The wave equation on Schwarzschild-de Sitter spacetimes
aa r X i v : . [ g r- q c ] S e p The wave equation on Schwarzschild-de Sitterspacetimes
Mihalis Dafermos ∗ Igor Rodnianski † October 29, 2018
Abstract
We consider solutions to the linear wave equation ✷ g φ = 0 on a non-extremal maximally extended Schwarzschild-de Sitter spacetime arisingfrom arbitrary smooth initial data prescribed on an arbitrary Cauchyhypersurface. (In particular, no symmetry is assumed on initial data,and the support of the solutions may contain the sphere of bifurcation ofthe black/white hole horizons and the cosmological horizons.) We provethat in the region bounded by a set of black/white hole horizons andcosmological horizons, solutions φ converge pointwise to a constant fasterthan any given polynomial rate, where the decay is measured with respectto natural future-directed advanced and retarded time coordinates. Wealso give such uniform decay bounds for the energy associated to theKilling field as well as for the energy measured by local observers crossingthe event horizon. The results in particular include decay rates along thehorizons themselves. Finally, we discuss the relation of these results toprevious heuristic analysis of Price and Brady et al. Contents J Xµ . . . . . . . . . . 9 ∗ University of Cambridge, Department of Pure Mathematics and Mathematical Statistics,Wilberforce Road, Cambridge CB3 0WB United Kingdom † Princeton University, Department of Mathematics, Fine Hall, Washington Road, Prince-ton, NJ 08544 United States J Yµ and J Yµ . . . . . 101.3.4 Comparison with the Schwarzschild case . . . . . . . . . . 111.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Note added . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . 13 r, t ) . . . . . . . . . . . . . . . . . . . 132.2 Regge-Wheeler coordinates ( r ∗ , t ) . . . . . . . . . . . . . . . . . . 142.3 Eddington-Finkelstein coordinates ( u, v ) . . . . . . . . . . . . . . 142.4 Useful formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 r i , R i . . . . . . . . . . . . . . . . . . . . . . . 174.2 Dependence of constants C , E , and ǫ on r i , R i . . . . . . . . . . 174.3 Cutoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3.1 The cutoffs η i . . . . . . . . . . . . . . . . . . . . . . . . . 184.3.2 The cutoffs χ i and χ i . . . . . . . . . . . . . . . . . . . . 18 Σ t and the region R ( t , t )
186 The main estimates 19 N , ˜ N and P . . . . . . . . . . . . . . . . . . . . 206.2 The quantities Z ˜ N,P , Z N and Q . . . . . . . . . . . . . . . . . . 206.3 Statement of the estimates . . . . . . . . . . . . . . . . . . . . . . 216.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 J X family of currents 22 J V,iµ for an arbitrary V = f ∂∂r ∗ . . . . . . . . . 227.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.3 The vector fields X ℓ . . . . . . . . . . . . . . . . . . . . . . . . . 247.3.1 The case ℓ = 0 . . . . . . . . . . . . . . . . . . . . . . . . 247.3.2 The case ℓ ≥ J X,i . . . . . . . . . . . . . . . . . . . . . . 257.3.4 Controlling the error . . . . . . . . . . . . . . . . . . . . . 267.4 Auxiliary currents . . . . . . . . . . . . . . . . . . . . . . . . . . 277.4.1 Auxiliary positive definite pointwise quantities . . . . . . 277.4.2 The current J X a , . . . . . . . . . . . . . . . . . . . . . . 287.4.3 The currents J X b , , J X b , . . . . . . . . . . . . . . . . . . 287.4.4 The current J X c , . . . . . . . . . . . . . . . . . . . . . . 297.4.5 The current J X d , . . . . . . . . . . . . . . . . . . . . . . 29 J Y and J Y Y and Y . . . . . . . . . . . . . . . . . . . . . . 308.2 Definition of the currents . . . . . . . . . . . . . . . . . . . . . . 318.3 Discussion and the choice of r , R . . . . . . . . . . . . . . . . . 31 J Θ J . . . . . . . . . 3410.3 Applications of the integral identity for currents . . . . . . . . . . 3810.4 Bounding Q φ from Z Nφ . . . . . . . . . . . . . . . . . . . . . . . . 4010.5 Bounding Z Nφ from Z ˜ N,Pφ , Q φ and Q Ω i φ . . . . . . . . . . . . . . 4110.6 Bounding Z Nφ ( t ) from Z Nφ ( t ) and Q φ . . . . . . . . . . . . . . . 42
11 Proof of Theorem 1.1 43
A Proof of Lemma 7.3.1 44B Proof of Lemma 11.1.1 45
The introduction of a positive cosmological constant in the Einstein equationsof general relativity gives rise to a wide variety of new interesting solution space-times, in particular, spacetimes containing both “black hole” and “cosmological”regions. As in the case of black-hole spacetimes with vanishing cosmological con-stant, the stability of these spacetimes as solutions to the Einstein equations is afundamental open problem of gravitational physics. Yet even the simplest ques-tions concerning the behaviour of linear waves on such spacetime backgroundstoday remain unanswered. In this paper, we initiate in the above context themathematical study of decay for solutions to the linear wave equation.The simplest family of black-hole spacetimes with positive cosmological con-stant is the so-called
Schwarzschild-de Sitter family. If the cosmological constantΛ > M , g ) to the Einstein vacuum equations R µν − g µν R = − Λ g µν , (1)with parameter M , called the mass . We shall consider only the non-extremalblack-hole case, corresponding to parameter values0 < M < √ Λ . (2)As with the Schwarzschild family, the first manifestation of the Schwarzschild-de Sitter family of solutions was an expression for the metric in local coordi-nates, in this case first published in 1918 by Kottler [17], and independently byWeyl [20], in the form − (cid:18) − Mr −
13 Λ r (cid:19) dt + (cid:18) − Mr −
13 Λ r (cid:19) − dr + r dσ S . (3)Here dσ S denotes the standard metric on the unit 2-sphere. The global struc-ture of maximal spherically symmetric vacuum extensions of such metrics wasonly understood much later [8, 18, 13] based on the methods of formal Pen-rose diagrams introduced by B. Carter. In fact, maximally extended sphericallysymmetric vacuum spacetimes ( M , g ) with various different topologies can beconstructed, all of which equally well merit the name “Schwarzschild-de Sitterwith parameter M and cosmological constant Λ”. Such solutions ( M , g ) allshare the property that the universal cover ˜ Q of the 2-dimensional Lorentzianquotient Q = M /SO (3) consists of an infinite chain of regions as depicted inthe Penrose diagram below: r = ∞ r = ∞ r = 0 r = 0 r = 0 r = 0 The results of this paper do not depend on the topology, and for definiteness,one may assume in what follows that the name “Schwarzschild de-Sitter” andthe notation ( M , g ) refer to the spacetime with quotient precisely the universalcover depicted above.It is then the wave equation ✷ g φ = 0 (4)on this background ( M , g ) whose mathematical study we wish to initiate here.There is already a rich body of heuristic work on this problem in the physicsliterature. (See Section 1.4 below for a discussion.) The motivation for the studyof (4) in the present context is multifold. In particular, as in the case of vanishingcosmological constant, studied in our previous [12], we believe that provingbounds on decay rates for solutions to (4) is a first step to a mathematicalunderstanding of non-linear stability problems for spacetimes containing blackholes, that is to say, to the problem of stability in the context of the dynamicsof (1). For a more detailed discussion, we refer the reader to the introductoryremarks of [12]. We are interested in solutions of (4) arising from suitably regular initial dataprescribed on a Cauchy surface Σ of M . For future applications to non-linearstability problems, it is crucial that all assumptions have a natural geometricinterpretation independent of special coordinate systems. Moreover, our pri-mary concern in this paper is the region D bounded by a set of black/white holehorizons H + ∪ H − and cosmological horizons H + ∪ H − : D = clos (cid:0) J − ( H + ∪ H + ) ∩ J + ( H − ∪ H − ) (cid:1) (5)as depicted below r = ∞ r = ∞ r = 0 r = 0 H − Σ H + H + D H − By causality, the global behaviour of φ in D can be understood independentlyof the behaviour near r = 0 and r = ∞ . The behaviour in say D ∩ J + (Σ)is completely determined by the behaviour of appropriate initial data on Σ ∩ J − ( D ). We review briefly in the next paragraph the solvability and domain ofdependence property for the initial value problem for (4).Let Σ ⊂ M be a smooth Cauchy surface and let n µ denote the future-directed unit normal of Σ. For s ≥
1, let ϕ be an H s loc (Σ) function and ˙ ϕ : Σ → R an H s − (Σ) function. Then there exists a unique global solution φ : M → R of ✷ g φ = 0 such that for all smooth spacelike hypersurfaces ˜Σ with futuredirected unit normal ˜ n , φ | Σ ′ ∈ H s loc , (˜ nφ ) | Σ ′ ∈ H s − , and φ | Σ = ϕ , nφ | Σ = ˙ ϕ .Moreover, if K ⊂ M is closed and φ , φ are two such solutions correspondingto data ( ϕ , ˙ ϕ ), ( ϕ , ˙ ϕ ) such that ϕ | Σ ′ ∩ K = ϕ | Σ ′ ∩ K , ˙ ϕ | Σ ′ ∩ K = ˙ ϕ | Σ ′ ∩ K ,then φ = φ on M\ ( J + (Σ \ K ) ∪ J − (Σ \ K )). In particular, setting K = J − ( D ),we obtain that φ = φ on J + (Σ) ∩ D . We employ in this paper the standard notation of Lorentzian geometry (e.g. J + , J − ,etc.), and Penrose diagrams. See [15]. Note, as depicted, that Σ ∩ J − ( D ) is not necessarily Σ ∩ D . Since our results will be quantitative, we need to introduce relevant norms onthe compact manifold with boundary Σ ∩ J − ( D ). Let k·k denote the Riemannian L norm on Σ ∩ J − ( D ). This induces a norm on sections of the tangent bundle,a norm we will denote also by k · k . If ϕ ∈ H (Σ), ˙ ϕ ∈ L (Σ), then let usdenote by φ the unique solution of ✷ g φ = 0 corresponding to initial data ( ϕ, ˙ ϕ ).Let us define now for all real s ≥ E s ( ϕ, ˙ ϕ ) . = k∇ Σ ϕ k + k ˙ ϕ k + X ℓ ≥ r s ℓ s ||∇ Σ φ ℓ || + r s ℓ s || nφ ℓ || , (6)where φ ℓ denotes the projection of φ to the ℓ ’th eigenspace of △ / , i.e. the ℓ ’thspherical harmonic of φ . The function r is discussed in Section 2. If Σ itselfis spherically symmetric, then we may replace φ ℓ , nφ ℓ be ϕ ℓ and ˙ ϕ ℓ , and theabove expression is a sum of integrals on initial data. For general Σ, a sufficientcondition for the finiteness of (6) is that ϕ ∈ H s +1loc (Σ), ˙ ϕ ∈ H s loc (Σ).In the case m ≥ E m geometrically asfollows. Let Ω i , i = 1 , . . . , so (3) associated to the spherical symmetry of ( M , g ). We call Ω i angular momentum operators . It easily follows that E m ( ϕ, ˙ ϕ ) ∼ X p ,...p m − =0 , X ≤ i ,...i m − ≤ k∇ Σ (Ω p i · · · Ω p m − i m − φ ) k + k n (Ω p i · · · Ω p m − i m − φ ) k . Again, if Σ itself is spherically symmetric, we may replace φ with ϕ in the firstterm, and remove the n from the second, replacing φ with ˙ ϕ . The main result of this paper is contained in the following
Theorem 1.1.
Let ( M , g ) denote the Schwarzschild-de Sitter spacetime withparameter M and cosmological constant Λ satisfying ( ) and let Σ be a Cauchysurface for M . Let D ⊂ M denote a region as defined in ( ) and let s ≥ .Then, there exist constants C s depending only on s , M , Λ , and the geometryof Σ ∩ J − ( D ) such that for all solutions φ of the wave equation ✷ g φ = 0 on M such that E s ( ϕ, ˙ ϕ ) is finite, where ϕ . = φ | Σ , ˙ ϕ . = nφ | Σ , and for all achronalhypersurfaces Σ ′ ⊂ D ∩ J + (Σ ′ ) , the bound Z Σ ′ T µν ( φ ) T µ n ν ≤ C s E s ( ϕ, ˙ ϕ )( v + (Σ ′ ) − s + u + (Σ ′ ) − s ) (7) holds, where u and v denote fixed Eddington-Finkelstein advanced and retardedcoordinates , u + . = max { u, } , u + (Σ) . = inf x ∈ Σ u + ( x ) , etc, T µ denotes the See Section 2. Although these coordinates are only defined in D o , the statements (7), (8)can be interpreted in all of D in view of conventions (19)–(22). Killing field coinciding in the interior of D with ∂∂t , T µν ( φ ) denotes the standardenergy-momentum tensor, and n ν is the future-directed unit normal wherever Σ ′ is spacelike, in which case the integral is taken with measure the induced volumeform. In addition, ( ) holds if T µ is replaced by the vector field N µ defined inSection 6. If s > , then the pointwise bound | φ − φ | ≤ C s E s ( ϕ, ˙ ϕ ) (cid:16) v − s +12 + + u − s +12 + (cid:17) (8) holds in J + (Σ) ∩ D , where φ is a constant satisfying | φ | ≤ sup x ∈ Σ | φ ( x ) | + C E ( ϕ, ˙ ϕ ) . In particular, the theorem applies to arbitrary smooth initial data ϕ ∈ C ∞ (Σ) , ˙ ϕ ∈ C ∞ (Σ) , where s can be taken arbitrarily large. There areno unphysical assumptions regarding vanishing of φ at the sphere of bifurcation of the horizons, i.e. at the sets H + ∩ H − and H + ∩ H − . The decay rates (8), (9)are uniform, i.e. they hold up to and including the horizons, setting u + = ∞ or v + = ∞ . In particular, Σ ′ in (9) can be taken (as depicted below) r = ∞ r = ∞ r = 0 r = 0 H − Σ H + H + D Σ ′ H − to contain subsets of H + and/or H + . The statement of Theorem 1.1 should be compared with the results of ourprevious [10, 12] concerning the wave equation on a Schwarzschild exterior.Recall that in the region r > M , the Schwarzschild metric is given by theexpression (3) for Λ = 0, M >
0. The Penrose diagramme of the closure of this A correct interpretation of n µ and the measure of integration for general achronal Σ ′ canbe derived by a limiting procedure. region in the maximally extended Schwarzschild spacetime is given below: i + i H + H − I + I − In [10], an analogue of (7) is proven for all s < ǫ , this result is expected to be sharp, as it not expectedto be true for s >
6, in view of heuristic arguments due to Price [19].In [12], an analogue of (7) is proven for s = 2 for arbitrary, not necessarilyspherically symmetric, initial data.In view of the fact that solutions of the wave equation vanish on I + , theresults of [12] allow one to obtain the uniform pointwise decay rate | φ | ≤ Cv − .As a uniform decay bound in v , this decay rate is in fact sharp. The loss of angular derivatives in the result of Theorem 1.1 can be more preciselyquantified by decomposing φ into spherical harmonics. Each spherical harmonic φ ℓ decays at least exponentially, but the bound on the exponential rate obtainedhere decreases inverse quadratically in the spherical harmonic number. We havethe following Theorem 1.2.
Let ( M , g ) , Σ , D be as in Theorem 1.1. Then there exists aconstant c depending only on M and Λ , and C depending only on M , Λ , andthe geometry of Σ ∩ J − ( D ) , such that for all φ ℓ solutions of the wave equationon M with spherical harmonic number ℓ with E ( ϕ ℓ , ˙ ϕ ℓ ) = k∇ ϕ ℓ k + k ˙ ϕ ℓ k < ∞ and all achronal hypersurfaces Σ ′ ⊂ J + (Σ) ∩ D , the bound Z Σ ′ T µν ( φ ℓ ) T µ n ν ≤ C E ( ϕ ℓ , ˙ ϕ ℓ ) (cid:16) e − cv + (Σ ′ ) /ℓ + e − cu + (Σ ′ ) /ℓ (cid:17) (9) holds for all ℓ ≥ , and, again as before, also with T µ above replaced with N µ defined in Section 6. In addition, the pointwise bounds | φ ℓ ( u, v ) | ≤ C E ( ϕ ℓ , ˙ ϕ ℓ )( e − cv + /ℓ + e − cu + /ℓ ) for ℓ ≥ , and | φ ( u, v ) − φ | ≤ C E ( ϕ , ˙ ϕ )( e − cv + /ℓ + e − cu + /ℓ ) , for ℓ = 0 , hold in J + (Σ) ∩ D , where φ is a constant satisfying | φ | ≤ inf x ∈ Σ | φ ( x ) | + C E ( ϕ , ˙ ϕ ) . The above theorem can easily be seen to imply Theorem 1.1.
In this paper, we insist on a framework of proof that in principle may haverelevance to the non-linear stability problem, that is to say, the problem of thedynamics of (1) starting from initial data close to those induced on a Cauchyhypersurface Σ of Schwarzschild-de Sitter. This leads us to try to exploit com-patible currents. In this section, we will describe this general approach, andthe natural relation of the currents we will define with various geometric andanalytical aspects of the problem at hand.
For quasilinear hyperbolic systems (like (1)) in 3 + 1 dimensions, all knowntechniques for studying the global dynamics are based on L estimates. In theLagrangian case, the origin of such estimates can be understood geometricallyin terms of compatible currents (see Christodoulou [9]). These are 1-forms J µ such that at each point x ∈ M , both J µ and the divergence K = ∇ µ J µ dependonly on the 1-jet of φ . An important class of these are the currents J Vµ obtainedby contracting the energy momentum tensor T µν with an arbitrary vector field V µ . See Section 3 for a discussion in the context of the linear wave equations ✷ g φ = 0 studied here. All estimates in this paper are obtained by exploitingthe integral identities Z R K = Z ∂ R J µ n µ (10)corresponding to compatible currents of the form J Vµ and straightforward mod-ifications thereof J µ = J Vµ + · · · , where the vector fields V are directly relatedto the geometry of the problem, and the region R is suitably chosen. J Xµ The timelike hypersurface r = 3 M is known as the photon sphere . This has theominous property of being spanned by null geodesics. If additional regularity isnot imposed, then it is clear by a geometric optics approximation that solutionsof the wave equation can concentrate their energy along such geodesics for ar-bitrary long times, and one can thus not achieve a quantitative bound for therate of decay in terms of initial energy alone. In particular, (7) cannot hold for s > E s is replaced by E .It is truly remarkable that this obstruction arising from geometrical opticsis captured, and quantified, by a current J µ associated to a vector field V f ( r ∗ ) ∂ r ∗ for a well-chosen function f . The story is not entirelystraightforward, however. The desired current is in fact not precisely of theform J Vµ , but a modification thereof, to be denoted J X, µ , which is associated ina well defined way to a collection of vector fields X ℓ = f ℓ ( r ∗ ) ∂ r ∗ . The currentis defined by summing over currents J X ℓ , which act on individual sphericalharmonics φ ℓ .The current J X, yields a nonnegative K X, , modulo an error term supportednear the horizons. In a first approximation, we may pretend that in fact K X, ≥
0, but degenerates (in regular coordinates) near the horizon. The identity (10)can then be used as an estimate for its left hand side, in view of the fact thatits right hand side will in fact be bounded by the flux of J T , for the Killing field T , which is conserved. The role of the photon sphere will be exemplified by thedegeneration at r = 3 M of the quantity controlled by this spacetime integral.In order to obtain decay results from the above, one would have to gain infor-mation about the quantity estimating the boundary terms–namely R ∂ R J Tµ n µ ,from the control of spacetime integral. The difficulty for this is that the space-time integral estimates one obtains degenerate at the photon sphere r = 3 M and at the horizons. This does not allow one to control J Tµ n µ there.The problem at the photon sphere is cured by applying the estimate also toangular derivatives. It is here that the argument “loses” an angular derivative.It is this loss that leads to the form of decay proven in (7).The problem on the horizon, on the other hand, turns out to be illusionary.The horizon is in fact a very favourable place for estimating the solution! Forthis, we will need to consider the “local observer” vector fields Y , Y , to bedescribed in the next section. J Yµ and J Yµ The heuristic mechanism ensuring decay near the horizons has been understoodfor many years, and is known as the red shift effect . This is typically describedin the language of geometric optics. If two observers A and B cross the eventhorizon at advanced times v A < v B , and A sends a signal to B at a certainfrequency, as he ( A ) measures it, then the frequency at which B receives it isexponentially damped in the quantity v B − v A .It turns out that this exponential damping property can be captured bythe integral identities (10) corresponding to the currents J Yµ and J Yµ associatedto vector fields Y , Y , defined in Section 8. These vector fields are supportednear the horizons H + , H + , respectively. The estimates (10) corresponding tothe currents J Yµ , J Yµ fulfill the double role of (a) correcting for the error regionwhere K X, < K Y + K Y and (b)controlling the spacetime integrated energy measured by local observers nearthe horizon. The choice of Y , Y is delicate, because there is an “error region”where K Y + K Y <
0, which must be controlled with the help of the currents of This insight, in the case of the wave equation on the Schwarzschild solution, is originallydue to Blue and Soffer [2]. See, however, [3]. X , Y and Y arethus strongly coupled. To see the above arguments in context, the reader may wish to compare withour previous [12], where versions of the currents J Xµ , J Yµ are also employed.The relation of our arguments with the physical mechanisms at play are in factmuch clearer in the present paper, than in [12]. This is due on the one hand tothe absence here of the Morawetz-type vector field (denoted K in [12]), and, onthe other hand, to the relative simplicity here in the construction of the current J Xµ . We give here some comments on these points.The Morawetz vector field employed in [12] is a highly unnatural quantityat the horizon from the geometric point of view. On the other hand, in viewof its weights, it somewhat magically captures a polynomial (as opposed to theproper exponential) version of the red shift. The pointwise decay rates achievedvia K at the horizon are worse than the decay rates away from the horizon, butsufficient if one is only interested in the behaviour of the solution away from thehorizon. (See also [4].) In our [12], uniform decay rates up to the horizon wereindeed obtained with the help of J Yµ . But these estimates could be obtained a posteriori . From the point of view of the non-linear stability problem, thisdecoupling appears to be an exceptional feature. It is in this sense that thescheme proposed in the present paper is perhaps more naturally connected tothe geometry of general black holes.The second point to be made here concerns the construction of J Xµ . In [12],positivity of the analogue of what we denote here K X, relied on an unmotivatedrecentring and rescaling of the derivatives of the functions f ℓ which obscuredperhaps the fundamental connection with the photon sphere. Here, this con-nection appears much more clear. Of course, this is at the expense of havingto bound − K X, from K Y and K Y . This should in no way be thought of as adisadvantage. The red-shift effect has a lot to offer. It should be used and notobscured. As noted above, the study of the asymptotic behaviour of solutions to ✷ g φ =0 on both Schwarzschild and Schwarzschild-de Sitter backgrounds has a longtradition in the physics literature. In the Schwarzschild case, the pioneeringheuristic study is due to Price [19]. See also [14]. For the Schwarzschild-deSitter case, there is numerical work of Brady et al [6], the subsequent [7], andreferences therein.The above studies are based entirely upon decomposition of φ into sphericalharmonics. The results of these heuristics or numerics are typically presentedin terms of the asymptotic behaviour of the tail: φ ℓ ( r, t ) ∼ t − ℓ − , φ ℓ ( u, v ) ∼ v − ℓ − , rφ ℓ ( u, v ) ∼ u − ℓ − (11)2for Schwarzschild, where 2 M < r < ∞ is fixed in the first formula, u ≥ v in thesecond, and v ≥ u in the third, and φ ℓ ( r, t ) ∼ e − cℓt , φ ℓ ( u, v ) ∼ e − cℓv , φ ℓ ( u, v ) ∼ e − cℓu (12)for Schwarzschild-de Sitter and ℓ ≥
1, where r b < r < r c is fixed in the firstformula, and u ≥ v in the second, and v ≥ u in the third.At first glance, statements (12) may appear stronger than what is actuallyproven in Theorem 1.2. As quantitative statements of decay, however, state-ments (11) and (12) are in fact much weaker than what has now been mathe-matically proven, here and in [12]. For, rewriting, in particular, the first formulaof (12) as | φ ℓ ( r, t ) | ≤ C ℓ ( r ) e − cℓt , (13)then there is no indication as to what C ℓ ( r ) depends on, indeed, if there isany bound on C ℓ provided by some norm of initial data, and if so, what is thebehaviour as ℓ → ∞ . This does not concern a mathematical pathology, butis intimately connected with the physical effect caused by the photon sphere.Indeed, a geometric optics approximation shows easily that if (13) is to holdand if C ℓ is to depend, say, on the initial energy of the spherical harmonic, then C ℓ → ∞ as ℓ → ∞ . It is the rate of this divergence that would then determinethe decay rate (if any) for φ . If one is interested in quantitative statements of decay, a statement like (12)provides no more information than the statementlim ( u,v ) → ( ∞ , ∞ ) φ ℓ ( u, v ) = 0 . (14)It is worth noting that the above statement at the level of individual sphericalharmonics, together with the (uniform) boundedness result | φ | ≤ C sup | ϕ | + C E s ( ϕ, ˙ ϕ ) , (15)can indeed be used to show, for fixed r , the statementlim ( u,v ) → ( ∞ , ∞ ) ( φ − φ )( u, v ) = 0 , (16)for the total φ . This can be termed the statement of (uniform) decay without arate.Thus it is truly only (16), and not the results of [12] or Theorem 1.1, thatcan be said to be suggested by heuristic and numerical studies.Results like (16) or even just (15) are sometimes referred to as “linear sta-bility” in the physics literature. One should keep in mind, however, that were The result (15) was shown for Schwarzschild in fundamental work of Kay and Wald [16].Our [12] gives an alternative proof not relying on the discrete symmetries of the maximal de-velopment. For Schwarzschild-de Sitter, the statement (15) of course follows from Theorem 1.1for any s >
1. We have not found another statement of this in the literature. Sometimes, even the statement ∀ r b < r < r c , lim t →∞ φ ( r, t ) = 0 is termed “linear sta-bility”. Such a result does not even imply (15). It is in fact entirely consistent with thestatement sup r ∈ ( r b ,r c ) t ∈ [0 , ∞ ) | φ ( r, t ) | = ∞ ! instability for Schwarzschildor Schwarzschild-de Sitter once one passes to the next order in perturbation the-ory. At very least, it would exclude all known techniques for proving non-linearstability for supercritical non-linear wave equations like (1). It is only quan-titative uniform decay bounds with decay rate sufficiently fast, suchthat moreover the bound depends only on a suitable norm of initialdata, which indeed can the thought of as suggestive of non-linear sta-bility.
One should thus be careful in associating the heuristic and numericaltradition exemplified in [19] with the conjecture that black holes are stable.
While the final version of this manuscript was being prepared, an interestingpreprint [5] appeared addressing a special case of the problem under considera-tion here with the methods of time-independent scattering theory. The specialcase where φ is not supported at H + ∩H − and H + ∩H − is considered and quan-titative exponential decay bounds are proven for R Σ ′ T µν T µ n ν in the coordinate t , where one must restrict to Σ ′ = { t } × [ r , R ], for r b < r , R < r c . Thebounds lose only an ǫ of an angular derivative, but depend on r , R , and theinitial support of φ in an unspecified way. The work [5] depends in an essentialway on a previous detailed analysis of S´a Barreto and Zworski [21] concerningresonances of an associated elliptic problem.For the special case of the data considered in [5], given that result, then theestimates of the present paper, in particular, those provided by the currents J Yµ , J Yµ , can be applied a posteriori to obtain uniform (i.e. holding up to thehorizons) exponential decay bounds. The authors thank the Massachusetts Institute of Technology for hospitality inthe Spring of 2006 where this research began. M.D. is supported by a ClayResearch Scholarship. I.R. is supported in part by NSF grant DMS-0702270.
We refer the reader to the references [8, 13, 18] for detailed discussions of thegeometry of Schwarzschild-de Sitter. ( r, t ) We recall that so-called Schwarzschild coordinates ( r, t ) map D o onto ( r b , r c ) × ( −∞ , ∞ ), in which the metric takes the form (3). Let the choice of the t coordinate be fixed. Here 0 < r b < r c denote the two positive roots of the4equation 1 − Mr − Λ3 r = 0 . (17)The function r can be given a geometric interpretation r ( p ) = p Area(ˆ π − (ˆ π ( p ))) / π, (18)where here ˆ π : M → Q is the natural projection. Thus r can be defined asa smooth function on all of M . It is known as the area-radius function . Thisfunction also clearly descends to Q .The ( r, t ) coordinates degenerate along the horizons H + ∪ H − and H + ∪ H − ,on which r = r b , r = r c , respectively.It is immediate from the explicit form of the metric that the vector field ∂∂t is Killing in D o . This extends to a globally defined Killing field T on ( M , g ),which is null along H + ∪ H − and H + ∪ H − , and vanishes along H + ∩ H − and H + ∩ H − . ( r ∗ , t ) We now proceed to define two related coordinate systems on D o . Let us denotethe unique negative root of (17) as r − , and let us set κ b = ddr (cid:18) − Mr − Λ3 r (cid:19) (cid:12)(cid:12) r = r b , and similarly κ c , κ − . We now set r ∗ . = − κ c log (cid:12)(cid:12)(cid:12)(cid:12) rr c − (cid:12)(cid:12)(cid:12)(cid:12) + 12 κ b log (cid:12)(cid:12)(cid:12)(cid:12) rr b − (cid:12)(cid:12)(cid:12)(cid:12) + 12 κ − log (cid:12)(cid:12)(cid:12)(cid:12) rr − − (cid:12)(cid:12)(cid:12)(cid:12) − C ∗ where C ∗ is a constant we may choose arbitrarily. For convenience, let uschoose C ∗ so that r ∗ = 0 when r = 3 M , the so-called photon sphere . We callthe coordinates ( r ∗ , t ) so-defined Regge-Wheeler coordinates . ( u, v ) From Regge-Wheeler coordinates ( r ∗ , t ), we can define now retarded and ad-vanced Eddington-Finkelstein coordinates u and v , respectively, by t = v + u and r ∗ = v − u. These coordinates turn out to be null: Setting µ = Mr + Λ r , the metric takesthe form − − µ ) dudv + r dσ S . r ∗ , t ) and ( u, v )in this paper. Note that in either, region D o is covered by ( −∞ , ∞ ) × ( −∞ , ∞ ).By appropriately rescaling u and v to have finite range, one can constructcoordinates which are in fact regular on H ± and ˜ H ± . By a slight abuse oflanguage, one can parametrize the future and past horizons in our present ( u, v )coordinate systems as H + = { ( ∞ , v ) } v ∈ [ −∞ , ∞ ) , (19) H − = { ( u, −∞ ) } u ∈ ( −∞ , ∞ ] , (20) H + = { ( u, ∞ ) } u ∈ [ −∞ , ∞ ) , (21) H + = { ( −∞ , v ) } v ∈ [ −∞ , ∞ ) . (22)Under these conventions, the statements of Theorem 1.1 can be applied up tothe boundary of D . Finally, we collect various formulas for future reference: µ = 2 Mr + 13 Λ r ,g uv = ( g uv ) − = − − µ ) ,∂ v r = (1 − µ ) , ∂ u r = − (1 − µ ) dt = dv + du, dr ∗ = dv − du,T = ∂∂t = 12 (cid:18) ∂∂v + ∂∂u (cid:19) ,∂∂r ∗ = 12 (cid:18) ∂∂v − ∂∂u (cid:19) ,dV ol M = 2 r (1 − µ ) du dv dA S ,dV ol t =const = r p − µ dr ∗ dA S , ✷ ψ = ∇ α ∇ α ψ = − (1 − µ ) − (cid:0) ∂ t ψ − r − ∂ r ∗ ( r ∂ r ∗ ψ ) (cid:1) + ∇ / A ∇ / A ψ. Here ∇ / denotes the induced covariant derivative on the group orbit spheres.6 As discussed in the introduction, the results of this paper will rely on L -basedestimates. Such estimates arise naturally in view of the Lagrangian structure ofthe wave equation. We review briefly here.Let φ be a solution of ✷ g φ = 0. In general coordinates, the energy-momentumtensor T αβ for φ is defined by the expression T αβ ( φ ) = ∂ α φ ∂ β φ − g αβ g γδ ∂ γ φ ∂ δ φ. The tensor T αβ is symmetric and divergence-free, i.e. we have ∇ α T αβ = 0 . (23)For the null coordinate system u, v, x A , x B we have defined, where x A , x B denote coordinates on S , we compute the components T uu = ( ∂ u φ ) ,T vv = ( ∂ v φ ) ,T uv = − g uv |∇ / φ | = (1 − µ ) |∇ / φ | . Here the notation |∇ / ψ | = g AB ∂ A ψ∂ B ψ = r − | dψ | dσ . Note moreover that |∇ / ψ | = r − P i =1 | Ω i ψ | .Let V α denote an arbitrary vector field. Let π αβV denote the deformationtensor of V , i.e., π αβV . = 12 ( ∇ α V β + ∇ β V α ) . (24)In local coordinates we have the following expression: T αβ ( φ ) π αβV = 14(1 − µ ) (cid:0) ( ∂ u φ ) ∂ v ( V v (1 − µ ) − ) + ( ∂ v φ ) ∂ u ( V u (1 − µ ) − )+ |∇ / φ | ( ∂ u V v + ∂ v V u ) (cid:1) − r ( V u − V v )( |∇ / φ | − φ α φ α ) . Set J Vα = T αβ V α . (25)The relations (23) and (24) give K V . = ∇ α J Vα . = T αβ ( φ ) π αβV , and the divergence theorem applied to an arbitrary region R gives (10).Identity (10) is particularly useful when the vector field V is Killing, forinstance the vector field T defined previously. For then, K T = 0 and oneobtains a conservation law for the boundary integrals. Moreover, when ∂ R J Tµ ( φ ) n µ on the right hand side of (26) when properly oriented are positive semi-definitein the derivatives of φ .Were the vector field T timelike in all of D , then by applying (10) to φ ,Ω i φ , etc., one could show the uniform boundedness of all derivatives of φ . Since T becomes null on H + ∪ H + , the integrand does not control all quantities onthe horizon. It is for this reason that even proving uniform boundedness forsolutions of ✷ g φ = 0 on D is non-trivial. (See [16].)For φ a solution to ✷ g φ = 0, the 1-form J Vµ ( φ ) defined above has the propertythat both it and its divergence ∇ µ J Vµ depend only on the 1-jet of φ . FollowingChristodoulou [9], we shall call one-forms J µ and their divergences K = ∇ µ J µ with the aforementioned property (thought of as form-valued and scalar valuedmaps on the bundle of 1-jets, respectively) compatible currents . r i , R i In the course of this proof we shall require special values r i , R i , i = 0 , , −∞ < r ∗ < r ∗ < r ∗ < r ∗ < < R ∗ < R ∗ < R ∗ < R < ∞ . Eventually, specific choices of these constants will be made, and these choiceswill depend only on M , Λ. Constants r , R are in fact only constrained byLemma 7.3.1. Constants r , R are constrained by the necessity of satisfyingProposition 8.3.1 of Section 8.3.To choose r , R , on the other hand, is more subtle, as we will have to keeptrack of a certain competition of constants as r , R vary. We adopt, thus, theconvention described in the next section. C , E , and ǫ on r i , R i In all formulas that follow in this paper, constants which can be chosen inde-pendently of r ∗ , R ∗ shall be denoted by C . Constants C will thus depend on M , Λ, r i , R i , for i = 1 ,
2, and, after r i , R i have been chosen, will depend onlyon M , Λ.Constants which depend on M , Λ, r ∗ and R ∗ and tend to 0 as r ∗ → −∞ , R ∗ → ∞ will be denoted by ǫ . Finally, all other constants depending on M ,Λ, r ∗ and R ∗ , will be denoted by E . Constants denoted by E in principlediverge as r ∗ → ∞ , R ∗ → ∞ . We will also use the convention A ≈ B whenever C − A ≤ B ≤ CA with aconstant C understood as above.In view of our above conventions, note finally the obvious algebra of con-stants: C ± C = C , ǫC = ǫ , CE = E , ǫE = E , etc.8 Associated to these special values of r , we will define a number of cutoff func-tions. It is convenient to introduce also the notation r ∗− . = 4 r ∗ , R ∗− . = 4 R ∗ . η i Let η : [0 , ∞ ) → R be a nonnegative smooth cutoff function which is equal to 1in [0 ,
1] and 0 outside [0 , i = − , , ,
2, define η i ( r ∗ ) = η ( r ∗ /r ∗ i ) for r ∗ ≤ η ( r ∗ /R ∗ i ) for r ∗ ≥ . Clearly η i has the property that η i = 1 in [ r ∗ i , R ∗ i ] and η i = 0 in ( −∞ , r ∗ i − ] ∪ ( R ∗ i − , ∞ ). Moreover, sup η ′ i → r ∗ i → −∞ , R ∗ i → ∞ . χ i and χ i Now let χ : ( −∞ , ∞ ) → R , χ : ( ∞ , ∞ ) → R be nonnegative smooth cutofffunctions such that χ is 1 in ( −∞ , −
1] and 0 in [ − , ∞ ), and χ is 1 in [1 , ∞ )and 0 in ( −∞ , ].Now for i = − , , χ i = χ ( − r ∗ /r ∗ i ) χ i = χ ( r ∗ /R ∗ i ) . Clearly, χ i has the property that χ i = 1 in ( −∞ , r ∗ i ], and 0 in [2 r ∗ i +1 , ∞ ).Similarly χ i has the property that χ i = 1 in [ R ∗ i , ∞ ), and 0 in ( −∞ , R ∗ i +1 ]. Σ t and the region R ( t , t ) Let r , R be the special values announced in Section 4.1. For all t , defineΣ t . = { t } × [ r , R ] ∪ { ( t − R ∗ ) / } × [( t + R ∗ ) / , ∞ ] ∪ [( t − r ∗ ) / , ∞ ] × { ( t + r ∗ ) / } and for any t > t , define R ( t , t ) = J + (Σ t ) ∩ J − (Σ t ) . r = r r = R Σ t t = t t = t v = v v = v u = u u = u We shall repeatedly apply the identity (10) in the region R ( t , t ). We recordbelow its explicit form:0 = Z R ( t ,t ) K ( φ ) + Z Σ t J µ ( φ ) n µ − Z Σ t J µ ( φ ) n µ (26)+ Z H + ∩{ v ≤ v ≤ v } J µ ( φ ) n µ + Z H − ∩{ u ≤ u ≤ v } J µ ( φ ) n µ . Here, n = (1 − µ ) − T whenever ( u, v ) belongs to the space-like portion of Σ t ,and the measure of integration, call it dm , is there understood to be the inducedvolume form. Let us moreover define n . = ∂∂u and n . = ∂∂v whenever ( u, v ) belongsto the v = const and u = const portions of Σ t respectively. With this choice,the measure of integration dm in the respective null segments is understood tobe given by dm v =const = r dA S du, dm u =const = r dA S dv. All integrals over Σ t that appear in the sections that follow are understoodto be with respect to the measure dm defined above. The integral over R ( t , t )is to be understood to be with respect to the spacetime volume form. SeeSection 2.4. In this section we will give a geometric statement of the main estimates.0 N , ˜ N and P Recall the cutoffs functions η i , χ , χ , from Section 4.3. Define the vector fields N . = χ − µ ∂∂u + χ − µ ∂∂v + T, (27) P i . = η i (1 − µ ) − / ∂∂r ∗ , (28)˜ N . = ( r − M ) N. (29)For convenience, let us denote P by P .The above vector fields will provide the fundamental directions in which theenergy momentum tensor T µν is to be contracted and/or φ is to be differentiatedin the definition of the fundamental quantities appearing in the main estimates.See Section 6.2 below.The coordinate-dependent definitions given above notwithstanding, the im-portant features of these vector fields can be understood geometrically. All threeare invariant with respect to the action of Ψ t , the one-parameter group of dif-ferentiable maps D → D generated by the Killing field T , N is future-directedtimelike on Σ t , ˜ N is future-directed timelike everywhere except r = 3 M , whereit vanishes quadratically, and P is supported away from the horizon and orthog-onal to T . ˜ N,P , Z N and Q Let T µν be the energy-momentum tensor defined in Section 3. Define the quan-tities Z ˜ N,Pφ ( t ) . = Z Σ t (cid:16) T µν ( φ ) ˜ N µ n ν + ( P φ ) (cid:17) , (30) Z Nφ ( t ) . = Z Σ t T µν ( φ ) N µ n ν , (31) Q φ ( t , t ) . = Z t t Z ˜ N,Pφ ( t ) dt. (32)The quantity Q φ is equivalent to the spacetime integral of the density q ( φ )defined by q ( φ ) . = (cid:18) T µν ( φ ) ˜ N µ n ν − µ + ( P φ ) (cid:19) , in the sense of the formula Q φ ( t , t ) ≈ Z R ( t ,t ) q ( φ ) , (33)understood with the conventions of Section 4.2. We will make use of this equiva-lence often in what follows. Recall also that the spacetime integral on the right1hand side of (33) is to be understood with respect to the volume form. SeeSection 2.4.Note that the quantity Z Nφ has integrand positive definite in dφ . It is infact precisely the flux through Σ t of the current J Nµ ( φ ). The quantity Z ˜ N,Pφ differs from Z Nφ in that control of the angular and t -derivatives degeneratesquadratically at r = 3 M . Similarly, the integrand of Q φ ( t , t ) also degeneratesat r = 3 M . This hypersurface r = 3 M is the so-called photon sphere discussedalready in the introduction. The main estimates of this paper are contained in the following
Theorem 6.1.
There exists a constant C depending only on M , Λ such thatfor all t > t and all sufficiently regular solutions φ of ✷ g φ = 0 in R ( t , t ) wehave Q φ ( t , t ) ≤ C Z Nφ ( t ) , (34) Z Nφ ( t ) ≤ C Z ˜ N,Pφ ( t ) + C ( t − t ) − (cid:16) Z Nφ ( t ) + Q φ ( t , t ) + X i =1 Q Ω i φ ( t , t ) (cid:17) , (35) Z Nφ ( t ) ≤ C (cid:0) Z Nφ ( t ) + Q φ ( t , t ) (cid:1) . (36) More generally than ( ) , if Σ ′ ⊂ R ( t , t ) is achronal then Z Σ ′ T µν ( φ ) N µ n ν ≤ C (cid:0) Z Nφ ( t ) + Q φ ( t , t ) (cid:1) . (37)These estimates will be used in Section 11 to prove Theorems 1.1 and 1.2.As described in the introduction, the proof of Theorem 6.1 will be accom-plished in Section 10 with the help of so called energy currents J µ associated tothe vector fields X ℓ , Y , Y and Θ. We turn in the next sections to the definitionof these currents. Were it Z ˜ N,P on the right hand side of (34), or alternatively, were Q definedas the time-integral of Z N , then inequality (34) would immediately lead toexponential decay in t for Q ( t, t ∗ ) (cf. Lemma 11.1.1).The appearance of Q Ω i φ on the right hand side of (35) signifies that the esti-mates “lose” an angular derivative. At the level of any fixed spherical harmonic φ ℓ , estimates (34) and (35) lead immediately to exponential decay for Q φ ℓ ( t, t ∗ )and Z Nφ . The nature of the loss of angular derivative in (35) means that for thetotal Q φ ( t, t ∗ ) and Z Nφ , one can only obtain polynomial decay in t , where thebound on the decay rate exponent is linear in the angular derivatives lossed.Exponential decay for φ would be retrieved if the “loss in angular derivatives”in estimate (35) were logarithmic. See the dependence in ℓ in Lemma 11.1.1.2 J X family of currents We define in this section a family of currents, all loosely based on vector fieldsparallel to ∂∂r ∗ . The role of these currents in capturing the role of the “photonsphere” has already been discussed in the introduction. J V,iµ for an arbitrary V = f ∂∂r ∗ Let f be a function of r ∗ and consider a vector field V = f ( r ∗ ) ∂∂r ∗ . (38)Define the currents J V, µ ( φ ) = T µν ( φ ) V ν ,J V, µ ( φ ) = T µν ( φ ) V ν + 14 (cid:18) f ′ + 2 1 − µr f (cid:19) ∂ µ ( φ ) − ∂ µ (cid:18) f ′ + 2 1 − µr f (cid:19) φ ,J V, µ ( φ ) = T µν ( φ ) V ν + 14 (cid:18) f ′ + 2 1 − µr f (cid:19) ∂ µ ( φ ) − ∂ µ (cid:18) f ′ + 2 1 − µr f (cid:19) φ − f ′ rf V µ φ ,J V, µ ( φ ) = T µν ( φ ) V ν + 14 (cid:18) f ′ + 2 1 − µr f (cid:19) ∂ µ ( φ ) − ∂ µ (cid:18) f ′ + 2 1 − µr f (cid:19) φ − f ′ rf V µ φ −
12 1 − M/rr f φ ∇ / µ φ,J V, µ ( φ ) = T µν ( φ ) V ν + 14 f ′ ∂ µ ( φ ) − ∂ µ f ′ φ , and the divergences K V,i = ∇ µ J V,iµ . Note the identities µ ′ − µ ) + 1 − µr = r − Mr , (39)12 r (cid:18) µ ′′ − µ − µ ′ r (cid:19) = Mr (cid:18) − Mr (cid:19) + M Λ3 r − r . We compute K V, ( φ ) = f ′ ( ∂ r ∗ φ ) − µ + |∇ / φ | (cid:18) µ ′ − µ ) + 1 − µr (cid:19) f − (cid:18) f ′ + 4 1 − µr f (cid:19) φ α φ α , (40)3 K V, ( φ ) = f ′ − µ ( ∂ r ∗ φ ) + |∇ / φ | (cid:18) µ ′ − µ ) + 1 − µr (cid:19) f − (cid:18) ✷ (cid:18) f ′ + 2 1 − µr f (cid:19)(cid:19) φ = f ′ − µ ( ∂ r ∗ φ ) + |∇ / φ | (cid:18) µ ′ − µ ) + 1 − µr (cid:19) f − (cid:18) − µ f ′′′ + 4 r f ′′ − µ ′ r (1 − µ ) f ′ + 2(1 − µ ) r (cid:18) µ ′ (1 − µ ) r − µ ′′ (cid:19) f (cid:19) φ ,K V, ( φ ) = f ′ (1 − µ ) r ( ∂ r ∗ ( rφ )) + 1 − M/rr f |∇ / φ | −
14 11 − µ f ′′′ φ + f (cid:18) Mr (cid:18) − Mr (cid:19) + M Λ3 r − r (cid:19) φ ,K V, ( φ ) = f ′ (1 − µ ) r ( ∂ r ∗ ( rφ )) − − M/rr f φ △ / φ −
14 11 − µ f ′′′ φ + f (cid:18) Mr (cid:18) − Mr (cid:19) + M Λ3 r − r (cid:19) φ ,K V, ( φ ) = f ′ − µ ( ∂ r ∗ φ ) + |∇ / φ | (cid:18) µ ′ − µ ) + 1 − µr (cid:19) f − − µr f φ α φ α −
14 ( ✷ f ′ ) φ = f ′ − µ ( ∂ r ∗ φ ) + |∇ / φ | (cid:18) µ ′ − µ ) + 1 − µr (cid:19) f − − µr f φ α φ α − (cid:18) − µ f ′′′ + 2 r f ′′ (cid:19) φ . The expression J V, µ is not a compatible current in the sense of Section 3,since K V, depends on the 2-jet of φ , but it can be treated as such when restrictedto eigenfunctions of △ / . The relation of the photon sphere to currents based on vector fields V of the form f ( r ∗ ) ∂ r ∗ is most clear upon examining the modified current J V, µ and noting thecoefficient of |∇ / φ | in K V, vanishes precisely at r = 3 M in view of (39). Thisindicates that if one is to have say K V, ≥
0, the function f must change signat r = 3 M , and the control of the angular derivatives must degenerate at leastquadratically.The task of choosing a suitable f is simplified by passing to the furthermodified current J V, which effectively “borrows” positivity from the ∂ r ∗ φ term.4Finally, one can take advantage of the further flexibility provided by chosingseparately V for each spherical harmonic, where now, after passing to the current J V, µ , the 0’th order terms are united with the angular derivative terms. InSection 7.3, we shall construct a current J X, µ ( φ ) . = J X µ ( φ ) + X ℓ J X ℓ , µ ( φ ℓ ) . We do not in fact ensure that K X, ≥ r ≤ r ≤ R . The region near the horizon will be handled with thehelp of the spacetime integral terms controlled by the currents J Y , J Y to bediscussed in the next section.Once an initial positive definite spacetime integral (albeit modulo an error)is constructed, other quantities can be controlled with the help of auxiliarycurrents. These are defined in Section 7.4. It is there where we also use thecurrent template J V, . X ℓ ℓ = 0Define f = − r − , and set X = f ∂ r ∗ . Let φ denote the 0’th spherical harmonic of a solution ✷ g φ = 0 of the wave equation. Consider J X , µ ( φ ), K X , ( φ ) as defined above.We compute K X , ( φ ) = 2 r − ( ∂ r ∗ φ ) − µ . (41)Note that K X , ( φ ) ≥ . ℓ ≥ K V, for a general vector field V of the form(38), applied to a spherical harmonic φ ℓ with spherical harmonic number ℓ ≥ K V, ( φ ℓ ) = f ′ (1 − µ ) r ( ∂ r ∗ ( rφ ℓ )) −
14 11 − µ f ′′′ φ ℓ (42)+ f (cid:18) ℓ ( ℓ + 1) 1 − M/rr + Mr (cid:18) − Mr (cid:19) + M Λ3 r − r (cid:19) φ ℓ . Define h ℓ ( r ) = ℓ ( ℓ + 1) 1 − M/rr + Mr (cid:18) − Mr (cid:19) + M Λ3 r − r . The following lemma is proven in Appendix A5
Lemma 7.3.1.
For all ℓ ≥ , there exists a unique zero r h ℓ of the function h ℓ ( r ) in [ r b , r c ] , and there exist constants r ∗ < , R ∗ > , depending only on M and Λ , such that r < r h ℓ < R . Moreover, lim ℓ →∞ r h ℓ → M .Let r ∗ , R ∗ now be fixed, chosen according to the above lemma. Were the middle term on the right hand side of (42) absent, then we wouldhave K V, ( φ ℓ ) ≥ f such that f ′ ≥ f ( r h ℓ ) = 0.The middle term of (42) vanishes if f ′′′ = 0, but, with the requirement that f ( r h ℓ ) = 0, in this case the function f cannot be bounded. By suitably cuttingoff a function linear in r ∗ , we can ensure that K V, ( φ ℓ ) ≥ r ≤ r ≤ R .Let η be the cutoff of Section 4.3.2. Define f ℓ ( r ∗ ) = Z r ∗ r ∗ hℓ η dr ∗ , and the vector field X ℓ by X ℓ = 12 f ℓ ∂∂v − f ℓ ∂∂u = f ℓ ∂∂r ∗ . (43)We have K X ℓ , ( φ ℓ ) ≥ r ∗ ≤ r ≤ R ∗ . Moreover, in this region we have in fact,( ∂ r ∗ ( rφ ℓ )) − µ + ( r − r h ℓ ) φ ℓ ≤ CK X ℓ , ( φ ℓ ) . (45)(Recall our conventions for constants C from Section 4.2.) J X,i
Finally, for i = 1 , , J X,iµ ( φ ) = J X , µ + X ℓ ≥ J X ℓ ,iµ ( φ ℓ ) , and their divergences K X ℓ ,i ( φ ℓ ) = ∇ µ J X ℓ ,iµ ( φ ℓ ) ,K X,i ( φ ) = K X , ( φ ) + X ℓ ≥ K X ℓ ,i ( φ ℓ ) = ∇ µ J X,iµ ( φ ) . Besides obtaining nonnegativity for K X, ( φ ) in the region r ≤ r ≤ R , we needto understand the error in the region r ≤ r and r ≥ R . It turns out thatthis error can be controlled by ǫ ˆ q ( φ ), where ˆ q ( φ ) is a slightly stronger quantitythan the energy density q ( φ ).Define the quantityˆ q ( φ ) . = (cid:18) T µν ( φ ) ˜ N µ n ν − µ + η − ( χ + χ )1 − µ | r ∗ | − δ − |∇ / φ | + η ( P φ ) (cid:19) . (46)Here η − , χ , χ are the cut-off functions defined in Section 4.3. Note thatˆ q ( φ ) ≈ χ ( ∂ u φ ) (1 − µ ) + χ ( ∂ v φ ) (1 − µ ) + η − ( χ + χ )1 − µ | r ∗ | − δ − |∇ / φ | (47)+ ( r − M ) (cid:0) ( ∂ t φ ) + |∇ / φ | (cid:1) + ( ∂ r ∗ φ ) . Compare with q ( φ ) ≈ χ ( ∂ u φ ) (1 − µ ) + χ ( ∂ v φ ) (1 − µ ) + ( ∂ r ∗ φ ) (48)+ ( r − M ) (cid:0) ( ∂ t φ ) + |∇ / φ | (cid:1) , where q ( φ ) is the density of the main quantity Q φ defined in (32).The inequality replacing (44) which holds globally is given by Lemma 7.3.2.
The inequality Z S K X, ( φ ) ≥ − ǫ Z S ˆ q ( φ ) (49) holds on all spheres of symmetry.Proof. It suffices to consider the regions r ≤ r and r ≥ R . Relation (47)implies C ˆ q ( φ ) ≥ (cid:18) η − ( χ + χ )1 − µ (cid:19) | r ∗ | − δ − |∇ / φ | , in these two regions, while Z S K X, ( φ ) ≥ X ℓ ≥ Z S K X ℓ , ( φ ℓ ) ≥ − X ℓ ≥ Z S f ′′′ ℓ − µ φ ℓ . The function f ′′′ ℓ is supported in the region [2 r ∗ , r ∗ ] ∪ [ R ∗ , R ∗ ] and obeys thepointwise bound | f ′′′ ℓ ( r ∗ ) | ≤ C | r ∗ | − , (50) Recall the conventions regarding constants ǫ in Section 4.2. C . Note that η − ( χ + χ ) = 1in the support of f ′′′ . Therefore, Z S K X, ( φ ) ≥ − C Z S − µ | r ∗ | − ( φ − φ ) holds on all spheres of symmetry. We obtain (49) with ǫ = C (cid:0) | r ∗ | − δ + | R ∗ | − δ (cid:1) . We will also need several “auxiliary” currents.
Let us first define, however, certain auxiliary positive definite quantities. Theauxiliary currents K will be seen to bound these quantities when integrated onspheres of symmetry in the region r ≤ r ≤ R .Define q i ( φ ) . = η i q ( φ ) = η i ( T µν ( φ ) ˜ N µ n ν (1 − µ ) − + ( P φ ) ) q ai ( φ ) . = η i ( P − φ ) ,q a ′ i ( φ ) . = η i ( φ − φ ) ,q bi ( φ ) . = η i ( r − M ) |∇ / φ | ,q di ( φ ) . = η i ( r − M ) ( T φ ) . Here η i are the cut-off functions defined in Section 4.3. Note that q ( φ ) ≈ C ( q a ( φ ) + q b ( φ ) + q d ( φ )) , (51) q xi +1 ≤ q xi for x = ∅ , a, a ′ , b, d , and that r − ℓ ( ℓ + 1) Z S ( r − M ) q a ′ i ( φ ℓ ) r dA S = Z S q bi ( φ ℓ ) r dA S for all spheres of symmetry.The currents to be described in what follows are motivated by the problemof bounding the positive definite quanitites whose latin superscripts they share.8 J X a , Define f aℓ ( r ∗ ) . = − η ( r ∗ )( r ∗ − r h ℓ ) , where η is as defined in Section 4.3.2.Note that ( f aℓ ) ′′′ = − r , R ], f aℓ ( r h ℓ ) = 0, and f aℓ = 0 for r ∗ ≤ r ∗ , r ∗ ≥ R ∗ .Set X aℓ = f aℓ ∂ r ∗ and define J X a , µ ( φ ) = X ℓ ≥ J X aℓ , µ ( φ ℓ ) ,K X a , ( φ ) = X ℓ ≥ K X aℓ , ( φ ℓ ) = ∇ µ J X a , µ ( φ ) . The current J X a , ( φ ) will allow us to bound the spacetime integrals of thequantities q a ( φ ) and q a ′ ( φ ). See Lemma 10.4.1. At this point, we can see from(42) and (45) that for each ℓ , the pointwise bound q a ( φ ℓ ) + q a ′ ( φ ℓ ) + ℓ ( ℓ + 1)( r − r h ℓ ) q a ′ ( φ ℓ ) ≤ CK X ℓ , ( φ ℓ ) + CK X aℓ , ( φ ℓ ) (52)holds in r ≤ r ≤ R . J X b , , J X b , Define f b ( r ∗ ) . = η ( r ∗ )( r − M )Set X b = f b ∂∂r ∗ and define as before the currents J X b , ( φ − φ ), J X b , ( φ ) and K X b , ( φ − φ ), K X b , ( φ ).The current J X b , ( φ − φ ) will allow us to bound–in addition to the previous–the spacetime integral of the quantity q b ( φ ). See Lemma 10.4.2. At this point,we can deduce from (41), (42) and (52) the bound Z S (cid:16) q a ( φ ) + q a ′ ( φ ) + q b ( φ ) (cid:17) r dA S (53) ≤ Z S (cid:16) CK X, ( φ ) + CK X a , ( φ ) + CK X b , ( φ − φ ) (cid:17) r dA S on each sphere of symmetry with r ∗ ∈ [ r ∗ , R ∗ ].The current J X b , ( φ ), in conjunction also with J X c , ( φ ) and J X d , ( φ − φ )to be defined below, will be used to help bound the quantity Z Nφ in Proposi-tion 10.5.1. For now, note that from (40), (41), (42) and (53), the estimate Z S (cid:16) q a ( φ ) + q a ′ ( φ ) + q b ( φ ) (54) − C (cid:0) η (1 − µ )(2( r − M ) /r + 1) + η ′ ( r − M ) (cid:1) ∂ µ φ ∂ µ φ (cid:17) r dA S , ≤ Z S (cid:16) CK X, ( φ ) + CK X a , ( φ ) + CK X b , ( φ − φ ) + CK X b , ( φ ) (cid:17) r dA S r ∗ ∈ [ r ∗ , R ∗ ]. Note that the left hand sideof (54) a priori does not have a sign, in view of the term containing ∂ µ φ∂ µ φ . Af-ter defining J X c , ( φ ) and J X d , ( φ − φ ) below, we shall be able to improve (54)with (57). J X c , Define f c ( r ∗ ) . = r . Set X c = f c ∂∂r ∗ , and define as before J X c , , K X c , .We have that K X c , ( φ ) = 2 r ( ∂ t φ ) . (55) J X d , Let us then finally define f d ( r ∗ ) . = η ( r ∗ )( r − M ) . Set X d = f d ∂∂r ∗ and define as before the currents J X d , and K X d , .The currents J X c , ( φ ) and J X d , ( φ − φ ) will allow us to bound the space-time integral of the quantity q d ( φ ). See Lemma 10.4.3. For now, it follows from(40), (53) and (55) that on each sphere of symmetry with r ∗ ∈ [ r ∗ , R ∗ ], we have Z S (cid:16) q a ( φ ) + q a ′ ( φ ) + q b ( φ ) + q d ( φ ) (cid:17) r dA S (56) ≤ Z S (cid:16) CK X, ( φ ) + CK X a , ( φ ) + CK X b , ( φ − φ ) + CK X c , ( φ )+ CK X d , ( φ − φ ) (cid:17) r dA S . The currents J X b , ( φ ), J X c , ( φ ) and J X d , ( φ − φ ) together allow us toestimate, on each sphere of symmetry for r ∗ ∈ [ r ∗ , R ∗ ], the quantity Z S Cη (2( r − M ) /r + 1)(( ∂ t φ ) − |∇ / φ | ) r dA S (57) ≤ Z S (cid:16) CK X, ( φ ) + CK X a , ( φ ) + CK X b , ( φ − φ ) + CK X b , ( φ )+ CK X c , ( φ ) + CK X d , ( φ − φ ) (cid:17) r dA S . Again, the left hand side of (57) does not have a sign.0 J Y and J Y In this section, we define the currents J Yµ and J Yµ , associated to two vector fields Y and Y , supported near the black hole and cosmological horizons, respectively.The role of these currents in capturing the “red-shift” effect has been discussedin the introduction. Y and Y Let 0 < δ < α ( r ∗ ), β ( r ∗ ) as follows.Recall the cutoff functions χ , η − from Section 4.3.2, and define α ( r ∗ ) = χ ( r ∗ )(2 − µ + η − ( r ∗ ) | r ∗ | − δ ) , (58) β ( r ∗ ) = 2 r − b (2 M/r b − r b / − χ ( r ∗ )(1 − µ + η − ( r ∗ ) | r ∗ | − δ ) . (59)We have α ≥ β ≥
0. To see the latter, note that ddr (1 − µ ) (cid:12)(cid:12)(cid:12) r = r b = (cid:18) Mr −
23 Λ r (cid:19) (cid:12)(cid:12)(cid:12) r = r b > r − < r b < r c are three non-degenerate roots of (1 − µ ), which is non-negative on [ r b , r c ]. The above expression approaches 0 as M and Λ tend to theextremal values.Similarly, define functions α , β as follows. Recall the cutoff χ from Sec-tion 4.3.2, and define α ( r ∗ ) = χ ( r ∗ )(2 − µ + η − | r ∗ | − δ ) , (60) β ( r ∗ ) = 2 r − c (2Λ r c / − M/r c ) − χ ( r ∗ )(1 − µ + η − | r ∗ | − δ ) . (61)Again, note that α ≥ β ≥ Y to be the vector field Y = α ( r ∗ )1 − µ ∂∂u + β ( r ∗ ) ∂∂v and Y to be Y = α ( r ∗ )1 − µ ∂∂v + β ( r ∗ ) ∂∂u . The application of cutoff η − is in fact not essential for the arguments of thepaper. The cutoff simply ensures that Y and Y have smooth extensions beyond H + and H + .1 Define the currents J Yµ ( φ ) = T µν Y ν ( φ ) ,J Yµ ( φ ) = T µν Y ν ( φ ) , and set K Y ( φ ) = ∇ µ J Yµ ( φ ) ,K Y ( φ ) = ∇ µ J Yµ ( φ ) . We have K Y ( φ ) = T µν ( φ ) π µνY = ( ∂ u φ ) − µ ) (cid:18) α (cid:18) Mr − r (cid:19) − α ′ (cid:19) + ( ∂ v φ ) − µ ) β ′ (62)+ 12 |∇ / φ | (cid:18) α ′ − µ − ( β (1 − µ )) ′ − µ (cid:19) − r (cid:18) α − µ − β (cid:19) ∂ u φ ∂ v φ and similarly K Y ( φ ) = T µν ( φ ) π µνY = ( ∂ v φ ) − µ ) (cid:18) − α (cid:18) Mr − r (cid:19) + α ′ (cid:19) − ( ∂ u φ ) − µ ) β ′ (63) − |∇ / φ | (cid:16) α ′ − (cid:0) β (1 − µ ) (cid:1) ′ (cid:17) + 1 r (cid:0) α − β (1 − µ ) (cid:1) ∂ u φ ∂ v φ. r , R In this section we exhibit the “red-shift” property of the currents J Y , J Y . Thisis essentially contained in Proposition 8.3.1 below. In the context of provingthis proposition, we will choose the constants r , R .Note that the polynomial powers in the definitions (58) (59) would be un-necessary had they not been present in the definition (46). (Their presence in(46) is in turn necessitated by our application of (50).)A slightly unpleasant feature of these polynomial decaying expressions isthat, if left bare, they would lead to vector fields Y , Y which fail to be C at H + , H + , respectively. This would not in fact pose a problem for the analysishere. We prefer, however, to introduce a cutoff η − in (58)–(61) to emphasizethe geometric nature of all objects involved in the proof. Proposition 8.3.1.
For r ∗ sufficiently small and R ∗ sufficiently large, depend-ing only on M , Λ we have CK Y ( φ ) ≥ ˆ q ( φ ) , CK Y ( φ ) ≥ ˆ q ( φ ) in r ≤ r and r ≥ R , respectively. Proof.
We give the proof only for K Y . The proof for K Y is similar.Recall the functions α , β from Section 8. Let us denote by γ b . = 2 r − b (2 M/r b − r b / − . In the region r ≤ r , we have α = 2 − µ + η − ( r ∗ ) | r ∗ | − δ , β = γ b (1 − µ + η − ( r ∗ ) | r ∗ | − δ ) . As a consequence, for all r ∗ < α ′ = (cid:18) Mr −
23 Λ r (cid:19) (1 − µ ) + η ′− | r ∗ | − δ + δη ′− | r ∗ | − δ ,β ′ = γ b (cid:18) Mr −
23 Λ r (cid:19) (1 − µ ) + γ b η ′− | r ∗ | − δ + γ b δη ′− | r ∗ | − − δ . Recall that (cid:18) Mr −
23 Λ r (cid:19) (cid:12)(cid:12)(cid:12) r = r b > η ′− ≥ r ≤ r .Recall now the expression (62) for K Y . In the region r ≤ r , we compute α (cid:18) Mr − r (cid:19) − α ′ = (1 + η − | r ∗ | − δ ) (cid:18) Mr −
23 Λ r (cid:19) − η ′− | r ∗ | − δ − δη − | r ∗ | − − δ ,α ′ − µ − ( β (1 − µ )) ′ − µ = ( µ + γ b (1 − µ ) − γ b η − ( r ∗ ) | r ∗ | − δ ) (cid:18) Mr −
23 Λ r (cid:19) + ((1 − µ ) − − γ b )( η ′− | r ∗ | − δ + δη − | r ∗ | − − δ ) , r (cid:18) α − µ − β (cid:19) = r − (cid:16) − µ ) − (1 + η − ( r ∗ ) | r ∗ | − δ − ) + 1 − γ b (1 − µ ) − γ b η − ( r ∗ ) | r ∗ | − δ (cid:17) . Note that r − (1 − µ ) − ≤
14 (2
M/r − r/ − µ ) − + r − (2 M/r − r/ − . (64)It follows that r can be chosen sufficiently close to r b , where the choice dependsonly on M , Λ, such that, in the region r ≤ r , we have CK Y ( φ ) ≥ (cid:18) ( ∂ u φ ) (1 − µ ) + (cid:18) η − − µ | r ∗ | − δ − (cid:19) (cid:0) ( ∂ v φ ) + |∇ / φ | (cid:1)(cid:19) . (65)In deriving (65), we have used the Cauchy-Schwarz inequality to bound the ∂ u φ∂ v φ term. (The fact that the constants work out is ensured by the limitinginequality (64). It is here that the presence of the constant factor γ b in thedefinition of β is paramount.)3It now follows from (47) that, in the region r ≤ r , CK Y ( φ ) ≥ ˆ q ( φ ) , as desired.Note finally, that, in conformance with our conventions of Section 4.2, theconstant C above does not depend on r , R , despite the appearance of thecutoff η − . Henceforth, let r , R be chosen so that the conclusion of the above propo-sition holds. The special values r , R , r , R now being fixed, constants denoted C nowdepend only on M , Λ. J ΘLet ζ : [0 , → [0 ,
1] be a cutoff function such that ζ ( x ) = 1 , x ≥ / ,ζ ( x ) = 0 , x ≤ / . Define ζ ( t ,t ) ( τ ) = ζ (( τ − t ) / ( t − t )) . Let θ be the Heaviside step-function and let Θ to be the vector fieldΘ . = (cid:0) θ ( r ∗ − r ∗ ) ζ ( t ,t ) (2 v − r ∗ ) + θ ( r ∗ − r ∗ ) θ ( R ∗ − r ∗ ) ζ ( t ,t ) ( t )+ θ ( r ∗ − R ∗ ) ζ ( t ,t ) (2 u + R ∗ ) (cid:1) T, and define as before the currents J Θ and K Θ . Despite the appearance of theHeaviside function, Θ is a C , vector field. It is of the form Θ = ξT , where ξ isa spacetime cut-off function adapted to the C , foliation Σ t , constant on eachleaf Σ t . We have K Θ ( φ ) = T µν ( φ ) π µν Θ = − − µ ) (cid:18) ( θ ( r ∗ − r ∗ ) ζ ′ ( t ,t ) (2 v − r ∗ ) (cid:18) ( ∂ u φ ) + 12 (1 − µ ) |∇ / φ | (cid:19) +2 θ ( r ∗ − r ∗ ) θ ( R ∗ − r ∗ ) ζ ( t ,t ) ( t ) (cid:0) ( ∂ t φ ) + ( ∂ r ∗ φ ) + (1 − µ ) |∇ / φ | (cid:1) + θ ( r ∗ − R ∗ ) ζ ′ ( t ,t ) (2 u + R ∗ ) (cid:18) ( ∂ v φ ) + 12 (1 − µ ) |∇ / φ | (cid:19)(cid:19) .
10 Proof of the main estimates
Let us define the auxiliary quantities Q xi,φ ( t , t ) . = Z R ( t ,t ) q xi ( φ )ˆ Q φ ( t , t ) . = Z R ( t ,t ) ˆ q ( φ ) F Tφ ( t , t ) . = Z H + ∩{ v ≤ v ≤ v } J T ( φ ) n µ + Z H + ∩{ u ≤ u ≤ u } J Tµ ( φ ) n µ Z Tφ ( t i ) . = Z Σ ti J Tµ n µ where q xi ( φ ) is as defined in Section 7.4.1, with x = ∅ , a, b, . . . , and ˆ q ( φ ) is asdefined in Section 7.3.4. For the boundary terms, note first that on the support of η , n = 1 √ − µ T. On the other hand, in the expression (27), each term gives a nonnegative con-tribution to J Nµ n µ , and we have on Σ t J Nµ n µ ≈ χ ( ∂ u φ ) − µ + χ ( ∂ v φ ) − µ + ( ∂ t φ ) + ( ∂ r ∗ φ ) + (1 − µ ) |∇ / φ | . (66)As a consequence, Z ˜ N,Pφ ( t i ) ≤ C Z Nφ ( t i ) , (67)and Z Tφ ( t i ) ≤ C Z Nφ ( t i ) . (68) J In this section we address questions of size of the boundary terms generated bythe currents J X , J X a , J X b , J X c , J X d from Section 7 and J Y , J Y from Section 8. Proposition 10.2.1.
For J = J X,i ( φ ) , J X a , ( φ ) , J X b , ( φ − φ ) , J X b , ( φ ) , J X c , ( φ ) , J X d , ( φ − φ ) we have (cid:12)(cid:12)(cid:12)(cid:12)Z Σ t J µ n µ (cid:12)(cid:12)(cid:12)(cid:12) ≤ E Z Tφ ( t ) , Z H + ∩{ v ≤ v ≤ v } J µ n µ + Z H + ∩{ u ≤ u ≤ u } J µ n µ ≤ E F Tφ ( t , t ) . Proof.
We shall here only prove the proposition in a representative case of thecurrent J X, ( φ ) = J X , ( φ ) + X ℓ ≥ J X ℓ , ( φ ℓ ) , defined in Section 7.3.3.Recall that Z Tφ ( t ) = Z Σ t T µν ( φ ) T µ n ν . Since on the space-like part of Σ t , we have n = (1 − µ ) − T , it follows that there T µν ( φ ) T µ n ν = 1 √ − µ T µν ( φ ) T µ T ν = 14 √ − µ ( T uu ( φ ) + 2 T uv ( φ ) + T vv ( φ )) , since in addition T = 1 / ∂/∂u + ∂/∂v ). On the other hand, since n = ∂/∂u and n = ∂/∂v on the null segments v =const and u =const of Σ t , respectively, T µν ( φ ) T µ n ν = T µν ( φ ) T µ (cid:18) ∂∂u (cid:19) ν = 12 (cid:0) T uu ( φ ) + T uv ( φ ) (cid:1) ,T µν ( φ ) T µ n ν = T µν ( φ ) T µ (cid:18) ∂∂v (cid:19) ν = 12 (cid:0) T vv ( φ ) + T uv ( φ ) (cid:1) . Since the space-like portion of Σ t corresponds to r -values r ≤ r ≤ R we havethat Z Tφ ( t ) ≈ Z Σ t ( T uv ( φ ) + (1 − χ ) T uu ( φ ) + (1 − χ ) T vv ( φ )) . Therefore, Z Tφ ( t ) ≈ Z Σ t (cid:0) (1 − χ )( ∂ u φ ) + (1 − χ )( ∂ v φ ) + (1 − µ ) |∇ / φ | (cid:1) . (69)On the other hand, F Tφ ( t , t ) = Z H + ∩{ v ≤ v ≤ v } J T ( φ ) n µ + Z H + ∩{ u ≤ u ≤ u } J Tµ ( φ ) n µ = 12 Z H + ∩{ v ≤ v ≤ v } ( ∂ v φ ) + 12 Z H + ∩{ u ≤ u ≤ u } ( ∂ u φ ) . We compare now (69) with the expression for R Σ t J X, µ n µ . Start with thecurrent J X , ν ( φ ) = − r T µν ( φ ) (cid:18) ∂∂r ∗ (cid:19) µ . ∂/∂r ∗ = 1 / ∂/∂v − ∂/∂u ), we infer that on Σ t Z Σ t | J X , ν ( φ ) n ν | ≤ C Z Σ t (cid:0) (1 − χ )( ∂ u φ ) + (1 − χ )( ∂ v φ ) + (1 − µ ) |∇ / φ | (cid:1) ≤ C Z Tφ ( t ) . For ℓ ≥
1, consider now J X ℓ , µ ( φ ℓ ) = − f ℓ T µν ( φ ℓ ) ( ∂r ∗ ) µ − (cid:18) f ′ ℓ + 2 1 − µr f ℓ (cid:19) ∂ µ φ ℓ + 14 ∂ µ (cid:18) f ′ ℓ + 2 1 − µr f ℓ (cid:19) φ ℓ . Recall that the functions f ′ ℓ ( r ∗ ) = 1 on the interval [ r ∗ , R ∗ ] and vanish for r ∗ ≤ r ∗ and r ∗ ≥ R ∗ . On the space-like portion of Σ t , i.e., for r ∈ [ r , R ],we have | J X ℓ , µ ( φ ℓ ) n µ | ≤ C (cid:0) T uv ( φ ℓ ) + T uu ( φ ℓ ) + T vv ( φ ℓ ) + φ ℓ (cid:1) , while on the null segments of Σ t , we have (cid:12)(cid:12)(cid:12)(cid:12) J X ℓ , µ ( φ ℓ ) (cid:18) ∂∂v (cid:19) µ (cid:12)(cid:12)(cid:12)(cid:12) ≤ E (cid:0) T uv ( φ ℓ ) + T uu ( φ ℓ ) + ( ∂ u φ ℓ ) + (1 − µ ) φ ℓ (cid:1) for v = const, and (cid:12)(cid:12)(cid:12)(cid:12) J X ℓ , µ ( φ ℓ ) (cid:18) ∂∂u (cid:19) µ (cid:12)(cid:12)(cid:12)(cid:12) ≤ E (cid:0) T uv ( φ ℓ ) + T vv ( φ ℓ ) + ( ∂ u φ ℓ ) + (1 − µ ) φ ℓ (cid:1) for u = const. Therefore, Z Σ t | J X ℓ , µ ( φ ℓ ) n µ | ≤ E Z Σ t (cid:0) T uv ( φ ℓ ) + (1 − χ ) T uu ( φ ℓ ) + (1 − χ ) T vv ( φ ℓ ) + (1 − µ ) φ ℓ (cid:1) ≤ E Z Σ t (cid:0) (1 − χ )( ∂ u φ ℓ ) + (1 − χ )( ∂ v φ ℓ ) + (1 − µ )( |∇ / φ ℓ | + φ ℓ ) (cid:1) . Summing over ℓ and using the identity ℓ ( ℓ + 1) Z S φ ℓ r r dA S = Z S |∇ / φ ℓ | r dA S , we obtain Z Σ t | J X, µ ( φ ) n µ | ≤ E Z Σ t (cid:0) (1 − χ )( ∂ u φ ) + (1 − χ )( ∂ v φ ) + (1 − µ ) |∇ / φ | (cid:1) ≤ E Z Tφ ( t ) , as desired. On the other hand, Z H + ∩{ v ≤ v ≤ v } J X, µ ( φ ) n µ = Z H + ∩{ v ≤ v ≤ v } J X , µ ( φ ) + X ℓ ≥ J X ℓ , µ ( φ ℓ ) (cid:18) ∂∂v (cid:19) µ . H + , we have (1 − µ ) = 0, and thus T uv = 0, f ′ ℓ = f ′′ ℓ = 0 and J X , µ ( φ ) (cid:18) ∂∂v (cid:19) µ = − r T µν ( φ ) (cid:18) ∂∂v (cid:19) µ = − r ( ∂ v φ ) ,J X ℓ , µ ( φ ℓ ) (cid:18) ∂∂v (cid:19) µ = (cid:18) − f ℓ T µν ( φ ℓ ) ( ∂r ∗ ) µ −
14 ( f ′ ℓ + 2 1 − µr f ℓ ) ∂ µ φ ℓ + 14 ∂ µ ( f ′ ℓ + 2 1 − µr f ℓ ) φ ℓ (cid:19) (cid:18) ∂∂v (cid:19) µ = − f ℓ ( ∂ v φ ℓ ) . As a consequence, since | f ℓ ( r ∗ ) | ≤ E , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z H + ∩{ v ≤ v ≤ v } J X, µ ( φ ) n µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z H + ∩{ v ≤ v ≤ v } r ( ∂ v φ ) + X ℓ ≥ | f ℓ | ( ∂ v φ ℓ ) ≤ E Z H + ∩{ v ≤ v ≤ v } ( ∂ v φ ) ≤ E F Tφ ( t , t ) , as desired. Similar arguments give the inequality on the horizon H + , as well asthe inequalities for the other currents of the statement of the proposition. Proposition 10.2.2.
For J = J Y ( φ ) , J Y ( φ ) , we have ≤ Z Σ t J µ n µ ≤ C Z ˜ N,Pφ ( t ) , Moreover Z H + ∩{ v ≤ v ≤ v } J Yµ ( φ ) n µ ≥ , Z H + ∩{ v ≤ v ≤ v } J Yµ ( φ ) n µ = 0 , Z H + ∩{ u ≤ u ≤ u } J Yµ ( φ ) n µ = 0 , Z H + ∩{ u ≤ u ≤ u } J Yµ ( φ ) n µ ≥ . Proof.
Once again, we shall here consider only the current J Y ( φ ). The consid-erations for J Y are practically identical.By the construction in Section 8, the support of J Y ( φ ) is contained in theregion r ∗ ≤ r ∗ . This immediately implies that J Y ( φ ) | H + = 0. Moreover Y is a bounded future-directed time-like vector field in the region r ∗ ≤ r ∗ < r ∗ ,which implies that there we have0 ≤ J Yµ ( φ ) n µ = T µν ( φ ) Y ν n µ ≤ CT µν ( φ ) T ν n µ = CJ Tµ ( φ ) n µ ≤ CJ ˜ Nµ ( φ ) n µ . Y , the remaining part of Σ t is contained in a nullsegment v = const. Thus, we have J Yµ ( φ ) n µ = T µν ( φ ) Y ν (cid:18) ∂∂v (cid:19) µ = α − µ T uu + βT uv = α − µ ( ∂ u φ ) + β (1 − µ ) |∇ / φ | . The functions α, β are non-negative and in the region r ≤ r are given by α = 2 − µ + η − ( r ∗ ) | r ∗ | − δ , β = γ b (1 − µ + η − ( r ∗ ) | r ∗ | − δ ) , which implies that0 ≤ J Yµ ( φ ) n µ ≤ C (cid:18) ( ∂ u φ ) − µ + (1 − µ ) |∇ / φ | (cid:19) . Comparing this to the expression for J Nµ ( φ ) n µ , given in (66), we see that0 ≤ J Yµ ( φ ) n µ ≤ CJ Nµ ( φ ) n µ on the null portion v = const of Σ t . Since on this portion we have N = ˜ N , wefinally obtain the desired inequality0 ≤ Z Σ t J Yµ ( φ ) n µ ≤ C Z ˜ N,Pφ ( t ) . On the other hand, on H + we have n = ∂∂v and J Yµ ( φ ) n µ = α − µ T uv + βT vv = 2 |∇ / φ | , since β = (1 − µ ) = 0 and T uv = (1 − µ ) |∇ / φ | on H + . Thus Z H + ∩{ v ≤ v ≤ v } J Yµ ( φ ) n µ = 2 Z H + ∩{ v ≤ v ≤ v } |∇ / φ | . In this section we exploit the divergence theorem for compatible currents torelate various integral quantities. From Proposition 10.2.1, Proposition 10.2.2and identity (26), the following two propositions follow immediately:
Proposition 10.3.1.
For J = J X,i ( φ ) , J X a , ( φ ) , J X b , ( φ − φ ) , J X b , ( φ ) , J X c , ( φ ) , J X d , ( φ − φ ) and K = ∇ µ J µ , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R ( t ,t ) K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E ( Z Tφ ( t ) + Z Tφ ( t ) + F Tφ ( t , t )) . Proposition 10.3.2.
We have Z R ( t ,t ) K Y ( φ ) + Z Σ t J Yµ ( φ ) n µ + Z H + ∩{ v ≤ v ≤ v } J Yµ ( φ ) n µ ≤ C Z ˜ N,Pφ ( t ) , (70) Z R ( t ,t ) K Y ( φ )+ Z Σ t J Yµ ( φ ) n µ + Z H + ∩{ u ≤ u ≤ u } J Yµ ( φ ) n µ ≤ C Z ˜ N,Pφ ( t ) . (71)Applying (26) with the energy current J Tµ gives us the following proposition Proposition 10.3.3.Z Tφ ( t ) + F Tφ ( t , t ) = Z Tφ ( t ) ≤ C Z Nφ ( t ) , Proof.
The statement follows immediately from K T = 0 and (67), (68).Applying (26) with the energy current J Tµ + J Yµ + J Yµ , we obtain Proposition 10.3.4.
Let Σ ′ ⊂ R t ,t be achronal. Then Z Σ ′ J Nµ ( φ ) n µ ≤ Z Nφ ( t ) + C Z R ( t ,t ) ∩ J − (Σ ′ ) − K Y ( φ ) − K Y ( φ ) In particular, Z Nφ ( t ) ≤ Z Nφ ( t ) + C Z R ( t ,t ) − K Y ( φ ) − K Y ( φ ) . Proof.
The vector field T + Y + Y is timelike and one sees easily (cid:16) J Tµ + J Yµ + J Yµ (cid:17) n µ ≈ J Nµ n µ , while certainly K T + K Y + K Y = K Y + K Y .We have an alternative bound on Z Nφ ( t ) as follows Proposition 10.3.5.Z Nφ ( t ) ≤ C Z ˜ N,Pφ ( t ) − C Z R ( t ,t ) K Θ ( φ ) . Proof.
Recall that by the construction given in Section 9, the vector field Θ isfuture timelike, Θ | Σ t = T and Θ | Σ t = 0. The result now follows from thestatement Z Nφ ( t ) ≤ C Z ˜ N,Pφ ( t ) + C Z Σ t J Θ µ ( φ ) n µ and the divergence theorem.0 φ from Z Nφ In this section we establish the first key part of Theorem 6.1, that is to say,statement (34). We begin with the following
Proposition 10.4.1.Q ,φ ( t , t ) ≤ E Z Nφ ( t ) + ǫ ˆ Q φ ( t , t ) Proof.
The proposition follows from the three Lemmas below:
Lemma 10.4.1.Q a ,φ + Q a ′ ,φ ≤ C Z R ( t ,t ) K X, ( φ ) + C Z R ( t ,t ) K X a , ( φ ) + ǫ ˆ Q φ . Lemma 10.4.2.Q b ,φ ≤ C Z R ( t ,t ) K X, ( φ )+ C Z R ( t ,t ) K X a , ( φ )+ C Z R ( t ,t ) K X b , ( φ − φ )+ ǫ ˆ Q φ . Lemma 10.4.3.Q d ,φ ≤ C Z R ( t ,t ) K X, ( φ ) + C Z R ( t ,t ) K X a , ( φ ) + C Z R ( t ,t ) K X b , ( φ − φ )+ C Z R ( t ,t ) K X c , ( φ ) + C Z R ( t ,t ) K X d , ( φ − φ ) + ǫ ˆ Q φ . Proof.
The statements of the above three Lemmas follow directly from (45),(52), (53) and (56), combined with the observation that, since the supports of thecurrents J X a , , J X b , and J X d , , as well as of the quantities q x ( φ ), are containedin the region { r ∗ ≤ r ∗ ≤ R ∗ } , since K X c , ( φ ) is nonnegative, and since K X, is nonnegative in { r ∗ ≤ r ∗ ≤ R ∗ } , it suffices to apply Lemma 7.3.2.Proposition 10.4.1 now follows from Proposition 10.3.1 and Proposition 10.3.3. Proposition 10.4.2. Z R ( t ,t ) K Y ( φ ) + K Y ( φ ) ≤ C Z ˜ N,Pφ ( t ) . Proof.
This follows immediately by adding (70), (71) of Proposition 10.3.2, inview also of Proposition 10.2.2.
Proposition 10.4.3. ˆ Q φ ( t , t ) ≤ C Q ,φ ( t , t ) + C Z R ( t ,t ) K Y ( φ ) + K Y ( φ )1 Proof.
The statement follows from the inequalityˆ q ( φ ) ≤ Cq ( φ ) + K Y ( φ ) + K Y ( φ ) , which is a direct consequence of (51), (47) and Proposition 8.3.1.It now immediately follows from Proposition 10.4.1, Proposition 10.4.2 andProposition 10.4.3 and our conventions regarding constants C , E and ǫ that thefollowing holds: Proposition 10.4.4. If r ∗ is chosen sufficiently small, and R ∗ is chosen suffi-ciently large, depending only on M , Λ , then ˆ Q φ ( t , t ) ≤ E Z Nφ ( t ) . Henceforth, let r , R be so chosen so that the conclusion of the above propo-sition holds. In particular, in what follows we shall need only make use of con-stants C depending only on M , Λ . Finally, since by (47), (48) Q φ ( t , t ) ≤ ˆ Q φ ( t , t ) , the statement (34) of Theorem 6.1 follows immediately. Nφ from Z ˜ N,Pφ , Q φ and Q Ω i φ Bound (35) of Theorem 6.1 is contained in the following
Proposition 10.5.1.
For all t < t Z Nφ ( t ) ≤ C Z ˜ N,Pφ ( t ) + ( t − t ) − C (cid:16) Z Nφ ( t ) + Q φ ( t , t ) + X i =1 Q Ω i φ ( t , t ) (cid:17) . Proof.
By Proposition 10.3.5 Z Nφ ( t ) ≤ C Z ˜ N,Pφ ( t ) − C Z R ( t ,t ) K Θ ( φ ) . The current K Θ ( φ ) was defined in Section 9. One easily sees − K Θ ( φ ) ≤ Ct − t (cid:18) χ ( ∂ u φ ) − µ + χ ( ∂ u φ ) − µ + |∇ / φ | + η (cid:0) ( ∂ t φ ) + ( ∂ r ∗ φ ) (cid:1)(cid:19) . Comparing this with (48), we infer that the statement of the proposition wouldfollow from the estimate Z R ( t ,t ) η (cid:0) ( ∂ t φ ) + |∇ / φ | (cid:1) ≤ C (cid:16) Z Nφ ( t )+ Q φ ( t , t )+ X i =1 Q Ω i φ ( t , t ) (cid:17) . (72)2From (48) we have that C Q φ ( t , t ) ≥ Z R ( t ,t ) ( r − M ) |∇ / φ | + ( ∂ r ∗ φ ) ,C X i =1 Q Ω i φ ( t , t ) ≥ Z R ( t ,t ) η | ∂ r ∗ ∇ / φ | . A one-dimensional Poincar´e inequality immediately implies that C Q φ ( t , t ) + X i =1 Q Ω i φ ( t , t ) ! ≥ Z R ( t ,t ) η |∇ / φ | . It thus follows from (57) that Z R ( t ,t ) (cid:16) CK X, ( φ ) + CK X a , ( φ ) + CK X b , ( φ − φ ) + CK X b , ( φ )+ CK X c , ( φ ) + CK X d , ( φ − φ ) (cid:17) ≥ Z R ( t ,t ) η ( ∂ t φ ) − C Q φ ( t , t ) + X i =1 Q Ω i φ ( t , t ) ! . The desired (72) now follows from Propositions 10.3.1 and 10.3.3. Nφ ( t ) from Z Nφ ( t ) and Q φ The final statements of Theorem 6.1 follows from
Proposition 10.6.1.
Let Σ ′ ⊂ R ( t , t ) be achronal. Then Z Σ ′ T µν ( φ ) N µ n ν ≤ C (cid:16) Z Nφ ( t ) + Q φ ( t , t ) (cid:17) . (73) In particular, Z Nφ ( t ) ≤ C (cid:16) Z Nφ ( t ) + Q φ ( t , t ) (cid:17) . (74) Proof.
By Proposition 10.3.4 Z Σ ′ J Nµ ( φ ) n µ ≤ Z Nφ ( t ) + C Z R ( t ,t ) ∩ J − (Σ ′ ) − K Y ( φ ) − K Y ( φ )Recall that the current K Y ( φ ) (respectively, K Y ( φ )) is positive for r ≤ r (respectively, r ≥ R ) and vanishes for r ∗ ≥ r ∗ (respectively, r ∗ ≤ R ∗ ).Moreover, comparing (48) and (62) easily implies the bound − K Y ( φ ) − K Y ( φ ) ≤ Cq ( φ )for r ∗ ≤ r ∗ ≤ r ∗ , and R ∗ ≤ r ∗ ≤ R ∗ . The result now follows immediately.3
11 Proof of Theorem 1.1
Proposition 11.1.1.
There exist constants C , c , depending only on M , Λ , suchthat for all φ ℓ solutions of ✷ g φ ℓ = 0 in J + (Σ ) ∩ D with spherical harmonicnumber ℓ , then Z Nφ ℓ ( t ) + Q φ ℓ ( t, t ∗ ) ≤ C Z Nφ ℓ (0) e − ct/ℓ for all t and all t ∗ ≥ t .Proof. This follows immediately from estimates (34) and (35), in view of thefollowing lemma, proved in Appendix B, applied to the functions f ( t ) = ˆ Z φ ℓ ( t )and h ( t ) = Z Nφ ℓ ( t ): Lemma 11.1.1.
Let k , k be positive constants and let f : [0 , ∞ ) → R , g ; [0 , ∞ ) → R be nonnegative continuous functions satisfying h ( t ) + Z t t f ( τ ) dτ ≤ k (cid:18) f ( t ) + ( t − t ) − (cid:18) h ( t ) + ℓ Z t t f ( τ ) dτ (cid:19)(cid:19) for all t > t > t ≥ , and R ∞ f ≤ k . Then there exists a constants c depending only on k , and a universal constant C such that h ( t ) + Z t ∗ t f (¯ t ) d ¯ t ≤ C (max { h (0) , k } ) e − ct/ℓ for all t and for all t ∗ ≥ t . Proposition 11.1.2.
There exist constants C , c , depending only on M , Λ , suchthat for all φ ℓ solutions of ✷ g φ ℓ = 0 in J + (Σ ) ∩ D with spherical harmonicnumber ℓ and for all achronal Σ ′ ⊂ D ∩ J + (Σ ) , Z Σ ′ T µν ( φ ℓ ) N µ n ν ≤ C Z Nφ ℓ (0) (cid:16) e − cv + (Σ ′ ) /ℓ + e − cu + (Σ ′ ) /ℓ (cid:17) , in particular Z Nφ ℓ ( t ) ≤ C Z Nφ ℓ (0) e − ct/ℓ . Proof.
This follows immediately from Propositions 11.1.1 and the inequality(37) of Theorem 6.1.The energy decay statements of Theorem 1.2 now follow from the following
Proposition 11.1.3.
Let Σ be a Cauchy surface for M , and let ϕ , ˙ ϕ , E ( ϕ, ˙ ϕ ) be as in the statement of Theorem 1.1 or Theorem 1.2. Then, there exist con-stants C , t ≥ depending only on M , Λ and the geometry of Σ ∩ J − ( D ) , suchthat Σ t ⊂ J + (Σ) ∩ D , and such that for all solutions φ to the wave equation ✷ g φ = 0 on J + (Σ) ∩ J − ( D ) , the estimate Z Nφ ( t ) ≤ C E ( ϕ, ˙ ϕ ) holds.Proof. This is completely standard. We give a sketch to emphasize here too therole of compatible currents based on vector fields! Extend say N from Σ toan arbitrary future-timelike vector field N in J − (Σ t ) ∩ J + (Σ), and consider anarbitrary spacelike foliation S τ of this region by manifolds with boundary, suchthat S − = Σ, S = Σ T , and ∂ S τ ⊂ J − (Σ T ) \ I − (Σ T ). Consider the analogueof (26) in J − ( S τ ) ∩ J + ( S − ). Set f ( τ ) = Z S τ J Nµ n µ . In view of the fact that N is future timelike, we obtain f ( τ ) ≤ f ( −
1) + Z J − ( S τ ) ∩ J + ( S − ) K. On the other hand, one easily sees that there exists a C depending on thegeometry of our chosen foliation of the compact set J − (Σ T ) ∩ J + (Σ) such thatfor all τ ∈ [ − , Z J − ( S τ ) ∩ J + ( S − ) K ≤ C Z τ − f (¯ τ ) d ¯ τ . It now follow that f (0) ≤ e C f ( − f ( − ≤ C E ( ϕ, ˙ ϕ ) , Z Nφ ( t ) = f (0) . The pointwise decay statements follow easily.
A Proof of Lemma 7.3.1
Consider the function A ( r ) . = r h ℓ ( r ) = ℓ ( ℓ + 1) (cid:18) − Mr (cid:19) + 3 Mr − M r + M Λ r − r . Recall that µ ( r ) = 2 Mr + Λ r , µ = 4 M r + 4Λ M r r . A ( r ) = ℓ ( ℓ + 1) (cid:18) − Mr (cid:19) + 3 Mr + 3Λ M r − µ . For ℓ ≥ A is concave in r : d Adr = − M ( ℓ + ℓ − r − − (cid:18) dµdr (cid:19) − µ d µdr , d µdr = 4 M r − + 23 Λ > . The value of A at r c is given by A ( r c ) = ( ℓ ( ℓ + 1) − (cid:18) − Mr c (cid:19) − Mr c + 3Λ M r c + 2 − µ ( r c )= ( ℓ ( ℓ + 1) − (cid:18) − Mr c (cid:19) − Mr c + 3Λ M r c , where we have used µ ( r c ) = 1. The same formula evidently holds at r = r b with r b replacing r c everywhere.Since r c > M , ℓ ( ℓ + 1) ≥ r c = 3 − Mr c , we have A ( r c ) ≥ Mr c (Λ r c −
1) = 3 Mr c (cid:18) − Mr c (cid:19) > . From r b <
3, we obtain similarly A ( r b ) ≤ Mr b (cid:18) − Mr b (cid:19) < . By concavity the function A ( r ) then has exactly one zero on the interval ( r b , r c ),and thus, so does h ℓ ( r ) = r − A ( r ).The second statement of the lemma is clear from the final, which in turnfollows immediately from the form of the function h ℓ . B Proof of Lemma 11.1.1
By replacing k with max { k, } , we may assume in what follows that k ≥
1. Let t , . . . , t i be a sequence with 18 k ( ℓ + 1) ≥ t i − t i − ≥ k ( ℓ + 1). We can choose t i +1 ∈ [ t i + 9 k ( ℓ + 1) , t i + 18 k ( ℓ + 1)]such that by pigeonhole principle f ( t i +1 ) ≤ k − (9 ℓ + 9) − Z t i +(18 ℓ +18) t i +(9 ℓ +9) f ( τ ) dτ. (75)Assumptions of the Lemma then imply that h ( t i ) + Z t i +1 t i f ( τ ) dτ ≤ kf ( t i ) + 19 h ( t i − ) + 19 Z t i t i − f ( τ ) d ( τ ) , Z t i +(18 ℓ +18) t i +(9 ℓ +9) f ( τ ) dτ ≤ kf ( t i ) + 19 h ( t i − ) + 19 Z t i t i − f ( τ ) d ( τ ) . Therefore, f ( t i +1 ) ≤ (9 ℓ + 9) − f ( t i ) + 19 k h ( t i − ) + 19 k Z t i t i − f ( τ ) d ( τ ) ! . Thus if f ( t i ) ≤ ¯ C − i − (76) h ( t i − ) + Z t i t i − f ( τ ) dτ ≤ ¯ C k − i +1 (77)then f ( t i +1 ) ≤ ¯ C (2 − i − + 3 − − i +2 ) ≤ ¯ C − i − h ( t i ) + Z t i +1 t i f ( τ ) dτ ≤ ¯ C ( k − i − + 3 − k − i +1 ) ≤ ¯ C k − i . Now we have that (76), (77) indeed hold for i = 1, where t = 0, and t isdefined so as to satisfy (75), with ¯ C = max { h (0) , k } . This proves that for thesequence t i , defined above, h ( t i ) + Z t i +1 t i f ( τ ) dτ ≤ ¯ C k − i ≤ ¯ C k e − ct i /kℓ . Using the assumptions of the Lemma again we immediately obtain the desiredstatement.
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