 since we do not have an (NEB) result
yet and the right hand side has an error term supported on Στ . However, the assumption
(3.3.1) implies that ∫
Στ∩[r+,Re]
JNµ [ψ]n
µ
Στ
∈ L1τ [0,∞).
By the pigeon-hole principle, there exists a constant C and a dyadic sequence τn → ∞
such that ∫
Στn∩[r+,Re]
JNµ [ψ]n
µ
Στn
≤ C
τn
. (3.6.10)
Since T is timelike in the region r ≥ Re we may apply its associated energy estimate∫
Στn
JNµ [ψ]n
µ
Στn
=
∫
Στn∩[r+,Re]
JNµ [ψ]n
µ
Στn
+B
∫
Στn∩[Re,∞)
JTµ [ψ]n
µ
Στn
≤ C
τn
+ C
∫
H+(0,τn)
JNµ [ψ]n
µ
H+ + C
∫
Σ0
JNµ [ψ]n
µ
Σ0
. (3.6.11)
Adding an -multiple of the estimate of Proposition 3.6.4 to (3.3.6) and then applying
(3.6.11), we have∫ τn
0
∫
Σs∩{r≥Re}
(
r−1χ\|∇/ψ|2 + r−1−δχ\(Tψ)2 + r−1−δ(Zψ)2 + r−3−δψ2
)
ds
+
∫ τn
0
∫
Σs∩{r≥Re}
(
r−1χ\|∇/ψ|2 + r−1−δχ\(Tψ)2 + r−1−δ(Zψ)2 + r−3−δψ2
)
ds
+ (b− )C
∫
H+(0,τn)
JNµ [ψ]n
µ
H+ + C
∫
I+
JNµ [ψ]n
µ
I+
≤ B (δ)
∫
Σ0
JNµ [ψ]n
µ
Σ0
+
C
τn
.
Taking small enough and letting n → ∞ yields (3.3.7). Since we proved (3.3.6) in
Proposition 3.6.3, this concludes the proof of the conditional Theorem 3.3.2.
3.7 The continuity argument
At this stage, we have proved Theorem 3.3.2 (and consequently Theorem 3.2.2) for solu-
tions of (2.2.3) which are assumed to be sufficiently integrable in the sense of (3.3.1). We
84
3.7. The continuity argument
now remove this assumption by proving the following:
Proposition 3.7.1. Let M > 0 and a2 + Q2 ≤ K20 < M . All solutions ψ to the wave
equation (2.2.3) (arising from smooth, compactly supported initial data on Σ0) are future
integrable.
We follow the strategy of [DRSR14] in first considering modes of fixed azimuthal
frequency. Since Φ is a Killing field, it commutes with the D’Alembertian g. Thus for
each azimuthal frequency m ∈ Z, the projection Pm of ψ to its mth azimuthal mode,
Pmψ(t, r, θ, φ) = ψ˜(t, r, θ)e
imφ, is well defined. Furthermore, g(Pmψ) = 0.
3.7.1 The reduction to fixed azimuthal frequency
Lemma 3.7.1. It suffices to prove Proposition 3.7.1 for solutions ψ to (2.2.3) supported
on a single fixed azimuthal frequency m ∈ Z.
Proof. Let ψ solve (2.2.3). The fundamental theorem of calculus implies that
B−1 sup
r∈[r+,A]
∫ ∞
0
∫
S2
∑
1≤i1+i2+i3≤j
|∇/ i1T i2(Z)i3ψ|2 sin θ dt dθ dφ
≤
∫
H+
∑
1≤i1+i2+i3≤j
|∇/ i1T i2(Z)i3ψ|2 +
∫ ∞
0
∫
Σs∩[r+,A]
∑
1≤i1+i2+i3≤j+1
|∇/ i1T i2(Z)i3ψ|2ds.
Suppose we have established Proposition 3.7.1 for solutions supported on any fixed az-
imuthal frequency. We may then use the orthogonality of the azimuthal modes to expand
ψ =
∑
m∈Z ψm. Since each ψm is future-integrable, we have (ILED) and (3.2.1) for each
ψm, verifying that (3.3.1) holds in the future of Σ0.
Now it remains to prove the following:
Proposition 3.7.2. Let M > 0, a2 + Q2 ≤ K20 < M2, m ∈ Z and ψ be a solution of
(2.2.3) that is supported only on the azimuthal frequency m. Then ψ satisfies (3.3.1).
Our main tool in the proof of is a version of (ILED) for fixed mode azimuthal mode
solutions, where we do not assume ψ is integrable a priori.
Lemma 3.7.2. Under the hypotheses of Proposition 3.7.2, for every τ ≥ 0, j ≥ 1 and
δ > 0,
85
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
∫ τ
0
∫
Σs
(
r−1χm\
(|∇/ψ|2 + r−δ |Tψ|2 )+ r−1−δ |Zψ|2 + r−3−δ |ψ|2) ds+ ∫
H+(0,τ)
JNµ [ψ]n
µ
H+
≤ B(δ,m)
(∫
Σ0
JN [ψ] · nΣ0 +
∫
Στ
JN [ψ] · nΣτ
)
(3.7.1)
and
∫ τ
0
∫
Σs
r−1−δ
∑
1≤i1+i2+i3≤j−1
(
|∇/ i1T i2(Z)i3+1ψ|2 + |∇/ i1T i2Zi3ψ|2
)
+χm\
∑
1≤i1+i2+i3≤j
|∇/ i1T i2(Z)i3ψ|2
ds
+
∫
H+(0,τ)
∑
1≤i1+i2+i3≤j
|∇/ i1T i2(Z)i3ψ|2
≤ B(δ, j,m)
∫
Σ0
∑
0≤i≤j−1
JN [N iψ] · nΣ0 +
∫
Στ
∑
0≤i≤j−1
JN [N iψ] · nΣτ
, (3.7.2)
where χm\ =
(
1− 1{(1+√2)M≤r≤R\}
)
.
Proof. For the first statement, we modify the cut-off γ of §3.3.1 and repeat the arguments
of §3.6. That is, let γ = 1 identically between Σ1 and Στ−1 and identically 0 to the past
of Σ0 and the future of Στ . This allows us to remove the assumption that ψ is future
integrable at the expense of picking up extra terms supported on Στ on the right hand
side of the estimates (3.7.1) and (3.7.2).
It is crucial that the degeneration of the estimate is encapsulated in χm\ rather than
χ\. This is due to Lemma 3.4.5, which tells us that for fixed m and large Λ the trapped
set is contained in
{
r ∈ [(1 +√2)M,∞)}.
The second statement follows from the first in the same way that (3.2.2) follows from
(3.2.1).
Remark Recall from Lemma 3.4.5 that the degeneration due to trapping lies outside
the ergoregion in this fixed azimuthal frequency case. This is extremely useful in the
subsequent argument.
The following corollary will be our final reduction of the problem.
86
3.7. The continuity argument
Corollary 3.7.3. Under the hypotheses of Proposition 3.7.2, ψ is future-integrable if
sup
τ≥0
∫
Στ
∑
1≤i1+i2+i3≤j
|∇/ i1T i2(Z)i3ψ|2 <∞ ∀ j ≥ 1. (3.7.3)
Proof. As in the proof of Lemma 3.7.1, observe that
[B(j)]−1 sup
r∈[r+,A]
∫ ∞
0
∫
S2
∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψ|2 sin θ dt dθ dφ
≤
∫
H+
∑
1≤i1+i2+i3≤j
|∇/ i1T i2(Z)i3ψ|2 +
∫ ∞
0
∫
Σs∩[r+,A]
∑
1≤i1+i2+i3≤j+1
|∇/ i1T i2(Z)i3ψ|2ds,
and apply (3.7.2) to the right hand side of this estimate.
3.7.2 The setting and non-emptiness
We are now ready to run a continuity argument in the parameter Q to prove Proposi-
tion 3.7.2. Proposition 3.7.1 then follows immediately by Corollary 3.7.3.
We fix M > 0 and a such that |a| < M and define for each m ∈ Z, the set
Qa,m := {Q2 ∈ [0,M2 − a2) : (3.7.3) holds for g = ga,Q,M}.
We will prove that Qa,m = [0,M2− a2) by showing that it is non-empty, open and closed.
Proposition 3.7.2 then follows by Corollary 3.7.3.
We begin with non-emptiness.
Proposition 3.7.4. For all m ∈ Z and a such that |a| < M , the set Qa,m is non-empty.
Proof. When Q = 0, the argument of [DRSR14, §11] shows that (3.7.3) holds. Thus
0 ∈ Qa,m.
Remark This appeal to the result of [DRSR14] is not necessary. We could just as well
have run the continuity argument in both parameters a and Q, reproving the result of
[DRSR14] in the process. In the interest of a clean and brief presentation, we just use the
continuity of the metric in Q here.
3.7.3 Openness
In this section, we prove
87
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
Proposition 3.7.5. For all m ∈ Z and a such that |a| < M , the set Qa,m is open. That
is, suppose Q˚ ∈ Qa,m. Then there exists > 0 such that∣∣∣Q− Q˚∣∣∣ < =⇒ Q ∈ Qa,m.
The proof is in two parts. We first prove a derivative gaining (NEB)-type estimates
in the spirit of §3.8.2 in §3.7.3. We then define a metric that interpolates between gM,a,Q
and gM,a,Q˚ and use the estimates of §3.7.1 and §3.7.3 to prove that if Q and Q˚ are close
enough and (3.7.3) holds for gM,a,Q, it also holds for gM,a,Q˚.
Gaining derivatives
We begin by defining the following smooth, locally Killing, globally timelike vector field
Definition 3.7.3. Let a2 +Q2 ≤ K20 < M2 and take 0 > 0 as in Lemma 2.2.3. Let α(r)
be a function such that V := T +α(r)Φ is a smooth vector field, timelike in M, satisfying
V = T +
a
r2+ + a
2
Φ, for r ∈ [r+, r+ + 0/2],
V = T +
a(r2 + a2 −∆)
(r2 + a2)2
Φ, for r ∈
[
r+ + 0,
M
(
7 +
√
2
)
4
]
,
V = T, for r ≥ M
(
3 +
√
2
)
2
.
Since 2M <
M(3+
√
2)
2 < M
(
1 +
√
2
)
, V is Killing in the region where trapping occurs
for fixed azimuthal frequency m (see Lemma 3.4.5). Because of this, the error terms arising
from the energy identity associated to V can be controlled by (3.7.1) and (3.7.2).
The following lemma is used as a converse to Lemma 3.7.2. However, we gain a
derivative in the sense that the spacetime integral term on the right hand side of (3.7.4)
is zeroth order.
Lemma 3.7.4. Let M > 0, a2+Q2 ≤ K20 < M2, m ∈ Z and ψ be a solution of (2.2.3) that
is supported only on the azimuthal frequency m. Then there exists a constant B = B(m)
such that for all τ ≥ 0,
∫
Στ
JNµ [ψ]n
µ
Στ
≤ B(m)
∫ τ
0
∫
Σs∩
{
r≤M(3+
√
2)
2
} |Φψ|2 ds+
∫
Σ0
JNµ [ψ]n
µ
Σ0
≤ B(m)
∫ τ
0
∫
Σs∩
{
r≤M(3+
√
2)
2
} |ψ|2 ds+
∫
Σ0
JNµ [ψ]n
µ
Σ0
. (3.7.4)
88
3.7. The continuity argument
Proof. Applying the energy identity associated to the vector field V , we have∫
Στ
JVµ [ψ]n
µ
Στ
≤ B
∫ τ
0
∫
Σs
∣∣KV [ψ]∣∣ ds+ ∫
Σ0
JVµ [ψ]n
µ
Σ0
.
It remains to control the spacetime integral term. We observe that KV [ψ] = 0 outside
supp(dαdr ) and ∣∣KV [ψ]∣∣ ≤ B (|∂rψ|2 + −1|Φψ|2) ,
and apply (3.7.1) to the first term:
∫ τ
0
∫
Σs∩supp( dαdr )
|∂rψ|2 ds ≤ B(m)
(∫
Στ
JNµ [ψ]n
µ
Στ
+
∫
Σ0
JNµ [ψ]n
µ
Σ0
)
.
Adding this estimate to the energy identity above, we have
∫
Στ
JVµ [ψ]n
µ
Στ
≤ B(m)
−1 ∫ τ
0
∫
Σs∩
{
r≤M(3+
√
2)
2
} |Φψ|2 ds+ ∫
Στ
JNµ [ψ]n
µ
Στ
+B(m)
∫
Σ0
JNµ [ψ]n
µ
Σ0
.
The proof is completed by applying the argument presented in [DRSR14, §11.2].
The proof above does not use the fact that the ergoregion and trapping region are
disjoint. Rather, the non-degeneracy of the (∂rψ)
2 term in (3.7.1) is used. Therefore, we
could prove the first line of (3.7.4) without the restriction to fixed m.
For fixed ψ supported on a fixed, azimuthal frequencym however, we have |Φψ| = |mψ|.
Since m is fixed, the presence of the ergoregion is only a low-frequency obstruction to
proving (NEB).
The proof of Lemma 3.7.4 is similar to Proposition 3.8.3 in that we obtain a nonde-
generate boundedness estimate without the use of a fully nondegenerate integrated local
energy decay estimate.
By combining (3.7.2) and (3.7.4), we obtain the following higher order version of
(3.7.4).
Lemma 3.7.5. Let M > 0, a2 + Q2 ≤ K20 < M2, m ∈ Z and ψ be a solution of (2.2.3)
89
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
that is supported only on the azimuthal frequency m. Then, for every j ≥ 1, and all τ ≥ 0,∫
Στ
∑
1≤i1+i2+i3≤j
|∇/ i1T i2(Z)i3ψ|2
≤ B(j,m)
∫ τ
0
∫
Σs∩
{
r≤M(3+
√
2)
2
} |ψ|2 ds+
∫
Σ0
∑
1≤i1+i2+i3≤j
|∇/ i1T i2(Z)i3ψ|2
. (3.7.5)
Proof. The additional Q-terms that arise in passing from the Kerr to the Kerr–Newman
case are harmless in the derivation of the higher order result, so the argument of [DRSR14,
§11.2] may be applied directly.
Combining (3.7.2) and (3.7.5), we obtain the following useful corollary.
Corollary 3.7.6. Let M > 0, a2 +Q2 ≤ K20 < M2, m ∈ Z and ψ be a solution of (2.2.3)
that is supported only on the azimuthal frequency m. Then, for every δ > 0, j ≥ 1, and
all τ ≥ 0,
sup
τ ′≤τ
∫
Στ ′
∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψ|2
+
∫ τ
0
∫
Σs
r−1−δ
r−2 |ψ|2 + 1[r+,(1+√2)M] ∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψ|2
ds
+
∫ τ
0
∫
Σs
r−1−δ
∑
1≤i1+i2+i3≤j−1
|∇/ i1T i2Zi3ψ|2 +
∑
1≤i1+i2+i3≤j−1
|∇/ i1T i2Zi3+1ψ|2
ds
≤ B(δ, j,m)
∫ τ
0
∫
Σs∩
{
r≤M(3+
√
2)
2
} |ψ|2 ds+ ∫
Σ0
∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψ|2
. (3.7.6)
The interpolating metric
We now prove Proposition 3.7.5.
Proof of Proposition 3.7.5. Consider fixed M > 0, fixed a2 ≤ a20 < M2 and fixed m ∈ Z.
Let Q˚ ∈ Qa,m. Then choose Q0 such that Q˚2 < Q20 < M2 − a2. Our aim is to find an
> 0 satisfying Q˚2 + 2 ≤ Q20 such that∣∣∣Q− Q˚∣∣∣ < =⇒ Q ∈ Qa,m.
Let ψ be a solution of (2.2.3) on gM,a,Q that is supported only on the azimuthal
frequency m and take Q such that
∣∣∣Q− Q˚∣∣∣ < for an to be determined.
90
3.7. The continuity argument
g˜τ := χτgM,a,Q˚ + (1− χτ ) gM,a,Q
g˜τ := gM,a,Q˚
g˜τ := gM,a,Q
Στ−δτ
ΣτH+ I+
i0
Figure 3.7.1: The interpolating metric.
Let ψ˜ be a solution of (2.2.3) on gM,a,Q˚, also supported only the single azimuthal
frequency m. Since Q˚ ∈ Qa,m, Corollary 3.7.3 implies that ψ˜ is future-integrable. We will
exploit this by introducing a metric g˜τ which interpolates between gM,aQ˚ and gM,a,Q:
Definition 3.7.6. Set τ ≥ 1. Let
χτ =
1 in the future of Στ
0 in the past of Στ−δτ ,
smooth between Στ−δτ and Στ
for sufficiently small δτ > 0. Now define the interpolating metric g˜τ by
g˜τ := χτgM,a,Q˚ + (1− χτ ) gM,a,Q. (3.7.7)
See Figure 3.7.1.
If is small enough, g˜τ is a Lorentzian metric on M.
With our interpolating metric defined, we define the solution to its wave equation.
Definition 3.7.7. Let ψ be the solution of gM,a,Qψ = 0 defined above. Let ψ˜τ be the
interpolating solution of g˜τ ψ˜τ = 0 with the same initial data as ψ on Σ0.
This is well defined since Στ is a past Cauchy hypersurface for the future of Στ with
respect to g˜τ (which is identically equal to gM,a,Q˚ in that region).
Now ψ˜τ = ψ in the past of Στ−δτ and gM,a,Q˚ψ˜τ = 0 in the future of Στ .
Since Φ is a Killing vector field for gM,a,Q and gM,a,Q˚ and χτ does not depend on Φ, it
is clear that Φ is Killing for the metric g˜τ . Therefore, the interpolating solution ψ˜τ and
91
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
the as the original solution ψ will be supported on the same azimuthal frequency m.
The assumption Q˚ ∈ Qa,m allows us to use Corollary 3.7.3 to conclude that ψ˜τ is
future-integrable with respect to Q˚.
Note that
gM,a,Q˚ψ˜τ =
(
gM,a,Q˚ −g˜τ
)
ψ˜τ .
A computation then shows that
r1+δ
∣∣∣(gM,a,Q˚ −g˜τ) ψ˜τ ∣∣∣2 ≤ B (δ−1τ ) ∣∣∣Q− Q˚∣∣∣2 r−2 ∑
1≤i1+i2+i3≤2
∣∣∣∇/ i1T i2Zi3ψ˜τ ∣∣∣2 . (3.7.8)
This structure is essential to proving the desired estimate. With (3.7.8) established, the
argument follows the same logic as in [DRSR14, §11.2].
In what follows, metric defined quantities refer to gM,a,Q˚.
The error term (3.7.8) is supported only in the past of Στ , so Proposition 3.3.1 followed
by (ILED) for gM,a,Q˚ imply∫ τ
0
∫
Σs∩
{
r≤M(3+
√
2)
2
} ( JNµ [ψ]nµΣs + |ψ|2) ds
≤
∫ τ−δτ
0
∫
Σs∩{r≤M(1+√2)}
(
JNµ [ψ]n
µ
Σs
+ |ψ|2
)
ds
≤ B(δτ ,m)
∣∣∣Q− Q˚∣∣∣2 ∫ τ
0
∫
Σs
∑
1≤i1+i2+i3≤2
r−2
∣∣∣∇/ i1T i2Zi3ψ˜τ ∣∣∣2 ds
+B (δτ ,m)
∣∣∣Q− Q˚∣∣∣2 ∫ ∞
τ
∫
Σs∩[r+,(1+
√
2)M ]
[∣∣∣T ψ˜τ ∣∣∣2 + |ψ|2] ds
+B(m)
∫
Σ0
[
JNµ [ψ]n
µ
Σ0
+ |ψ|2
]
, (3.7.9)
for δτ sufficiently small.
Again by Proposition 3.3.1 (applied to second order derivatives of ψ˜τ and those of ψ
respectively),∫ τ
τ−δτ
∫
Σs
∑
1≤i1+i2+i3≤2
r−2
(∣∣∣∇/ i1T i2Zi3ψ˜τ ∣∣∣2 + ∣∣∣∇/ i1T i2Zi3ψ∣∣∣2) ds
≤ B
∫
Στ−δτ
∑
1≤i1+i2+i3≤2
∣∣∣∇/ i1T i2Zi3ψ∣∣∣2 . (3.7.10)
Since ψ˜τ is future integrable, we can apply (3.2.2), and Proposition 3.3.1 again, to
92
3.7. The continuity argument
arrive at∫ ∞
τ
∫
Σs∩[r+,(1+
√
2)M ]
[∣∣∣T ψ˜τ ∣∣∣2 + |ψ|2] ≤ B ∫
Στ−δτ
∑
1≤i1+i2+i3≤2
∣∣∣∇/ i1T i2Zi3ψ∣∣∣2 . (3.7.11)
Combining (3.7.9), (3.7.10) and (3.7.11) gives∫ τ
0
∫
Σs∩
{
r≤M(3+
√
2)
2
} ( JNµ [ψ]nµΣs + |ψ|2) ds
≤B(m)
∣∣∣Q− Q˚∣∣∣2 ∫
Στ−δτ
∑
1≤i1+i2+i3≤2
∣∣∣∇/ i1T i2Zi3ψ˜τ ∣∣∣2
+B(m)
∣∣∣Q− Q˚∣∣∣2 ∫ τ
0
∫
Σs
∑
1≤i1+i2+i3≤2
r−2
∣∣∣∇/ i1T i2Zi3ψ∣∣∣2 ds (3.7.12)
+B(m)
∫
Σ0
[
JNµ [ψ]n
µ
Σ0
+ |ψ|2
]
.
Applying Corollary 3.7.6 followed by (3.7.12):
sup
τ ′≤τ
∫
Στ ′
∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψ|2
+
∫ τ
0
∫
Σs
r−1−δ
r−2 |ψ|2 + 1[r+,(1+√2)M] ∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψ|2
ds
+
∫ τ
0
∫
Σs
r−1−δ
∑
1≤i1+i2+i3≤j−1
|∇/ i1T i2Zi3ψ|2 +
∑
1≤i1+i2+i3≤j−1
|∇/ i1T i2Zi3+1ψ|2
ds
≤B(δ, j,m)
∫ τ
0
∫
Σs∩
{
r≤M(3+
√
2)
2
} |ψ|2 ds+
∫
Σ0
∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψ|2
≤B(δ, j,m)
∣∣∣Q− Q˚∣∣∣2 ∫ τ
0
∫
Σs
∑
1≤i1+i2+i3≤2
|∇/ i1T i2Zi3ψ|2ds
+B(δ, j,m)
∣∣∣Q− Q˚∣∣∣2 ∫ τ
0
∫
Σs
r−2
∑
1≤i1+i2+i3≤2
|∇/ i1T i2Zi3ψ|2 ds
+B(δ, j,m)
∫
Σ0
∑
0≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψ|2.
Taking j ≥ 3 and letting be small enough that we can absorb the
∣∣∣Q− Q˚∣∣∣2 term on the
left hand side. We conclude that
sup
τ ′≤τ
∫
Στ ′
∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψ|2 ≤ B(j,m)
∫
Σ0
∑
0≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψ|2 <∞.
93
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
Since τ was chosen arbitrarily and the right hand side of this estimate is independent of
τ , this is the sufficient condition for integrability of Corollary 3.7.3. We have therefore
shown that there exists an small enough that
∣∣∣Q− Q˚∣∣∣ < =⇒ Q ∈ Qa,m.
3.7.4 Closedness
To close the continuity argument and complete the proof of Proposition 3.7.2, it remains
to prove
Proposition 3.7.7. The set Qa,m is closed in [0,M2 − a2). That is, if the sequence
{Qk}∞k=1 ⊂ Qa,m and Q2k < M2 − a2, then Q ∈ Qa,m.
Proof. As in the statement of Proposition 3.7.2, let ψ be a solution the wave equation
gM,a,Qψ = 0 supported on a fixed azimuthal frequency m. Set Q0 < M such that
Q2 < Q20. Without loss of generality, we assume that Q
2
k ≤ Q20 for all k.
Now define the sequence of functions ψk to be solutions of gM,a,Qkψk = 0 with the
same initial data as ψ. Then by Lemma 2.2.1,∫
Στ
∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψ|2 = lim
k→∞
∫
Στ
∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψk|2 (3.7.13)
for every τ ≥ 0, j ≥ 1. Since each Qk ∈ Qa,m, each ψk is future integrable. We may then
apply (3.2.3) to each ψk: for every j ≥ 1,
sup
τ≥0
∫
Στ
∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψk|2 ≤ B(j,m)
∫
Σ0
∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψk|2. (3.7.14)
Combining (3.7.13) and (3.7.14), we conclude that
sup
τ≥0
∫
Στ
∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψ|2 ≤ B(j,m) lim
k→∞
∫
Σ0
∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψk|2
= B(j,m)
∫
Σ0
∑
1≤i1+i2+i3≤j
|∇/ i1T i2Zi3ψ|2,
which is (3.7.3), so Q ∈ Qa,m.
3.7.5 Proof of (ILED)
We have now proved Proposition 3.7.2, the integrability result for solutions supported
on fixed azimuthal frequency. By the reduction given by Lemma 3.7.1, Proposition 3.7.1
94
3.8. Proof of (NEB)
follows directly. We have therefore shown that any solution of (2.2.3) on a subextremal
Kerr–Newman exterior spacetime is future integrable. This allows us to appeal to the
conditional Theorem 3.3.2 for the full range of solutions. From this we conclude that
(ILED) in Theorem 3.2.1 holds unconditionally.
3.8 Proof of (NEB)
At this point we have proved that for a2 +Q2 < M2, every solution ψ of (2.2.3) is future-
integrable and moreover satisfies the integrated decay statements (ILED) and (3.2.1). We
now prove (NEB). Naively, one may apply the energy identity for N ,∫
Στ
JNµ [ψ]n
µ
Στ
≤
∫
Σ0
JNµ [ψ]n
µ
Σ0
−
∫ τ
0
∫
Σt
KN [ψ] dt∗ (3.8.1)
and attempt to control the last term on the right hand side by (3.2.1). In the case that
a2 +Q2 M2, this approach works. This is because one has a small parameter to exploit,
which allows the vector field N to be chosen in such a way that KN [ψ] is not supported
in the physical space projection of the trapped set. See [DR11a] for the details.
In general, this approach fails due to the degeneracies of (3.2.1). For the full subex-
tremal Kerr–Newman case, we turn to a more sophisticated approach which employs phase
space localisation of ψ and specific features of the Kerr–Newman geometry.
We begin by recalling from Lemmas 3.4.1 and 3.5.2 that there exists a constant R\
such that the degeneration of estimate (3.5.3) due to trapping only occurs in R(0, τ) ∩
{re < r < R\} where
re < R\ Re.
Theorem 3.2.1 ensures that we can fix Re large enough in (ILED) to satisfy the inequality
above. Cover M by the sets
A˜H = M∩ {r+ ≤ r < re − } ,
A˜trap = M∩ {re − 2 < r < R\ + 2}
and A˜R = M∩ {r > R\ + } ,
where the presence of > 0 ensures that the trapping region is strictly contained in A˜trap.
Let us take a smooth partition of unity {χH(r), χtrap(r), χR(r)} subordinate to the cover{
A˜H , A˜trap, A˜R
}
with
supp χ{H,trap,R}(r) = A˜{H,trap,R}.
95
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
Now for a solution of (2.2.3) we can write
ψ = ψH + ψtrap + ψR, (3.8.2)
where
ψ{H,trap,R} = χ{H,trap,R} · ψ.
Clearly each ψ{H,trap,R} has the same smoothness and integrability properties as ψ. Fur-
thermore,
gψ{H,trap,R} = G{H,trap,R}, (3.8.3)
where
G{H,trap,R} = ((χ{H,trap,R}))ψ + 2∇µ(χ{H,trap,R})∇µψ.
Outside the trapping region A˜trap, the (NEB) statement may be extracted from (ILED)
in a direct manner. We will consider this first.
3.8.1 (NEB) outside the trapping region
(NEB) near the horizon
Applying the energy identity for N to ψH , we have∫
Στ
JNµ [ψH ]n
µ
Στ
≤
∫
Σ0
JNµ [ψH ]n
µ
Σ0
−
∫ τ
0
∫
Σt∩{r+≤r≤re−}
(KN [ψH ] + EN [ψH ]) dt∗.
Since χH is smooth with compactly supported derivatives,∣∣EN [ψH ]∣∣ = |[N(χH)ψ + (χH)(Nψ)][(χH)ψ + 2∇µ(χH)(∇µψ)]|
≤ Cψ2 + C|∂ψ|2.
So∫ τ
0
∫
Σt∩{r+≤r≤re−}
∣∣KN [ψH ]∣∣+ ∣∣EN [ψH ]∣∣ dt∗ ≤ C ∫ τ
0
∫
Σt∩{r+≤r≤re−}
ψ2 + |∂ψ|2 dt∗.
The estimate (3.6.8) does not degenerate in A˜H , so we may apply it to the spacetime
integral above to obtain∫
Στ
JNµ [ψH ]n
µ
Στ
≤
∫
Σ0
JNµ [ψH ]n
µ
Σ0
+ C
∫
Σ0
JNµ [ψ]n
µ
Σ0
.
96
3.8. Proof of (NEB)
The zeroth order terms in JNµ [ψH ]n
µ
Σ0
can be controlled using the Hardy inequality (2.2.8)
in r (we exploit the smoothness of the cut-offs and the compact support of ψH). Hence∫
Στ
JNµ [ψH ] ≤ (CH + C)
∫
Σ0
JNµ [ψ]n
µ
Σ0
. (3.8.4)
(NEB) for large r
Here we apply the energy identity for T to ψR,∫
Στ
JTµ [ψR]n
µ
Στ
≤
∫
Σ0
JTµ [ψR]n
µ
Σ0
−
∫ τ
0
∫
Σt∩{r≥R\+}
ET [ψR] dt∗.
Note that∫
Σ0
JTµ [ψR]n
µ
Σ0
≤ C
∫
Σ0
|∇(χRψ)|2
≤ C
∫
Σ0
|χR(∇ψ) + (∂rχR)(ψ)|2
≤ C
∫
Σ0
|(∂ψ)|2 + C
∫
Σ0∩{R\+≤r≤R\+2}
(∂rχR)
2(ψ)2
≤ C(R\ + 2)2
∫ τ
0
∫
Σt∩{R\+≤r≤R\+2}
r−2ψ2 + |∂ψ|2 dt∗
≤ C
∫
Σ0
JNµ [ψ]n
µ
Σ0
,
where we have used the Hardy inequality (2.2.8) in r. The spacetime integrand is
ET [ψR] = [T (χR)ψ + (χR)(Tψ)][(χR)ψ + 2∇µ(χR)(∇µψ)]
= (χR)(Tψ)[(χR)ψ + 2∇µ(χR)(∇µψ)]
=
(
∂r(∆∂rχR)
ρ2 sin θ
)
(ψ)(χR)(Tψ) + 2(g
rr)(χR)(∂rχR)(∂rψ)(Tψ)
Observe that each term in ET [ψR] contains factors of (∂rχR) or (∂2rχR) and is conse-
quently supported in {R\ + ≤ r ≤ R\ + 2}. Applying Cauchy Schwarz and using the
boundedness of the metric components and cut-offs,∫ τ
0
∫
Σt∩{r≥R\+}
∣∣ET [ψR]∣∣ dt∗ = ∫ τ
0
∫
Σt∩{R\+≤r≤R\+2}
∣∣ET [ψR]∣∣ dt∗
≤ C
∫ τ
0
∫
Σt∩{R\+≤r≤R\+2}
ψ2 + |∂ψ|2 dt∗
≤ C
∫
Σ0
JNµ [ψ]n
µ
Σ0
,
97
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
where we have used (3.6.8) in the last inequality (note that this estimate does not degen-
erate in the region under consideration). We conclude that
∫
Στ
JNµ [ψR] ≤ CR
∫
Σ0
JNµ [ψ]n
µ
Σ0
. (3.8.5)
3.8.2 The set up for the proof of (NEB) in the trapping region
Recall the decomposition (3.8.2),
ψ(t∗, r, θ, φ) = (χH(r) + χR(r) + χtrap(r))ψ(t∗, r, θ, φ).
We have proved (NEB) for (χH + χR)ψ in (3.8.4) and (3.8.5). The proof of (NEB) for
χtrapψ requires a more sophisticated approach which employs features of the Kerr–Newman
geometry and localisation of ψ in phase space.
The idea is to project to finitely many wave packets for which the degeneration of
(3.6.8) is contained in a particular bounded r−interval. Then a bespoke energy current
can be constructed for each wave packet so that, upon application of the associated energy
identity, we obtain (NEB) for each wave packet. It then remains to sum over the wave
packets to obtain (NEB) for the full solution of (2.2.3).
We first describe the relevant geometric features. The localisation depends on these
features and follows in §3.8.3.
It will be convenient to extend the solution ψ of (2.2.3) from the future of Σ0 to the
entire domain of outer communications.
Extending to the past
The initial data in (2.2.3) only determine the solution ψ in the future of Σ0. We extend
the solution ψ to the entire domain of outer communications as follows:
Consider the maximal globally hyperbolic extension of the Kerr–Newman manifold
Me (see for example [Car73]). Denote the extension of Σ0 as a spacelike hypersurface to
Me by Σe0. Now Σe0 is a Cauchy hypersurface for Me.
We can extend the initial data on Σ0 to Σ
e
0 as C
1 functions in such a way that∫
Σe0
JNµ [ψ]n
µ
Σe0
≤ C
∫
Σ0
JNµ [ψ]n
µ
Σ0
. (3.8.6)
Since Σe0 is a Cauchy hypersurface for Me, we can solve the initial value problem both
forwards and backwards in such a way that the solution is C2(M) and agrees with the
solution ψ of (2.2.3) in M∩ J+(Σ0).
98
3.8. Proof of (NEB)
From here on we consider the extended solution, which we continue to denote by ψ.
Let Σ˜e0 be the image of Σ
e
0 under the map t 7→ −t, where t is the Boyer–Lindquist
coordinate of §2.1.1. Then applying the energy identity for nΣe0 between Σe0 and Σ˜e0, the
proof of Proposition 3.3.1 and the analogue of (3.8.6) for Σe0 we have∫
Σ˜0
JNµ [ψ]n
µ
Σ˜0
≤ C
∫
Σ0
JNµ [ψ]n
µ
Σ0
. (3.8.7)
We now extend (ILED) to the past of Σ0.
Corollary 3.8.1. Let ψ solve (2.2.3). For any r+ < r0 < R0 ≤ Re there exists a positive
constant Cr0,R0 such that∫ ∞
−∞
∫
Στ∩{r0≤r≤R0}
(
χ\|∂ψ|2 + |ψ|2
)
dt∗ ≤ Cr0,R0
∫
Σ0
|∂ψ|2, (3.8.8)
Here χ\ is a cut-off function that vanishes in a neighbourhood of the physical space pro-
jection of the trapped set, see §3.5.3.
Proof. Write∫ ∞
−∞
∫
Σs∩{r0≤r≤R0}
(
χ\|∂ψ|2 + |ψ|2
)
ds
=
(∫
J+(Σ0)∩{r0≤r≤R0}
+
∫
{J−(Σ0)∩J+(Σ˜0)}∩{r0≤r≤R0}
)(
χ\|∂ψ|2 + |ψ|2
)
+
∫
J−(Σ˜0)∩{r0≤r≤R0}
(
χ\|∂ψ|2 + |ψ|2
)
.
For the first integral, we simply apply (ILED). For the last integral, we note that mapping
(t, a) 7→ (−t,−a) is an isometry, so (ILED) holds with Σ0 replaced by Σ˜0 and the integrals
(0, τ) replaced by integrals over (−τ, 0). The remaining term is controlled Proposition
3.3.1 and (3.8.7).
Locally Killing, globally timelike vector fields
Let us first recall the following property of subextremal Kerr–Newman spacetimes: For
any r˜ ≥ r+, there exists a constant c = c(r˜) such that the smooth vector field
Qc = T + cΦ
is timelike and Killing at r˜. By continuity of the metric, there exists an open set of the
form A = {ra < r < RA} containing this r˜ for which Qc remains timelike and Killing in
99
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
A. We extend Qc to a globally defined smooth vector field by defining
QA = T + bA(r)Φ
where bA(r) is a smooth function chosen such that QA is smooth and globally timelike in
R(0, τ) with the further property that bA(r) = c in A.
Covering the trapping region
Our goal is to overcome the degeneration of (3.6.8) in a neighbourhood of the physical
space projection of the trapped set
{
r : r ∈ ⋂∞L=1⋃`≥L r(aω)m` } . This degeneration arose
from the fact that the estimate (3.5.3) must degenerate on
{
r = r
(aω)
m`
}
(ω,m,`)
for each
trapped mode. Recall from Lemmas 3.4.1 and 3.5.2 that these r
(aω)
m` must lie in R(0, τ) ∩
{re < r < R\} ⊂ A˜trap.
By compactness and the construction in §3.8.2, A˜trap can be covered by finitely many
open sets, say {An}N˜n=1, for which the smooth vector field
QAn = T + cnΦ
is timelike and Killing in An. We now use Q
An to define the vector field
Qn = T + bn(r)Φ
where bn(r) is a smooth function chosen such that Qn is smooth and globally timelike in
R(0, τ) with the further property that
Qn =
N for R(0, τ) ∩ {r < re − 3} ,
QAn in An,
T for R(0, τ) ∩ {r > R\ + 3} .
(3.8.9)
We emphasise that Qn is smooth and globally timelike and in particular, a local Killing
vector field in An.
For n = 0 we just let Q0 = N .
A refinement
Ultimately we want to apply energy currents JQn to appropriately localised solutions of
(3.3.4), which we will call wave packets (the precise definition is given later in (3.8.10)).
We want to construct each wave packet in such a way that (3.5.11) applied to the nth
wave packet is nondegenerate in R(0, τ) \An.
100
3.8. Proof of (NEB)
To ensure this, we will need a refinement
{
A˜n
}N˜
n=1
of {An}N˜n=1 such that the degener-
ation of (3.5.11) applied to the nth wave packet is strictly contained in A˜n ⊂ An. Denote
each An by
An = (rAn , RAn) where rAn < rAn+1 < RAn < RAn+1 .
Observe that the intersections of adjacent An are nonempty open sets of the form
An−1 ∩An = (rAn , RAn−1).
Hence we can define
A˜n = (rAn + n, RAn − n)
where each n is chosen small enough that the intersections of adjacent A˜n are nonempty
open sets.
3.8.3 Construction of wave packets
We now construct the wave packets by localising solutions of (3.8.3) appropriately in phase
space. First recall our decomposition (3.8.2):
ψ = ψH + ψR + ψtrap.
We apply the decomposition (3.3.8) to ψtrap = χtrapψ:
ψtrap(t, r, θ, φ) =
1√
2pi
∫ ∞
−∞
∑
m∈Z
∑
`>|m|
χtrap(r)R
(aω)
m` (r)S
(aω)
m` (cos θ)e
imφe−iωt dω.
Let {χn(r)}N˜n=1 be a smooth partition of unity subordinate to the cover
{
A˜n
}N˜
n=1
such
that
supp χn(r) = A˜n.
Define the cut-off functions as follows:
αn(ω,m, `) = χn(r
(aω)
m` ).
We use these to localise in phase space by defining, for (ω,m,Λ) ∈ F\:
101
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
ψn(t, r, θ, φ) =
1√
2pi
∫ ∞
−∞
∑
m∈Z
`≥|m|
αn(ω,m, `)χtrap(r)R
(aω)
m` (r)S
(aω)
m` (cos θ)e
imφe−iωt dω.
(3.8.10)
To get the full solution of (3.8.3) upon summation, we need one more wave packet
that takes into account (ω,m,Λ) /∈ F\. Define
ψ0(t, r, θ, φ) =
1√
2pi
∫ ∞
−∞
∑
m∈Z
∑
`>|m|
α0(ω,m, `)χtrap(r)R
(aω)
m` (r)S
(aω)
m` (cos θ)e
imφe−iωt dω
where
α0(ω,m, `) = 1−
N˜∑
n=1
αn(ω,m, `).
Note that the estimate (3.5.11) for α0(ω, r,m, `)u
(aω)
m` will not degenerate as the phase-
space support of this wave packet lies outside the trapping regime. Therefore the (NEB)
statement in A˜trap for these modes is proved by the same argument as in §3.8.1, so∫
Στ
JNµ [ψ0] ≤ C0
∫
Σ0
JNµ [ψ]n
µ
Σ0
. (3.8.11)
Remark Note that in this section, R
(aω)
m` and u
(aω)
m` denote the solutions of (3.3.10) and
(3.3.12) respectively with F
(aω)
m` and H
(aω)
m` arising from the cut-off χtrap(r) rather than
the cut-off γ(t∗).
Proposition 3.8.2. Each wave packet ψn defined by (3.8.10) enjoys the following prop-
erties.
1. Each ψn is sufficiently integrable in the sense of (3.3.1).
2. The estimate (3.5.11) for αn(ω,m, `)u
(aω)
m` will degenerate only in A˜n as desired.
3. Every (ω,m,Λ) ∈ F\ belongs to the phase-space support of some wave packet ψ̂n.
Proof. 1. This follows immediately from Plancherel and the assumption that ψ satisfies
(3.3.1).
2. We have constructed αn such that
supp αn(ω,m, `) =
{
(ω,m, `) | r(aω)m` ∈ A˜n
}
.
102
3.8. Proof of (NEB)
3. This follows from the item above and the fact that
{
A˜n
}N˜
n=1
covers A˜trap.
3.8.4 (ILED) for wave packets
We first prove that the analogue of (ILED) holds for each wave packet.
Lemma 3.8.1. There exists a positive constant C depending only on M , re, Re and P
such that for any wave packet ψn, n ≥ 1, and any τ > 0,∫ τ
0
∫
Σt∩[r∗e ,R∗e ]
[
(∂r∗ψn)
2 + ψ2n + ξn(r)
(
(Tψn)
2 + |∇/ψn|2
)]
dt∗ ≤ C
∫
Σ0
JNµ [ψ]n
µ
Σ0
,
(3.8.12)
where ξn degenerates only in A˜n.
Proof. Note that |αn(ω,m, `)| ≤ 1 and |χtrap| ≤ 1. Since χtrap(r) and χn(r) are compactly
supported smooth functions in r,
|∂rχn|2 + |∂rχtrap|2 ≤ max
1≤n≤N˜
sup
r∈[rAn ,RAn ]
|∂rχn|2 + sup
r∈[re,Re]
|∂rχtrap|2 := C.
This constant is independent of ω, m and `. Repeating the proof of Proposition 3.5.9 and
recalling that J
(aω)
m` is not supported in the trapping regime F\,∫ ∞
−∞
∑
m∈Z
`≥|m|
∫ R∗e
r∗e
(∣∣∣∣ ddr∗ (αnχtrapu(aω)m` )
∣∣∣∣2 + ∣∣∣αnχtrapu(aω)m` ∣∣∣2
+(r − r(aω)m` )2(ω2 + Λ)
∣∣∣αnχtrapu(aω)m` ∣∣∣2) dr∗dω
≤ 4C
∫ ∞
−∞
∑
m∈Z
`≥|m|
∫ R∗e
r∗e
∣∣∣∣ ddr∗u(aω)m`
∣∣∣∣2 + ∣∣∣u(aω)m` ∣∣∣2 + (r − r(aω)m` )2(ω2 + Λ)∣∣∣u(aω)m` ∣∣∣2dr∗dω
≤
∫ ∞
−∞
∑
m∈Z
`≥|m|
∫
A˜trap
C
(aω)
m` (r
∗)Re
[
(u
(aω)
m` )
′H¯(aω)m`
]
+D
(aω)
m` (r
∗)Re
[
u
(aω)
m` H¯
(aω)
m`
]
dr∗dω
+ B
∫ ∞
−∞
∑
m∈Z
`≥|m|
∫
A˜trap
ωIm
[
u
(aω)
m` H¯
(aω)
m`
]
dr∗dω,
where we have absorbed constants,
H
(aω)
m` (r) :=
∆G
(aω)
m` (r)
(r2 + a2)1/2
103
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
and
G
(aω)
m` :=
∫
S2
Ĝtrap(ω, r, θ,m) · S(aω)m` (cos θ)eimφdVS2 .
By construction, χn(r
(aω)
m` ) is only supported in A˜n for 1 ≤ n ≤ N˜ . Hence (r − r(aω)m` )2
vanishes only in A˜n for 1 ≤ n ≤ N˜ and χ0 = 1. From the identities in §3.3.3 and (3.8.10),
we have for any τ > 0
∫ ∞
−∞
∑
m∈Z
`≥|m|
∫ R∗e
r∗e
(∣∣∣∣ ddr∗ (αnχtrapu(aω)m` )
∣∣∣∣2 + ∣∣∣αnχtrapu(aω)m` ∣∣∣2
+(r − r(aω)m` )2(ω2 + Λ)
∣∣∣αnχtrapu(aω)m` ∣∣∣2) dr∗dω
≥ c
∫ ∞
−∞
∫
S2
∫ Re
r+
[
(∂r∗ψn)
2 + ψ2n + ξn(r)
(
(Tψn)
2 + (∇/ψn)2
)]
ρ2 sin θ dθ dφ dr dt∗
≥ c
∫ τ
0
∫
Σt∩[r∗e ,R∗e ]
[
(∂r∗ψn)
2 + ψ2n + ξn(r)
(
(Tψn)
2 + (∇/ψn)2
)]
dt∗,
where c is a positive constant depending on M , re and Re and ξn(r) is a strictly positive
function outside of A˜n which vanishes on the physical space projection of r
(aω)
m` for r
(aω)
m` ∈
A˜n.
So we have
c
∫ τ
0
∫
Σt∩[r∗e ,R∗e ]
[
(∂r∗ψn)
2 + ψ2n + ξn(r)
(
(Tψn)
2 + (∇/ψn)2
)]
dt∗
≤
∫ ∞
−∞
∑
m∈Z
`≥|m|
∫
A˜trap
C
(aω)
m` (r
∗)Re
[
(u
(aω)
m` )
′H¯(aω)m`
]
+D
(aω)
m` (r
∗)Re
[
u
(aω)
m` H¯
(aω)
m`
]
dr∗dω
+ B
∫ ∞
−∞
∑
m∈Z
`≥|m|
∫
A˜trap
ωIm
[
u
(aω)
m` H¯
(aω)
m`
]
dr∗dω.
It thus remains to prove that the right hand side of this estimate is controlled by initial
data. Recall that
Gtrap = (χtrap)ψ + 2∇µ(χtrap)∇µψ
=
1
ρ2 sin θ
∂r(∆∂rχtrap)ψ + 2
∆
ρ2
∂r(χtrap)(∂rψ). (3.8.13)
By the identities in §3.3.3,
B
∫ ∞
−∞
∫ ∞
r+
∑
m∈Z
`≥|m|
ωIm
[
u
(aω)
m` H¯
(aω)
m`
]
dr∗dω = B
∫ ∞
−∞
∫ ∞
r+
∫
S2
(Tψ)(Gtrap) ∆drdgS2dt
104
3.8. Proof of (NEB)
Observe that each term in Gtrap contains factors of (∂rχtrap) or (∂
2
rχtrap) and conse-
quently
ζ := supp Gtrap ⊂ {re − 2 ≤ r ≤ re − } ∪ {R\ + ≤ r ≤ R\ + 2} .
Note that supp H
(aω)
m` ⊂ ζ as well. Recall from §3.6 that C(aω)m` (r), D(aω)m` (r) and χtrap(r)
are smooth functions, bounded uniformly with respect to ω,m,Λ, so by (3.8.8) there exists
a positive frequency independent constant C such that
∫ ∞
−∞
∑
m∈Z
`≥|m|
∫
A˜trap
C
(aω)
m` (r
∗)Re
[
(u
(aω)
m` )
′H¯(aω)m`
]
+D
(aω)
m` (r
∗)Re
[
u
(aω)
m` H¯
(aω)
m`
]
dr∗dω
+ B
∫ ∞
−∞
∑
m∈Z
`≥|m|
∫
A˜trap
ωIm
[
u
(aω)
m` H¯
(aω)
m`
]
dr∗dω
≤ C
∫ ∞
−∞
∑
m∈Z
`≥|m|
∫
ζ
∣∣∣(u(aω)m` )′∣∣∣2 + ∣∣∣u(aω)m` ∣∣∣2 + ω2∣∣∣u(aω)m` ∣∣∣2 + ∣∣∣H(aω)m` ∣∣∣2 dr∗dω.
≤ C
∫ ∞
−∞
∫
Σt∩ζ
|ψ|2 + |∂rψ|2 + |∂tψ|2 dt.
Now ζ is disjoint from the trapping region, so we may apply (3.8.8) to obtain the result.
3.8.5 (NEB) for wave packets
Proposition 3.8.3. For each wave packet ψn, we have (NEB). That is, there exists a
constant Cn, depending on M , re, Re, P (from Proposition 3.3.1), λ2, ω1 and ω2 such
that ∫
Στ
JQnµ [ψn]n
µ
Στ
≤ Cn
∫
Σ0
JQnµ [ψ]n
µ
Σ0
. (3.8.14)
Proof. In light of (3.8.11), we need only consider the wave packets supported in F\, that
is, ψn for n ≥ 1.
Each wave packet ψn is sufficiently integrable in the sense of (3.3.1) so for each n there
exists a dyadic sequence τ
(n)
j → −∞ and a constant kn such that∫
Σ
τ
(n)
j
JQnµ [ψn]n
µ
Σ
τ
(n)
j
≤ −kn
τ
(n)
j
. (3.8.15)
105
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
Applying the energy identity for Qn to ψn between a Στ (n)j
and Στ for 1 ≤ n ≤ N˜ we have
∫
Στ
JQnµ [ψn]n
µ
Στ
=
∫
Σ
τ
(n)
j
JQnµ [ψn]n
µ
Σ
τ
(n)
j
−
∫ τ
τ
(n)
j
∫ Re
re−2
∫
S2
(EQn [ψn] +KQn [ψn]) ρ2 sin θ dr dθ dφ dt.
By (3.8.15), taking j →∞, we have∫
Στ
JQnµ [ψn]n
µ
Στ
≤
∣∣∣∣∫ τ−∞
∫ Re
re−2
∫
S2
(EQn [ψn] +KQn [ψn]) ρ2 sin θ dr dθ dφ dt∣∣∣∣. (3.8.16)
Since Qn is a timelike Killing vector field in An and Qn = T for r > R\ + 3, the bulk
term KQn [ψn] vanishes in these regions, so∫ τ
−∞
∫
r∈[rAn ,RAn ]∪{rR\+3}
∫
S2
KQn [ψn] ρ2 sin θ dr dθ dφ dt = 0.
Moreover, (3.8.12) has no degeneracies outside of A˜n ⊂ An, so∫ τ
−∞
∫
{re−2≤r≤R\+3}\[rAn ,RAn ]
∫
S2
∣∣KQn [ψn] ρ2 sin θ∣∣ dr dθ dφ dt ≤ C ∫
Σ0
JQnµ [ψ]n
µ
Σ0
.
It remains to control the error term. Recalling (3.8.13),∫ τ
−∞
∫ Re
re−2
∫
S2
EQn [ψn] ρ2 sin θ dr dθ dφ dt
=
∫ τ
−∞
∫ Re
re−2
∫
S2
(Qnψn)(Gtrap) ρ
2 sin θ dr dθ dφ dt
=
∫ τ
−∞
∫
r∈ζ
∫
S2
(Qnψn)(Gtrap) ρ
2 sin θ dr dθ dφ dt
≤ C(re, Re, , χtrap)
∫ τ
−∞
∫
r∈ζ
∫
S2
ψ2 + |∂ψ|2 dr dθ dφ dt,
where
ζ := supp Gtrap ⊂ {re − 2 ≤ r ≤ re − } ∪ {R\ + ≤ r ≤ R\ + 2} .
Recall that the degeneracy of (3.8.12) is contained in the region {re < r < R\} which is
disjoint from ζ. Therefore, the right hand side of the estimate above may now be controlled
by applying (3.8.12).
106
3.8. Proof of (NEB)
3.8.6 (NEB) for the full solution
We now need to sum up the energies of the wave packets to carry this result over to the
full solution ψ of (2.2.3).
Proposition 3.8.4. For each vector field Qn, 0 ≤ n ≤ N˜ and τ ≥ 0, there exists a
constant cn > 0 such that
∫
Στ
JQnµ [ψ]n
µ
Στ
≤ cn
N˜∑
k=0
∫
Στ
JQkµ [ψk]n
µ
Στ
+ cn
∫
Στ
JQnµ [ψH ]n
µ
Στ
+ cn
∫
Στ
JQnµ [ψR]n
µ
Στ
.
(3.8.17)
Proof. From (2.2.7) we see that
∫
Στ
JQnµ [ψ]n
µ
Στ
=
∫
Στ
JQnµ [
N˜∑
k=0
ψk + ψH + ψR]n
µ
Στ
≤ (cn/2)
∫
Στ
∣∣∣∣∣∣∂
N˜∑
k=0
ψk + ψH + ψR
∣∣∣∣∣∣
2
≤ cn
N˜∑
k=0
∫
Στ
|∂ψk|2 + cn
∫
Στ
|∂ψH |2 + cn
∫
Στ
|∂ψR|2
which is the result modulo a constant which can be absorbed into cn.
We now complete the proof of (NEB). First apply Proposition 3.8.4. Then apply the
(NEB)-type results Proposition 3.8.3, (3.8.4), (3.8.5) and (3.8.11).
∫
Στ
JQnµ [ψ]n
µ
Στ
≤ cn
N˜∑
k=0
∫
Στ
JQkµ [ψk]n
µ
Στ
+ cn
∫
Στ
JQnµ [ψH ]n
µ
Στ
+ cn
∫
Στ
JQnµ [ψR]n
µ
Στ
≤ cn(C0 + CH + CR)
∫
Σ0
JQnµ [ψ]n
µ
Σ0
+
N˜∑
n=1
Cn
∫
Σ1
JQnµ [ψ]n
µ
Σ1
.
This concludes the proof of Theorem 3.2.1.
107
Stability of subextremal Kerr–Newman spacetimes for linear scalar perturbations
108
Chapter 4
Quantitative mode stability for
the wave equation on the
Kerr–Newman spacetime
109
Quantitative mode stability for the wave equation on the Kerr–Newman spacetime
4.1 Introduction
In this chapter, mode stability results for solutions of the wave equation on a subextremal
Kerr–Newman spacetime are proved. Both the qualitative results of the type proved in
[Whi89] for the Kerr case and the extended, qualitative results of the type proved in
[SR13] for the Kerr case are shown to hold in the Kerr–Newman case. In particular, the
quantitative mode stability result is used to prove an energy estimate for low superradiant
frequencies, required in the proof of Proposition 3.6.1.
As in the Kerr case, one of the major difficulties in understanding the wave equation on
a Kerr–Newman background is that of superradiance, the fact that the conserved ∂t energy
is not positive definite and thus does not control the solution ψ. After an appropriate
frequency localisation in the frequency parameters ω and m (corresponding to the Killing
fields ∂t and ∂φ respectively), the superradiant frequencies are seen to be those satisfying
0 ≤ mω ≤ am
2
2Mr+ −Q2 . (4.1.1)
In particular, the ∂t energy identity does not preclude finite-energy exponentially growing
mode solutions (with explicit t, φ dependence e−iωteimφ ) associated with the frequencies
(4.1.1), with ω in the upper half-plane. The statement that such modes do not exist is
known as mode stability. In the Kerr case, this has indeed been proven by Whiting in the
celebrated [Whi89].
The proof of quantitative boundedness and decay for solutions of (1.2.1) in the Kerr
case given in [DRSR14] in fact depended on a quantitative refinement of Whiting’s [Whi89].
The necessary refinement was proved very recently by Shlapentokh-Rothman in [SR13]
by first extending [Whi89] to exclude resonances on the real axis and then refining this
qualitative statement to a quantitative estimate.1
Turning to the Kerr–Newman spacetimes, even the analogue of Whiting’s mode sta-
bility is absent in the literature. In the present chapter, we will prove for these spacetimes
both the qualitative mode stability results (in the upper half-plane and on the real axis) as
well as the quantitative estimate in the spirit of [SR13]. In particular, the latter result is
needed for the general boundedness and decay results presented in Chapter 3. The precise
mode stability results are stated here in §4.5 and the estimate needed in in the proof of
Proposition 3.6.1 is presented here in Theorem 4.8.2.
In the Kerr case, the crucial ingredients in the proof of mode stability given in [Whi89]
1In the case |a| M , one need not appeal to Whiting’s [Whi89] (or its refinement [SR13]) as the small
parameter may be exploited to deal directly with superradiance. A boundedness result had been obtained
for |a| M in [DR11b] followed by decay results in [AB09], [DR09] and [TT11]. For the situation in the
extremal case |a| = M , see [Are12a] and [Are12b]. For the case where Λ > 0, see [Dya11] and for the Λ < 0
case, see [Gan12], [HS13a] and [HS13b].
110
4.2. Mode solutions of the wave equation
and [SR13] are the remarkable transformation properties of the radial ODE satisfied by the
modes. Miraculously, all the essential elements of this structure are preserved in passing
from the Kerr to the Kerr–Newman solution. In particular, we show that the radial ODE
can be represented as a confluent Heun equation (See §4.4). We then define the Whiting
transform for u(ω,m, λ, r) with Im(ω) ≥ 0 (see (4.6.3) for the definition). The Whiting
transform takes the solution u∗ of a confluent Heun equation to u˜ which solves another
confluent Heun equation with different coefficients (See Proposition 4.6.1). There are three
pivotal facts about this transform:
(a) The potential in the confluent Heun equation satisfied by u˜ possesses certain positivity
properties. (See Proposition 4.7.1.)
(b) u˜ has ‘good’ asymptotics near the horizon and near null infinity. (See Propositions
4.6.2 and 4.6.3.)
(c) For ω 6= 0 on the real axis, the limit of u at the horizon is a positive multiple of the
limit of u˜ at r →∞. (See Proposition 4.6.3.)
The statements above were shown to be true for the Kerr case in [Whi89] and [SR13];
there is no a priori reason why one would expect these properties to carry over to the
Kerr–Newman case. It is thus a fortunate fact that the potential and ∆ parameter for the
Kerr–Newman case differ from those in the Kerr case in such a way that the conditions
(a), (b) and (c) still hold. This is discussed further in §4.6.
4.2 Mode solutions of the wave equation
A general subextremal Kerr–Newman metric possesses only the two Killing fields ∂t and ∂φ.
Nonetheless, Carter discovered in [Car68] that the wave equation (2.1.8) can be formally
separated. This is related to the existence of an additional hidden symmetry. We use this
to make the following definition:
Definition 4.2.1. Let (M, g) be a subextremal Kerr–Newman spacetime. A smooth so-
lution ψ of the wave equation (2.1.8) is called a mode solution if there exist (ω,m, `) ∈
C \ {0} × Z× {Z : ` ≥ |m|} such that
ψ(t, r, θ, φ) = R
(aω)
m` (r)S
(aω)
m` (θ)e
imφe−iωt,
where
111
Quantitative mode stability for the wave equation on the Kerr–Newman spacetime
1. S
(aω)
m` solves the following Sturm-Liouville problem
1
sin θ
d
dθ
(
sin θ
dS
(aω)
m`
dθ
)
−
(
m2
sin2 θ
− a2ω2 cos2 θ
)
S
(aω)
m` + λ
(aω)
m` S
(aω)
m` = 0 (4.2.1)
with the boundary condition that
eimφS
(aω)
m` (θ) extends smoothly to S
2, (4.2.2)
with S
(aω)
m` an eigenfunction with corresponding eigenvalue λ
(aω)
m` of the angular ODE
(4.2.1).2
2. R solves the radial equation[
∂r(∆∂r)− ω2
(
a2 − (a
2 + r2)2
∆
)
+
a2m2
∆
− 2amω(2Mr −Q
2)
∆
− λ(aω)m`
]
R = 0.(4.2.3)
3. R(r)(r − r+)−
i(am−(2Mr+−Q2)ω)
r+−r− is smooth at r = r+.
3
4. There exist constants {Ck}∞k=0 such that for any N ≥ 1,
R(r∗) =
eiωr
∗
r
N∑
k=0
Ckr
−k +O(r−N−2),
for large r.4
The boundary conditions (4.2.2) and in points 3 and 4 above are uniquely determined
by requiring that ψ extends smoothly to the horizon H+ and has finite energy along
asymptotically flat hypersurfaces for Im(ω) > 0 and along hyperboloidal hypersurfaces
for Im(ω) ≤ 0. See the discussion in [SR13, Appendix D] for details, cf. [Dya11] and
[War13].
It is convenient to define
u
(aω)
m` (r
∗) =
√
r2 + a2R
(aω)
m` (r) (4.2.4)
2The Sturm–Liouville problem admits a set of eigenfunctions
{
S
(aω)
m`
}∞
`=|m|
and real eigenvalues{
λ
(aω)
m`
}∞
`=|m|
. The eigenfunctions
{
S
(aω)
m`
}
are called “oblate spheroidal harmonics” and define an or-
thonormal basis for L2(sin θdθ).
3We will subsequently denote this as R(r) ∼ (r − r+)
i(am−(2Mr+−Q2)ω)
r+−r− at r = r+.
4We will subsequently denote this as R(r∗) ∼ r−1eiωr∗ as r →∞.
112
4.3. The Wronskian
which satisfies the radial Carter ODE:
d2
(dr∗)2
u
(aω)
m` (r
∗) +
(
ω2 − V (aω)m` (r)
)
u
(aω)
m` = 0. (4.2.5)
Note that even though R
(aω)
m` is complex-valued, the potential V
(aω)
m` is real (see (3.3.12)
for more details).
We will often drop the indices ω,m, ` when there is no risk of confusion. We will also
adopt the convention that u′ denotes a derivative with respect to r∗.
4.3 The Wronskian
Through asymptotic analysis of (4.2.5), one can make the following definitions:
Definition 4.3.1. Let uhor(r
∗, ω,m, `) be the unique function satisfying
1. u′′hor + (ω
2 − V )uhor = 0.
2. uhor ∼ (r − r+)
i(am−(2Mr+−Q2)ω)
r+−r− as r∗ → −∞.
3.
∣∣∣∣∣
(
(r(r∗)− r+)−
i(am−(2Mr+−Q2)ω)
r+−r− uhor
)
(−∞)
∣∣∣∣∣
2
= 1.
Definition 4.3.2. Let uout(r
∗, ω,m, `) be the unique function satisfying
1. u′′out + (ω2 − V )uout = 0.
2. uout ∼ eiωr∗ as r∗ →∞.
3.
∣∣(uoute−iωr∗) (∞)∣∣2 = 1.
One then defines the Wronskian
W (ω,m, `) = uhor(r
∗)u′out(r
∗)− u′hor(r∗)uout(r∗). (4.3.1)
The Wronskian can be evaluated at any fixed r∗. The Wronskian W will vanish if the
solutions are linearly dependent. Then W = 0 implies
∣∣W−1∣∣ = ∞. The quantitative
mode stability result will be an explicit upper bound for
∣∣W−1∣∣, so that uout and uhor are
linearly independent and any solution of the Carter ODE (4.2.5) can be expressed as a
superposition of those solutions.
113
Quantitative mode stability for the wave equation on the Kerr–Newman spacetime
4.4 The inhomogeneous equation
In the proof of Theorem 4.5.1, we will consider the following inhomogeneous form of
(4.2.3),[
∂r(∆∂r)− ω2
(
a2 − (a
2 + r2)2
∆
)
+
a2m2
∆
− 2amω(2Mr −Q
2)
∆
− λ(aω)m`
]
R
(aω)
m` = F,
(4.4.1)
where F is a compactly supported smooth function on (r+,∞). The corresponding inho-
mogeneous version of (4.2.5) is then
d2
(dr∗)2
u
(aω)
m` (r
∗) +
(
ω2 − V (aω)m` (r)
)
u
(aω)
m` = H :=
∆F
(r2 + a2)1/2
. (4.4.2)
4.5 Statement of mode stability results
For a subextremal Kerr–Newman spacetime (M, g), we have the following results.
Theorem 4.5.1 (Quantitative mode stability on the real axis). Let
F ⊂ {(ω,m, `) ∈ R× {Z× Z | ` ≥ |m|}}
be a frequency range for which
CF := sup
(ω,m,`)∈F
(
|ω|+ |ω|−1 + |m|+
∣∣∣λ(aω)m` ∣∣∣) <∞.
Then the Wronskian W given by (4.3.1) satisfies
sup
(ω,m,`)∈F
∣∣W−1∣∣ ≤ G(CF , a,Q,M).
where the function G can, in principle, be given explicitly.
In proving the quantitative result above, we will also obtain the following qualitative
results.
Theorem 4.5.2 (Mode Stability on the real axis). There exist no non-trivial mode solu-
tions corresponding to ω ∈ R \ {0}.
Theorem 4.5.3 (Mode Stability). There exist no non-trivial mode solutions corresponding
to Im(ω) > 0.
Theorem 4.5.3 is the analogue of Whiting’s original mode stability result [Whi89].
Theorem 4.5.2 is the analogue of Shlapentokh-Rothman’s extension of Whiting’s mode
114
4.6. The Whiting transform
stability result [Whi89] to the real axis. Theorem 4.5.1 is the quantitative refinement
of Theorem 4.5.2 needed in Chapter 3 for the proof of linear stability of subextremal
Kerr–Newman black holes.
Note that for non-superradiant frequencies ω, m, i.e. those outside of the range (4.1.1),
Theorem 4.5.2 and Theorem 4.5.3 follow immediately from the energy identity (see [SR13,
§1.5 & §1.6]). In what follows, we will not however make a distinction between superradiant
and non-superradiant frequencies.
4.6 The Whiting transform
The problem with trying to derive energy estimates for the Carter ODE (4.2.5) is that
the boundary condition at r∗ = −∞ may give a non-positive term due to superradiance.
To deal with this, we will first cast (4.2.5) as a confluent Heun equation (4.6.2). Applying
the Whiting transform (4.6.3) to (4.6.2), we will obtain a new confluent Heun equation
(4.6.4) with different coefficients and boundary conditions that allow for a useful energy
estimate.
4.6.1 The confluent Heun equation
We rescale R as follows. Let
u∗ := eiωr(r − r−)−η(r − r+)−ξR(r) (4.6.1)
where
η := − i
(
am− ω (2Mr− −Q2))
r+ − r− and ξ :=
i
(
am− ω (2Mr+ −Q2))
r+ − r− .
Then u∗ satisfies the following Confluent Heun equation:
(r − r+)(r − r−)d
2u∗
dr2
+ (γ(r − r+) + δ(r − r−) + p(r − r+)(r − r−)) du
∗
dr
+ (αp(r − r−) + σ)u∗ = G (4.6.2)
115
Quantitative mode stability for the wave equation on the Kerr–Newman spacetime
where
γ := 2η + 1,
δ := 2ξ + 1,
p := −2iω,
α := 1,
σ := 2amω − 2ωr−i− λ(aω)m` − a2ω2
and G := eiωr(r − r−)−η(r − r+)−ξF.
This can be verified by a direct calculation, generalising the analogous computation in
[Whi89].
Note that, as in the (subextremal) Kerr case, r+ and r− are distinct roots of ∆. If ∆
had more roots, or if these roots were not distinct, the Carter ODE would lie in a different
class of equations.
4.6.2 The transformed equation
We now generalise the Whiting transformation to the Kerr–Newman case.
Proposition 4.6.1. Let Im(ω) ≥ 0, ω 6= 0, and let R solve (4.4.1) with the boundary
conditions of Definition 4.2.1. Define x∗ analogously to r∗ by
dx∗
dx
=
x2 + a2
(x− r+)(x− r−) , x
∗(3M) = 0,
Then define u˜ by
u˜(x∗) := (x2 + a2)1/2(x− r+)−2iMωe−iωx
×
∫ ∞
r+
e
2iω
r+−r− (x−r−)(r−r−)(r − r−)η(r − r+)ξe−iωrR(r)dr (4.6.3)
where
η := − i
(
am− ω (2Mr− −Q2))
r+ − r− and ξ :=
i
(
am− ω (2Mr+ −Q2))
r+ − r− .
Then u˜(x) is smooth on (r+,∞) and satisfies the following confluent Heun equation:
u˜′′ + Φu˜ = H˜, (4.6.4)
116
4.6. The Whiting transform
where primes denote derivatives with respect to x∗,
H˜(x∗) :=
(x− r+)(r − r−)
(x2 + a2)2
G˜(x),
G˜(x) :=
(x2 + a2)1/2
(x− r+)2iMω e
−iωx
∫ ∞
r+
e
2iω
r+−r− (x−r−)(r−r−)(r − r−)η(r − r+)ξe−iωrF (r)dr,
Φ(x∗) :=
(x− r−)(x− r+)
(a2 + x2)4
((
2x2 − a2) (r−r+)− 2Mx(x2 − 2a2)− 3a2x2)
+
(x− r−)(x− r+)
(a2 + x2)2
(
4am(x−M)ω
r− − r+ − λ
(aω)
m` − a2ω2
+
8M2(x−M)(x− r−)ω2
(r− − r+)(r+ − x) +
(x− r−)
(
(r+ − r−)(x− r+)− 4Q2
)
ω2
r+ − r−
)
Proof. It turns out that the proof is a direct modification of the computations in [SR13,
§4]. Let us remark on the fortuitous structure of the Kerr–Newman spacetimes that makes
this so. We have already remarked in §4.6.1 that (4.6.2) is a confluent Heun equation and
thus (at least formally) admits non-trivial transformations. The exponents η and ξ are
obtained from indicial equation associated to (4.2.5). They are the unique exponents that
give the correct asymptotics at r+ and r−.
The definitions of η, ξ, r+ and r− for the Kerr–Newman case differ from those in the
Kerr case, but the potential V
(aω)
m` , the parameter ∆ and the asymptotics of the solutions
of mode solutions of (4.2.5), have the same structure. The convergence of the integral in
(4.6.3) thus follows as in [SR13, §4].
Remark. The Whiting transform is a shifted, rescaled Fourier transform of a rescaled
version of R. This fact will be crucial in showing that the vanishing of u˜ forces R to vanish.
4.6.3 Asymptotics of the transformed solution
The good asymptotic properties of u˜ (c.f. (b) and (c) of the introduction) are encapsulated
in the following two propositions.
Proposition 4.6.2. Let ω and u˜ be as in the statement Proposition 4.6.1. If Im(ω) > 0
then
1. u˜ = O
(
(x− r+)2MIm(ω)
)
as x→ r+.
2. u˜′ = O
(
(x− r+)2MIm(ω)
)
as x→ r+.
3. u˜ = O
(
e−Im(ω)x1+2MIm(ω)
)
as x→∞.
117
Quantitative mode stability for the wave equation on the Kerr–Newman spacetime
4. u˜′ = O
(
e−Im(ω)x1+2MIm(ω)
)
as x→∞.
Proposition 4.6.3. Let ω and u˜ be as in the statement Proposition 4.6.1. If ω ∈ R \ {0}
then
1. u˜ and u˜′ are uniformly bounded.
2. |u˜(∞)|2 = (r+−r−)2|Γ(2ξ+1)|2
4(2Mr+−Q2)ω2 |u(−∞)|
2, where Γ(z) :=
∫∞
0 e
−ttz−1dt is the Gamma func-
tion.
3. u˜′ − iωu˜ = O (x−1) as x∗ →∞.
4. u˜′ + iωr−1+ (r+ − r−)u˜ = O (x− r+) as x∗ → −∞.
The proofs of these propositions are direct modifications of the computations in [SR13,
§4].
For all the results above, except Proposition 4.6.3.2, the difference between the Kerr
and Kerr–Newman case is encapsulated within the different definitions of r+ and r−.
Proposition 4.6.3.2 is exceptional in that we see an explicit difference from the Kerr
case. This is due to the presence of (2Mr+−Q2) in the null generator of the Kerr–Newman
horizon.
Proposition 4.6.3.2 is crucial in proving the quantitative result Theorem 4.5.1 as it
provides a correspondence between the horizon asymptotics of the solution of the Carter
ODE and the large r∗ asymptotics of the transformed solution. This correspondence is
what allows for the quantitative estimate of the horizon flux in terms of the inhomogeneity
F (see the proof of Proposition 4.7.2).
We can now prove the qualitative Theorems 4.5.2 and 4.5.3.
4.7 Proofs of mode stability
4.7.1 Qualitative results
The final element of the structure necessary to prove mode stability for the Kerr–Newman
spacetimes is the following positivity property (c.f. (a) of the introduction):
Proposition 4.7.1. Under the conditions of Proposition 4.6.1,
Im(Φω¯) ≥ 0.
If ω ∈ R \ {0}, then Φ is real-valued.
118
4.7. Proofs of mode stability
Proof. The second statement is clear from the definition of Φ. A (tedious) computation
shows that
Im(Φω¯)
=
(x− r−)(x− r+)
(a2 + x2)2
Im
(
(−λ(aω)m` − a2ω2)ω¯
)
+
(x− r−)2(x− r+)2ωI |ω|2
(a2 + x2)2
+
(x− r−)2(x− r+)ωI |ω|2
(a2 + x2)2
(
8M2(x−M)− 4Q2(x− r+) + (r+ − r−)(x− r+)2
)
(r+ − r−)(x− r+)
+
(x− r−)(x− r+)
(a2 + x2)4
(ωI)
[
x2(r+ − a2 −Q2) + r−(x2 + a2)(x− r+)
+2xa2(x+ r− − r+)
]
.
To see that Im
(
(−λ(aω)m` − a2ω2)ω¯
)
≥ 0, multiply (4.2.1) by ωS(aω)m` sin θ and integrate by
parts over [0, pi].
The positivity of the other terms follows from the following chain of inequalities
0 ≤ r− ≤M ≤ r+ ≤ x
and the subextremal condition a2 +Q2 < M2.
We define the microlocal energy current
Q˜T := Im(u˜
′ωu˜).
Proof of Theorem 4.5.3 (Mode stability in the upper half-plane). Let ω = ωR + iωI and
Im(ω) = ωI > 0 and consider a mode solution of (2.1.8) with (u
(aω)
m` , S
(aω)
m` , λ
(aω)
m` ). Define
u˜ to be the (4.6.3) of u
(aω)
m` . Then Proposition 4.6.2 implies that Q˜T (±∞) = 0 so
0 = −
∫ ∞
−∞
(Q˜T )
′dr∗ =
∫ ∞
−∞
ωI
∣∣u˜′∣∣2 + Im(Φω¯)|u˜|2dr∗
Since Proposition 4.7.1 guarantees that Im(Φω¯) ≥ 0, we conclude that, u˜, the Whiting
transform of u vanishes. Hence
R˜(x) :=
∫ ∞
r+
e
2iω
r+−r− (x−r−)(r−r−)(r − r−)η(r − r+)ξe−iωrR(r)dr = 0.
Extending R by 0, we see that the Fourier transform of (r− r−)η(r− r+)ξe−iωrR(r) is (up
119
Quantitative mode stability for the wave equation on the Kerr–Newman spacetime
to a change of variable)
Rˆ(z) :=
∫ ∞
−∞
e2i|ω|
2(z−r−)(r − r−)η(r − r+)ξe−iωrR(r)dr.
The function R is supported in [0,∞), so by the Paley–Wiener Theorem, Rˆ can be ex-
tended holomorphically into the upper half plane. Since R = 0 for x ∈ (−∞, r+) and
R˜ = 0 for x ∈ (r+,∞), Rˆ = 0 on the real line. We can therefore use the Schwartz
reflection principle to extend Rˆ holomorphically to all of C.
Furrthermore, the vanishing of R˜ on x ∈ (r+,∞) implies that Rˆ = 0 on the line
{z = ω¯(x− r+)/(r+ − r−) | x ∈ (r+,∞)} .
The Identity Theorem for holomorphic functions then implies that Rˆ vanishes everywhere.
This forces R to vanish everywhere, completing the proof.
Lemma 4.7.1 (Unique continuation [SR13]). Suppose that we have a solution u(r∗) :
(−∞,∞)→ C to
u′′ + (ω2 − V )u = 0
such that
1. ω ∈ R \ {0},
2. u is uniformly bounded and (|u′|2 + |u|2)(∞) = 0,
3. V is real, uniformly bounded, V = O(r−1) as r →∞ and V ′ = O(r−2) as r →∞.
Then u is identically 0.
Proof. This follows exactly as in [SR13, §6]
Proof of Theorem 4.5.2 (Mode stability on the real axis). Let ω ∈ R \ {0} and consider a
mode solution of (2.1.8) with (u
(aω)
m` , S
(aω)
m` , λ
(aω)
m` ). Define u˜ by (4.6.3). By Proposition
4.6.1, Φ is real, so (Q˜T )
′ = 0 . Hence Q˜T (∞) − QT (−∞) = 0. The boundary conditions
from Proposition 4.6.3 then imply that
ω2|u˜(∞)|2 + ∣∣u˜′(∞)∣∣2 + ω2 r+ − r−
r+
|u˜(−∞)|2 + r+
r+ − r−
∣∣u˜′(−∞)∣∣2 = 0.
By Lemma 4.7.1, we conclude that u˜ vanishes.
120
4.7. Proofs of mode stability
Extending R by 0, we see that
R˜(y) :=
∫ ∞
−∞
e
2iω
r+−r− (y−r−)(r−r−)(r − r−)η(r − r+)ξe−iωrR(r)dr
vanishes for {y ∈ (r+,∞)}. Now repeating the closing argument of the proof of Theorem
4.5.3, we conclude that R must vanish everywhere.
4.7.2 Quantitative results
The strategy is to express u˜ in terms of the functions uout and uhor and W defined in §4.3
and obtain an an estimate for W−1 in terms of u(−∞). This quantity is then estimated
using the ODE (4.4.2).
Proposition 4.7.2. Define F as in Theorem 4.5.1. For (ω,m, `) ∈ F let u solve (4.4.2)
with H(x∗) a smooth, compactly supported function. Then for sufficiently small > 0,
there exists a positive constant C := C(F , a,Q,M) such that
|u(−∞)|2 ≤ C
(
−1
∫ ∞
r+
|F (r)|2r4dr
)
.
Proof. Since (Q˜T )
′ = ωIm(H˜u¯),∫ ∞
−∞
ωIm(H˜u¯)dr∗ = Q˜T (∞)− Q˜T (−∞).
The boundary conditions from Proposition 4.6.3 imply that
ω2|u˜(∞)|2 + ∣∣u˜′(∞)∣∣2 + ω2 r+ − r−
r+
|u˜(−∞)|2 + r+
r+ − r−
∣∣u˜′(−∞)∣∣2 = ∫ ∞
−∞
ωIm(H˜u¯)dr∗.
So changing variables, applying the Plancherel identity and the Cauchy Schwarz inequality,
we have
ω2|u˜(∞)|2 ≤
∫ ∞
−∞
ωIm(H˜u¯)dr∗ ≤ C
(
−1
∫ ∞
r+
|F (r)|2r4dr +
∫ ∞
r+
|R(r)|2dr
)
.
Then by Proposition 4.6.3
|u(−∞)|2 = 4ω
2(2Mr+ −Q2)
|Γ(2ξ + 1)|2 |u˜(∞)|
2 ≤ C
(
−1
∫ ∞
r+
|F (r)|2r4dr +
∫ ∞
r+
|R(r)|2dr
)
.
Finally,
∫ ∞
r+
|R(r)|2dr ≤ C
∫ ∞
r+
|F (r)|2r4dr,
121
Quantitative mode stability for the wave equation on the Kerr–Newman spacetime
by the same argument as found in [SR13, §5].
For the quantitative result, we construct mode solutions solutions to the Carter ODE
from the Wronskian and apply the proposition above.
Lemma 4.7.2. Let H(x∗) be compactly supported. For any (ω,m, `) ∈ F (where F is as
defined in Theorem 4.5.1), the function
u(r∗) = W (ω,m, `)−1
(
uout(r
∗)
∫ r∗
−∞
uhor(x
∗)H(x∗)dx∗
+uhor(r
∗)
∫ ∞
r∗
uout(x
∗)H(x∗)dx∗
)
satisfies
u′′ + (ω2 − V )u = H
and the boundary conditions of a mode solution (see Definition 4.2.1).
Proof. This is verified by a direct calculation.
Proof of Theorem 4.5.1 (Quantitative mode stability on the real axis). Take u˜ as defined
in Lemma 4.7.2. Then
|u(−∞)|2 = ∣∣W−2∣∣∣∣∣∣∫ ∞−∞ uout(x∗)H(x∗)dx∗
∣∣∣∣2.
Rearranging this expression and applying Proposition 4.7.2 we find that
∣∣W−2∣∣ = |u(−∞)|2∣∣∣∫∞−∞ uout(x∗)H(x∗)dx∗∣∣∣2 ≤ C
∫∞
−∞
∣∣(r2 + a2)1/2∆−1H(r∗)∣∣2r4dr∣∣∣∫∞−∞ uout(x∗)H(x∗)dx∗∣∣∣2 .
Note that by Proposition 4.6.3, for sufficiently large x,
∣∣uout(x)− eiωx∣∣ < Cx−1 for an
explicit C. Since W is independent of H we choose a compactly supported H for which
the right hand side of the estimate above is finite. We thus have a quantitative estimate
for
∣∣W−2∣∣.
4.8 Application: Integrated local energy decay
We now apply Theorem 4.5.1 to prove Theorem 4.8.2, which provides a quantitative energy
decay estimate for solutions of the wave equation (2.1.8) on subextremal Kerr–Newman
spacetimes which are supported in a compact range of superradiant frequencies. This is
122
4.8. Application: Integrated local energy decay
the estimate appealed to in the proof of Proposition 3.6.1 to control the horizon term
|u(aω)m` (−∞)|2 in the bounded superradiant frequency region.
We wish to apply Carter’s separation to the solution of (2.1.8). In order to perform
this separation, we must be able to take the Fourier transform in time. We therefore deal
with solutions of (2.1.8). which belong to the following class of functions.
Definition 4.8.1. A smooth function f(t, r, θ, φ) is said to be admissible if for any multi-
indices α, β s.t. |α| ≥ 1, |β| ≥ 0, we have
1.
∫
r>r0
∫
S2
|∂αf |2|t=0r2 sin θdr dθ dφ <∞ for sufficiently large r0.
2.
∫ ∞
0
∣∣∣∂βf ∣∣∣2dt <∞ for any (r, θ, φ) ∈ (r+,∞)× S2.
3.
∫ ∞
0
∫
K
∣∣∣∂βf ∣∣∣2 sin θ dr dθ dφ dt <∞ for any compact K ∈ (r+,∞)× S2.
For an admissible function f we also define
|∂f |2 := |(∂t + ∂r∗)f |2 + ∆|(∂t − ∂r∗)f |2 + r−2
(
sin−2 θ|∂φf |2 + |∂θf |2
)
.5 (4.8.1)
The main application of Theorem 4.5.1 in §3.6 is to admissible solutions ψ of (2.1.8)
which are cut off as follows.
Definition 4.8.2. Let Σ0 be a spacelike hyperboloidal hypersurface connecting the horizon
H+ and future null infinity. Let Σ1 be the time 1 image of Σ0 under the flow generated by
∂t. Then define a smooth cut-off γ which is identically 0 in the past of Σ0 and identically 1
in the future of Σ1. We define ψQ := γψ, which satisfies the inhomogeneous wave equation
gψQ = F, where F = (γ)ψ + 2∇µγ∇µψ. (4.8.2)
Proposition 4.8.1 (Carter’s separation). Admissible solutions f of (2.1.8) and (4.8.2)
can be expressed as
f(t, r, θ, φ) =
Fourier transform︷ ︸︸ ︷
1√
2pi
∫ ∞
−∞
∑
m,`≥|m|
R
(aω)
m` (r) · S(aω)m` (cos θ)eimφ︸ ︷︷ ︸
Oblate spheroidal expansion
e−iωt dω. (4.8.3)
The function R
(aω)
m` corresponding to f = ψ solves (4.2.3). The function R
(aω)
m` correspond-
ing to f = ψQ satisfies the inhomogeneous equation (4.4.1) with F = F (aω)m` , the Fourier
5The apparent degeneration of this energy as r → ∞ is due to the hyperboloidal nature of Σ0. The
term ∆|(∂t − ∂r∗)f |2 converges to the transversal derivative at the horizon.
123
Quantitative mode stability for the wave equation on the Kerr–Newman spacetime
transform of F projected to the oblate spheroidal harmonic corresponding to λ
(aω)
m` . The
rescaled function u
(aω)
m` satisfies (4.4.2) with H = H
(aω)
m` := ∆(r
2 +a2)−1/2F (aω)m` , where this
equality is to be understood in the sense of L2ω∈B`
2
m,`∈C. Note moreover that this H is not
compactly supported.
Proof. See §3.3.3.
Theorem 4.8.2. Let ψQ be an admissible solution of (4.8.2) and let B ⊂ R and
C ⊂ {(m, `) ∈ Z× Z | ` ≥ |m|}
such that
CB := sup
ω∈B
(
|ω|+ |ω|−1
)
<∞ and CC := sup
m,`∈C
(
|m|+
∣∣∣λ(aω)m` ∣∣∣) <∞.
There exists a constant K := K(r0, r1, CB, CC , a,Q,M) such that∫
B
∑
m,`∈C
((∣∣∣u(aω)m` (−∞)∣∣∣2 + ∣∣∣u(aω)m` (∞)∣∣∣2)+ ∫ r1
r0
∣∣∣∂r∗u(aω)m` ∣∣∣2 + ∣∣∣u(aω)m` ∣∣∣2 dr∗) dω
≤ K
∫
Σ0
|∂ψ|2, (4.8.4)
where |∂ψ|2 is defined by (4.8.1), u(aω)m` =
√
r2 + a2R
(aω)
m` and each R
(aω)
m` solves (4.4.1) for
ω ∈ B and (m, `) ∈ C.
Proof. For u satisfying the hypotheses of the theorem, we have for any r∗ ∈ (−∞,∞),
u(r∗) = W (ω,m, `)−1
(
uout(r
∗)
∫ r∗
−∞
uhor(x
∗)H(x∗)dx∗
+uhor(r
∗)
∫ ∞
r∗
uout(x
∗)H(x∗)dx∗
)
, (4.8.5)
u′(r∗) = W (ω,m, `)−1
(
u′out(r
∗)
∫ r∗
−∞
uhor(x
∗)H(x∗)dx∗
+u′hor(r
∗)
∫ ∞
r∗
uout(x
∗)H(x∗)dx∗
)
, (4.8.6)
where the inequalities above hold in the sense of L2ω∈B`
2
m,`∈C (see [SR13, §3] for the full
derivation of this representation).6
6Roughly speaking, this is the converse of Lemma 4.7.2.
124
4.8. Application: Integrated local energy decay
By the construction of uhor and uout, there exists a positive K := K(CB, CC , a,Q,M)
such that
sup
r∗∈R,ω∈B,(m,`)∈C
(|uhor|+ |uout|) < K <∞, (4.8.7)
Evaluating (4.8.5) at r∗ = −∞ and taking (4.8.7) into account,
∫
B
∑
m,`∈C
∣∣∣u(aω)m` (−∞)∣∣∣2 dω ≤ K lim sup
r∗→−∞
∫
B
∑
m,`∈C
W−2
∣∣∣∣∫ ∞
r∗
uout(x
∗)H(aω)m` (x
∗)dx∗
∣∣∣∣2 dω.
(4.8.8)
For the term |u(∞)|2 we apply the microlocal energy current:
ω2
∣∣∣u(aω)m` (∞)∣∣∣2 = QT (∞) = QT (−∞) + ∫ ∞−∞(QT )′dr∗
= ω(am− (2Mr+ −Q2)ω)
∣∣∣u(aω)m` (−∞)∣∣∣2 + ω ∫ ∞−∞ Im(H(aω)m` u¯(aω)m` )dr∗
So by (4.8.8),
∫
B
∑
m,`∈C
∣∣∣u(aω)m` (∞)∣∣∣2 dω ≤ K ∫B ∑m,`∈CW−2
∣∣∣∣∫ ∞−∞ uout(x∗)H(aω)m` (x∗)dx∗
∣∣∣∣2 dω
+
∫
B
∑
m,`∈C
ω
∫ ∞
−∞
Im(H
(aω)
m` u¯
(aω)
m` )dr
∗ dω. (4.8.9)
For the integral term, we begin by taking R1 much larger than r1 and applying (4.8.5):
∫
B
∑
m,`∈C
sup
r∗∈(r0,r1)
∣∣∣u(aω)m` ∣∣∣2 dω
≤ K
∫
B
∑
m,`∈C
W−2
sup
r∗∈[r0,r1]
∣∣∣∣∣
∫ r∗
−∞
uhor(x
∗)H(aω)m` (x
∗)dx∗
∣∣∣∣∣
2
+ sup
r∗∈[r0,r1]
∣∣∣∣∫ R1
r∗
uout(x
∗)H(aω)m` (x
∗)dx∗
∣∣∣∣2
+
∣∣∣∣∫ ∞
R1
uout(x
∗)H(aω)m` (x
∗)dx∗
∣∣∣∣2
)
dω
≤ K
∫
B
∑
m,`∈C
W−2
(∫ R1
r+
|F |2dr +
∣∣∣∣∫ ∞
R1
uout(x
∗)H(aω)m` (x
∗)dx∗
∣∣∣∣2
)
dω.
125
Quantitative mode stability for the wave equation on the Kerr–Newman spacetime
This estimate may be integrated over (r0, r1) to obtain∫
B
∑
m,`∈C
∫ r1
r0
∣∣∣u(aω)m` ∣∣∣2 dω ≤ K ∫B ∑m,`∈CW−2
∫ R1
r+
|F |2dr
+K
∫
B
∑
m,`∈C
W−2
∣∣∣∣∫ ∞
R1
uout(x
∗)H(aω)m` (x
∗)dx∗
∣∣∣∣2 dω. (4.8.10)
The same argument, with (4.8.5) replaced with (4.8.6) yields∫
B
∑
m,`∈C
∫ r1
r0
∣∣∣(u(aω)m` )′∣∣∣2 dω ≤ K ∫B ∑m,`∈CW−2
∫ R1
r+
|F |2dr
+K
∫
B
∑
m,`∈C
W−2
∣∣∣∣∫ ∞
R1
uout(x
∗)H(aω)m` (x
∗)dx∗
∣∣∣∣2 dω. (4.8.11)
Collecting (4.8.8), (4.8.9), (4.8.10) and (4.8.11) and applying Theorem 4.5.1 to control
W−2, we have∫
B
∑
m,`∈C
((∣∣∣u(aω)m` (−∞)∣∣∣2 + ∣∣∣u(aω)m` (∞)∣∣∣2)+ ∫ r1
r0
∣∣∣∂r∗u(aω)m` ∣∣∣2 + ∣∣∣u(aω)m` ∣∣∣2 dr∗) dω
≤ KG
∫
B
∑
m,`∈C
[∣∣∣∣∫ ∞
R1
uout(x
∗)H(aω)m` (x
∗)dx∗
∣∣∣∣2 + ∫ R1
r+
|F |2dr
+ω
∫ ∞
−∞
Im(H
(aω)
m` u¯
(aω)
m` )dr
∗
]
dω.
It remains to control the right hand side of this estimate by
∫
Σ0
JN [ψ] · nΣ0 . The control
of the first term is achieved using the proof of [SR13, Lemma 3.3]. The remaining terms
are controlled using the argument presented in §3.6.
Remark We can replace the hyperboloidal hypersurface Σ0 with an asymptotically flat
hypersurface in Theorem 4.8.2 as follows. Let Σ∗0 be an asymptotically flat hypersurface
that agrees with Σ0 for {r ≤ R} and which lies in the past of Σ0. Choosing R large enough
that T is timelike in {r ≤ R}, applying the T energy estimate immediately implies that∫
Σ0
|∂ψ|2 ≤ C
∫
Σ∗0
∣∣∣∇gΣ∗0ψ∣∣∣2 + ∣∣nΣ∗0ψ∣∣2,
so we can then replace the right hand side of (4.8.4) by this integral over an asymptotically
flat hypersurface.
126
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