Linear Waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes Volker Schlue King’s College University of Cambridge Department of Pure Mathematics and Mathematical Statistics April 2012 This dissertation is submitted for the degree of Doctor of Philosophy Declaration of Originality The research presented in this thesis was conducted at the Department of Pure Math- ematics and Mathematical Statistics, University of Cambridge in the period between October 2008 and April 2012. The work contained in this thesis is original and was entirely performed by myself. The content of Chapter 1 has been submitted in part to the Smith-Rayleigh-Knight Prize in January 2010 and in full for publication in a jour- nal; a shortened version is also available online at http://arxiv.org/abs/1012.5963. The content of Chapter 2 is unpublished in any form at the time of submission. This dissertation has not been submitted for any degree or other qualification. Summary I study linear waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes. In the first part of this thesis two decay results are proven for general finite energy solutions to the linear wave equation on higher dimensional Schwarzschild black holes. I establish uniform energy decay and improved interior first order energy decay in all dimensions with rates in accordance with the 3 + 1-dimensional case. The method of proof departs from earlier work on this problem. I apply and extend the new physical space approach to decay of Dafermos and Rodnianski. An integrated local energy decay estimate for the wave equation on higher dimensional Schwarzschild black holes is proven. In the second part of this thesis the global study of solutions to the linear wave equation on expanding de Sitter and Schwarzschild de Sitter spacetimes is initiated. I show that finite energy solutions to the initial value problem are globally bounded and have a limit on the future boundary that can be viewed as a function on the standard cylinder. Both problems are related to the Cauchy problem in General Relativity. Acknowledgments I would like to thank Mihalis Dafermos for his advice and encouragement. I am very grateful to him for suggesting the problems resolved in this thesis and for many insightful conversations which have been a genuine source of inspiration and motivation to me. I would like to thank him for his generous support and the freedom he has given me in my time here. I was very fortunate to have him as my advisor. I would also like to thank Demetrios Christodoulou who has sparked my initial interest in General Relativity for his support and encouragement to continue studies at Cambridge, and I am indebted especially to Martin Hyland and Mihalis Dafermos for giving me the opportunity to do so. I am grateful to the UK Engineering and Physical Sciences Research Council (EPSRC), the Cambridge European Trust (CET) and the European Research Council (ERC) for their financial support. I am grateful to many friends and my family. Cambridge, April 2012 Volker Schlue Contents 1 Decay of linear waves on higher dimensional Schwarzschild black holes 13 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.1 Statement of the Theorems . . . . . . . . . . . . . . . . . . . . . . 14 1.1.2 Overview of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2 Global causal geometry of the higher dimensional Schwarzschild solution . 20 1.3 The Red-shift effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4 Integrated Local Energy Decay . . . . . . . . . . . . . . . . . . . . . . . . 35 1.4.1 Radial multiplier vectorfields . . . . . . . . . . . . . . . . . . . . . . 37 1.4.2 High angular frequencies . . . . . . . . . . . . . . . . . . . . . . . . 43 1.4.3 Low angular frequencies and commutation . . . . . . . . . . . . . . 55 1.4.4 Boundary terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 1.5 The Decay Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 1.5.1 Uniform Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . 74 1.5.2 Energy decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 1.5.3 Improved interior decay of the first order energy . . . . . . . . . . . 85 1.5.4 Digression: Conformal energy decay . . . . . . . . . . . . . . . . . . 98 1.6 Pointwise bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2 Linear waves on expanding Schwarzschild de Sitter spacetimes 121 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 2.2 Linear Waves on de Sitter . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.2.1 Global geometry of de Sitter . . . . . . . . . . . . . . . . . . . . . . 127 2.2.1.1 Static region . . . . . . . . . . . . . . . . . . . . . . . . . 129 2.2.1.2 Cosmological horizon . . . . . . . . . . . . . . . . . . . . . 130 7 2.2.1.3 Expanding region . . . . . . . . . . . . . . . . . . . . . . . 131 2.2.2 Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 2.2.2.1 Energy estimates and global redshift in the expanding region133 2.2.2.2 Proof of the energy estimates in the expanding region . . . 135 2.2.2.3 Integrated local energy decay and local redshift effect in the static region . . . . . . . . . . . . . . . . . . . . . . . 140 2.2.2.4 Proof of integrated local energy decay in the static region 144 2.2.2.5 The redshift effect on the cosmological horizon . . . . . . 158 2.2.2.6 Pointwise estimates on the timelike future boundary . . . 161 2.3 Linear Waves on Schwarzschild de Sitter . . . . . . . . . . . . . . . . . . . 163 2.3.1 Geometry of the expanding region of Schwarzschild de Sitter . . . . 168 2.3.2 Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 2.3.2.1 Energy estimate in the expanding region . . . . . . . . . . 173 2.3.2.2 Redshift vectorfield on the cosmological horizon . . . . . . 176 A Improved interior decay of higher order energy for the wave equation on 3 + 1-dimensional Minkowski space 179 B Reference for Chapter 1 199 B.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 B.2 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 B.2.1 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 B.2.2 Radial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 B.3 Boundary Integrals and Hardy Inequalities . . . . . . . . . . . . . . . . . . 210 C Reference for Chapter 2 216 C.1 Decomposition formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 C.2 Coercivity inequality on the sphere . . . . . . . . . . . . . . . . . . . . . . 217 Preface In the dynamical formulation of General Relativity Theory we are presented with a Cauchy problem describing the evolution of an initial geometric data set under the laws of grav- itation, namely Einstein’s field equations; (an interesting introduction to the Cauchy problem in General Relativity is for example to be found in [12]). In the absence of symmetry this problem has only been fully understood for initial data sufficiently close to Minkowski space in the seminal work of Christodoulou and Klainerman [11], and for initial data close to de Sitter space by Friedrich [28]. The resolution of the Cauchy problem with initial data close to a given black hole space- time has remained elusive for many years. Our expectations for the global behaviour of these solutions are the content of the nonlinear black hole stability conjecture and its proof is one of the major open problems in the field; (for a rough formulation of this conjecture see e.g. [18]). The motivation for the problems treated in this thesis is largely drawn from their relation to the nonlinear stability problem and my results can be viewed as linear stability state- ments. Instead of studying the dynamics of the spacetime itself (which is governed by equations that are hyperbolic in nature) we fix a spacetime manifold and study solutions to the linear wave equation on this background. (For an introduction to the linear theory in the context of black hole spacetimes see [17].) We are interested in uniform bounded- ness and decay properties of general finite energy solutions and aim at proving them in a suitably robust way. It is important to recognize here the role of the method used in the proof. While in this thesis we always take an explicitly known solution of the Einstein equations as the back- ground spacetime manifold the methods we use and develop manifestly lend themselves to the application of our proofs to the study of the wave equation on small perturbations of the background geometry. It is mainly due to this fact that our work has relevance to the question of black hole stability. We shall be concerned with the study of linear waves on two explicitly known black hole spacetimes: (i) higher dimensional Schwarzschild black holes and (ii) Schwarzschild de Sitter spacetimes. The treatment of linear waves on black hole spacetimes in 3 + 1 dimensions has been completed in the much larger class of rotating Kerr black holes; (for a 9 10 review of the black hole stability problem for linear scalar perturbations see [19]). In view of recent advances in the analysis of hyperbolic partial differential equations (see e.g. [17] and references therein) which have lead to more robust methods (in particular a new physical space approach to proving decay [22]) it is natural to apply these methods to the study of linear waves on higher dimensional black holes. (Black hole spacetimes in high dimensions are also of independent interest for high energy physics; more on the relevance for theories in high energy physics can be found in [25].) Moreover I have advanced these methods which has lead to more robust proofs of already known results in 3+1 dimensions. Our interest in (ii) stems from the fact that the expansion of a spacetime is believed to introduce another decay mechanism into the problem which may bring the resolution of the nonlinear stability problem in the context of expanding black hole spacetimes within reach. At any rate, our present understanding of cosmology justifies the choice made in this thesis of expanding spacetimes as background manifolds corresponding to a positive cosmological constant; (for a historical account of the discovery of the expanding universe see [5]). Let us give an informal statement of the results in this thesis. Result 1 (Decay of linear waves on higher dimensional Schwarzschild black holes). In Chapter 1 we consider solutions to the linear wave equation on higher dimensional Schwarzschild black hole spacetimes and prove robust nondegenerate energy decay esti- mates that are in principle required in a nonlinear stability problem. More precisely, it is shown that for solutions φ to the wave equation on the domain of outer communications of the Schwarzschild spacetime manifold (Mnm, g) (where n ≥ 3 is the spatial dimension, and m > 0 is the mass of the black hole) the associated energy flux E[φ](Στ ) through a foliation of hypersurfaces Στ (terminating at future null infinity and crossing the event horizon to the future of the bifurcation sphere, obtained by timelike translations along the Killing vectorfield T = ∂t where ∂tτ = 1) decays, E[φ](Στ ) ≤ CD τ 2 , where C is a constant depending on n and m, and D <∞ is a suitable higher order initial energy on Σ0 (i.e. an energy involving commutations with the Killing vectorfields of Mnm, in particular T ); moreover we improve the decay rate for the first order energy to E[∂tφ](Σ R τ ) ≤ CDδ τ 4−2δ for any δ > 0 where ΣRτ denotes the hypersurface Στ truncated at an arbitrarily large fixed radius R < ∞ provided the higher order energy Dδ on Σ0 is finite. We conclude our treatment by interpolating between these two results to obtain the pointwise estimate |φ|ΣRτ ≤ CD′δ τ 3 2 −δ . 11 A precise statement of Result 1 is given in Section 1.1.1. The decay argument developed in Section 1.5 is sufficiently robust to imply similar decay results for solutions to the wave equation on a wide class of asymptotically flat spacetimes. We elaborate on our argument for Minkowski space in 3 + 1 dimensions in Appendix A. Result 2 (Decay of linear waves on asymptotically flat spacetimes). Improved interior first order energy decay as stated in Result 1 holds for solutions to the linear wave equation in all dimensions on a wide class of asymptotically flat black hole exteriors, whenever an integrated local energy decay estimate is available. In particular, it holds on small perturbations of Minkowski space. Note that our decay results are obtained in the domain of outer communications or in other words the exterior of the black hole up to and including the event horizon. This is precisely the region of spacetime which is expected to be stable. In my recent work I turned to cosmological black hole spacetimes and investigated what is expected to be the stable region of Schwarzschild de Sitter spacetimes. The exterior of the black hole does here not only contain a static region but also what shall be referred to as the expanding region lying to the future of the static domain beyond a cosmological horizon. While the linear theory for the static region (namely the domain enclosed by the event and cosmological horizons) has already been addressed in [16, 4, 38], we make use of an additional stability mechanism in the expanding region. Result 3 (Global boundedness of linear waves on Schwarzschild de Sitter spacetimes). In Chapter 2 I describe the global study of linear waves on de Sitter and Schwarzschild de Sitter spacetimes. We prove that solutions to the Cauchy problem for the linear wave equation posed in the past of the future boundary Σ+ of the expanding region of (subex- tremal) Schwarzschild de Sitter spacetimes (M(m)Λ , g) are globally bounded and have a limit on Σ+ which can be viewed as a function on the standard cylinder R× S2 provided their energy is initally finite. In fact, if Σ is a spacelike hypersurface in the past of Σ+ (such that Σ+ is in the domain of dependence of Σ) and the energy D[φ] of a solution φ to the linear wave equation on Σ is finite, then∫ Σ+ ∣∣ ◦∇ φ∣∣2 dµ◦g ≤ C D[φ] where C is a constant that only depends on Σ, m and Λ, and ◦ g and ◦ ∇ denote the standard metric and derivatives on the cylinder. Moreover, we show that on de Sitter spacetimes this limit is identically zero for solutions to the Klein-Gordon equation gφ = mΛφ (pro- vided mΛ ≥ 2Λ3 ). The precise formulation of Result 3 is given in Section 2.2 for de Sitter spacetimes and in Section 2.3 for Schwarzschild de Sitter spacetimes. The fact that a global solution to the linear wave equation on expanding black hole spacetimes has a limit on the future 12 boundary as a function on the cylinder for which we have an explicit global integral bound is a nontrivial statement which is in agreement with our expectations for the nonlinear stability problem. Chapter 1 Decay of linear waves on higher dimensional Schwarzschild black holes 1.1 Overview The study of the wave equation on black hole spacetimes has generated considerable interest in recent years. As discussed in the preface this stems mainly from its role as a model problem for the nonlinear black hole stability problem [18, 19], and more recent advances in the analysis of linear waves [17]. In this Chapter we study the linear wave equation on higher dimensional Schwarzschild black holes. The motivation for this problem lies — apart from the above mentioned relation to the nonlinear stability problem (which is expected to be simpler in the higher dimensional case [9]; for work on the 5-dimensional case under symmetry see also [15, 30]) — on one hand in the purely mathematical curiosity of dealing with higher dimensions and on the other hand in its interest for theories of high energy physics [25]. In the philosophy of [11] it is understood that the resolution of the nonlinear stability problem requires an understanding of the linear equations in a sufficiently robust setting. In particular, we require a proof of the uniform boundedness and decay of solutions to the linear wave equation based on the method of energy currents which (ideally) only uses properties of the spacetime that are stable under perturbations, and does not rely heavily on the specifics of the unperturbed metric; (for an introduction in the context of black hole spacetimes see [17]). Correspondingly we establish on higher dimensional Schwarzschild spacetime backgrounds boundedness and decay results analogous to the current state of the art in the 3 + 1-dimensional case [36]. The decay argument presented here departs from earlier work that either makes use of multipliers with weights in the temporal variable (notably [10, 2, 7, 20, 36]) which in one 13 14 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES I + : future null infinity ι− : timelike past infinity ι+ : timelike future infinity H+ : event horizon bifurcation sphere ι0 : spacelike infinity D : domain of outer communications Σ0 r = R Στ Figure 1.1: The hypersurface Σ0 in the domain of outer communications D. form or the other are due to Morawetz [39], or that relies on the exact stationarity of the spacetime (such as [8, 46, 24] based on Fourier analytic methods). Here we follow the new physical-space approach to decay of Dafermos and Rodnianski [22], which only uses multipliers with weights in the radial variable. Thus my work — especially the improvement of Section 1.5.3 — is of independent interest for the 3 + 1-dimensional Schwarzschild and Minkowski case and also for a wider class of spacetimes including Kerr black hole exteriors. 1.1.1 Statement of the Theorems We consider solutions to the wave equation gφ = 0 (1.1.1) on higher dimensional Schwarzschild black hole spacetimes; these backgrounds are a family of n+1-dimensional Lorentzian manifolds (Mnm, g) parametrized by the mass of the black hole m > 0, (n ≥ 3). They arise as spherically symmetric solutions of the vacuum Einstein equations, the governing equations of General Relativity, and are discussed as such in Section 1.2; for the relevant concepts see also [17, 29]. More precisely, we consider solutions to (1.1.1) on the domain of outer communications D of M — which comprises the exterior up to and including the event horizons of the black hole — with initial data prescribed on a hypersurface Σ0 consisting of an incoming null segment crossing the event horizon to the future of the bifurcation sphere, a spacelike segment and an outgoing null segment emerging from a larger sphere of radius R termi- nating at future null infinity; see figure 1.1 (the exact parametrization – which is chosen merely for technical reasons – is given in Section 1.4). In the exterior of the black hole the metric g takes the classical form in (t, r)-coordinates 1.1. OVERVIEW 15 [45], g = −(1− 2m rn−2 ) dt2 + ( 1− 2m rn−2 )−1 dr2 + r2 ◦ γn−1 , (1.1.2) where r > n−2 √ 2m, t ∈ (−∞,∞), and ◦γn−1 denotes the standard metric on the unit n−1-sphere; however this coordinate system breaks down on the horizon r = n−2√2m and we shall for that reason introduce in Section 1.2 the global geometry of (Mnm, g) using a double null foliation, from which we derive an alternative double null coordinate system for the exterior of the black-hole, g = −4(1− 2m rn−2 ) du∗ dv∗ + r2 ◦ γn−1 , (1.1.3) so called Eddington-Finkelstein coordinates. In this work both the conditions on the initial data and the statements on the decay of the solutions are formulated using the concepts of energy and the energy momentum tensor associated to (1.1.1) in particular (see Section 1.1.2 and also Appendix B.2): Tµν [φ] = ∂µφ ∂νφ− 1 2 gµν ∂ αφ∂αφ . (1.1.4) The corresponding 1-contravariant-1-covariant tensorfield fulfills the physical requirement that the linear transformation −T : TM→ TM maps the hyperboloid of future-directed unit timelike vectors into the closure of the open future cone at each point. Physically, −T · u ∈ TpM is the energy-momentum density relative to an observer at p ∈ M with 4-velocity u ∈ TpM, and it is for this reason that we refer to ε = g(T · u, u) = T (u, u) ≥ 0 as the energy density at p ∈ M relative to the observer with 4-velocity u ∈ TpM. One may think of a spacelike hypersurface as a collection of locally simultaneous observers with a 4-velocity given by the normal. The hypersurfaces relative to which we establish energy decay are simply defined by Στ . = ϕτ (Σ0 ∩ D), where ϕτ denotes the 1-parameter group of isometries generated by ∂ ∂t . The energy flux through the hypersurface Στ is then given by E[φ](Στ ) . = ∫ Στ ( JN [φ], n ) (1.1.5) where (JN [φ], n) . = T [φ](N, n), n is the normal1 to Στ and N is a timelike ϕτ -invariant future directed vectorfield which is constructed in Section 1.3 for the purpose of turning εN . = T (N,N) into a nondegenerate energy up to and including the horizon. Note that the energy E[φ](Στ ) in particular bounds a suitably defined H˙ 1-norm on Στ . 1On spacelike segments of Στ the vector n is indeed timelike; however, on the null segments of the hypersurfaces Στ the “normal” n is in fact a null vector, but the notation is kept for convenience; see Appendix B.1. 16 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES The classes of solutions to (1.1.1) to which our results apply are formulated in terms of finite energy conditions on the initial data, for the purpose of which we list the following quantities: D (2) 2 (τ0) . = ∫ ∞ τ0+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 1∑ k=0 r2 (∂(r n−12 ∂kt φ) ∂v∗ )2∣∣∣ u∗=τ0 + ∫ Στ0 ( 2∑ k=0 JN [∂kt φ], n ) (1.1.6) D (4−δ) 5 (τ0) . = ∫ ∞ τ0+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { 1∑ k=0 r4−δ (∂2(r n−12 ∂kt φ) ∂v∗2 )2 + 4∑ k=0 r2 (∂(r n−12 ∂kt φ) ∂v∗ )2 + 3∑ k=0 n(n−1) 2∑ i=1 r2 (∂r n−12 Ωi∂kt φ ∂v∗ )2}∣∣∣ u∗=τ0 + ∫ Στ0 ( 5∑ k=0 JN [∂kt φ] + 4∑ k=0 n(n−1) 2∑ i=1 JN [Ωi∂ k t φ], n ) (1.1.7) D (4−δ) 7+[n 2 ](τ0) . = ∫ ∞ τ0+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { 2∑ k=0 ∑ |α|≤[n 2 ]+1 r4−δ (∂2(r n−12 Ωα∂kt φ) ∂v∗2 )2 + 5∑ k=0 ∑ |α|≤[n 2 ]+1 r2 (∂r n−12 Ωα∂kt φ ∂v∗ )2 + 4∑ k=0 ∑ |α|≤[n 2 ]+2 r2 (∂r n−12 Ωα∂kt φ ∂v∗ )2}∣∣∣ u∗=τ0 + ∫ Στ0 ( 6∑ k=0 ∑ |α|≤[n 2 ]+1 JN [Ωα∂kt φ] + 5∑ k=0 ∑ |α|≤[n 2 ]+2 JN [Ωα∂kt φ], n ) (1.1.8) Here Ωi : i = 1, . . . , n(n−1) 2 are the generators of the spherical isometries of the spacetime M, α is a multiindex, and for any radius R we denote by R∗ the corresponding Regge- Wheeler radius (1.2.22). (See also Section 1.4.2.) Among the propositions on linear waves on higher dimensional Schwarzschild black hole spacetimes proven in this thesis, we wish to highlight the following conclusions2. Theorem 1 (Energy decay). Let φ be a solution of the wave equation gφ = 0 on D ⊂Mnm, where n ≥ 3 and m > 0, with initial data prescribed on Στ0 (τ0 > 0). • If D .= D(2)2 (τ0) <∞ then there exists a constant C(n,m) such that E[φ](Στ ) ≤ C D τ 2 (τ > τ0) . (1.1.9) 2The “redshift” proposition, and the “integrated local energy decay” proposition are to be found on page 28 in Section 1.3 and page 35 in Section 1.4 respectively. 1.1. OVERVIEW 17 • Furthermore if for some 0 < δ < 1 2 and R > n−2 √ 8nm/δ also D′ .= D(4−δ)5 (τ0) < ∞ then there exists a constant C(n,m, δ, R) such that E[∂tφ](Σ ′ τ ) ≤ C D′ τ 4−2δ (τ > τ0) , (1.1.10) where Σ′τ . = Στ ∩ {r ≤ R}. While each of these energy decay statements lend themselves to prove pointwise estimates for φ and ∂tφ respectively (see Section 1.6) we would like to emphasize that using the (refined) integrated local energy decay estimates of Section 1.4 an interpolation argument allows us to improve the pointwise bound on φ directly in the interior.3 Theorem 2 (Pointwise decay). Let φ be a solution of the wave equation as in Theorem 1. If for some 0 < δ < 1 4 , D . = D (4−δ) 7+[n 2 ](τ0) < ∞ (τ0 > 1) then there exists a constant C(n,m, δ, R) such that r n−2 2 |φ| ∣∣∣ Σ′τ ≤ C D τ 3 2 −δ ( n−2√ 2m ≤ r < R, τ > τ0) (1.1.11) where Σ′τ and R are as in Theorem 1. Remark 3 (Decay rates and method of proof). Theorems 1 and 2 extend the presently known decay results for linear waves on 3 + 1-dimensional Schwarzschild black holes to higher dimensions n > 3; for 3+ 1-dimensional Schwarzschild black holes (1.1.9) was first established in [20], and (1.1.10, 1.1.11) more recently in [36]. However, both proofs use multipliers with weights in t, [20] by using the conformal Morawetz vectorfield in the decay argument, and [36] by using in addition the scaling vectorfield. Here we extend (1.1.9) to higher dimensions n > 3 in the spirit of [22] only using multipliers with weights in r, and provide a new proof of the improved decay results (1.1.10) and (1.1.11) in the n = 3-dimensional case in particular. 1.1.2 Overview of the Proof In this section we give an overview of the work in this part of my thesis, and present some of the ideas in the proof that lead to Theorem 1; references to previous work is made when useful, but for a more detailed account of previous work on the wave equation on Schwarzschild black hole spacetimes see §1.3 in [23] and references therein. 3In this work we use the term “interior” to refer to a region of finite radius, i.e. the term “interior region” is used interchangeably with “a region of compact r (including the horizon)”, and is of course not meant to refer to the interior of the black hole, which is not considered in my present work. 18 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Energy Identities. Let us recall that the wave equation (1.1.1) arises from an action principle and that the corresponding energy momentum tensor is conserved. Indeed, here we find (1.1.4) and by virtue of the wave equation (1.1.1) ∇µTµν = (gφ)(∂νφ) = 0 . (1.1.12) Moreover, the energy momentum tensor (1.1.4) satisfies the positivity condition, namely T (X, Y ) ≥ 0 for all future-directed causal vectors X, Y at a point. Now let X be a vectorfield on M. We define the energy current JX [φ] associated to the multiplier X by JXµ [φ] . = Tµν [φ]X ν . (1.1.13) Then KX . = ∇µJXµ = (X)πµνTµν (1.1.14) where we have used that Tµν is conserved and symmetric. Here (X)π(Y, Z) . = 1 2 (LXg)(Y, Z) = 1 2 g(∇YX,Z) + 1 2 g(Y,∇ZX) (1.1.15) is the deformation tensor of X. Remark 1.1. If X is a Killing field, i.e. X generates an isometry of g, (X)π = 0, then KX = 0, i.e. JX is conserved. In the following we shall refer to ∫ R KX dµg = ∫ ∂R ∗JX (1.1.16) as the energy identity for JX (or simply X) on R, where R ⊂ M; (this is of course the content of Stokes’ Theorem, and ∗J denotes the Hodge-dual of J , see also Appendix B.2). Moreover we refer to X in (1.1.16) as the multiplier vectorfield. We will largely be concerned with the construction of vectorfields X, associated currents JX and their modifications, and the application of (1.1.16) and various derived energy inequalities to appropriately chosen domains R ⊂ D. The new approach [22] to obtaining robust decay estimates requires us to first establish (i) uniform boundedness of energy, (ii) an integrated local energy decay estimate and (iii) good asymptotics towards null infinity. Redshift effect. The reason (i) is nontrivial as compared to Minkowski space is that the energy corresponding to the multiplier ∂t degenerates on the horizon (the vectorfield ∂t becomes null on the horizon and no control on the angular derivatives is obtained, c.f. [17]); it was recognized in [20], and formulated more generally in [17], that the redshift property of Killing horizons is the key to obtaining an estimate for the nondegenerate energy (i.e. an 1.1. OVERVIEW 19 energy with respect to a strictly timelike vectorfield up to the horizon, which controls all derivatives tangential to the horizons). An explicit construction of a suitable timelike vectorfield N is given in Section 1.3 which allows us to state the redshift property in the language of multipliers and energy currents, and a proof of the uniform boundedness of the nondegenerate energy is given (independently of other calculations in this work) in Section 1.5.1. Integrated local energy decay. Section 1.4 is devoted to establishing (ii). This is achieved by the use of radial multiplier vectorfields of the form f(r∗)∂r∗ (see Section 1.4.1). In Section 1.4.2 a construction of a positive definite current for the high angular frequency regime is given using a decomposition on the sphere. In Section 1.4.3 a more general construction of a current is given using a commutation with the angular momentum operators. We wish to emphasize that the decay results of Section 1.5 — albeit with a higher loss of differentiability — could be obtained solely on the basis of the latter current, without the recourse in Section 1.4.2 to the Fourier expansion on the sphere. However, the dependence on the initial data is significantly improved by virtue of the integrated local energy decay estimate Prop. 1.11; here (see Section 1.4.4) the results of Section 1.4.2 and Section 1.4.3 are combined in order to replace the commutation with the angular momentum operators by a commutation with the vectorfield ∂t only. The difficulty in both constructions lies in overcoming the “trapping” obstruction, which is the insight that it is impossible to prove an integrated local energy decay estimate on spacetime regions that contain the photon sphere without losing derivatives (see [17]). In the context of the Schwarzschild spacetime the need for vectorfields whose associated currents give rise to positive definite spacetime integrals was first recognised and used in [6, 20], and such estimates have since then been extended by many authors [37, 1]. p-hierarchy. In Section 1.5.2 we use a multiplier of the form rp∂v∗ that gives rise to a weighted energy inequality which we consequently exploit in a hierarchy of two steps; this approach — which yields the corresponding quadratic decay rate in (1.1.9) — is pioneered in [22] for a large class of spacetimes, including the 3 + 1-dimensional Schwarzschild and Kerr black hole spacetimes. In Section 1.5.3 a further commutation with ∂v∗ is carried out, which allows us to extend the hierarchy of commuted weighted energy inequalities to four steps, yielding the correponding decay rate for the first order energy. The argument involves dealing with an (arbitrarily small) degeneracy of the first order energy density at infinity which corresponds to the δ-loss in the decay estimate (1.1.10). In both cases (iii) is ensured by the imposition of higher order finite energy conditions on the initial data. Interpolation. The pointwise decay of Theorem 2 then follows from Theorem 1 and the (refined) integrated local energy decay estimates of Section 1.4.4 by a simple interpolation argument given in Section 1.6. 20 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Final Comments. The currents in Section 1.4.2 and Section 1.4.3 and the correspond- ing integrated local energy decay result already appeared in the Smith-Rayleigh-Knight essay [42]. Independently a version of integrated local energy decay was subsequently obtained in [35]. [42] also contained an alternative proof of (1.1.9) of Theorem 1 using the conformal Morawetz vectorfield, which is included in this thesis in Section 1.5.4. 1.2 Global causal geometry of the higher dimensional Schwarzschild solution In this Section, we give a discussion (in the spirit of §3 of [13]) of the global geometry of the n + 1-dimensional Schwarzschild black hole spacetime [45], the underlying manifold on which the wave equation is studied in this thesis. The n+1-dimensional Schwarzschild spacetime manifoldM (n ≥ 3, n ∈ N) is spherically symmetric, i.e. SO(n) acts by isometry. The group orbits are (n − 1)-spheres, and the quotient Q=M/SO(n) is a 2-dimensional Lorentzian manifold with boundary. The metric g on M assumes the form g = Q g +γr = Q g +r2 ◦ γn−1 (1.2.1) where Q g is the Lorentzian metric on Q to be discussed below, ◦γn−1 is the standard metric on Sn−1, and r is the area radius; or more precisely in local coordinates xa : a = 1, 2 on Q, and local coordinates yA : A = 1, . . . , n− 1 on Sn−1 g(x,y) = gab(x) dx a dxb + r2(x)( ◦ γn−1)AB dyA dyB . Remark 1.2 (Area radius). Since det γr = (r 2)n−1 det ◦ γn−1 the area radius r(x) is related to the area of the (n− 1)-sphere at x by Area(x) = ∫ S(x) dµγr = ∫ Sn−1 rn−1 dµ◦ γn−1 = ωnr n−1 where ωn = 2π n 2 Γ(n 2 ) is the area of the unit (n− 1)-sphere. The n + 1-dimensional Schwarzschild spacetime is a solution of the vacuum Einstein equations, which in other words means that its Ricci curvature vanishes identically. This implies in particular (see derivation below) that the area radius function r satisfies the Hessian equations ∇a∂br = (n− 2) 2r [ 1− (∂cr)(∂cr) ] gab , (1.2.2) 1.2. GLOBAL CAUSAL GEOMETRY 21 as a result of which the mass function m on Q defined4 by 1− 2m rn−2 = gab ∂ar ∂br (1.2.3) is constant; we take this parameter m > 0 to be positive. Hessian equations. The non-vanishing connection coefficients of g are Γcab, Γ C AB = ◦ Γ C AB and ΓaAB = −r gab ∂br ( ◦ γn−1)AB ΓAaB = 1 r ∂ar δ A B . Since the components of the Ricci curvature are given by Rµν = ∂αΓ α µν − ∂νΓαµα + ΓαβαΓβµν − ΓαβνΓβµα we obtain the decomposition Rab = Q Rab −∂bΓAaA + ΓAdAΓdab − ΓABbΓBaA = K gab − (n− 1) 1 r ∇a∂br where we have used that Q Rab= K gab , K being the Gauss curvature of the 2-dimensional manifold Q, and similarly RaA = 0 , RAB = ∂aΓ a AB + Γ a baΓ b AB + Γ D aDΓ a AB − ΓaDBΓDAa − ΓDaBΓaAD+ ◦ RAB = ( ◦ γn−1)AB [ (n− 2)− (n− 3) (∂ar)(∂ar)−∇a(r∂ar) ] , because the Ricci curvature of Sn−1 is simply ◦ RAB= (n− 2)( ◦ γn−1)AB . The vacuum Einstein equations Rµν = 0 therefore reduce to the following system on Q : K gab − (n− 1) 1 r ∇a∂br = 0 (1.2.4a) (n− 2)− (n− 3) (∂ar)(∂ar)−∇a(r∂ar) = 0 . (1.2.4b) 4We choose the normalization of the mass function to be independent of the dimension n; this is motivated by a consideration of the mass equations in the presence of matter, see Remark 1.3. 22 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Taking the trace of (1.2.4a) gives 2K − (n− 1) 1 r ∇a∂ar = 0 and substituting for ∇a∂ar from (1.2.4b) yields K = n− 1 2r ∇a∂ar = n− 1 2r2 [∇a(r∂ar)− (∂ar)(∂ar)] = (n− 1)(n− 2) 2r2 [ 1− (∂ar)(∂ar) ] . Then returning to (1.2.4a), we arrive at the Hessian equations (1.2.2). The mass function defined by (1.2.3), m = rn−2 2 [ 1− (∂ar)(∂ar) ] is thus constant on Q : ∂am = n− 2 2 rn−3 [ 1− (∂br)(∂br) ] ∂ar − 1 2 rn−2∇a [ (∂br)(∂br) ] = 0 . We take m > 0. Moreover K = (n− 1)(n− 2)m rn . Remark 1.3 (Normalization of the mass). One may prefer a normalization of the mass function that depends on the dimension n, i.e. favour the definition 1− anm rn−2 = gab ∂ar ∂br where an is a constant depending on n. However, in the presence of matter the Einstein equations in (n+ 1) dimensions necessarily take the form Rµν − 1 2 gµν R = (n− 1)ωn Tµν (in units where Newton’s constant G = 1 and the speed of light c = 1) and we fix the an by taking the prefactor in the mass equations ∂am = ωnr n−1 (Tab − gab trT )∂br to be the area of the (n− 1)-sphere. The analogous calculation of the Hessian equations in (n + 1) dimensions in the presence of matter shows that this is precisely satisfied for an = 2 , independently of n. 1.2. GLOBAL CAUSAL GEOMETRY 23 On Q we choose functions u, v whose level sets are outgoing and incoming null curves respectively, which are increasing towards the future. These functions define a null system of coordinates, in which the metric Q g takes the form Q g= −Ω2 du dv . (1.2.5) Note. The only non-vanishing connection coefficients are Γuuu = 2 Ω ∂Ω ∂u Γvvv = 2 Ω ∂Ω ∂v . The Gauss curvature is given by K = − 2 Ω2 Q Ruv= 4 Ω2 ∂2 log Ω ∂u ∂v . The Hessian equations (1.2.2) in null coordinates read ∂2r ∂u2 − 2 Ω ∂Ω ∂u ∂r ∂u = 0 (1.2.6a) ∂2r ∂u ∂v + n− 2 r ∂r ∂u ∂r ∂v = −n− 2 4r Ω2 (1.2.6b) ∂2r ∂v2 − 2 Ω ∂Ω ∂v ∂r ∂v = 0 , (1.2.6c) and the defining equation for the mass function (1.2.3) is 1− 2m rn−2 = − 4 Ω2 ∂r ∂u ∂r ∂v . (1.2.7) The system ((1.2.6b), (1.2.7)) can be rewritten as the partial differential equation ∂2r∗ ∂u∂v = 0 (1.2.8) for a new radial function r∗(r) that is related to r by dr∗ dr = 1 1− 2m rn−2 . (1.2.9) Remark 1.4. Indeed, by substituting for Ω2 from (1.2.7) in (1.2.6b), r ∂2r ∂u ∂v = (n− 2) 2m rn−2 − 2m ∂r ∂u ∂r ∂v , we see that r∗ = ∫ 1 1− 2m rn−2 dr (1.2.10) satisfies ∂2r∗ ∂u ∂v = −(n− 2) 2m rn−1( 1− 2m rn−2 )2 ∂r∂u ∂r∂v + 11− 2m rn−2 ∂2r ∂u ∂v = 0 . 24 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES A solution of ((1.2.8),(1.2.9)) is given by r∗ = 1 (n− 2) n−2√ 2m log |uv| , (1.2.11) or |uv| = e (n−2) r∗ n−2√2m = e (n−2) r n−2√2m exp [∫ n− 2 xn−2 − 1 dx ∣∣ x= rn−2√2m ] . (1.2.12) Remark 1.5. The general solution is r∗ = f(u) + g(v) . Since r∗ = r + ∫ 2m rn−2 − 2m dr = r + n−2√ 2m ∫ 1 xn−2 − 1 dx ∣∣∣∣ x= rn−2√2m we require the representation in terms of null coordinates to be such that r∗ = −∞ is contained in the (u, v) plane and the metric to be non-degenerate at r = n−2 √ 2m and take f(u) = n−2√ 2m log |u| 1n−2 g(v) = n−2√ 2m log |v| 1n−2 so that r∗ = (n− 2)−1 n−2 √ 2m log |uv| . (1.2.13) We find more explicitly, uv =  e r 2m ( 1− r 2m ) , n = 3 e 2r√ 2m ( 1− r√ 2m )( 1 + r√ 2m ) , n = 4 e (n−2) r n−2√2m ( 1− r n−2√2m ) 1 , n odd(1 + rn−2√2m)−1 , n even × [n−3 2 ]∏ j=1 ( r2 (2m) 2 n−2 − 2 cos( 2πj n− 2 ) r (2m) 1 n−2 + 1 )cos(2πj n−3 n−2 ) × [n−3 2 ]∏ j=1 exp [ 2 sin ( 2πj n− 3 n− 2 ) arctan ( r n−2√2m − cos( 2πj n−2) sin( 2πj n−2) )] , n ≥ 5 (1.2.14) Note, in particular that the u = 0 and v = 0 lines are the constant r = n−2 √ 2m curves, and that all other curves of constant radius are hyperbolas in the (u, v) plane — timelike for r > n−2 √ 2m, spacelike for r < n−2 √ 2m. This outlines the well-known global causal geometry of the Schwarzschild solution (see figure 1.2). 1.2. GLOBAL CAUSAL GEOMETRY 25 u = 0 v = 0 r = n−2 √ 2m r = n−2 √ 2m r = 0 r = 0 r > n−2 √ 2m T ∗ : v < 0 0 0, v > 0 R : u < 0, v > 0 Figure 1.2: Global causal geometry of the Schwarzschild solution. Remark 1.6. The rational function in (1.2.12) can be integrated using elementary methods (see Appendix B.2.1) to obtain: |uv| = e (n−2) r n−2√2m  ∣∣∣ r 2m − 1 ∣∣∣ , n = 3 | r√ 2m − 1| | r√ 2m + 1| , n = 4 ∣∣∣∣ rn−2√2m − 1 ∣∣∣∣ 1 , n odd| rn−2√2m + 1|−1 , n even × [n−3 2 ]∏ j=1 ∣∣∣ r2 (2m) 2 n−2 − 2 cos( 2πj n− 2 ) r (2m) 1 n−2 + 1 ∣∣∣cos(2πj n−3n−2 ) × [n−3 2 ]∏ j=1 exp [ 2 sin ( 2πj n− 3 n− 2 ) arctan ( r n−2√2m − cos( 2πj n−2) sin( 2πj n−2) )] , n ≥ 5 Note that | r 2 (2m) 2 n−2 −2 cos( 2πj n− 2) r (2m) 1 n−2 +1| ≥ |1−cos2( 2πj n− 2)| > 0 (j = 1, . . . , [ n− 3 2 ], n ≥ 5) . The condition that u, v are increasing towards the future selects the sign, and we finally 26 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES obtain (1.2.14). We have uv  < 0 = 0 > 0 ⇐⇒ r  > n−2 √ 2m = n−2 √ 2m < n−2 √ 2m Note also that r = 0 is the spacelike hyperbola uv = 1 , n = 3, 4;n ≥ 6 even∏n−32 j=1 exp [−2 sin(2πj n−3 n−2) arctan cot( 2πj n−2) ] , n ≥ 5 odd ≤ 1 . It is easy to see that for (1.2.11) the trapped region, the apparent horizon, the exterior, and the antitrapped regions respectively are given by T . = { (u, v) ∈ Q : ∂r ∂u < 0, ∂r ∂v < 0 } = { (u, v) ∈ Q : u > 0, v > 0 } A . = { (u, v) ∈ Q : ∂r ∂u < 0, ∂r ∂v = 0 } = { (u, v) ∈ Q : u = 0, v > 0 } R . = { (u, v) ∈ Q : ∂r ∂u < 0, ∂r ∂v > 0 } = { (u, v) ∈ Q : u < 0, v > 0 } T ∗ .= { (u, v) ∈ Q : ∂r ∂u > 0 } = { (u, v) ∈ Q : v < 0 } . Indeed, using the solution (1.2.11) ∂r∗ ∂u = 1 n− 2 n−2√2m u ∂r∗ ∂v = 1 n− 2 n−2√2m v and on the other hand with (1.2.9) ∂r∗ ∂u = 1 1− 2m rn−2 ∂r ∂u ∂r∗ ∂v = 1 1− 2m rn−2 ∂r ∂v we deduce ∂r ∂u = ( 1− 2m rn−2 ) n−2√2m (n− 2)u (1.2.15a) ∂r ∂v = ( 1− 2m rn−2 ) n−2√2m (n− 2)v (1.2.15b) Note this forms a partition of Q = T ∪A ∪R ∪ T ∗, and that in view of (1.2.7) r < n−2√2m in T , r = n−2√2m in A and r > n−2√2m in R. We shall refer to D .= R = { (u, v) ∈ Q : u ≤ 0, v ≥ 0 } (1.2.16) as the domain of outer communications. 1.2. GLOBAL CAUSAL GEOMETRY 27 Finally, Ω2 = −4(1− 2m rn−2 )−1 ∂r ∂u ∂r ∂v by (1.2.7) = −( 2 n− 2 )2( 1− 2m rn−2 )(2m) 2n−2 uv by (1.2.15) (1.2.17a) =  4 (2m)3 r e− r 2m , n = 3 (2m r )2( r√ 2m + 1 )2 e − 2r√ 2m , n = 4 ( 2 n− 2 )2 (2m) nn−2 rn−2 1 , n odd( rn−2√2m + 1)2 , n even × [n−3 2 ]∏ j=1 ( r2 (2m) 2 n−2 − 2 cos( 2πj n− 2) r n−2√2m + 1 )1−cos(2πj n−3 n−2 ) × [n−3 2 ]∏ j=1 exp [ −2 sin(2πjn− 3 n− 2) arctan ( rn−2√2m − cos( 2πjn−2) sin( 2πj n−2) )] × e− (n−2) r n−2√2m , n ≥ 5 (1.2.17b) One may now think of r as a function of u, v implicitly defined by (1.2.14). In R where r > n−2 √ 2m recall from (1.2.12) that this relation is uv = − exp [∫ n− 2 xn−2 − 1 dx ∣∣ x= rn−2√2m ] e (n−2) r n−2√2m (1.2.18) and may be complemented in this region R where v − u > |u + v| by the coordinate t defined by t = 2 n− 2 n−2√ 2m artanh (u+ v v − u ) ; (1.2.19) we will denote by Σt the corresponding level sets in D. We find dt = 1 n− 2 n−2√ 2m ( 1 v dv − 1 u du) (1.2.20) and from (1.2.18) v du+ u dv = − rn−2 2m rn−2 2m − 1 exp [∫ n− 2 xn−2 − 1 dx ∣∣ x= rn−2√2m ] e (n−2) r n−2√2m n− 2 n−2√2m dr . Alternatively (1.2.20) can be written as v du− u dv = −uv n− 2 n−2√2m dt = exp [ ∫ n− 2 xn−2 − 1 dx ∣∣ x= rn−2√2m ] e (n−2) r n−2√2m n− 2 n−2√2m dt . 28 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Hence −Ω2 du dv = Ω 2 4uv [( v du− u dv)2 − (v du+ u dv)2] = −(1− 2m rn−2 ) dt2 + 1( 1− 2m rn−2 ) dr2 . We have arrived at the classic expression for the Schwarzschild metric in its original coordinates for the exterior region: g = −(1− 2m rn−2 ) dt2 + ( 1− 2m rn−2 )−1 dr2 + r2 ◦ γn−1 . (1.2.21) In Regge-Wheeler coordinates (t, r∗), where r∗ is centered at the photon sphere r = n−2√nm, r∗ = ∫ r (nm) 1 n−2 1 1− 2m r′n−2 dr′ , (1.2.22) the metric obviously takes the conformally flat form g = ( 1− 2m rn−2 ) (− dt2 + dr∗2) + r2 ◦γn−1 . (1.2.23) We shall also use the Eddington-Finkelstein coordinates u∗ = 1 2 (t− r∗) v∗ = 1 2 (t+ r∗) (1.2.24) which are again double null coordinates: g = −4(1− 2m rn−2 ) du∗ dv∗ + r2 ◦ γn−1 . (1.2.25) The two systems of null coordinates in R are related by − 1 n− 2 n−2√2m u du = du∗ 1 n− 2 n−2√2m v dv = dv∗ or u = −e− (n−2) u∗ n−2√2m v = e (n−2) v∗ n−2√2m . (1.2.26) 1.3 The Red-shift effect In this section we prove a manifestation of the local redshift effect in the Schwarzschild geometry of Section 1.2 in the framework of multiplier vectorfields. Proposition 1.7 (local redshift effect). Let φ be a solution of the wave equation (1.1.1), then there exists a ϕt-invariant future-directed timelike smooth vectorfield N on D, two radii n−2 √ 2m < r (N) 0 < r (N) 1 , and a constant b > 0 such that KN(φ) ≥ b (JN(φ), N) ( n−2 √ 2m ≤ r < r(N)0 ) (1.3.1) and N = T (r ≥ r(N)1 ). The vectorfield N will be constructed explicitly with the following vectorfields. 1.3. THE RED-SHIFT EFFECT 29 T -vectorfield. Here ϕt is the 1-parameter group of diffeomorphisms generated by the vectorfield T = 1 2 n− 2 n−2√2m ( v ∂ ∂v − u ∂ ∂u ) ; (1.3.2) note that in R where r > n−2 √ 2m (recall (1.2.20)) T = ∂ ∂t . T is a Killing vectorfield, (T )π = 0 . (1.3.3) For, (T )πuu = 0 (T )πvv = 0 (T )πuv = 1 4 n− 2 n−2√2mguv ( v ∂r ∂v − u∂r ∂u )∂ log Ω2 ∂r = 0 (T )πaA = 0 (1.3.4) (T )πAB = 1 2 n− 2 n−2√2m ( v ∂r ∂v − u∂r ∂u ) r ( ◦ γn−1)AB = 0 . T is timelike in the exterior, spacelike in the interior of the black hole and null on the horizon, g(T, T ) = 1 4 (n− 2)2 (2m) 2 n−2 uvΩ2 = −(1− 2m rn−2 )  < 0 r > n−2 √ 2m = 0 r = n−2 √ 2m > 0 r < n−2 √ 2m (1.3.5) In particular, T |H+ = 1 2 n− 2 n−2√2m v ∂ ∂v (1.3.6) T |H+∩H− = 0 . Y -vectorfield. Let us also define a vectorfield Y on H+ conjugate to T (that is to say Y is null and orthogonal to the sections of H+ and normalized by (1.3.8)): Y |H+ = − 2∂r ∂u ∂ ∂u (1.3.7) Indeed, g(T, Y )|H+ = −2 (1.3.8) because Ω2|H+ = −4 n−2√2m n− 2 1 v ∂r ∂u . Furthermore, as a consequence of (1.2.6b) ∂2r ∂u ∂v ∣∣∣∣ H+ = − n− 2 4r Ω2 ∣∣∣∣ H+ = 1 v ∂r ∂u ∣∣∣∣ H+ 30 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES we have [T, Y ]|H+ = [T, Y ]u ∂ ∂u ∣∣∣∣ H+ + [T, Y ]v ∂ ∂v ∣∣∣∣ H+ = n− 2 n−2√2m 1 ∂r ∂u [ v 1 ∂r ∂u ∂2r ∂u ∂v − 1 ] ∂ ∂u ∣∣∣∣ H+ = 0 . (1.3.9) EA-vectorfields. We denote by EA : A = 1, . . . , n − 1 an orthonormal frame field tangential to the orbits of the spherical isometry, g(EA, EB) = δAB = 1 , A = B0 , A 6= B (1.3.10a) g(EA, Y )|H+ = 0 , g(EA, T ) = 0|H+ , (A = 1, . . . , n− 1) . (1.3.10b) We can now prove that the surface gravity of the event horizon is positive; this is essential for the existence of the redshift effect, (see more generally [17], and also [3] for work where this is not the case). Lemma 1.8 (surface gravity). On H+ ∇TT = κnT (1.3.11) with κn = 1 2 n− 2 n−2√2m > 0 . (1.3.12) κn is called the surface gravity. Note. T = κn(v ∂ ∂v − u ∂ ∂u ) Proof. Since (T )π = 0, we have g(∇XT, Y ) = −g(∇Y T,X) for all vectorfields X, Y . Therefore, g(∇TT, T ) = −g(∇TT, T ) = 0 g(∇TT,EA) = −g(∇EAT, T ) = − 1 2 EA · g(T, T ) = 0 : A = 1, . . . , n− 1, because g(T, T ) = 0 on H+, and similarly g(∇TT, Y ) = −1 2 Y · g(T, T ) . Now, g(T, T ) = − n− 2 n−2√2m u ∂r ∂u 1.3. THE RED-SHIFT EFFECT 31 so Y · g(T, T )|H+ = 2 n− 2n−2√2m . We obtain ∇TT = −1 2 g(∇TT, T )Y − 1 2 g(∇TT, Y )T + n−1∑ j=1 g(∇TT,EA)EA = 1 2 n− 2 n−2√2m T . Alternatively, κn is characterized by ∇TY = −κnY (1.3.13) on H+. Clearly g(∇TY, Y ) = 1 2 T · g(Y, Y ) = 0 since Y is null along H+, and g(∇TY, T ) (1.3.9)= g(∇Y T, T ) (1.3.3)= −g(∇TT, Y ) = 2κn ; also g(∇TY,EA) (1.3.9)= g(∇Y T,EA) (1.3.3)= −g(∇EAT, Y ) = 0 : A = 1, . . . , n− 1, because ∇EAT = 0. Note, for later use, ∇EAY = − 2 n−2√2m EA (1.3.14) on H+. We defined Y on H+ conjugate to T . Next we extend Y to a neighborhood of the horizon by ∇Y Y = −σ(Y + T ) , (σ ∈ R) , where in fact we will shall assume σ > 16 n− 2(2m) 3 n−2 , and then we extend Y to R by Lie-transport along the integral curves of T : [T, Y ] = 0 . Proposition 1.9 (redshift). For the future-directed timelike vectorfield N = T + Y (1.3.15) there is a b > 0 such that on H+ KN ≥ b (JN , N) . (1.3.16) 32 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Proof. Let us calculate KY =(Y )πµνTµν = 1 4 { (Y )π(T, T )T (Y, Y ) + 2(Y )π(T, Y )T (Y, T ) + (Y )π(Y, Y )T (T, T ) } − n−1∑ A=1 { (Y )π(EA, Y )T (EA, T ) + (Y )π(EA, T )T (EA, Y ) } + n−1∑ A,B=1 (Y )π(EA, EB)T (EA, EB) Now, on one hand, on H+, (Y )π(T, T ) = g(∇TY, T ) = 2κn (Y )π(T, Y ) = 1 2 g(∇TY, Y ) + 1 2 g(T,∇Y Y ) = σ (Y )π(Y, Y ) = g(∇Y Y, Y ) = 2σ (Y )π(EA, Y ) = 1 2 g(∇EAY, Y ) + 1 2 g(EA,∇Y Y ) = 0 (Y )π(EA, T ) = 1 2 g(∇EAY, T ) + 1 2 g(EA,∇TY ) = 0 (Y )π(EA, EB) = 1 2 g(∇EAY,EB) + 1 2 g(EA,∇EBY ) = − 2 n−2√2mδAB . Thus KY = 1 2 κn T (Y, Y ) + 1 2 σ T (Y + T, T )− 2 n−2√2m n−1∑ A=1 T (EA, EA) . On the other hand, on H+, T (Y, Y ) = ( 2 ∂r ∂u ∂φ ∂u )2 T (Y, T ) = ∣∣∇/ φ∣∣2 r2 ◦ γn−1 T (T, T ) = ( κnv ∂φ ∂v )2 and, on H+, T (EA, EB) = (EA · φ)(EB · φ)− 1 2 (2m) 2 n−2 δAB ∣∣∇/ φ∣∣ r2 ◦ γn−1 − 1 2 (n− 2)(2m) 1n−2 v ∂r ∂u δAB( ∂φ ∂u )( ∂φ ∂v ) . 1.3. THE RED-SHIFT EFFECT 33 Using Cauchy’s inequality, on H+, − 2 n−2√2m n−1∑ A=1 T (EA, EA) = =(n− 3)(2m) 1n−2 ∣∣∇/ φ∣∣2 r2 ◦ γn−1 + (n− 2)(n− 1) v ∂r ∂u ( ∂φ ∂u )( ∂φ ∂v ) ≥(n− 3)(2m) 1n−2T (Y, T )− 1 4 κnT (Y, Y ) − 1 κn 2(n− 1) (n− 2) (2m) 2 n−2 T (T, T ) ≥− 1 4 κnT (Y, Y )− n− 1 κ2n (2m) 1 n−2T (T, T ) . Since we have chosen σ > 2n−1 κ2n (2m) 1 n−2 , KY has a sign, KY ≥ 1 4 κn T (Y, Y ) + σ ′ T (Y + T, T ) for 0 < σ′ < σ 2 − n−1 κ2n (2m) 1 n−2 , or KY ≥ b T (Y + T, Y + T ) for 0 < b < min{κn 4 , σ ′ 2 }. This yields the result KN = KY ≥ b T (N,N) = b (JN , N) . Finally, we find an explicit expression for Y . Consider the vectorfield Yˆ = − 2 ∂r ∂u ∂ ∂u on R ∪A formally defined by the expression for Y on H+. In R Yˆ = 2 1− 2m rn−2 ∂ ∂u∗ . Yˆ generates geodesics, this being a consequence of the Hessian equations (1.2.6a), ∇Yˆ Yˆ = ( 2 ∂r ∂u )2[− 1 ∂r ∂u ∂2r ∂u2 + 2 Ω ∂Ω ∂u ] ∂ ∂u = 0 , and is Lie-transported by T : [T, Yˆ ] = 2 ( ∂r ∂u )2 ( [T, ∂ ∂u ] · r) ∂ ∂u − 2 ∂r ∂u [T, ∂ ∂u ] = −κnYˆ + κnYˆ = 0 because [T, ∂ ∂u ] = κn ∂ ∂u . Y as constructed above coincides with Y = α(r)Yˆ + β(r)T (1.3.17) 34 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES where α(r) = 1 + σ 4κn ( 1− 2m rn−2 ) β(r) = σ 4κn ( 1− 2m rn−2 ) . Indeed, on H+, Y |H+ = Yˆ |H+ = − 2∂r ∂u ∂ ∂u ∣∣∣∣ H+ and ∇Y Y |H+ = ∇Yˆ Y |H+ = (Yˆ · α)Yˆ |H+ + ∇Yˆ Yˆ ∣∣∣ H+ + (Yˆ · β)T |H+ = −σ (Y + T )|H+ since Yˆ · α|H+ = σ 4κn (n− 2) 2m rn−1 Yˆ · r|H+ = −σ Yˆ · β|H+ = −σ and Y remains Lie-transported by T : [T, Y ] = (T · α)Yˆ + (T · β)T + α [T, Yˆ ] + β [T, T ] = 0 since T · α = 0 = T · β . Thus the vectorfield Y is given explicitly by Y = − 2 ∂r ∂u ∂ ∂u on H+[ 1 + σ 4κn ( 1− 2m rn−2 )] 2 1− 2m rn−2 ∂ ∂u∗ + σ 4κn ( 1− 2m rn−2 ) ∂ ∂t in R (1.3.18) Clearly, by continuity, we can choose two values n−2 √ 2m < r (N) 0 < r (N) 1 <∞ and set N = T + Y n−2√2m ≤ r ≤ r(N)0 T r ≥ r(N)1 with a smooth ϕt-invariant transition of the timelike vectorfield N in r (N) 0 ≤ r ≤ r(N)1 , such that (1.3.16) extends to the neighborhood n−2 √ 2m < r < r (N) 0 of the event horizon. Remark 1.10 (Interpretation of Prop. 1.9). Consider a small strip along the horizon S = ⋃ v∗∈[v∗1 ,v∗2 ] V (v∗) = ⋃ v∗∈[v∗1 ,v∗2 ] [u∗1,∞]× {v∗} . 1.4. INTEGRATED LOCAL ENERGY DECAY 35 Then the energy identity for N in S becomes (upon neglecting for u∗1 large enough any difference in the contributions from {u∗1} × [v∗1, v∗2] and the corresponding segment on the horizon) the inequality∫ V (v∗2 ) (JN , n) + b ∫ S (JN , n) ≤ ∫ V (v∗1 ) (JN , n) where we have replaced N by the normal n to V (v∗) in Prop. 1.9, which implies the energy decay reminiscent of the redshift:∫ V (v∗2 ) (JN , n) ≤ e−b(v∗2−v∗1 ) ∫ V (v∗1 ) (JN , n) Note that the decay is determined from the surface gravity. 1.4 Integrated Local Energy Decay In this Section we prove several integrated local energy decay statements, i.e. estimates on the energy density of solutions to (1.1.1) integrated on (bounded) space-time regions; this in an essential ingredient for the decay mechanism employed in Section 1.5. LetRr0,r1(t0, t1, u∗1, v∗1) be the region composed of a trapezoid and characteristic rectangles as follows, (see figure 1.3): Rr0,r1(t0, t1, u∗1, v∗1) .= { (t, r) : t0 ≤ t ≤ t1, r0 ≤ r ≤ r1 } (1.4.1) ∪ { (t, r) : r ≤ r0, 1 2 (t− r∗) ≤ u∗1, t0 + r∗0 ≤ t+ r∗ ≤ t1 + r∗0 } ∪ { (t, r) : r ≥ r1, 1 2 (t+ r∗) ≤ v∗1, t0 − r∗1 ≤ t− r∗ ≤ t1 − r∗1 } We denote by R∞r0,r1(t0) . = ⋃ t1≥t0 ⋃ u∗1≥ 12 (t1−r∗0) ⋃ v∗1≥ 12 (t1+r∗1) R(t0, t1, u∗1, v∗1) (1.4.2) and its past boundary by Στ0 . = ∂−R∞r0,r1(t0) τ0 = 1 2 (t0 − r∗1) . (1.4.3) We shall first state the central estimate. Proposition 1.11 (Integrated local energy decay estimate). Let φ be a solution of the wave equation gφ = 0. Then there exist (2m) 1 n−2 < r0 < r1 and a constant C(n,m) depending on the dimension n and the mass m, such that∫ R∞r0,r1 (t0) { 1 rn ( ∂φ ∂r∗ )2 + 1 rn+1 (∂φ ∂t )2 + 1 r3 ( 1− 2m rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 } dµg ≤ C(n,m) ∫ Στ0 ( JT (φ) + JT (T · φ), n ) (1.4.4) for any t0 ≥ 0, where τ0 = 12(t0 − r∗1). 36 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES t0 r0 r1 t1 u∗1 v ∗ 1 Figure 1.3: The region Rr0,r1(t0, t1, u∗1, v∗1). The degeneracy at infinity can in fact be improved: Proposition 1.12 (Improved integrated local energy decay estimate). Let φ be a solution of the wave equation gφ = 0, then there exists a constant C(n,m, δ) for each 0 < δ < 1 such that∫ R∞r0,r1 (t0) { 1 r1+δ ( ∂φ ∂r∗ )2 + 1 r1+δ (∂φ ∂t )2 + 1 r ( 1− 2m rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 } dµg ≤ ≤ C(n,m, δ) ∫ Στ0 ( JT (φ) + JT (T · φ), n ) (1.4.5) for any t0 ≥ 0, where r0 < r1 are as above, and τ0 = 12(t0 − r∗1). As a consequence of the redshift effect of Section 1.3, and the uniform boundedness of the nondegenerate energy (which is proven independently in Section 1.5.1), we can infer in a more geometric formulation: Corollary 1.13 (nondegenerate integrated local energy decay). Let φ be a solution of (1.1.1), then for any R > n−2 √ 2m there exists a constant C(n,m,R) such that∫ τ τ ′ dτ ∫ Σ′τ ( JN (φ), n ) ≤ C(n,m,R) ∫ Στ ′ ( JN (φ) + JT (T · φ), n ) , (1.4.6) for all τ ′ < τ , where Σ′τ . = Στ ∩ {r ≤ R}. Proof. Let (we use standard notation for causal sets, see e.g.[29]) R′(τ ′, τ) .= J−(Σ′τ ) ∩ J+(Στ ′) . In R′(τ ′, τ) ∩ {r < r(N)0 } we have by Prop. 1.7( JN (φ), n ) ≤ 1 b KN(φ) , and in R′(τ ′, τ)∩{r ≥ r(N)1 } trivially (JN(φ), n) ≤ (JT (φ), n). Therefore using the energy identity for N onR′(τ ′, τ) the estimate (1.4.6) follows from Prop. 1.35 and Prop. 1.11. 1.4. INTEGRATED LOCAL ENERGY DECAY 37 In the above, no control is obtained on a spacetime integral of φ2 itself; however, all that is needed for the decay argument of Section 1.5 is an estimate for the integral of φ2 on timelike boundaries. Proposition 1.14 (zeroth order terms on timelike boundaries). Let φ be solution of the wave equation (1.1.1), and R > n−2 √ 8nm. Then there is a constant C(n,m,R) such that ∫ 2τ+R∗ 2τ ′+R∗ dt ∫ Sn−1 dµ◦ γn−1 φ2|r=R ≤ ≤ C(n,m,R) ∫ 2τ+R∗ 2τ ′+R∗ dt ∫ Sn−1 dµ◦ γn−1 {( ∂φ ∂r∗ )2 + ∣∣∇/ φ∣∣2}∣∣ r=R + C(n,m,R) ∫ Στ ′ ( JT (φ), n ) (1.4.7) for all τ ′ < τ . The central result of Prop. 1.11 combines results for two different regimes, that of high angular frequencies and that of low angular frequencies. First we will use radial multiplier vectorfields to construct positive definite currents to deal with the former regime, and then a more general current using a commutation with angular momentum operators for the latter. Remark 1.15. The specific parametrization (1.4.3) has technical advantages, but Στ can in principle be replaced by a foliation of strictly spacelike hypersurfaces terminating at future null infinity and crossing the event horizon to the future of the bifurcation sphere. 1.4.1 Radial multiplier vectorfields A radial multiplier is a vectorfield of the form X = f(r∗) ∂ ∂r∗ . (1.4.8) We would like the associated current to be positive, however we find in general, as it is shown below: KX = f ′ 1− 2m rn−2 ( ∂φ ∂r∗ )2 + f r ( 1− nm rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 − 1 2 [ f ′ + (n− 1)f r ( 1− 2m rn−2 )] ∂αφ ∂αφ (1.4.9) Note. The prefactor to the angular derivatives vanishes at the photon sphere at r = n−2√nm. 38 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Calculation of the deformation tensor (X)π. It is convenient to work in Eddington- Finkelstein coordinates X = 1 2 f(r∗) ∂ ∂v∗ − 1 2 f(r∗) ∂ ∂u∗ . (1.4.10) For the connection coefficients of (1.2.25) one obtains ∇ ∂ ∂u∗ ∂ ∂u∗ = −(n− 2) 2m rn−1 ∂ ∂u∗ ∇ ∂ ∂v∗ ∂ ∂v∗ = (n− 2) 2m rn−1 ∂ ∂v∗ ∇EAEB = ∇/ EAEB + r 2 ( ◦ γn−1)AB ∂ ∂u∗ − r 2 ( ◦ γn−1)AB ∂ ∂v∗ (1.4.11) ∇ ∂ ∂u∗ EB = −1 r ( 1− 2m rn−2 ) EB ∇ ∂ ∂v∗ EB = 1 r ( 1− 2m rn−2 ) EB . Therefore (X)πu∗u∗ = g(∇ ∂ ∂u∗ X, ∂ ∂u∗ ) = ( 1− 2m rn−2 ) f ′ (X)πv∗v∗ = g(∇ ∂ ∂v∗ X, ∂ ∂v∗ ) = ( 1− 2m rn−2 ) f ′ (X)πu∗v∗ = 1 2 g(∇ ∂ ∂u∗ X, ∂ ∂v∗ ) + 1 2 g( ∂ ∂u∗ ,∇ ∂ ∂v∗ X) = −(1− 2m rn−2 )( f ′ + (n− 2) 2m rn−1 f ) (1.4.12) (X)πaA = 0 (X)πAB = 1 2 g(∇EAX,EB) + 1 2 g(EA,∇EBX) = f r ( 1− 2m rn−2 ) ( ◦ γn−1)AB The formula for KX above (1.4.9) is now obtained by writing out (see also Appendix B.2) KX = (X)παβ Tαβ and rearranging the terms so as to complete ( ∂φ ∂u∗ ) 2+( ∂φ ∂v∗ ) 2 to ( ∂φ ∂r∗ ) 2. This rearrangement is also related to the following modification of currents; for observe that (φ2) = 2(∂αφ)(∂αφ) (1.4.13) if φ = 0. First modified current. Denoting by JX,0µ = TµνX ν (1.4.14) 1.4. INTEGRATED LOCAL ENERGY DECAY 39 define the first modified current by JX,1µ = J X,0 µ + 1 4 ( f ′ + (n− 1)f r ( 1− 2m rn−2 )) ∂µ(φ 2) − 1 4 ∂µ ( f ′ + (n− 1)f r ( 1− 2m rn−2 )) φ2 . (1.4.15) Consequently the divergences are KX,0 = ∇µJX,0µ = KX (1.4.16) KX,1 = ∇µJX,1µ = KX + 1 4 ( f ′ + (n− 1)f r ( 1− 2m rn−2 )) (φ2) − 1 4  ( f ′ + (n− 1)f r ( 1− 2m rn−2 )) φ2 = f ′ 1− 2m rn−2 ( ∂φ ∂r∗ )2 + f r ( 1− nm rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 − 1 4  ( f ′ + (n− 1)f r ( 1− 2m rn−2 )) φ2 (1.4.17) Since for any function w (w) = (g−1)µν∇µ∂νw = − 1 1− 2m rn−2 ∂u∗∂v∗w − n− 1 2r ( ∂u∗w − ∂v∗w ) +△/ r2 ◦ γn−1 w , (1.4.18) a straight-forward calculation for w = f ′ + (n− 1)f r ( 1− 2m rn−2 ) (1.4.19) shows  ( f ′ + (n− 1)f r ( 1− 2m rn−2 )) = = 1 1− 2m rn−2 f ′′′ + 2(n− 1)f ′′ r + (n− 1) [ (n− 3) + (n− 1) 2m rn−2 ]f ′ r2 + (n− 1) [( (n− 1)(n− 2)− (n− 3) )( 2m rn−2 )2 − n 2m rn−2 − (n− 3) ] f r3 . (1.4.20) Thus we finally obtain KX,1 = f ′ 1− 2m rn−2 ( ∂φ ∂r∗ )2 + f r ( 1− nm rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 − 1 4 f ′′′ 1− 2m rn−2 φ2 − n− 1 2 f ′′ r φ2 − n− 1 4 [ (n− 3) + (n− 1) 2m rn−2 ]f ′ r2 φ2 − n− 1 4 [ (n− 1)2( 2m rn−2 )2 − n 2m rn−2 − (n− 3) ] f r3 φ2 . (1.4.21) 40 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Applications of the first modified current. The proofs of Prop. 1.12 and Prop. 1.14 are applications of this formula, as it appears in the energy identity for JX,1 on RDτ2τ1 , see Appendix B.2. Proof of Prop. 1.14. Choose f = 1 identically, then KX,1 = 1 r ( 1− nm rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 + n− 1 4 [ (n− 3) + n 2m rn−2 − (n− 1)2( 2m rn−2 )2] 1 r3 φ2 . (1.4.22) Since precisely g ( JX,1, ∂ ∂r∗ ) = 1 4 ( ∂φ ∂v∗ )2 + 1 4 ( ∂φ ∂u∗ )2 − 1 2 ( 1− 2m rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 + n− 1 2r ( 1− 2m rn−2 ) φ ∂φ ∂r∗ + n− 1 4r2 [ 1− (n− 1) 2m rn−2 ]( 1− 2m rn−2 ) φ2 (1.4.23) we deduce from the energy identity for J ∂ ∂r∗ ,1 in RDττ ′ that∫ R∗+2τ R∗+2τ ′ dt ∫ Sn−1 dµ◦ γn−1 rn−1× × { 1 4 ( ∂φ ∂v∗ )2 + 1 4 ( ∂φ ∂u∗ )2 + n− 1 4R2 [1 2 − (n− 1) 2m Rn−2 ]( 1− 2m Rn−2 ) φ2 } |r=R + ∫ RDτ τ ′ n− 1 4r [ (n− 3) + n 2m rn−2 − (n− 1)2( 2m rn−2 )2] 1 r2 φ2 dµg ≤ ≤ ∫ R∗+2τ R∗+2τ ′ dt ∫ Sn−1 dµ◦ γn−1 rn−1× × { 1 2 ( 1− 2m rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 + n− 1 2 ( 1− 2m rn−2 )( ∂φ ∂r∗ )2}|r=R + C(n,m) ∫ Στ ′ ( JT (φ), n ) , (1.4.24) where we have used Prop. B.5 for the boundary terms on ∂RDττ ′ \ {r = R}; note that (n− 3) + n 2m rn−2 − (n− 1)2( 2m rn−2 )2 > 0 (R > n−2√ 8nm) . Proof of Prop. 1.12. On one hand we need f ′ = O( 1 r1+δ ) in view of (1.4.21) while on the other we already know from the proof of Prop. 1.14 that f = 1 generates a positive bulk term for r large enough. We choose f = 1− (R r )δ (1.4.25) (where R > n−2 √ 2m is chosen suitably in the last step of the proof) and indeed find KX,1 = δ Rδ r1+δ ( ∂φ ∂r∗ )2 + f r ( 1− nm rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 1.4. INTEGRATED LOCAL ENERGY DECAY 41 + { n− 1 4 (n− 3) [ 1− (R r )δ (1 + δ) ] + 1 4 (R r )δ[ 2(n− 1)− (2 + δ) ] δ(1 + δ) + [ n− 1 4 n [ 1− (R r )δ]− δ 4 (R r )δ[ n(n + δ)− 2(1 + δ)2 ]] 2m rn−2 − [ (n− 1)3 4 [ 1− (R r )δ]− δ 4 (R r )δ[( n− (1 + δ))(n− 1)− δ2]]( 2m rn−2 )2} 1 r3 φ2 ≥ 0 (1.4.26) for r ≥ R1 > R, R1 = R1(n,m) > n−2 √ 2m chosen large enough. This gives control on ∂φ ∂r∗ and the angular derivatives:∫ R1Dτ2τ1 { δ Rδ r1+δ ( ∂φ ∂r∗ )2 + f(R1) r ( 1− nm rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 } ≤ ∫ R1Dτ2τ1 KX,1 Here and in the following τ2 > τ1 > 1 2 ( t0 − R∗). For ∂φ∂t we use the auxiliary current (see also Appendix B.3) Jauxµ = 1 2 ( 1− 2m rn−2 ) δ Rδ r1+δ ∂µ(φ 2) to find easily∫ R1Dτ2τ1 δ Rδ r1+δ (∂φ ∂t )2 ≤ ∫ R1Dτ2τ1 { δ(n + δ) Rδ r1+δ ( ∂φ ∂r∗ )2 + δ Rδ r1+δ ( 1− 2m rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 + δ(n + δ) Rδ r3+δ φ2 +Kaux } Note that for r ≥ R1 in particular 1 4 [ 2(n− 1)− (2 + δ) ] δ(1 + δ) Rδ r3+δ φ2 ≤ KX,1 hence∫ R1Dτ2τ1 δ Rδ r1+δ {(∂φ ∂t )2 + ( ∂φ ∂r∗ )2} ≤ C(n,m, δ) ∫ R1Dτ2τ1 { KX,1 +Kaux } ≤ ≤ C(n,m, δ) ∫ RDτ2τ1 { KX,1 +Kaux } + C(n,m, δ) ∫ RDτ2τ1∩{R 0 r > n−2 √ nm . Since f should also be bounded one may guess that f(r∗) = arctan ((n− 1) r∗ n−2√nm ) is a good choice; while it can ensure positivity at the photon sphere, it fails to do so away from the photon sphere in the intermediate regions near the horizon and in the asymptotics. After having briefly recalled the decomposition into spherical harmonics, we will therefore give a more refined construction of f , nonetheless guided by the overall characteristics of this function, which will in particular allow us to track the dependence of the lowest spherical harmonic number (for which we can establish non-negativity) on the dimension n. Fourier expansion on the sphere Sn−1. We recall that by considering homogeneous harmonic polynomials on Rn (see e.g. discussion of spherical harmonics in [44]) we find that all eigenvalues of − ◦ △/ n−1 + (n− 2 2 )2 on Sn−1 are given by ( l + n− 2 2 )2 (l ≥ 0) . Let El, l ≥ 0, be the corresponding eigenspace in L2(Sn−1). Recall dimC El = ( l + n− 2 2 )2 l ( n− 2 + l − 1 l − 1 ) and furthermore L2(Sn−1) = ⊕ l≥0 El . Denote by πl the orthogonal projection of L 2(Sn−1) onto El, then for φ ∈ L2(Sn−1) φ = ∑ l≥0 πlφ . (1.4.27) This is the Fourier expansion on the sphere Sn−1. We find ◦ △/ n−1 πlφ = −l(l + n− 2)πlφ . (1.4.28) 44 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Now, l(l + n− 2) ∫ Sn−1 (πlφ) 2 dµ◦ γn−1 = − ∫ Sn−1 (πlφ)( ◦ △/ n−1 πlφ) dµ◦γn−1 = ∫ Sn−1 | ◦ ∇/ n−1 πlφ|2 dµ◦γn−1 and assuming that πlφ = 0 (0 ≤ l < L) for some L > 0, L(L+ n− 2) 1 r2 ∫ Sr φ2 dµγr ≤ 1 r2 ∑ l≥L l(l + n− 2) ∫ Sn−1 (πlφ) 2rn−1 dµ◦ γn−1 = 1 r2 ∑ l≥L ∫ Sn−1 | ◦ ∇/ n−1 πlφ|2 dµ◦γn−1r n−1 = 1 r2 ∑ l≥L ∫ Sn−1 |πl ◦ ∇/ n−1 φ|2 dµ◦γn−1r n−1 = ∫ Sn−1 1 r2 | ◦ ∇/ n−1 φ|2 rn−1 dµ◦γn−1 = ∫ Sr ∣∣∇/ φ∣∣2 r2 ◦ γn−1 dµγr , where for the commutation of πl with ◦ ∇/ n−1 we have used that the projection is of the form (πlφ)(rξ) = ∫ Sn−1 πl(〈ξ, ξ′〉)φ(rξ′) dµ◦γn−1(ξ ′) . (1.4.29) We have proven the following Poincare´-type inequality: Lemma 1.16 (Poincare´ inequality). Let φ ∈ H1(Sr), Sr = (Sn−1, r2 ◦ γn−1), have vanishing projection to El, 0 ≤ l < L, for some L ∈ N, i.e. πlφ = 0 (0 ≤ l < L) , then ∫ Sr ∣∣∇/ φ∣∣2 dµγr ≥ L(L+ n− 2) 1r2 ∫ Sr φ2 dµγr . Construction of the multiplier function for high angular frequencies. The idea is to prescribe the 3rd derivative of f and to find its 2nd and 1st derivatives by integration with boundary values and parameters that ensure that f remains bounded. Let α = n− 1 (nm) 1 n−2 (1.4.30) 1.4. INTEGRATED LOCAL ENERGY DECAY 45 and γ ≥ 2, γ ∈ N. Consider f IIIγ,α(r ∗) =  −1 , |r∗| ≤ 1 γα 1 , 1 γα < |r∗| ≤ bγ,α(bγ,α r∗ )6 , |r∗| ≥ bγ,α (1.4.31) where bγ,α = 5 6 2 γα . (1.4.32) Note that bγ,α is chosen so that ∫ ∞ 0 f IIIγ,α(r ∗) dr∗ = 0 . (1.4.33) Now define f IIγ,α(r ∗) = ∫ r∗ 0 f IIIγ,α(t) dt . (1.4.34) Obviously f IIγ,α(−r∗) = −f IIγ,α(r∗) and in explicit form f IIγ,α(r ∗) =  −r∗ |r∗| ≤ 1 γα r∗ − 2 γα 1 γα < r∗ ≤ bγ,α r∗ + 2 γα −bγ,α ≤ r∗ < − 1γα − b 6 γ,α 5r∗5 |r∗| ≥ bγ,α . (1.4.35) The functions f IIγ,α and f III γ,α are sketched in figure 1.4. Next define f Iγ,α = ∫ r∗ −∞ f IIγ,α(t) dt . (1.4.36) Here we find f Iγ,α(r ∗) =  b6γ,α 20r∗4 r∗ ≤ −bγ,α b2γ,α 20 + 1 2 (r∗2 − b2γ,α) + 2 γα (r∗ + bγ,α) −bγ,α ≤ r∗ ≤ − 1γα 13 12 1 (γα)2 − r ∗2 2 − 1 γα ≤ r∗ ≤ 0 (1.4.37) and f Iγ,α(r ∗) = f Iγ,α(−r∗), as sketched in figure 1.5. Finally define f 0γ,α(r ∗) = ∫ r∗ 0 f Iγ,α(t) dt . (1.4.38) Here again f 0γ,α(−r∗) = −f 0γ,α(r∗) and in particular f 0γ,α( 1 γα ) = ∫ 1 γα 0 (13 12 1 (γα)2 − t 2 2 ) dt = 11 12 1 (γα)3 . (1.4.39) 46 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES f IIIγ,α r∗1 γα bγ,α 1 1 − 1 γα −bγ,α −1 1 γα −1 3 1 γα − 1 γα 1 3 1 γα f IIγ,α r∗1 γα − 1 γα c Figure 1.4: Sketch of the functions f IIγ,α and f III γ,α, and the adjusted functions (dot-dashed) for r∗ ≤ 0. b2γ,α 20 r∗bγ,α1γα 13 12 1 (γα)2 f Iγ,α 7 12 1 (γα)2 − 1 γα −bγ,α 1 γα r∗ 3 2 1 (γα)3 − 1 γα f 0γ,α 11 12 1 (γα)3 −11 12 1 (γα)3 Figure 1.5: Sketch of the functions f Iγ,α and f 0 γ,α, and the adjusted functions (dot-dashed) for r∗ ≤ 0. 1.4. INTEGRATED LOCAL ENERGY DECAY 47 Moreover the calculus yields f(bγ,α) > 1 (γα)3 lim r∗→∞ f 0γ,α(r ∗) < 3 2 1 (γα)3 . (1.4.40) The function f 0γ,α is sketched in figure 1.5. While this function would suffice in the region r∗ ≥ − 1 γα it does not fall-off fast enough as r∗ → −∞. Lemma 1.17. With r∗ defined by (1.2.22) we have for all n ≥ 3 lim r∗→−∞ ( 1− 2m rn−2 ) (−r∗) = 0 . In fact, ( 1− 2m rn−2 ) ≤ (2m) 1n−2 (−r∗) for all r∗ < 0. Proof. See Appendix B.2.2. It is easy to convince oneself that one can make an adjustment to f III on r∗ ≤ 0 that introduces faster decay while keeping the area under the graph of f III and f II fixed. In other words, there are constants bγ,α ≤ b ≤ 4 γα 1 4 ≤ c ≤ 1 (1.4.41) such that if we redefine f IIIγ,α for r ∗ ≤ 0 as f IIIγ,α(r ∗) =  −1 − 1 γα ≤ r∗ ≤ 0 c −b ≤ r∗ ≤ 1 γα( 1− 2m rn−2 )6( b (2m) 1 n−2 )6 r∗ ≤ −b (1.4.42) then ∫ −∞ 0 f IIIγ,α(r ∗) dr∗ = 0∫ 0 −∞ ∫ r∗ 0 f IIIγ,α(t) dt dr ∗ = ∫ 0 −∞ ∫ −r∗ 0 (−f IIIγ,α(t)) dt dr∗ . Remark 1.18 (Existence of constants). This is because the area that is lost under the graph of f III after the replacement to faster decay is less than∫ −bγ,α −∞ ( b r∗ )6 dr∗ = 1 5 bγ,α 48 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES which can be accounted for by a shift of bγ,α to b ′ < 2 γα . The resulting lost of area under the graph of f II is less than ∫ −bγ,α −∞ (−1 5 b6γ,α r∗5 ) dr∗ = 1 20 b2γ,α which can be compensated by an adjustment of the slope from 1 to c ≤ 1 on the interval −b′ ≤ r∗ ≤ − 1 γα (while keeping the area fixed). The gain of area under the graph of f II on the interval (−bγ,α,− 1γα) alone is enough,∫ bγ,α 1 γα (1− c)(t− 1 γα ) dt = 1 20 b2γ,α if we choose c = 3 8 ≥ 1 4 . The upper bound on b then follows easily to be b ≤ 4 γα . The adjusted functions in comparison the the old are also sketched in figures 1.4 and 1.5. Note in particular that for r∗ ≤ 0 f II(r∗) ≤ 1 γα (1.4.43) f I(r∗) ≤ f I(−r∗) ≤ 13 12 1 (γα)2 (1.4.44) and for r∗ ≤ − 1 γα 11 12 1 (γα)3 ≤ |f 0(r∗)| ≤ f 0(−r∗) < 3 2 1 (γα)3 . (1.4.45) Remark 1.19. In order to deal with smooth functions one could use (e.g. at the level of second derivatives) a convolution with a Gaussian on the scale given by γα (or finer). I.e. one could define f ′′γ,α(r ∗) = γα√ π ∫ ∞ −∞ e−(γα) 2(r∗−t)2f IIγ,α(t) dt and find f ′′′γ,α = d dr∗f ′′ γ,α by differentiation, and f ′ γ,α and fγ,α by integration with the boundary values f ′γ,α(−∞) = 0, fγ,α(0) = 0 as above. However, I choose not to do so (as it does not give further insight) and work directly with the step-functions, i.e. define f ′′γ,α = f III γ,α . We are now in the position to prove a non-negativity property of the terms occuring in (1.4.21) which we will denote by 0KX,1, KX,1 = f ′ 1− 2m rn−2 ( ∂φ ∂r∗ )2 + 0KX,1 . (1.4.46) Proposition 1.20 (Positivity of the current JXγ,α,1). For n ≥ 3, Xγ,α = fγ,α ∂ ∂r∗ (where we choose γ = 12) , 1.4. INTEGRATED LOCAL ENERGY DECAY 49 and φ ∈ H1(S) satisfy ∫ S 0KXγ,α,1 dµγ ≥ 0 provided πlφ = 0 (0 ≤ l < L) for a fixed L ≥ (6γn)2. Proof. By Lemma 1.16∫ S 0KXγ,α,1 dµγ ≥ ∫ S { L(L+ n− 2)fγ,α r3 ( 1− nm rn−2 ) − 1 4 f ′′′γ,α 1− 2m rn−2 − n− 1 2 f ′′γ,α r − n− 1 4 [ (n− 3) + (n− 1) 2m rn−2 ]f ′γ,α r2 − n− 1 4 [ (n− 1)2( 2m rn−2 )2 − n 2m rn−2 − (n− 3) ]fγ,α r3 } φ2 dµγ . (1.4.47) We divide into the five regions −∞ < − 4 γα < − 1 γα < 1 γα < bγ,α <∞ . Step 1. (near the photon sphere, |r∗| < 1 γα ) Lemma 1.21. In the region |r∗| < 1 γα the corresponding value of r lies in the interval n−2√ δnm < r < n α where δ = max{1 3 , 4 3 2 n }. Recalling the graphs of fγ,α and its derivatives we then find in the region |r∗| < 1γα :∫ S 0KXγ,α,1 dµγ ≥ ≥ ∫ S { 1 4 − 1 2 α δ 1 n−2 1 γα − 1 4 α2 δ 2 n−2 1 n− 1 [ (n− 3) + 1 δ (n− 1) 2 n ]13 12 1 (γα)2 − 1 4 α3 δ 3 n−2 2 δn 3 2 1 (γα)3 } φ2 dµγ ≥ ∫ S { 1 4 − 1 2 3 γ − 3 4 13 12 (3 γ )2 − 3 4 (3 γ )3} φ2 dµγ ≥ ∫ S 1 4 1 8 φ2 dµγ because γ = 12. 50 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Proof of the Lemma. For the lower bound write 1 γα = ∫ (nm) 1n−2 r(r∗=− 1 γα ) 1 1− 2m rn−2 dr ≤ (nm) 1 n−2 − r(r∗ = − 1 γα ) 1− 2m r(r∗=− 1 γα )n−2 . If we assume the form r(r∗ = − 1 γα ) = n−2 √ δnm for some 2 n < δ < 1, then this inequality becomes 1 γ(n− 1) ≤ 1− n−2√δ 1− 2 n 1 δ which is satisfied for example with δ = max{1 3 , 4 3 2 n } (γ ≥ 2). The upper bound immediatly follows from 1 γα = ∫ r(r∗= 1 γα ) (nm) 1 n−2 1 1− 2m rn−2 dr ≥ r(r∗ = 1 γα )− n−2√nm . Step 2. (in the intermediate region, 1 γα ≤ r∗ ≤ 5 6 2 γα ) Lemma 1.22. In the region 1 γα ≤ r∗ ≤ 5 6 2 γα we for the corresponding value of r,( 1 + 1 3γ(n− 1) ) (nm) 1 n−2 ≤ r ≤ n α . Collecting the first and the last term, we find in this region,∫ S 0KXγ,α,1 dµγ ≥∫ S { α3 n3 11 12 1 (γα)3 [( 1− nm rn−2 ) L(L+ n− 2)− n− 1 4 (n− 1)2(2 n )2 + 1 4 (n− 1)(n− 3) ] − 1 4 1 1− 2 n + α 3 1 3 1 γα − 1 4 α2 1 n− 1 [ (n− 3) + (n− 1) 2 n ] 1 (γα)2 } φ2 dµγ ≥ ∫ S { 11 12 1 (nγ)3 [ 1 6γ(n− 1)L(L+ n− 2)− (n− 1) + 1 4 (n− 1)(n− 3) ] − 3 4 − 1 4 1 γ2 } φ2 dµγ ≥ ∫ S { 11 12 1 6 ((6γn)2 γ2n2 )2 − 1 } φ2 dµγ ≥ ∫ S φ2 dµγ because L ≥ (6γn)2, where we have used that for 1 γα ≤ r∗ ≤ 5 6 2 γα , 1− nm rn−2 ≥ 1 6γ(n− 1) . Proof of the Lemma. For the lower bound note 1 γα = ∫ r(r∗= 1 γα ) (nm) 1 n−2 1 1− 2m rn−2 dr ≤ 1 1− 2 n ( r − n−2√nm) 1.4. INTEGRATED LOCAL ENERGY DECAY 51 to find that r(r∗ = 1 γα ) ≥ ( n− 1 + 1 γ ( 1− 2 n )) 1 α ≥ ( 1 + 1 3γ(n− 1) ) (nm) 1 n−2 . The upper bound again follows easily from 5 6 2 γα ≥ r(r∗ = 5 6 2 γα )− n−2√nm since γ ≥ 2. Step 3. (in the asymptotics, r∗ ≥ bγ,α) Given the general fact Prop. B.1 we here only need the weaker statement Lemma 1.23. For r∗ ≥ 5 6 2 γα , r r∗ ≤ 2γn . Here∫ S 0KXγ,α,1 dµγ ≥∫ S { 1 (γα)3 [ 1 6γ(n− 1)L(L+ n− 2)− 3 2 (n− 1) ] 1 r3 − 1 4 1 1− 2 n (5 6 2 γαr∗ )6 − 1 4 α2 1 n− 1 [ (n− 3) + (n− 1) 2 n ] 1 20 ( 5 6 2 γα )6 r∗4 } φ2 dµγ ≥ ∫ S [ L2 6γ4n + L 6γ4n (n− 2)− 3 2 1 γ3 (n− 1)− 3 4 ( r r∗ )3(5 6 2 γ )6 1 (αr∗)3 − 1 4 1 20 ( r r∗ )3(5 6 2 γ )6 1 αr∗ ] 1 (αr)3 φ2 dµγ ≥ ∫ S [ (6n)3 − (4n)3 ] 1 (αr)3 φ2 dµγ ≥ ∫ S ( n αr )3φ2 dµγ ≥ ∫ S ( (nm) 1 n−2 r )3 φ2 dµγ where in the third bound we have again used L ≥ (6γn)2 and the Lemma. Proof of the Lemma. Since r∗ ≥ r − n−2√nm we directly arrive at r r∗ ≤ 1 + (nm) 1 n−2 5 6 2 γα ≤ 2γn . 52 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Step 4. (in the intermediate region, − 4 γα ≤ r∗ ≤ − 1 γα ) Recall γ = 12. Lemma 1.24. For k ≤ γ, k ∈ N,( 1− 2m rn−2 )−1 |r∗=− k γα ≤ 17 and consequently − ( 1− nm rn−2 ) |r∗=− 1 γα ≥ 1 20 1 2γ . In the region − 4 γα ≤ r∗ ≤ − 1 γα we directly apply the Lemma to see that,∫ S 0KXγ,α,1 dµγ ≥∫ S { L(L+ n− 2) 1 (nm) 3 n−2 11 12 1 (γα)3 1 20 1 2γ − 1 4 17− n− 1 2 1 (2m) 1 n−2 1 γα − n− 1 2 [ (n−3)+(n−1) ] 1 (2m) 2 n−2 13 12 1 (γα)2 − n− 1 4 [ n+(n−3) ] 1 (2m) 3 n−2 2 1 (γα)3 } φ2 dµγ ≥ ∫ S { 1 (3γ)4 1 (n− 1)3L(L+ n− 2) − 17 4 − 3 2 1 2γ − 13 12 1 γ2 (n 2 ) 2 n−2 − 1 n− 1 1 γ3 (n 2 ) 3 n−2 } φ2 dµγ ≥ ∫ S { 24n− 23 4 } φ2 dµγ ≥ ∫ S φ2 dµγ because L ≥ (6γn)2. Proof of the Lemma. According to Prop. B.4 we have for the value of r corresponding to r∗ = − k γα (recall (B.8)): k γα ≥ (2m) 1 n−2 n− 2 log q0 (( n 2 ) 1 n−2 ) q0 ( r (2m) 1 n−2 ) Thus q0 ( r (2m) 1 n−2 ) ≥ e− kγ n−2n−1 (n2 ) 1 n−2 q0 ((n 2 ) 1 n−2 ) ≥ 1√ e3 ( n 2 ) 1 n−2 − 1( n 2 ) 1 n−2 + 1 ≥ 1 5 2 5 1 n− 2 log (n 2 ) Take r(r∗ = − k γα ) in the form r = n−2 √ 2m(1 + β) then we find for β β ≥ (2 5 )2 1 n− 2 log (n 2 ) 1.4. INTEGRATED LOCAL ENERGY DECAY 53 since β ≥ 0. Therefore( 1− 2m rn−2 ) |r∗=− k γα ≥ 1− 1( 1 + (2 5 )2 1 n−2 log n 2 )n−2 ≥ 1− e− 320 log n2 ≥ 1− ( 2 n ) 3 20 and ( 1− 2m rn−2 )−1 |r∗=− k γα ≤ ( n 2 ) 3 20( n 2 ) 3 20 − 1 ≤ 17 . Consequently 1 γα = ∫ (nm) 1n−2 r(r∗=− 1 γα ) 1 1− 2m rn−2 dr ≤ 1 1− ( 2 n ) 3 20 ( n−2√nm− r ) or r(r∗ = − 1 γα ) ≤ ( 1− 1 17 1 γ(n− 1) ) (nm) 1 n−2 which yields the desired lower bound on − ( 1− nm rn−2 ) |r∗=− 1 γα ≥ − ( 1− 1( 1− 1 17 1 γ(n−1) )n−2) ≥ −1 + e 120 n−2γ(n−1) ≥ 1 20 n− 2 γ(n− 1) ≥ 1 20 1 2γ . Step 5. (near the horizon, r∗ ≤ −b) Finally we see for r∗ ≤ −b, recalling the adjustment to faster fall-off,∫ S 0KXγ,α,1 dµγ ≥∫ S { L(L+ n− 2) 1 (nm) 3 n−2 11 12 1 (γα)3 1 20 1 2γ − 1 4 ( 1− 2 n )5 − n− 1 2 1 (2m) 1 n−2 1 γα − (n− 1) 2 4 1 (2m) 1 n−2 1 (γα)2 − n− 1 4 [ n+ (n− 3) ] 1 (2m) 3 n−2 2 1 (γα)3 } φ2 dµγ ≥ ∫ S { 1 (3γ)4 1 (n− 1)3L(L+ n− 2) − 1 4 − 1 2γ (n 2 ) 1 n−2 − 1 (2γ)2 (n 2 ) 2 n−2 − 4 n− 1 1 (2γ)3 (n 2 ) 3 n−2 } φ2 dµγ ≥ ∫ S { 24n− 5 4 } φ2 dµγ ≥ ∫ S φ2 dµγ where we have used that here f ′′′ 1− 2m rn−2 = ( 1− 2m rn−2 )5( b (2m) 1 n−2 )6 ≤ (1− 2 n )5 ≤ 1 . 54 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES In fact, we have shown more, because all lower bounds in Step 1-5 are minorized by 1 4 1 8 (2m) 3 n−2 r3 . Corollary 1.25. Let φ ∈ H2 be a solution of the wave equation, gφ = 0 , satisfying πlφ = 0 (0 ≤ l < L) on the standard sphere S = (Sn−1, r2 ◦ γn−1) for a fixed L ≥ (6γn)2. Then∫ S { 1 4 1 8 (2m) 3 n−2 r3 φ2 + 1 (20γ2)3 1 (n− 2)2(n− 1)6 ( 1− 2m rn−2 )5 (2m) 6n−2 r4 ( ∂φ ∂r∗ )2} dµγ ≤ ∫ S KXγ,α,1 dµγ Proof. It remains to be shown that 1 20 1 (4 · 5(n− 2))2 ( 1− 2m rn−2 )6 b6γ,α r4 ≤ f ′γ,α . (∗) First ∫ r∗ −∞ ( 1− 2m rn−2 )6 dr∗ = ∫ r (2m) 1 n−2 ( 1− 2m rn−2 )5 dr because dr∗/ dr = ( 1− 2m rn−2 )−1 . Now choose n−2 √ 2m < r0 < r so close to r as to satisfy r − r0 r0 = 1 2 1 5(n− 2) ( 1− 2m rn−2 ) then by the mean value theorem∫ r (2m) 1 n−2 ( 1− 2m rn−2 )5 dr ≥ (1− 2m rn−20 )5 (r − r0) ≥ (1− 2m rn−2 )5[ 1− 5(n− 2) 1 1− 2m rn−2 r − r0 r0 ] (r − r0) ≥ 1 4 1 5(n− 2) ( 1− 2m rn−2 )6 (2m) 1 n−2 . We conclude for r∗ ≤ −b, f ′γ,α(r ∗) = ∫ r∗ −∞ ∫ s∗ −∞ ( 1− 2m rn−2 )|r∗=s∗( b (2m) 1 n−2 )6 ds∗ dr∗ ≥ 1 4 1 5(n− 2) ∫ r∗ −∞ ( 1− 2m rn−2 )6 dr∗ b6 (2m) 5 n−2 ≥ (1 4 1 5(n− 2) )2( 1− 2m rn−2 )6 b6 (2m) 4 n−2 ≥ 1 (4 · 5(n− 2))2 ( 1− 2m rn−2 )6 b6γ,α r4 . 1.4. INTEGRATED LOCAL ENERGY DECAY 55 Second for r∗ ≥ 0 1 (4 · 5(n− 2))2 1 r4 = 1 (4 · 5(n− 2))2 (r∗ r )4 1 r∗4 ≤ 1 r∗4 Since, thirdly, bγ,α r ≤ 1 , we have established (∗) for the regions r∗ ≤ −b, r∗ ≥ bγ,α, −b ≤ r∗ ≤ bγ,α, respectively. Remark 1.26. This estimate of the zeroth order term φ2 suffices to obtain an estimate for all derivatives using a commutation with the vectorfield T ; see Proof of Prop. 1.11 in Section 1.4.4. 1.4.3 Low angular frequencies and commutation While the current constructed in Section 1.4.2 required a decomposition into spherical har- monics we will now altogether avoid a recourse to the Fourier expansion on the sphere. The key to the positivity property was Poincare´’s inequality which states in more gener- ality (see e.g. [31]): Lemma 1.27 (Poincare´ inequality). Let (S, γ) be a compact Riemannian manifold, and φ ∈ H1(S) a function on S with mean value φ¯ = 1∫ S dµγ ∫ S φ dµγ . Then ∫ S (φ− φ¯)2 dµγ ≤ 1 λ1(S) ∫ S |∇/φ|2 dµγ where λ1(S) is the first nonzero eigenvalue of the negative Laplacian, −△/ = −∇/ a∇/ a, on S. (∇/ denotes covariant differentiation on S.) Now let (S, γ) = (Sn−1, ◦ γn−1) then we read off from (1.4.28) here λ1(S n−1) = n− 1 . (1.4.48) Choose a basis of the Lie algebra of SO(n), Ωi : i = 1, . . . , n(n− 1) 2 , (1.4.49) and apply Lemma 1.27 to the functions Ωiφ of vanishing mean:∫ Sn−1 Ωiφ dµ◦γn−1 = 0 . (1.4.50) Then we obtain ∫ Sn−1 |∇/Ωiφ|2 dµ◦γn−1 ≥ (n− 1) ∫ Sn−1 (Ωiφ) 2 dµ◦ γn−1 (1.4.51) 56 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES or on (S, γ) = (Sr, γr) = (S n−1, r2 ◦ γn−1):∫ Sr |∇/Ωiφ|2 dµγr ≥ n− 1 r2 ∫ Sr (Ωiφ) 2 dµγr . (1.4.52) Also note n(n−1) 2∑ i=1 (Ωiφ) 2 = r2 ∣∣∇/ φ∣∣2 r2 ◦ γn−1 . (1.4.53) Second modified current. Recall we are considering vectorfields of the form X = f(r∗) ∂ ∂r∗ . Define JX,2µ = J X,1 µ + f ′ f(1− 2m rn−2 ) β Xµ φ 2 (1.4.54) where β = β(r∗) is a function to be chosen below. Then KX,2 = KX,1 +∇µ ( f ′ f(1− 2m rn−2 ) β Xµ φ 2 ) = f ′ 1− 2m rn−2 ( ∂φ ∂r∗ + βφ )2 + f r ( 1− nm rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 − 1 4 f ′′′ 1− 2m rn−2 φ2 + f ′′ 1− 2m rn−2 [ β − n− 1 2r ( 1− 2m rn−2 )] φ2 − f ′ 1− 2m rn−2 [ β2 − β ′ − n− 1 r β ( 1− 2m rn−2 ) + n− 1 4r2 ( (n− 3) + (n− 1) 2m rn−2 )( 1− 2m rn−2 )] φ2 − n− 1 4 [ (n− 1)2( 2m rn−2 )− n 2m rn−2 − (n− 3) ] f r3 φ2 (1.4.55) Now choose β = n− 1 2r ( 1− 2m rn−2 ) + δ (1.4.56) then β2−β ′− n− 1 r β ( 1− 2m rn−2 ) + n− 1 4r2 ( (n−3)+(n−1) 2m rn−2 )( 1− 2m rn−2 ) = −δ′+δ2 (1.4.57) and KX,2 = f ′ 1− 2m rn−2 ( ∂φ ∂r∗ + βφ )2 + f r ( 1− nm rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 − 1 1− 2m rn−2 { 1 4 f ′′′ − δf ′′ + (δ2 − δ′)f ′}φ2 − n− 1 4 [ (n− 1)2( 2m rn−2 )2 − n 2m rn−2 − (n− 3) ] f r3 φ2 (1.4.58) Note. Suppose outside a compact interval [−α, α] ⊂ R f ′ is of the form f ′(r∗) = 1 r∗2 (|r∗| > α). Then we could choose δ = − 1 r∗ (|r∗| > α) so that δf ′′ = 2r∗4 ≥ 0 and −δ′ + δ2 = 0 . 1.4. INTEGRATED LOCAL ENERGY DECAY 57 Definition of the current J (α). Let α > 0 and introduce a shifted coordinate x = r∗ − α−√α . (1.4.59) The modification we choose is δ = − x α2 + x2 (1.4.60) so that −δ′ + δ2 = α 2 (α2 + x2)2 . (1.4.61) Let fa = − C α2rn−1 (C > 0) (1.4.62) and (f b) ′ = 1 α2 + x2 (f b)(r∗) = ∫ r∗ 0 1 α2 + x(t∗)2 dt∗ . (1.4.63) Note that then (fa)′ + (n− 1)f a r ( 1− 2m rn−2 ) = 0 (1.4.64) and 1 4 (f b) ′′′ − δ(f b)′′ + (δ2 − δ′)(f b)′ = −1 2 x2 − α2 (x2 + α2)3 . (1.4.65) Our current is built from the multiplier vectorfields Xa = fa ∂ ∂r∗ Xb = f b ∂ ∂r∗ (1.4.66) by setting J (α)µ (φ) . = JX a,0 µ (φ) + n(n−1) 2∑ i=1 JX b,2 µ (Ωiφ) (1.4.67) and will be shown to have the property that its divergence K(α) . = ∇µJ (α)µ (1.4.68) is nonnegative upon integration over the spheres. Proposition 1.28 (Positivity of the current J (α)). For n ≥ 3, and φ ∈ H1(S)∫ S K(α) dµγ ≥ 0 provided α is chosen sufficently large, and C(n,m, α) set to be (∗) below. Proof. In view of (1.4.64) and (1.4.65) K(α) ≥ (f a)′ 1− 2m rn−2 ( ∂φ ∂r∗ )2 + fa r ( 1− nm rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 58 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES + n(n−1) 2∑ i=1 f b r ( 1− nm rn−2 )∣∣∇/Ωiφ∣∣2r2 ◦γn−1 + n(n−1) 2∑ i=1 F (Ωiφ) 2 + n(n−1) 2∑ i=1 n− 1 4r3 [ (n− 3) + n 2m rn−2 − (n− 1)2( 2m rn−2 )2] f b (Ωiφ) 2 (1.4.69) where F . = 1 2 1 1− 2m rn−2 x2 − α2 (x2 + α2)3 . (1.4.70) So by Poincare´’s inequality (1.4.52) and (1.4.53)∫ S K(α) dµγ ≥ ∫ S { C(n− 1) α2rn ( ∂φ ∂r∗ )2 + + [ (n− 1)f b r ( 1− nm rn−2 ) + F r2 + 1 r H ]∣∣∇/φ∣∣2 r2 ◦ γn−1 } dµγ (1.4.71) where H . = n− 1 4 [ (n− 3) + n 2m rn−2 − (n− 1)2( 2m rn−2 )2] f b − C α2rn−1 ( 1− nm rn−2 ) . (1.4.72) Step 1. H ≥ 0 It is equivalent to show that H˘(r) . = rn−1H(r) rn−2 2m is nonnegative. We consider H˘ to be a function of ρ . = rn−2 2m so H˘ = n− 1 4 (2mr) [ (n− 3)ρ2 + nρ− (n− 1)2 ] f b − C α2 ( ρ− n 2 ) Note that r = n−2 √ nm ⇐⇒ ρ = n 2 ⇐⇒ r∗ = 0 and H˘( n 2 ) = 0 . Moreover we choose the constant C such that dH˘ dρ |ρ=n 2 = 0 . dH˘ dρ = n− 1 4 (2mr) [(n− 3)(2n− 3) n− 2 ρ+ n− 1 n− 2n− (n− 1)2 n− 2 1 ρ ] f b + n− 1 4(n− 2) 2mr2 ρ− 1 [ (n− 3)ρ2 + nρ− (n− 1)2 ] (f b) ′ − C α2 1.4. INTEGRATED LOCAL ENERGY DECAY 59 where we have used dr dρ = r (n− 2)ρ dr∗ dρ = 1 ρ− 1 r n− 2 . Hence we choose C = (n− 1)2 4(n− 2) (n 2 )2 − (n− 1) n 2 − 1 2m (nm) 2 n−2 α2 α2 + (α + √ α)2 . (∗) Note that then also dH dr |r= n−2√nm = 0 . Now returning to the expression for H˘ let us denote by 1 ≤ ρ0 ≤ n2 the value of ρ for which (n− 3)ρ0 + n− (n− 1)2 1 ρ0 = 0 , i.e. ρ0 = 2(n− 1)2 n + √ n2 + 4(n− 1)2(n− 3) . We divide into the four regions 1 < ρ0 < n 2 < ρ∗ <∞ where ρ∗ is to be chosen large enough below. Step 1a. (near the horizon, 1 ≤ ρ ≤ ρ0) Clearly H˘ ≥ 0 termwise, because f b ≤ 0. Step 1b. (near the photon sphere, ρ0 ≤ ρ ≤ n2 ) We show H = H(r) is convex on r0 ≤ r ≤ n−2 √ nm where r0 = n−2 √ 4(n− 1)2m n+ √ n2 + 4(n− 1)2(n− 3) . Differentiating twice yields d2H dr2 = n− 1 4 1( 1− 2m rn−2 )2 (f b)′′[(n− 3) + n 2mrn−2 − (n− 1)2( 2mrn−2 )2] + n− 1 2 1 1− 2m rn−2 (f b) ′ (n− 2) [ 2(n− 1)2 2m rn−2 − n ] 2m rn−1 − n− 1 4 1( 1− 2m rn−2 )2 (f b)′(n− 2)[(n− 3) + n 2mrn−2 − (n− 1)2( 2mrn−2 )2] 2mrn−1 + n− 1 4 (f b) [ (n− 2)(n− 1)n2m rn − 2(2n− 3)(n− 2)(n− 1)2( 2m rn−1 )2] − (n− 1)nC αn−1rn+1 ( 1− nm rn−2 ) + 3 (n− 1)(n− 2)C αn−1rn nm rn−1 60 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Since (f b) ′′ ≥ 0 we further have in this region the bound d2H dr2 ≥ n− 1 2 1 1− 2m rn−2 × × [ 2(n− 1)2 2m rn−2 − n− 1 2 1 1− 2m rn−20 ( (n− 3) + n 2m rn−2 − (n− 1)2( 2m rn−2 )2)]× × 2m rn−1 (n− 2)(f b)′ + n− 1 4 2m rn−2 [ 1− 2(2n− 3)(n− 1) n ( 2m rn−2 )](f b) r2 Since for n ≥ 3 2(n− 1)2 2 n − n − 1 2 2(n− 1)2 2(n− 1)2 − n−√n2 + 4(n− 1)2(n− 3) ( (n− 3) + 2− (2n− 1 n )2) ≥ 1 and 1− 2(2n− 3)(n− 1) n 2 n ≤ −1 we finally obtain in this region d2H dr2 ≥ (n− 1)(n− 2) 2r 1 ρ− 1(f b) ′ > 0 . Step 1c. (in the intermediate region, n 2 ≤ ρ ≤ ρ∗) We show H˘ = H˘(ρ) is convex on n 2 ≤ ρ ≤ ρ∗ for r∗(ρ = ρ∗) ≤ α. d2H˘ dρ2 = (n− 1)2 4(n− 2)2 2mr ρ2 [ (n− 3)(2n− 3)ρ2 + nρ+ (n− 3)(n− 1) ] (f b) + (n− 1)2 4(n− 2)2 2mr2 (ρ− 1)2× × [ 3(n− 3)ρ2 − 3(n− 5)ρ+ (n− 1)(n− 5)− n2n− 1 n− 1 + 3(n− 1) 1 ρ ] (f b) ′ + n− 1 4(n− 2)2 2mr3 (ρ− 1)2 [ (n− 3)ρ2 + nρ− (n− 1)2 ] (f b) ′′ Since for ρ ≥ n 2 , and n ≥ 3, 3(n− 3)ρ(ρ− 1) + 6ρ+ (n− 1)(n− 5)− n2n− 1 n− 1 + 3(n− 1) 1 ρ ≥ 1 and (n− 3)ρ2 + nρ− (n− 1)2 ≥ 0 we have d2H˘ dρ2 ≥ (n− 1) 2 4(n− 2)2 2mr2 (ρ− 1)2 (f b) ′ > 0 because (f b) ≥ 0 for r∗ ≥ 0, and (f b)′′ ≥ 0 for x ≤ 0. 1.4. INTEGRATED LOCAL ENERGY DECAY 61 Step 1d. (in the asymptotics, ρ ≥ ρ∗) We show directly H(r) > 0 for r∗ ≥ R∗ .= r∗(ρ = ρ∗) and ρ∗ chosen large enough. Let r∗ ≥ R∗, R∗ ≤ α then f b ≥ ∫ R∗ 0 (f b) ′ dr∗ = 1 α ∫ R∗−α−√α α − ( 1+ 1√ α ) 1 1 + t∗2 dt∗ ≥ R ∗ 5α2 (1.4.73) provided α ≥ 1, and of course f b ≤ 1 α arctan t∗|0−(1+ 1√ α ) ≤ π 2α . Thus H = (n− 1)(n− 3) 4 + [(n− 1)n 4 f b − C α22m 1 r ] 2m rn−2 − [(n− 1)3 4 f b − Cn α24m 1 r ]( 2m rn−2 )2 ≥ 1 α2 [(n− 1)n 4 R∗ 5 − C 2m 1 r ] 2m rn−2 − (n− 1) 3 4 π 2α ( 2m rn−2 )2 > 0 for R∗ (and consequently α) chosen large enough. Step 2. (1.4.74) Since ( 1− nm rn−2 ) f b ≥ 0 and F ≥ 0 for |x| ≥ α we need to show (n− 1)(f b)(1− nm rn−2 ) + F r3 ≥ 0 (1.4.74) for −α ≤ x ≤ α ⇐⇒ √α ≤ r∗ ≤ √α + 2α . In this whole region, in view of Prop. B.1, lim α→∞ r∗ r = 1 lim α→∞ ( 1− 2m rn−2 ) = lim α→∞ ( 1− nm rn−2 ) = 1 . n ≥ 4: Since f b(r∗) ≥ ∫ r∗ √ α 1 α2 + x2 dr∗ ≥ x+ α 2α2 (1.4.75) it suffices to show (n− 1)x+ α 2α2 + 1 2 x2 − α2 (x2 + α2)3 r3 ≥ 0 (1.4.76) which is implied by α− x n− 1 (x+ α + √ α)3 (x2 + α2)2 ≤ 1 . (1.4.77) 62 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES For −α ≤ x ≤ 0 (x+ α + √ α)3 ≤ α3(1 + 1√ α )3 ≤ 4 3 α3 for α large enough, thus α− x n− 1 (x+ α + √ α)3 (x2 + α2)2 ≤ 1 n− 1 2α α4 4 3 α3 ≤ 8 9 . (1.4.78) For 0 ≤ x ≤ α we have to show α n− 1 (x+ α + √ α)3 (x2 + α2)2 ≤ 1 . Since (x+ α + √ α)3 ≤ 2 32 (1 + 1√ α )3(x2 + α2) 3 2 we have for α large enough α n− 1 (x+ α + √ α)3 (x2 + α2)2 ≤ α n− 1 2 3 2 (1 + 1√ α )3 (x2 + α2) 1 2 ≤ 2 3 2 3 (1 + 1√ α )3 < 1 . (1.4.79) n = 3: We see that (1.4.78) and (1.4.79) fail in the case n = 3, as a consequence of which also (1.4.77) fails to hold. In the case n = 3, we have to use a better approximation of (1.4.75), see [21] for details. Note also that in view of (1.4.77) the positivity property (1.4.74) is “easily” satisfied for large values of n, which indicates that there may be yet another simplified proof in higher dimensions. Given the strict inequalities proven in Step 2 of the proof of Prop. 1.28 for α chosen large enough we can keep a fraction of the manifestly nonnegative |∇/Ωiφ|2 term in (1.4.69). Furthermore we have obtained control on the |∇/ φ|2 term from (1.4.71). Corollary 1.29. Let φ ∈ H2(S) be a solution of the wave equation (1.1.1). Then there exists a constant C(n,m) and a current K such that∫ S { 1 rn ( ∂φ ∂r∗ )2 + 1 rn+1 (∂φ ∂t )2 + r ( 1− nm rn−2 )2∣∣∇/ 2φ∣∣2 r2 ◦ γn−1 + r2 (1− 2m rn−2 )(1 + r ∗2)2 ∣∣∇/ φ∣∣2 r2 ◦ γn−1 } dµγ ≤ C(n,m) ∫ S K dµγ . (1.4.80) Proof. Set K = K(α) +Kaux and choose α large enough. Here we retrieve the time derivatives with the auxiliary current Kaux = ∇µJauxµ ; Jaux = JX aux,0; Xaux = f aux ∂ ∂r∗ , 1.4. INTEGRATED LOCAL ENERGY DECAY 63 where f aux = − 1 rn satisfies (f aux)′ + (n− 1)f aux r ( 1− 2m rn−2 ) = 1 rn+1 ( 1− 2m rn−2 ) ; for in view of (1.4.9) 1 rn+1 (∂φ ∂t )2 ≤ 2Kaux + 3 1 rn+1 ∣∣∇/ φ∣∣2 r2 ◦ γn−1 . 1.4.4 Boundary terms In this section we first prove Prop. 1.11 and then a refinement thereof for finite regions, which requires to estimate the boundary terms of the currents introduced in Section 1.4.2 and 1.4.3. Proof of Prop. 1.11 We can now combine our ealier results Cor. 1.25 and Cor. 1.29 to prove the integrated local energy decay estimate (1.4.4); note that there is no restriction on the spherical harmonic number, and that no commutation with angular momentum operators is required. Proof of Prop. 1.11. Write φ = π n−2√2m. Then there exists a constant C(n,m,R), such that∫ RPτ2τ1 {( ∂φ ∂r∗ )2 + (∂φ ∂t )2 + ( 1− 2m rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 } dµg ≤ C(n,m,R) ∫ Σ τ2 τ1 ( JT (φ) + JT (T · φ), n ) (1.4.107) for any τ2 > τ1. In view of the remarks above the proof of Prop. 1.32 is of course identical to the proof of Prop. 1.11 given in Section 1.4.4 by replacing the unbounded domain R∞r0,r1(2τ1 +R∗) by the bounded domain RPτ2τ1 ∪ RD\ τ2τ1 . However, this estimate does not include the zeroth order term, which we have covered seperately in Prop. 1.14. Proposition 1.33 (Refinement for zeroth order terms on timelike boundaries). Let φ be solution of the wave equation (1.1.1), and R > n−2 √ 8nm. Then there is a constant 74 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES C(n,m,R) such that∫ 2τ+R∗ 2τ ′+R∗ dt ∫ Sn−1 dµ◦ γn−1 φ2|r=R ≤ ≤ C(n,m,R) {∫ 2τ+R∗ 2τ ′+R∗ dt ∫ Sn−1 dµ◦ γn−1 {( ∂φ ∂r∗ )2 + ∣∣∇/ φ∣∣2}∣∣ r=R + ∫ Στ τ ′ ( JT (φ), n ) + ∫ Sn−1 dµ◦ γn−1 rn−2φ2|(τ ′,R∗+τ) } (1.4.108) for all τ ′ < τ . The proof remains the same as for Prop. 1.14 on page 40 with the exception that we consider the energy identity for JX,1 on RD\ ττ ′ in place of RDττ ′ and use Prop. B.9 instead of Prop. B.5. 1.5 The Decay Argument We will here prove energy decay for solutions to the wave equation and higher order energy decay of their time derivatives in the interior based on the integrated local energy decay statements of Section 1.4, following the new physical-space approach to decay of [22]. Remark 1.34. Instead one could use the conformal Morawetz vectorfield Z = u∗2 ∂ ∂u∗ + v∗2 ∂ ∂v∗ to prove energy decay of solutions to the wave equation with a rate corresponding to the weights in Z; this method predates the approach followed here, but is included for completeness in Section 1.5.4 (see also [42]). Similary the use of the scaling vectorfield S = v∗ ∂ ∂v∗ + u∗ ∂ ∂u∗ should provide an alternative approach to prove higher order energy decay [36]. Here however, we shall avoid the use of multipliers with weights in t. 1.5.1 Uniform Boundedness A preliminary feature of the solutions to the wave equation (1.1.1) that is necessary to employ the decay mechanism of [22] is the uniform boundedness of their (nondegenerate) energy; this is a consequence of the conservation of the degenerate energy associated to the multiplier T , and the redshift effect of Section 1.3, which allows us to control the nondegenerate energy on the horizon. 1.5. THE DECAY ARGUMENT 75 r = R Σ′ Σ Σ′τ Σττ ′ N Figure 1.7: The construction of the surfaces Σ′τ from Σ. Let Σ be a (spherically symmetric) spacelike hypersurface in M, Σ′ = Σ ∩ {r ≤ R} and N the outgoing null hypersurface emerging from ∂Σ′; moreover let Στ = ϕτ ( (Σ′ ∪N ) ∩ D ) and Σ′τ = Στ ∩ {r ≤ R} , Σττ ′ = Στ ′ ∩ J−(Σ′τ ) . Proposition 1.35 (Uniform Boundedness). Let φ be a solution of the wave equation (1.1.1) with initial data on Σ0, then there exists a constant C(Σ0) such that∫ Σ′τ ( JN(φ), n ) ≤ C ∫ Στ0 ( JN (φ), n ) (τ > 0) . (1.5.1) Proof. One can proceed in analogy to the local observer’s energy estimate of [17]; indeed, from the energy identity for N on the domain R(τ ′, τ) = ∪τ ′≤τ≤τΣττ it follows∫ Σ′τ (JN , n) + ∫ R(τ ′,τ) KN ≤ ∫ Στ τ ′ (JN , n) (1.5.2) since (JN , nH) ≥ 0, and (JN , nN ) ≥ 0. By Prop. 1.9, namely the redshift effect, KN is bounded from below by (JN , n) near the horizon, and from above by (JT , n) away from the horizon; since also the lapse of the foliation of R is bounded from above and below we conclude that there are constants 0 < b < B only depending on Σ and N such that∫ Σ′τ (JN , n) + b ∫ τ τ ′ dτ ∫ Σττ (JN , n) ≤ B ∫ τ τ ′ dτ ∫ Σττ (JT , n) + ∫ Στ τ ′ (JN , n) ≤ ≤ B(τ − τ ′) ∫ Στ τ ′ (JT , n) + ∫ Στ τ ′ (JN , n) (1.5.3) 76 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES where in the last step we have used the energy identity for T on R(τ ′, τ ) and KT = 0. Thus the desired energy bound follows from the elementary Lemma 1.36. Lemma 1.36. Let f : R → R be a nonnegative function, f ≥ 0, such that for all t1 ≤ t2 and two positive constants 0 < c < C f(t2) + c ∫ t2 t1 f(t) dt ≤ C(t2 − t1) + f(t1) , then f(t2) ≤ f(t1) + C c (t2 ≥ t1) . Proof. Define F (t2, t1) . = ∫ t2 t1 ( c f(t)− B) dt then f(t2) + F (t2, t1) ≤ f(t1) . Consider − d dt ( F (t2, t)e −c(t2−t) ) = − ( Ft1 + c F ) e−c(t2−t) ≥ c(f(t2)− C c ) e−b(t2−t) because Ft1 = − ( c f(t1)− B ) . Upon integrating on [t1, t2] f(t2)− C c ≤ 1 ec(t2−t1) − 1F (t2, t1) we infer with F (t2, t1) ≤ f(t1)− f(t2) that f(t2) ≤ f(t1)e−c(t2−t1) + C c ( 1− e−c(t2−t1) ) ≤ f(t1) + C c . 1.5.2 Energy decay In this Section we prove quadratic decay of the nondegenerate energy. Let Στ0 . = ∂−R∞r0,R(t0) τ0 = 1 2 (t0 − R∗) (1.5.4) with R > n−2 √ 8nm, t0 > 0 and r0 . = r (N) 0 according to Prop. 1.7. Proposition 1.37 (Energy decay). Let φ be a solution of the wave equation (1.1.1) with initial data on Στ0 satisfying D . = ∫ ∞ τ0+R∗ dv ∫ Sn−1 dµ◦ γn−1 1∑ k=0 r2 (∂r n−12 ∂kt φ ∂v∗ )2∣∣∣ u=τ0 + ∫ Στ0 ( 2∑ k=0 JN(T k · φ), n ) <∞ , (1.5.5) 1.5. THE DECAY ARGUMENT 77 then there exists a constant C(n,m,R) such that∫ Στ ( JN (φ), n ) ≤ C D τ 2 (τ > τ0) . (1.5.6) The proof is based on a weighted energy inequality, derived from the energy identity for the current (1.5.8) on the domain RDτ2τ1 = { (u∗, v∗) : τ1 ≤ u∗ ≤ τ2, v∗ − u∗ ≥ R∗ } . (1.5.7) Weighted energy identity. Consider the current r Jµ (φ) = Tµν(ψ)V ν (1.5.8) where ψ = r n−1 2 φ (1.5.9) V = rq ∂ ∂v∗ , q = p+ 1− n , p ∈ {1, 2} . (1.5.10) This may also be viewed as the current to the multiplier vectorfield rp ∂ ∂v∗ , modified by the following terms: r Jµ (φ) = Tµν(φ) r p ( ∂ ∂v∗ )ν + (n− 1 2 )2 rp−2 ( 1− 2m rn−2 ) (∂µr)φ 2 + 1 2 n− 1 2 rp−1(∂µr) ∂φ2 ∂v∗ + 1 2 n− 1 2 rp−1 ( 1− 2m rn−2 )( ∂µ φ 2 ) − 1 2 (n− 1 2 )2 rp−2 ( 1− 2m rn−2 )( ∂ ∂v∗ ) µ φ2 − 1 2 n− 1 2 ( ∂ ∂v∗ ) µ rp−1 ∂φ2 ∂r∗ If gφ = 0 then we calculate gψ = − ( 1− 2m rn−2 )−1 ∂u∗∂v∗ψ + n− 1 r ∂ψ ∂r∗ + 1 r2 ◦ △/ n−1 ψ = n− 1 2 (n− 3 2 + n− 1 2 2m rn−2 ) 1 r2 ψ + n− 1 r ∂ ∂r∗ ψ . (1.5.11) So the wave equation for φ gφ = 0 is equivalent to the following equation for ψ: − ∂u∗∂v∗ψ + ( 1− 2m rn−2 ) 1 r2 ◦ △/ n−1 ψ − n− 1 2 (n− 3 2 + n− 1 2 2m rn−2 ) 1 r2 ( 1− 2m rn−2 ) ψ = 0 . (1.5.12) Now, r K (φ) = ∇µ r Jµ (φ) = g(ψ)V · ψ +KV (ψ) , (1.5.13) 78 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES where KV (ψ) = (V )πµν Tµν(ψ) . Since (V )πu∗u∗ = 2 q r q−1(1− 2m rn−2 )2 (V )πv∗v∗ = 0 (V )πu∗v∗ = − ( 1− 2m rn−2 ) rq−1 [ q + (n− q − 2) 2m rn−2 ] (1.5.14) (V )πaA = 0 (V )πAB = r q−1(1− 2m rn−2 ) gAB we find r K ·rn−1 = n− 1 4 (n− 3 2 + n− 1 2 2m rn−2 )rp r2 ∂ψ2 ∂v∗ + p 2 rp−1 ( ∂ψ ∂v∗ )2 + 1 2 rp−1 [ (2− p) + (p− n) 2m rn−2 ]∣∣∇/ψ∣∣2 r2 ◦ γn−1 . (1.5.15) One may integrate the first term by parts to obtain:∫ ∞ u∗+R∗ dv∗ r K ·rn−1 = n− 1 4 (n− 3 2 + n− 1 2 2m rn−2 )rp r2 ψ2 ∣∣∣∞ u∗+R∗ + ∫ ∞ u∗+R∗ dv∗ {[n− 1 4 (2− p)n− 3 2 + n− 1 2 (n− p) 2m rn−2 ]rp r3 ( 1− 2m rn−2 ) ψ2 + p 2 rp−1 ( ∂ψ ∂v∗ )2 + 1 2 rp−1 [ (2− p) + (p− n) 2m rn−2 ]∣∣∇/ψ∣∣2 r2 ◦ γn−1 } (1.5.16) We can now write down the energy identity for the current r J (see also Appendix B.2):∫ RDτ2τ1 r K dµg = ∫ ∂RDτ2τ1 ∗ r J Dropping the positive zeroth order terms, we obtain:∫ ∞ τ2+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 1 2 rp ( ∂ψ ∂v∗ )2|u∗=τ2 + ∫ τ2 τ1 du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × × { p 2 rp−1 ( ∂ψ ∂v∗ )2 + 1 2 rp−1 [ (2− p) + (p− n) 2m rn−2 ]∣∣∇/ψ∣∣2 r2 ◦ γn−1 } + ∫ τ2 τ1 du∗ ∫ Sn−1 dµ◦ γn−1 1 2 rp ∣∣∇/ψ∣∣2 r2 ◦ γn−1 |v∗→∞ ≤ ≤ (1− 2m Rn−2 )−1{∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 1 2 rp ( ∂ψ ∂v∗ )2|u∗=τ1 + ∫ 2τ2+R∗ 2τ1+R∗ dt ∫ Sn−1 dµ◦ γn−1 [1 4 rp ( ∂ψ ∂v∗ )2 + 1 4 rp ∣∣∇/ψ∣∣2 r2 ◦ γn−1 + n− 1 4 1 2 rp (n− 3 2 + n− 1 2 2m Rn−2 ) 1 r2 ψ2 ] |r=R } (1.5.17) 1.5. THE DECAY ARGUMENT 79 Note that the powers of r that appear in the bulk term are 1 less than those that appear in the boundary terms. This allows for a hierarchy of inequalities (1.5.17) for different values of p, the so called p-hierarchy. Proof of Prop. 1.37: In a first step the decay of the solutions at future null infinity will be deduced from the weighted energy inequality, and in a second step the continuation to the event horizon will be inferred from the redshift effect. Step 1. The p-hierarchy consists of two steps which exploits (1.5.17) first with p = 2, then with p = 1; but in a zeroth step we need to obtain control on the angular derivatives from (1.5.17) with p = 1: Since 1− (n− 1) 2m rn−2 > 1 2 (r > R) we have from the weighted energy inequality for p = 1 on the domain r ′ 0Dττ ′ for τ > τ ′ ≥ τ0 . = 1 2 (t0 −R∗),∫ τ τ ′ du∗ ∫ ∞ u∗+r′0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 1 4 ∣∣∇/ψ∣∣2 r2 ◦ γn−1 ≤ ≤ (1− 2m Rn−2 )−1 ∫ ∞ τ ′+r′0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 1 2 r ( ∂ψ ∂v∗ )2∣∣∣ u∗=τ ′ + C(n,m,R) ( 1− 2m Rn−2 )−1 ∫ Στ0 ( JT (φ) + JT (T · φ), n ) ; (1.5.18) here, we have estimated the boundary integrals as follows: Choose r′0 ∈ (R∗, R∗ + 1) such that∫ R∗+1 R∗ dr∗ ∫ ∞ t0+(r∗−R∗) dt ∫ Sn−1 dµ◦ γn−1 ( 1− 2m rn−2 ) rn−1× × { 1 rn ( ∂φ ∂r∗ )2 + 1 rn+1 (∂φ ∂t )2 + 1 r3 ( 1− 2m rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 } = ∫ ∞ t0+(r′0 ∗−R∗) dt ∫ Sn−1 dµ◦ γn−1 ( 1− 2m r′0 n−2 ) r′0 n−1× × { 1 r′0 n ( ∂φ ∂r∗ )2 + 1 r′0 n+1 (∂φ ∂t )2 + 1 r′0 3 ( 1− 2m r′0 n−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 } then∫ 2τ+r′0∗ 2τ ′+r′0 ∗ dt ∫ Sn−1 dµ◦ γn−1 [ 1 4 rp ( ∂ψ ∂v∗ )2 + 1 4 rp ∣∣∇/ψ∣∣2 r2 ◦ γn−1 + n− 1 4 1 2 rp (n− 3 2 + n− 1 2 2m rn−2 ) 1 r2 ψ2 ] |r=r′0 ≤ 80 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES ≤ ∫ 2τ+r′0∗ 2τ ′+r′0 ∗ dt ∫ Sn−1 dµ◦ γn−1 r′0 p−2 [ 1 2 (n− 1 2 )2 φ2 + 1 2 r′0 2( ∂φ ∂v∗ )2 + 1 4 r′0 2∣∣∇/ φ∣∣2 r2 ◦ γn−1 + n− 1 4 1 2 (n− 3 2 + n− 1 2 2m Rn−2 ) φ2 ]∣∣∣ r=r′0 rn−1 ≤ ≤ C(n,m,R) ∫ Στ0 ( JT (φ) + JT (T · φ), n ) because by (1.4.24) ∫ r′0∗+2τ r′0 ∗+2τ ′ dt ∫ Sn−1 dµ◦ γn−1 rn−1 [ 1 4 ( ∂φ ∂v∗ )2 + n− 1 (4r′0)2 ( 1− 2m rn−2 ) φ2 ]∣∣∣ r=r′0 ≤ ≤ ∫ r′0∗+2τ r′0 ∗+2τ ′ dt ∫ Sn−1 dµ◦ γn−1 rn−1× × [ 1 2 ( 1− 2m rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 + n− 1 2 ( 1− 2m rn−2 )( ∂φ ∂r∗ )2]∣∣∣ r=r′0 + C(n,m,R) ∫ Στ0 ( JT (φ), n ) and by Prop. 1.11 (and the choice of r′0):∫ ∞ t0+(r′0 ∗−R∗) dt ∫ Sn−1 dµ◦ γn−1 rn−1× × [ 1 r′0 n ( ∂φ ∂r∗ )2 + 1 r′0 3 ( 1− 2m r′0 n−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 ]∣∣∣ r=r′0 ≤ ≤ C(n,m) ∫ Στ0 ( JT (φ) + JT (T · φ), n ) . Note that for the use of (1.4.24) that with our choice of R (n− 3) + n 2m rn−2 − (n− 1)( 2m rn−2 )2 > 0 (r > R) . p = 2: For p = 2 (1.5.17) reads ∫ τ τ ′ du∗ ∫ ∞ u∗+r′0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 [ r ( ∂ψ ∂v∗ )2 − 1 2 r(n− 2) 2m rn−2 ∣∣∇/ψ∣∣2 r2 ◦ γn−1 ] ≤ ≤ (1− 2m Rn−2 )−1 ∫ ∞ τ ′+r′0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 1 2 r2 ( ∂ψ ∂v∗ )2|u∗=τ ′ + C(n,m,R) ∫ Στ0 ( JT (φ) + JT (T · φ), n ) . Thus, with the previous estimate (1.5.18),∫ τ τ ′ du∗ ∫ ∞ u∗+r′0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 r ( ∂ψ ∂v∗ )2 ≤ 1.5. THE DECAY ARGUMENT 81 ≤ C(n,m,R))−1{∫ ∞ τ ′+r′0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 1 2 r2 ( ∂ψ ∂v∗ )2|u∗=τ ′ + ∫ Στ0 ( JT (φ) + JT (T · φ), n )} . (1.5.19) Let us define τj+1 = 2τj (j ∈ N0) τ0 = 1 2 (t0 −R∗) , then there is a sequence (τ ′j)j∈N0 with τ ′ j ∈ (τj , τj+1) (j ∈ N0) such that∫ ∞ τ ′j+r ′ 0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 r ( ∂ψ ∂v∗ )2|u∗=τ ′j ≤ ≤ 1 τj C(n,m,R) [∫ ∞ τj+r′0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 r2 ( ∂ψ ∂v∗ )2|u∗=τj + ∫ Στ0 ( JT (φ) + JT (T · φ), n )] and again by (1.5.17)∫ ∞ τj+r′0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 1 2 r2 ( ∂ψ ∂v∗ )2|u∗=τj ≤ ≤ C(n,m,R) [∫ ∞ τ0+r′0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 r2 ( ∂ψ ∂v∗ )2|u∗=τ0 + ∫ Στ0 ( JT (φ) + JT (T · φ), n )] . Since 1 τj ≤ 1 τ ′j τj+1 τj = 2 τ ′j we have∫ ∞ τ ′j+r ′ 0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 r ( ∂ψ ∂v∗ )2|u∗=τ ′j ≤ ≤ C(n,m,R) τ ′j [∫ ∞ τ0+r′0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 r2 ( ∂ψ ∂v∗ )2|u∗=τ0 + ∫ Στ0 ( JT (φ) + JT (T · φ), n )] . (1.5.20) p = 1: In order to deal with the timelike boundary integrals analogously to the above choose r′′j ∗ ∈ (r′0∗, r′0∗ + 1) such that∫ 2τ ′j+1+r′′j ∗ 2τ ′j+r ′′ j ∗ dt ∫ Sn−1 dµ◦ γn−1 rn−1× × [ 1 rn ( ∂φ ∂r∗ )2 + 1 rn+1 (∂φ ∂t )2 + 1 r3 ( 1− 2m rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 ] |r=r′′j ≤ 82 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES ≤ C(n,m) ∫ Στ ′ j ( JT (φ) + JT (T · φ), n ) . Then, proceeding as before,∫ 2τ ′j+1+r′′j ∗ 2τ ′j+r ′′ j ∗ dt ∫ Sn−1 dµ◦ γn−1 × × [ 1 4 r ( ∂ψ ∂v∗ )2 + 1 4 r ∣∣∇/ψ∣∣2 r2 ◦ γn−1 + n− 1 4 1 2 r (n− 3 2 + n− 1 2 2m rn−2 ) 1 r2 ψ2 ] |r=r′′j ≤ ≤ C(n,m,R) ∫ Στ ′ j ( JT (φ) + JT (T · φ), n ) . (1.5.21) Now apply (1.5.17) to the region r ′′ jDτ ′ j+1 τ ′j to obtain: ∫ τ ′j+1 τ ′j du∗ ∫ ∞ u∗+r′′j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × × [ 1 2 ( ∂ψ ∂v∗ )2 + 1 4 ∣∣∇/ψ∣∣2 r2 ◦ γn−1 ] ≤ ≤ (1− 2m Rn−2 )−1 ∫ ∞ τ ′j+r ′′ j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 1 2 r ( ∂ψ ∂v∗ )2|u∗=τ ′j + C(n,m,R) ∫ Στ ′ j ( JT (φ) + JT (T · φ), n ) By virtue of the result (1.5.20) from the case p = 2, this yields∫ τ ′j+1 τ ′j du∗ ∫ ∞ u∗+r′′j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 [ 1 2 ( ∂ψ ∂v∗ )2 + 1 4 ∣∣∇/ψ∣∣2 r2 ◦ γn−1 ] ≤ ≤ C(n,m,R) τ ′j [∫ ∞ τ0+r′0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 r2 ( ∂ψ ∂v∗ )2|u∗=τ0 + ∫ Στ0 ( JT (φ) + JT (T · φ), n )] + C(n,m,R) ∫ Στ ′ j ( JT (φ) + JT (T · φ), n ) . (1.5.22) Step 2. Our aim is to prove decay for the non-degenerate energy. Let us first find an estimate for∫ τ ′j+1 τ ′j dτ ∫ Στ ( JN(φ), n ) = = ∫ τ ′j+1 τ ′j dτ ∫ Στ∩{r≤r′′j } ( JN(φ), n ) + ∫ τ ′j+1 τ ′j dτ ∫ Στ∩{r≥r′′j } ( JT (φ), n ) . The estimate of the first term is exactly the content of Cor. 1.13, and for the second term∫ τ ′j+1 τ ′j dτ ∫ Σ′τ∩{r≥r′′j } ( JT (φ), n ) = 1.5. THE DECAY ARGUMENT 83 = ∫ τ ′j+1 τ ′j du∗ ∫ ∞ u∗+r′′j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 rn−1 [ 1 2 ( ∂φ ∂v∗ )2 + 1 2 ( 1− 2m rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 ] we can use (1.5.22) once we have turned it into an estimate for the derivatives of φ. Note that∫ ∞ u∗+r′0 ∗ dv∗ ( ∂ψ ∂v∗ )2 = = ∫ ∞ u∗+r′0 ∗ dv∗ [ n− 1 2 ( 1− 2m rn−2 ) r n−3 2 ∂ ∂v∗ ( r n−1 2 φ2 ) + rn−1 ( ∂φ ∂v∗ )2] = −n− 1 2 ( 1− 2m rn−2 ) rn−2 φ2|v∗=u∗+r′0∗ + ∫ ∞ u∗+r′0 ∗ dv∗ { −n− 1 2 ( 1− 2m rn−2 )[ (n− 2)2m rn + n− 3 2 ( 1− 2m rn−2 ) 1 r2 ] φ2+ ( ∂φ ∂v∗ )2} rn−1 and by Lemma B.6∫ ∞ u∗+r′0 ∗ dv∗ 1 r2 φ2 rn−1 ≤ ≤ C(n,m) ∫ ∞ u∗+r′0 ∗ dv∗ ( ∂ψ ∂v∗ )2 + C(n,m) rn−1φ2|(u∗,v∗=u∗+r′0∗) . Thus ∫ ∞ u∗+r′0 ∗ dv∗ ( ∂φ ∂v∗ )2 rn−1 ≤ C(n,m,R) [ φ2|(u∗,u∗+r′0∗) + ∫ ∞ u∗+r′0 ∗ dv∗ ( ∂ψ ∂v∗ )2] , and finally in view of (1.5.21) ∫ τ ′j+1 τ ′j du∗ ∫ ∞ u∗+r′′j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 ( ∂φ ∂v∗ )2 rn−1 ≤ ≤ C(n,m,R) ∫ τ ′j+1 τ ′j du∗ ∫ ∞ u∗+r′′j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 ( ∂ψ ∂v∗ )2 + C(n,m,R) ∫ Στ ′ j ( JT (φ) + JT (T · φ), n ) . (1.5.23) Therefore, putting the estimates for the two terms back together, ∫ τ ′j+1 τ ′j dτ ∫ Στ ( JN (φ), n ) ≤ ≤ C(n,m,R) ∫ τ ′j+1 τ ′j du∗ ∫ ∞ u∗+r′′j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 {( ∂ψ ∂v∗ )2 + ∣∣∇/ψ∣∣2 r2 ◦ γn−1 } + C(n,m) ∫ Στ ′ j ( JN(φ) + JT (T · φ), n ) ≤ 84 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES ≤ C(n,m,R) τ ′j [∫ ∞ τ0+r′0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 r2 ( ∂ψ ∂v∗ )2|u∗=τ0 + ∫ Στ0 ( JT (φ) + JT (T · φ), n )] + C(n,m,R) ∫ Στ ′ j ( JN(φ) + JT (T · φ), n ) (1.5.24) where we have now used (1.5.22). The same inequality holds for τ ′j+2 in place of τ ′ j+1, by adding the inequalities corresponding to the intervals [τ ′j , τ ′ j+1] and [τ ′ j+1, τ ′ j+2] and using Prop. 1.35 for the last term. So there is a sequence (τ ′′j )j∈N τ ′′ j ∈ ( τ ′j , τ ′ j+2 ) such that ∫ τ ′j+2 τ ′j dτ ∫ Στ ( JN (φ), n ) ≥ τj+1 ∫ Στ ′′ j ( JN(φ), n ) and since 1 τj+1 ≤ 1 τ ′′j τj+3 τj+1 = 4 τ ′′j we have ∫ Στ ′′ j ( JN(φ), n ) ≤ 4 τ ′′j ∫ τ ′j+2 τj dτ ∫ Σ′τ ( JN (φ), n ) . (1.5.25) Now for any given τ > τ0 we may choose j∗ = max{j ∈ N : τ ′′j ≤ τ} so that by (1.5.1) ∫ Στ ( JN (φ), n ) ≤ C ∫ Στ ′′ j∗ ( JN(φ), n ) with τ τ ′′ j∗ ≤ τ ′′ j∗+1 τ ′′ j∗ ≤ 24. In particular we may estimate the last integral in (1.5.24) ∫ Στ ′ j ( JN(φ) + JT (T · φ), n ) ≤ C τ ′j ∫ τ ′j+1 τ ′j−1 dτ ∫ Στ ( JN(φ) + JN(T · φ), n ) to see that in fact we have∫ τ ′j+2 τ ′j dτ ∫ Στ ( JN(φ), n ) ≤ ≤ C(n,m,R) τ ′j [∫ ∞ τ0+r′0 ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 { r2 (∂r n−12 φ ∂v∗ )2 + r2 (∂r n−12 ∂φ ∂t ∂v∗ )} + ∫ Στ0 ( JN (φ) + JN (T · φ) + JT (T 2φ), n )] . (1.5.26) Again with the sequence (τj ′′)j∈N∫ Στ ′′ j ( JN(φ), n ) ≤ 1 τj+1 ∫ τ ′j+2 τ ′j dτ ∫ Στ ( JN(φ), n ) 1.5. THE DECAY ARGUMENT 85 and since 1 τj+1 1 τ ′j ≤ 25 τ ′′j 2 we obtain by virtue of Prop. 1.35 our final result: ∫ Στ ( JN (φ), n ) ≤ ≤ C(n,m,R) τ 2 [∫ ∞ τ0+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 { r2 (∂r n−12 φ ∂v∗ )2 + r2 (∂r n−12 ∂φ ∂t ∂v∗ )2} + ∫ Στ0 ( JN(φ) + JN(T · φ) + JT (T 2 · φ), n )] (1.5.27) 1.5.3 Improved interior decay of the first order energy In this Section we prove an energy estimate for the first order energy which improves the decay rate as compared to Prop. 1.37 in a bounded radial region. Remark 1.38. The argument largely depends on the asymptotic properties of the space- time, and is similar and slightly easier in Minkowski space, see Appendix A. Proposition 1.39 (Improved interior first order energy decay). Let 0 < δ < 1 2 , R > n−2 √ 8nm δ , and let φ be a solution of the wave equation (1.1.1) with initial data on Στ1 (τ1 > 0) satisfying D . = ∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { 1∑ k=0 r4−δ (∂(T k · χ) ∂v∗ )2 + 4∑ k=0 r2 (∂(T k · ψ) ∂v∗ )2 + 3∑ k=0 n(n−1) 2∑ i=1 r2 (∂T kΩiψ ∂v∗ )2}∣∣∣ u∗=τ1 + ∫ Στ1 ( 5∑ k=0 JN (T k · φ) + 4∑ k=0 n(n−1) 2∑ i=1 JN(T kΩiφ), n ) <∞ . (1.5.28) Then there exists a constant C(n,m, δ, R) such that∫ Σ′τ ( JN (T · φ), n ) ≤ C D τ 4−2δ (τ > τ1) (1.5.29) where Σ′τ = Στ ∩ {r ≤ R}. In addition to the weighted energy identity arising from the multiplier rp ∂ ∂v∗ that was used to prove Prop. 1.37 we will here also use a commutation with ∂ ∂v∗ to obtain the energy decay for ∂φ ∂t of Prop. 1.39. 86 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Weighted energy and commutation. Consider the current v Jµ (φ) . = Tµν(χ)V ν , (1.5.30) where now χ . = ∂v∗ψ = ∂(r n−1 2 φ) ∂v∗ V = rq ∂ ∂v∗ (1.5.31) q = p− (n− 1) 2 < p < 4 δ = 4− p . Notation. To make the dependence on p explicit, we denote by v Kp (φ) . = ∇µ vJµ (φ) . (1.5.32) The error terms for v K arise from the fact that χ is not a solution of (1.1.1); here, similarly to (1.5.11), we find: gχ = −n− 1 2r3 { (n− 3) + n 2m rn−2 − (n− 1)2 ( 2m rn−2 )2} ψ + n− 1 4r2 [ (n− 3) + (n− 1) 2m rn−2 ] χ+ 1 r [ 2− n 2m rn−2 ] △/ ψ + n− 1 r ∂χ ∂r∗ (1.5.33) Hence v Kp (φ) =(χ)V · χ+KV (χ) = 1 2 p rq−1 ( ∂χ ∂v∗ )2 + 1 2 [ (2− p)− (n− p) 2m rn−2 ] rq−1 ∣∣∇/χ∣∣2 (1.5.34) − n− 1 2 rq−3 [ (n− 3) + n 2m rn−2 − (n− 1)2 ( 2m rn−2 )2] ψ ∂2ψ ∂v∗2 (1.5.35) + n− 1 8 rq−2 [ (n− 3) + (n− 1) 2m rn−2 ]∂χ2 ∂v∗ (1.5.36) + rq−1 [ 2− n 2m rn−2 ](△/ ψ)( ∂χ ∂v∗ ) (1.5.37) which is not positive definite. However, we have 1 4 p rp−1 ( ∂χ ∂v∗ )2 ≤ v Kp (φ) · rn−1 + 1 2 [ (p− 2) + (n− p) 2m rn−2 ] rp−1 ∣∣∇/χ∣∣2 + (n− 1)2(n− 2)2 2 r(p−2)−1 1 r2 ψ2 + 4 p r(p−2)−1r2 ( △/ ψ )2 − n− 1 8 rp−2 [ (n− 3) + (n− 1) 2m rn−2 ]∂χ2 ∂v∗ , (1.5.38) where we have used that n− 2 > n− 3 + n 2m rn−2 − (n− 1)2 ( 2m rn−2 )2 ≥ n− 3 (1.5.39) 1.5. THE DECAY ARGUMENT 87 (is decreasing) on r > n−2 √ 4nm. The key insight here is that we are able to control all other terms on the right hand side of (1.5.38) by the current r J of Section 1.5.2 with p− 2 in the role of p, i.e.∫ τ2 τ1 du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 rp−1 ( ∂χ ∂v∗ )2 ≤ ≤ C(n,m, δ, p, R) ∫ RDτ2τ1 { v Kp (φ)+ r Kp−2 (φ) + n(n−1) 2∑ i=1 r Kp−2 (Ωiφ) } +C(n,m, δ, p, R) ∫ 2τ2+R∗ 2τ1+R∗ dt ∫ Sn−1 dµ◦ γn−1 { ψ2+ ( ∂ψ ∂v∗ )2 + n(n−1) 2∑ i=1 ( Ωiψ )2 + n(n−1) 2∑ i=1 ∣∣∇/Ωiψ∣∣2}|r=R . (1.5.40) Indeed, the first term |∇/ ∂v∗ψ|2 can be integrated by parts twice (such that we can absorb the resulting ∂v∗χ term in the left hand side):∫ τ2 τ1 du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 rp−1 ∣∣∇/χ∣∣2 = = − ∫ τ2 τ1 du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 rp−1△/ ∂v∗ψ · ∂v∗ψ = − ∫ τ2 τ1 du∗ ∫ Sn−1 dµ◦ γn−1 rp−1△/ ψ ∂ψ ∂v∗ ∣∣∣∞ u∗+R∗ + ∫ τ2 τ1 du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 { (p− 1)rp−2(1− 2m rn−2 ) (△/ ψ)( ∂ψ ∂v∗ ) + rp−1(△/ ψ) ∂χ ∂v∗ + rp−1 2 r ( 1− 2m rn−2 ) (△/ ψ) ∂ψ ∂v∗ } ≤ ≤ ∫ τ2 τ1 du∗ ∫ Sn−1 dµ◦ γn−1 rp−1 (△/ ψ) ( ∂ψ ∂v∗ ) ∣∣∣ v∗=u∗+R∗ + ∫ τ2 τ1 du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 {( p− 1 + n− 2 p + 2 ) r(p−2)−1 ( △/ ψ )2 r2 + (p− 1 + 2)r(p−2)−1 ( ∂ψ ∂v∗ )2 + 1 2 p 4 2 n− 2r p−1 ( ∂χ ∂v∗ )2} (1.5.41) The second term in (1.5.38) is controlled by the Hardy inequality 1 2 ∫ ∞ u∗+R∗ dv∗ r(p−2)−1 1 r2 ψ2 ≤ 1 4− p 1 R4−p 1 1− 2m Rn−2 ψ2|(u∗,u∗+R∗) + 2 (4− p)2(1− 2m Rn−2 ) 2 ∫ ∞ u∗+R∗ dv∗ r(p−2)−1 ( ∂ψ ∂v∗ )2 , (1.5.42) and the third term simply by the following commutation with Ωi: Lemma 1.40. For any function φ ∈ H2(Sr) we have △/ r2 ◦ γn−1 φ ∈ L2(Sr), and there exists a constant C > 0 such that it holds∫ Sn−1 ( △/ ψ )2 r2 dµ◦ γn−1 ≤ C ∫ Sn−1 {n(n−1)2∑ i=1 ∣∣∣∇/ (Ωiψ)∣∣∣2 + ∣∣∣∇/ψ∣∣∣2} dµ◦γn−1 . (1.5.43) 88 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES The last term in (1.5.38) we can rearrange as follows: − n− 1 8 rp−2 [ n− 3 + (n− 1) 2m rn−2 ]∂χ2 ∂v∗ = = − ∂ ∂v∗ { n− 1 8 rp−2 [ (n− 3) + (n− 1) 2m rn−2 ]( ∂ψ ∂v∗ )2} + n− 1 8 r(p−2)−1 [ (p− 2)(n− 3) + (n− 1) ( (p− 2) + (n− 2) ) 2m rn−2 ]( 1− 2m rn−2 )( ∂ψ ∂v∗ )2 (1.5.44) Therefore (see also Appendix B.2)∫ τ2 τ1 du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × 1 8 p rp−1 ( ∂χ ∂v∗ )2 ≤ ≤ 1 2 1 1− 2m Rn−2 ∫ RDτ2τ1 v Kp (φ) dµg + C(n, p, δ, R) ∫ τ2 τ1 du∗ ∫ Sn−1 dµ◦ γn−1 { rp−2 ( △/ ψ )2 r2 + rp−2 ( ∂ψ ∂v∗ )2 + ψ2 } |v∗=u∗+R∗ + C(p, n, δ, R) ∫ τ2 τ1 du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × × { r(p−2)−1 n(n−1) 2∑ i=1 ∣∣∇/Ωiψ∣∣2 + r(p−2)−1∣∣∇/ψ∣∣2 + r(p−2)−1( ∂ψ ∂v∗ )2} − n− 1 8 ∫ τ2 τ1 du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × rp−2 [ n+ 3 + (n− 1) 2m rn−2 ]∂χ2 ∂v∗ . (1.5.45) Now, recall (1.5.15), and note that δ − (n− (2− δ)) 2m rn−2 > δ 2 (r > n−2 √ 4nm δ ) , (1.5.46) to see that r(p−2)−1 n(n−1) 2∑ i=1 ∣∣∇/ r n−12 Ωiφ∣∣2 ≤ 4 δ n(n−1) 2∑ i=1 r Kp−2 (Ωiφ)rn−1 − 4 δ n(n−1) 2∑ i=1 n− 1 4 [n− 3 2 + n− 1 2 2m rn−2 ] r(p−2)−2 ∂ ( r n−1 2 Ωiφ )2 ∂v∗ . (1.5.47) So∫ τ2 τ1 du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × p rp−1 ( ∂χ ∂v∗ )2 ≤ ≤ C(n,m, δ, p, R) ∫ RDτ2τ1 { v Kp (φ)+ r Kp−2 (φ) + n(n−1) 2∑ i=1 r Kp−2 (Ωiφ) } − C(n,m, δ, p) ∫ τ2 τ1 du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × n− 1 8 [ n− 3 + (n− 1) 2m rn−2 ] × × { rp−2 r2 ∂ψ2 ∂v∗ + rp−2 r2 n(n−1) 2∑ i=1 ∂ ( Ωiψ )2 ∂v∗ + rp r2 ∂χ2 ∂v∗ } 1.5. THE DECAY ARGUMENT 89 + C(n,m, δ, p, R) ∫ 2τ2+R∗ 2τ1+R∗ dt ∫ Sn−1 dµ◦ γn−1 {n(n−1) 2∑ i=1 ∣∣∇/Ωiφ∣∣2 + ∣∣∇/ φ∣∣2 + ( ∂ψ ∂v∗ )2 + ψ2 } |r=R (1.5.48) which upon integration by parts yields (1.5.40); note that the ∂v∗ψ 2 and ∂v∗(Ωiψ) 2 terms generate boundary terms at infinity and zeroth order bulk terms with the right sign by (1.5.16) while the ∂v∗χ 2 is reduced to a (∂v∗ψ) 2 term by (1.5.44). By virtue of Stokes’ theorem (B.5) and in view of (B.6) we conclude ∫ ∞ τ2+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { rp ( ∂χ ∂v∗ )2 + rp−2 ( ∂ψ ∂v∗ )2 + n(n−1) 2∑ i=1 rp−2 (∂Ωiψ ∂v∗ )2}∣∣∣ u∗=τ2 + ∫ τ2 τ1 du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × rp−1 ( ∂χ ∂v∗ )2 ≤ ≤ C(n,m, δ, p, R) ∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × × { rp ( ∂χ ∂v∗ )2 + rp−2 ( ∂ψ ∂v∗ )2 + n(n−1) 2∑ i=1 rp−2 (∂Ωiψ ∂v∗ )2}∣∣∣ u∗=τ1 + C(n,m, δ, p, R) ∫ 2τ2+R∗ 2τ1+R∗ dt ∫ Sn−1 dµ◦ γn−1 { ψ2 + ( ∂ψ ∂v∗ )2 + ( ∂2ψ ∂v∗2 )2 + n(n−1) 2∑ i=1 [( Ωiψ )2 + (∂Ωiψ ∂v∗ )2] + ∣∣∇/χ∣∣2 + ∣∣∇/ψ∣∣2 + n(n−1)2∑ i=1 ∣∣∇/Ωiψ∣∣2}∣∣∣ r=R . (1.5.49) Proof of Prop. 1.39. We shall use this weighted energy inequality for χ to proceed in a hierarchy of four steps. p = 4− δ: Let τ1 > 0, and τj+1 = 2τj (j ∈ N). In a first step we use (1.5.49) with p = 4 − δ and (1.5.17) with p = 2 as an estimate for the spacetime integral of ∂v∗χ, ∂v∗ψ, and ∂v∗(Ωjψ) on RDτj+1τj , and in a second step as an estimate for the corresponding integral on the future boundary of RDτjτ1 : ∫ τj+1 τj du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r3−δ ( ∂χ ∂v∗ )2 + r ( ∂ψ ∂v∗ )2 + n(n−1) 2∑ i=1 r (∂Ωiψ ∂v∗ )2} ≤ ≤ C(n,m, δ, R) ∫ ∞ τj+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r4−δ ( ∂χ ∂v∗ )2 +r2 ( ∂ψ ∂v∗ )2 + n(n−1) 2∑ i=1 r2 (∂Ωjψ ∂v∗ )2}∣∣∣ u∗=τj + C(n,m, δ, R) ∫ 2τj+1+R∗ 2τj+R∗ dt ∫ Sn−1 dµ◦ γn−1 × { ψ2 + ( ∂ψ ∂v∗ )2 + ( ∂2ψ ∂v∗2 )2 + ∣∣∇/ψ∣∣2 + ∣∣∇/ ∂ψ ∂v∗ ∣∣2 + n(n−1)2∑ i=1 [( Ωiψ )2 + (∂Ωiψ ∂v∗ )2 + ∣∣∇/Ωiψ∣∣2]}∣∣∣ r=R ≤ 90 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES ≤ C(n,m, δ, R) ∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r4−δ ( ∂χ ∂v∗ )2 +r2 ( ∂ψ ∂v∗ )2 + n(n−1) 2∑ i=1 r2 (∂Ωiψ ∂v∗ )2}∣∣∣ u∗=τ1 + C(n,m, δ, R) ∫ 2τj+1+R∗ 2τ1+R∗ dt ∫ Sn−1 dµ◦ γn−1 × { ψ2 + ( ∂ψ ∂v∗ )2 + ( ∂2ψ ∂v∗2 )2 + ∣∣∇/ψ∣∣2 + ∣∣∇/ ∂ψ ∂v∗ ∣∣2 + n(n−1)2∑ i=1 [( Ωiψ )2 + (∂Ωiψ ∂v∗ )2 + ∣∣∇/Ωiψ∣∣2]}∣∣∣ r=R (1.5.50) Thus by the mean value theorem of integration we obtain a sequence τ ′j ∈ (τj , τj+1) (j ∈ N) such that the corresponding integral from the left hand side on u∗ = τ ′j is bounded by τ−1j × the right hand side of (1.5.50). p = 3− δ: Next we shall use (1.5.49) with p = 3− δ on RjDτ ′ 2j+1 τ ′2j−1 , (with R∗j ∈ (R∗, R∗ + 1) (j ∈ N) chosen appropriately below). However, the quantity we are actually interested in is not ∂v∗χ, but rather(∂r n−12 T · φ ∂v∗ )2 = (∂T · r n−12 φ ∂v∗ )2 = (1 2 ∂2ψ ∂v∗2 + 1 2 ∂2ψ ∂u∗∂v∗ )2 = (1.5.12) = (1 2 ∂2ψ ∂v∗2 + 1 2 ( 1− 2m rn−2 )△/ ψ − 1 2 n− 1 2 (n− 3 2 + n− 1 2 2m rn−2 ) 1 r2 ( 1− 2m rn−2 ) ψ )2 ≤ ( ∂2ψ ∂v∗2 )2 + ( △/ ψ )2 + n− 1 4 2(n− 2) 1 r4 ψ2 . (1.5.51) Using the simple Hardy inequality 1 2 ∫ ∞ u∗+R∗ dv∗ r2−δ 1 r4 ψ2 ≤ ≤ 1 1− 2m Rn−2 1 r1+δ ψ2(u∗, u∗ +R∗) + 2( 1− 2m Rn−2 )2 ∫ ∞ u∗+R∗ dv∗ 1 rδ ( ∂ψ ∂v∗ )2 (1.5.52) and again the commutation introduced in Lemma 1.40 we obtain∫ τ ′2j+1 τ ′2j−1 du∗ ∫ ∞ u∗+Rj∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × r2−δ (∂r n−12 T · φ ∂v∗ )2 ≤ ≤ ∫ τ ′2j+1 τ ′2j−1 du∗ ∫ ∞ u∗+Rj∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r2−δ ( ∂χ ∂v∗ )2 + C rδ n(n−1) 2∑ i=1 ∣∣∇/Ωiψ∣∣2 + C rδ ∣∣∇/ψ∣∣2 + (n− 1)(n− 2) 2 2( 1− 2m Rn−2 )2 r−δ( ∂ψ∂v∗)2 } + 1 1− 2m Rn−2 ∫ 2τ ′2j+1+R∗j 2τ ′2j−1+R ∗ j dt ∫ Sn−1 dµ◦ γn−1 × { 1 r1+δ ψ2 }∣∣∣ r=Rj ≤ ≤ C(n,m, δ) ∫ τ ′2j+1 τ ′2j−1 du∗ ∫ ∞ u∗+Rj∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r2−δ ( ∂χ ∂v∗ )2 + r K1−δ (φ)rn−1 + n(n−1) 2∑ i=1 r K1−δ (Ωiφ)rn−1 } 1.5. THE DECAY ARGUMENT 91 + C(n,m, δ) ∫ 2τ ′2j+1+R∗j 2τ ′2j−1+R ∗ j dt ∫ Sn−1 dµ◦ γn−1 × { ψ2 + n(n−1) 2∑ i=1 ( Ωiψ )2}∣∣∣ r=Rj (1.5.53) where in the last step we have again used (1.5.16). Furthermore, by now applying (1.5.49) with p = 3− δ,∫ τ ′2j+1 τ ′2j−1 du∗ ∫ ∞ u∗+Rj∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × r2−δ (∂r n−12 T · φ ∂v∗ )2 ≤ ≤ C(n,m, δ, R) ∫ ∞ τ ′2j−1+R ∗ j dv∗ ∫ Sn−1 dµ◦ γn−1 × { r3−δ ( ∂χ ∂v∗ )2 + r ( ∂ψ ∂v∗ )2 + n(n−1) 2∑ i=1 r (∂Ωiψ ∂v∗ )2}∣∣∣ u∗=τ ′2j−1 + C(n,m, δ, R) ∫ 2τ ′2j+1+R∗j 2τ ′2j−1+R ∗ j dt ∫ Sn−1 dµ◦ γn−1 × { ψ2 + ( ∂ψ ∂v∗ )2 + ( ∂2ψ ∂v∗2 )2 + ∣∣∇/ψ∣∣2 + ∣∣∇/ ∂ψ ∂v∗ ∣∣2 + n(n−1)2∑ i=1 [( Ωiψ )2 + (∂Ωiψ ∂v∗ )2 + ∣∣∇/Ωiψ∣∣2]}∣∣∣ r=Rj , (1.5.54) we obtain a sequence τ ′′j ∈ (τ ′2j−1, τ ′2j+1) (j ∈ N) such that in view of the previous step:∫ ∞ τ ′′j +R ∗ j dv∗ ∫ Sn−1 dµ◦ γn−1 × { r2−δ (∂(r n−12 T · φ) ∂v∗ )2}∣∣∣ u∗=τ ′′j ≤ ≤ C(n,m, δ, R) τ2j τ2j−1 ∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r4−δ ( ∂χ ∂v∗ )2 + r2 ( ∂ψ ∂v∗ )2 + r2 n(n−1) 2∑ i=1 (∂Ωiψ ∂v∗ )2}∣∣∣ u∗=τ1 + C(n,m, δ, R) τ2j τ2j−1 ∫ 2τ2j+1+R∗ 2τ1+R∗ dt ∫ Sn−1 dµ◦ γn−1 × { ψ2 + ( ∂ψ ∂v∗ )2 + ( ∂2ψ ∂v∗2 )2 + ∣∣∇/ψ∣∣2 + ∣∣∇/ ∂ψ ∂v∗ ∣∣2 + n(n−1)2∑ i=1 [( Ωiψ )2 + (∂Ωiψ ∂v∗ )2 + ∣∣∇/Ωiψ∣∣2}∣∣∣ r=R + C(n,m, δ, R) ∫ 2τ ′2j+1+R∗j 2τ ′2j−1+R ∗ j dt ∫ Sn−1 dµ◦ γn−1 × { ψ2 + ( ∂ψ ∂v∗ )2 + ( ∂2ψ ∂v∗2 )2 + ∣∣∇/ψ∣∣2 + ∣∣∇/ ∂ψ ∂v∗ ∣∣2 + n(n−1)2∑ i=1 [( Ωiψ )2 + (∂Ωiψ ∂v∗ )2 + ∣∣∇/Ωiψ∣∣2}∣∣∣ r=Rj (1.5.55) Now, by writing out the derivatives of ψ = r n−1 2 φ, and using (1.5.12), we calculate that∫ Sn−1 dµ◦ γn−1 { ψ2 + ( ∂ψ ∂v∗ )2 + ( ∂2ψ ∂v∗2 )2 + ∣∣∇/ψ∣∣2 + ∣∣∇/ ∂ψ ∂v∗ ∣∣2 + n(n−1)2∑ i=1 [( Ωiψ )2 + (∂Ωiψ ∂v∗ )2 + ∣∣∇/Ωiψ∣∣2]}∣∣∣ r=R ≤ 92 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES ≤ C(R) ∫ Sn−1 dµ◦ γn−1 { φ2 + ( ∂φ ∂v∗ )2 + ( ∂φ ∂u∗ )2 + (∂T · φ ∂v∗ )2 + ∣∣∇/ φ∣∣2 + n(n−1)2∑ i=1 [∣∣∇/Ωiφ∣∣2 + (∂Ωiφ ∂v∗ )2]}∣∣∣ r=R ; (1.5.56) by applying Prop. 1.11 first to the domain r1Dτ2j+1τ1 ⊂ R∞r0,r1(2τ1+ r∗1) where r1 > n−2 √ 4nm δ to fix the radius R and then to the domain r(r ∗=R∗+1)Dτ ′ 2j+1 τ ′2j−1 \RDτ ′ 2j+1 τ ′2j−1 ⊂ R∞r0,R(2τ ′2j−1+R∗) to fix the radii Rj (j ∈ N) by using the mean value theorem for the integration in r∗ this yields (see also Appendix B.2)∫ ∞ τ ′′j +R ∗ j dv∗ ∫ Sn−1 dµ◦ γn−1 × { r2−δ (∂r n−12 T · φ ∂v∗ )2}∣∣∣ u∗=τ ′′j ≤ ≤ C(n,m, δ, R) (τ ′′j )2 {∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r4−δ ( ∂χ ∂v∗ )2 + r2 ( ∂ψ ∂v∗ )2 + r2 n(n−1) 2∑ i=1 (∂Ωiψ ∂v∗ )2}∣∣∣ u∗=τ1 + ∫ Στ1 ( JT (φ) + JT (T · φ) + JT (T 2 · φ) + n(n−1) 2∑ i=1 [ JT (Ωiφ) + J T (T · Ωiφ) ] , n )} + C(n,m, δ, R) ∫ Στ2j−1 ( JT (φ) + JT (T · φ) + JT (T 2 · φ) + n(n−1) 2∑ i=1 [ JT (Ωiφ) + J T (T · Ωiφ) ] , n ) . (1.5.57) Therefore, by Prop. 1.37:∫ ∞ τ ′′j +R ∗ j dv∗ ∫ Sn−1 dµ◦ γn−1 × { r2−δ (∂r n−12 T · φ ∂v∗ )2}∣∣∣ u∗=τ ′′j ≤ ≤ C(n,m, δ, R) (τ ′′j )2 {∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r4−δ ( ∂χ ∂v∗ )2 + 3∑ k=0 r2 (∂T k · ψ ∂v∗ )2 + 2∑ k=0 n(n−1) 2∑ i=1 r2 (∂T kΩiψ ∂v∗ )2}∣∣∣ u∗=τ1 + ∫ Στ1 ( 4∑ k=0 JN(T k · φ) + 3∑ k=0 n(n−1) 2∑ i=1 JN(T kΩiφ), n )} (1.5.58) Remark 1.41. This statement should be compared to the assumptions of Prop. 1.37 (1.5.5), from which all that one can deduce with (1.5.17) is∫ ∞ τ+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r2 (∂(r n−12 T · φ) ∂v∗ )2}∣∣∣ u∗=τ <∞ (τ > τ0) . (1.5.59) 1.5. THE DECAY ARGUMENT 93 We shall now proceed along the lines of the proof of Prop. 1.37 in Section 1.5.2, just that we have (1.5.58) as a starting point for the solution T · φ of (1.1.1), (and (1.5.6)); however, as opposed to Prop. 1.37 the hierarchy does not descend from p = 2 but p < 2, which introduces a degeneracy in the last step, and requires the refinement of Prop. 1.11 to Prop. 1.32, and Prop. 1.14 to Prop. 1.33, see Section 1.4.4. Lemma 1.42 (Pointwise decay under special assumptions). Let φ be a solution of the wave equation (1.1.1), with initial data on Στ1 (τ1 > 0) satisfying D . = ∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r4−δ ( ∂χ ∂v∗ )2 + 3∑ k=0 r2 (∂T kψ ∂v∗ )2 + 2∑ k=0 n(n−1) 2∑ i=1 r2 (∂T kΩiψ ∂v∗ )2}∣∣∣ u∗=τ1 + ∫ Στ1 ( 4∑ k=0 JN (T k · φ) + 3∑ k=0 n(n−1) 2∑ i=1 JN(T kΩiφ), n ) <∞ for some δ > 0 and∫ ∞ τ ′+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × r2−δ (∂T · ψ ∂v∗ )2 |u∗=τ ′ ≤ C(n,m, δ, R)D τ ′2 (∗) for some τ ′ > τ1. Then there is a constant C(n,m, δ, R) such that∫ Sn−1 dµ◦ γn−1 rn−1− δ 2 ( T · φ)2|(u∗=τ ′,v∗=R∗+τ) ≤ C D τ ′2 for all τ > τ ′. Remark 1.43. Note the gain in powers of r in comparison to the boundary term arising in Prop. 1.33. Proof. First, integrating from infinity, (T · φ)(τ ′, R∗ + τ ′) = − ∫ ∞ τ ′+R∗ ∂(T · φ) ∂v∗ dv∗ and then by Cauchy’s inequality,∫ Sn−1 dµ◦ γn−1 (T ·φ)2(τ ′, R+τ ′) ≤ ∫ ∞ R∗+τ ′ 1 rn−1 dv∗× ∫ ∞ R+τ ′ ∫ Sn−1 dµ◦ γn−1 (∂(T · φ) ∂v∗ )2 rn−1 dv∗ ≤ 1 2 ( 1− 2m rn−2|(u∗=τ ′,v∗=R∗+τ ′) )−1 1 n− 2 1 rn−2 × × C(m,n) ∫ ∞ R∗+τ ′ dv∗ ∫ Sn−1 dµ◦ γn−1 rn−1 (∂(T · φ) ∂v∗ )2 . 94 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Therefore, by Prop. 1.37,∫ Sn−1 dµ◦ γn−1 ( rn−2(T · φ)2)(τ ′, R∗ + τ ′) ≤ C(n,m) 1− 2m Rn−2 ∫ Στ ′ ( JT (T · φ), n ) ≤ C(n,m,R) τ ′2 D . (∗∗) Now rn−1 ∫ Sn−1 dµ◦ γn−1 (T · φ)2 (τ ′, R∗ + τ) = = ∫ Sn−1 dµ◦ γn−1 ( rn−1(T · φ)2)(τ ′, R∗ + τ ′) + ∫ R∗+τ R∗+τ ′ dv∗ ∫ Sn−1 dµ◦ γn−1 2T · ψ∂T · ψ ∂v∗ ≤ ≤ Rn−1 ∫ Sn−1 dµ◦ γn−1 (T · φ)2(τ ′, R∗ + τ ′) + 2r δ 2 |(u∗=τ ′, v∗=R∗+τ) √∫ ∞ R∗+τ ′ dv∗ ∫ Sn−1 dµ◦ γn−1 1 r2 (T · φ)2rn−1× × √∫ ∞ R∗+τ ′ dv∗ ∫ Sn−1 dµ◦ γn−1 r2−δ (∂T · ψ ∂v∗ )2 , which proves the pointwise estimate of the Lemma in view of the Hardy inequality of Lemma B.6, Prop. 1.37, the assumption (∗) and (∗∗). p = 2− δ: By the weighted energy inequality with p = 2− δ and r n−12 T · φ in the role of ψ, see (1.5.16) in particular,∫ τ ′′2j+1 τ ′′2j−1 du∗ ∫ ∞ u∗+R′j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × r1−δ (∂T · ψ ∂v∗ )2 ≤ ≤ C(n,m) ∫ τ ′′2j+1 τ ′′2j−1 du∗ ∫ ∞ u∗+R′j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × rK2−δ (T · φ) + C(n,m) ∫ 2τ ′′2j+1+R′j∗ 2τ ′′2j−1+R ′ j ∗ dt ∫ Sn−1 dµ◦ γn−1 × {( T · ψ )2}∣∣∣ r=R′j ≤ ≤ C(n,m,R) ∫ ∞ τ ′′2j−1+R ′ j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r2−δ (∂T · ψ ∂v∗ )2}∣∣∣ u∗=τ ′′2j−1 + C(n,m) ∫ 2τ ′′2j+1+R′j∗ 2τ ′′2j−1+R ′ j ∗ dt ∫ Sn−1 dµ◦ γn−1 × {( T · ψ )2 + (∂T · ψ ∂v∗ )2 + ∣∣∇/ T · ψ∣∣2}∣∣∣ r=R′j (1.5.60) where we choose R′j ∗ ∈ (R∗ + 1, R∗ + 2) such that Prop. 1.11 applied to the domain r(r∗=R∗+2)Dτ ′′ 2j+1 τ ′′2j−1 \r(r∗=R∗+1)Dτ ′′ 2j+1 τ ′′2j−1 yields an estimate for the integral on the timelike bound- ary above in terms of the first and second order energies on Στ ′′2j−1 which in turn decays by Prop. 1.37. Therefore there exists a sequence τ ′′′j ∈ (τ ′′2j−1, τ ′′2j+1) (j ∈ N) such that∫ ∞ τ ′′′j +R ′ j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r1−δ (∂T · ψ ∂v∗ )2}∣∣∣ u∗=τ ′′′j ≤ 1.5. THE DECAY ARGUMENT 95 ≤ C(n,m, δ, R) (τ ′′′j )3 {∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r4−δ ( ∂χ ∂v∗ )2 + 3∑ k=0 r2 (∂(T k · ψ) ∂v∗ )2 + 2∑ k=0 n(n−1) 2∑ i=1 r2 (∂T kΩiψ ∂v∗ )2}∣∣∣ u∗=τ1 + ∫ Στ1 ( 4∑ k=0 JN(T k · φ) + 3∑ k=0 n(n−1) 2∑ i=1 JN(T kΩiφ), n )} . (1.5.61) p = 1− δ: Since, by integrating by parts,∫ ∞ u∗+R∗ dv∗ 1 rδ ( ∂ψ ∂v∗ )2 = = ∫ ∞ u∗+R∗ dv∗ 1 rδ {n− 1 2r r n−1 2 ( 1− 2m rn−2 )∂(r n−12 φ2) ∂v∗ + rn−1 ( ∂φ ∂v∗ )2} = 1 rδ n− 1 2r ( 1− 2m rn−2 ) ψ2|∞u∗+R∗ + ∫ ∞ u∗+R∗ dv∗ { δ r1+δ n− 1 2r ( 1− 2m rn−2 )2 ψ2 + 1 rδ n− 1 2r2 ( 1− 2m rn−2 ) ψ2 [ (n− 2) + (1− 2m rn−2 )n− 3 2 ] + 1 rδ ( ∂φ ∂v∗ )2 rn−1 } (1.5.62) we have by (1.5.15) that also (with R′′j ∗ ∈ (R∗ + 2, R∗ + 3)),∫ τ ′′′2j+1 τ ′′′2j−1 du∗ ∫ ∞ u∗+R′′j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × 1 rδ {(∂T · φ ∂v∗ )2 + ∣∣∇/ T · φ∣∣2}rn−1 ≤ ≤ C(n,m) {∫ τ ′′′2j+1 τ ′′′2j−1 du∗ ∫ ∞ u∗+R′′j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × rK1−δ (T · φ) · rn−1 + ∫ 2τ ′′′2j+1+R′′j ∗ 2τ ′′′2j−1+R ′′ j ∗ dt ∫ Sn−1 dµ◦ γn−1 × {( T · ψ )2}∣∣∣ r=R′′j } . (1.5.63) By virtue of Stokes theorem (B.5), (B.6) and our previous result (1.5.61) we obtain∫ τ ′′′2j+1 τ ′′′2j−1 du∗ ∫ ∞ u∗+R′′j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × 1 rδ ( JT (T · φ), ∂ ∂v∗ ) rn−1 ≤ ≤ C(n,m) {∫ ∞ τ ′′′2j−1+R ′′ j ∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r1−δ (∂(T · ψ) ∂v∗ )2}∣∣∣ u∗=τ ′′′2j−1 + ∫ 2τ ′′′2j+1+R′′j ∗ 2τ ′′′2j−1+R ′′ j ∗ dt ∫ Sn−1 dµ◦ γn−1 × {(∂(T · ψ) ∂v∗ )2 + ∣∣∇/ T · ψ∣∣2 + (T · ψ)2}∣∣∣ r=R′′j } ≤ C(n,m, δ, R) (τ ′′′j )3 {∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r4−δ ( ∂χ ∂v∗ )2 + 3∑ k=0 r2 (∂(T k · ψ) ∂v∗ )2 + 2∑ k=0 n(n−1) 2∑ i=1 r2 (∂T kΩiψ ∂v∗ )2}∣∣∣ u∗=τ1 + ∫ Στ1 ( 4∑ k=0 JN(T k · φ) + 3∑ k=0 n(n−1) 2∑ i=1 JN(T kΩiφ), n )} 96 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES + C(n,m,R) {∫ Σ τ ′′′ 2j+1 τ ′′ 2(2j−1)−1 ( JT (T · φ) + JT (T 2 · φ), n ) + ∫ Sn−1 dµ◦ γn−1 rn−2 ( T · φ )2 |(u∗=τ ′′ 2(2j−1)−1, v∗=R′′j ∗+τ ′′′2j+1) } , (1.5.64) where in the last inequality we have used Prop. 1.33, and then chosen R′′j (j ∈ N) suitably by Prop. 1.32; furthermore the inequality still holds if we add the integral of the nonde- generate energy on R ′′ jPτ ′′′ 2j+1 τ ′′′2j−1 on the left hand side and replace JT by JN in the first term of the integral on Σ τ ′′′2j+1 τ ′′ 2(2j−1)−1 on the right hand side. The last two terms on the right hand side of (1.5.64) in fact decay with almost the same rate as the first; for first note here that we could have used Prop. 1.14 and Cor. 1.13 instead, and then employ Prop. 1.37 to obtain in any case that ∫ τ ′′′2j+1 τ ′′′2j−1 dτ ∫ Στ 1 rδ ( JN(T · φ), n ) ≤ ≤ C(n,m, δ, R) (τ ′′′2j−1)2 {∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r4−δ ( ∂χ ∂v∗ )2 + 3∑ k=0 r2 (∂(T k · ψ) ∂v∗ )2 + 2∑ k=0 n(n−1) 2∑ i=1 r2 (∂T kΩiψ ∂v∗ )2}∣∣∣ u∗=τ1 + ∫ Στ1 ( 4∑ k=0 JN(T k · φ) + 3∑ k=0 n(n−1) 2∑ i=1 JN (T kΩiφ), n )} . (1.5.65) It then follows that there exists a sequence τ ′′′′j ∈ (τ ′′′2j−1, τ ′′′2j+1) such that∫ Σ τ ′′′ 2(j+2)+1 τ ′′′′ j ( JN (T · φ), n ) ≤ rδ|(u∗=τ ′′′′j , v∗=R′′j+2 ∗+τ ′′′ 2(j+2)+1 ) ∫ Στ ′′′′ j 1 rδ ( JN (T · φ), n ) ≤ ≤ C(n,m, δ, R) (τ ′′′′j )3−δ {∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r4−δ ( ∂χ ∂v∗ )2 + 3∑ k=0 r2 (∂(T k · ψ) ∂v∗ )2 + 2∑ k=0 n(n−1) 2∑ i=1 r2 (∂T kΩiψ ∂v∗ )2}∣∣∣ u∗=τ1 + ∫ Στ1 ( 4∑ k=0 JN(T k · φ) + 3∑ k=0 n(n−1) 2∑ i=1 JN(T kΩiφ), n )} (1.5.66) because τ ′′′′j (τ ′′′ 2(j+2)+1 − τ ′′′′j )−1 ≤ 1. And second the assumptions of Lemma 1.42 are satisfied in view of (1.5.58) on u∗ = τ ′′j (j ∈ N) which yields∫ Sn−1 dµ◦ γn−1 rn−2 ( T · φ )2 |(u∗=τ ′′ 2(2j−1)−1 ,v ∗=R′′j ∗+τ ′′′2j+1) ≤ 1.5. THE DECAY ARGUMENT 97 ≤ C(n,m, δ, R) (τ ′′′2j−1) 3− δ 2 {∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r4−δ ( ∂χ ∂v∗ )2 + 3∑ k=0 r2 (∂T k · ψ ∂v∗ )2 + 2∑ k=0 n(n−1) 2∑ i=1 r2 (∂T kΩiψ ∂v∗ )2}∣∣∣ u∗=τ1 + ∫ Στ1 ( 4∑ k=0 JN (T k · φ) + 3∑ k=0 JN (T kΩiφ), n )} (1.5.67) because also τ ′′′2j−1(τ ′′′ 2j+1 − τ ′′2(2j−1)−1)−1 ≤ C. We shall now return to (1.5.64) (and its extension that includes the nondegenerate energy on R ′′ jPτ ′′′ 2j+1 τ ′′′2j−1 ) to find that, after inserting (1.5.66) and using Prop. 1.35,∫ Σ τ ′′′ 2j+1 τ ′′ 2(2j−1)−1 ( JN(T · φ) + JT (T 2 · φ), n ) ≤ C ∫ Σ τ ′′′ 2j+1 τ ′′′ (2j−1)−1 ( JN(T · φ) + JT (T 2 · φ), n ) ≤ ≤ C ∫ Σ τ ′′′ 2j+1 τ ′′′′ j−2 ( JN(T · φ) + JT (T 2 · φ), n ) ≤ ≤ C(n,m, δ, R) (τ ′′′′j−2)3−δ {∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { r4−δ ( ∂χ ∂v∗ )2 + r4−δ (∂(T · χ) ∂v∗ )2 + 4∑ k=0 r2 (∂(T k · ψ) ∂v∗ )2 + 3∑ k=0 n(n−1) 2∑ i=1 r2 (∂T kΩiψ ∂v∗ )2}∣∣∣ u∗=τ1 + ∫ Στ1 ( 5∑ k=0 JN(T k · φ) + 4∑ k=0 n(n−1) 2∑ i=1 JN (T kΩiφ), n )} (1.5.68) and using (1.5.67), that there exists (another) sequence τ ′′′′j ∈ (τ ′′′2j−1, τ ′′′2j+1) (j ∈ N) such that∫ Στ ′′′′ j 1 rδ ( JN (T · φ), n ) ≤ ≤ C(n,m, δ, R) (τ ′′′′j )4−δ {∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { 1∑ k=0 r4−δ (∂(T k · χ) ∂v∗ )2 + 4∑ k=0 r2 (∂(T k · ψ) ∂v∗ )2 + 3∑ k=0 n(n−1) 2∑ i=1 r2 (∂T kΩiψ ∂v∗ )2}∣∣∣ u∗=τ1 + ∫ Στ1 ( 5∑ k=0 JN(T k · φ) + 4∑ k=0 n(n−1) 2∑ i=1 JN(T kΩiφ), n )} . (1.5.69) So for any τ > τ1 we can choose j ∈ N such that τ ∈ (τ ′′′2j−1, τ ′′′2j+1) to obtain finally by Prop. 1.35 that∫ Στ∩{r≤R} ( JN(T · φ), n ) ≤ 98 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES ≤ ∫ Σ τ ′′′ 2j+1 τ ′′′′ j−1 ( JN(T · φ), n ) ≤ rδ|(u∗=τ ′′′′j−1, v∗=R∗+τ ′′′2j+1) ∫ Στ ′′′′ j−1 1 rδ ( JN (T · φ), n ) ≤ C(n,m, δ, R) τ 4−2δ {∫ ∞ τ1+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { 1∑ k=0 r4−δ (∂(T k · χ) ∂v∗ )2 + 4∑ k=0 r2 (∂(T k · ψ) ∂v∗ )2 + 3∑ k=0 n(n−1) 2∑ i=1 r2 (∂T kΩiψ ∂v∗ )2}∣∣∣ u∗=τ1 + ∫ Στ1 ( 5∑ k=0 JN(T k · φ) + 4∑ k=0 n(n−1) 2∑ i=1 JN (T kΩiφ), n )} . (1.5.70) Remark 1.44. Note that for the removal of the restriction to dyadic sequences in the last step of the proof, (1.5.69) - (1.5.70), we could have equally obtained a decay estimate for the energy flux through Στ ∩{r∗ ≤ R∗+ τk} (with k ∈ N) by replacing Στ ′′′ 2j+1 τ ′′′′j−1 by Σ τ ′′′′j−1+τ k τ ′′′′j−1 in the first estimate in (1.5.70); if δ > 0 for a chosen k ∈ N is restricted to δ < (1+k)−1 we then still obtain a decay rate of τ 4−(1+k)δ for the energy flux through Στ ∩{r∗ ≤ R∗+ τk}. 1.5.4 Digression: Conformal energy decay In this section we present an alternative proof of energy decay using the conformal Morawetz vectorfield. This is the conventional way to prove energy decay — it was first applied in the context of the Schwarzschild spacetime in [20] —, but is now super- seded by the physical space approach [22] applied in Section 1.5.2. In particular it has the disadvantage of introducing weights in t into the argument, and makes the dependence on the initial data less transparent. Let Σ˜ be a spacelike hypersurface in J+(u∗ = 1, v∗ = 1) (see figure 1.8). And Σ˜τ . = ϕτ (Σ˜ ∩D) . Proposition 1.45 (Conformal energy decay). Let φ be a solution of the wave equation (1.1.1) with initial data on the spacelike hypersurface t = 0 such that D(Z) . = ∫ Σ0 ( JN (φ) + n(n−1) 2∑ i=1 JN (Ωiφ) + n(n−1) 2∑ i,j=1 JN(ΩiΩjφ) + n(n−1) 2∑ i,j,k=1 JT (ΩiΩjΩkφ), n ) + ∫ Σ0 ( JZ,1(φ) + n(n−1) 2∑ i=1 JZ,1(Ωiφ) + n(n−1) 2∑ i,j=1 JZ,1(ΩiΩjφ), n ) <∞ , and let Σ˜ be chosen as above. Then there exists a constant C(n,m) such that∫ Σ˜τ ( JN(φ), n ) ≤ C D τ 2 (τ > 0) . 1.5. THE DECAY ARGUMENT 99 (u∗ = 1, v∗ = 1) Σ˜0 Figure 1.8: A spacelike hypersurface Σ˜0. Morawetz vectorfield Z. In Minkowski space Z = u∗2 ∂ ∂u∗ + v∗2 ∂ ∂v∗ (1.5.71) is a conformal Killing vectorfield which has the weights that allow us to prove energy decay. Here the trace-free part of the deformation tensor (Z)πˆµν = (Z)πµν − 1 n+ 1 gµν tr (Z)π does not vanish, but we may still write: ∇µJZµ = (Z)πˆµνTµν + 1 n+ 1 tr (Z)π trT . (1.5.72) Calculation of (Z)π. First, (Z)πu∗u∗ = 0 (Z)πv∗v∗ = 0 (Z)πu∗v∗ = −2t ( 1− 2m rn−2 )− (n− 2) t r∗ 2m rn−1 ( 1− 2m rn−2 ) (Z)πAB = tr∗ r ( 1− 2m rn−2 ) r2( ◦ γn−1)AB . Second, with tr (Z)π = 2t+ (n− 2)tr∗ 2m rn−1 + (n− 1)t r ∗ r ( 1− 2m rn−2 ) and denoting by (Z)π˜ = 2t (r∗ r (1− 2m rn−2 )− 1)− (n− 2)tr∗ 2m rn−1 (1.5.73) we find (Z)πˆab = −1 2 n− 1 n + 1 (Z)π˜ gab (a, b = u ∗, v∗) (Z)πˆAB = 1 n+ 1 (Z)π˜ gAB (A,B = 1, . . . , n− 1) . 100 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES In the above trT = −n− 1 4 (φ2) so we are lead to consider ∇µ ( JZµ + 1 4 n− 1 n+ 1 tr (Z)π ∂µ ( φ2 )− 1 4 n− 1 n+ 1 ∂µ ( tr (Z)π ) φ2 ) = = (Z)πˆµνTµν − 1 4 n− 1 n+ 1  ( tr (Z)π ) φ2 . Now (Z)πˆµνTµν = 1 2 n− 1 n+ 1 (Z)π˜ 1− 2m rn−2 ( ∂φ ∂u∗ )( ∂φ ∂v∗ ) + 1 n + 1 (Z)π˜ ∣∣∇/ φ∣∣2 r2 ◦ γn−1 and thus we see (Z)πˆµνTµν + 1 2 n− 1 n+ 1 (Z)π˜ ∂αφ ∂αφ = 1 2 (Z)π˜ ∣∣∇/ φ∣∣2 r2 ◦ γn−1 . So if we define JZ,1µ . = JZµ + 1 4 n− 1 n+ 1 ( tr (Z)π + (Z)π˜ ) ∂µ ( φ2 )− 1 4 n− 1 n + 1 ∂µ ( tr (Z)π + (Z)π˜ ) φ2 then ∇µJZ,1µ = 1 2 (Z)π˜ ∣∣∇/ φ∣∣2 r2 ◦ γn−1 − 1 4 n− 1 n+ 1  ( tr (Z)π + (Z)π˜ ) φ2 . Note that finally tr (Z)π + (Z)π˜ = (n+ 1) tr∗ r ( 1− 2m rn−2 ) and  (tr∗ r ( 1− 2m rn−2 )) = = [ − 1 1− 2m rn−2 ( ∂ ∂t − ∂ ∂r∗ )( ∂ ∂t + ∂ ∂r∗ )− n− 1 2r (−2 ∂ ∂r∗ )](tr∗ r ( 1− 2m rn−2 )) = −(n− 3) t r2 ( 1− 2m rn−2 )(r∗ r ( 1− 2m rn−2 )− 1)− (n− 2) t r 2m rn−1 ( 3 r∗ r ( 1− 2m rn−2 )− 2) + (n− 2)2( 2m rn−1 )2 tr∗ r . We conclude with JZ,1µ = J Z µ + n− 1 4 t r∗ r ( 1− 2m rn−2 ) ∂µ ( φ2 )− n− 1 4 ∂µ (t r∗ r ( 1− 2m rn−2 )) φ2 (1.5.74) we have KZ,1 = 1 2 [ 2t (r∗ r ( 1− 2m rn−2 )− 1)− (n− 2)t r∗ 2m rn−1 ]∣∣∇/ φ∣∣2 r2 ◦ γn−1 + { (n− 3)(n− 1) 4 t r2 ( 1− 2m rn−2 )(r∗ r ( 1− 2m rn−2 )− 1) + (n− 2)(n− 1) 4 t r 2m rn−1 ( 3 r∗ r ( 1− 2m rn−2 )− 2) − (n− 1)(n− 2) 2 4 ( 2m rn−1 )2 t r∗ r } φ2 . (1.5.75) 1.5. THE DECAY ARGUMENT 101 Remark 1.46 (Minkowski space). If we set m = 0 and r∗ = r then KZ,1 = 0. While KZ,1 does not vanish identically the following Lemma shows that at least near the horizon and at infinity it has a sign. Lemma 1.47. There are constant values of the radius n−2√ 2m < r0(n,m) < R(n,m) <∞ such that KZ,1 ≥ 0 (r ≤ r0) (1.5.76) and KZ,1 ≥ n− 1 8 2mt rn φ2 (r ≥ R) . (1.5.77) Proof. A suitable rearrangement of the terms in (1.5.75) is in powers of 2m rn−2 : KZ,1 = −t [ 1 + r∗ 2r ( n 2m rn−2 − 2)]|∇/φ|2 − n− 1 4 t r2 [ (n− 1) 2m rn−2 + (n− 3) + r ∗ r ( (n2 + 1) ( 2m rn−2 )2 − n 2m rn−2 − (n− 3) )] φ2 (1.5.78) (1.5.76) is clear by inspection as r∗ → −∞ (r → n−2√2m). For (1.5.77) note that the coefficient to φ2 tends to −n− 1 4 t r2 [ − 2m rn−2 + (n2 + 1) ( 2m rn−2 )2]ց 0 (r →∞) . It remains to be shown the nonnegativity of the coefficient to |∇/ φ|2 in the asymptotic region r →∞. Here we insert r∗ = r − (nm) 1n−2 + (2m) 1n−2 ∫ r n−2√2m (n 2 ) 1 n−2 1 xn−2 − 1 dx to obtain the limit −t [ 1 + r∗ 2r ( n 2m rn−2 − 2 )] (r→∞)−→ lim y→∞ t y ∫ y (n 2 ) 1 n−2 1 xn−2 − 1 dx which approaches zero from above. The next proposition shows that the current JZ,1 gives rise to nonnegative boundary terms on t-const surfaces with the desired weights. Proposition 1.48. For n ≥ 4,∫ R (JZ,1, T ) rn−1 dr∗ ≥ ∫ R 1 4 { 1 5 ( u∗2 ( ∂φ ∂u∗ )2 + v∗2 ( ∂φ ∂v∗ )2) + ( t2 + r∗2 )( 1− 2m rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 + n− 1 2 3n− 10 7 ( 1− 2m rn−2 )r∗2 + t2 r2 φ2 } rn−1 dr∗ . 102 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Proof. We compute (JZ , T ) = 1 2 u∗2 ( ∂φ ∂u∗ )2 + 1 2 v∗2 ( ∂φ ∂v∗ )2 + 1 2 ( u∗2 + v∗2 )( 1− 2m rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 . Consider the vectorfields S = v∗ ∂ ∂v∗ + u∗ ∂ ∂u∗ = t ∂ ∂t + r∗ ∂ ∂r∗ S = v∗ ∂ ∂v∗ − u∗ ∂ ∂u∗ = t ∂ ∂r∗ + r∗ ∂ ∂t . Then ∫ R (JZ,1, T ) rn−1 dr∗ = ∫ R { (JZ , T ) + n− 1 4 tr∗ r ( 1− 2m rn−2 ) 2φ ∂φ ∂t − n− 1 4 r∗ r ( 1− 2m rn−2 ) φ2 } rn−1 dr∗ = ∫ R 1 4 { (S · φ)2 + (S · φ)2 + (t2 + r∗2)(1− 2m rn−2 )|∇/φ|2 + n− 1 2 r∗ r ( 1− 2m rn−2 ) 2φ(S · φ) − n− 1 2 r∗ r ( 1− 2m rn−2 ) 2φ r∗ ∂φ ∂r∗ + n− 1 2 r∗ r ( 1− 2m rn−2 ) 2φ t r∗ ( S · φ) − n− 1 2 r∗ r ( 1− 2m rn−2 ) 2φ t2 r∗ ∂φ ∂r∗ − (n− 1)r ∗ r ( 1− 2m rn−2 ) φ2 } rn−1 dr∗ . We can integrate by parts − ∫ R r∗ r ( 1− 2m rn−2 ) 2φ r∗ ∂φ ∂r∗ rn−1 dr∗ = ∫ R ∂ ∂r∗ ( r∗2rn−2 ( 1− 2m rn−2 )) φ2 dr∗ = ∫ R ( 1− 2m rn−2 ){ 2 r∗ r + (n− 2)(r∗ r )2} rn−1φ2 dr∗ and − ∫ R r∗ r ( 1− 2m rn−2 ) 2φ t2 r∗ ∂φ ∂r∗ rn−1 dr∗ = ∫ R ∂ ∂r∗ ( rn−2t2 ( 1− 2m rn−2 )) φ2 dr∗ = ∫ R ( 1− 2m rn−2 ) (n− 2) t 2 r2 φ2 rn−1 dr∗ to obtain∫ R (JZ,1, T ) rn−1 dr∗ = 1.5. THE DECAY ARGUMENT 103 = ∫ R 1 4 { 2m rn−2 [ (S · φ)2 + (S · φ)2 ] + ( t2 + r∗2 )( 1− 2m rn−2 )|∇/φ|2 + ( 1− 2m rn−2 )[ (S · φ)2 + (n− 1)(S · φ)r ∗ r φ+ n− 1 2 (n− 2)r ∗2 r2 φ2 + (S · φ)2 + (n− 1)(S · φ) t r φ+ n− 1 2 (n− 2) t 2 r2 φ2 ]} rn−1 dr∗ . By Cauchy’s inequality (S · φ)2 + (n− 1)(S · φ)r ∗ r φ+ n− 1 2 (n− 2)r ∗2 r2 φ2 ≥ ≥ (1− 2ǫ)(S · φ)2 + n− 1 2 ( (n− 2)− 1 2ǫ n− 1 2 )r∗2 r2 φ2 ; (1.5.79) here we choose ǫ = 7 16 . So∫ R (JZ,1, T ) rn−1 dr∗ ≥ ≥ ∫ R 1 4 { 2m rn−2 ( (S · φ)2 + (S · φ)2 ) + ( t2 + r∗2 )( 1− 2m rn−2 )|∇/ φ|2 + ( 1− 2m rn−2 )1 8 [ (S · φ)2 + (S · φ)2 + n− 1 2 8 7 (3n− 10)r ∗2 + t2 r2 φ2 ]} rn−1 dr∗ and the stated inequality follows. Remark 1.49 (Case n = 3 in Prop 1.48). If we set ǫ = 1 2 in (1.5.79) we merely obtain in the n = 3-dimensional case∫ R (JZ,1, T ) r2 dr∗ ≥ ≥ ∫ R 1 4 { 2 2m r ( u∗2 ( ∂φ ∂u∗ )2 + v∗2 ( ∂φ ∂v∗ )2) + ( t2 + r∗2) ( 1− 2m rn−2 )|∇/ φ|2}r2 dr∗ . The degeneracy at infinity can however be removed by splitting the t∂φ ∂t -term differently before the integration by parts. (See [20] Prop 10.10.) The last fact related to the Morawetz vectorfield is that the bulk term in a t-const slab can be controlled with the J (α) current (see Cor. 1.29) in a finite annular region. Proposition 1.50. For n ≥ 3, there are constant radial values n−2√ 2m < r0(n,m) < r1(n,m) <∞ and a constant C(n,m) > 0 such that − ∫ R KZ,1 rn−1 ( 1− 2m rn−2 ) dr∗ ≤ ≤ C(n,m) t ∫ r∗1 r∗0 { 1 rn ( ∂φ ∂r∗ )2 + r2 (1− 2m rn−2 )(1 + r ∗2)2 |∇/ φ|2 } rn−1 ( 1− 2m rn−2 ) dr∗ . 104 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Lemma 1.51. Let φ ∈ C1(R) with 1√ x φ(x), dφ dx (x) ∈ L2 loc (R), and let a, b ∈ R with a < b. Then for any ε > 0, there exist c(ε, b, a) > b,C(ε, b, a) > 0 such that∫ b a φ2 dx ≤ ε ∫ c b 1 (y − a) + (b− a)φ 2(y) dy + C ∫ c a ( dφ dx )2 dx . Proof of the Lemma. Take x ∈ [1, 2], y ∈ [2,∞) then φ2(x)− φ2(y) ≤ 2 ∫ y x |φ dφ dz | dz ≤ 2 (∫ y x 1 z φ2(z) dz ) 1 2 (∫ y x z ( dφ dz )2 dz ) 1 2 . Now apply Cauchy’s inequality and integrate over x ∈ [1, 2]:∫ 2 1 φ2(x) dx ≤ 2ǫ ∫ y 1 1 z φ2(z) dz + 1 2ǫ ∫ y 1 z ( dφ dz )2 dz + φ2(y) Next divide by y, choose d < (log 2)−1 and integrate over y ∈ [2, e 1d ]: ∫ e 1d 2 1 y dy ∫ 2 1 φ2(x) dx ≤ ≤ ǫ ∫ e 1d 1 1 x φ2(x) dx+ 1 4ǫ e 2 d ∫ e 1d 1 x ( dφ dx )2 dx+ ∫ e 1d 2 1 y φ2(y) dy where we have replaced ǫe 1 d by ǫ. Writing∫ e 1d 1 1 x φ2(x) dx ≤ ∫ 2 1 φ2(x) dx+ ∫ e 1d 2 1 x φ2(x) dx and absorbing the first term in the left hand side we obtain with a choice of d ≤ 1 4 log 2 + 2 ǫ ≤ 1 2∫ 2 1 φ2(x) dx ≤ 2d ∫ e 1d 2 1 y φ2(y) dy + de 3 d 3 ∫ e 1d 1 ( dφ dx )2 dx . Hence∫ b a φ2(x) dx = ∫ 2 1 φ2(t(b− a) + 2a− b)(b− a) dt ≤ ≤ 2d(b− a) ∫ e 1d (b−a)+2a−b b 1 (y − a) + (b− a)φ 2(y) dy + de 3 d 3 (b− a)2 ∫ e 1d (b−a)+2a−b a ( dφ dx )2 dx . So given 0 < ε < 2(b− a)(4 log 2 + 2) we choose d = ε (2(b− a))−1 and then have in fact c = (b− a)e 2(b−a)ε + 2a− b C = ε(b− a) 6 e 6(b−a) ε . 1.5. THE DECAY ARGUMENT 105 Proof. In view of Lemma 1.47 and (1.5.78) we have∫ R KZ,1 rn−1 ( 1− 2m rn−2 ) dr∗ ≥ ≥ −B(n,m) t ∫ R∗ r∗0 { |∇/φ|2 + φ2 } rn−1 ( 1− 2m rn−2 ) dr∗ + n− 1 8 2mt ∫ ∞ R∗ 1 rn φ2 rn−1 ( 1− 2m rn−2 ) dr∗ for some constant B(n,m). Using the Lemma we infer that ∫ R∗ r∗0 φ2 rn−1 ( 1− 2m rn−2 ) dr∗ ≤ ≤ Rn−1(1− 2m Rn−2 ) ε ∫ r∗1 R∗ r (r∗ − r∗0) + (R∗ − r∗0) 1 r φ2 dr∗ +Rn−1C ∫ r∗1 r∗0 ( ∂φ ∂r∗ )2 dr∗ where r1 = r1(ε, r0, R) > R(n,m) and C(ε, r0, R) > 0. In fact choose ε = 2m 8 1 B(n,m)Rn−1 then since supr≥R r r∗−r∗0+R∗−r∗0 ≤ 1 (if not choose r0 smaller and R bigger) we can com- pensate with the last term in the region r ≥ R:∫ R KZ,1 rn−1 ( 1− 2m rn−2 ) dr∗ ≥ ≥ −B(n,m)(1 +R ∗2)2 r20 t ∫ R∗ r∗0 r2 (1 + r∗2)2 ∣∣∇/ φ∣∣2 r2 ◦ γn−1 rn−1 ( 1− 2m rn−2 ) dr∗ + n− 2 8 ∫ r∗1 R∗ 2mt rn φ2 rn−1 ( 1− 2m rn−2 ) dr∗ − Rn−1r1B(n,m)C ( 1− 2m rn−20 )−1 t ∫ r∗1 r∗0 1 rn ( ∂φ ∂r∗ )2 rn−1 ( 1− 2m rn−2 ) dr∗ where r∗1 = (R ∗ − r∗0)(n,m)e 2m 8 B(n,m)R(n,m)n−12(R∗−r∗0)(n,m) + 2r∗0 −R∗ and C = 1 6 2m 8 (R∗ − r∗0)(n,m) B(n,m)R(n,m)n−1 e 8 2m B(n,m)R(n,m)n−16(R∗−r∗0)(n,m) . Proof of Prop. 1.45. Step 1. 1 t -decay on t-const hypersurfaces. We begin our argument using a t-const foliation (see figure 1.9) Σt = { (u, v) : u ≤ 0, v ≥ 0, 2 n− 2(2m) 1 n−2 artanh (u+ v v − u ) = t } 106 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Στ ′ Σt R(τ ′, t) Figure 1.9: The t-const foliation. and denote the corresponding spacetime slab by R(t′, t) = ⋃ t′≤t≤t Σt . Recall the current (1.4.67). The energy identity for J (α) reads∫ Σt (J (α), n) + ∫ R(t′,t) K(α) = ∫ Σt′ (J (α), n) (1.5.80) where n = 1( 1− 2m rn−2 ) 1 2 ∂ ∂t . Note in general that ∫ Σt (J, n) = ∫ ∞ −∞ ∫ Sn−1 (J, T ) rn−1 dr∗ dµ◦ γn−1 because dµg|Σt = ( 1− 2m rn−2 ) 1 2 rn−1 dr∗ ∧ dµ◦ γn−1 . We conclude from (1.5.80) in conjunction with Lemma 1.30 — where we have already addressed the boundary terms of the J (α)-current on t-const hypersurfaces — that ∫ R(t′,t) K(α) ≤ C(n,m, α) ∫ Σt′ ( JT (φ) + n(n−1) 2∑ i=1 JT (Ωiφ), n ) (τ ′ > t) . (1.5.81) Choose n−2 √ 2m < r˜1 < r˜2 <∞ such that 1 2 ( t0 − r˜∗2 ) ≥ 1 1 2 ( t0 + r˜ ∗ 1 ) ≥ 1 (1.5.82) 1.5. THE DECAY ARGUMENT 107 r˜1 r˜2 r∗ = 0 t = 0 Σt0 Σt Figure 1.10: Construction of the triple (t0, r˜1, r˜2). for some fixed t0 > 0 (see figure 1.10). Then for t ≥ t0∫ r˜∗2 r˜∗1 ( ∂φ ∂u∗ )2 dr∗ ≤ ( 2 t− r˜∗2 )2 ∫ r˜∗2 r˜∗1 u∗2 ( ∂φ ∂u∗ )2 dr∗ and ∫ r˜∗2 r˜∗1 ( ∂φ ∂v∗ )2 ≤ ( 2 t+ r˜∗1 )2 ∫ r˜∗2 r˜∗1 v∗2 ( ∂φ ∂v∗ )2 dr∗ . Therefore by Prop. 1.48∫ r˜∗2 r˜∗1 (JT , T ) rn−1 dr∗ ≤ ( 20 (t− r˜∗2)2 + 20 (t+ r˜∗1)2 + 2 t2 ) × × ∫ r˜∗2 r˜∗1 1 4 { 1 5 ( u∗2 ( ∂φ ∂u∗ )2 + v∗2 ( ∂φ ∂v∗ )2) + ( t2 + r∗2 )( 1− 2m rn−2 )|∇/ φ|2}rn−1 dr∗ ≤ C(t0) t2 ∫ R (JZ,1, T ) rn−1 dr∗ (t > t0) . Note that the larger t0 > 0, the larger one may choose the interval [r˜ ∗ 1, r˜ ∗ 2] ⊂ R. On the other hand Prop. 1.50 yields when combined with Cor. 1.29 − ∫ R(t0,t1) KZ,1 = − ∫ t1 t0 ∫ R ∫ Sn−1 KZ,1 ( 1− 2m rn−2 ) rn−1 dt dr∗ dµ◦ γn−1 ≤ C(n,m) ∫ R(t0,t1)∩{r0≤r≤r1} tK(α) . (1.5.83) Therefore by the energy identity for JZ,1:∫ Σt∩{r˜1≤r≤r˜2} (JT , n) ≤ C(t0) t2 ∫ Σt (JZ,1, n) ≤ C(t0) t2 [∫ Σt0 (JZ,1, n) + C(n,m) t ∫ R(t0,t) K(α) ] 108 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES ≤ C(n,m, α, t0) t [∫ Σt0 (JZ,1, n) + ∫ Σt0 ( JT (φ) + n(n−1) 2∑ i=1 JT (Ωiφ), n )] (t > t0) (1.5.84) Step 2. 1 τ -decay on the hypersurfaces Σ˜τ∩{r ≤ r2(τ)}. The spacelike hypersurfaces relative to which we expect the local observer’s energy to decay terminate at future null infinity (and on the horizon to the future of the bifurcation sphere). Let Σ˜0 be a spacelike hypersurface in J+(u∗ = 1, v∗ = 1) and first choose n−2 √ 2m < r˜1 ≤ r0(n,m) and then t0 and r˜2 such that (t0, r˜ ∗ 1) ∈ Σ˜0, (t0, r˜∗2) ∈ Σ0. Clearly, (t0, r˜1, r˜2) satisfy (1.5.82) by construction. Note that by definition of Σ˜τ = ϕτ (Σ˜0) , Σ˜τ and Σt0+τ will intersect at r = r˜1 and r = r˜2 for all τ ≥ 0. Also denote R˜(τ ′, τ) = ⋃ τ ′≤τ≤τ Σ˜τ . By conservation of JT flux,∫ Σ˜τ∩{r˜1≤r≤r˜2} (JT , n) = ∫ Σt0+τ∩{r˜1≤r≤r˜2} (JT , n) . (1.5.85) We may assume that KN ≥ b (JN , n) on Σ˜τ ∩ {r ≤ r˜1} for all τ ≥ 0, as well as r˜2 ≥ r1(n,m); for otherwise choose r˜1 closer to the horizon, see also Prop. 1.7. Then∫ R˜(τ ′,τ)∩{r≤r˜1} (JN , n) ≤ 1 b ∫ R˜(τ ′,τ)∩{r≤r˜1} KN ≤ 1 b ∫ Σ˜τ ′∩{r≤Rτ ′,τ} (JN , n) + 1 b ∫ R(0,t0+τ)∩{r≥r˜1} |KN | (1.5.86) where (t0 + τ, R ∗ τ ′,τ) ∈ Σ˜τ ′ (see figure 1.11). We can use the J (α) current to control the spacetime integral:∫ R(0,t0+τ)∩{r≥r˜1} |KN | ≤ C(n,m, σ, r˜1) ∫ R(0,t0+τ) { K(α) +Kaux } ≤ C(n,m, α, σ, r˜1) ∫ Σ0 ( JT (φ) + n(n−1) 2∑ i=1 JT (Ωiφ), n ) By virtue of the uniform boundedness result of Section 1.5.1 applied to the first term in (1.5.86) (recall also N = T for r ≥ rN1 ) ∫ R˜(τ ′,τ)∩{r≤r˜1} (JN , n) ≤ C(n,m, α, b, σ, r˜1) ∫ Σ0 ( JN(φ) + n(n−1) 2∑ i=1 JT (Ωiφ), n ) (1.5.87) 1.5. THE DECAY ARGUMENT 109 Rτ ′,τ Σt0+τ Σ˜τ ′ r˜1 Figure 1.11: Illustration of (1.5.86). Lemma 1.52. On Σt ∩ {r˜1 ≤ r ≤ r˜2},∫ S |KN | dµγ ≤ C(n,m, σ, r˜1, rN1 ) ∫ S { K(α) +Kaux } dµγ . Proof. The statement is immediate from Cor. 1.29. In fact for rN0 ≤ r ≤ rN1 |KN | = |(N)πµνTµν | ≤ |(N)πµν ||∂µφ||∂νφ|+ 1 2 | tr (N)π||∂αφ||∂αφ| ≤ ≤ B (JT , n) , (1.5.88) whereas KN = KT = 0 for r ≥ rN1 . For r ≤ rN0 one may use N = [ 1 + σ 4κ ( 1− 2m rn−2 )]( 2 1− 2m rn−2 ∂ ∂u∗ + ∂ ∂t ) to calculate KN = (n− 2) σ 4κ 2m rn−1 (∂φ ∂t )( ∂φ ∂r∗ ) + (n− 2) 2m rn−1 ( 1− 2m rn−2 )−1( ∂φ ∂u∗ )2 + (n− 2) σ 4κ 2m rn−1 |∇/ φ|2 + n− 3 r [ 1 + σ 4κ ( 1− 2m rn−2 )]|∇/ φ|2 − n− 1 r [ 1 + σ 4κ ( 1− 2m rn−2 )] 1 1− 2m rn−2 ( ∂φ ∂u∗ )( ∂φ ∂v∗ ) . to find the bound |KN | ≤ n− 2 2 σ 4κ (2m)− 1 n−2 [(∂φ ∂t )2 + ( ∂φ ∂r∗ )2] + 2(n− 2)(2m)− 1n−2(1− 2m r˜n−21 )−1[(∂φ ∂t )2 + ( ∂φ ∂r∗ )2] + (n− 2) σ 4κ (2m)− 1 n−2 |∇/ φ|2 + n− 3 (2m) 1 n−2 [ 1 + σ 4κ ] |∇/ φ|2 + n− 1 (2m) 1 n−2 [ 1 + σ 4κ ]( 1− 2m r˜n−21 )−1[(∂φ ∂t )2 + ( ∂φ ∂r∗ )2] because r ≥ r˜1. 110 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES The spacetime integral estimate (1.5.87) near the horizon is the starting point of the following pigeonhole argument. Let τ1 > 0, τn+1 = 2 τn (n ∈ N) then there is a sequence (τ ′n)n∈N, τ ′n ∈ (τn, τn+1) such that∫ τn+1 τn dτ ∫ Σ˜τ∩{r≤r˜1} (JN , n) = τn ∫ Σ˜τ ′n∩{r≤r˜1} (JN , n) and we have for all n ∈ N ∫ τn+1 τn dτ ∫ Σ˜τ∩{r≤r˜1} (JN , n) ≤ C(n,m, α, b, σ, r˜1, Σ˜0) ∫ Σ0 ( JN(φ) + n(n−1) 2∑ i=1 JT (Ωiφ), n ) . Therefore∫ Σ˜τ ′n∩{r≤r˜1} (JN , n) ≤ ≤ C(n,m, α, b, σ, r˜1, Σ˜0) τ ′n ∫ Σ0 ( JN (φ) + n(n−1) 2∑ i=1 JT (Ωiφ), n ) (n ≥ 2) . (1.5.89) Recall (1.5.85) which implies∫ Σ˜τ∩{r˜1≤r≤r˜2} (JN , n) ≤ ≤ C(n,m, α, t0, r˜1) (t0 + τ) {∫ Σt0 ( JZ,1(φ), n ) + ∫ Σt0 ( JT (φ) + n(n−1) 2∑ i=1 JT (Ωiφ), n )} (1.5.90) Next define for τ ≥ 0, r2(τ) by ( t0 + τ + τ , r ∗ 2(τ ) ) ∈ Σ˜τ (see figure 1.12) then∫ Σ˜τ∩{r˜2≤r≤r2(τ)} (JT , n) ≤ ∫ Σt0+τ+τ∩J+(Σ˜τ ) (JT , n) ≤ ≤ C(t0) (t0 + τ)2 ∫ Σt0+τ+τ (JZ,1(φ), n) (1.5.91) because min Σt0+τ+τ∩J+(Σ˜τ ) u∗2 ≥ min Σ˜τ u∗2 min Σt0+τ+τ∩J+(Σ˜τ ) v∗2 ≥ min Σ˜τ v∗2 . However again∫ Σt0+τ+τ ( JZ,1, n ) ≤ ≤ C(n,m, α, t0) (t0 + τ + τ) [∫ Σt0 ( JZ,1, n ) + ∫ Σt0 ( JT (φ) + n(n−1) 2∑ i=1 JT (Ωiφ), n )] . (1.5.92) 1.5. THE DECAY ARGUMENT 111 r˜1 r˜2 r2(τ) Σ˜τ Σt0+τ Σt0+τ+τ Figure 1.12: Extension to the region r˜2 ≤ r ≤ r2(τ). Now choose τ = c (t0 + τ) then with (1.5.90)∫ Σ˜τ∩{r˜1≤r≤r2(τ)} (JN , n) ≤ ≤ C(n,m, α, t0, r˜1) (t0 + τ) ∫ Σt0 ( JZ,1(φ) + JT (φ) + n(n−1) 2∑ i=1 JT (Ωiφ), n ) ; (1.5.93) note we can arrange for r∗2(τ) ≥ r˜∗2 + τ . For any τ ≥ τ3 we may choose τ ′ = max { τ ′n : τ ′n ≤ τ } so by the local observer’s energy estimate∫ Σ˜τ∩{r≤r˜1} (JN , n) ≤ C ∫ Σ˜τ ′∩{r≤r2(τ ′)} (JN , n) and using (1.5.89) and (1.5.93) we finally obtain∫ Σ˜τ∩{r≤r2(τ)} (JN , n) ≤ C(n,m, α, b, σ, t0, r˜1, Σ˜0) τ × × {∫ Σt0 ( JZ,1(φ), n ) + ∫ Σ0 ( JN (φ) + n(n−1) 2∑ i=1 JT (Ωiφ), n )} (τ ≥ 4τ1) . (1.5.94) Step 3. 1 τ2 -decay on the hypersurfaces Σ˜τ . The aim is here to improve (1.5.94) to the extend that 1 τ will be replaced by 1 τ2 and the restriction on r ≤ r2(τ) will be removed. Recall the regions (1.4.1), and consider in particular R(t0, t1, u∗1) .= Rr0,r1(t0, t1, u∗1, 1 2 (t1 + r ∗ 1) R∞(t0, t1) .= ⋃ u∗1≥ 12 (t1−r∗0) R(t0, t1, u∗1) . 112 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES As we have already seen in Step 2a of the proof of Prop. 1.11 in Section 1.4.4 we obtain ∫ R∞(t0,t1) K(α) ≤ C(n,m, α) ∫ ∂−R∞(t0,t1) ( JT (φ) + n(n−1) 2∑ i=1 JT (Ωiφ), n ) (1.5.95) where ∂−R∞(t0, t1) is the past boundary of R∞(t0, t1). Now cover the region {r0(n,m) ≤ r ≤ r1(n,m)} dyadically with R∞(tn, tn+1) (n ∈ N) where t1 > 0, tn+1 = 2 tn (n ∈ N) and also denote Σ˜n . = ∂−R∞(tn, tn+1) . Then for N ∈ N large enough Σ˜n ⊂ J+(u∗ = 1, v∗ = 1) (n ≥ N + 1) thus by the previous estimate (1.5.95) and the earlier 1 τ decay result (1.5.94) (recall r˜1 ≤ r0(n,m), r˜2 ≥ r1(n,m), r∗2(τ) ≥ r˜∗2 + τ) and choose τ1 < tN4 :∫ {tn≤t≤tn+1}∩{r0≤r≤r1} K(α) ≤ CD (Z) 2 (tn − tN ) (n ≥ N + 1) (1.5.96) where C = C(n,m, α, b, σ, t0, Σ˜0) and D (Z) 2 = ∫ Σt0 ( JZ,1(φ) + n(n−1) 2∑ i=1 JZ,1(Ωiφ), n ) + ∫ Σ0 ( JN (φ) + n(n−1) 2∑ i=1 JN(Ωiφ) + n(n−1) 2∑ i,j=1 JT (ΩiΩjφ), n ) . (1.5.97) By summing over n ∈ N,∫ {t0≤t≤t′1}∩{r0≤r≤r1} tK(α) ≤ ≤ tN ∫ {t0≤t≤tN+1} K(α) + [t′1]∑ n=N+1 2 ∫ {tn≤t≤tn+1} tnK (α) ≤ tN CD(Z)2 + 2CD(Z)2 [t′1] where [t′1] ∈ N is the smallest number such that t[t′1] ≥ t′1; observe that since t[t′1] = 2 [t′1]t1 [t′1] = log t[t′1] − log t1 2 ≤ 1 + log t ′ 1 log 2 if t1 > 1 . Hence ∫ {t0≤t≤t1}∩{r0≤r≤r1} tK(α) ≤ (1 + log t1)CD(Z)2 , (1.5.98) with C,D (Z) 2 as above, and we immediatly improve (1.5.83) to − ∫ R(t0,t1) KZ,1 ≤ (1 + log t1)CD(Z)2 , (1.5.99) 1.5. THE DECAY ARGUMENT 113 with the consequence that also (1.5.84) is replaced by∫ Σt∩{r˜1≤r≤r˜2} (JT , n) ≤ 1 + log t t2 CD (Z) 2 (1.5.100) with C,D (Z) 2 given as above by (1.5.97). With the same argument as before we extend this to ∫ Σ˜τ∩{r˜1≤r≤r2(τ)} (JN , n) ≤ 1 + log(t0 + τ) (t0 + τ)2 CD (Z) 2 . (1.5.101) Next we refine (1.5.87). By Cor. 1.29, (1.5.95) and the 1 τ -decay result (1.5.94)∫ R˜(τ ′,τ)∩{r≤r˜1} (JN , n) ≤ ≤ 1 b ∫ R˜(τ ′,τ)∩{r≤r˜1} ∩{v∗≤ 1 2 (t0+τ ′+r˜∗1)} KN + ∫ {r≤r˜1}∩ { 1 2 (t0+τ ′+r˜∗1)≤v∗≤ 12 (t0+τ+r˜∗1)} (JN , n) ≤ 1 b ∫ Σ˜τ ′∩{r≤r˜1} (JN , n) + C(n,m) ∫ R∞(t0+τ ′,t0+τ) { K(α) +Kaux } ≤ C(n,m, α, b) ∫ Σ˜τ ′∩{r≤r2(τ ′)} ( JN (φ) + n(n−1) 2∑ i=1 JT (Ωiφ), n ) ≤ C(n,m, α, b, σ, t0, Σ˜0) τ ′ {∫ Σt0 ( JZ,1(φ) + n(n−1) 2∑ i=1 JZ,1(Ωiφ), n ) + ∫ Σ0 ( JN (φ) + n(n−1) 2∑ i=1 JN (Ωiφ) + n(n−1) 2∑ i=1 JN (ΩiΩjφ), n )} (1.5.102) or with C,D (Z) 2 as above ∫ R˜(τ ′,τ)∩{r≤r˜1} (JN , n) ≤ CD (Z) 2 τ ′ . (1.5.103) Proceeding as above let τ1 > 0, τn+1 = 2 τn (n ∈ N), then there is a sequence (τ ′n)n∈N, τ ′n ∈ (τn, τn+1) such that∫ Σ˜τ ′n∩{r≤r˜1} (JN , n) = 1 τn ∫ τn+1 τn dτ ∫ Σ˜τ∩{r≤r˜1} (JN , n) ≤ 1 τ 2n CD (Z) 2 ≤ 16 τ ′n 2CD (Z) 2 (1.5.104) and for any τ ≥ τ3 we may choose τ ′ = max{τ ′n : τ ′n ≤ τ} so that∫ Σ˜τ∩{r≤r˜1} (JN , n) ≤ C ∫ Σ˜τ ′∩{r≤r2(τ ′)} (JN , n) . Thus ∫ Σ˜τ∩{r≤r2(τ)} (JN , n) ≤ 1 + log(t0 + τ) (t0 + τ)2 C D (Z) 2 . (1.5.105) 114 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES One can repeat this step to remove the 1+ log(t0+ τ)-term and the restriction r ≤ r2(τ). Namely, now using (1.5.105) yields in place of (1.5.96)∫ {tn≤t≤tn+1} ∩{r0≤r≤r1} K(α) ≤ 1 + log(tn − tN ) (tn − tN)2 C D (Z) 3 (n ≥ N + 1) (1.5.106) where C = C(n,m, α, b, σ, t0, Σ˜0) and D (Z) 3 . = ∫ Σt0 ( JZ,1(φ) + n(n−1) 2∑ i=1 JZ,1(Ωiφ) + n(n−1) 2∑ i,j=1 JZ,1(ΩiΩjφ), n ) + ∫ Σ0 ( JN(φ) + n(n−1) 2∑ i=1 JN(Ωiφ) + n(n−1) 2∑ i,j=1 JN(ΩiΩjφ) + n(n−1) 2∑ i,j,k=1 JT (ΩiΩjΩkφ), n ) . (1.5.107) In other words ∫ {t0≤t≤t1} ∩{r0≤r≤r1} tK(α) ≤ C D(Z)3 (1.5.108) because by summing over n ∈ N∫ {t0≤t≤t′1} ∩{r0≤r≤r1} tK(α) ≤ tN+1 ∫ {t0≤t≤tN+1} ∩{r0≤r≤r1} K(α) + 2 ∞∑ i=N+1 1 + log(tn − tN ) (tn − tN) C D3 with the last sum being finite ∞∑ N+1 1 + log(tn − tN ) tn − tN = ∞∑ k=1 1 + log(2k − 1) + (N − 1) log 2 + log t1 (2k − 1) 2N−1 t1 <∞ since ∞∑ k=1 log 2k 2k = (log 2) ∞∑ k=1 ( k√k 2 )k <∞ as k √ k → 1 (k →∞). This immediatly improves (1.5.83) to − ∫ R(t0,t1) KZ,1 ≤ C D(Z)3 . (1.5.109) Therefore ∫ Σt∩{r˜1≤r≤r˜2} (JT , n) ≤ 1 t2 CD (Z) 3 ; (1.5.110) moreover no power of (t0 + τ + τ) is lost in (1.5.92) anymore and (1.5.91) is replaced by∫ Σ˜τ∩{r≥r˜2} (JT , n) ≤ 1 (t0 + τ)2 CD (Z) 3 (1.5.111) as we let τ →∞. We arrive when combined with (1.5.104) at the final result:∫ Σ˜τ (JN , n) ≤ 1 τ 2 CD (Z) 3 (1.5.112) 1.6. POINTWISE BOUNDS 115 where C,D (Z) 3 are as above given by (1.5.107). Note that again by (1.5.83) and (1.5.81) D (Z) 3 ≤ C(n,m, α, t0)× × ∫ Σ0 ( JN(φ) + n(n−1) 2∑ i=1 JN(Ωiφ) + n(n−1) 2∑ i,j=1 JN(ΩiΩjφ) + n(n−1) 2∑ i,j,k=1 JT (ΩiΩjΩkφ), n ) + ∫ Σ0 ( JZ,1(φ) + n(n−1) 2∑ i=1 JZ,1(Ωiφ) + n(n−1) 2∑ i,j=1 JZ,1(ΩiΩjφ), n ) . 1.6 Pointwise bounds In this Section we first prove pointwise estimates on |φ| and |∂tφ| separately based on the energy decay results Prop. 1.37 and Prop. 1.39 in Section 1.5. Then we give the interpolation argument to improve the pointwise decay on |φ|. As we shall see in view of the nondegenerate energy estimates of Section 1.5 we may restrict ourselves in the first place to a radial region away from the horizon. Recall the definition (1.4.3) of Στ , (r1 . = R > n−2 √ 8nm). Proposition 1.53 (Pointwise decay). (i) Let φ be a solution of the wave equation (1.1.1), with initial data on Στ0 (τ0 > 0) such that D . = ∫ ∞ τ0+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 [n 2 ]+1∑ k=0 r2 (∂T k · ψ ∂v∗ )2 |u∗=τ0 + ∫ Στ0 ([n2 ]+2∑ k=0 JN (T k · φ), n ) <∞ . (1.6.1) Then there is a constant C(n,m) such that for r0 < r < R, |φ(t, r)| ≤ C(n,m) √ D τ (τ = 1 2 (t− R∗) > τ0) . (1.6.2) (ii) If moreover, the initial data satisfies D . = ∫ ∞ τ0+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × {[n 2 ]+1∑ k=0 r4−δ (∂(T k · χ) ∂v∗ )2 + [n 2 ]+4∑ k=0 r2 (∂(T k · ψ) ∂v∗ )2 + [n 2 ]+3∑ k=0 n(n−1) 2∑ i=1 r2 (∂T kΩiψ ∂v∗ )2}∣∣∣ u∗=τ0 + ∫ Στ0 ([n2 ]+5∑ k=0 JN (T k · φ) + [n 2 ]+4∑ k=0 n(n−1) 2∑ i=1 JN (T kΩiφ), n ) <∞ (1.6.3) 116 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES for some 0 < δ < 1 4 , and R > n−2 √ 8nm δ , then there is a constant C(n,m, δ, R) such that for r0 < r < R, |∂tφ(t, r)| ≤ C √ D τ 2−2δ (τ = 1 2 (t− R∗) > τ0) . (1.6.4) The pointwise bounds are obtained from the energy estimates of Section 1.5 using Sobolev inequalties and elliptic estimates; the former provide the link between pointwise and integral quantities, and the latter allow for the expression of these integral quantities in terms of higher order energies. Sobolev embedding. By the extension theorem applied to the Sobolev embedding Hs(Rn) ⊂ L∞(Rn) (s > n 2 ) [27] we have, for r0 < r < R, |φ(t, r)|2 ≤ C(n) ∫ R∗ r∗0 dr∗ ∫ Sn−1 dµ◦ γn−1 { φ2 + |α|≤[n 2 ]+1∑ |α|≥1 ∣∣∇αφ∣∣2}rn−1∣∣∣ t=t (1.6.5) where ∇ denote the tangential derivatives to the hypersurface Σt, and α denotes a multi- ndex of order n. Elliptic estimates. Note that for any solution φ of the wave equation T 2 · φ = ∂ 2φ ∂r∗2 + ( 1− 2m rn−2 )n− 1 r ∂φ ∂r∗ + ( 1− 2m rn−2 )△/ r2 ◦ γn−1 φ . = L · φ (1.6.6) where the operator L = ( 1− 2m rn−2 ) gij∇i∂j (1.6.7) is clearly elliptic, (here gt = g|Σt denotes the restriction of g to the spacelike hypersurfaces Σt, a Riemannian metric on Σt, and i, j = 1, . . . , n). In view of the standard higher order interior elliptic regularity estimate (c.f. [27]), ‖φ‖Hm+2(bΣt) ≤ C ( ‖L · φ‖Hm(bΣt) + ‖φ‖L2(bΣt) ) Σ̂t . = Σt ∩ {r0 < r < R} , (1.6.8) we conclude with (1.6.5) that in the case where [n 2 ] + 1 is even, |φ|2 ≤ C(n,m) ∫ R∗ r∗0 dr∗ ∫ Sn−1 dµ◦ γn−1 [n 2 ]+1∑ l=0 ( T l · φ )2 rn−1 ; (1.6.9) in general we have: Lemma 1.54 (Pointwise estimate in terms of higher order energies). Let φ be a solution of the wave equation (1.1.1), and n ≥ 3. Then there exists a constant C(n,m) such that for all r0 < r < R: |φ(t, r)|2 ≤ C(n,m) [ ‖φ‖2 L2(bΣt) + ∫ bΣt [n 2 ]∑ l=0 ( JT (T l · φ), n )] (1.6.10) 1.6. POINTWISE BOUNDS 117 Proof of Prop. 1.53. In view of the Lemma 1.54 and the energy decay estimates of Section 1.5 it remains to control the zeroth order term ‖φ‖L2(bΣt); we multiply the integrand by (R r )2 ≥ 1 and extend the integral to u∗ = τ = 1 2 (t− R∗), v∗ ≥ 1 2 (t+R∗). (i) By Lemma B.6 we can then estimate ‖φ‖2 L2(bΣt) by the energy flux through Στ= 12 (t−R∗), and apply Prop. 1.37 to the higher order energies of Lemma 1.54. (ii) Here we extend the integral only to τ +R∗ ≤ v∗ ≤ τ +R∗+ τ 3 and apply Lemma B.8 to obtain∫ R∗ r∗0 dr∗ ∫ Sn−1 dµ◦ γn−1 (∂tφ) 2rn−1 ≤ C(n,m)R2 ∫ Στ∩{r∗≤R∗+τ3} ( JT (∂tφ), n ) + C(n,m) R2 r ∫ Sn−1 rn−1(∂tφ)2|(u∗=τ,v∗=τ+R∗+τ3) . (1.6.11) As in the proof of Lemma 1.42 we obtain by integrating from infinity and Cauchy’s inequality that∫ Sn−1 dµ◦ γn−1 rn−2(∂tφ)2(τ, τ +R∗) ≤ C(n,m) 1− 2m Rn−2 ∫ Στ ( JT (∂tφ), n ) (1.6.12) which decays by Prop. 1.37 with a rate τ−2. Moreover, as in the proof of Lemma 1.42,∫ Sn−1 dµ◦ γn−1 rn−1(∂tφ)2|(u∗=τ,v∗=τ+R∗+τ3) = = ∫ Sn−1 dµ◦ γn−1 rn−1(∂tφ)2|(u∗=τ,v∗=τ+R∗) + ∫ τ+R∗+τ3 τ+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 2 ∂tψ ∂∂tψ ∂v∗ |u∗=τ (1.6.13) and∫ τ+R∗+τ3 τ+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 ∂tψ ∂∂tψ ∂v∗ |u∗=τ ≤ ≤ √∫ ∞ τ+R∗ ∫ Sn−1 dµ◦ γn−1 1 r2 (∂tφ)2rn−1 × √∫ ∞ τ+R∗ ∫ Sn−1 dµ◦ γn−1 r2 (∂r n−12 ∂tφ ∂v∗ )2 , (1.6.14) the first factor decaying with a rate τ−1 by Lemma B.6 and Prop. 1.37, and the second factor bounded by the weighted energy inequality for r n−1 2 ∂tφ in place of ψ with p = 2. Therefore ∫ Sn−1 rn−1(∂tφ)2|(u∗=τ,v∗=τ+R∗+τ3) ≤ C(n,m) 1− 2m Rn−2 D τ . (1.6.15) By virtue of Prop. 1.39, compare in particular Remark 1.44 on page 98, the first term on the right hand side of (1.6.11) decays with a rate of τ 4−4δ, and this is matched by the second term in view of the prefactor r−1 = (R∗ + τ 3)−1, which is the result of our choice of powers of τ in the extension of the integral (1.44). Lemma 1.54 applied to the solution ∂tφ of (1.1.1) then yields the pointwise decay result (1.6.4) after having applied Prop. 1.39 to the higher order energies on the right hand side of (1.6.10). 118 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES Interpolation. We shall now interpolate between the results Prop. 1.53 (i) and (ii) to improve the pointwise estimate for |φ|. Our argument can in some sense be compared to the proof of improved decay in [36]. The basic observation underlying this argument is that for r0 < r < R and t1 > t0 rn−2φ2(r, t1) = rn−2φ2(r, t0) + ∫ t1 t0 2φ(t, r) ∂φ ∂t (t, r) rn−2 dt ≤ rn−2φ2(r, t0) + 1 t1−2δ0 ∫ t1 t0 φ2(t, r) rn−2 dt+ t1−2δ0 ∫ t1 t0 (∂φ ∂t )2 (t, r) rn−2 dt . (1.6.16) Moreover, as a consequence of Lemma 1.55, rn−2φ2(t, r) ≤ Rn−2φ2(t, R) + (1− 2m rn−20 )−1 ∫ R∗ r∗ ( ∂φ ∂r∗ )2 rn−1 dr∗ , (1.6.17) we obtain an estimate for the timelike integrals in terms of the corresponding integrals at r = R and spacetime integrals, using the Sobolev inequality on the sphere: ∫ t1 t0 rn−2φ2(t, r) dt ≤ ∫ t1 t0 dt ∫ Sn−1 dµ◦ γn−1 ∑ |α|≤[n 2 ]+1 Rn−2 ( Ωαφ )2 (t, R) + ( 1− 2m rn−20 )−1 ∫ t1 t0 dt ∫ R∗ r∗ dr∗ ∫ Sn−1 dµ◦ γn−1 rn−1 ∑ |α|≤[n 2 ]+1 (∂Ωαφ ∂r∗ )2 (t, r) (1.6.18) Lemma 1.55. Let a < b ∈ R and φ ∈ C1([a, b]) then an−2φ2(a) ≤ bn−2φ2(b) + ∫ b a ( dφ dx )2 xn−1 dx (1.6.19) for all n ≥ 3. Proof. Since, by integration by parts, ∫ b a 2φ(x) dφ dx (x)xn−2 dx = 2φ2(x)xn−2|ba − ∫ b a 2φ(x) dφ dx (x)xn−2 dx− ∫ b a 2φ2(x)(n− 2)xn−3 dx , it clearly follows, with Cauchy’s inequality, an−2φ2(a) ≤ bn−2φ2(b) + ∫ b a ( dφ dx )2 xn−1 dx + [ 1− (n− 2)] ∫ b a 1 x2 φ2(x)xn−1 dx . 1.6. POINTWISE BOUNDS 119 Proposition 1.56 (Improved interior pointwise decay). Let φ be a solution of the wave equation (1.1.1), with initial data on Στ0 (τ0 > 1) satisfying D . = ∫ ∞ τ0+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 × { 2∑ k=0 ∑ |α|≤[n 2 ]+1 r4−δ (∂(T k · Ωαχ) ∂v∗ )2 + 5∑ k=0 ∑ |α|≤[n 2 ]+1 r2 (∂T kΩαψ ∂v∗ )2 + 4∑ k=0 ∑ |α|≤[n 2 ]+2 r2 (∂T kΩαψ ∂v∗ )2}∣∣∣ u∗=τ0 + ∫ Στ0 ( 6∑ k=0 ∑ |α|≤[n 2 ]+1 JN(T kΩαφ) + 5∑ k=0 ∑ |α|≤[n 2 ]+2 JN(T kΩαφ), n ) <∞ . (1.6.20) for some 0 < δ < 1 4 , where R > n−2 √ 8nm δ , n ≥ 3. Then there exists a constant C(n,m, δ, R) such that for n−2 √ 2m < r0 < r < R, r n−2 2 |φ|(t, r) ≤ C D t 3 2 −δ . (1.6.21) Proof. Let t¯0 = 2(τ0+τ0)+R ∗ and t¯1 = t¯0+2τ0 then by (1.6.18), Prop. 1.14 and Prop. 1.11∫ t¯1 t¯0 φ2(t, r)rn−2 dt ≤ C(n,m,R) ∫ Σ2τ0 ( 1∑ k=0 ∑ |α|≤[n 2 ]+1 JT [T kΩαφ], n ) ; (1.6.22) hence by Prop. 1.37 there exists t′0 ∈ (t¯0, t¯1) such that rn−2φ2(t′0, r) ≤ C(n,m,R)D t¯30 . (1.6.23) Now set τ ′0 = 1 2 (t′0 − R∗) and τ ′j = 2τ ′j−1 (j ∈ N), and t′j = 2τ ′j + R∗ (j ∈ N); note that t′j+1 − t′j = 12(t′j − R∗). Now consider (1.6.16) with t1 = t′j+1, t0 = t′j ; since by (1.6.18), together with Prop. 1.11 and Prop. 1.14,∫ t′j+1 t′j rn−2φ2(t, r) dt ≤ C(n,m,R) ∫ Στ ′ j ( 1∑ k=0 ∑ |α|≤[n 2 ]+1 JT [T kΩαφ], n ) , (1.6.24) and by Prop. 1.32 and Prop. 1.33, ∫ t′j+1 t′j rn−2(∂tφ)2(t, r) dt ≤ ≤ C(n,m,R) {∫ Στ ′ j ∩{r∗≤R∗+(τ ′j)3} ( 2∑ k=1 ∑ |α|≤[n 2 ]+1 JT [T kΩαφ], n ) + ∫ Sn−1 dµ◦ γn−1 ∑ |α|≤[n 2 ]+1 rn−2(Ωα∂tφ)2|(u∗=τ ′j ,v∗=R∗+τ ′j+(τ ′j)3) } , (1.6.25) 120 1. HIGHER DIMENSIONAL SCHWARZSCHILD BLACK HOLES which decays with the rate τ 4−4δ as is shown in the proof of Prop. 1.53 (ii), we obtain rn−2φ2(r, t′j+1) ≤ ≤ rn−2φ2(r, t′j) + C(n,m,R) (t′j)1−2δ D (τ ′j)2 + C(n,m, δ, R)(t′j) 1−2δ D (τ ′j)4−4δ ≤ ≤ rn−2φ2(r, t′j) + C(n,m, δ, R)D (t′j)3−2δ . (1.6.26) In fact, by induction on j ∈ N using (1.6.23) for j = 0, we have shown rn−2φ2(r, t′j) ≤ C(n,m, δ, R)D (t′j)3−2δ (j ∈ N ∪ {0}) . (1.6.27) Finally for any t ≥ t′0 we may choose j ∈ N ∪ {0} such that t ∈ (t′j, t′j+1) and conclude the proof by applying (1.6.27) and (1.6.26) which holds with t in place of t′j+1. Extension to the horizon. Note that for n−2 √ 2m ≤ r < r0 the same interpolation (1.6.16) by integration along lines of constant radius r < r0 can be carried out. However, on the right hand sides of (1.6.17) and (1.6.18) a new term results from the integration on v∗ = 1 2 (t0 + r ∗ 0) from the radius r < r0 to r = r0; but we infer from the explicit construction (1.3.18) that the resulting integrand( 2 1− 2m rn−2 ∂φ ∂u∗ )2 ≤ T [φ](Y, Y ) ≤ ( JN [φ], N ) (1.6.28) is controlled by Cor. 1.13 and the proof of Prop. 1.56 above extends to that of Thm. 2 by replacing JT by JN on the right hand sides of (1.6.22), (1.6.24) and (1.6.25). Chapter 2 Linear waves on expanding Schwarzschild de Sitter spacetimes 2.1 Overview It is the purpose of the work presented in this Chapter to initiate the global study of linear waves on cosmological spacetimes. A common feature of expanding spacetimes is suggested by the global causal geometry of the simplest explicitly known solutions to the vacuum Einstein equations with positive cosmological constant. The Schwarzschild de Sitter family exhibits a region of spacetime which is bounded in the past by two cosmological horizons and in the future by a spacelike hypersurface of unbounded area. We provide a suitably robust approach to the analysis of linear waves in these regions. More precisely, we establish uniform energy estimates for general solutions to the linear wave equation in the expanding regions of de Sitter and Schwarzschild de Sitter spacetimes which extend by a stable redshift mechanism to a global estimate. Statement of the Main Result. Let Σ+ be the timelike future boundary of a chosen expanding region in a subextremal Schwarzschild de Sitter spacetime (M, g). Σ+ is a spacelike hypersurface with topology R × S2 endowed with the standard metric ◦g of the cylinder. Let Σ ⊂ J−(Σ+) be a spacelike hypersurface in the past of Σ+ such that Σ+ is in the domain of dependence of Σ and such that Σ crosses the cosmological and event horizons to the future of the bifurcation spheres (see figure 2.1). We consider the Cauchy problem gψ = 0 (2.1.1) 121 122 2. SCHWARZSCHILD DE SITTER SPACETIMES Σ+ Σ C¯+C+ Figure 2.1: Cauchy problem (2.1.1) with initial data on Σ. The expanding region is bounded in the past by the cosmological horizons C¯+ ∪ C+ and in the future by the spacelike hypersurface Σ+. with initial data prescribed on Σ. Let T be the energy momentum tensor of (2.1.1) and n the normal to Σ. We show if the energy of a solution to (2.1.1) is initially finite D[ψ] . = ∫ Σ T (n, n) <∞ , (2.1.2) then it is globally bounded in the expanding region and has a limit on Σ+. Furthermore, the limit of ψ on Σ+ as a function on R× S2 satisfies∫ Σ+ ∣∣ ◦∇ ψ∣∣2◦g dµ◦g ≤ C(M,Σ)D[ψ] (2.1.3) where C is a constant that only depends on the given manifoldsM and Σ, and ◦∇ denotes the gradient on the standard cylinder R× S2. Remark 2.1. The expanding region can be foliated by spacelike hypersurfaces (Σr, gr) which are conformal to the standard cylinder; in fact gr = r 2 ◦ g +O(1 r ). Since the future boundary can be viewed as the set Σ+ = ⋂ r J+(Σr) , (2.1.4) the statement (2.1.3) is a decay result for the induced derivatives on Σr as they approach Σ+. Remark 2.2. The wave equation on “asymptotically de Sitter-like spaces” was previously studied by [47] however with results that are local in nature. The precise statement of the main result and an overview of our proof is given in Sec- tion 2.3. The reader is advised however to first familiarize herself with our treatment of linear waves on de Sitter spacetimes in Section 2.2 where many of the ideas for our approach originate and their application is seen more clearly. Moreover, Section 2.2 offers a self-contained global analysis of linear waves on de Sitter spacetimes including results for the Klein-Gordon equation. 2.2. LINEAR WAVES ON DE SITTER 123 2.2 Linear Waves on de Sitter In the following we consider solutions to the linear wave equation on de Sitter space- time. The de Sitter spacetime is the simplest solution to the vacuum Einstein equations with positive cosmological constant and as such the primary example of an expanding spacetime. In the context of the stability problems for the Einstein equations it is generally expected that a positive cosmological constant introduces a stability mechanism. This may in particular be true for the dynamics of black hole exteriors. Accordingly my treatment aims at an understanding of linear waves on Schwarzschild-de Sitter spacetimes which should be viewed as a first model problem for the nonlinear stability problem of black hole exteriors in expanding cosmological spacetimes. Remark 2.3. The favorable role of the positive cosmological constant for a stability prob- lem was first recognized by Friedrich [28] who was able to reduce the global stability problem for initial data close to de Sitter to a local problem for a quasi-linear symmetric hyperbolic system. Ringstro¨m has extended these ideas to scalar field models in [40], and the stabilizing effect of the cosmological constant has independently been confirmed for perfect fluids in [41, 43] (the case of pure radiation perfect fluids which is not in the scope of this work has been resolved using the conformal method in [33]). However, it is not in the scope of these approaches to accomodate for initial data close to Schwarzschild -de Sitter for which the global geometry is very different; (in particular the spacetime is not foliated by compact Cauchy hypersurfaces). It is for this reason that I emphasize the global aspect of my approach in this Section which at no point makes use of the homo- geneity (or even local homogeneity) of the spacetime; (which is lost in Schwarzschild de Sitter spacetimes). It would be possible to localize the linear problem near the future boundary of (Schwarzschild-)de Sitter but this approach would not yield a global result in the absence of homogeneity. Problem. We are interested in a full understanding of the global behaviour of solutions to the linear wave equation gψ = mΛψ (2.2.1) on de Sitter spacetime (MΛ, g) where mΛ ≥ 0. We take the point of view on de Sitter as a member of the Schwarzschild de Sitter family with mass parameter m = 0. This is equivalent to an a priori choice of a timelike geodesic Γ in (MΛ, g) as the center of symmetry. (De Sitter spacetime is homogeneous, in particular spherically symmetric with respect to any chosen timelike geodesic.) In Section 2.2.1 I describe the global causal geometry of de Sitter in terms of the past, the past boundary, and its complement of any chosen timelike geodesic Γ. 124 2. SCHWARZSCHILD DE SITTER SPACETIMES It is shown that the intersection of the past and future of Γ is in fact a static region of spacetime. Its boundaries are described in analogy to black hole event horizons as cosmological horizons with positive surface gravity. The complement of these domains is the expanding region of spacetime which exhibits a global redshift effect. In Section 2.2.2 it is shown how these crucial geometric properties are used to establish energy estimates for solutions to (2.2.1). My approach in particular yields explicit estimates on the spacelike future boundary of the spacetime in terms of initial data prescribed on any given spacelike hypersurface. Main Results. Let Γ be a timelike geodesic in de Sitter (MΛ, g), where Λ > 0 is the cosmological constant. We denote by r the area radius of the round spheres defined by the orbits of the SO(3) subgroup of the isometry group that leaves Γ invariant, and by Ω(i) the generators of this group action. Consider the level sets Σr of the area radius function r on (MΛ, g). These are timelike hypersurfaces for r < √ 3 Λ , and null hypersurfaces for r = √ 3 Λ . The domain r < √ 3 Λ is the static region of spacetime in the intersection of the timelike future and past of Γ, and as such endowed with a timelike Killing vectorfield T , which is tangential to Σr. We denote by C+ the future boundary of the past of Γ, and by C− the past boundary of the future of Γ; these are the cosmological horizons of Γ. The level sets Σr are spacelike hypersurfaces for r > √ 3 Λ . More precisely, the hyper- surfaces Σr are topologically cylinders that foliate the expanding region r > √ 3 Λ , and T extends to a global Killing vectorfield tangential to Σr. In this domain the metric then takes the form g = −φ2 dr2 + gr = − 1 Λ 3 r2 − 1 dr 2 + gr , (2.2.2) where gr is the induced metric on Σr, a Riemannian metric conformal to the standard cylinder; in fact lim r→∞ 1 r2 gr = 1 4 dλ2+ ◦ γ .= ◦ g , (2.2.3) where λ ∈ (−∞,∞) and ◦γ denotes the standard metric on the unit sphere S2. We denote by Σ+ the timelike future boundary of the expanding region ⋃ r> √ 3 Λ Σr endowed with the rescaled metric ◦ g. Theorem 4. Let Σ be a spacelike hypersurface with normal n in de Sitter (MΛ, g), and Γ a timelike geodesic with cosmological horizons C+ and C−, and assume that Σ crosses the horizons to the future of C+ ∩C− (see figure 2.2). Let moreover ψ be a solution to the linear wave equation (2.2.1) for either mΛ = 0 or mΛ ≥ 2Λ3 with initial data prescribed on Σ such that D[ψ] . = ∫ Σ { Jn[ψ] · n +mΛψ2 } <∞ . (2.2.4) 2.2. LINEAR WAVES ON DE SITTER 125 Σr r =∞ Γ C+ C− Σ Figure 2.2: Penrose diagram of de Sitter depicted as a spherically symmetric spacetime with respect to any chosen timelike geodesic Γ. Then the energy of ψ is globally bounded, and satisfies∫ Σ+ {( T · ψ)2 + 3∑ i=1 ( Ω(i)ψ )2 +mΛ ( rψ )2} dµ◦ g ≤ C(Λ,Σ)D[ψ] (2.2.5) on the future boundary Σ+, where C is a constant that only depends on Λ and the initial hypersurface Σ. Note that the tangent space to Σ+ is spanned by Killing vectorfields and that those tangential derivatives are controlled in L2 by (2.2.5). By the classical Sobolev inequalities this immediately implies the following pointwise estimates. Theorem 5. Let Σ and Γ be as in Theorem 4, and let ψ be a solution to (2.2.1) for either mΛ = 0 or mΛ ≥ 2Λ3 with initial data prescribed on Σ such that in addition to (2.2.4) we have D . = D[ψ] + 3∑ i=1 D[Ω(i)ψ] + 3∑ i,j=1 D[Ω(i)Ω(j)ψ] +D[Tψ] + 3∑ i=1 D[Ω(i)Tψ] + 3∑ i,j=1 D[Ω(i)Ω(j)Tψ] <∞ . (2.2.6) Then sup p,q∈Σ+ {∣∣∣(∂ψ ∂λ )2 (p)− (∂ψ ∂λ )2 (q) ∣∣∣+ ∣∣∣∣∣ ◦∇/ ψ∣∣2◦γ(p)− ∣∣ ◦∇/ ψ∣∣2◦γ(q)∣∣∣ +mΛ ∣∣∣(rψ)2(p)− (rψ)2(q)∣∣∣} ≤ C(Λ)D , (2.2.7) 126 2. SCHWARZSCHILD DE SITTER SPACETIMES where ∂λ denotes a coordinate derivative on Σ + (c.f. (2.2.3)) and ◦ ∇/ the covariant deriva- tive on the standard sphere (S2, ◦ γ), and C is a constant that only depends on Λ. The result in particular states that solutions to the linear wave equation (2.2.1) have a limit on Σ+ which can be viewed as a function on R × S2 satisfying (2.2.5) in the case mΛ = 0 or vanishing identically in the case mΛ ≥ 2Λ3 . Proof and Overview. In our approach the study of solutions to the wave equation on de Sitter spacetime is carried out in three separate domains: the expanding region, the cosmological horizon and the static region. In Section 2.2.2.1 we construct a global redshift vectorfield that captures the expansion of the spacetime in the region r > √ 3 Λ . In fact, in Proposition 2.13 we establish that for any solution of (2.2.1) with mΛ ≥ 0 we have∫ Σr2 φ { φ2 ( T · ψ)2 + |∇/ψ|2 +mΛψ2 } dµgr2 ≤ ≤ ∫ Σr1 φ { φ2 ( T · ψ)2 + 1 φ2 (∂ψ ∂r )2 + |∇/ψ|2 +mΛψ2 } dµgr1 (2.2.8) for all r2 > r1 > √ 3 Λ , where φ is the lapse function of the foliation (Σr) given by (2.2.38). In Section 2.2.2.5 we explicitly exploit the positive surface gravity of the cosmological horizons to construct a local redshift vectorfield. Given a spacelike hypersurface Σ that crosses the cosmological horizon C+ to the future of C+∩C−, we prove in Proposition 2.28 that the right hand side of (2.2.8) is bounded by C(r1,Λ) ∫ C+0 { Λ 3 u ( ∂ ∂u ψ )2 + 1 u ∣∣∇/ψ∣∣2 + mΛ u ψ2 } + C(r1) ∫ Σ′ Jn;mΛ · n , (2.2.9) where Σ′ is the segment of Σ truncated by Σr1 and C+, and C+0 denotes the segment of C+ truncated by Σ, provided r > √ 3 Λ is chosen small enough. Note that Σr1 is an asymptote hypersurface to the the null hypersurface C+. The bound (2.2.9) holds for all solutions of (2.2.1) with mΛ ≥ 0, where C(r1,Λ) is a constant that only depends on the fixed value r1 and Λ. In Section 2.2.2.3 we finally obtain control on the nondegenerate energy on the cosmolog- ical horizon as the boundary of the static region. More precisely we establish in Proposi- tions 2.23 and 2.26 that if mΛ = 0 or mΛ ≥ 2Λ3 then there exists a timelike vectorfield N such that ∫ C+0 1 u {∣∣∇/ψ∣∣2 +mΛψ2} ≤ C(Λ) ∫ Σ { JN [ψ] · n+mΛψ2 } ; (2.2.10) here and in the above (u, v) are the double null coordinates introduced in Section 2.2.1 and |∇/ψ| denotes the angular derivatives on the sphere in the induced norm on the spheres 2.2.1. GLOBAL GEOMETRY 127 of constant area radius r, and . The estimate (2.2.10) relies on an integrated local energy estimate in the static region which is established in Proposition 2.20 using Morawetz type vectorfields. It is in this part of our proof that the condition mΛ ≥ 2Λ3 is introduced; (mΛ = 2 Λ 3 is the conformal value of the mass, see also Remark 2.16). It arises because our proof in the homogeneous case mΛ = 0 relies on a Poincare´ inequality, and we expect that an alternative proof can avoid a lower bound on mΛ (and establish the required estimate for all mΛ ≥ 0), which in view of (2.2.8) should not be essential to our result. Given initial data to the Cauchy problem of (2.2.1) of finite energy on a spacelike hyper- surface Σ crossing the cosmological horizons to the future of the sphere C+ ∩ C− we have thus established that also the energy flux through Σr1 on the right hand side of (2.2.8) is finite and bounded by that initial energy, (provided r1 > √ 3 Λ is chosen small enough and mΛ = 0 or mΛ ≥ 2Λ3 ). Since φ dµgr = 1 2 √ 3 Λ r2 dλ ∧ dµ◦ γ , (2.2.11) and by the coercivity equality on the sphere (see Appendix C.2) r2 ∣∣∇/ψ∣∣2 = 3∑ i=1 ( Ω(i)ψ )2 , (2.2.12) we are now allowed to take the limit in (2.2.8) as r2 tends to infinity, to conclude on the finiteness of the following energy: lim r→∞ ∫ Σr φ { φ2 ( T · ψ)2 + |∇/ψ|2 +mΛψ2 } dµgr2 = = 1 2 √ 3 Λ ∫ Σ+ { 3 Λ ( T · ψ)2 + 3∑ i=1 ( Ω(i)ψ )2 +mΛ ( rψ )2} dλ ∧ dµ◦ γ . (2.2.13) Here Σ+ is the future boundary of the expanding region ⋃ r> √ 3 Λ Σr, a spacelike hyper- surface with topology R × S2 endowed with the standard volume form of the cylinder dλ ∧ dµ◦ γ . 2.2.1 Global geometry of de Sitter The de Sitter spacetime is the simplest solution to the Einstein equations Rµν − 1 2 gµνR + Λgµν = 0 , (2.2.14) with positive cosmological constant Λ > 0. We take the point of view of de Sitter as the member of the Schwarzschild-de Sitter family with m = 0. Let us make this more precise, with a few comments on the mass function in spherically symmetric cosmological spacetimes. See also [26]. 128 2.2. LINEAR WAVES ON DE SITTER Spherical Symmetry in cosmological spacetimes. Similarly to the expositions in [13], and Section 1.2 it is useful to decompose (2.2.14) into its spherical part and Q .= M/SO(3); here the spherical orbits are centred at a fixed timelike geodesic Γ. The metric takes the form g = −Ω2 du dv + r2 ◦γ , (2.2.15) where g is the standard metric of the unit sphere S2, and we can think of (2.2.14) as partial differential equations for the area radius r (and the conformal factor Ω) in the double null coordinates u, v. Indeed, it is easily deduced from (2.2.14) that the area radius r satisfies the Hessian equations ∂2r ∂u2 − 2 Ω ∂Ω ∂u ∂r ∂u = 0 (2.2.16a) ∂2r ∂u ∂v + 1 r ∂r ∂u ∂r ∂v = −Ω 2 4r + Ω2 4 Λr (2.2.16b) ∂2r ∂v2 − 2 Ω ∂Ω ∂v ∂r ∂v = 0 . (2.2.16c) It is a key insight that by virtue of these equations the mass function m defined by 1− 2m r − Λr 2 3 = − 4 Ω2 ∂r ∂u ∂r ∂v (2.2.17) is constant (c.f. [13] in the case Λ = 0), and is precisely the quantity that parametrizes the Schwarzschild-de Sitter family for any fixed Λ > 0. Here of course, m = 0 . (2.2.18) It is also useful to introduce the “tortoise coordinate” r∗ = ∫ 1 1− Λr2 3 dr = 1 2 √ 3 Λ log ∣∣∣1 + √ Λ 3 r 1− √ Λ 3 r ∣∣∣ (2.2.19) which satisfies – again as a consequence of (2.2.16) – the simple partial differential equation ∂2r∗ ∂u ∂v = 0 . (2.2.20) In the coordinate system that covers the entire manifold (with boundary) Q we have r∗ = 1 2 √ 3 Λ log 1 |uv| , (2.2.21) and we can express r explicitly as a function of u, v: r = √ 3 Λ 1 + uv 1− uv . (2.2.22) Finally, using (2.2.17), Ω2 = 3 Λ ( 1 + √ Λ 3 r )2 , (2.2.23) 2.2.1. GLOBAL GEOMETRY 129 r = 0 uv = −1 √ 3 Λ < r <∞ 0 < r < √ 3 Λ u = 0v = 0 r = √ 3 Λ r =∞ uv = 1 Σr Figure 2.3: Global causal geometry of de Sitter. and the de Sitter metric takes the form: g = − 3 Λ ( 1 + √ Λ 3 r )2 du dv + r2 ◦ γ . (2.2.24) The causal geometry of the spacetime can thus be depicted as in figure 2.3. For future reference, note also ∂r ∂u = √ 3 Λ 2v (1− uv)2 ∂r ∂v = √ 3 Λ 2u (1− uv)2 (2.2.25) and 1 1 + √ Λ 3 r = 1− uv 2 . (2.2.26) 2.2.1.1 Static region In the patch S .= {(u, v) : u > 0, v < 0, uv > −1} we may complement (2.2.22) by t = 1 2 √ 3 Λ log u −v . (2.2.27) (And similarly in the region S ′ = {(u, v) : u < 0, v > 0, uv > −1}.) Then the metric takes the familiar static form g = − ( 1− Λr 2 3 ) dt2 + ( 1− Λr 2 3 )−1 dr2 + r2 ◦ γ ( 0 < r < √ 3 Λ ) . (2.2.28) 130 2.2. LINEAR WAVES ON DE SITTER We shall also make use of “Eddington-Finkelstein”-coordinates u∗ = 1 2 ( t− r∗) (2.2.29a) v∗ = 1 2 ( t+ r∗ ) (2.2.29b) which yields as well g = −4 ( 1− Λr 2 3 ) du∗ dv∗ + r2 ◦ γ . (2.2.30) Remark 2.4. It is worth noting that this region S is “small”: it is the intersection of the past of a point with the future of a point. To see this we calculate the arclength of the spacelike curve γλ = {(u, v) : u2 − v2 = λ2} ∩ S to find limλ→∞ L[γλ] = 0. In other words the point (u =∞, v = 0) is indeed a point on the future boundary ofM. This should be compared to the past of a point (u, v) with uv = 1; as discussed below it turns out that this region of M is in fact the past of a sphere, and in this sense “large”. 2.2.1.2 Cosmological horizon This is the null hypersurface r = √ 3 Λ , but we will refer more specifically to the future boundary of S as the cosmological horizon, namely C .= {(u, v) : v = 0, u ≥ 0}. It has positive surface gravity κC = √ Λ 3 , (2.2.31) and thus plays an analogeous role to the event horizon in black hole spacetimes. The globally defined vectorfield T = √ Λ 3 ( u ∂ ∂u − v ∂ ∂v ) (2.2.32) coincides with ∂t in S, and has the following crucial properties: Lemma 2.5 (Properties of the vectorfield T ). (i) T is future directed timelike in S, null on C, and spacelike for r > √ 3 Λ . (ii) T is globally Killing, (T )π . = LTg = 0 . (2.2.33) (iii) On C, ∇TT = √ Λ 3 T . (2.2.34) Note that (iii) is the defining equation for (2.2.31). 2.2.1. GLOBAL GEOMETRY 131 2.2.1.3 Expanding region Let us denote the “expanding” or “cosmological” region by R .= {(u, v) : u > 0, v > 0, uv < 1}. Note that it is foliated by the level sets of the area radius R = ⋃ r> √ 3 Λ Σr ; Σr . = { (u, v) : √ 3 Λ 1 + uv 1− uv = r } . (2.2.35) Time function. We can use the radial function as a time function in R, because in the expanding region the area radius is increasing towards the future. We denote the gradient vectorfield by V µ = −(g−1)µν∂νr (2.2.36) which is given by V = √ Λ 3 ( u ∂ ∂u + v ∂ ∂v ) . (2.2.37) Therefore we obtain for the lapse function of the foliation (Σr) φ = 1√−g(V, V ) = 1√Λ 3 r2 − 1 , (2.2.38) and the decomposition of the metric in R takes the form g = − 1 Λ 3 r2 − 1 dr 2 + gr , (2.2.39) where gr denotes the induced metric on Σr. Also note that while the normal to Σr is n = φV , we have for ∂r = φ 2V . Coarea formula. ∫ R f dµg = ∫ ∞ √ 3 Λ [∫ Σr φ f dµgr ] dr (2.2.40) Induced metric on Σr. It is useful, for future reference, to give here an explicit ex- pression for gr. For this purpose, we also introduce in R the coordinate σ . = √ 3 Λ u v . (2.2.41) A short calculation of dr, and dσ in terms of du and dv then reveals that gr = 1 4 3 Λ (Λ 3 r2 − 1 ) 1 σ2 dσ2 + r2 ◦ γ . (2.2.42) The coordinate vectorfield ∂ ∂σ = 1 2 √ Λ 3 v2 1 v ∂ ∂u − 1 2 √ Λ 3 v2 1 u ∂ ∂v (2.2.43) 132 2.2. LINEAR WAVES ON DE SITTER does have the advantage that it is Lie transported by ∂ ∂r = 1 2 √ Λ 3 (1− uv)2 2v ∂ ∂u + 1 2 √ Λ 3 (1− uv)2 2u ∂ ∂v ; (2.2.44) indeed [ ∂ ∂r , ∂ ∂σ ] = 0 . (2.2.45) This is a convenient frame for the decomposition of the Einstein equations and the wave equation relative to the foliation (Σr), as is discussed in Appendix C.1. Alternatively, we can introduce λ . = log σ , (2.2.46) and since then dλ = 1 σ dσ (2.2.47) the volume form on Σr takes the form dµgr = 1 2 √ 3 Λ (Λ 3 r2 − 1 ) dλ ∧ dµ r2 ◦ γ . (2.2.48) It is important to note that ∂f ∂λ ∣∣ r = σ ∂f ∂σ ∣∣ r = 1 2 √ 3 Λ T · f . (2.2.49) Together with the vectorfields Ω(i) (discussed in Appendix C.2) this implies that the tangent space to Σr is spanned by vectorfields that generate isometries of the spacetime. This is relevant for the Sobolev inequality on Σr, see Section 2.2.2.6. Remark 2.6. We call the regionR (to the future of the cosmological horizons) “expanding” because it is the past of spheres of infinite radius rather than the past of a point. Indeed, a curve γr = {(u, v) : uv = √ Λ 3 r−1√ Λ 3 r+1 , u ≤ 1, v ≤ 1} has arclength lim r→∞ L[γr] = 2 √ 3 Λ (2.2.50) in Q, and thus defines an ideal point on the boundary representing a sphere of infinite radius. 2.2.2 Energy estimates In this section I will discuss energy estimates for solutions to the linear homogeneous wave equation gψ = 0 (2.2.51) as well as for the inhomogeneous wave equation (or simply the “Klein-Gordon equation”) gψ = mΛψ (mΛ > 0) . (2.2.52) 2.2.2. ENERGY ESTIMATES 133 Our main result concerns the expanding region and it applies as is discussed in Sec- tion 2.2.2.1 to the general case mΛ ≥ 0; the construction relies on a “global redshift” vectorfield, and lends itself to generalization in other expanding spacetimes. In Sec- tion 2.2.2.3 we prove a version of integrated local energy decay in the static region with an argument that we learn from [16] that dealt with the corresponding region in the Schwarzschild-de Sitter case. In Section 2.2.2.5 we will finally discuss the redshift effect on the cosmological horizon that shall allow us to turn our estimate in Section 2.2.2.1 into a global energy estimate. 2.2.2.1 Energy estimates and global redshift in the expanding region In this section I present an argument to prove energy estimates in the expanding region, which is complemented by Section 2.2.2.2 where those calculations are to be found which are omitted here in favour of a clear exposition. Recall the area radius r; its level sets Σr are spacelike hypersurfaces in R and define a foliation with lapse (2.2.38), and the metric takes the form (2.2.39). Homogeneous wave equation. Consider the energy current JM associated to the multiplier M = Y¯ + Y , (2.2.53) where Y¯ = 1 ∂r ∂u ∂ ∂u , Y = 1 ∂r ∂v ∂ ∂v . (2.2.54) We can show that for any solutions of (2.2.51), this is a positive current, i.e. ∇ · JM ≥ 0 . (2.2.55) Therefore, by the energy identity for JM ,∫ Σr2 JM · n dµgr2 ≤ ∫ Σr1 JM · n dµgr1 (2.2.56) which yields our energy estimate for solutions of (2.2.51). Proposition 2.7 (Global Energy Boundedness, Homogeneous Case). Let ψ be a solution of the homogeneous wave equation (2.2.51), then∫ Σr2 φ { φ2 ( T · ψ)2 + |∇/ψ|2} dµgr2 ≤ ∫ Σr1 JM · n dµgr1 (2.2.57) for any r2 > r1 > √ 3 Λ . 134 2.2. LINEAR WAVES ON DE SITTER Note that φ dµgr = 1 2 √ 3 Λ r2 dλ ∧ dµ◦ γ . It is equivalent to consider the multiplier M = 1 1 + √ Λ 3 r ( Y¯ + Y ) . (2.2.58) The associated energy current has the “redshift” property φ∇ · JM ≥ √ Λ 3 1 + √ Λ 3 r JM · n (2.2.59) for any solution of (2.2.51); (here n denotes the unit normal to Σr). The energy identity for JM then implies, in view of the coarea formula, ∫ Σr2 JM · n dµgr2 + ∫ r2 r1 dr √ Λ 3 1 + √ Λ 3 r ∫ Σr JM · n dµgr ≤ ≤ ∫ Σr1 JM · n dµgr1 . (2.2.60) By a Gronwall-type inequality we obtain ( 1 + √ Λ 3 r2 ) ∫ Σr2 JM · n dµgr2 ≤ ( 1 + √ Λ 3 r1 ) ∫ Σr1 JM · n dµgr1 , (2.2.61) which precisely reproduces (2.2.57). In fact, as we shall see below, the argument applies to any multiplier M = ( 1 + √ Λ 3 r )α ( Y¯ + Y ) (α < 0) ; (2.2.62) while for all α < 0 we can establish the redshift property, we “only” have the positivity property for α = 0. In that sense, Prop. 2.7 is sharp. Inhomogeneous wave equation. Let us now consider the modified energy current JM ;mΛ . = JM − mΛ 2 ψ2M ♭ (2.2.63) where M ♭ is simply the 1-form corresponding to the vectorfield M , i.e. M ♭ ·X = g(M,X), and M = ( 1 + √ Λ 3 r )α( Y¯ + Y ) , (α < 0) . (2.2.64) For any solution of the inhomogeneous wave equation (2.2.52) with mΛ > 0, this current then has the “redshift” property, in analogy to (2.2.59), φ∇ · JM ;mΛ ≥ |α| √ Λ 3 1 + √ Λ 3 r JM ;mΛ · n , (2.2.65) 2.2.2. ENERGY ESTIMATES 135 which then similarly to the above leads to ( 1 + √ Λ 3 r2 )|α| ∫ Σr2 JM ;mΛ · n dµgr2 ≤ ( 1 + √ Λ 3 r1 )|α| ∫ Σr1 JM ;mΛ · n dµgr1 . (2.2.66) Proposition 2.8 (Global Energy Boundedness, Inhomogeneous Case). Let ψ be a solution of the inhomogeneous wave equation (2.2.52), then∫ Σr2 φ { φ2 ( T · ψ)2 + |∇/ψ|2 +mΛψ2 } dµgr2 ≤ ≤ ( 1 + √ Λ 3 r1 )|α| ∫ Σr1 JM ;mΛ · n dµgr1 (2.2.67) for any r2 > r1 > √ 3 Λ . 2.2.2.2 Proof of the energy estimates in the expanding region In the following I shall use the double null coordinates (2.2.24) to prove the crucial prop- erties of the currents (2.2.63). However, the basic properties of (2.2.63) that are needed for the argument presented above in Section 2.2.2.1, that lead to Prop. 2.7 and 2.8, do not rely on the specific coordinate system used. In fact, in Section 2.3 the multiplier (2.2.58) will find a suitable geometric generalization for the expanding region of the Schwarzschild de Sitter spacetime and the properties of the corresponding current will be demonstrated in an entirely different coordinate system. Moreover, the following serves as a reference for the argument already presented, and shall not repeat it. Homogeneous wave equation. Recall the definition of the standard energy current associated to the multiplier M , JMµ [ψ] = Tµν [ψ]M ν (2.2.68) where Tµν denotes the standard energy momentum tensor associated to (1.1.1), Tµν [ψ] = ∂µψ ∂νψ − 1 2 gµν ∂ αψ∂αψ . (2.2.69) For any solution of (2.2.51), (2.2.69) is conserved, and therefore ∇ · JM .= ∇µJMµ = (M)πµνTµν [ψ] .= KM [ψ] (2.2.70) where (M)π . = 1 2 LMg is the deformation tensor. It is the content of the divergence theorem combined with the coarea formula that we then have∫ Σr2 JM · n dµgr2 + ∫ r2 r1 dr ∫ Σr φKM dµgr = ∫ Σr1 JM · n dµgr1 , (2.2.71) 136 2.2. LINEAR WAVES ON DE SITTER for any vectorfield M , (all r2 > r1 > √ 3 Λ ); we also refer to (2.2.71) as the energy identity for JM . In order to establish (2.2.56) it is thus enough to show KM ≥ 0. For a general vectorfield M =Mu ∂ ∂u +Mv ∂ ∂v (2.2.72) we calculate KM =− Λ 3 2 (1 + √ Λ 3 r)2 ∂Mu ∂v (∂ψ ∂u )2 − Λ 3 2 (1 + √ Λ 3 r)2 ∂Mv ∂u (∂ψ ∂v )2 − 1 2 ∂µM µ |∇/ψ|2 − 1 2 2 1 + √ Λ 3 r √ Λ 3 (M · r) |∇/ψ|2 + 1 r (M · r) Λ 3 ( 2 1 + √ Λ 3 r )2∂ψ ∂u ∂ψ ∂v . (2.2.73) Here we first set M . = Y¯ + Y = 1 ∂r ∂u ∂ ∂u + 1 ∂r ∂v ∂ ∂v . (2.2.74) This choice should be viewed as a global redshift vectorfield. Indeed, Y¯ and Y separately exploit the redshift effect locally at the cosmological horizons; (Y¯ at u = 0 and Y at v = 0). While it is clear that Y¯ and Y (without a suitable extension) in and by themselves do not give rise to a positive current, the following result shows that the estimate can be symmetrized. Proposition 2.9 (Positivity Property). Let M be defined by (2.2.74), then KM ≥ 0 . (2.2.75) Proof. Here Mu = √ Λ 3 (1− uv)2 2v Mv = √ Λ 3 (1− uv)2 2u . (2.2.76) Thus − ∂M u ∂u − ∂M v ∂v − 2 1 + √ Λ 3 r √ Λ 3 M · r = = √ Λ 3 (1− uv) + √ Λ 3 (1− uv)− 2 √ Λ 3 (1− uv) = 0 , (2.2.77) and ∂Mu ∂v = − √ Λ 3 (1 + uv)(1− uv) 2v2 , (2.2.78) ∂Mv ∂u = − √ Λ 3 (1 + uv)(1− uv) 2u2 . (2.2.79) 2.2.2. ENERGY ESTIMATES 137 Now, 2 r (M · r) ∂ψ ∂u ∂ψ ∂v ≥ −2 r u v (∂ψ ∂u )2 − 2 r v u (∂ψ ∂v )2 , (2.2.80) and therefore KM ≥ Λ 3 1( 1 + √ Λ 3 r )2 {√ Λ 3 [ (1 + uv)(1− uv) v2 − 4u v 1− uv 1 + uv ](∂ψ ∂u )2 + √ Λ 3 [ (1 + uv)(1− uv) u2 − 4v u 1− uv 1 + uv ](∂ψ ∂u )2} = Λ 3 1( 1 + √ Λ 3 r )2 √ Λ 3 (1− uv)3 (1 + uv) [ 1 v2 (∂ψ ∂u )2 + 1 u2 (∂ψ ∂v )2] ≥ 0 . (2.2.81) The energy identity (2.2.71) immediately implies: Corollary 2.10 (Boundary Terms). For any r2 > r1 > √ 3 Λ and all solutions ψ of (1.1.1) we have∫ Σr2 φ { φ2 ( T · ψ)2 + |∇/ψ|2 } dµgr2 ≤ ≤ ∫ Σr1 φ { φ2 ( T · ψ)2 + 1 φ2 (∂ψ ∂r )2 + |∇/ψ|2 } dµgr1 . (2.2.82) It is useful to note that the lapse function may be rewritten as φ = 1 (uv) 1 2 1 1 + √ Λ 3 r . (2.2.83) Proof. To verify (2.2.82) note that JM · n = T (M,φV ), and T (M,V ) = 1 2 Λ 3 ( 1− uv)2[u v (∂ψ ∂u )2 + v u (∂ψ ∂v )2] + ∣∣∇/ψ∣∣2 . (2.2.84) Moreover, ( T · ψ)2 + (V · ψ)2 = 2Λ 3 u2 (∂ψ ∂u )2 + 2 Λ 3 v2 (∂ψ ∂v )2 , (2.2.85) and V = φ−2∂r. We have remarked that it is equivalent to consider the vectorfield M = 1 1 + √ Λ 3 r ( Y¯ + Y ) . (2.2.86) The crucial property (2.2.59) should here be seen as obtained by prescribing a derivative near the cosmological horizons. Indeed we can think of (2.2.86) as obtained by multiplying Y¯ + Y by (−uv) and then adding Y¯ + Y to ensure that the vectorfield remains timelike. We shall give a proof of the “redshift” property (2.2.59) of the multiplier (2.2.86) and its consequences directly in the more general inhomogeneous case. 138 2.2. LINEAR WAVES ON DE SITTER Inhomogeneous wave equation. Let JM ;mΛµ [ψ] . = JMµ [ψ]− mΛ 2 Mµ ψ 2 , (2.2.87) where M . = ( 1 + √ Λ 3 r )α( Y¯ + Y ) (α < 0) . (2.2.88) Then we have for any solution ψ of (2.2.52), ∇ · JM ;mΛ [ψ] = (ψ)M · ψ + Tµν [ψ](M)πµν − mΛ 2 ∇µMµ ψ2 −mΛψM · ψ = KM [ψ]− mΛ 2 ∇ ·M ψ2 . (2.2.89) Proposition 2.11 (Global Redshift Property). Let ψ be a solution of (2.2.52), and let M be defined by (2.2.88), then φ∇ · JM ;mΛ ≥ |α| √ Λ 3 1 + √ Λ 3 r JM ;mΛ · n (r >√ 3 Λ ) , (2.2.90) where n denotes the normal to Σr. Note here, JM ;mΛ · n = JM · n+ mΛ 2 φ[−g(M,V )]ψ2 . (2.2.91) Since with (2.2.88) −g(M,V ) = 2(1 +√Λ 3 r )α , (2.2.92) we obtain control on the energy density JM ;mΛ · n = φ { 2 Λ 3 1( 1 + √ Λ 3 r )2−α[uv (∂ψ∂u )2 + vu(∂ψ∂v )2] + ( 1 + √ Λ 3 r )α|∇/ψ|2 +mΛ(1 +√Λ 3 r )α ψ2 } . (2.2.93) Proof. Here Mu = √ Λ 3 2 v (1− uv 2 )2−α (2.2.94) Mv = √ Λ 3 2 u (1− uv 2 )2−α . (2.2.95) Firstly, −∂M u ∂u − ∂M v ∂v − 2 1 + √ Λ 3 r √ Λ 3 M · r = (−2α) √ Λ 3 ( 1 + √ Λ 3 r )α−1 , (2.2.96) 2.2.2. ENERGY ESTIMATES 139 and secondly, by symmetrizing as in (2.2.80), − 2 Ω2 ∂Mu ∂v (∂ψ ∂u )2 − 2 Ω2 ∂Mv ∂u (∂ψ ∂v )2 + 1 r 4 Ω2 ∂ψ ∂u ∂ψ ∂v ≥ ≥ 2 √ Λ 3 3(1− uv 2 )3−α[(1− uv)2 − α(1 + uv)uv (1 + uv) ]{ 1 v2 (∂ψ ∂u )2 + 1 u2 (∂ψ ∂v )2} ≥ (−2α) √ Λ 3 3 1( 1 + √ Λ 3 r )3−α [uv (∂ψ∂u )2 + vu(∂ψ∂v )2] . (2.2.97) In accordance with (2.2.96) we also find −∇ ·M = √ Λ 3 (−2α)( 1 + √ Λ 3 r )1−α , (2.2.98) and thus φ∇µJM ;mΛµ = φ Tµν [ψ](M)πµν − mΛ 2 φ∇ ·M ψ2 = −αφ √ Λ 3 1 + √ Λ 3 r 2 Λ 3 1( 1 + √ Λ 3 r )2−α [ u v (∂ψ ∂u )2 + v u (∂ψ ∂v )2] − αφ √ Λ 3 1 + √ Λ 3 r ( 1 + √ Λ 3 r )α∣∣∇/ψ∣∣2 − αφmΛ √ Λ 3 1 + √ Λ 3 r ( 1 + √ Λ 3 r )α ψ2 , (2.2.99) which yields (2.2.90) by comparison with (2.2.93). By virtue of the energy identity ∫ Σr2 JM ;mΛ · n dµgr2 + |α| ∫ r2 r1 dr √ Λ 3 1 + √ Λ 3 r ∫ Σr JM ;mΛ · n dµgr ≤ ≤ ∫ Σr1 JM ;mΛ · n dµgr1 , (2.2.100) and the following Gronwall inequality, we can conclude that( 1 + √ Λ 3 r2 )|α| ∫ Σr2 JM ;mΛ · n dµgr2 ≤ ( 1 + √ Λ 3 r1 )|α| ∫ Σr1 JM ;mΛ · n dµgr1 . (2.2.101) Lemma 2.12 (Gronwall Inequality for decreasing functions). Let α < 0, and f, g ∈ C1([r1, r2]) with g 6= 0, satisfying the inequality f ′ ≤ αg ′ g f (2.2.102) on the interval [r1, r2]. Then f(r2) ≤ |g(r1)| |α| |g(r2)||α|f(r1) . (2.2.103) 140 2.2. LINEAR WAVES ON DE SITTER Proof. By (2.2.102), d dr [ f(r) exp [ −α ∫ r r1 g′(r′) g(r′) dr′ ]] ≤ 0 , (2.2.104) which yields (2.2.103) upon integration on the interval [r1, r2]. As an immediate consequence of (2.2.101) we obtain – in view of (2.2.93) – the following estimate, which is precisely the content of Prop. 2.8. Proposition 2.13. Let ψ be a solution of (2.2.52), then ∫ Σr2 φ { φ2 ( T · ψ)2 + 1 φ2 (∂ψ ∂r )2 + |∇/ψ|2 +mΛψ2 } dµgr2 ≤ ≤ ∫ Σr1 φ { φ2 ( T · ψ)2 + 1 φ2 (∂ψ ∂r )2 + |∇/ψ|2 +mΛψ2 } dµgr1 (2.2.105) for all r2 > r1 > √ 3 Λ . 2.2.2.3 Integrated local energy decay and local redshift effect in the static region It is the principal purpose of this Section to establish integrated local energy estimates in the static region. They are required for our proof because the energy flux in the vicinity of the cosmological horizon that we are presented with in Section 2.2.2.1 can only be controlled with a redshift vectorfield that generates error terms away from the horizon. In the following we construct currents that control these error terms using “Morawetz”- type vectorfields. Homogeneous wave equation. It useful to approach the problem first with the aim of contructing an integrated energy estimate in the static region without a redshift com- ponent. We present a slightly simplified argument of [16] for the corresponding region in Schwarzschild de Sitter. Recall that in this region we can use Eddington-Finkelstein coordinates (2.2.30); let us denote by µ . = Λr2 3 . (2.2.106) We shall construct a current based on a multiplier of the form X = f(r) ∂ ∂r∗ . (2.2.107) Let us illustrate the strategy (which may be referred to as the “Morawetz vectorfield method”) by first choosing f(r) = r∗(r) . (2.2.108) 2.2.2.3. INTEGRATED LOCAL ENERGY DECAY 141 The current we then wish to consider is the following modification of the standard energy current associated to X: JX,1µ = J X µ + 1 4 [ 1 + 2 r r∗ ( 1− µ)]∂µ[ψ2]− 1 4 ∂µ [ 1 + 2 r r∗ ( 1− µ)]ψ2 (2.2.109) Note that lim r→ √ 3 Λ r∗(r) ( 1− µ(r) ) = 0 , (2.2.110) because r∗(r) = ∫ r 0 1 1− µ(r) dr = 1 2 √ 3 Λ log ∣∣∣1 + √ Λ 3 r 1− √ Λ 3 r ∣∣∣ . (2.2.111) The divergence of the current (2.2.109) satisfies for any solution ψ of the wave equation (2.2.51) of vanishing spherical mean ψ = 1 4πr2 ∫ S ψ dµ r2 ◦ γ = 0 (2.2.112) the integral inequality∫ S ∇µJX,1µ [ψ] dµr2 ◦γ ≥ ≥ ∫ S { (1− µ)(∂ψ ∂r )2 + 2 r2 r∗(1− µ) r (1 + µ)ψ2 + 2Λ 3 ψ2 } dµ r2 ◦ γ . (2.2.113) By Stokes theorem the energy density on the right-hand side is thus controlled in the static region by the corresponding boundary integrals of (2.2.109) which on the cosmological horizon C take the form ∫ du∗ ∫ S {( JX , ∂ ∂u∗ ) + 1 4 ∂ψ2 ∂u∗ } dµ r2 ◦ γ . (2.2.114) While the terms arising from the modification are finite on the boundary, the main con- tribution to the energy density from T (X, ∂u∗) is of course not finite, because f = r∗ →∞ as r → √ 3 Λ . (2.2.115) It is for this reason that we have to introduce a cut-off in (2.2.108), which in turn neces- sitates the construction of a red-shift vectorfield. For the multiplier (2.2.107) we will construct more generally in Section 2.2.2.4 a current for which the divergence reads ∇µJX,1µ [ψ] = f (1) 1− µ ( ∂ψ ∂r∗ )2 + f r ∣∣∇/ψ∣∣2 + { −1 4 1 1− µf (3) − 1 r f (2) + 2µ r2 f (1) − 2µ 2 r3 f } ψ2 , (2.2.116) 142 2.2. LINEAR WAVES ON DE SITTER where f (i) = d if dr∗i ; note that for f ≥ 0, f (1) ≥ 0, and f (2) ≤ 0 (2.2.113) is replaced by∫ S ∇µJX,1µ [ψ] dµr2 ◦γ ≥ ≥ ∫ S { f (1) 1− µ ( ∂ψ ∂r∗ )2 + 2f r3 (1− µ2)ψ2 − 1 4 1 1− µ d3f dr∗3 ψ2 } dµ r2 ◦ γ , (2.2.117) which holds as a consequence of Poincare´’s inequality if ψ is assumed to verify (2.2.112). As a result the error that is introduced by the cut-off is precisely of the form of the last term in (2.2.117). Now, on very general grounds we know that in view of √ Λ 3 > 0 (2.2.31) the cosmological horizon exhibits a redshift effect and allows for the construction of a multiplier that gives rise to a positive current on the horizon [17]. However, while in the case of black hole horizons it suffices to provide a construction on the event horizon and then to extend the multiplier suitably into the neighborhood by a continuity argument, here an explicit redshift vectorfield is needed which extends beyond the immediate vicinity of the cosmo- logical horizon, namely into the region where the cut-off is introduced (and the last term in the above inequality becomes indefinite). In analogy to [20] we choose as a starting point for the construction of a vectorfield that captures the red-shift effect Y = ( 1 + σ(1− µ))Yˆ + σ(1− µ)T (2.2.118) where Yˆ = 2 1− µ ∂ ∂v∗ , (2.2.119) note that σ > 0 is precisely the parameter that appears in the general construction [17]. (There we construct Y transversal to the horizon and require ∇Y Y = −σ(Y + T ) for the extension to the vicinity of the null hypersurface). This multiplier gives rise to a positive current near the cosmological horizon, in the sense that KY ≥ 0 (1− µ≪ 1) (2.2.120) and controls all derivatives. However, it does not yield control on the error term unless σ is chosen very large; (note that (2.2.120) only holds for 1− µ≪ 1 while the error term may be of order (1 − µ)−1). But we cannot add KY to KX,1 if the magnitude of Y is large, for the latter is required to control the former in the region where the positivity of (2.2.120) fails. We obtain a vectorfield with the desired properties by adding yet another term Y = Y + κ η r∗δ ( Yˆ + T ) , (2.2.121) where κ > 0, δ > 0 and η is a suitable cut-off function supported away from the horizon. Let us also denote by Y c = χY where χ is a cut-off function such that χ = 0 for µ ≪ 1 and χ = 1 for 1− µ ≪ 1. We are then able to show that only a “small” contribution of the redshift effect is required in this construction. 2.2.2.3. INTEGRATED LOCAL ENERGY DECAY 143 A precise version of this statement is given in Proposition 2.20 on page 148. We have seen that the proof of an integrated energy estimate compels the construction of a redshift vectorfield; alternatively we may say that the boundary terms on the cosmological horizon arising from a redshift vectorfield can only be controlled in conjuction with a positive current arising from a Morawetz vectorfield. The construction which is described above and carried out in more detail in the following Section 2.2.2.4 leads us to the result: Proposition 2.14. Let Σ0 be a spacelike hypersurface with normal n in the static region of de Sitter crossing the cosmological horizon C+ to the future of the sphere C+ ∩ C− (in (u, v) coordinates this is the sphere (0, 0)), and denote by C+0 = J+(Σ0)∩C+ the segment of the cosmological horizon to the future of Σ0. Then there exist a strictly timelike vectorfield N (normalized in the sense that g(T,N) is constant on C+0 ) and a constant C(Λ,Σ0) such that for all solutions of the homogeneous wave equation (2.2.51) we have∫ C+0 ∗JN ≤ C(Λ,Σ0) ∫ Σ0 ( Jn, n ) . (2.2.122) Inhomogeneous wave equation. In the inhomogeneous case we can rely on the same vectorfields but our identitites are based on modified currents. Namely, the current JX,1 is replaced by JX,1;mΛ = JX,1 − mΛ 2 X♭ψ2 , (2.2.123) where X♭ · Y = g(X, Y ), and instead of (2.2.116) we have ∇µJX,1;mΛµ [ψ] = f (1) 1− µ ( ∂ψ ∂r∗ )2 + f r ∣∣∇/ψ∣∣2 +mΛ f r ψ2 + { −1 4 1 1− µf (3) − 1 r f (2) + 2µ r2 f (1) − 2µ 2 r3 f } ψ2 . (2.2.124) In our analysis of the inhomogeneous case the zeroth order term steps into the role of the angular derivatives term of the homogeneous case. Indeed, while in (2.2.116) we have conluded by Poincare´’s inequality that the last terms of each line taken together are always nonnegative after integration over the spheres, c.f. (2.2.117), we argue here that mΛ f r ψ2 − 2µ 2 r3 fψ2 = [ mΛ − 2µΛ 3 ]f r ψ2 ≥ 0 if mΛ ≥ 2Λ 3 . (2.2.125) This leads us to the following result. Proposition 2.15. Let Σ0 and C+ be as in Prop. 2.14. Consider solutions ψ to the inhomogeneous wave equation (2.2.52) with mΛ ≥ 2Λ3 . There exists a strictly timelike vectorfield N (as in Prop. 2.14) and a constant C(Λ,Σ0) such that for all solutions ψ we have ∫ C+0 { ∗JN [ψ] + ψ2 } ≤ C(Λ,Σ0) ∫ Σ0 {( Jn[ψ], n ) + ψ2 } . (2.2.126) 144 2.2. LINEAR WAVES ON DE SITTER Remark 2.16 (Conformal Invariance of the wave equation). We expect that the lower bound on the mass mΛ is a shortcoming of the proof of Prop. 2.15, in particular we expect that (2.2.126) holds for all mΛ > 0. However, the value mΛ = 2 Λ 3 does have mathematical meaning, for in this case (2.2.52) is conformal to the wave equation on Minkowski space. More precisely, recall that we can cast the de Sitter spacetime as the subset M = { x ∈ R3+1 : 〈x, x〉 = −(x0)2 + (x1)2 + (x2)2 + (x3)2 > −4 κ } , (2.2.127) endowed with the metric g = Ω2 〈 dx, dx〉 = Ω2 η , (2.2.128) where Ω = 1 1 + κ 4 〈x, x〉 , and κ = Λ 3 . (2.2.129) Then (M, g) is a solution to (2.2.14), namely Rµν = Λ gµν . (2.2.130) Lemma 2.17. Let ψ be a solution to gψ = 2κψ = 2 Λ 3 ψ, then ψ˜ = Ωψ is a solution to g˜ψ˜ = 0 where g˜ = Ω −2g = η is the Minkowski metric. This well known result demonstrates the significance of the specific value mΛ = 2 Λ 3 , and allows us to read off qualitatively our main result in this special case. For a solution of the classical wave equation ηψ˜ = 0 will assume nonvanishing values on the boundary 〈x, x〉 = −4/κ. The decay of ψ is then entirely due to the conformal factor Ω, i.e. the “expansion” of the spacetime. 2.2.2.4 Proof of integrated local energy decay in the static region In this section the argument that was motivated in Section 2.2.2.3 will be carried out in more detail. Homogeneous wave equation. We start with the construction of a suitable current for the multiplier (2.2.107). Morawetz current. Consider the vectorfield X = f(r) ∂ ∂r∗ . (2.2.131) Note that ∂ ∂r∗ = 1 2 ∂ ∂v∗ − 1 2 ∂ ∂u∗ (2.2.132) 2.2.2.3. INTEGRATED LOCAL ENERGY DECAY 145 and ∂f ∂u∗ = −(1− µ)f ′ ∂f ∂v∗ = (1− µ)f ′ . (2.2.133) For the (non-vanishing) connection coefficients of (2.2.30) we find Γu ∗ u∗u∗ = 2Λ r 3 Γv ∗ v∗v∗ = − 2Λ r 3 (2.2.134a) Γu ∗ AB = r 2 ◦ γAB Γ v∗ AB = − r 2 ◦ γAB (2.2.134b) ΓBu∗C = − 1 r ( 1− Λr 2 3 ) δBC Γ B u∗C = 1 r ( 1− Λr 2 3 ) δBC . (2.2.134c) And thus we readily calculate the (non-vanishing) components of the deformation tensor (X)π = LXg (2.2.135) to be (X)πu∗u∗ = (1− µ)2f ′ (X)πv∗v∗ = (1− µ)2f ′ (2.2.136a) (X)πu∗v∗ = −(1− µ)2f ′ + (1− µ)2µ r f (2.2.136b) (X)πAB = f r (1− µ)gAB . (2.2.136c) Therefore KX [ψ] . = (X)παβTαβ [ψ] = = 1 4 f ′ [( ∂ψ ∂u∗ )2 + ( ∂ψ ∂v∗ )2]2 − 1 2 f ′(1− µ)∣∣∇/ψ∣∣2 + 1 2 2µ r f ∣∣∇/ψ∣∣2 + f r ∂ψ ∂u∗ ∂ψ ∂v∗ , (2.2.137) because Tu∗u∗ [ψ] = ( ∂ψ ∂u∗ )2 Tv∗v∗ [ψ] = ( ∂ψ ∂v∗ )2 (2.2.138a) Tu∗v∗ [ψ] = (1− µ) ∣∣∇/ψ∣∣2 (2.2.138b) (g−1)ABTAB = 1 1− µ ∂ψ ∂u∗ ∂ψ ∂v∗ . (2.2.138c) Next, consider the modified current JX,1µ [ψ] = J X µ + 1 4 ( f ′ + 2 r f )( 1− µ)∂µ[ψ2] − 1 4 ∂µ [( f ′ + 2 r f )( 1− µ)]ψ2 (2.2.139) which clearly satisfies KX,1 . = ∇µJX,1µ = KX + 1 2 ( f ′ + 2 r f )( 1− µ)∂µψ∂µψ − 1 4 g [( f ′ + 2 r f )( 1− µ)]ψ2 (2.2.140) 146 2.2. LINEAR WAVES ON DE SITTER for any solution of the homogeneous wave equation (2.2.51). Thus KX,1[ψ] = f ′ ( ∂ψ ∂r∗ )2 + f r ∣∣∇/ψ∣∣2 − 1 4 g [( f ′ + 2 r f )( 1− µ)]ψ2 . (2.2.141) While the modification (2.2.139) is chosen such that the indefinite term in (2.2.137) is cancelled, its usefulness only reveals itself in the precise form that the last term in (2.2.141) takes; here find1 KX,1[ψ] = f ′ ( ∂ψ ∂r∗ )2 + f r ∣∣∇/ψ∣∣2 − 1 4 ( 1− µ)2f ′′′ψ2 − 1 2 ( 2− 5µ)(1− µ)f ′′ r ψ2 + ( 9− 11µ)µ 2 f ′ r2 ψ2 − 2µ2 f r3 ψ2 . (2.2.142) This follows from an elementary albeit lengthy calculation using gψ = − 1 1− µ ∂2ψ ∂t2 + ∂ ∂r [( 1− µ)∂ψ ∂r ] + 2 r ( 1− µ)∂ψ ∂r +△/ ψ . (2.2.143) Note that KX,1 = 2 r3 ( ∂ψ ∂r∗ )2 ≥ 0 for f = − 1 r2 if ∣∣∇/ψ∣∣2 = 0 . (2.2.144) The above formula (2.2.142) becomes more transparent if we instead express it in terms of derivatives f (i) . = dif dr∗i : i = 1, . . . , 3 (2.2.145) Indeed, we then have KX,1[ψ] = f (1) 1− µ ( ∂ψ ∂r∗ )2 + f r ∣∣∇/ψ∣∣2 + { −1 4 1 1− µf (3) − 1 r f (2) + 2µ r2 f (1) − 2µ 2 r3 f } ψ2 . (2.2.146) As explained in Section 2.2.2.3 we have to choose the function f such that it remains bounded as µ → 1, as a consequence of which its derivatives cannot vanish identically. For this reason we discuss the following current which will allow us to control the error terms. Redshift current. The redshift vectorfield is based on Yˆ = 2 1− µ ∂ ∂v∗ . (2.2.147) Since (Yˆ )πu∗u∗ = 4 2µ r (Yˆ )πv∗v∗ = 0 (Yˆ )πu∗v∗ = 0 (2.2.148a) (Yˆ )πAB = 2 r gAB (2.2.148b) 1Note that no boundary terms at the origin r = 0 are generated if limr→0 f r = 1. 2.2.2.3. INTEGRATED LOCAL ENERGY DECAY 147 we immediately obtain K Yˆ [ψ] . = (Yˆ )πµνTµν [ψ] = 1 (1− µ)2 2µ r ( ∂ψ ∂v∗ )2 + 2 r 1 1− µ ∂ψ ∂u∗ ∂ψ ∂v∗ . (2.2.149) Next let us introduce Y = ( 1 + σ(1− µ))Yˆ + σ(1− µ)T (2.2.150) with a parameter σ > 0 to be chosen below. This has the effect that in contrast to (2.2.148) we have (Y )πu∗u∗ = 4 2µ r − σ2µ r (1− µ)2 (Y )πv∗v∗ = σ2µ r (1− µ)2 (2.2.151a) (Y )πu∗v∗ = 2σ 2µ r (1− µ) (2.2.151b) (Y )πAB = 2 r ( 1 + σ(1− µ))gAB (2.2.151c) which in particular recovers the missing derivatives in (2.2.149): KY = 1 4 σ 2µ r ( ∂ψ ∂u∗ )2 + 1 (1− µ)2 2µ r ( ∂ψ ∂v∗ )2 − 1 4 σ 2µ r ( ∂ψ ∂v∗ )2 + σ 2µ r ∣∣∇/ψ∣∣2 + 2 r ( 1 + σ(1− µ)) 1 1− µ ∂ψ ∂u∗ ∂ψ ∂v∗ (2.2.152) Finally, let us also define Y = ( 1 + σ(1− µ) + κ η r∗δ ) Yˆ + ( σ(1− µ) + κ η r∗δ ) T (2.2.153) and Y c = χY , (2.2.154) where η, χ ∈ C∞(R) are cut-off functions which will be specified below, and κ > 0. Lemma 2.18. Let σ ≥ 8, then KY ≥ 1 4 2 r ( ∂ψ ∂u∗ )2 + 2 r 1 4 3 8 1 (1− µ)2 ( ∂ψ ∂v∗ )2 + σ 2µ r ∣∣∇/ψ∣∣2 (2.2.155) for 7 8 ≤ µ ≤ 1. Proof. By Cauchy’s inequality, we have from (2.2.152) for any µ0 > 0 KY ≥ 1 4 2 r ( σ(µ− µ0)− 1 µ0 )( ∂ψ ∂u∗ )2 + 2 r ( µ− µ0 − σ ( 1 µ0 + µ 4 ) (1− µ)2 ) 1 (1− µ)2 ( ∂ψ ∂v∗ )2 + σ 2µ r ∣∣∇/ψ∣∣2 . (2.2.156) For 1−µ ≤ 1 8 , all coefficients are bounded below as given in (2.2.155), if we choose µ0 = 1 2 and σ = 8. 148 2.2. LINEAR WAVES ON DE SITTER Auxiliary current. Choose f = r2 then KX = 2r (∂ψ ∂t )2 + ( 2µ− 1)r∣∣∇/ψ∣∣2 (2.2.157) by (2.2.137). Cut-off parameters. Recall the choice (2.2.108) which corresponds to f (1) = 1. Here we will have instead f (1) = 1, r ≤ R10, r ≥ R2 (2.2.158) with suitable values R1 < R2 <∞. Lemma 2.19. Let ε > 0, and R1 < ∞. Then there exists a finite R2 > R1 and g ∈ C∞([0,∞)) such that g(x) = 1 for x ≤ R1, g(x) = 0 for x ≥ R2 and |g′(x)| ≤ ε x , |g′′(x)| ≤ ε x2 , for x ∈ [R1, R2] . (2.2.159) Given R1 < R2 we choose R0 < R1, R3 > R2 and η ∈ C∞c ( (R0, R3) ) such that η = 1 on [R1, R2]; moreover let χ ∈ C∞ ( [0,∞)) such that χ = 0 for r ≤ R0 and χ = 1 for r ≥ R1. Proposition 2.20. Let J be a current of the form J = JX,1 + JYc + JXa (2.2.160) where X = f(r) ∂r∗, Xa = fa(r) ∂r∗ and J X,1 is given by (2.2.139), and denote the diver- gence of J by K = ∇µJµ. There exist bounded functions f, fa : [0, √ 3 Λ ]→ R (2.2.161) of the area radius r and a future-directed causal vectorfield Yc defined in the static region (with a regular extension to the cosmological horizon) such that for any solution ψ of the homogeneous wave equation (2.2.51) with the vanishing mean property (2.2.112) we have∫ Sr K dµ r2 ◦ γ ≥ 0 (2.2.162) on all spheres (Sr, r 2 ◦ γ) in the static region with 0 < r < √ 3 Λ . Proof. For any given R1 > 0, ε > 0 let f (1) be chosen according to Lemma 2.19 such with R2(R1, ε) > R1 f (1) = 1, r ≤ R10, r ≥ R2 (2.2.163) 2.2.2.3. INTEGRATED LOCAL ENERGY DECAY 149 and f (1) ≥ 0, f (2) ≤ 0 on r ∈ [R1, R2] with |f (2)| ≤  ε|r∗| , R1 ≤ r ≤ R20, r ≤ R1 or r ≥ R2 (2.2.164) |f (3)| ≤  ε|r∗|2 , R1 ≤ r ≤ R20, r ≤ R1 or r ≥ R2 . (2.2.165) We then define f(r) = ∫ r∗(r) 0 f (1) dr∗ ; (2.2.166) note that f ≤ R∗2 <∞. In the following it will be shown that we can choose R1 > 0 large enough and ε > 0 small enough for (2.2.162) to hold with Yc = ε Y c. By Poincare´’s inequality ∫ Sr 1 r ∣∣∇/ψ∣∣2 dµ r2 ◦ γ ≥ ∫ Sr 2 r3 ψ2 dµ r2 ◦ γ (2.2.167) and thus ∫ S KX,1 dµ r2 ◦ γ ≥ 0 ( r ∈ (0, R1] ∪ [R2, √ 3 Λ ) ) . (2.2.168) Y. Recall η ∈ C∞c ( (R0, R3) ) such that η = 1 on [R1, R2]. Since (Y )πu∗u∗ = (Y )πu∗u∗ + κ r∗δ ( 4 + (1− µ))(η(1) − δ η r∗ ) + 4κ η r∗δ 2µ r (2.2.169a) (Y )πv∗v∗ = (Y )πv∗v∗ − κ r∗δ ( η(1) − δ η r∗ ) (1− µ) (2.2.169b) (Y )πu∗v∗ = (Y )πu∗v∗ − 2 κ r∗δ ( η(1) − δ η r∗ ) (2.2.169c) (Y )πAB = (Y )πAB + 2 r κ η r∗δ gAB (2.2.169d) we find (cf. Lemma 2.18) KY = (Y )πµνTµν [ψ] ≥ ≥ 1 4 [( σ(µ− µ0)− 1 µ0 ( 1 + κ η r∗δ ))2 r + ( δ η r∗ − η(1)) κ r∗δ 1 1− µ ]( ∂ψ ∂u∗ )2 + {[ (µ− µ0) ( 1 + κ η r∗δ )− σ( 1 µ0 + µ 4 ) (1− µ)2 ] 2 r − (1 + 1 4 (1− µ))(δ η r∗ − η(1) ) κ r∗δ } 1 (1− µ)2 ( ∂ψ ∂v∗ )2 + [ σ 2µ r + ( δ η r∗ − η(1)) κ r∗δ 1 1− µ ]∣∣∇/ψ∣∣2 , (2.2.170) where we choose in fact µ0 = 1 2 . R1 ≤ r ≤ R2 : Observe that on [R1, R2] we have gained better control for ∣∣∇/ψ∣∣2 in 150 2.2. LINEAR WAVES ON DE SITTER (2.2.170) than in our previous (2.2.155). Here η = 1 and η(1) = 0. In view of (2.2.167) the final term in (2.2.170) is precisely the positive quantity that allows us to control the error introduced by the cut-off of f . For here∫ S KX,1[ψ] dµ r2 ◦ γ ≥ −1 4 ∫ S 1 1− µf (3)ψ2 dµ r2 ◦ γ , (2.2.171) and − 1 4 1 1− µf (3) ≥ −1 4 1 1− µ ε |r∗|2 = = −ε1 4 1 |r∗|1+δ 1 |r∗|1−δ 1 1− µ ≥ −ε κδ |r∗|1+δ 1 1− µ 2 r2 (2.2.172) if 1 4 1 R∗1 1−δ ≤ 2κδ Λ 3 ; (2.2.173) the latter being a condition on the largeness of R1(κ, δ,Λ) <∞. It remains to show that all other coefficients in KY are positive on [R1, R2] as well. This is in fact true up to and including the cosmological horizon. r∗ ≥ R∗ 1 : Let σ ≥ 12, and let R1 in addition to (2.2.173) be chosen large enough so that for all r ≥ R1(σ) 1− µ(r) ≤ 1 σ . (2.2.174) Then µ− µ0 − σ ( 1 µ0 + µ 4 ) (1− µ)2 ≥ 1 6 , (2.2.175) and also[ (µ− µ0) ( 1 + κ η r∗δ )− σ( 1 µ0 + µ 4 ) (1− µ)2 ] 2 r − (1 + 1 4 (1− µ))(δ η r∗ − η(1) ) κ r∗δ ≥ 1 3 1 r − κ(1 + 1 4 (1− µ))δ η|r∗|1+δ ≥ 16 1r , (2.2.176) because κ ≥ 0, η(1) ≤ 0, if 1 |R∗1|1+δ ≤ 1 6 4 5 1 δκ √ Λ 3 (2.2.177) which is satisfied for R1(κ, δ, σ,Λ) chosen large enough. Finally, with σ ≥ 12, µ0 = 12 , σ(µ− µ0)− 1 µ0 ( 1 + κ η r∗δ ) ≥ 2 (2.2.178) if 1 |R∗1|δ ≤ 1 2κ . (2.2.179) Choose R1(κ, δ, σ,Λ) < ∞ large enough such that (2.2.173), (2.2.174), (2.2.177), and (2.2.179) are satisfied, then we have shown by (2.2.172), (2.2.178), and (2.2.176) that for 2.2.2.3. INTEGRATED LOCAL ENERGY DECAY 151 R1 ≤ r < √ 3 Λ :∫ S { KX,1 + εKY } dµ r2 ◦ γ ≥ ≥ ∫ S { ε r ( ∂ψ ∂u∗ )2 + 1 6 ε r 1 (1− µ)2 ( ∂ψ ∂v∗ )2 + ε σ 2µ r ∣∣∇/ψ∣∣2 } dµ r2 ◦ γ (2.2.180) Note that we may choose ε > 0 arbitrarily small, for in this construction this will only result in a sufficiently large choice of R2(ε) > R1(κ, δ, σ,Λ) (cf. Lemma 2.19). Yc. Recall χ = 1 on r ≥ R1, and χ = 0 on r ≤ R0. We define a cut-off of Y by Y c = χY , (2.2.181) whence (Y c)πu∗u∗ = 4χ (1) ( 1 + σ(1− µ))+ κ η r∗δ ) +χ(1) ( σ(1− µ) + κ η r∗δ ) (1− µ) + χ (Y )πu∗u∗ (2.2.182a) (Y c)πv∗v∗ = −χ(1) ( σ(1− µ) + κ η r∗δ ) (1− µ) + χ (Y )πv∗v∗ (2.2.182b) (Y c)πu∗v∗ = −2χ(1) ( 1 + σ(1− µ) + κ η r∗δ ) + χ (Y )πu∗v∗ (2.2.182c) (Y c)πAB = χ (Y )πAB . (2.2.182d) R0 ≤ r ≤ R1: Note that Y c = Y on [R1, R2]. For r ∈ (R0, R1) we only have χ(1) ≥ 0 (and 1 ≥ χ ≥ 0) and can infer from (2.2.182) the estimate KY c ≥ χ KY − 1 4 ( σ(1− µ) + κ η r∗δ ) χ(1) 1− µ ( ∂ψ ∂u∗ )2 − ( 1 + σ(1− µ) + κ η r∗δ ) χ(1) 1− µ ∣∣∇/ψ∣∣2 . (2.2.183) Let in fact R0 be chosen such that µ(R0) = µ0 = 1 2 . (2.2.184) Recall also (2.2.157), to see that then KXa ≥ λ 1− µ (∂ψ ∂t )2 on [R0, R1] , (2.2.185) for any 1 ≥ λ > 0 where fa = λ 1− µ(R1) 1 2R0 r2 , Xa = fa ∂ ∂r∗ . (2.2.186) Thus on [R0, R1] (where f (1) = 1, f (2) = f (3) = 0, and also η(1) ≥ 0) we have using (2.2.146) as well as (2.2.170) and (2.2.183) that∫ S { KX,1 + εKY c +KXa } dµ r2 ◦ γ ≥ 152 2.2. LINEAR WAVES ON DE SITTER ≥ ∫ S { λ 1− µ [( ∂ψ ∂r∗ )2 + (∂ψ ∂t )2] + 2r∗ r3 ( 1− µ2)ψ2 − 1 4 ε χ [ 1 µ0 ( 1 + κ η R∗0 ) 2 R0 + η(1) κ R∗0 1 1− µ(R1) ]( ∂ψ ∂u∗ )2 − 1 4 ε χ(1) (σ 2 + κ η R∗0 ) 1 1− µ(R1) ( ∂ψ ∂u∗ )2 − ε χ [ σ 5 4 1 4 2 R0 + 5 4 δ η κ R∗0 1+δ ] 1 1− µ(R1) 1 1− µ ( ∂ψ ∂v∗ )2 − ε χ η(1) κ R∗0 δ 1 1− µ(R1) 2 R20 ψ2 − ε χ(1) ( 1 + σ 2 + κ R∗0 1+δ ) 1 1− µ(R1) 2 R20 ψ2 } dµ r2 ◦ γ ≥ ≥ ∫ S { 1 4 λ 1− µ [( ∂ψ ∂u∗ )2 + ( ∂ψ ∂v∗ )2] + r∗ r3 ( 1− µ2)ψ2} dµ r2 ◦ γ (2.2.187) provided ε = ε(σ, κ, δ, λ, R0, R1) is chosen sufficiently small. 0 < r < R0: Note finally that ∫ S { KX,1 +KXa } dµ r2 ◦ γ ≥ 0 (2.2.188) is ensured by choosing λ > 0 suitable small. Timelike Killing vectorfield. In the static region (2.2.32) is timelike and (T )π = 0 . (2.2.189) Moreover it will allow us to control all boundary terms of the general current (2.2.160); note here ( JT , n ) = 1 4 1√ 1− µ [( ∂ψ ∂u∗ )2 + ( ∂ψ ∂v∗ )2 + 2(1− µ)∣∣∇/ψ∣∣2] (2.2.190a) ( JT , ∂ ∂v∗ ) = 1 2 ( ∂ψ ∂v∗ )2 + 1 2 (1− µ)∣∣∇/ψ∣∣2 , (2.2.190b) where n = 1√ 1− µT (2.2.191) is the normal to the surfaces of constant t (2.2.27). Foliation. Recall that the proof of Prop. 2.20 provides us with a function 0 ≤ f < ∞ and values 0 < R1 < R2 < √ 3 Λ such that f (1) ≤ 1 for r ≤ R2 and f (1) = 0 for r ≥ R2. We define hypersurfaces Σt by Σt . = ({ (u, v) : 1 2 √ 3 Λ log u −v = t } ∩ { (u, v) : 0 ≤ √ 3 Λ 1 + uv 1− uv ≤ R2 }) ∪ ({ (u, v) : u∗ = 1 2 (t− R∗2) } ∩ { (u, v) : R2 ≤ √ 3 Λ 1 + uv 1− uv ≤ √ 3 Λ }) ; (2.2.192) 2.2.2.3. INTEGRATED LOCAL ENERGY DECAY 153          Ct2t1 r = 0 r = R2 Σt1 Rt2t1 Figure 2.4: Foliation of the static region by hypersurfaces Σt. in other words, Σt coincides with the hypersurface of constant t for r ≤ R2, and with the outgoing null hypersurfaces from the sphere (t, R2) for r ≥ R2. Also denote by Rt2t1 = ⋃ t1≤t≤t2 Σt (2.2.193) the spacetime region enclosed by Σt1 , Σt2 , and the incoming null segment Ct2t1 = { (u, 0) : 1 2 (t1 − R∗2) ≤ u∗ ≤ 1 2 (t2 − R∗2) } ; (2.2.194) compare also figure 2.4. In any of these regions R we can use Stokes’ theorem to derive the energy identities for any of the currents J : ∫ R K dµg = ∫ ∂R ∗J (2.2.195) Here K = ∇µJµ, and dµg = 2(1− µ) r2 du∗ ∧ dv∗ ∧ dµ◦γ . (2.2.196) Lemma 2.21. Let f be as above, and let ψ satisfy ψ = 0. Then for all −∞ < t <∞∫ Σt ( JX,1[ψ], nΣ ) ≤ C(max|f |) ∫ Σt ( JT [ψ], nΣ ) , (2.2.197) where nΣ = n on the spacelike segment, and nΣ = ∂ ∂v∗ on the null segment of Σt. 154 2.2. LINEAR WAVES ON DE SITTER Proof. Recall (2.2.190). r ≤ R2. Clearly, ∣∣∣(JX , n)∣∣∣ ≤ 2|f |(JT , n) , (2.2.198)∣∣∣(JX,1, n)∣∣∣ ≤ 2|f |(JT , n)+ 1 4 ( 1 + 2 r |f | )[ ψ2 + (∂ψ ∂t )2] . (2.2.199) Thus by Poincare´’s inequality (2.2.167)∫ S ∣∣∣(JX,1, n)∣∣∣ dµ r2 ◦ γ ≤ ≤ [ 2|f |+ 1 2 ( 1 + 2 r |f | )( 1 + r2 2 1√ 1− µ(R2) )]∫ S ( JT , n ) dµ r2 ◦ γ . (2.2.200) r ≥ R2. Again, ∣∣∣(JX , ∂ ∂v∗ )∣∣∣ ≤ |f |(JT , ∂ ∂v∗ ) . (2.2.201) Here however,∣∣∣(JX,1, ∂ ∂v∗ )∣∣∣ ≤ ∣∣∣(JX , ∂ ∂v∗ )∣∣∣ + 1 r |f |(1− µ)|ψ||∂v∗ψ|+ 1 4 ∣∣∣∣ ∂∂v∗ [2r f(1− µ)] ∣∣∣∣ψ2 . (2.2.202) Thus∫ S ∣∣∣(JX,1, ∂ ∂v∗ )∣∣∣dµ r2 ◦ γ ≤ ≤ [ |f |+ 1 2 r|f |+ 1 r |f |+ 1 2 + µ|f | ]∫ S ( JT , ∂ ∂v∗ ) dµ r2 ◦ γ . (2.2.203) Since by construction f is bounded and r−1f → 1 as r → 0, (2.2.197) follows from the estimates above with a constant only depending on R2 and max|f | <∞. Note that Lemma 2.21 suffices to estimate the boundary terms arising from the current (2.2.160) on Σt; for we have already shown∣∣∣(JXa , nΣ)∣∣∣ ≤ 2|fa|(JT , nΣ) , (2.2.204) and since Yc and nΣ are both causal we obtain( JYc , nΣ ) = T (Yc, nΣ) ≥ 0 . (2.2.205) Lemma 2.22. On C+, (i) ∣∣∣(JX , ∂∂u)∣∣∣ ≤ |f |(JT , ∂∂u) 2.2.2.3. INTEGRATED LOCAL ENERGY DECAY 155 (ii) ∣∣∣(JX,1, ∂∂u)∣∣∣ ≤ |f |(JT , ∂∂u) . Moreover, on the cosmological horizon C+,( JT+Y c , ∂ ∂u ) ≥ √ Λ 3 u (∂ψ ∂u )2 + √ 3 Λ 1 u ∣∣∇/ψ∣∣2 . (2.2.206) Recall here the coordinate system (u, v), in particular the relation (2.2.21) which shows that ∂ ∂r∗ has a regular extension to the horizon is given by ∂ ∂r∗ = − √ Λ 3 u ∂ ∂u − √ Λ 3 v ∂ ∂v . (2.2.207) We also recall (2.2.32). Proof. Since T |C+ = √ Λ 3 u ∂ ∂u and ∂ ∂r∗ |C+ = − √ Λ 3 u ∂ ∂u we clearly have (i). And by (2.2.139) we have with f bounded and f (1) = 0 simply |(JX,1, ∂u)| = |(JX , ∂u)| on C+, and thus (ii). Now, Y c|C+ = Yˆ |C+ = 2∂r ∂v ∣∣∣ v=0 ∂ ∂v = √ Λ 3 1 u ∂ ∂v , (2.2.208) which yields ( JYc , ∂ ∂u )∣∣∣ C+ = √ Λ 3 1 u Tuv ∣∣ r= √ 3 Λ = √ 3 Λ 1 u ∣∣∇/ψ∣∣2 . We are now able to prove the main conclusion of this section. Proposition 2.23. There exists a timelike vectorfield N , and a constant C only depending on Λ, such that for all solutions ψ of the wave equation (2.2.51) we have∫ Ct2t1 1 u ∣∣∇/ψ∣∣2 ≤ C(Λ) ∫ Σt1 ( JN [ψ], n ) (2.2.209) for all t2 > t1 > 0. Proof. Since ∇/ψ = 0 we may assume without loss of generality that ψ = 0. With the currents used in the proof of Prop. 2.20 we know in particular from (2.2.187) that there exist vectorfields X, Xa, Y c and ε > 0, λ > 0 such that 0 ≤ ∫ Rt2t1 { λ 2 [( ∂ψ ∂u∗ )2 + ( ∂ψ ∂v∗ )2] r2 + 2 r∗ r (1− µ)2(1 + µ)ψ2 } du∗ ∧ dv∗ ∧ dµ◦ γ ≤ ≤ ∫ Rt2t1 { KX,1 + εKY c +KXa } dµg . (2.2.210) Therefore by (2.2.195)∫ Σt2 ( JX,1 + ε JY c + JXa , n ) + ∫ Ct2t1 ∗ ( JX,1 + ε JY c + JXa ) ≤ ≤ ∫ Σt1 ( JX,1 + ε JY c + JXa , n ) . (2.2.211) 156 2.2. LINEAR WAVES ON DE SITTER More precisely, we have with u∗i = 1 2 (ti −R∗2), i = 1, 2, obtained an estimate on: ε ∫ u2 u1 ∫ S ( JY c , ∂ ∂u ) dµ r2 ◦ γ du ≤ 2∑ i=1 ∣∣∣∣∫ Σti ( JX,1 + JXa , n )∣∣∣∣ + ∣∣∣∣∫Ct2t1 ∗ ( JX,1 + JXa )∣∣∣∣+ ε ∫ Σt1 ( JY c , n ) . (2.2.212) In view of Lemma 2.21 and Lemma 2.22 we immediately obtain ε ∫ u2 u1 ∫ S √ 3 Λ 1 u ∣∣∇/ψ∣∣2 dµ r2 ◦ γ du ≤ ≤ C(max{|f |, |fa|}, R2,Λ) ∫ Σt1 ( JT + εJY c , n ) (2.2.213) by (2.2.189). The statement of the Proposition thus follows with N = T + εY c. Remark 2.24. In view of (2.2.206) and (2.2.189) we have in fact proven that there exists a constant C such that∫ u2 u1 ∫ S {√ Λ 3 u (∂ψ ∂u )2 + ε √ 3 Λ 1 u ∣∣∇/ψ∣∣2}r2 dµ r2 ◦ γ du ≤ ≤ C(ε,Λ) ∫ Σt1 ( JT + εJY c , n ) , (2.2.214) for all 0 < u1 < u2 <∞. Inhomogeneous wave equation. As discussed in Section 2.2.2.3 our analysis of the inhomogeneous wave equation is based on the same vectorfields but modified currents. This is motivated by the following insight. Lemma 2.25. Let (M, g) be a spherically symmetric spacetime, g = gab dx a dxb + r2 ◦ γ= −Ω2 du dv + r2 ◦γ , (2.2.215) and Y a vectorfield on the quotient plane, Y = Y u∂u + Y v∂v. Let the modified current of JY be defined by JY ;mΛ [ψ] = JY [ψ]− mΛ 2 Y ♭ψ2 , (2.2.216) then for all solutions ψ of (2.2.52) we have ∇ · JY ;mΛ = 2∑ a,b=1 Kab∂aψ ∂bψ +K/ ∣∣∇/ψ∣∣2 +mΛKψ2 , (2.2.217) where K = K/ = −1 2 tr (Y )π , (2.2.218) 2.2.2.3. INTEGRATED LOCAL ENERGY DECAY 157 and Kuu = − 2 Ω2 ∂Y u ∂v Kvv = − 2 Ω2 ∂Y v ∂u (2.2.219) Kuv = Kvu = − 2 Ω2 1 r ( Y · r) . (2.2.220) Note that JY ;mΛ reduces to JY if mΛ = 0. The relevance of Lemma 2.25 lies however in the fact that if mΛ > 0 then the coefficients to the zeroth order term are the same as for the angular derivatives term, as expressed in (2.2.218). We have already stated an instance of this result for the Morawetz current, namely in (2.2.123) and (2.2.124). But we shall also apply this modification to the currents JYc and JT to obtain: Proposition 2.26. There exists a timelike vectorfield N , and a constant C only depending on Λ, such that for all solutions ψ of the inhomogeneous wave equation (2.2.52) we have∫ Ct2t1 1 u {∣∣∇/ψ∣∣2 +mΛψ2} ≤ C(Λ) ∫ Σt1 ( JN ;mΛ [ψ], n ) (2.2.221) for all t2 > t1 > 0, provided mΛ ≥ 2Λ3 . Proof. Let ε > 0, R1 > 0, and let f , fa, X, Xa, and Y c be constructed as in the proof of Prop. 2.20. Let JX,1;mΛ be defined by (2.2.123), then we have on one hand ∇ · JX,1;mΛ ≥ [ mΛ − 2µΛ 3 ]f r ψ2 ≥ 0 , on r ∈ (0, R1] ∪ [R2,√ 3 Λ ] , (2.2.222) since mΛ ≥ 2Λ3 , and ∇ · JX,1;mΛ [ψ] ≥ −1 4 1 1− µf (3)ψ2 , on r ∈ [R1, R2] . (2.2.223) On the other hand, by Lemma 2.25 and (2.2.170) we have ∇ · JY c;mΛ ≥ δκ r∗1+δ 1 1− µψ 2 , on r ∈ [R1, R2] , (2.2.224) which shows that the modified redshift current associated to Yc = εY c cancels the error (2.2.223) similarly to (2.2.172) provided R1 is chosen large enough. By Lemma 2.25 and the proof of Prop. 2.20 we also have that ∇ · JY c;mΛ is positive on R1 ≤ r < √ 3 Λ , and that ∇ · JX,1;mΛ + ε∇ · JY ;mΛ +∇ · JXa;mΛ ≥ 0 , on 0 < r < R1 , (2.2.225) since ε can be chosen arbitrarily small. The boundary terms of the currents JX,1:mΛ and JXa;mΛ are controlled by the current JT ;mΛ = JT − mΛ 2 T ♭ψ2 (2.2.226) 158 2.2. LINEAR WAVES ON DE SITTER which is conserved by (2.2.52) because T is Killing, ∇ · JT ;mΛ = KT − mΛ 2 ∇ · T ψ2 , (2.2.227) KT = (T )πµνTµν = 0 ∇ · T = tr (T )π = 0 . (2.2.228) Note for example, for the boundary terms on surfaces of constant t, JT ;mΛ · T = 1 4 ( ∂ψ ∂u∗ )2 + 1 4 ( ∂ψ ∂u∗ )2 + 1 2 (1− µ)∣∣∇/ψ∣∣2 + mΛ 2 (1− µ)ψ2 . (2.2.229) Finally, since Yc ∣∣∣ C+ = εYˆ ∣∣∣ C+ = √ Λ 3 ε u ∂ ∂v , (2.2.230) we obtain for the boundary terms of the redshift vectorfield on the cosmological horizon: JYc;mΛ · ∂ ∂u ∣∣∣ C+ = √ 3 Λ 2ε u ∣∣∇/ψ∣∣2 +mΛ√ 3 Λ ε u ψ2 . (2.2.231) The result then follows with N = T + εY c. Remark 2.27. Taking into account the contribution from the vectorfield T we have in fact shown that if mΛ ≥ 2Λ3 then there exists a constant C such that∫ ∞ u′ du ∫ S2 dµ◦ γ { Λ 3 u (∂ψ ∂u )2 + 1 u ∣∣∇/ψ∣∣2 +mΛ 1 u ψ2 }∣∣∣ v=0 ≤ C(Λ) ∫ Σt′ ( JN ;mΛ , n ) , (2.2.232) where 2(u′)∗ = t−R∗1. 2.2.2.5 The redshift effect on the cosmological horizon In this Section we tie our results for the expanding region in Section 2.2.2.1 to our results for the static region in Section 2.2.2.3. The energy estimates of Section 2.2.2.1 provide control on solutions to the linear wave equation up to and including the spacelike future boundary of the spacetime in terms of energies on spacelike hypersurfaces that can be chosen arbitrarily close to the cosmological horizon. In Section 2.2.2.3 we have in particu- lar established that we can control the nondegenerate energy on the cosmological horizon. Here we prove that as a consequence of the redshift effect the latter controls the energies arising in Section 2.2.2.1. It is in particular a consequence of this Section that our results in Thm. 4 and 5 can expressed in terms of initial data prescribed on an arbitrary spacelike hypersurface. Redshift vectorfield. In Section 2.2.2.1 we have found a redshift vectorfield based on the combination Y¯ + Y . The positivity property of the associated current depends on a symmetrization argument that is only valid for r > √ 3 Λ . Here we construct a redshift 2.2.2. ENERGY ESTIMATES 159 vectorfield on the cosmological horizon and its vicinity based on the vectorfields Y¯ and Y separately. We will address the general case mΛ ≥ 0 directly. Let us recall the vectorfield T from (2.2.32) and its properties summarized in Lemma 2.5, and set Y ∣∣∣ C+ = 2 ∂r ∂v ∣∣∣ v=0 ∂ ∂v . (2.2.233) We have g(T, Y ) ∣∣∣ C+ = −2 , (2.2.234) and [ T, Y ]∣∣∣ C+ = 0 . (2.2.235) Extend, as in the construction for black hole horizons [17], the vectorfield Y into the expanding region by ∇Y Y = −σ(Y + T ) , (2.2.236) where σ > 0. It is then easy to show (cf. [17]) that KY = Tµν [ψ] (Y )πµν ≥ 1 4 √ Λ 3 ( Y · ψ)2 + [σ 2 − 4 √ Λ 3 ]( T · ψ)2 + σ 2 ∣∣∇/ψ∣∣2 , (2.2.237) which is positive for σ > 8 √ Λ 3 . However, in contrast to the construction for black hole horizons the vectorfield Y defined by the extension (2.2.236) does not remain causal in the expanding region.2 But, as we shall see next T + Y remains timelike in the vicinity of C+. The condition (2.2.236) is easily shown to be equivalent to dY u dv ∣∣∣ v=0 = −σu2 (2.2.238) dY v dv ∣∣∣ v=0 = −2 √ Λ 3 − σ . (2.2.239) Also, Y ∣∣∣ C+ = √ Λ 3 1 u ∂ ∂v T ∣∣∣ C+ = √ Λ 3 u ∂ ∂u . (2.2.240) For σ = 8 √ Λ 3 we thus obtain Y + T ∣∣∣ (u,v) = √ Λ 3 ( 1− 8 uv)u ∂ ∂u + √ Λ 3 ( 1 uv − 11) v ∂ ∂v +O(v2) . (2.2.241) Therefore we can choose σ > 8 √ Λ 3 and r0(σ) > √ 3 Λ sufficiently small such that KN ≥ 0 and T + Y is timelike on √ 3 Λ ≤ r ≤ r0 to the future of a given u0 > 0. 2On the level of Penrose diagrams the extension of the vectorfield Y from the cosmological horizon the the expanding region corresponds in the black hole case to the extension from the event horizon to the interior of the black hole. 160 2.2. LINEAR WAVES ON DE SITTER Redshift current. We consider in the general case mΛ ≥ 0 the current JN ;mΛ = JN − mΛ 2 N ♭ψ2 , (2.2.242) associated to the multiplier N ∣∣∣ (u,v) = √ Λ 3 ( 1− 10 uv)u ∂ ∂u + √ Λ 3 ( 1 uv − 13) v ∂ ∂v ; (2.2.243) (this vectorfield arises in the above construction as N = T + Y with σ = 10 √ Λ 3 ). Proposition 2.28. Let Σ be a spacelike hypersurface with normal n crossing the cosmo- logical horizon C+ to the future of the sphere C+∩C−. Let ψ be a solution to (2.2.52) with mΛ > 0 or mΛ = 0, and r1 > √ 3 Λ a fixed radius close enough to the horizon such that√ Λ 3 r1 − 1√ Λ 3 r1 + 1 ≤ 1 15 , (2.2.244) and denote by Σ′ the segment of Σ truncated by Σr1 and C+, and by C+0 the segment of C+ truncated by Σ. Then∫ Σr1∩J+(Σ′) φ { φ2 ( T · ψ)2 + 1 φ2 (∂ψ ∂r )2 + ∣∣∇/ψ∣∣2 +mΛψ2 } dµgr1 ≤ ≤ C(r1,Λ) ∫ C+0 { Λ 3 u ( ∂ ∂u ψ )2 + 1 u ∣∣∇/ψ∣∣2 + mΛ u ψ2 } + C(r1) ∫ Σ′ Jn;mΛ · n , (2.2.245) where C is a constant that only depends on the chosen value of r1, and Λ. Remark 2.29. Note that the energy on the right hand side of (2.2.245) is precisely of the form controlled in (2.2.232). Proof. We have ∇ · JN ;mΛ = KN − mΛ 2 ∇ ·N ψ2 , (2.2.246) and by construction on C+: KN ≥ 1 4 √ Λ 3 ( Y · ψ)2 +√Λ 3 ( T · ψ)2 + 5√Λ 3 ∣∣∇/ψ∣∣2 . (2.2.247) By continuity the positivity of KN ≥ 0 holds up to a fixed radius r1 > √ 3 Λ . Using the explicit expression (2.2.243) we find ∇ ·N = ∇µNµ = −12 √ Λ 3 − 2 √ Λ 3 10 uv + 2 1− uv √ Λ 3 − 2 uv 1− uv √ Λ 3 (12 + 10uv) , (2.2.248) 2.2.2.6. POINTWISE ESTIMATES 161 and thus on uv ≤ 1 2 : −∇ ·N ≥ 8 √ Λ 3 . (2.2.249) It remains to calculate the boundary terms of the current JN ;mΛ using the expression (2.2.243). Let in addition uv ≤ 1 15 then on Σr JN ;mΛ · n = φ JN ;mΛ · V = φ T (N, V ) + mΛ 2 φ [−g(N, V )]ψ2 ≥ φ uv [1 3 Λ 3 u v (∂ψ ∂u )2 + 2 Λ 3 v u (∂ψ ∂v )2 + 1 2 ( 1 + √ Λ 3 r )2∣∣∇/ψ∣∣2] + mΛ 2 φ 1 2 1 10 ( 1 + √ Λ 3 r )2 ψ2 ≥ 1 6 φ−1 [ φ2 ( T · ψ)2 + 1 φ2 (∂ψ ∂r )2 + ∣∣∇/ψ∣∣2]+ 1 6 mΛ 8 ( 1 + √ Λ 3 r )2 ψ2 , (2.2.250) and on C+ JN ;mΛ · ∂ ∂u = √ 3 Λ 1 u JN ;mΛ · T = √ 3 Λ 1 u T (T + Y, T )− √ 3 Λ 1 u mΛ 2 g(T + Y, T )ψ2 = √ 3 Λ 1 u [( T · ψ)2 + ∣∣∇/ψ∣∣2 +mΛψ2] . (2.2.251) The estimate of the Proposition then follows from the energy identity for JN ;mΛ on the domain bounded by C+, Σr1 and Σ provided r1 > √ 3 Λ is chosen small enough (in particular such that uv ≤ 1/15 on Σr1), with C(Λ, r1) = 6 8 Λ 3 r21 − 1 . (2.2.252) 2.2.2.6 Pointwise estimates on the timelike future boundary Pointwise estimates follow from energy estimates by Sobolev inequalities. Here we are interested in the relevant inequality on Σr. Proposition 2.30 (Sobolev inequality on Σr). Let ψ ∈ H3(Σr), r > √ 3 Λ , then √ 3 Λ (Λ 3 r2 − 1 ) r2 sup p,q∈Σr |ψ2(p)− ψ2(q)| ≤ ≤ C(Λ) ∫ Σr { ψ2 + 3∑ i=1 ( Ω(i)ψ )2 + 3∑ i,j=1 ( Ω(i)Ω(j)ψ )2 + ( Tψ )2 + 3∑ i=1 ( Ω(i)Tψ )2 + 3∑ i,j=1 ( Ω(i)Ω(j)Tψ )2} dµgr . (2.2.253) 162 2.2. LINEAR WAVES ON DE SITTER The angular derivatives on the sphere are here estimated using the generators of the spherical isometries Ω(i) : i = 1, 2, 3. This fact is known as the coercivity inequality on the sphere which is discussed in Appendix C.2, where also the precise definition of the vectorfields Ω(i) can be found. Proof. Recall the induced metric on Σr derived in Section 2.2.1.3, gr = 1 4 3 Λ (Λ 3 r2 − 1 ) dλ2 + r2 ◦ γ . (2.2.254) Let λ , λ0 ∈ (−∞,∞), and ξ ∈ S2, then |ψ2(λ; ξ)− ψ2(λ0; ξ)| ≤ 2 ∫ λ λ0 |ψ||∂ψ ∂λ | dλ (2.2.255) and by (2.2.49) ∫ λ λ0 |∂ψ ∂λ |2 dλ = ∫ λ λ0 1 4 3 Λ ( Tψ )2 dλ . (2.2.256) Therefore by Corollary C.3 |r2ψ2(λ)− r2ψ2(λ0)| ≤ ≤ C ∫ ∞ −∞ ∫ Sr |ψ|2 + 3∑ i=1 ( Ω(i)ψ )2 + 3∑ i,j=1 ( Ω(i)Ω(j)ψ )2 dµγr dλ + C ∫ ∞ −∞ 1 4 3 Λ ∫ Sr |Tψ|2 + 3∑ i=1 ( Ω(i)Tψ )2 + 3∑ i,j=1 ( Ω(i)Ω(j)Tψ )2 dµγr dλ , (2.2.257) which proves the stated inequality in view of (2.2.48). For a solution ψ to the wave equation Proposition 2.30 applied to the functions Tψ and Ω(i)ψ : i = 1, 2, 3 yields quantities on the right hand side which are monotone by our results of Section 2.2.2.1. As a consequence we obtain the following pointwise bounds. Proposition 2.31 (Pointwise estimates). Let ψ be a solution to the linear wave equation (1.1.1) with mΛ ≥ 0, and assume that D[ψ; Σr1] . = ∫ Σr1 φ { φ2 ( T · ψ)2 + 1 φ2 (∂ψ ∂r )2 + |∇/ψ|2 +mΛψ2 } dµgr1 <∞ (2.2.258) for some r1 > √ 3 Λ ; moreover assume that also D[Ω(i)ψ,Σr1 ] <∞ : i = 1, 2, 3 , D[Ω(i)Ω(j)ψ,Σr1 ] <∞ : i, j = 1, 2, 3 as well as D[Tψ,Σr1] <∞, and D[Ω(i)Tψ,Σr1] <∞ : i = 1, 2, 3 , D[Ω(i)Ω(j)Tψ,Σr1] <∞ : i, j = 1, 2, 3 . 2.3. LINEAR WAVES ON SCHWARZSCHILD DE SITTER 163 Then for all r > r1 also r2 sup p,q∈Σr ∣∣∣|∇/ψ(p)|2γr − |∇/ψ(q)|2γr ∣∣∣ <∞ , (2.2.259) and (rφ)2 sup p,q∈Σr ∣∣∣(Tψ)2(p)− (Tψ)2(q)∣∣∣ <∞ , (2.2.260) and if mΛ > 0 also sup p,q∈Σr ∣∣∣(rψ)2(p)− (rψ)2(q)∣∣∣ <∞ , (2.2.261) and there exists a constant C that only depends on Λ such that for all r > r1 sup p,q∈Σr { (rφ)2 ∣∣∣(Tψ)2(p)− (Tψ)2(q)∣∣∣ + r2 ∣∣∣|∇/ψ(p)|2γr − |∇/ψ(q)|2γr ∣∣∣+mΛ∣∣∣(rψ)2(p)− (rψ)2(q)∣∣∣} ≤ ≤ C(Λ) [ D[ψ; Σr1] + 3∑ i=1 D[Ω(i)ψ; Σr1 ] + 3∑ i,j=1 D[Ω(i)Ω(j)ψ; Σr1 ] +D[Tψ; Σr1 ] + 3∑ i=1 D[Ω(i)Tψ; Σr1] + 3∑ i,j=1 D[Ω(i)Ω(j)Tψ; Σr1 ] ] . (2.2.262) Proof. Apply Proposition 2.30 to the functions φ 3 2T · ψ and φ 12 1 r Ω(i)ψ : i = 1, 2, 3 as well as (φmΛψ) 1 2 and use Propositions 2.7 and 2.9. Note that [T,Ω(i)] = 0 : i = 1, 2, 3. We may now proceed as before in Section 2.2.2.5 to estimate the energies on the right hand side of (2.2.262) in terms of energies on a spacelike hypersurface that crosses the cosmological horizon. This yields Theorem 5. 2.3 Linear Waves on Schwarzschild de Sitter This Section applies the ideas and constructions of Section 2.2 to the global study of solutions to gψ = 0 (2.3.1) on Schwarzschild de Sitter spacetimes (M(m)Λ , g). We have chosen to view the homogeneous de Sitter spacetime as a member of the spher- ically symmetric Schwarzschild de Sitter family corresponding to the parameter m = 0. This shall allow us to proceed analogously to the analysis in Section 2.2 in the case m > 0. We recall in Section 2.3.1 that the causal geometry of the expanding region is qualitatively the same as in the case m = 0, namely it is bounded in the timelike future by a spacelike hypersurface along which the area of every sphere is infinitely large, and in the past by the 164 2.3. LINEAR WAVES ON SCHWARZSCHILD DE SITTER cosmological horizons C+∪C¯+. Moreover this region is foliated by spacelike hypersurfaces of constant area radius. The adjacent domains beyond the cosmological horizons are as in the case m = 0 static regions of spacetime, which are in turn bounded by black hole event horizons and the interior domains of black holes. However, for our analysis the region of interest is the expanding region and its extension across the cosmological horizons; correspondingly we derive in Section 2.3.1 a double null foliation which covers this domain. For definiteness, the global picture described above is correct for any choice of Λ > 0 and 0 < m < (3 √ Λ)−1. The Schwarzschild de Sitter spacetimes (M(m)Λ , g) are spherically symmetric and have topology Q(m)Λ ×SO(3) where Q(m)Λ is a 1+1-dimensional Lorentzian manifold; as discussed in Section 2.3.1 they are unique as solutions to (2.2.14) with that topology for Λ > 0. We denote as above by r the area radius of the orbits of the SO(3) group action. It turns out that the expanding region corresponds to r > rC where rC(Λ, m) is a root of a polynomial of degree 3 with coefficients in Λ and m, and that the metric on this domain takes the form g = −φ2 dr2 + gr = − 1 Λr2 3 + 2m r − 1 dr 2 + gr , (2.3.2) where gr is the Riemannian metric of a cylinder on the level sets Σr of r as a function on Q(m)Λ ; (this expression should be compared to (2.2.2)). The cosmological horizons are the null hypersurfaces r = rC, and have positive surface gravity κC > 0 as in the case m = 0. Global Redshift Vectorfield. The crucial estimate for our analysis is given in Section 2.3.2.1 where we establish the global redshift property of the vectorfield M = 1 r ∂ ∂r ; (2.3.3) here ∂r = φn, where n is the normal to Σr. Indeed, according to Proposition 2.34 we have for all solutions of (2.3.1) the lower bound φ∇ · JM [ψ] ≥ 1 r JM [ψ] · n , (2.3.4) where JM is the energy current associated to the multiplier M ; (recall the discussion in Section 2.2.2.2). By the energy identity for JM on Rr2r1 . = ⋃ r1≤r≤r2 Σr , (2.3.5) we thus obtain∫ Σr2 JM · n dµgr2 + ∫ r2 r1 dr 1 r ∫ Σr JM · n dµgr ≤ ≤ ∫ Σr2 JM · n dµgr2 + ∫ Rr2r1 KM dµg = ∫ Σr1 JM · n dµgr1 , (2.3.6) 165 which implies (compare the discussion of the inhomogeneous wave equation in Section 2.2.2.2, in particular Lemma 2.12): r2 ∫ Σr2 JM · n dµgr2 ≤ r1 ∫ Σr1 JM · n dµgr1 (r2 > r1 > rC) . (2.3.7) This is the content of Corollary 2.35. Local Redshift Effect. In Lemma 2.33 we state the strict positivity of the surface gravity of the cosmological horizons C¯+ ∪ C+ in (M(m)Λ , g). While the global redshift captured above can be attributed to the expansion of the spacetime, a local redshift is due to the stated property of the horizons. In Section 2.3.2.2 we apply the general construction provided by [17] to the cosmological horizons, to estimate the right hand side of (2.3.7). More precisely, we construct in Proposition 2.36 a vectorfield N = T + Y with Y ∣∣∣ C+ = 2 ∂r ∂v ∣∣∣ v=0 ∂ ∂v (2.3.8) which is timelike and gives rise to a positive current ∇ · JN ≥ 0 (2.3.9) in a neighborhood of C+ which is of the form{ rC ≤ r ≤ r0 } ∩ J+(Σ) (2.3.10) where Σ is an achronal hypersurface that crosses C¯+ ∪ C+ to the future of C¯+ ∩ C+, and r0 only depends on Λ, m and our choice of Σ. Let now r1 ≤ r0 in (2.3.7). It follows in particular from (2.3.91) and the energy identity for JN on (2.3.10) that for the right hand side of (2.3.7) we have r1 ∫ Σr1∩J+(Σ) JM · n dµgr1 ≤ ≤ C(r0) ∫ C+∩J+(Σ) ∗JT+Y + C(r0) ∫ Σ∩ {rC≤r≤r0} JN · n dµg . (2.3.11) Given any spacelike hypersurface Σ the energy on the left hand side of (2.3.7) is thus bounded for all r2 such that Σr2 ⊂ J+(Σ) by the energy on Σ and the nondegenerate energy with respect to N on the segments of the cosmological horizons lying to the future of Σ. 166 2.3. LINEAR WAVES ON SCHWARZSCHILD DE SITTER Static regions. A large part of our analysis of linear waves on de Sitter is preoccupied with proving an integrated local energy estimate in the static regions which allows us to control the nondegenerate energy with the respect to the redshift vectorfield on the cosmological horizon; see Section 2.2.2.3. The corresponding proof for the static regions of Schwarzschild de Sitter is more involved due to the presence of trapping ; however the required result to close our estimate has already been obtained in [16]. Proposition 2.32 (Prop. 10.3.2, [16]). Let Σ be a spacelike hypersurface with normal n in the static region rH ≤ r ≤ rC of (M(m)Λ , g) crossing the horizons to the future of the bifurcation spheres, and let ψ be a solution (2.3.1). Then there exists a constant C (only depending on Λ, m and Σ) such that∫ C+∩J+(Σ) ∗JY [ψ] ≤ C ∫ Σ Jn[ψ] · n dµg . (2.3.12) While Y is constructed in [16] using Eddington Finkelstein coordinates for the static region, it coincides with our definition (2.3.8) on the cosmological horizon; in fact the definiton of Y in [16] is in the same spirit as our construction of Y c for the static region of de Sitter (see discussion of the redshift vectorfield in Section 2.2.2.3, in particular (2.2.118), (2.2.119) and (2.2.121)). Hence (also in view of the fact that T is globally a Killing vectorfield) we are able to control the energy flux through the cosmological horizon in (2.3.11) in terms of a naturally prescribed energy on an initial spacelike hypersurface Σ. Future Boundary. The induced metric on the level sets Σr in (2.3.2) is in fact given by gr = (Λr2 3 + 2m r − 1 ) dt2 + r2 ◦ γ , (2.3.13) and we find φ dµgr = r 2 dt ∧ dµ◦ γ . (2.3.14) In view of the results stated above we are then allowed to take the limit on the left hand side of (2.3.7) to conclude on the finiteness of the following integral: lim r→∞ r ∫ Σr JM · n dµgr = ∫ Σ+ { 3 Λ (∂ψ ∂t )2 + r2 ∣∣∇/ψ∣∣2} dt ∧ dµ◦ γ . (2.3.15) This is the main result of our work. Theorem 6. Let Σ+ be the future boundary of the expanding region in (M(m)Λ , g) endowed with the volume form dt∧ dµ◦ γ of the standard cylinder, and let Σ ⊂ J−(Σ+) be a spacelike hypersurface with normal n in the past of Σ+ (and Σ+ in the domain of dependence of Σ) crossing the horizons to the future of the bifurcation spheres (see figure 2.5). Then there 167 Σ+ Σ C¯+C+H+ H¯+ C+ ∩ C¯+ Figure 2.5: Cauchy Problem for Theorem 6. exists a constant C that only depends on Λ, m and Σ such that for all solutions to (2.3.1) with initial data prescribed on Σ such that D[ψ] = ∫ Σ Jn[ψ] · n dµg <∞ , (2.3.16) we have that the energy is globally bounded in the domain of dependence of Σ and satisfies the bound ∫ Σ+ {(∂ψ ∂t )2 + ∣∣ ◦∇/ ψ∣∣2} dt ∧ dµ◦ γ ≤ C(Λ, m,Σ)D (2.3.17) on the future boundary Σ+. Alternatively we can write the result (2.3.17) using the coercivity equality on the sphere (see Appendix C.2) as ∫ Σ+ {( T · ψ)2 + 3∑ i=1 ( Ω(i)ψ )2} ≤ C D , (2.3.18) and note that the tangent space to Σr is spanned by the Killing vectorfields T , and Ω(i) : i = 1, 2, 3. This immediately implies the following pointwise estimates, (similarly to Section 2.2.2.6 in the de Sitter case). Theorem 7. Let Σ+ and Σ be as in Thm. 6. There exists a constant C(Λ, m,Σ) such that for all solutions to (2.3.1) which satisfy in addition to (2.3.16) the condition Dc[ψ] = D[ψ] + 3∑ i=1 D[Ω(i)ψ] + 3∑ i,j=1 D[Ω(i)Ω(j)ψ] +D[Tψ] + 3∑ i=1 D[Ω(i)Tψ] + 3∑ i,j=1 D[Ω(i)Ω(j)Tψ] <∞ (2.3.19) we have the pointwise estimates sup p,q∈Σ+ ∣∣∣∣∣∣∣(∂ψ∂t )2(p)− (∂ψ∂t )2(q) ∣∣∣+ ∣∣∣∣∣ ◦∇/ ψ∣∣2(p)− ∣∣ ◦∇/ ψ∣∣2(q)∣∣∣∣∣∣∣ ≤ C(Λ, m,Σ)Dc (2.3.20) on the future boundary Σ+. We conclude that solutions to the linear wave equation on Schwarzschild de Sitter space- times decay in the expanding region to a function on the spacelike future boundary given by a function on R× S2 with bounded derivatives. 168 2.3. LINEAR WAVES ON SCHWARZSCHILD DE SITTER 2.3.1 Geometry of the expanding region of Schwarzschild de Sitter In this section we shall discuss the Schwarzschild de Sitter spacetimes as solutions to (2.2.14) with positive Λ > 0 and m > 0. It is here useful to recall our discussion of the global geometry of de Sitter in Section 1.2. Spherically symmetric cosmological spacetimes with positive mass. Recall that the mass function m on the quotient Q =M/SO(3) of a spherically symmetric solution (M, g) to (2.2.14) defined by 1− 2m r − Λr 2 3 = − 4 Ω2 ∂r ∂u ∂r ∂v (2.3.21) is constant by virtue of the Hessian equations (2.2.16). The massm parametrizes precisely the Schwarzschild de Sitter family for any given Λ > 0 and correspondingly we choose here m > 0 . (2.3.22) We have also seen that as a consequence of (2.2.16) the function r∗ = ∫ 1 1− 2m r − Λr2 3 dr (2.3.23) is a solution to (2.2.20). Thus we have that in a given double null coordinate system (u, v) on Q the “tortoise” coordinate has the form r∗ = f(u) + g(v) , (2.3.24) where f , g may be any functions on R. The different charts of the manifold (M, g) are obtained in explicit form for different choices of centering (2.3.23) and suitable choices of f , g in (2.3.24). In particular the charts that cover the horizons (where the right hand side of (2.3.21) vanishes) are found in this manner. In contrast however to the case m = 0, there is no single chart that covers the entire spacetime manifold for m > 0. In the following we shall briefly discuss a double null coordinate system that covers the cosmological horizons and extends to the adjacent regions, in particular to the future into the expanding region. Extension across the cosmological horizons. The polynomial on the left hand side of (2.3.21) has three distinct real roots in r provided 3m < 1√ Λ . (2.3.25) In fact, as discussed in [34], if we set cos ξ = −3m √ Λ (π < ξ < 3π 2 ) , (2.3.26) 2.3.1. GEOMETRY OF THE EXPANDING REGION 169 then the three roots are given by rH = 2√ Λ cos ξ 3 (2.3.27a) rC = 2√ Λ cos ξ + 4π 3 (2.3.27b) rC = 2√ Λ cos ξ + 2π 3 , (2.3.27c) and satisfy rC < 0 < 2m < rH < 3m < rC . (2.3.28) In particular we have r − 2m− Λr 3 3 = −Λ 3 (r − rH)(r − rC)(r + |rC|) , (2.3.29) with −|rC|+ rC + rH = 0 (2.3.30a) rC|rC|+ rH|rC| − rHrC = 3 Λ (2.3.30b) rHrC|rC| = 6m Λ , (2.3.30c) and by decomposition into partial fractions 1 1− 2m r − Λr2 3 = 3 Λ rH (rC − rH)(rH + |rC|) 1 r − rH − 3 Λ rC (rC − rH)(rC + |rC|) 1 r − rC + 3 Λ |rC| (|rC|+ rH)(|rC|+ rC) 1 r + |rC| . (2.3.31) Let (2.3.23) be centred at r = 3m, r∗(r) = ∫ r 3m 1 1− 2m r − Λr2 3 dr , (2.3.32) and choose f(u) = − 3 Λ rC (rC − rH)(rC + |rC|) log |u| A (2.3.33a) g(v) = − 3 Λ rC (rC − rH)(rC + |rC|) log |v| A (2.3.33b) in (2.3.24) with A2 = (rC − 3m)(3m− rH)− rH rC rC+|rC| rH+|rC| (3m+ |rC|)− |rC| rC rC−rH |rC|+rH . (2.3.34) 170 2.3. LINEAR WAVES ON SCHWARZSCHILD DE SITTER r =∞ C+ C¯+ Σr uv < 0 uv < 0 Figure 2.6: Causal geometry of the spacetime (M(m)Λ , g) for 3m √ Λ < 1 in the domain rH < r <∞. Then integrating (2.3.32) using (2.3.31) we obtain by (2.3.24) in view of (2.3.33) the following relation between the chosen null coordinates (u, v) and the radius function r: uv = r − rC (r − rH) rH rC rC+|rC| rH+|rC| (r + |rC|) |rC| rC rC−rH |rC|+rH . (2.3.35) Note that the null hypersurfaces u = 0 and v = 0 are the cosmological horizons r = rC, while the surfaces of constant r 6= rC are spacelike hyperbolas in the uv-plane for r > rC and timelike hyperbolas for r < rC. Moreover the future timelike boundary r = ∞ is identified with the spacelike hyperbola uv = 1. (See figure 2.6.) Let us also denote by C+ = { (u, 0) : u ≥ 0 } (2.3.36a) C¯+ = { (0, v) : v ≥ 0 } (2.3.36b) the two components of the past boundary of the expanding region r > rC. Since ∂r∗ ∂u = 1 1− 2m r − Λr2 3 ∂r ∂u = − 3 Λ rC (rC − rH)(rC + |rC|) 1 u (2.3.37a) ∂r∗ ∂v = 1 1− 2m r − Λr2 3 ∂r ∂v = − 3 Λ rC (rC − rH)(rC + |rC|) 1 v (2.3.37b) we can solve (2.3.21) for Ω2 to obtain: Ω2 = 4 r 3 Λ r2C (rC − rH)2(rC + |rC|)2× × (r − rH)1+ rHrC rC+|rC|rH+|rC| (r + |rC|)1+ |rC|rC rC−rH|rC|+rH (2.3.38) 2.3.1. GEOMETRY OF THE EXPANDING REGION 171 The metric g on the chart that covers the region rH < r < ∞ and extends across the cosmological horizon r = rC thus takes in double null coordinates (u, v) the form g = −4 r 3 Λ r2C (rC − rH)2(rC + |rC|)2× × (r − rH)1+ rHrC rC+|rC|rH+|rC| (r + |rC|)1+ |rC|rC rC−rH|rC|+rH du dv + r2 ◦γ (2.3.39) where r is a function of (u, v) implicity given by (2.3.35). Expanding Region. In the domain uv > 0 it is convenient to introduce the coordinate t = − 3 Λ rC (rC + |rC|)(rC − rH) log| v u | . (2.3.40) The metric can then alternatively be expressed in (t, r) coordinates as g = − 1 Λr2 3 + 2m r − 1 dr 2 + (Λr2 3 + 2m r − 1 ) dt2 + r2 ◦ γ . (2.3.41) From the expressions for the differentials that follow from (2.3.35) and (2.3.40) we can also conclude that ∂ ∂t = 1 2 Λ 3 (rC + |rC|)(rC − rH) rC ( u ∂ ∂u − v ∂ ∂v ) , (2.3.42) and ∂r ∂u = 1 4 Λ 3 (rC + |rC|)(rC − rH) rC Ω2 v (2.3.43a) ∂r ∂v = 1 4 Λ 3 (rC + |rC|)(rC − rH) rC Ω2 u . (2.3.43b) We shall think of (2.3.41) as the decomposition of the metric g in the expanding region uv > 0 with respect to the level sets of the time function r. Indeed, if we set V µ = −gµν∂νr , (2.3.44) or in other words V = 1 2 Λ 3 (rC − rH)(rC + |rC|) rC ( u ∂ ∂u + v ∂ ∂v ) , (2.3.45) then we find for the lapse function φ . = 1√−g(V, V ) = 1√Λr2 3 + 2m r − 1 (2.3.46) and the metric takes the form (2.3.41), namely g = −φ2 dr2 + gr . (2.3.47) Moreover, the normal to the level sets Σr = { (u, v) : (2.3.35) } (r > rC) (2.3.48) is given by n = φV , (2.3.49) and ∂r = φ 2V . 172 2.3. LINEAR WAVES ON SCHWARZSCHILD DE SITTER Cosmological Horizon. The vectorfield (2.3.42) extends to a global vectorfield that characterizes the cosmological horizon as a Killing horizon with positive surface gravity. Lemma 2.33 (Positive surface gravity of the cosmological horizons). The vectorfield T . = κC ( u ∂ ∂u − v ∂ ∂v ) (2.3.50) is globally Killing, i.e. (T )π . = 1 2 LTg = 0 , (2.3.51) and satisfies ∇TT = κCT on C+ , (2.3.52) where κC = 1 2 Λ 3 (rC − rH)(rC + |rC|) rC > 0 (2.3.53) is the surface gravity of the cosmological horizons. The result is obtained with the help of the vectorfield Y ∣∣∣ C+ = 2 ∂r ∂v ∂ ∂v (2.3.54) which is conjugate to T along the horizon: g(T, Y ) ∣∣∣ C+ = −2 . (2.3.55) Indeed, by (2.3.51), g(∇TT, Y ) ∣∣∣ C+ = −g(∇Y T, T ) ∣∣∣ C+ = −1 2 Y · g(T, T ) ∣∣∣ C+ = d dr ( 1− 2m r − Λr 2 3 )∣∣∣ r=rC . (2.3.56) Alternatively κC is characterized by ∇Y T = ∇TY = −κCY on C+ ; (2.3.57) note that this in particular implies that Y is Lie transported by T along the horizon: [T, Y ] ∣∣∣ C+ = 0 . (2.3.58) 2.3.2 Energy estimates We develop here the energy estimates for the linear wave equation (2.3.1) on Schwarzschild de Sitter. We study the expanding region in Section 2.3.2.1, and the cosmological horizons in Section 2.3.2.2, while the static region is already dealt with in [16]. 2.3.2. ENERGY ESTIMATES 173 2.3.2.1 Energy estimate in the expanding region It is the purpose of this section to show that the vectorfield M = 1 r ∂ ∂r (2.3.59) captures the global redshift effect in the expanding region of Schwarzschild de Sitter. Proposition 2.34. Let JM be the current associated to the multiplier (2.3.59), and n the normal to Σr (r > rC). Then for any solution ψ to the wave equation (2.3.1) we have φ∇ · JM [ψ] ≥ 1 r JM [ψ] · n . (2.3.60) Proof. It is equivalent to establish KM . = (M)πµνTµν [ψ] ≥ 1 r2 1 φ2 Trr[ψ] . (2.3.61) Recall from (2.3.41) that grr = − 3 Λ r (r − rH)(r − rC)(r + |rC|) (2.3.62a) gtt = 1 r Λ 3 (r − rH)(r − rC)(r + |rC|) (2.3.62b) gAB = r 2 ◦ γAB , (2.3.62c) and thus the non-vanishing connection coefficients are: Γrrr = 1 2 (g−1)rr∂rgrr = 1 2 1 r rH (rC − rH)(rH + |rC|) (r − rC)(r + |rC|) r − rH − 1 2 1 r rC (rC − rH)(rC + |rC|) (r − rH)(r + |rC|) r − rC + 1 2 1 r |rC| (|rC|+ rH)(|rC|+ rC) (r − rH)(r − rC) r + rH (2.3.63a) Γtrt = 1 2 (g−1)tt∂rgtt = 1 2 [ −1 r + 1 r − rH + 1 r − rC + 1 r + |rC| ] , (2.3.63b) and Γrtt as well as ΓrAB = Λ 3 (r − rH)(r − rC)(r + |rC|) ◦ γAB (2.3.64a) ΓBrA = 1 r δBA . (2.3.64b) Let us first consider M ′ = ∂ ∂r . (2.3.65) 174 2.3. LINEAR WAVES ON SCHWARZSCHILD DE SITTER We have (M ′)πrr = (g−1)rrΓrrr (M ′)πtt = (g−1)ttΓtrt (2.3.66a) (M ′)πAB = 1 r (g−1)AB , (2.3.66b) and KM ′ = (M ′)πrrTrr + (M ′)πttTtt + (M ′)πABTAB . (2.3.67) Now, Trr = 1 2 (∂ψ ∂r )2 + 1 2 ( 3 Λ )2 r2 (r − rH)2(r − rC)2(r + |rC|)2 (∂ψ ∂t )2 + 1 2 3 Λ r (r − rH)(r − rC)(r + |rC|) ∣∣∇/ψ∣∣2 , (2.3.68) Ttt = 1 2 (∂ψ ∂t )2 + 1 2 1 r2 (Λ 3 )2 (r − rH)2(r − rC)2(r + |rC|)2 (∂ψ ∂r )2 − 1 2 1 r Λ 3 (r − rH)(r − rC)(r + |rC|) ∣∣∇/ψ∣∣2 , (2.3.69) and gABTAB = Λ 3 1 r (r − rH)(r − rC)(r + |rC|) (∂ψ ∂r )2 − 3 Λ r (r − rH)(r − rC)(r + |rC|) (∂ψ ∂t )2 ; (2.3.70) also note that 1 φ2 Trr = 1 2 1 r Λ 3 (r − rH)(r − rC)(r + |rC|) (∂ψ ∂r )2 + 1 2 3 Λ r (r − rH)(r − rC)(r + |rC|) (∂ψ ∂t )2 + 1 2 ∣∣∇/ψ∣∣2 . = 1 2 Lr (∂ψ ∂r )2 + 1 2 Lt (∂ψ ∂t )2 + 1 2 ∣∣∇/ψ∣∣2 . (2.3.71) We then find that KM ′ = 1 2 [ K0 +K1 +K2 ] Lr (∂ψ ∂r )2 + 1 2 [ K0 +K1 −K2 ] Lt (∂ψ ∂t )2 + 1 2 [ K0 −K1 ]∣∣∇/ψ∣∣2 , (2.3.72) where K0 = −1 2 1 r rH (rC − rH)(rH + |rC|) (r − rC)(r + |rC|) r − rH + 1 2 1 r rC (rC − rH)(rC + |rC|) (r − rH)(r + |rC|) r − rC − 1 2 1 r |rC| (|rC|+ rH)(|rC|+ rC) (r − rH)(r − rC) r + |rC| 2.3.2. ENERGY ESTIMATES 175 = −1 2 (Λr2 3 + 2m r − 1 ) ∂ ∂r 1 Λr2 3 + 2m r − 1 = 1 2 3 Λ 2Λr 2 3 − 2m r (r − rH)(r − rC)(r + |rC|) , (2.3.73) and K1 = 1 2 1 r ( −1 + r r − rH + r r − rC + r r + |rC| ) (2.3.74) and finally K2 = 2 1 r . (2.3.75) Next we observe that K0 −K1 = 1 2 1 r 1 (r − rH)(r − rC)(r + |rC|)× × [ 2r3 − 6m Λ + (r − rH)(r − rC)(r + |rC|) − r(r2 + |rC|r − rCr − rC|rC|) − r(r2 + |rC|r − rHr − rH|rC|) − r(r2 − rCr − rHr + rHrC)] = 0 (2.3.76) by (2.3.30), and thus K0 +K1 −K2 = 2K1 −K2 = 1 r ( −3 + r r − rH + r r − rC + r r + |rC| ) = 1 r 1 (r − rH)(r − rC)(r + |rC|) [ 2rH|rC|r + 2rC|rC|r − 2rHrCr − 3rC|rC|rH ] = 3 Λ 2 r 1 (r − rH)(r − rC)(r + |rC|) [ r − 3m ] ≥ 3 Λ 2 r 1 (r − rH)(r + |rC|) , (2.3.77) again using the relations (2.3.30); and finally K0 +K1 +K2 ≥ 2K2 = 41 r . (2.3.78) We have shown in particular KM ′ ≥ 0 . (2.3.79) Since (M)πrr = − 1 r2 grr + 1 r (M ′)πrr (2.3.80a) (M)πtt = 1 r (M ′)πtt (2.3.80b) (M)πAB = 1 r (M ′)πAB , (2.3.80c) we conclude KM = − 1 r2 (g−1)rrTrr + 1 r KM ′ = Λ 3 1 r3 (r − rH)(r − rC)(r + |rC|)Trr + 1 r KM ′ ≥ 1 r2 1 φ2 Trr . (2.3.81) 176 2.3. LINEAR WAVES ON SCHWARZSCHILD DE SITTER As a immediate consequence of the energy identity for JM we obtain: Corollary 2.35. Let ψ be a solution to (2.3.1) then for all r2 > r1 > rC we have∫ Σr2 φ { φ2 (∂ψ ∂t )2 + ∣∣∇/ψ∣∣2} dµgr2 ≤ ≤ ∫ Σr1 φ { 1 φ2 (∂ψ ∂r )2 + φ2 (∂ψ ∂t )2 + ∣∣∇/ψ∣∣2} dµgr1 . (2.3.82) Proof. Note here that JM · n = 1 r 1 φ Trr = 1 2 1 r2 φ (Λr3 3 + 2m− r )(∂ψ ∂r )2 + 1 2 φ 1 Λr3 3 + 2m− r (∂ψ ∂t )2 + 1 2 1 r φ ∣∣∇/ψ∣∣2 . (2.3.83) 2.3.2.2 Redshift vectorfield on the cosmological horizon In this Section we construct a vectorfield that captures the local redshift effect of the cosmological horizon. As in Section 2.2.2.5 we follow [17], however using the double null coordinates introduced in Section 2.3.1. Extension of the Null Frame. Recall the null frame (T, Y, EA : A = 1, 2) from our discussion of the cosmological horizon in Section 2.3.1. We have seen that, also using (2.3.43), T ∣∣∣ C+ = κC u ∂ ∂u , (2.3.84a) Y ∣∣∣ C+ = 2 ∂r ∂v |v=0 ∂ ∂v = ιC 1 u ∂ ∂v , ιC . = 4 κC 1 Ω2 ∣∣∣ C+ , (2.3.84b) are conjugate null vectors on the cosmological horizon g(T, Y ) ∣∣∣ C+ = ιCκC guv = −2 , (2.3.85a) g(T,EA) ∣∣∣ C+ = 0 g(Y,EA) ∣∣∣ C+ = 0 , (2.3.85b) and satisfy the commutation relation (2.3.58). Here T is globally given by (2.3.50). Let us now extend the vectorfield Y away from the horizon by ∇Y Y = −σ(Y + T ) , (2.3.86) 2.3.2. ENERGY ESTIMATES 177 where σ > 0. It is well known from [17] that this extension gives rise to a positive current by virtue of Lemma 2.33. Indeed, as in Section 2.2.2.5, we find using (2.3.57) and (2.3.86) that KY ∣∣∣ C+ . = (Y )πµνTµν ∣∣∣ C+ = 1 2 σ ( T · ψ)2 + 1 2 κC ( Y · ψ)2 + 1 2 σ ∣∣∇/ψ∣∣2 + 2 r ( T · ψ)(Y · ψ) , (2.3.87) and thus KY ∣∣∣ C+ ≥ 1 4 κC ( Y · ψ)2 + [1 2 σ − ( 2 rC )2 1 κC ]( T · ψ)2 + 1 2 σ ∣∣∇/ψ∣∣2 ≥ 0 , (2.3.88) if σ > (2/rC)22/κC. By construction Y is Lie transported by T , [T, Y ] = 0 . (2.3.89) However, it remains to show that T + Y is timelike in the vicinity of the cosmological horizon. Proposition 2.36. Let N = T + Y with Y as constructed above, M = ∂ ∂r and let ψ be a solution (2.3.1). There exists r0 > rC (for any given u0 > 0, only depending on Λ and m) such that KN [ψ] ≥ 0 on rC ≤ r ≤ r0 (and u ≥ u0) , (2.3.90) and there exists a constant C that only depends on r0 such that JM [ψ] · n ≤ C(r0) JN [ψ] · n on r = r0 (and u ≥ u0) . (2.3.91) Proof. In view of (2.3.88) and (2.3.51), by continuity KN = KY ≥ 0 (2.3.92) in a neighborhood of v = 0, u ≥ u0; since Y is invariant under the flow of T , the positivity property is preserved uniformly in rC ≤ r ≤ r0, for r0 > rC chosen small enough, as stated in (2.3.90). Next we shall find an explicit expression for the extension of N to verify that N remains timelike in such a domain. The condition (2.3.86) on C+ is equivalent to the ordinary differential equations ∂Y u ∂v ∣∣∣ v=0 = −σκC ιC u2 (2.3.93a) ∂Y v ∂v ∣∣∣ v=0 = −σ − 2 Ω2 ∂Ω2 ∂r ∣∣∣ r=rC . (2.3.93b) Since, by differentiating (2.3.38), 1 Ω2 ∂Ω2 ∂r ∣∣∣ r=rC = Λ 3 1 r2C 1 κC ( r2C + rH|rC| ) , (2.3.94) 178 2.3. LINEAR WAVES ON SCHWARZSCHILD DE SITTER we obtain with say σ = k ( 2 rC )2 2 κC , k ≥ 1 , (2.3.95) furthermore that ∂Y u ∂v ∣∣∣ v=0 = −k( 2 rC )3 u2 (2.3.96a) ∂Y v ∂v ∣∣∣ v=0 = −( 2 rC )2 2 κC [ k + 1 4 Λ 3 ( r2C + rH|rC| )] , (2.3.96b) and thus N = T + Y = κC [ 1− k κC ( 2 rC )3 uv ] u ∂ ∂u + ιC [ 1 uv − κC ιC − 1 ιC ( 2 rC )2 2 κC [ k + 1 4 Λ 3 ( r2C + rH|rC| )]] v ∂ ∂v +O(v2) , (2.3.97) which is timelike for uv small enough, i.e. in view of (2.3.35) on r ≤ r0 for r0(k,Λ, m) chosen small enough. Let now r0 be chosen according to the two smallness conditions above, so that K Y ≥ 0 and N timelike on rC ≤ r ≤ r0. Recall (2.3.49), namely that the normal to Σr is given by n = φV where V = κC ( u ∂ ∂u + v ∂ ∂v ) . (2.3.98) Since also here M = ∂ ∂r = φ2V , (2.3.99) we obtain on one hand JM · n = T (M,n) = φ3 T (V, V ) = = 1 2κC φ { φ2 (∂ψ ∂t )2 + 1 φ2 (∂ψ ∂r )2 + ∣∣∇/ψ∣∣2} . (2.3.100) On the other hand, we can assume by (2.3.97) that on r = r0 (u ≥ u0) N has the form N = κCαu ∂ ∂u + ιCβ v ∂ ∂v , (2.3.101) where 0 < α ≤ 1, β ≥ 1 are constants on Σr0 . Therefore JN · n = T (N, n) = αφκ2C u2Tuu + φ ( ακC + βιC ) κC uv Tuv + βφιCκC v2Tvv ≥ 1 2κC 1 φ min { ακC, βιC }{ φ2 (∂ψ ∂t )2 + 1 φ2 (∂ψ ∂r )2 + ∣∣∇/ψ∣∣2} , (2.3.102) which yields JM · n ≤ C(r0) JN · n , (2.3.103) as desired, with C(r0) = 1 min { ακC, βιC} φ2(r0) . Appendix A Improved interior decay of higher order energy for the wave equation on 3 + 1-dimensional Minkowski space In this Appendix we prepare the argument of Section 1.5.3 in the simpler framework of Minkowski space. Here of course, much more is known than is proven in this Appendix. However, the discussion that follows has implications for the study of the wave equation on perturbations of Minkowski space. In this Appendix we give as an exercise a proof of “improved” interior decay of the first order energy for the wave equation on 3+1 dimensional Minkowski space. More precisely, given a solution φ of the wave equation φ = 0 with finite inital higher order energy D <∞, we show that ∫ Στ∩{r≤R} (JT (T · φ), n) ≤ C(δ, R) τ 4−δ D (A.1) for any δ > 0, where C is a constant, and Στ is the hypersurface coinciding with t = R+2τ for r ≤ R and with the corresponding outgoing null hypersurface for r ≥ R. We follow the new physical-space approach of [22], and introduce in addition a commutation with the vectorfield ∂v. The wave equation on Minkowski space Minkowski space. The background spacetime (M, g) is here the flat 3+1 dimensional Minkowski space. g = − dt2 + dr2 + r2 ◦γ (A.2) In null coordinates u = 1 2 (t− r) v = 1 2 (t+ r) (A.3) 179 180 A. INTERIOR DECAY ON MINKOWSKI SPACE the metric takes the form g = −4 du dv + r2 ◦γ (A.4) where ◦ γ is the standard metric on the unit sphere. Remark A.1 (Connection Coefficients). The connection coefficients of (A.4), which we may write in local coordinates as g = gab dx a dxb + r2 ◦ γAB dy A dyB , (A.5) are easily calculated: Γuuu = Γ v vv = 0 Γuuv = Γ v uv = Γ A uv = 0 ΓuAB = r 2 ◦ γAB Γ v AB = − r 2 γAB (A.6) ΓAab = Γ b aA = 0 ΓBAv = 1 r δBA Γ B Au = − 1 r δBA ΓCAB = ◦ Γ C AB Wave equation. The wave equation on (M, g) takes the classical form φ = (g−1)αβ∇α∂βφ = −∂u∂vφ+ 2 r ∂φ ∂r + 1 r2 ◦ △/ φ = 0 . (A.7) Note that given a solution φ the function ψ = rφ satisfies ∂u∂vψ = △/ ψ . (A.8) Energy currents. The Lagrangian theory of (A.7) provides us with a conserved energy momentum tensor Tµν(φ) = ∂µφ ∂νφ− 1 2 gµν ∂ αφ ∂αφ , (A.9) which allows us to construct energy currents with multiplier vectorfields V : JVµ (φ) = Tµν(φ)V ν (A.10) Recall the basic identities for the construction of energy estimates for (A.7): KV (φ) . = ∇µJVµ (φ) = (V )πµνTµν(φ) ; (V )π = 1 2 LV g (A.11)∫ D KV dµg = ∫ ∂D ∗JV (A.12) Here D ⊂M, and ∗JVαβγ = (JV )µ ǫµαβγ is the Hodge dual with respect to the volume form ǫ = dµg. WAVE EQUATION ON MINKOWSKI SPACE 181 Remark A.2 (Components of the energy momentum tensor). The null decomposition of the energy momentum tensor is: Tuu = (∂φ ∂u )2 Tvv = (∂φ ∂v )2 Tuv = ∣∣∇/ φ∣∣2 (A.13) TAB = EAφEBφ− 1 2 gAB∂ αφ ∂αφ Cauchy Problem. Let RPτ2τ1 = { (u, v) : v − u ≤ R ,R + 2τ1 ≤ v + u ≤ R + 2τ2 } (A.14) RDτ2τ1 = { (u, v) : τ1 ≤ u ≤ τ2 , v − u ≥ R } (A.15) and Στ1 the past boundary of ⋃ τ2≥τ1 RPτ2τ1 ∪ RDτ2τ1 : Στ = { (u, v) : u+ v = 2τ +R , v − u ≤ R } ∪ { (u, v) : u = τ , v − u ≥ R } (A.16) Initial data for (A.7) is prescribed on Στ1 , and our aim is here to prove estimates for the energy flux through Στ (τ > τ1). First, we have the classical conservation of energy, related to the timelike Killing vector- field T = ∂ ∂t : Proposition A.3 (Boundedness of energy). Let φ be solution of (A.7) then∫ Στ2 ( JT (φ), n ) ≤ ∫ Στ1 ( JT (φ), n ) (τ2 > τ1) (A.17) Proof. Choose D = RPτ2τ1 ∪ RDτ2τ1 and V = T in (A.12), and observe that (T )π = 0. Second, we have spacetime integral estimate, originally due to [39]. Assumption A.4 (Integrated local energy decay). Let φ be a solution to the wave equa- tion (A.7), then there exists a constant C(R) such that∫ RPτ2τ1 {(∂φ ∂u )2 + (∂φ ∂v )2 + ∣∣∇/ φ∣∣2} ≤ C(R) ∫ Στ1 ( JT (φ), n ) (A.18) for all τ2 > τ1. Notation (Integration). The volume form is usually omitted. Moreover we write ∫ Σ (JV , n) for the boundary terms arising in (A.12) if Σ ⊂ ∂D is spacelike or null; in fact for Στ as defined in (A.16) we have∫ Στ ( JV (φ), n ) . = ∫ Στ∩{r≤R} ( JV (φ), ∂ ∂t ) + ∫ ∞ τ+R dv ∫ S2 dµ◦ γ r2 ( JV (φ), ∂ ∂v ) (A.19) 182 A. INTERIOR DECAY ON MINKOWSKI SPACE Here it is stated as an assumption, because we are not concerned with the proof of (A.18), but rather with its implications for energy decay. Theorem 8. Let φ be a solution of the wave equation (A.7) and D . = ∫ Στ1 ( JT (φ), n ) + ∫ ∞ τ1+R dv ∫ S2 dµ◦ γ r2 (∂(rφ) ∂v )2 <∞ (A.20) for some fixed τ1 > 0. Then there exists a constant C(R) such that∫ Στ ( JT (φ), n ) ≤ C(R) τ 2 D (A.21) for all τ > τ1. The derivation of this energy decay result from the assumption A.4 on integrated local decay (and boundedness of energy) is subject of [22], and will also be apparent from the following. Interior decay of the first order energy We are going to show in this note the following result on interior decay of the first order energy: Proposition A.5 (Interior first order energy decay). Let φ be a solution of the wave equation (A.7), and D . = ∫ ∞ τ1+R dv ∫ S2 dµ◦ γ [ r4−δ (∂2(rφ) ∂v2 )2 + r2 (∂(rφ) ∂v )2 + r2 (∂(rT · φ) ∂v )2 + 3∑ j=1 r2 (∂(rΩjφ) ∂v )2 + 3∑ j=1 r2 (∂(rΩjT · φ) ∂v )2 + 3∑ j,k=1 r2 (∂(rΩjΩkT · φ) ∂v )2] |u=τ1 + ∫ Στ1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωjφ) + 3∑ j=1 JT (ΩjT · φ) + 3∑ j,k=1 JT (ΩjΩkT · φ), n ) (A.22) be finite, for any chosen R > 0, τ1 > 0, and 0 < δ < 1 2 . Then there exists a constant C(δ, R) such that ∫ Στ∩{r≤R} ( JT (T · φ), n ) ≤ C(δ, R) τ 4−2δ D (A.23) for all τ > τ1. This holds under the same assumptions as Thm. 8, namely Prop. A.3 and Assumption A.4, with the exception that we also need a refinement of the latter: Let RD\ τ2τ1 = { (u, v) : τ1 ≤ u ≤ τ2 , v − u ≥ R , v ≤ τ2 +R } (A.24) IMPROVED DECAY OF THE FIRST ORDER ENERGY 183 Στ1 Στ2 Στ1,τ2 RPτ2τ1 Figure A.1: The hypersurfaces (A.16) and (A.25) depicted in the Penrose diagram of Minkowski space. and Στ1,τ2 the past boundary of RPτ2τ1 ∪ RD\ τ2τ1 (see also figure A.1): Στ1,τ2 = { (u, v) : u+ v = 2τ +R , v − u ≤ R } ∪ { (u, v) : u = τ , R + τ1 ≤ v ≤ R + τ2 } (A.25) Assumption A.6 (Integrated local energy decay for finite regions). Let φ be a solution of (A.7), then there is a constant C(R) such that∫ RPτ2τ1 {(∂φ ∂u )2 + (∂φ ∂v )2 + |∇/φ|2 } ≤ ≤ C(R) [∫ Στ1,τ2 ( JT (φ), n ) + ∫ S2 dµ◦ γ (rφ2)(τ1, R + τ2) ] (A.26) for all τ2 > τ1. Notation. In the following we use the short hand notation∫ Σ′τ1,τ2 ( JT (φ), n ) . = ∫ Στ1,τ2 ( JT (φ), n ) + ∫ S2 dµ◦ γ (rφ2) ∣∣ (u=τ1,v=R+τ2) . (A.27) Sketch of Proof (of the refinement of Assumption A.4). The estimate (A.18) is readily proven using radial multiplier vectorfields. The modified currents associated to these multipliers 184 A. INTERIOR DECAY ON MINKOWSKI SPACE give rise to zeroth order terms on the boundary, which are estimated by the energy using a Hardy inequality. For example, on the past boundary of RDτ2τ1 , which is the null segment u = τ1, r ≥ R, we would employ the estimate∫ ∞ τ1+R dv r2 × 1 r2 φ2|u=τ1 ≤ 4 ∫ ∞ τ1+R (∂φ ∂v )2 r2 dv|u=τ1 . (A.28) In order to prove (A.26), we would instead infer the following inequality, on the past boundary of RD\ τ2τ1 , 1 2 ∫ R+τ2 R+τ1 dv r2 × 1 r2 φ2|u=τ1 ≤ (rφ2)(τ1, R + τ2) + 2 ∫ R+τ2 R+τ1 (∂φ ∂v )2 r2 dv , (A.29) which follows from a simple integration by parts: ∫ R+τ2 R+τ1 dv r2 × 1 r2 φ2 = ∫ R+τ2 R+τ1 dv d dv ( v − τ1 ) φ2 ≤ ≤ (R + τ2 − τ1)φ2|v=R+τ2 + ∫ R+τ2 R+τ1 dv {1 2 φ2 + 2 (∂φ ∂v )2 r2 } (A.30) Similarly for the future boundary of RD\ τ2τ1 . The refinement introduces the new boundary term in (A.26). Lemma A.7 (Pointwise decay). Let φ be a solution of the wave equation (A.7), with initial data on Στ0 (τ0 > 0) satisfying D . = ∫ ∞ τ0+R dv ∫ S2 dµ◦ γ [ r2 (∂(rφ) ∂v )2 + 3∑ j=1 r2 (∂(rΩjφ) ∂v )2 + 3∑ k,j=1 r2 (∂(rΩkΩjφ) ∂v )2] |u=τ0 + ∫ Στ0 ( JT (φ) + 3∑ j=1 JT (Ωjφ) + 3∑ k,j=1 JT (ΩkΩjφ), n ) <∞ . (A.31) Then there is a constant C(R) such that∫ S2 dµ◦ γ r2φ2|(τ1,R+τ2) ≤ C(R) τ1 D (A.32) for any τ2 > τ1 > τ0. Remark A.8. Note that by comparison to (A.26) we gain a power in r. Proof. First, integrating from infinity, φ(τ1, R + τ1) = − ∫ ∞ τ1+R ∂φ ∂v dv (A.33) IMPROVED DECAY OF THE FIRST ORDER ENERGY 185 and then by Cauchy’s inequality, and the Sobolev inequality on the sphere, φ2(τ1, R + τ1) ≤ ∫ ∞ R+τ1 1 r2 dv × ∫ ∞ R+τ1 (∂φ ∂v )2 r2 dv ≤ 1 r |v=R+τ1 ∫ ∞ R+τ1 dv ∫ S2 dµ◦ γ r2 {(∂φ ∂v )2 + 3∑ j=1 ( Ωj∂vφ )2 + 3∑ j,k=1 ( ΩkΩj∂vφ )2} . (A.34) Therefore, by Thm. 8, (rφ2)(τ1, R + τ1) ≤ ∫ Στ1 ( JT (φ) + 3∑ j=1 JT (Ωjφ), n ) ≤ C(R) τ 21 D . (A.35) Now r2 ∫ S2 dµ◦ γ φ2 (τ1, R + τ2) = ∫ S2 dµ◦ γ (rφ)2(τ1, R + τ1) + ∫ R+τ2 R+τ1 dv ∫ S2 dµ◦ γ ∂(rφ)2 ∂v ≤ ≤ R2 ∫ S2 dµ◦ γ φ2(τ1, R + τ1) + 2 √∫ ∞ R+τ1 dv ∫ S2 dµ◦ γ φ2 √∫ ∞ R+τ1 dv r2 (∂(rφ) ∂v )2 , (A.36) which proves (A.32) in view of the Hardy inequality (A.28), Thm. 8 and (A.35). Remark A.9. The decay rate in (A.32) is in fact not sufficient. However, we see from (A.36) that if a solution is already known to satisfy∫ ∞ τ+R dv ∫ S2 dµ◦ γ r2 (∂(rφ) ∂v )2 |u=τ ≤ C D τ 2 (τ > τ0) then indeed also ∫ S2 dµ◦ γ r2φ2|(τ,R+τ ′) ≤ C D τ 2 (τ ′ > τ) ; this is the case for the solution T · φ (on “good” slices with a loss in the power of r) as shown below, and will be used in the argument. Furthermore, it is important to point out that Assumption A.6 also allows us to control zeroth order terms on timelike boundaries. Lemma A.10 (Control of zeroth order terms). Let φ be a solution of (A.7), then there is a constant C > 0 such that∫ 2τ2+R 2τ1+R dt ∫ S2 dµ◦ γ r2 × 1 r2 φ2|r=R ≤ ≤ C {∫ Σ′τ1,τ2 ( JT (φ), n ) + ∫ 2τ2+R 2τ1+R dt ∫ S2 dµ◦ γ [(∂φ ∂u )2 + (∂φ ∂v )2 + ∣∣∇/ φ∣∣2]} (A.37) for all τ2 > τ1. 186 A. INTERIOR DECAY ON MINKOWSKI SPACE The proof of this fact is in the same vein as Assumption A.4, A.6 and not at the centre of our interest here, but we include the proof of a simplified version of Lemma A.10 without the refinement to finite regions; the difference to the proof given here amounts to keeping track of the boundary terms in the Hardy inequalities, (cf. Sketch of Proof of Assumption A.6). Proposition A.11 (Example of an X-estimate). Let φ be a solution of (A.7), then there is a constant C > 0 such that∫ 2τ2+R 2τ1+R dt ∫ S2 dµ◦ γ r2 × 1 r2 φ2|r=R ≤ ≤ C {∫ Στ1 ( JT (φ), n ) + ∫ 2τ2+R 2τ1+R dt ∫ S2 dµ◦ γ [(∂φ ∂u )2 + (∂φ ∂v )2 + ∣∣∇/ φ∣∣2]} (A.38) for all τ2 > τ1. Proof. Consider the vectorfield X = ∂ ∂r . (A.39) We find KX = (X)πµνTµν(φ) = 1 r (g−1)ABTAB(φ) = 1 r |∇/ φ|2 − 1 r ∂αφ ∂αφ (A.40) and are thus led to the modified current JX,1µ = J X µ + 1 2 1 r ∂µφ 2 − 1 2 ∂µ (1 r ) φ2 , (A.41) with nonnegative divergence: ∇µJX,1µ = 1 r ∣∣∇/ φ∣∣2 (A.42) Here we used that φ2 = 2∂αφ ∂αφ for any solution of (A.7), and  ( 1 r ) = 0 away from the origin. Now apply (A.12) with (A.41) to the domain RDτ2τ1 . Since∫ ∂RDτ2τ1 ∗JX,1 = =− ∫ ∞ R+τ2 dv ∫ S2 dµ◦ γ r2 × { T ( ∂ ∂v , ∂ ∂r ) + 1 r φ ∂φ ∂v + 1 2 1 r2 φ2 } |u=τ2 + ∫ ∞ R+τ1 dv ∫ S2 dµ◦ γ r2 × { T ( ∂ ∂v , ∂ ∂r ) + 1 r φ ∂φ ∂v + 1 2 1 r2 φ2 } |u=τ1 − ∫ 2τ2+R 2τ1+R dt ∫ S2 dµ◦ γ r2 × { T ( ∂ ∂r , ∂ ∂r ) + 1 r φ ∂φ ∂r + 1 2 1 r2 φ2 } |r=R (A.43) we have (using Cauchy’s inequality for the mixed term φ ∂φ/∂r) an estimate for ∫ 2τ2+R 2τ1+R dt ∫ S2 dµ◦ γ r2 × 1 4 1 r2 φ2 ≤ IMPROVED DECAY OF THE FIRST ORDER ENERGY 187 ≤ ∫ ∞ R+τ1 dv ∫ S2 dµ◦ γ r2 × {(∂φ ∂v )2 + 1 2 ∣∣∇/ φ∣∣2 + 1 r2 φ2 } |u=τ1 + ∫ 2τ2+R 2τ1+R dt ∫ S2 dµ◦ γ r2 × { 1 4 (∂φ ∂v )2 + 1 2 ∣∣∇/ φ∣∣2 + 1 4 (∂φ ∂u )2 + (∂φ ∂r )2} |r=R (A.44) which implies (A.38) in view of the Hardy inequality (A.28) and the fact that ( JT (φ), ∂ ∂v ) = 1 2 (∂φ ∂v )2 + 1 2 ∣∣∇/ φ∣∣2 KT (φ) = 0 . Corollary A.12. Let φ be a solution of (A.7), R0 > 0 and τ2 > τ1 . Then there is a constant C(R0) such that∫ 2τ2+R 2τ1+R dt ∫ S2 dµ◦ γ {(∂φ ∂u )2 + (∂φ ∂v )2 + ∣∣∇/ φ∣∣2+φ2}|r=R ≤ C(R0) ∫ Σ′τ1,τ2 ( JT (φ), n ) (A.45) for some R0 < R < R0 + 1. Proof. Apply Assumption A.6 to the domain R0+1Pτ2τ1 \ R0Pτ2τ1 and use the mean value theorem for the integration on (R0, R0 + 1). Proof of Proposition A.5. The method of proof is based on weighted energy identities. While [22] uses a weighted energy identity arising from the multiplier rp ∂ ∂v to prove Thm. 8, here we also use a commutation with ∂ ∂v to obtain energy decay of the solution ∂φ ∂t of (A.7). Weighted energy identity. We shall first consider the current that yields the weighted energy identity of [22]: r Jµ (φ) . = Tµν(ψ)V ν (A.46) ψ = rφ V = rq ∂ ∂v q = p− 2 0 < p ≤ 2 Since (V )πuu = 2q r q−1 (V )πuv = −q rq−1 (V )πvv = 0 (V )πAB = r q−1gAB (A.47) we find KV (φ) = (V )παβTαβ(φ) = 1 2 q rq−1 (∂φ ∂v )2 − 1 2 q rq−1 ∣∣∇/ φ∣∣2 + rq−1(∂uφ)(∂vφ) . (A.48) Notation. To make the dependence on p explicit, we denote by r Kp (φ) . = ∇µ rJµ (φ) . (A.49) 188 A. INTERIOR DECAY ON MINKOWSKI SPACE We calculate r Kp (φ) = (ψ)V · ψ +KV (ψ) = 2 r ∂ψ ∂r rq ∂ψ ∂v + 1 2 q rq−1 (∂ψ ∂v )2 − 1 2 q rq−1 ∣∣∇/ψ∣∣2 + rq−1∂ψ ∂u ∂ψ ∂v (A.50) = 1 2 p rq−1 (∂ψ ∂v )2 + 1 2 (2− p)rq−1∣∣∇/ψ∣∣2 which is nonnegative for p ≤ 2; by (A.12) this immediately implies∫ ∞ τ2+R dv ∫ S2 dµ◦ γ 1 2 rp (∂ψ ∂v )2∣∣∣ u=τ2 + ∫ τ2 τ1 du ∫ ∞ u+R dv ∫ S2 dµ◦ γ × p 2 rp−1 (∂ψ ∂v )2 ≤ ≤ ∫ ∞ τ1+R dv ∫ S2 dµ◦ γ 1 2 rp (∂ψ ∂v )2∣∣∣ u=τ1 + ∫ 2τ2+R 2τ1+R dt ∫ S2 dµ◦ γ × {(∂ψ ∂v )2 + ∣∣∇/ψ∣∣2}∣∣∣ r=R (A.51) for p = 1, 2 in particular, (and all τ2 > τ1). Weighted energy and commutation. Now consider the current: v Jµ (φ) . = Tµν(χ)V ν (A.52) χ = ∂vψ = ∂(rφ) ∂v V = rq ∂ ∂v (A.53) q = p− 2 0 < p < 4 δ = 4− p Notation. We denote similarly by v Kp (φ) . = ∇µ vJµ (φ) . (A.54) The error terms for v K, and r K, – in comparison to (A.11) – arise from the fact that χ, and ψ respectively, are not solutions of (A.7); here we find: χ = 2 r ∂χ ∂r + 2 r △/ ψ (A.55) Hence v Kp (φ) = (χ)V · χ+KV (χ) = 1 2 p rq−1 (∂χ ∂v )2 + 2 rq−1 (△/ ψ) ∂χ ∂v + 1 2 (2− p) rq−1 ∣∣∇/χ∣∣2 , (A.56) which is not positive definite. However, we have 1 4 p rp−1 (∂2(rφ) ∂v2 )2 ≤ ≤ v Kp (φ) r 2 + 4 p r(p−2)−1 ( △/ (rφ) )2 r2 + 1 2 (p− 2)rp−1|∇/ ∂v(rφ)|2 , (A.57) and are able to bound the second term using a commutation with Ωi, and the third term with an integration by parts argument. IMPROVED DECAY OF THE FIRST ORDER ENERGY 189 Lemma A.13. For any function φ ∈ H2(Sr) we have △/ φ ∈ L2(Sr), and there exists a constant C > 0 such that it holds∫ S2 ( △/ (rφ) )2 r2 dµ◦ γ ≤ C ∫ S2 { 3∑ i=1 ∣∣∣∇/ (rΩiφ)∣∣∣2 + ∣∣∣∇/ (rφ)∣∣∣2} dµ◦γ . (A.58) If p ≤ 2 (A.57) holds without the last term; and if p > 2 we can “interchange” the derivatives of |∇/ ∂vψ|2 by integrating by parts twice, (such that we can absorb the resulting ∂vχ term in the left hand side):∫ τ2 τ1 du ∫ ∞ u+R dv ∫ S2 dµ◦ γ × rp−1|∇/ ∂v(rφ)|2 = = − ∫ τ2 τ1 du ∫ ∞ u+R dv ∫ S2 dµ◦ γ × { rp−1∂v (△/ (rφ)) ∂v(rφ) + rp−12 r △/ (rφ) ∂v(rφ) } = − ∫ τ2 τ1 du ∫ S2 dµ◦ γ × rp−1△/ (rφ) ∂v(rφ) ∣∣∣∞ u+R + ∫ τ2 τ1 du ∫ ∞ u+R dv ∫ S2 dµ◦ γ × { (p− 1)rp−2△/ (rφ) ∂v(rφ) + rp−1△/ (rφ) ∂2v(rφ) + 2rp−2△/ (rφ) ∂v(rφ) } ≤ ∫ τ2 τ1 du ∫ S2 dµ◦ γ × rp−1△/ (rφ) ∂v(rφ) ∣∣ v=u+R + ∫ τ2 τ1 du ∫ ∞ u+R dv ∫ S2 dµ◦ γ × {[ 2 + (p− 1) + 2(p− 2)8 p ] r(p−2)−1 ( △/ (rφ) )2 r2 + [ 2 + (p− 1)]r(p−2)−1(∂(rφ) ∂v )2 + 2 p− 2 p 8 rp−1 (∂2(rφ) ∂v2 )2} (A.59) Therefore∫ τ2 τ1 du ∫ ∞ u+R dv ∫ S2 dµ◦ γ × rp−1 (∂2(rφ) ∂v2 )2 ≤ ≤ C(p, δ) ∫ RDτ2τ1 dµg { v Kp (φ)+ r Kp−2 (φ) + 3∑ j=1 r Kp−2 (Ωjφ) } + C ∫ 2τ2+R 2τ1+R dt ∫ S2 dµ◦ γ rp−2 { 3∑ j=1 |∇/ rΩjφ|2 + (∂(rφ) ∂v )2} |r=R . (A.60) Note that we need p ≤ 4 in order to control the (∂vψ)2 term by (A.50) with p ≤ 2, and strictly p < 4 to control the (△/ ψ)2 term by (A.50) using Lemma A.13. Thus after turning the divergences into boundary terms using (A.12), we arrive at the following weighted energy inequality for χ:∫ ∞ τ2+R dv ∫ S2 dµ◦ γ × { rp (∂χ ∂v )2 + rp−2 (∂ψ ∂v )2 + 3∑ j=1 rp−2 (∂Ωjψ ∂v )2}∣∣∣ u=τ2 + ∫ τ2 τ1 du ∫ ∞ u+R dv ∫ S2 dµ◦ γ × rp−1 (∂χ ∂v )2 190 A. INTERIOR DECAY ON MINKOWSKI SPACE ≤ C(p, δ) ∫ ∞ τ1+R dv ∫ S2 dµ◦ γ × { rp (∂χ ∂v )2 + rp−2 (∂ψ ∂v )2 + 3∑ j=1 rp−2 (∂Ωjψ ∂v )2}∣∣∣ u=τ1 + C(p, δ) ∫ 2τ2+R 2τ1+R dt ∫ S2 dµ◦ γ × { rp (∂χ ∂v )2 + rp−2 (∂ψ ∂v )2 + 3∑ j=1 rp−2 (∂Ωjψ ∂v )2 + rp ∣∣∇/χ∣∣2 + rp−2∣∣∇/ψ∣∣2 + 3∑ j=1 rp−2 ∣∣∇/Ωjψ∣∣2}∣∣∣ r=R (τ2 > τ1) (A.61) The fact that this inequality holds for up to p < 4 – which appears as the weight rp in the boundary terms – is directly related to the decay rate in (A.23). Correspondingly we will proceed in a hierarchy of four steps: p = 4− δ: Let τ1 > 0, and τj+1 = 2τj (j ∈ N). In a first step we use (A.61) with p = 4−δ and (A.51) with p = 2 as an estimate for the spacetime integral of ∂vχ, ∂vψ, and ∂v(Ωjψ) on RDτj+1τj , and in a second step as an estimate for the corresponding integral on the future boundary of RDτjτ1 :∫ τj+1 τj du ∫ ∞ u+R dv ∫ S2 dµ◦ γ × { r3−δ (∂χ ∂v )2 + r (∂ψ ∂v )2 + 3∑ j+1 r (∂Ωjψ ∂v )2} ≤ ≤ C(δ) ∫ ∞ τj+R dv ∫ S2 dµ◦ γ × { r4−δ (∂χ ∂v )2 + r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2}∣∣∣ u=τj + C(δ) ∫ 2τj+1+R 2τj+R dt ∫ S2 dµ◦ γ × { r4−δ (∂χ ∂v )2 + r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2 + r4−δ ∣∣∇/χ∣∣2 + r2∣∣∇/ψ∣∣2 + 3∑ j=1 r2 ( ∇/Ωjψ )2}∣∣∣ r=R ≤ ≤ C(δ) ∫ ∞ τ1+R dv ∫ S2 dµ◦ γ × { r4−δ (∂χ ∂v )2 + r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2}∣∣∣ u=τ1 + C(δ) ∫ 2τj+1+R 2τ1+R dt ∫ S2 dµ◦ γ × { r4−δ (∂χ ∂v )2 + r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2 + r4−δ ∣∣∇/χ∣∣2 + r2∣∣∇/ψ∣∣2 + 3∑ j=1 r2 ( ∇/Ωjψ )2}∣∣∣ r=R (A.62) Observe that, (∂ψ ∂v )2 = (∂(rφ) ∂v )2 ≤ 2φ2 + 2r2 (∂φ ∂v )2 (A.63)(∂χ ∂v )2 = (∂2(rφ) ∂v2 )2 ≤ 4 (∂φ ∂v )2 + 2r2 (∂2φ ∂v2 )2 , (A.64) and using first the wave equation (A.7), ∂2φ ∂v2 = 2 ∂T · φ ∂v − 1 r ∂φ ∂v + 1 r ∂φ ∂u − 1 r2 ◦ △/ φ (A.65) IMPROVED DECAY OF THE FIRST ORDER ENERGY 191 and then Lemma A.13,∫ S2 r2 (∂2φ ∂v2 )2 dµ◦ γ ≤ ≤ C ∫ S2 [ r2 (∂(T · φ) ∂v )2 + (∂φ ∂v )2 + (∂φ ∂u )2 + 3∑ j=1 ∣∣∇/Ωjφ∣∣2 + ∣∣∇/ φ∣∣2] ; (A.66) moreover, ∣∣∇/χ∣∣2 = ∣∣∇/ ∂v(rφ)∣∣2 ≤ 2∣∣∇/ φ∣∣2 + 2 3∑ j=1 (∂Ωjφ ∂v )2 . (A.67) Therefore, by virtue of Assumption A.4 and Proposition A.11, we can choose a R0 ∈ (R,R + 1) such that∫ 2τj+1+R0 2τ1+R0 dt ∫ S2 dµ◦ γ × {(∂χ ∂v )2 + (∂ψ ∂v )2 + 3∑ j=1 (∂Ωjψ ∂v )2 + ∣∣∇/χ∣∣2 + ∣∣∇/ψ∣∣2 + 3∑ j=1 ( ∇/Ωjψ )2}∣∣∣ r=R0 ≤ ≤ C(R) ∫ Στ1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωjφ), n ) (A.68) and henceforth there exists a sequence τ ′j ∈ (τj , τj+1) such that∫ ∞ τ ′j+R0 dv ∫ S2 dµ◦ γ × { r3−δ (∂χ ∂v )2 + r (∂ψ ∂v )2 + 3∑ j+1 r (∂Ωjψ ∂v )2}∣∣∣ u=τ ′j ≤ ≤ C(δ) τj ∫ ∞ τ1+R0 dv ∫ S2 dµ◦ γ × { r4−δ (∂χ ∂v )2 + r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2}∣∣∣ u=τ1 + C(δ, R) τj ∫ Στ1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωjφ), n ) . (A.69) p = 3− δ: Here we apply (A.61) with p = 3 − δ and (A.51) with p = 1 to the domain RjDτ ′ 2j+1 τ ′2j−1 , ∫ τ ′2j+1 τ ′2j−1 du ∫ ∞ u+Rj dv ∫ S2 dµ◦ γ × { r2−δ (∂χ ∂v )2 + (∂ψ ∂v )2 + 3∑ j=1 (∂Ωjψ ∂v )2} ≤ ≤ C ∫ ∞ τ ′2j−1+Rj dv ∫ S2 dµ◦ γ × { r3−δ (∂χ ∂v )2 + r (∂ψ ∂v )2 + 3∑ j=1 r (∂Ωjψ ∂v )2}∣∣∣ u=τ ′2j−1 + C ∫ 2τ ′2j+1+Rj 2τ ′2j−1+Rj dt ∫ S2 dµ◦ γ × { r3−δ (∂χ ∂v )2 + r (∂ψ ∂v )2 + 3∑ j=1 r (∂Ωjψ ∂v )2 + r3−δ ∣∣∇/χ∣∣2 + r∣∣∇/ψ∣∣2 3∑ j=1 r ∣∣∇/Ωjψ∣∣2}∣∣∣ r=Rj (A.70) 192 A. INTERIOR DECAY ON MINKOWSKI SPACE where Rj ∈ (R0, R0 + 1) is chosen by Assumption A.4 and Prop. A.11 such that∫ 2τ ′2j+1+Rj 2τ ′2j−1+Rj dt ∫ S2 dµ◦ γ × {(∂χ ∂v )2 + (∂ψ ∂v )2 + 3∑ j=1 (∂Ωjψ ∂v )2 + ∣∣∇/χ∣∣2 + ∣∣∇/ψ∣∣2 + 3∑ j=1 ∣∣∇/Ωjψ∣∣2}∣∣∣ r=Rj ≤ ≤ C(R) ∫ Στ ′ 2j−1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωjφ), n ) . (A.71) Therefore, there exists a sequence τ ′′j ∈ (τ ′2j−1, τ ′2j+1) such that∫ ∞ τ ′′j +Rj dv ∫ S2 dµ◦ γ × { r2−δ (∂χ ∂v )2 + (∂ψ ∂v )2 + 3∑ j=1 (∂Ωjψ ∂v )2}∣∣∣ u=τ ′′j ≤ ≤ C(δ) (τ ′′j )2 ∫ ∞ τ1+R0 dv ∫ S2 dµ◦ γ × { r4−δ (∂χ ∂v )2 + r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2}∣∣∣ u=τ1 + C(δ, R) (τ ′′j )2 ∫ Στ1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωjφ), n ) + C(R) ∫ Στ2j−1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωj), n ) (A.72) where we have used (A.69) for the first term in (A.70), However, by virtue of Thm. 8 the last term decays with the same rate, and we obtain∫ ∞ τ ′′j +Rj dv ∫ S2 dµ◦ γ × { r2−δ (∂χ ∂v )2 + (∂ψ ∂v )2 + 3∑ j=1 (∂Ωjψ ∂v )2}∣∣∣ u=τ ′′j ≤ ≤ C(δ, R) (τ ′′j )2 ∫ ∞ τ1+R0 dv ∫ S2 dµ◦ γ × { r4−δ (∂χ ∂v )2 +r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2 +r2 (∂T · ψ ∂v )2}∣∣∣ u=τ1 + C(δ, R) (τ ′′j )2 ∫ Στ1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωjφ), n ) . (A.73) In fact, r2−δ (∂(rT · φ) ∂v )2 = r2−δ (∂(T · ψ) ∂v )2 = r2−δ (1 2 ∂2(rφ) ∂v2 + 1 2 ∂2(rφ) ∂u∂v )2 ≤ ≤ 1 2 r2−δ (∂2(rφ) ∂v2 )2 + 1 2 r2−δ ( △/ (rφ) )2 (A.74) so by Lemma A.13∫ τ ′2j+1 τ ′2j−1 du ∫ ∞ u+Rj dv ∫ S2 dµ◦ γ × r2−δ (∂rT · φ ∂v )2 ≤ ≤ C ∫ τ ′2j+1 τ ′2j−1 du ∫ ∞ u+Rj dv ∫ S2 dµ◦ γ × { r2−δ (∂χ ∂v )2 + r K1−δ (φ) r2 + 3∑ j=1 r K1−δ (Ωjφ) r2 } (A.75) IMPROVED DECAY OF THE FIRST ORDER ENERGY 193 generates boundary terms that are already present in (A.70), (cf. (A.50); consequently we have∫ ∞ τ ′′j +Rj dv ∫ S2 dµ◦ γ × { r2−δ (∂rT · φ ∂v )2}∣∣∣ u=τ ′′j ≤ ≤ C(δ, R) (τ ′′j )2 ∫ ∞ τ1+R0 dv ∫ S2 dµ◦ γ × { r4−δ (∂χ ∂v )2 +r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2 +r2 (∂T · ψ ∂v )2}∣∣∣ u=τ1 + C(δ, R) (τ ′′j )2 ∫ Στ1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωjφ), n ) . (A.76) Remark A.14. This statement should be compared to Thm. 8, where all that one can assert is ∫ ∞ τ+R dv ∫ S2 dµ◦ γ × { r2 (∂(rφ) ∂v )2}∣∣∣ u=τ <∞ (τ > τ1) . (A.77) We will now proceed along the lines of the proof of Thm. 8, just that we have (A.76) as a starting point for the solution T ·φ of (A.7), (and (A.21)); however, as opposed to Thm. 8 the hierarchy does not descend from p = 2 but p < 2, which introduces a degeneracy in the last step, and requires the refinement of Assumption A.4 to Assumption A.6 (and Prop. A.11 to Prop. A.10), summarized in Corollary A.12. p = 2− δ: By (A.50),∫ τ ′′2j+1 τ ′′2j−1 du ∫ ∞ u+R′j dv ∫ S2 dµ◦ γ × r1−δ (∂r T · φ ∂v )2 ≤ ≤ ∫ τ ′′2j+1 τ ′′2j−1 du ∫ ∞ u+R′j dv ∫ S2 dµ◦ γ × 2 rK2−δ (T · φ) r2 ≤ ≤ ∫ ∞ τ ′′2j−1+R ′ j dv ∫ S2 dµ◦ γ × { r2−δ (∂(r T · φ) ∂v )2}∣∣∣ u=τ ′′2j−1 + ∫ 2τ ′′2j+1+R′j 2τ ′′2j−1+R ′ j dt ∫ S2 dµ◦ γ × 1 2 r2−δ {(∂(r T · φ) ∂v )2 + ∣∣∇/ (r T · φ)∣∣2}∣∣∣ r=R′j ≤ ≤ C(δ, R) (τ ′′2j−1)2 ∫ ∞ τ1+R0 dv ∫ S2 dµ◦ γ × { r4−δ (∂χ ∂v )2 + r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2 + r2 (∂(r T · φ) ∂v )2}∣∣∣ u=τ1 + C(δ, R) (τ ′′2j−1)2 ∫ Στ1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωjφ), n ) (A.78) where in the last step we have chosen R′j ∈ (R0+1, R0+2) suitably using Assumption A.4 and Prop. A.10, and subsequently applied Thm. 8 to the flux through the past boundary Στ ′′2j−1 . Therefore there exists a sequence τ ′′′ j ∈ (τ ′′2j−1, τ ′′2j+1) (j ∈ N) such that∫ ∞ τ ′′′j +R ′′ j dv ∫ S2 dµ◦ γ × { r1−δ (∂(r T · φ) ∂v )2}∣∣∣ u=τ ′′′j ≤ 194 A. INTERIOR DECAY ON MINKOWSKI SPACE ≤ C(δ, R) (τ ′′′j )3 ∫ ∞ τ1+R0 dv ∫ S2 dµ◦ γ × { r4−δ (∂χ ∂v )2 + r2 (∂(r T · φ) ∂v )2 + r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2}∣∣∣ u=τ1 + C(δ, R) (τ ′′′j )3 ∫ Στ1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωjφ), n ) . (A.79) p = 1− δ: Since ∫ ∞ u+R dv 1 rδ (∂ψ ∂v )2 = ∫ ∞ u+R dv 1 rδ { φ2 + 2rφ ∂φ ∂v + r2 (∂φ ∂v )2} = ∫ ∞ u+R dv 1 rδ { ∂(rφ2) ∂v + r2 (∂φ ∂v )2} = r1−δφ2 ∣∣∣∞ u+R + ∫ ∞ u+R dv { δ r1+δ rφ2 + r2−δ (∂φ ∂v )2} (A.80) we can estimate the degenerate energy density of T · φ by: ∫ τ ′′′2j+1 τ ′′′2j−1 du ∫ ∞ u+R′′j dv ∫ S2 dµ◦ γ × 1 rδ {(∂(T · φ) ∂v )2 + ∣∣∇/ (T · φ)∣∣2}r2 ≤ ≤ ∫ τ ′′′2j+1 τ ′′′2j−1 du ∫ ∞ u+R′′j dv ∫ S2 dµ◦ γ × 1 rδ {(∂(r T · φ) ∂v )2 + ∣∣∇/ (r T · φ)∣∣2} + ∫ 2τ ′′′2j+1+R′′j 2τ ′′′2j−1+R ′′ j dt ∫ S2 dµ◦ γ × { r1−δ ( T · φ )2}∣∣∣ r=R′′j (A.81) On one hand we have, ∫ τ ′′′2j+1 τ ′′′2j−1 du ∫ ∞ u+R′′j dv ∫ S2 dµ◦ γ × 1 rδ {(∂(r T · φ) ∂v )2 + ∣∣∇/ (r T · φ)∣∣2} ≤ ≤ C(δ) ∫ τ ′′′2j+1 τ ′′′2j−1 du ∫ ∞ u+R′′j dv ∫ S2 dµ◦ γ × rK1−δ (T · φ) r2 (A.82) and on the other hand, now by Cor. A.12 (recall also the notation introduced on pg. 183), we can choose R′′j such that∫ 2τ ′′′2j+1+R′′j 2τ ′′′2j−1+R ′′ j dt ∫ S2 dµ◦ γ × { r1−δ ( T · φ )2}∣∣∣ r=R′′j ≤ C(R) ∫ Σ′ τ ′′′ 2j−1,τ ′′′ 2j+1 ( JT (T · φ), n ) . (A.83) Therefore, by virtue of (A.79), ∫ τ ′′′2j+1 τ ′′′2j−1 du ∫ ∞ u+R′′j dv ∫ S2 dµ◦ γ × 1 rδ {(∂(T · φ) ∂v )2 + ∣∣∇/ (T · φ)∣∣2}r2 ≤ IMPROVED DECAY OF THE FIRST ORDER ENERGY 195 ≤ C(δ) ∫ ∞ τ ′′′2j−1+R ′′ j dv ∫ S2 dµ◦ γ × { r1−δ (∂(r T · φ) ∂v )2}∣∣∣ u=τ ′′′2j−1 + C(R) ∫ 2τ ′′′2j+1+R′′j 2τ ′′′2j−1+R ′′ j dt ∫ S2 dµ◦ γ × {(∂(T · φ) ∂u )2 + (∂(T · φ) ∂v )2 + ∣∣∇/ (T · φ)∣∣2}∣∣∣ r=R′′j + C(R) ∫ Σ′ τ ′′′ 2j−1,τ ′′′2j+1 ( JT (T · φ), n ) ≤ C(δ, R) (τ ′′′2j−1)3 ∫ ∞ τ1+R0 dv ∫ S2 dµ◦ γ × { r4−δ (∂χ ∂v )2 + r2 (∂(r T · φ) ∂v )2 + r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2}∣∣∣ u=τ1 + C(δ, R) (τ ′′′2j−1)3 ∫ Στ1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωjφ), n ) + C(R) ∫ Σ′ τ ′′′ 2j−1,τ ′′′ 2j+1 ( JT (T · φ), n ) . (A.84) Remark A.15. Moreover, by Assumption A.6, we can extend the integral on the left hand side to R ′′ jPτ ′′′ 2j+1 τ ′′′2j−1 while leaving the right hand side unchanged; i.e. (A.84) holds with the left hand side replaced by ∫ τ ′′′2j+1 τ ′′′2j−1 dτ ∫ Στ ζδ(r) ( JT (T · φ), n ) , where ζδ(r) = 1 r < R′′j(R′′j r )δ r > R′′j . This inequality can in itself be improved so as to show the same decay rate for all terms on the right hand side; for certainly by Thm. 8 applied to the last term1,∫ τ ′′′2j+1 τ ′′′2j−1 du ∫ ∞ u+R′′j dv ∫ S2 dµ◦ γ × 1 rδ {(∂(T · φ) ∂v )2 + ∣∣∇/ (T · φ)∣∣2}r2 ≤ ≤ C(δ, R) (τ ′′′2j−1)2 {∫ ∞ τ1+R0 dv ∫ S2 dµ◦ γ × [ r4−δ (∂χ ∂v )2 + r2 (∂(r T · φ) ∂v )2 + r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2]∣∣∣ u=τ1 + ∫ Στ1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωjφ), n )} . (A.85) But then, for a sequence τ ′′′′j ∈ (τ ′′′2j−1, τ ′′′2j+1), we obtain∫ τ ′′′ 2(j+1)+1 +R′′j+1 τ ′′′′j +R ′′ j dv ∫ S2 dµ◦ γ r2 × {(∂(T · φ) ∂v )2 + ∣∣∇/ (T · φ)∣∣2}∣∣∣ u=τ ′′′′j ≤ 1more precisely by applying Theorem 8 to the last term in the analogue of (A.84) that results from not using the refinement in (A.83), but Assumption A.4 and Proposition A.11 as above. 196 A. INTERIOR DECAY ON MINKOWSKI SPACE ≤ (τ ′′′2(j+1)+1 +R′′j+1 − τ ′′′′j )δ ∫ ∞ τ ′′′′j +R ′′ j dv ∫ S2 dµ◦ γ r2 × 1 rδ {(∂(T · φ) ∂v )2 + ∣∣∇/ (T · φ)∣∣2}∣∣∣ u=τ ′′′′j ≤ C(δ, R) (τ ′′′′j )3−δ {∫ ∞ τ1+R0 dv ∫ S2 dµ◦ γ × [ r4−δ (∂χ ∂v )2 + r2 (∂(r T · φ) ∂v )2 + r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2]∣∣∣ u=τ1 + ∫ Στ1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωjφ), n )} ; (A.86) since by Lemma A.7 (compare Remark A.9)∫ S2 dµ◦ γ r(T · φ)2 ∣∣∣ (u=τ ′′′2j−1,v=R ′′ j+τ ′′′ 2j+1) ≤ ≤ R (τ ′′′2j+1 +R ′′ j − τ ′′′2j−1) C(R) (τ ′′′2j−1)2 × × {∫ ∞ τ1+R0 dv ∫ s dµ◦ γ [ r2 (∂(r T · φ) ∂v )2 + 3∑ j=1 r2 (∂(rΩjT · φ) ∂v )2 + 3∑ j,k=1 r2 (∂(rΩjΩkT · φ) ∂v )2] + ∫ Στ1 ( JT (T · φ) + 3∑ j=1 JT (ΩjT · φ) + 3∑ j,k=1 JT (ΩjΩkT · φ), n )} + 2 (τ ′′′2j+1 +R ′′ j − τ ′′′2j−1)1−δ √∫ ∞ R′′j+τ ′′′ 2j+1 dv ∫ S2 dµ◦ γ (T · φ)2 √∫ ∞ R′′j+τ ′′′ 2j+1 dv r2−δ (∂(rTφ) ∂v )2 (A.87) this shows in particular (recall here our previous Remark on pg. 195) that∫ Σ′ τ ′′′ 2j−1,τ ′′′2j+1 ( JT (T · φ), n ) ≤ ≤ ∫ Στ ′′′′ j−1,τ ′′′2j+1 ( JT (T · φ), n ) + ∫ S2 dµ◦ γ ( r(T · φ)2 )∣∣∣ (τ ′′′2j−1,R ′′ j+τ ′′′ 2j+1) ≤ C(δ, R) (τ ′′′2j−1)3−δ {∫ ∞ τ1+R0 dv ∫ S2 dµ◦ γ × [ r4−δ (∂χ ∂v )2 + r2 (∂(T · ψ) ∂v )2 + r2 (∂ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2 + 3∑ j=1 r2 (∂(ΩjT · ψ) ∂v )2 + 3∑ j,k=1 r2 (∂(ΩjΩkT · ψ) ∂v )2]∣∣∣ u=τ1 + ∫ Στ1 ( JT (φ) + JT (T · φ) + 3∑ j=1 JT (Ωjφ) + 3∑ j=1 JT (ΩjTφ) + 3∑ j,k=1 JT (ΩjΩkTφ) } . (A.88) Returning to (A.84) we can now conclude that there is a sequence τ ′′′′j ∈ (τ ′′′2j−1, τ ′′′2j+1) such that∫ ∞ τ ′′′′j +R ′′ j dv ∫ S2 dµ◦ γ r2 × 1 rδ {(∂(T · φ) ∂v )2 + ∣∣∇/ (T · φ)∣∣2} ≤ C(δ, R) (τ ′′′′j )4−δ ×D (A.89) IMPROVED DECAY OF THE FIRST ORDER ENERGY 197 u = τ ′′′2j−1 u = τ ′′′′j−1 v = R + τ ′′′2j+1 u = τ ′′′2j+1 Στ ∩ {r ≤ R} Figure A.2: Removal of the restriction to the dyadic sequence (τ ′′′′j ). where D = ∫ ∞ τ1+R0 dv ∫ S2 dµ◦ γ × { r4−δ (∂χ ∂v )2 + r2 (∂ψ ∂v )2 + r2 (∂T · ψ ∂v )2 + 3∑ j=1 r2 (∂Ωjψ ∂v )2 + 3∑ j=1 r2 (∂ΩjT · ψ ∂v )2 + 3∑ j,k=1 r2 (∂ΩjΩkT · ψ ∂v )2}∣∣∣ u=τ1 + ∫ Στ1 ( JT (φ) + JT (T · φ) + 3∑ j=1 (Ωjφ) + 3∑ j=1 JT (ΩjT · φ) + 3∑ j,k=1 JT (ΩjΩkT · φ), n ) ; (A.90) or alternatively,∫ τ ′′′2j−1+R′′j τ ′′′′j−1+R ′′ j−1 dv ∫ S2 dµ◦ γ r2 × {(∂(T · φ) ∂v )2 + ∣∣∇/ (T · φ)∣∣2} ≤ C(δ, R) (τ ′′′′j )4−2δ ×D . (A.91) Remark A.16. As previously noted, (as a consequence of Assumption A.6 used for the extension to R ′′ jPτ ′′′ 2j+1 τ ′′′2j−1 in (A.84)) the left hand side of (A.89) may be replaced by∫ Στ ′′′′ j ζδ ( JT (T · φ), n ) while leaving the right hand side unchanged. 198 A. INTERIOR DECAY ON MINKOWSKI SPACE Therefore ∫ Στ∩{r≤R} ( JT (T · φ), n ) ≤ C(δ, R) τ 4−2δ ×D , (A.92) because for any given τ > τ1 we can choose j ∈ N such that τ ∈ (τ ′′′2j−1, τ ′′′2j+1) (compare also figure A.2) and∫ Στ∩{r≤R} ( JT (T · φ), n ) ≤ ∫ Στ ′′′′ j−1,τ ′′′ 2j+1 ( JT (T · φ), n ) . (A.93) Appendix B Reference for Chapter 1 B.1 Notation Contraction We sum over repeated indices. Also we use interchangeably g(V,N) . = (V,N) . = VµN µ J ·N .= (J,N) .= JµNµ , (B.1) where V , N are vectorfields, and J is a 1-form. Integration Let D inM be a domain bounded by two homologous hypersurfaces, Σ1 and Σ2 being its past and future boundary respectively. We then write ∫ Σ1 (J, n) for the boundary terms on Σ1 arising from a general current J in the expression ∫ ∂D ∗J . If S ⊂ Σ1 is spacelike, then (J, n) = g(J, n) is in fact the inner product of J with the timelike normal n to Σ1; e.g. on constant t-slices Σt (see Section 1.2) we have n = (1− 2mrn−2 )− 1 2 ∂ ∂t . If U ⊂ Σ1 is an outgoing null segment then ∫ U(J, n) denotes an integral of the form ∫ dv ∫ S dµγg(J, ∂ ∂v ); e.g. on the outgoing null segments of the hypersurfaces Στ (see Section 1.4) we have∫ Στ∩{r≥R} (J, n) . = ∫ ∞ τ+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 rn−1 ( J, ∂ ∂v∗ ) . (B.2) The volume form is usually omitted,∫ D f . = ∫ D f dµg (D ⊂M) . 199 200 APPENDIX B. REFERENCE FOR CHAPTER 1 B.2 Formulas In this appendix we summarize a few formulas for reference. The wave equation The d’Alembert operator can we written out in any coordinate system according to gφ = (g −1)µν∇µ∂νφ (B.1) where ∇ denotes the covariant derivative of the Levi-Civita connection of g. Components of the energy momentum tensor The components of the energy momentum tensor Tµν(φ) = ∂µφ ∂νφ− 1 2 gµν ∂ αφ ∂αφ tangential to Q are given in (u∗, v∗)-coordinates by Tu∗u∗ = ( ∂φ ∂u∗ )2 (B.2a) Tv∗v∗ = ( ∂φ ∂v∗ )2 (B.2b) Tu∗v∗ = ( 1− 2m rn−2 )∣∣∇/ φ∣∣2 r2 ◦ γn−1 . (B.2c) We also refer to (B.2) as the null decomposition of the energy momentum tensor. Note here that ∂αφ ∂αφ = − 1 1− 2m rn−2 ( ∂φ ∂u∗ )( ∂φ ∂v∗ ) + ∣∣∇/ φ∣∣2 r2 ◦ γn−1 and 1 r2 ◦ γ AB n−1 TAB = ∣∣∇/ φ∣∣2 r2 ◦ γn−1 − 1 2 (n− 1)∂αφ ∂αφ . Integration A typical domain of integration that we use is RDτ2τ1 = { (u∗, v∗) : τ1 ≤ u∗ ≤ τ2 , v∗ − u∗ ≥ R∗ } . (B.3) In local coordinates we have, by calculating the volume form from (1.2.25), that∫ RDτ2τ1 dµg = ∫ τ2 τ1 du∗ ∫ ∞ u∗+R∗ dv∗ ∫ Sn−1 dµ◦ γn−1 2 ( 1− 2m rn−2 ) rn−1 . (B.4) B.2. FORMULAS 201 For a general current J the energy identity on this domain reads∫ RDτ2τ1 KX dµg = ∫ ∂RDτ2τ1 ∗J , (B.5) where the right hand side is given more explicitly by∫ ∂RDτ2τ1 ∗J =− ∫ ∞ R∗+τ2 dv∗ ∫ Sn−1 dµ◦ γn−1 rn−1 g(J, ∂ ∂v∗ )|u∗=τ2 − ∫ τ2 τ1 du∗ ∫ Sn−1 dµ◦ γn−1 rn−1g(J, ∂ ∂u∗ )|v∗→∞ + ∫ ∞ R∗+τ1 dv∗ ∫ Sn−1 dµ◦ γn−1 rn−1 g(J, ∂ ∂v∗ )|u∗=τ1 − ∫ R∗+2τ2 R∗+2τ1 dt ∫ Sn−1 rn−1g(J, ∂ ∂r∗ )|r=R . (B.6) Dyadic sequences In our argument, Section 1.5.3 in particular, we construct a hierarchy of dyadic sequences, beginning with a sequence of real numbers (τj)j∈N where τ1 > 0 and τj+1 = 2τj (j ∈ N). We then obtain (by the mean value theorem of integration) a sequence (τ ′j)j∈N with τ ′ j in the interval (τj , τj+1) of length τj for all j ∈ N. We then built up on these values another sequence (τ ′′j )j∈N which takes values (as selected by the mean value theorem) in the intervals (τ ′2j−1, τ ′ 2j+1) ∋ τ ′′j ; note that their length is at least τ ′2j+1 − τ ′2j−1 ≥ τ2j+1 − τ2j = τ2j . In the same fashion the sequence (τ ′′′j )j∈N is built upon (τ ′′j )j∈N, etc. B.2.1 Rational functions For completeness, we include in this appendix the elementary integration of the rational function (xn − 1)−1 needed in Section 1.2. Consider the real polynomial of degree n ∈ N, p(z) = zn − 1 (z ∈ C) . The zeros p(zj) = 0 are zj = e 2πi j n (j ∈ Z) where |j| ≤ n− 1 2 if n is odd |j| ≤ n− 2 2 , j = n 2 if n is even. Therefore in factor representation p(z) = ∏ |j|≤[n−1 2 ] (z − zj) 1 , n odd(z + 1) , n even 202 APPENDIX B. REFERENCE FOR CHAPTER 1 By division into partial fractions there are numbers cj ∈ C such that 1 p(z) = ∑ |j|≤[n−1 2 ] cj z − zj + 0 , n oddcn2 z+1 , n even . In fact cj = lim z→zj z − zj p(z) = ( nzn−1j )−1 = 1 n e−2πij n−1 n , so in particular c0 = 1 n cn 2 = −1 n (n even). Note that c−j = cj . Now, for x ∈ R, 1 xn − 1 = [n−1 2 ]∑ j=1 ( cj x− zj + cj x− zj ) + c0 x− 1 + 0 , n oddcn2 x+1 , n even = [n−1 2 ]∑ j=1 (cj + cj)x− (cjzj + cjzj) x2 − (zj + zj)x+ |zj|2 + c0 x− 1 + 0 , n oddcn2 x+1 , n even = [n−1 2 ]∑ j=1 2 n cos(2πj n−1 n )x− 1 x2 − 2 cos(2πj n )x+ 1 + 1 n 1 x− 1 + 0 , n odd− 1 n 1 x+1 , n even since ℜ[cj ] = 1 n cos(2πj n− 1 n ) ℜ[cjzj ] = 1 n . We may now integrate ∫ 1 xn − 1 dx = = 1 n [n−1 2 ]∑ j=1 cos ( 2πj n− 1 n ) ∫ 2x− 2 cos(2πj n ) x2 − 2 cos(2πj n )x+ 1 dx + 2 n [n−1 2 ]∑ j=1 ( cos ( 2πj n− 1 n ) cos (2πj n )− 1)∫ 1 x2 − 2 cos(2πj n )x+ 1 dx + 1 n ∫ 1 x− 1 dx+ 0 , n odd− 1 n ∫ 1 x+1 dx , n even B.2. FORMULAS 203 = 1 n [n−1 2 ]∑ j=1 cos ( 2πj n− 1 n ) log ∣∣x2 − 2 cos(2πj n )x+ 1 ∣∣ + 2 n [n−1 2 ]∑ j=1 sin ( 2πj n− 1 n ) arctan (x− cos(2πj n ) sin ( 2πj n ) ) + 1 n log |x− 1|+ 0 , n odd− 1 n log |x+ 1| , n even because ∫ 1 x2 − 2 cos(2πj n ) + 1 dx = = ∫ 1 t2 + sin2(2πj n ) dt ∣∣∣ t=x−cos( 2pij n ) = 1 sin(2πj n ) arctan (x− cos(2πj n ) sin ( 2πj n ) ) . Therefore exp [∫ n xn − 1 dx ] = = |x− 1| 1 , n odd|x+ 1|−1 , n even × [n−1 2 ]∏ j=1 ∣∣∣x2 − 2 cos(2πj n ) x+ 1 ∣∣∣cos(2πj n−1n ) × [n−1 2 ]∏ j=1 exp [ 2 sin ( 2πj n− 1 n ) arctan (x− cos(2πj n ) sin(2πj n ) )] . B.2.2 Radial functions In this appendix we prove some statements on the relation between r∗ = ∫ r (nm) 1 n−2 1 1− 2m rn−2 dr (B.7) and r. The rational function that essentially appears is integrated already in Appendix B.2.1, 204 APPENDIX B. REFERENCE FOR CHAPTER 1 and we obtain r∗ = r − (nm) 1n−2 + (2m) 1n−2 ∫ r n−2√2m (n 2 ) 1 n−2 1 xn−2 − 1 dx = r − (nm) 1n−2 + (2m) 1n−2× × [ 1 n− 2 log |x− 1|+ 0 n odd− 1 n−2 log |x+ 1| n even + 1 n− 2 [n−3 2 ]∑ j=1 cos ( 2πj n− 3 n− 2 ) log ∣∣∣x2 − 2 cos( 2πj n− 2 ) + 1 ∣∣∣ + 2 n− 2 [n−3 2 ]∑ j=1 sin ( 2πj n− 3 n− 2 ) arctan (x− cos( 2πj n−2) sin( 2πj n−2) )]x= rn−2√2m (n 2 ) 1 n−2 It is useful to rewrite the last two terms as follows: [n−3 2 ]∑ j=1 cos ( 2πj n− 3 n− 2 ) log ∣∣∣x2 − 2 cos( 2πj n− 2 ) x+ 1 ∣∣∣ = [ 1 2 [n−3 2 ]]∑ j=1 cos ( 2π (n− 3)− (j − 1) n− 2 ) log ∣∣∣x2 − 2 cos( 2πj n− 2 ) x+ 1 ∣∣∣ + [ 1 2 [n−3 2 ]]∑ j=1 cos ( 2π [n−2 2 ] + j n− 2 ) log ∣∣∣x2 + 2 cos( 2π n− 2 ( j − { 1/2 n odd 0 n even )) x+ 1 ∣∣∣ +  0 [n−3 2 ] even cos ( 3 2 π { 1− 1 3(n−2) n odd 1 n even ) log ∣∣∣x2 − 2 cos(π2 { n−1 n−2 n odd 1 n even ) x+ 1 ∣∣∣ [n−32 ] odd = [ 1 2 [n−3 2 ]]∑ j=1 cos ( 2πj n− 2 ) log ∣∣∣x2 − 2 cos( 2πj n− 2 ) x+ 1 ∣∣∣ − [ 1 2 [n−3 2 ]]∑ j=1 cos ( 2π n− 2 ( j − { 1/2 n odd 0 n even )) log ∣∣∣x2 + 2 cos( 2π n− 2 ( j − { 1/2 n odd 0 n even )) x+ 1 ∣∣∣ +  0 [n−3 2 ] even − cos ( π 2 { n−3 n−2 n odd 1 n even ) log ∣∣∣x2 − 2 cos(π2 { n−1 n−2 n odd 1 n even ) x+ 1 ∣∣∣ [n−32 ] odd B.2. FORMULAS 205 [n−3 2 ]∑ j=1 sin ( 2πj n− 3 n− 2 ) arctan (x− cos( 2πj n−2) sin( 2πj n−2) ) =− [ 1 2 [n−3 2 ]]∑ j=1 sin ( 2πj n− 2 ) arctan (x− cos( 2πj n−2) sin( 2πj n−2) ) − [ 1 2 n−3 2 ]]∑ j=1 sin ( 2π n− 2 ( j − { 1/2 n odd 0 n even )) arctan (x+ cos( 2πn−2(j − { 1/2 n odd 0 n even )) sin ( 2π n−2 ( j − { 1/2 n odd 0 n even )) ) +  0 [n−3 2 ] even − sin ( π 2 { n−3 n−2 n odd 1 n even ) arctan (x−cos(pi2 8>< >: n−1 n−2 n odd 1 n even ) sin ( pi 2 8>< >: n−1 n−2 n odd 1 n even ) ) [n−32 ] odd It is now easy to see that for x ≥ 1 these sums are in fact negative or at most zero, and are bounded below by [n−3 2 ]∑ j=1 cos ( 2πj n− 3 n− 2 ) log ∣∣∣x2 − 2 cos( 2πj n− 2 ) x+ 1 ∣∣∣ ≥ [ 1 2 [n−3 2 ]]∑ j=1 log ∣∣sin2( 2πj n−2) ∣∣cos( 2pijn−2 )∣∣(x+ 1)2∣∣ − cos(π2 n− 3n− 2) log∣∣(x+ 1)2∣∣ ≥ n− 3 2 log sin 2π n− 2 − n + 1 2 log(1 + x) [n−3 2 ]∑ j=1 sin ( 2πj n− 3 n− 2 ) arctan (x− cos( 2πj n−2) sin( 2πj n−2) ) ≥ −n− 1 4 π for x ≥ 1, and of course n ≥ 5 (= 0 for n = 3, 4). Proposition B.1. For all n ≥ 3, lim r n−2√2m →∞ r∗ r = 1 . Proof. Step 1. (Upper bound on r ∗ r ) Let r∗ ≥ 0, r ≥ n−2√nm. Then using the insights from 206 APPENDIX B. REFERENCE FOR CHAPTER 1 above r∗ ≤ r − (nm) 1n−2 + (2m) 1n−2× × [ 1 n− 2 log ∣∣ r n−2√2m − 1 ∣∣∣∣(n 2 ) 1 n−2 − 1∣∣ + 0 n odd1 n−2 log ∣∣(n 2 ) 1 n−2 + 1 ∣∣ n even +  0 n = 3, 4 1 n−2 ( −n−3 2 log sin 2π n−2 + n+1 2 log ( 1 + (n 2 ) 1 n−2 )) + 2 n−2 n−1 4 π n ≥ 5 ] ≤ r − (nm) 1n−2 + (2m) 1n−2× × [ 1 n− 2 log ∣∣ r n−2√2m − 1 ∣∣∣∣(n 2 ) 1 n−2 − 1∣∣ + log 52 n = 3, 43 log 5 2 + π − 1 2 log sin 2π n−2 n ≥ 5 ] Since lim x→∞ log x x = 0 we have lim r n−2√2m →∞ r∗ r ≤ 1 . Step 2. (Lower bound on r ∗ r ) Since ( 1− 2m rn−2 ) ≤ 1, r∗ = ∫ r (nm) 1 n−2 1 1− 2m rn−2 dr ≥ r − n−2√nm . Hence r ≤ n−2√nm+ r∗ and lim r∗→∞ r r∗ ≤ 1 . While this fact concerns the region r∗ ≥ 0 and is essentially due to limx→∞ log xx = 0, the next concerns r∗ ≤ 0 and is similarly due to limx→0 x log x = 0. Proposition B.2. For all n ≥ 3, lim r n−2√2m →1 ( 1− 2m rn−2 ) (−r∗) = 0 . B.2. FORMULAS 207 Proof. Using the formulas from above for r∗ ≤ 0,( 1− 2m rn−2 ) (−r∗) = (1− 2m rn−2 )( n−2√nm− r) + ( 1− 2m rn−2 ) (2m) 1 n−2 ∫ (n 2 ) 1 n−2 r n−2√2m 1 xn−2 − 1 dx ≤ (1− 2m rn−2 )( ( n 2 ) 1 n−2 − 1)(2m) 1n−2 + (1− 2m rn−2 ) (2m) 1 n−2× × [ 1 n− 2 log ∣∣(n 2 ) 1 n−2 − 1∣∣∣∣ r n−2√2m − 1 ∣∣ + 0 n = 3, 4−1 2 log sin 2π n−2 + log( r n−2√2m + 1) + π n ≥ 5 ] Now recall xn−2 − 1 = (x− 1)× [n−3 2 ]∏ j=1 ( x2 − 2 cos( 2πj n− 2)x+ 1 )× 1 n odd(x+ 1) n even so for x ≥ 1 xn−2 − 1 ≤ (x− 1)(x+ 1)× [n−3 2 ]∏ j=1 (x2 + 2x+ 1) ≤ (x− 1)(x+ 1)n−2 . Therefore ( 1− 2m rn−2 ) log ∣∣(n 2 ) 1 n−2 − 1∣∣∣∣ r n−2√2m − 1 ∣∣ ≤ (1 + rn−2√2m )n−2 × × [ −( r n−2√2m − 1 ) log ∣∣ r n−2√2m − 1 ∣∣+ ( r n−2√2m − 1 ) log ∣∣(n 2 ) 1 n−2 − 1∣∣] and ( 1− 2m rn−2 )( −r∗ (2m) 1 n−2 ) ≤ (1− 2m rn−2 )( ( n 2 ) 1 n−2 − 1) − 1 n− 2 ( 1 + ( n 2 ) 1 n−2 )n−2( r n−2√2m − 1 ) log ∣∣ r n−2√2m − 1 ∣∣ + ( 1− 2m rn−2 )0 n = 3, 41 2 log n−2 4 + π + log(1 + (n 2 ) 1 n−2 ) n ≥ 5 We see clearly lim r n−2√2m →1 ( 1− 2m rn−2 ) (−r∗) ≤ 0 because lim x→0 x log x = 0 . 208 APPENDIX B. REFERENCE FOR CHAPTER 1 While the previous propositions could be proven with fairly rough bounds on r∗, the fol- lowing propositions concerning the region r∗ ≤ 0 require the (error) terms to be arranged more carefully. With q0(x) = x− 1 x+ 1 (B.8) qα,β(x) = (x2 − 2αx+ 1)α (x2 + 2βx+ 1)β (B.9) we see in view of the above that −r∗ =(nm) 1n−2 − r + (2m) 1 n−2 n− 2 log ∣∣(n 2 ) 1 n−2 − 1∣∣∣∣ r n−2√2m − 1 ∣∣ +  0 n odd − (2m) 1 n−2 n−2 log ∣∣(n 2 ) 1 n−2+1 ∣∣∣∣ r n−2√2m +1 ∣∣ n even + (2m) 1 n−2 n− 2 [ 1 2 [n−3 2 ]]∑ j=1 log q cos( 2pij n−2 ),cos ( 2pi n−2 (j− { 1/2 n odd 0 n even ) )((n2 ) 1n−2) q cos( 2pij n−2 ),cos ( 2pi n−2 (j− { 1/2 n odd 0 n even ) )( rn−2√2m) +  0 [n−3 2 ] even −(2m) 1 n−2 n− 2 cos (π 2 { n−3 n−2 n odd 1 n even ) × × log ∣∣∣(n2 ) 2n−2 − 2 cos(π2 { n−1 n−2 n odd 1 n even ) (n 2 ) 1 n−2 + 1 ∣∣∣ ∣∣∣ r2 (2m) 2 n−2 − 2 cos(π 2 { n−1 n−2 n odd 1 n even ) r (2m) 1 n−2 + 1 ∣∣∣ [n−3 2 ] odd −2(2m) 1 n−2 n− 2 [ 1 2 [n−3 2 ]]∑ j=1 {∫ (n 2 ) 1 n−2 r n−2√ 2m 1 1 + (y−cos( 2pij n−2 ) sin( 2pij n−2 ) )2 dy + ∫ (n 2 ) 1 n−2 r n−2√ 2m 1 1 + ( y+cos( 2pin−2 (j−{ 1/2 n odd 0 n even ) ) sin ( 2pi n−2 (j− { 1/2 n odd 0 n even ) ) )2 dy } +  0 [n−3 2 ] odd −2(2m) 1 n−2 n− 2 ∫ (n 2 ) 1 n−2 r n−2√2m 1 1 + (y−cos(pi2{ n−1n−2 n odd 1 n even ) sin ( pi 2 { n−1 n−2 n odd 1 n even ) )2 dy [n−3 2 ] even B.2. FORMULAS 209 In addition to Prop. B.2 we in fact have: Proposition B.3. For r∗ < 0, ( 1− 2m rn−2 ) ≤ (2m) 1 n−2 (−r∗) . This being an upper bound on (−r∗) we will also need a lower bound: Proposition B.4. For r∗ ≤ 0, (−r∗) ≥ (2m) 1 n−2 n− 2 log q0 ( (n 2 ) 1 n−2 ) q0 ( r n−2√2m ) . We only give a proof for the case where n is even because qα,α(x) = (x2 − 2αx+ 1 x2 + 2αx+ 1 )α is monotone increasing which simplifies the proof; indeed q′α,α(x) = 4α 2 ( x2 − 2αx+ 1)α−1( x2 + 2αx+ 1 )α+1 (x2 − 1) ≥ 0 for 0 < α < 1, x ≥ 1, and qα,α(1) = (1− α 1 + α )α lim x→∞ qα,α(x) = 1 . Proof (of Prop.B.3, n even). From the above we get the upper bound (−r∗) (2m) 1 n−2 ≤ (n 2 ) 1 n−2 − r (2m) 1 n−2 + 1 n− 2 log ∣∣(n 2 ) 1 n−2 − 1∣∣∣∣ r n−2√2m − 1 ∣∣ + 1 n− 2 [ 1 2 [n−3 2 ]]∑ j=1 log [(1 + cos( 2πj n−2 ) 1− cos( 2πj n−2 ))cos( 2pijn−2 )qcos( 2pij n−2 ),cos( 2pij n−2 ) ( ( n 2 ) 1 n−2 )] Since (n 2 ) 1 n−2 → 1 the last term tends to zero; in fact it is bounded by 1 100 . Along the lines of the proof of Prop. B.2 we now recall that for n even xn−2 − 1 = (x− 1)(x+ 1)× [n−4 4 ]∏ j=1 ( x2 − 2 cos( 2πj n− 2)x+ 1 )× 1 n−42 evenx2 + 1 n−4 2 odd = (x− 1)× qn−2(x) thus 1− 1 xn−2 = 1 xn−2 (x− 1)qn−2(x) 210 APPENDIX B. REFERENCE FOR CHAPTER 1 and lim x→1 1 xn−2 qn−2(x) = lim x→1 1 xn−2 xn−2 − 1 x− 1 = n− 2 ; in fact 1 n− 2 ( 1− 1 xn−2 ) ≤ x− 1 . Therefore, ( 1− 2m rn−2 ) (−r∗) (2m) 1 n−2 ≤((n 2 ) 1 n−2 − 1)(1− 2 n ) − ( r n−2√2m − 1 ) log | rn−2√2m − 1| |(n 2 ) 1 n−2 − 1| + 1 100 ( 1− 2 n ) ≤1 2 + 1 4 + 1 100 ≤ 1 Proof (of Prop. B.4, n even). Substracting the last terms from the first we get, (−r∗) ≥ 1 n− 2 ( (nm) 1 n−2 − r)+ (2m) 1n−2 n− 2 log q0 ( (n 2 ) 1 n−2 ) q0 ( r n−2√2m ) + (2m) 1 n−2 n− 2 [ 1 2 [n−3 2 ]]∑ j=1 log qcos( 2pij n−2 ),cos( 2pij n−2 ) ( (n 2 ) 1 n−2 ) qcos( 2pij n−2 ),cos( 2pij n−2 ) ( r n−2√2m ) ≥(2m) 1 n−2 n− 2 log q0 ( (n 2 ) 1 n−2 ) q0 ( r n−2√2m ) because qα,α is monotone increasing. B.3 Boundary Integrals and Hardy Inequalities In this appendix we prove appropriate Hardy inequalities that are needed in our argument to estimate boundary terms that typically arise in the energy identities. X-type currents. Let X = f(r∗) ∂ ∂r∗ and recall the modification (1.4.15). Proposition B.5 (Boundary terms near null infinity). Let f = O(1), f ′ = O(1 r ), and f ′′ = O( 1 r2 ), then there exists a constant C(n,m) such that∫ ∂RDτ2τ1\{r=R} ∗JX,1 ≤ C(n,m) ∫ Στ1 ( JT (φ), n ) . (B.1) B.3. BOUNDARY INTEGRALS AND HARDY INEQUALITIES 211 Proof. For the boundary integrals on the null segments u∗ = τ1, τ2 we find∣∣∣∣∫ ∞ R∗+τi dv∗ ∫ Sn−1 dµ◦ γn−1 g(JX,1, ∂ ∂v∗ ) rn−1 ∣∣∣∣ ≤ ≤ C(n) ∫ ∞ R∗+τi dv∗ ∫ Sn−1 dµ◦ γn−1 rn−1 {( ∂φ ∂v∗ )2 + ∣∣∇/ φ∣∣2 + [ |f | r2 + |f ′| r + |f ′|2 + |f ′′| ] φ2 } (B.2) and in view of the Hardy inequality Lemma B.6∫ ∞ R∗+τi dv∗ ∫ Sn−1 dµ◦ γn−1 1 r2 φ2 rn−1|u∗=τi ≤ C(n,m) ∫ Στi ( JT (φ), n ) ; (B.3) note that the corresponding zero order terms vanish at future null infinity, cf. Remark B.7. Then (B.1) follows from the energy identity for T on RDτ2τ1 . Lemma B.6 (Hardy inequality). Let φ ∈ C1([a,∞)), a > 0, with |φ(a)| <∞ and lim x→∞ x n−2 2 φ(x) = 0 , (B.4) then a constant C(n) > 0 exists such that∫ ∞ a 1 x2 φ2(x) xn−1 dx ≤ C(n) ∫ ∞ a ( dφ dx )2 xn−1 dx . (B.5) Proof. This is a consequence of the Cauchy-Schwarz inequality, after integration by parts∫ ∞ a 1 x2 φ2(x) xn−1 dx = ∫ ∞ a g′(x)φ2(x) dx with g(x) = ∫ x a yn−3 dy . Remark B.7. The conditions of the Lemma on φ are in fact satisfied for any solution of the wave equation (1.1.1). By a density argument we may assume without loss of generality that the initial data is compactly supported. Then for a fixed τ , and v∗ large enough φ(τ, v∗) = 0 and for u∗ ≥ τ φ(u∗, v∗) = ∫ u∗ τ ∂φ ∂u∗ du∗ . Thus φ(u∗, v∗) ≤ (∫ u∗ τ ( ∂φ ∂u∗ )2 rn−1 du∗ ) 1 2 (∫ u∗ τ 1 rn−1 du∗ ) 1 2 . On one hand ∫ u∗ τ ∫ Sn−1 ( ∂φ ∂u∗ )2 rn−1 dµ◦ γn−1 du∗ ≤ ∫ Στ ( JT (φ), n ) <∞ , 212 APPENDIX B. REFERENCE FOR CHAPTER 1 whereas on the other hand∫ u∗ τ 1 rn−1 du∗ = 1 n− 2 ∫ u∗ τ ( 1− 2m rn−2 )−1 ∂ ∂u∗ ( 1 rn−2 ) du∗ ≤ 1 n− 2 ( 1− 2m Rn−2 )−1( 1− (r(u∗, v∗) r(τ, v∗) )n−2) 1 rn−2 if we restrict u∗ ≥ τ to r(u∗, v∗) ≥ R. Hence lim v∗→∞ r n−2 2 φ = 0 . Instead of (B.5) which requires (B.4) one can prove the corresponding Hardy inequality for finite intervals: Lemma B.8 (Hardy inequality for finite intervals). Let 0 < a < b, and φ ∈ C1((a, b)) then 1 2 ∫ b a 1 x2 φ2(x)xn−1 dx ≤ 1 n− 2b n−2φ2(b) + 2 ( 2 n− 2 )2 ∫ b a ( dφ dx )2 xn−1 dx . (B.6) Proof. Let g(x) = ∫ x a yn−3 dy = 1 n− 2y n−2|xa then by integration by parts and using Cauchy’s inequality∫ b a 1 x2 φ2(x)xn−1 dx = g φ2|ba − ∫ b a g(x)2φ(x) dφ dx dx ≤ ≤ g(b)φ2(b) + 2ǫ ∫ b a 1 x2 φ2(x)xn−1 dx+ 1 2ǫ ∫ b a g(x)2 xn−3 ( dφ dx )2 dx where ǫ > 0; (B.6) follows for ǫ = 1 4 because g(b) ≤ 1 n− 2b n−2 g(x)2 xn−3 ≤ 2 n− 2 ( 1 + (a x )2(n−2)) xn−1 . Recall the domain (1.4.105); by using Lemma B.8 instead of Lemma B.6 we can prove the following refinement of Prop. B.5 to bounded domains: Proposition B.9 (Boundary terms on bounded domains). Let f = O(1), f ′ = O(1 r ), and f ′′ = O( 1 r2 ), then there exists a constant C(n,m) such that∫ ∂RD\ τ2τ1\{r=R} ∗JX,1 ≤ ≤ C(n,m) {∫ Σ τ2 τ1 ( JT (φ), n ) + ∫ Sn−1 dµ◦ γn−1 rn−2φ2|(u∗=τ1,v∗=R∗+τ2) } . (B.7) B.3. BOUNDARY INTEGRALS AND HARDY INEQUALITIES 213 Recall the domain (1.4.2). Proposition B.10 (Boundary terms near the event horizon). Let f = O(1), f ′ = O( 1|r∗|4 ), and f ′′ = O( 1|r∗|5 ), and πlφ = 0 (0 ≤ l < L) , for some L ∈ N, then there exists a constant C(n,m, L) such that∫ ∂R∞r0,r1(t0) ∗JX,1 ≤ C(n,m, L) ∫ Στ0 ( JT (φ), n ) . (B.8) where τ0 = 1 2 (t0 − r∗1). The proof is given in Section 1.4.4 in the special case f = fγ,α using the following Lemma. Lemma B.11 (Hardy inequality). Let a > 0, φ ∈ C1([a,∞)) with lim x→∞ |φ(x)| <∞ . Then∫ ∞ a 1 1 + x2 φ2(x) dx ≤ ≤ 81 + a 2 a2 ∫ ∞ a ( dφ dx )2 dx+ 2π ∫ a+1 a { φ2 + ( dφ dx )2} dx . (B.9) Proof. Let us first assume that φ(a) = 0. Define g(x) = − ∫ ∞ x 1 1 + y2 dy then∫ ∞ a 1 1 + x2 φ2(x) dx = ∫ ∞ a g′(x)φ2(x) dx = g(x)φ2(x)|∞a − 2 ∫ ∞ a g(x)φ(x) dφ dx dx ≤ 2 (∫ ∞ a g(x)2 g′(x) ( dφ dx )2 dx ) 1 2 (∫ ∞ a g′(x)φ2(x) dx ) 1 2 . Since |g(x)| ≤ 1 x we have g(x)2 g′(x) ≤ 1 + x 2 x2 ≤ 1 + a 2 a2 and therefore∫ ∞ a 1 1 + x2 φ2(x) dx ≤ 4 ∫ ∞ a g(x)2 g′(x) ( dφ dx )2 dx ≤ 41 + a 2 a2 ∫ ∞ a ( dφ dx )2 dx . 214 APPENDIX B. REFERENCE FOR CHAPTER 1 Without the assumption φ(a) = 0 this applied to the function φ(x)− φ(a) yields∫ ∞ a 1 1 + x2 φ2(x) dx ≤ ≤ 2 ∫ ∞ a 1 1 + x2 ( φ(x)− φ(a))2 dx+ 2 ∫ ∞ a 1 1 + x2 φ(a)2 dx ≤ 81 + a 2 a2 ∫ ∞ a ( dφ dx )2 dx+ πφ(a)2 . We conclude the proof with the following pointwise bound: On one hand for some a′ ∈ (a, a + 1) ∫ a+1 a φ(x)2 dx = φ(a′)2 and on the other hand φ(a′)2 − φ(a)2 = ∫ a′ a d dx φ(x)2 dx ≤ ∫ a′ a { φ(x)2 + ( dφ dx )2} dx . Hence φ(a)2 ≤ ∫ a′ a { φ(x)2 + ( dφ dx )2} dx+ ∫ a+1 a φ(x)2 dx ≤ 2 ∫ a+1 a { φ(x)2 + ( dφ dx )2} dx . Auxiliary currents. For auxiliary currents of the form Jauxµ = 1 2 h(r)∂µ(φ 2) (B.10) we have the same results. Proposition B.12. Let h = O(1 r ), then there exists a constants C(n,m) such that∫ ∂RDτ2τ1\{r=R} ∗Jaux ≤ C(n,m) ∫ Στ1 ( JT (φ), n ) , (B.11) and moreover for a constant C(n,m) we have the refinement∫ ∂RD\ τ2τ1\{r=R} ∗Jaux ≤ ≤ C(n,m) {∫ Σ τ2 τ1 ( JT (φ), n ) + ∫ Sn−1 dµ◦ γn−1 rn−2φ2|(τ1,R∗+τ2) } . (B.12) Proof. Note that here, in comparison to the proof of Prop. B.5,∣∣∣g(Jaux, ∂ ∂v∗ ) ∣∣∣ ≤ h2φ2 + ( ∂φ ∂v∗ )2 . B.3. BOUNDARY INTEGRALS AND HARDY INEQUALITIES 215 Proposition B.13. Let h = O( 1|r∗|), then there exists a constant C(n,m) such that∫ ∂R∞r0,r1 (t0) ∗Jaux ≤ C(n,m) ∫ Στ0 ( JT (φ), n ) . (B.13) where τ0 = 1 2 (t0 − r∗1). Remark B.14. Note that in view of Prop. B.3 the function h = 1 r ( 1 − 2m rn−2 ) satisfies the assumption of the Proposition. Appendix C Reference for Chapter 2 C.1 Decomposition formulas In this appendix we shall carry out several decompositions relative to the foliation of R by level sets of the area radius. Computations are significantly simplified by the use of the coordinates (r, σ) introduced in Section 2.2.1.3 which defines a “convenient frame” in the sense that [ ∂ ∂r , ∂ ∂σ ] = 0 . (C.1) Recall the first fundamental form gσσ = 1 4 3 Λ (Λ 3 r2 − 1 ) 1 σ2 (C.2a) gσA = 0 gAB = r 2 ◦ γAB , (C.2b) and by the first variational formula we have kij = 1 2φ ∂gij ∂r (C.3) for the second fundamental form k of the foliation (Σr). Thus it is easily checked that tr k = gijr kij = Λ 3 φ r + 2 r 1 φ (C.4a) |k|2 = gimr gjnr kmnkij = (Λ 3 φ r )2 + 2 (rφ)2 , (C.4b) and ∂ tr k ∂r = Λφ− φ|k|2 (C.5) in agreement with the general second variation formula. 216 C.2. COERCIVITY INEQUALITY ON THE SPHERE 217 Wave equation. Let us derive the explicit form of the wave equation gψ = 0 (C.6) for a metric of the form (2.2.39), i.e. g = −φ2 dr2 + gr . (C.7) Since Γrrr = 1 φ ∂φ ∂r (C.8a) Γirr = φ gr ij ∂jφ (C.8b) Γrij = 1 2 1 φ2 ∂rgrij = 1 φ kij (C.8c) we obtain gψ = (g −1)rr∇r∂rψ + (g−1)ij∇i∂jψ = = − 1 φ2 ∂2ψ ∂r2 + 1 φ [ 1 φ2 ∂φ ∂r − tr k ]∂ψ ∂r + 1 φ ∇φ · ∇ψ +△ψ . (C.9) For the lapse (2.2.38) we have 1 φ2 ∂φ ∂r = −Λ 3 φ r (C.10) ∇φ = 0 , (C.11) so that here gψ = − 1 φ2 ∂2ψ ∂r2 − 2 [Λ 3 ( rφ ) + 1 (rφ) ]1 φ ∂ψ ∂r +△ψ . (C.12) C.2 Coercivity inequality on the sphere In this appendix we shall prove the coercivity formula for the standard sphere, and recall the classic Sobolev inequality on the sphere. Let us denote by Sr = { x ∈ R3 : |x| = r } (C.1) the sphere of radius r in Euclidean space, a submanifold of( R 3, e = ( dx1)2 + ( dx2)2 + ( dx3)2 ) . (C.2) We denote the metric of the round sphere Sr by γr = e ∣∣ TSr = r2 ◦ γ , (C.3) 218 APPENDIX C. REFERENCE FOR CHAPTER 2 where ◦ γ is the standard metric on the unit sphere S2. Let Ω(i) = ǫijk x j ∂ ∂xk i = 1, 2, 3 , (C.4) where ǫ is the volume form of e. We have 3∑ i=1 Ωm(i)Ω n (i) = |x|2δmn − xnxm , (C.5) and thus for all x ∈ R3\{0} and u : R3 → R differentiable: 3∑ i=1 ( Ω(i)u )2 (x) = |x|2 [ |∇u|2 − 〈 x|x| ,∇u〉2] = |x|2|∇/ u|2 . (C.6) This is the coercivity formula on the sphere. Here ∇/ = Π∇, and Πba(ξ) = δ b a − ξa ξb , ξ ∈ S2 , (C.7) is the projection to the sphere; by uniqueness ∇/ is the connection of γr. Lemma C.1 (Coercivity inequalities on the sphere). Let u be a smooth function on Sr, then r2|∇/ u|2γr ≤ 3∑ i=1 ( Ω(i)u )2 , (C.8) and r4|∇/ 2u|2γr ≤ 3∑ i,j=1 ( Ω(i)Ω(j)u )2 . (C.9) We have already shown the first inequality. For the second inequality we can use that more generally for any Sr-1-form (a 1-form on R 3 such that θ ·X = θ · ΠX) it holds 3∑ i=1 ∣∣L/Ω(i)θ∣∣2γr = r2∣∣∇/ θ∣∣2γr + ∣∣θ∣∣2γr , (C.10) where L/ denotes the Lie derivative on Sr. (To prove (C.10) one can proceed analogeously to Lemma 11.2 in [14].) By substituting θ = d/u . = du|TSr , (C.11) we then obtain r2 ∣∣∇/ 2u∣∣2 γr = r2 ∣∣∇/ d/u∣∣2 γr ≤ ≤ 3∑ i=1 ∣∣L/ Ω(i) d/u∣∣2γr = 3∑ i=1 ∣∣ d/L/ Ω(i)u∣∣2γr = 3∑ i=1 ∣∣∇/ (Ω(i))∣∣2γr , (C.12) because Lie derivatives commute with exterior derivatives. Inequality (C.9) then follows from (C.12) using (C.8). We recall the classical Sobolev inequality on the sphere. C.2. COERCIVITY INEQUALITY ON THE SPHERE 219 Lemma C.2 (Sobolev embedding on S2). Let u ∈ H2(S2), then u ∈ L∞(S2) and ‖u‖L∞(S2) ≤ C ‖u‖H2(S2) . (C.13) This can be shown similarly to the Sobolev embedding on R2 using the fact that ◦ γ is conformal to the Euclidean metric on the plane. Indeed, given any x ∈ S2 ⊂ R3, we can introduce stereograhic coordinates y ∈ R2, such that y(x) = 0 and ◦γ takes the form ◦ γ= 1 (1 + 1 4 |y|2)2 | dy| 2 . (C.14) It is then easy to verify Morrey’s inequality, namely that W1,p(S2) ⊂ L∞(S2) for all p > 2. On the other hand, using two stereographic charts to cover the sphere, it is easy to check that also the classic embedding W1,1(S2) ⊂ L2(S2) is valid, and therefore also W1,q(S2) ⊂ Lp(S2) for 1/p = 1/q − 1/2. Thus in particular W1, 32 (S2) ⊂ L6(S2), while W1,6(S2) ⊂ L∞(S2) and ‖u‖∞ ≤ C ( ‖u‖L6(S2) + ‖∇/ u‖L6(S2) ) ≤ ≤ C ( ‖u‖ L 3 2 (S2) + ‖∇/ u‖ L 3 2 (S2) + ‖∇/ 2u‖ L 3 2 (S2) ) ≤ C‖u‖H2(S2) . (C.15) Given a function u on Sr we can apply Lemma C.2 to u ◦ h, where h : S2 → Sr, ξ 7→ rξ . (C.16) Since, in the coordinates γr = r 2 ◦ γ= r2 ◦ γAB dy A dyB , (C.17) h is the identity mapping, we get | ◦ ∇/ (u ◦ h)|2◦ γ = r2|(∇/ u) ◦ h|2γr , | ◦ ∇/ 2 (u ◦ h)|2◦ γ = r4|(∇/ 2u) ◦ h|2γr , (C.18) and thus r|u| ∣∣∣ Sr ≤ C (∫ Sr |u|2 dµγr ) 1 2 + C (∫ Sr r2|∇/u|2 dµγr ) 1 2 + C (∫ Sr r4|∇/ 2u|2 dµγr ) 1 2 . (C.19) Corollary C.3. Let u ∈ H2(Sr), then r|u| ∣∣∣ Sr ≤ C (∫ Sr |u|2 dµγr ) 1 2 + C (∫ Sr 3∑ i=1 ( Ω(i)u )2 dµγr ) 1 2 + C (∫ Sr 3∑ i,j=1 ( Ω(i)Ω(j)u )2 dµγr ) 1 2 . (C.20) Bibliography [1] S. 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