Morawetz estimate for linearized gravity in Schwarzschild
MMORAWETZ ESTIMATE FOR LINEARIZED GRAVITY INSCHWARZSCHILD
LARS ANDERSSON, PIETER BLUE, AND JINHUA WANG
Abstract.
The equations governing the perturbations of the Schwarzschildmetric satisfy the Regge-Wheeler-Zerilli-Moncrief system. Applying the tech-nique introduced in [2], we prove an integrated local energy decay estimate forboth the Regge-Wheeler and Zerilli equations. In these proofs, we use someconstants that are computed numerically. Furthermore, we make use of the r p hierarchy estimates [13, 32] to prove that both the Regge-Wheeler and Zerillivariables decay as t − in fixed regions of r . Contents
1. Introduction 11.1. Regge-Wheeler and Zerilli equations 31.2. Statement of main results 41.3. Comment on the proof 62. Preliminaries 72.1. Notations 72.2. Energy estimate for wave equation 72.3. Hypergeometric Differential Equation 83. Integrated local energy decay and Uniform bound 83.1. Morawetz Vector Field 83.2. Integrated decay estimate 103.3. Non-degenerate energy 224. Decay estimate 244.1. Energy decay 254.2. Improved decay estimate 35References 401.
Introduction
The Schwarzschild spacetime is an 1 + 3 − dimensional Lorentzian manifold withthe Lorentz metric taking the following form in Boyer-Lindquist coordinates ( x α ) =( t, r, θ, φ ), g µν d x µ d x ν = − (cid:18) − Mr (cid:19) d t + (cid:18) − Mr (cid:19) − d r + r d σ S , in the exterior region which is given by M = R × [2 M, ∞ ) × S . For notationalconvenience, we let η = 1 − µ, µ = 2 Mr , ∆ = r − M r. (1.1)We use r ∗ to denote the Regge-Wheeler tortoise coordinate r ∗ = r + 2 M log( r − M ) − M − M log M, (1.2) a r X i v : . [ m a t h . A P ] A ug L. ANDERSSON, P. BLUE, AND J. WANG and use the retarded and advanced Eddington-Finkelstein coordinates u and v defined by u = t − r ∗ , v = t + r ∗ .In the region near the event horizon H + , located at r = 2 M , or inside the blackhole, we are also going to consider the coordinate system ( v, r, θ, φ ), where v and r are defined as above. In the ( v, r, θ, φ ) coordinate system the metric is g µν d x µ d x ν = − (1 − µ )d v + 2d r d v + r d σ S . The study of the equations governing the perturbations of the vacuum Schwarzschildmetric was initiated by Regge-Wheeler [30] and later completed by Vishveshwara[36] and Zerilli [37]. In fact, perturbations of odd and even parity were treatedseparately. The perturbations of odd parity are governed by the Regge-Wheelerequation, which is similar to the wave equation for scalar field on the Schwarzschildmanifold. Later, Zerilli considered the even case and showed, by decomposing intospherical harmonics (belonging to the different (cid:96) values), that the even parities aregoverned by the Zerilli equations. A gauge-invariant formulation was also carriedout by Moncrief [26, 27] and Clarkson-Barrett [9]. In [9], Clarkson-Barrett ex-tended the 1 + 3 covariant perturbation formalism to a ‘1 + 1 + 2 covariant sheet’formalism by introducing a radial unit vector in addition to the timelike congruence,and decomposing all covariant quantities with respect to this. On the other hand,Dafermos-Holzegel-Rodnianski [10] used the double null foliation of Schwarzschildspacetime to derive the 1 + 1 + 2 covariant perturbation formalism. Bardeen andPress [3] analyzed the perturbation equations using the Newman-Penrose formal-ism. Teukolsky [35] extended this to the Kerr family and found that the extremeNewman-Penrose components satisfy the
Teukolsky equation . The Bardeen Pressequation is Teukolsky equation restricted to Schwarzschild case. More relationsbetween the Bardeen-Press, Regge-Wheeler, Zerilli, and Teukolsky equations wereestablished by Chandrasekhar [7, 8].We shall prove boundedness, an integrated local energy decay estimate, andpointwise decay for solutions to Regge-Wheeler equation and Zerilli equations, bothof which take the form of (cid:50) g ψ − V g ψ = 0 , (1.3)in the exterior region of the Schwartzchild spacetime. Here, V g is the Regge-Wheeleror Zerilli potential.We briefly recall some earlier results about linear wave on black hole spacetimes.Integrated local energy decay estimates were proved for the wave equation outsideSchwarzschild black holes [4, 6, 14]. The existence of a uniformly bounded energyand integrated local energy decay estimates were proved for | a | (cid:28) M [2, 11, 33] andmore recently for all | a | < M [16]. In addition, there is related work by Finster-Kamran-Smoller-Yau [18] in the | a | < M range. The red-shift effect was first usedto control linear waves near the event horizon in [14] (also see [15]). Furthermore,Dafermos and Rodnianski [13] introduced an r p hierarchy, weighted estimates toprove energy decay. Using this method, Schlue [32] improved the decay rate forlinear wave in fixed regions of r outside a Schwarzschild black hole to t − / δ ,and for time derivative to t − δ . This could be compared to an earlier resultby Luk [22], where he introduced a commutator that is analogous to the scalingand derived similar decay rate. Moschidis [28] performed the r p -weighted energymethod to general asymptotically flat spacetimes with hyperboloidal foliation, andproved the decay rate for wave is τ − (where τ is the hyperboloidal time function)providing that an integrated local energy decay statement holds. There is muchmore work using different methods to improve the decay rate of linear wave. In fact,the local uniform decay rate for linear waves can be improved as t − , see Tataru,Donninger-Schlag-Soffer, etc. [12, 17, 25, 34]. Besides, we should also mention ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 3 the results by Pasqualotto [29], where he proved pointwise decay for the Maxwellsystem in Schwarzschild spacetime and by Ma [23], where a uniform boundednessof energy and a Morawetz estimate are proved for each extreme Newman-Penrosecomponent on slowly rotating Kerr background.There has also been a lot of work on energy bounds for the linearized Ein-stein equation on a Schwarzschild black hole. Integrated energy decay estimateswere proved for the Regge-Wheeler equaion [5]. More recently, Dafermos-Holzegel-Rodnianski [10] exhibited the decay t − for solutions to Teukolsky equations, whichare deduced by exploiting the associated quantities satisfying the Regge-Wheelertype equation. Hung, Keller and Wang [19, 20] worked with the Regge-Wheeler-Zerilli-Moncrief system, and established the decay estimate t − for solutions toRegge-Wheeler and Zerilli equations. In the recent work of Ma [24], Morawetzestimates for the extreme components are proved in Kerr spacetime.In this paper, we begin with the Regge-Wheeler-Zerilli-Monciref system, usingthe techniques in [2] to prove the integrated decay estimate for both Regge-Wheelerand Zerilli variables. Based on this, we apply the r p hierarchy estimate [13, 28, 32]to prove that solutions to Regge-Wheeler or Zerilli equations uniformly decay as t − , and the time derivatives decay as t − . Hence, the pointwise decay could beimproved as t − / for finite r . Since both Regge-Wheeler and Zerilli variables haveangular frequence (cid:96) ≥
2, it should be possible to improve the decay rate in thispaper to at least t − / by vector field methods and to the rate given by the Pricelaw, t − , by other methods.1.1. Regge-Wheeler and Zerilli equations.
The Regge-Wheeler equation isgiven by (cid:50) g ψ − V RWg ψ = 0 , V RWg = − Mr . (1.4)The Zerilli equation is given for each spherical harmonic mode by (cid:50) g ψ − V Zg ψ = 0 , V Zg = − Mr (2¯ λ + 3)(2¯ λr + 3 M ) r λr + 3 M ) , (1.5)where 2¯ λ = ( (cid:96) − (cid:96) + 2) ≥
4. These two equations are related by Chandrasekhartransformation [7, 8]. We note that, decomposed into spherical harmonics, (cid:96) ≥ (cid:96) = 0 and (cid:96) = 1 spherical harmonic modes.These correspond to perturbations of the mass (corresponding to moving from oneSchwarzschild solution to another) and of the angular momentum (corresponds tochanging the non-rotating Schwarzschild background to a rotating Kerr solutionand to gauge transformations), see [21, 31]. For this reason, we only considersolution to (1.4) or (1.5) with support on (cid:96) ≥ Remark 1.1 (Modes (cid:96) ≥ . We only consider solution to Regge-Wheeler (1.4) orZerilli equations (1.5) with modes (cid:96) ≥ . For these, the spectrum of − ˚ (cid:52) / S actingon functions with (cid:96) ≥ is (cid:96) ( (cid:96) + 1) ≥ . Hence upon integrating over S ( t, r ) , (cid:90) S ( t,r ) |∇ / ψ | ≥ (cid:90) S ( t,r ) r | ψ | , (1.6) where ∇ / is the induced covariant derivative on the sphere S ( t, r ) of constant r and t . L. ANDERSSON, P. BLUE, AND J. WANG H + r ≥ r NH r < r NH Σ iτ Σ eτ Figure 1.
The hypersurfaces Σ τ = Σ iτ ∪ Σ eτ Statement of main results.
We now state our main results.We shall make of the following hypersurfaces, which are illustrated in Figure1. Near the event horizon H + , we fix r NH > M with corresponding tortoisecoordinate value r ∗ NH and letΣ iτ . = { v = τ + r ∗ NH } ∩ { r < r NH } . (1.7)While away from the horizon, we use the usual time function t and letΣ eτ . = { t = τ } ∩ { r ≥ r NH } . (1.8)The hypersurfaces Σ τ is given by Σ τ = Σ iτ ∪ Σ eτ . (1.9)Let d µ Σ denote the volume form of Σ . n α Σ is the future normal vector of Σ. Wewill drop the subscript Σ on n α Σ when there is no confusion. We use D to denote theLevi-Civita connection associated with the Schwarzschild metric g µν . Let { Ω i } , i =1 , , S ( t, r ) with constant t and r , and ∇ / the induced covariant derivative on S ( t, r ).We shall allow us to use the short hand notation Ω k for (cid:80) i + i + i ≤ k Ω i Ω i Ω i for all k ∈ N . We fix a globally defined time-like vector field N in the future developmentof the initial hypersurface J + (Σ τ ), such that N = ∂ t for r ≥ r NH > M . Thiswould be defined in Section 2.The energy-momentum tensor and the corresponding momentum vector for (1.3)is T αβ ( ϕ ) = D α ϕD β ϕ − g αβ ( D γ ϕD γ ϕ + V g ϕ ) ,P ξα ( ϕ ) = T αβ · ξ β . (1.10)for a vector field ξ µ . We take ξ = N or ξ = ∂ t to define the energy class. Thenon-degenerate energy E N ( ϕ, Σ τ ) associated with N is given by E N ( ϕ, Σ τ ) . = (cid:90) Σ τ P Nα ( ϕ ) n α Σ τ d µ Σ τ ∼ (cid:90) Σ iτ (cid:18) | ∂ r ϕ | + |∇ / ϕ | + | ϕ | r (cid:19) r d r d σ S + (cid:90) Σ eτ (cid:18) | ∂ t ψ | + | ∂ r ϕ | + |∇ / ϕ | + | ϕ | r (cid:19) r d r ∗ d σ S , (1.11) ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 5 where the integral in the second line of (1.11) is written in ( v, r, θ, φ ) coordinate.For ∂ t , the associated energy E ∂ t ( ϕ, τ ) is given by E ∂ t ( ϕ, τ ) . = (cid:90) { t = τ } P ∂ t α ( ϕ ) n α d µ { t = τ } ∼ (cid:90) { t = τ } (cid:18) | ∂ t ϕ | + | ∂ r ∗ ψ | + (1 − µ ) (cid:18) |∇ / ϕ | + | ϕ | r (cid:19)(cid:19) r d r ∗ d σ S . (1.12)Note that this energy is defined as an integral on the hypersurface { t = τ } not Σ τ .We prove the uniform boundedness, integrated decay estimates and pointwisedecay for solution to the Regge-Wheeler or Zerilli equations. Theorem 1.2 (Uniform Boundedness of the Energy) . Let ψ be a solution to theRegge-Wheeler (1.4) or Zerilli equations (1.5) , then for τ > τ , E N ( ψ, Σ τ ) (cid:46) E N ( ψ, Σ τ ) . (1.13)The uniform boundedness of higher order energies is stated in Corollary 3.9. Theorem 1.3 (Integrated Decay Estimate) . Given
M > , there are positiveconstants ¯ r and a function r (cid:54) (cid:104) M that is identically one for | r − M | > ¯ r and zerootherwise, such that for all smooth function ψ solving the Regge-Wheeler equation (1.4) or Zerilli equations (1.5) , we have (cid:90) ττ d t (cid:90) ¯Σ t (cid:18) ∆ r | ∂ r ψ | + | ψ | r + r (cid:54) (cid:104) M r (cid:0) | ∂ t ψ | + | r ∇ / ψ | (cid:1)(cid:19) r d r d σ S (cid:46) E ∂ t ( ψ, τ ) , (1.14) where ¯Σ τ = { t = τ } . Remark 1.4. 1 r (cid:54) (cid:104) M is due to the trapped null geodesic at r = 3 M . However, witha loss of regularity, commuting with T , we have, (cid:90) ττ d t (cid:90) ¯Σ t (cid:18) ∆ r | ∂ r ψ | + | ψ | r + | ∂ t ψ | r + | r ∇ / ψ | r (cid:19) r d r d σ S (cid:46) E ∂ t ( ψ, τ ) + E ∂ t ( T ψ, τ ) . (1.15) Alternatively, the sharp cut-off r (cid:54) (cid:104) M can be replaced by a function that vanishesquadratically in ( r − M ) . Combining the red shift effect, the uniform boundedness of the energy (Theorem1.2), and the integrated decay estimate (Theorem 1.3), one can use the globallytime-like vector field N to obtain a non-degenerate local integrated decay estimate(Corollary 3.7). This can be generalized to the high order derivative cases (Corollary3.8).To state the decay estimate, we introduce additional notation. Let R > M bea large constant. We define the interior regionΣ (cid:48) τ = Σ τ ∩ { r ≤ R } . (1.16) Theorem 1.5 (Energy Decay and Pointwise Decay) . Let
R > M , and define theweighted derivatives D = { r∂ v , (1 − µ ) − ∂ u , r ∇ / } . Let ψ be a solution of Regge-Wheeler (1.4) or Zerilli equations (1.5) , with initial data (setting u = τ − R ∗ and v = τ + R ∗ ) satisfying I . = sup n ∈ N (cid:90) ∞ v d v (cid:90) S (cid:88) i ≤ n (cid:88) j ≤ ,k + l ≤ | D i ( r∂ v ) j Ω l ∂ kt ψ | r d σ S (cid:12)(cid:12) u = u + (cid:90) Σ (cid:48) τ (cid:88) k + l ≤ n +9 P Nµ ( ∂ kt Ω l ψ ) n µ d µ Σ (cid:48) τ < ∞ (1.17) L. ANDERSSON, P. BLUE, AND J. WANG for all n ∈ N . Then we have the energy decay estimate sup n ∈ N (cid:90) ∞ τ + R ∗ d v (cid:90) S P Nα ( D n ψ ) ∂ αv · r d σ S (cid:12)(cid:12) u = τ − R ∗ + sup k,l ∈ N (cid:90) Σ (cid:48) τ P Nµ ( ∂ kt Ω l ψ ) n µ d µ Σ (cid:48) τ (cid:46) Iτ , (1.18) and sup n ∈ N (cid:90) ∞ τ + R ∗ d v (cid:90) S P Nα ( D n ∂ t ψ ) ∂ αv · r d σ S (cid:12)(cid:12) u = τ − R ∗ + sup k,l ∈ N (cid:90) Σ (cid:48) τ P Nµ ( ∂ kt Ω l ∂ t ψ ) n µ d µ Σ (cid:48) τ (cid:46) Iτ . (1.19) For t = τ ≥ τ , u ≥ u , we have the uniform pointwise decay estimate r sup n ∈ N | D n ψ | ( τ, r ) (cid:46) Iτ , r sup n ∈ N | D n ∂ t ψ | ( τ, r ) (cid:46) Iτ , (1.20) and the improved interior pointwise decay estimate sup n ∈ N | D n ψ | ( τ, r ) (cid:46) Iτ , for r < R. (1.21)1.3. Comment on the proof.
In this section we present some of the idea in theproof leading to the main results.
Integrated local energy decay.
The proof is based on the techniques in An-dersson and Blue [2]. We use the radial multiplier vector field as in [2] (reducing tothe Schwarzschild case), which is explained in Section 3 in details. Using the factthat (cid:96) ≥ V of the po-tential in Morawetz estimates. This allows us to find a positive C hypergeometricfunction which further gives a Hardy type estimate. In the Zerilli case, there aresome difficulties in getting a smooth lower bound for the Morawetz potential in thewhole region [2 M, ∞ ) and in applying the same strategy as in Regge-Wheeler case.However, we are allowed to relax the requirement of smooth lower bound, and finda continuous lower bound V joint for the Morawetz potential. After that, similar tothe Regge-Wheeler case, we could work on the second order ordinary differentialequation with the potential V being replaced by V joint , and find a positive C solu-tion. Then the Hardy type estimate follows. The crucial part would be in finding apositive C solution to the hypergeometric differential equation (associated to theZerilli case). This could be done by further analysis for the two Frobenius solutionsto the hypergeometric differential equation. Especially, the asymptotic expansionat the singularity r = ∞ is needed to prove the positivity. r p hierarchy. In section 4, we use a multiplier of the form r p ∂ v which gives the r p hierarchy of estimates and this yields the energy decay. This approach, originated inthe context of wave equation [13], can also be adapted to Regge-Wheeler and Zerilliequations. Proceeding to the high order case, we further commute the equationswith r∂ v and derive a first order r p weighted inequality for all 0 < p ≤
2. (For thewave equation, the range 0 < p < p = 2was reached in [28].) Based on this, the r p hierarchy of estimate yields the decayrate t − / for ψ ( t, r ) with finite r , which should be compared with [32] (or [22]),where the decay is as t − / δ in a compact region. Technically, when p = 2, onewould lose the control for the angular derivatives (see the r p inequality (4.9) wherethe coefficient of the angular derivative terms in the spacetime integral vanishes).To achieve the first order r p weighted inequality for p = 2, we additionally performthe integration by parts twice on the leading error term which involves angular ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 7
Laplacian (see (4.46), (4.47) and (4.48)). In doing this, we gain the additional factor(2 − p ) , which further entails that the leading angular derivative term vanishes when p = 2. In summary, this approach adapts some idea in [28] to the Regge-Wheelerand Zerilli equations and improves the decay estimates from t − / δ to t − / infinite radius region.The paper is organized as follows: We begin in Section 2 with preliminaries,introducing basic notation and background. Section 3 is devoted to the integratedlocal energy decay estimate and uniform boundedness. The energy decay and point-wise decay are proved in Section 4. Acknowledgements
We are grateful to Steffen Aksteiner, Siyuan Ma and Vin-cent Moncrief for many helpful discussions and suggestions. J.W. was supportedby a Humboldt Foundation post-doctoral fellowship at the Albert Einstein Insti-tute during the period 2014-16, when part of this work was done. She is alsosupported by Fundamental Research Funds for the Central Universities (Grant No.20720170002). 2.
Preliminaries
In this section, we present some more notations and basic estimates that we shalluse throughout the paper.2.1.
Notations.
Let us introduce the notation T to denote the coordinate vectorfield ∂ t with respect to ( t, r ) coordinates. It is time-like only when r > M . Theglobally defined time-like vector field N could be defined as N = ∂ t + y ( r )1 − µ ∂ u + y ( r ) ∂ v , where y , y > r ≤ r NH , and y =1 , y = 0 at the event horizon. Notice that we can also write N in the ( v, r, θ, φ )coordinates as N = (1 + 2 y ( r )) ∂ v − ( y ( r ) − y ( r )(1 − µ )) ∂ r . We shall let D denote the covariant derivative associated with the Schwarzschildmetric, ∂ the derivative in terms of coordinates, and ∇ / the induced covariant de-rivative on the sphere S ( t, r ) of constant t and r . (cid:52) / is the induced Laplacian on S ( t, r ). For volume forms, we denote the spacetime volume form by d µ g , volumeform on Σ τ by d µ Σ τ . And let S be the unit 2-sphere with ˚ (cid:52) / S the induced Lapla-cian on S .The notation x (cid:46) y means x ≤ cy for a universal constant c , and the notation x ∼ y means x (cid:46) y and y (cid:46) x . All objects are smooth unless otherwise stated.2.2. Energy estimate for wave equation.
We would like to study the solutionsto the wave equation (1.3) on Schwarzschild spacetime. The energy-momentumtensor for (1.3) is T αβ ( ϕ ) = D α ϕD β ϕ − g αβ ( D γ ϕD γ ϕ + V g ϕ ) . (2.1)Given a vector field ξ µ , the corresponding momentum vector is defined by P ξα ( ϕ ) = T αβ · ξ β . (2.2)The corresponding energy on a hypersurface Σ τ is E ξ ( ϕ, Σ τ ) = − (cid:90) Σ τ P ξα ( ϕ ) n α Σ τ d µ Σ τ , (2.3) L. ANDERSSON, P. BLUE, AND J. WANG where n α Σ τ is outward normal to Σ τ . The energy identity takes the form E ξ ( ϕ, Σ τ ) − E ξ ( ϕ, Σ τ ) = − (cid:90)(cid:90) D D α P ξα d µ g (2.4)where D is the region enclosed between Σ τ and Σ τ . The associated current K ξ ( ϕ )is defined as K ξ ( ϕ ) = D α P ξα ( ϕ ) . (2.5)In applications, ξ will be taken as the vector field ∂ t or N. Hypergeometric Differential Equation.
We refers to [1]( §
15) for morebackground of hypergeometric functions . z (1 − z ) d wd z + ( c − ( a + b + 1) z ) dwdz − abw = 0 . (2.6)This is the hypergeometric differential equation .2.3.1. Fundamental Solutions.
Solution of the hypergeometric differential equation(2.6) has regular singularities at z = 0 , , ∞ with corresponding exponent pairs { , − c } , { , c − a − b } , { a, b } respectively [1] ( § c, c − a − b, a − b is an integer,we have the following pairs f ( z ) , f ( z ) of fundamental solutions. They are alsonumerically satisfactory in the neighborhood of the corresponding singularity. • Adapted to
Singularity z = 0 f ( z ) = F ( a, b ; c ; z ) ,f ( z ) = z − c F ( a − c + 1 , b − c + 1; 2 − c ; z ) . (2.7) • Adapted to
Singularity z = 1 f ( z ) = F ( a, b ; a + b + 1 − c ; 1 − z ) ,f ( z ) = (1 − z ) c − a − b F ( c − a, c − b ; c − a − b + 1; 1 − z ) . (2.8) • Adapted to
Singularity z = ∞ f ( z ) = z − a F ( a, a − c + 1; a − b + 1; 1 z ) ,f ( z ) = z − b F ( b, b − c + 1; b − a + 1; 1 z ) . (2.9)2.3.2. Integral Representations.
The hypergeometric function F ( a, b ; c ; z ) has thefollowing integral representation [1] ( § < (cid:60) b < (cid:60) c and z (cid:54)∈ [1 , ∞ ) F ( a, b ; c ; z ) = Γ( c )Γ( b )Γ( c − b ) (cid:90) t b − (1 − t ) c − b − (1 − zt ) − a d t. (2.10)And F is symmetric in its first two arguments, F ( a, b ; c ; z ) = F ( b, a ; c ; z ).3. Integrated local energy decay and Uniform bound
Morawetz Vector Field.
In this section, we consider momentum of the formassociated to solution ψ of the wave equation (1.3), P α = P α [ ψ, ξ, q ] = T αβ · ξ β + qD α ψ · ψ − D α q · ψ . We shall consider a generalized Morawetz vector field A of the form: A = − zwf ∂ r , f = ∂ r z,q reduced = 12 ( ∂ r z ) wf, q = 12 ( ∂ r A r ) + q reduced . ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 9
Defining the functions w, z : w = r , ∆ = r − M r, z = ∆ r , we have D α P α [ ψ, A , q ] = A ( ∂ r ψ ) + U αβ ∂ α ψ∂ β ψ + V| ψ | , (3.1)with A = − z / ∆ / ∂ r (cid:18) w z / ∆ / f (cid:19) , V = 14 ∂ r (∆ ∂ r ( z∂ r ( wf ))) + 12 wf ∂ r ( zV ) , and U αβ ∂ α ψ∂ β ψ involves angular derivatives U αβ ∂ α ψ∂ β ψ = U| r ∇ / ψ | + ∂ r (cid:18) r (cid:19) zwf ( ∂ t ψ ) , U = 12 wf . With the choice of z and w , we have A = M ∆ r . As in [2], we define B α = UL αβ ∂ β ψ · ψ, where the second order operator L = ∂ t + ˚ (cid:52) / S . With the boundary term B α , wehave D α ( P α [ ψ, A , q ] − B α ) = A ( ∂ r ψ ) + U αβ ∂ α ψ∂ β ψ + V| ψ | . (3.2)Here U αβ = UL αβ , (we still denote it by U αβ without confusion) and A , V are the same as above.For the junk term U αβ ∂ α ψ∂ β ψ in (3.2), recalling that U αβ = 12 wf L αβ , we have U αβ ∂ α ψ∂ β ψ ≥ wf (cid:0) | r ∇ / ψ | + ( ∂ t ψ ) (cid:1) . Hence upon integrating over S ( t, r ), using the fact that the spectrum of − ˚ (cid:52) / S acting on functions with (cid:96) ≥ (cid:96) ( (cid:96) + 1) ≥ , we have (cid:90) S ( t,r ) U αβ ∂ α ψ∂ β ψ ≥ (cid:90) S ( t,r ) U | ψ | , (3.3)where U is U = 13 ( r − M ) r . (3.4)Putting this together we have for some constant (cid:15) > D α ( P α [ ψ, A , q ] − B α ) ≥ (cid:15) (cid:0) A ( ∂ r ψ ) + ( V + 6 U ) | ψ | (cid:1) + (1 − (cid:15) ) UL αβ ∂ α ψ∂ β ψ. (3.5)Hence, the Morawetz estimate reduces to prove the Hardy inequality (cid:90) ∞ M (cid:0) A | ∂ r ϕ | + V | ϕ | (cid:1) d r ≥ (cid:15) Hardy (cid:90) ∞ M (cid:18) ∆ r | ∂ r ϕ | + ϕ r (cid:19) d r. (3.6) As in the proof of Lemma 3.12 in [2], one has to show that there is a positive C solution of the ordinary differential equation − ∂ r ( A∂ r ) φ + V φ = 0 . Integrated decay estimate.
We would like to introduce a lemma relatingthe above ODE and the hypergeometric functions. Recall that∆ = r − M r.
Lemma 3.1.
Let A = M ∆ r and V = Mr ( V r + V M r + V M ) .Let α = 12 + √ V + 2 V + V + 12 ,β = 12 − √ V ,a = 1 + √ V + 2 V + V − √ V − √ V ,b = 1 + √ V + 2 V + V − √ V + √ V ,c =1 + (cid:112) V + 2 V + V . Assume none of c , c − a − b , and a − b are integers.For the ODE − ∂ r ( A∂ r φ ) + V φ = 0 , (3.7) a pair of fundamental solutions, which we call the Frobenius solutions adapted to r = 2 M , is ( r − M ) α r β F (cid:18) a, b, c, − r − M M (cid:19) , ( r − M ) α r β (cid:18) − r − M M (cid:19) − c F (cid:18) a − c + 1 , b − c + 1 , − c, − r − M M (cid:19) . The second can also be expressed as ( r − M ) α r β (cid:18) − r − M M (cid:19) − c (cid:16) r M (cid:17) c − a − b F (cid:18) − a, − b, − c, − r − M M (cid:19) . Another pair of fundamental solutions, which we call the Frobenius solutions adapted r = ∞ , is ( r − M ) α r β (cid:18) − r − M M (cid:19) − a F (cid:18) a, a − c + 1 , a − b + 1 , − Mr − M (cid:19) , ( r − M ) α r β (cid:18) − r − M M (cid:19) − b F (cid:18) b, b − c + 1 , b − a + 1 , − Mr − M (cid:19) . Proof.
We follow the argument from [2]. Let v = A / u,x = r − M. (3.8)The ODE (3.7) then becomes and find that the resulted ordinary differential equa-tion − ∂ x v + W v = 0 , (3.9) ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 11 and W = VA + 12 ∂ x AA − ( ∂ x A ) A = V r + ( V − M r + ( V + 8) M r (2 M − r ) = V x + (4 V + V − M x + (4 V + 2 V + V ) M x ( x + 2 M ) . Now, let ˜ v be such that v = x α ( x + d ) β ˜ v , so that the above ODE becomes x α − ( x + d ) β − P,P = − x ( x + d ) ∂ x ˜ v − x ( x + d )(( α + β ) x + αd ) ∂ x ˜ v + (cid:0) − ( α ( α − x + d ) + 2 αβx ( x + d ) + β ( β − x ) + W x ( x + d ) (cid:1) ˜ v. Let d = 2 M , so that W x ( x + d ) = V x + (4 V + V − M x + (4 V + 2 V + V ) M , and the coefficient of ˜ v in P above is( − α ( α − − αβ − β ( β −
1) + V ) x + ( − α ( α − − αβ + 4 V + V − M x + ( − α ( α −
1) + 4 V + 2 V + V ) M . We choose α so that the M coefficient vanishes, i.e.0 = − α ( α −
1) + 4 V + 2 V + V = − α + 4 α + (4 V + 2 V + V ) ,α = 12 ± √ V + 2 V + V + 12 . From the second line above, we also have the identity − α ( α −
1) = − (4 V + 2 V + V ) /
4. We now choose β so that the ratio of the M x coefficient to the x coefficientis 2, i.e. 2 = − α ( α − − αβ + 4 V + V − − α ( α − − αβ − β ( β −
1) + V − α ( α − − αβ − β ( β −
1) + 2 V = − α ( α − − αβ + 4 V + V − − β ( β −
1) = − α ( α −
1) + 2 V + V − (cid:18) − V − V − V (cid:19) + 2 V + V − − V − , β − β + ( − V − β = 12 ± √ V . We choose the + sign in α and − sign in β .The substitution x = − M z now yields the ODE0 = z (1 − z ) ∂ z ˜ v + (2 α − (2 α + 2 β ) z ) ∂ z ˜ v − (2 αβ + 1 + V / V / v. Observe that equation (3.23) in [2] is missing a minus sign in front of x ( x + d ) ∂ x ˜ v , but therest of the argument there is correct. This is now in the form of a standard hypergeometric differential equations, andthe corresponding parameters thus satisfy c =2 α,a + b + 1 =2 α + 2 β,ab =(2 αβ + 1 + V / V / , which implies the parameters a and b are a = α + β − − (cid:112) α ( α −
1) + 4 β ( β − − V − V − ,b = α + β −
12 + (cid:112) α ( α −
1) + 4 β ( β − − V − V − . Thus, the parameters are a = 1 + √ V + 2 V + V − √ V − √ V ,b = 1 + √ V + 2 V + V − √ V + √ V ,c =1 + (cid:112) V + 2 V + V . Several forms for solutions to the hypergeometric function are given in [1]. Re-versing all the substitutions made so far in the proof, one finds a pair of fundamentalsolutions adapted to z = 0 (i.e. r = 2 M ) is( r − M ) α r β F (cid:18) a, b, c, − r − M M (cid:19) , ( r − M ) α r β (cid:18) − r − M M (cid:19) − c F (cid:18) a − c + 1 , b − c + 1 , − c, − r − M M (cid:19) . An alternative way of writing the second solution is( r − M ) α r β (cid:18) − r − M M (cid:19) − c (cid:16) r M (cid:17) c − a − b F (cid:18) − a, − b, − c, − r − M M (cid:19) . Another pair of fundamental solutions adapted to z = ∞ (i.e. r approaching thepoint at infinity on the Riemann sphere except along the negative real axis) is( r − M ) α r β (cid:18) − r − M M (cid:19) − a F (cid:18) a, a − c + 1 , a − b + 1 , − Mr − M (cid:19) , ( r − M ) α r β (cid:18) − r − M M (cid:19) − b F (cid:18) b, b − c + 1 , b − a + 1 , − Mr − M (cid:19) . (cid:3) In this section, we will prove the integrated decay estimate for both Regge-Wheeler and zerilli equations.3.2.1.
The Regge-Wheeler case.Proof of Theorem 1.3 for
Regge-Wheeler case . We first prove the statement forRegge-Wheeler case, where we have A = M ∆ r , (3.10) V = − M r − + 15 M r − − M r − . (3.11) ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 13
Denoting V = V + 6 U · Mr , we have the lower bound for the Morawetz potential V + 6 U ≥ V and V = 32 M r − − M r − + 13 M r − . (3.12)Recalling the A (3.10) and V (3.12), we let A = A , V = V . (3.13)Now we apply the same transformations (3.8) and find that the resulted ordinarydifferential equation (3.9) has a solution taking the following form u = A − ( r − M ) α r β F ( a ; b ; c ; z ) , (3.14)where z = − r − M M , and F ( a ; b ; c ; z ) is the associated hypergeometric function. Wefollows Lemma 3.1 to choose the parameters, α = 1 ± √ , β = 1 ± √ . We take the + sign in α and the − sign in β . Then we can immediately read offsome quantities in terms of α and β , c = 1 + √ ,a + b = 1 + √ − √ ,ab = 1 + √ − √ − √
112 + 4 . The remaining two parameters a, b could be solved { a, b } = { √ − √ ± √ } . We choose a < b such that a < − . < < . < b < . < . < c. We can use the integral representation (2.10) to show that the hypergeometricfunction F ( a ; b ; c ; z ) with z < (cid:90)(cid:90) M ∆ r | ∂ r ψ | + | ψ | r + r (cid:54) (cid:104) M (cid:18) | ∂ t ψ | r + | r ∇ / ψ | r (cid:19) d t d r d σ S (cid:46) E T ( ψ, τ ) . That is, (cid:90)(cid:90) M ∆ r | ∂ r ψ | + | ψ | r + r (cid:54) (cid:104) M (cid:18) | ∂ t ψ | r + | r ∇ / ψ | r (cid:19) d µ g (cid:46) E T ( ψ, τ ) . (3.15)Therefore we complete the proof for the Regge-Wheeler case. (cid:3) The Zerilli case.
Let V RWg denote the potential of Regge-Wheeler equation,and V Zg the Zerilli potential. They are related by V Zg = V RWg (1 + ζ ) , with ζ = 2¯ λ + 34¯ λ (cid:32) (cid:18) − M Λ (cid:19) − (cid:33) − . (3.16)Here Λ = ¯ λr + 3 M, λ = (cid:96) ( (cid:96) + 1) − . (3.17)Noting that (cid:96) ≥
2, we have ¯ λ ≥
2. We calculate ∂ r ζ = 9 M (cid:0) λ + 3 (cid:1) (cid:18) − M Λ (cid:19) Λ − . (3.18)It is easy to check that ∂ r ζ > . Before proceeding to the proof for Zerilli case, we first state the main idea andmain steps for the proof. In the Zerilli case, it would be difficult to find a smoothlylower bound for the Morawetz potential V + 6 U (3.5) in the whole region [2 M, ∞ ).The key point is: we are allowed to relax the requirement of smoothly lower bound,and find a continuously lower bound V joint for the Morawetz potential. Then,similar to the Regge-Wheeler case, we could work on the second order ordinarydifferential equation (3.7) with the potential V being replaced by V joint , and find apositive C solution. The Hardy inequality then follows and hence the integrateddecay estimate.The first step is to separate the estimate in the two regions [2 M, M ] and(3 M, ∞ ) and find a C lower bound V joint for the Morawetz potential. Note that,the lower bounding potential V joint is chosen such that the second order ordinarydifferential equation (3.7) could be transformed to hypergeometric differential equa-tions in each of the two intervals. In step two, we will analyze the hypergeometricdifferential equation associated to the ODE (3.7), and find out the Frobenius so-lutions (adapted to x = 0) in [2 M, M ] and (3 M, ∞ ). The Frobenius solutions(adapted to x = ∞ ) follow by making some transformations on the old Frobeniussolutions (adapted to x = 0). At last, in step three, we will construct a C solutionto the hypergeometric differential equation, which comes from linear combinationsof the Frobenius solutions (adapted to x = 0). We will show that this solution is C and positive by finding a positive lower bound G ( x ). The construction of G ( x )is based on an observation that the ratio of two Frobenius solutions (adapted to x = 0) has a limitation at infinity. Remarkably, further analysis for the asymptoti-cally expansion at infinity shows that G ( x ) is proportional to one of the Frobeniussolutions adapted to x = ∞ . We can further make use of the integral representation(2.10) to show the positivity of G ( x ). Step I: The continuous lower bound for the Morawetz potential.Lemma 3.2 (Lower bound for the Morawetz potential) . In the Zerilli case, wehave the lower bound for the Morawetz potential V + 6 U in (3.5) as the following:For case I, r ≤ M V + 6 U ≥ V r ≤ M . = 5 M r − (cid:18)
13 + 23 (cid:19) M r + 18 M r . (3.19) For case II, r ≥ M V + 6 U ≥ V r ≥ M . = M r − M r + 7 M r . (3.20) In particular, we have (cid:0) V r ≤ M − V r ≥ M (cid:1) (cid:12)(cid:12) r =3 M = 0 . (3.21) Proof of Lemma 3.2.
As in the Regge-Wheeler case, we still have the formulation(3.5) with A = M ∆ r being the same as in (3.10). Now we have to estimate thelower bound for V + 6 U in (3.5). First, we recall the formula for V , V = 14 ∂ r (∆ ∂ r ( z∂ r ( wf ))) + 12 wf ∂ r ( zV Z ) , (3.22)where the first term ∂ r (∆ ∂ r ( z∂ r ( wf ))) is the same as in Regge-Wheeler case. Nowwe consider the second term wf ∂ r ( zV Z ), which is given by12 wf ∂ r ( zV Z ) = 12 wf ∂ r ( zV RW )(1 + ζ ) + 12 wf zV RW ∂ r ζ. (3.23)Notice that f = − r − M ) r . ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 15
In case I: M ≤ r ≤ M. M (2¯ λ + 3) ≤ ≤ M (¯ λ + 1) , (2¯ λ + 2) r ≤ ≤ (2¯ λ + 3) r. (3.24)And − ≤ ζ (2 M ) = − λ + 3) ≤ ζ ≤ ζ (3 M ) = − λ + 1) < . (3.25)Submitting (3.24) into (3.18), we have514 r ≤ ∂ r ζ ≤ r . (3.26)Since 2 M ≤ r ≤ M, we have wf ≥ . Moreover, in view of (3.26) and the factthat zV RW <
0, we have the estimate for the second term in (3.23)12 wf zV RW ∂ r ζ < . (3.27)Let us turn to the first term wf ∂ r ( zV RW )(1 + ζ ) in (3.23). However, ∂ r ( zV RW )has no sign. Hence we separate ∂ r ( zV RW ) into positive and negative part: ∂ r ( zV RW ) = 8 M r − (3 r − M ) = 24 M r − ( r − M ) + 8 M r − , ignoring the negative part, we use the bound of ζ in (3.25) to obtain,12 wf ∂ r ( zV RW ) ζ ≥ − wf M r . Again, using the bound of ∂ r ζ in (3.26) to get the lower bound for (3.27), we obtain12 wf ∂ r ( zV Z ) = 12 wf (cid:0) ∂ r ( zV RW )(1 + ζ ) + zV RW ∂ r ζ (cid:1) ≥ wf (cid:18) (3 r − M ) 8 Mr − M r − ( r − M ) 6 Mr (cid:19) . (3.28)With the above estimate (3.28) for the case 2 M ≤ r ≤ M, the lower bound ofthe potential V + 6 U in the Morawetz estimate is given by V + 6 U ≥ V + 6 U Mr ≥ M r − (cid:18)
13 + 57 (cid:19) M r + (cid:18)
18 + 17 (cid:19) M r . (3.29)Additionally, using the fact that ≤ Mr ≤ , we finally get the lower bound forthe potential for r ≤ M : V + 6 U ≥ V r ≤ M . = 5 M r − (cid:18)
13 + 23 (cid:19) M r + 18 M r . (3.30) In case II: r ≥ M. We have wf ≤ , and ∂ r ( zV RW ) > . Additionally, ζ ≤ λ ≤ . Hence in (3.23) the first term has the lower bound12 wf ∂ r ( zV RW )(1 + ζ ) ≥ wf ∂ r ( zV RW ) . In view of the fact that zV RW < ∂ r ζ ≥
0, the second term12 wf zV RW ∂ r ζ ≥ . Thus, we have the lower bound12 wf ∂ r ( zV Z ) ≥ wf ∂ r ( zV RW ) . (3.31) We can estimate V + 6 U for r ≥ M as V + 6 U ≥ V + 6 U Mr ≥ V r ≥ M . = M r − M r + 7 M r . (3.32)In summary, for case I, wf ≥
0, we had found a lower bound L r ≤ M for ∂ r ( zV Z ),such that V + 6 U ≥ ∂ r (∆ ∂ r ( z∂ r ( wf ))) + 12 wf L r ≤ M + 6 U Mr = V r ≤ M . (3.33)For case II, wf ≤
0, we had found a upper bound U r ≥ M for ∂ r ( zV Z ), such that V + 6 U ≥ ∂ r (∆ ∂ r ( z∂ r ( wf ))) + 12 wf U r ≥ M + 6 U Mr = V r ≥ M . (3.34)Both wf (cid:12)(cid:12) r =3 M = 0 and U (cid:12)(cid:12) r =3 M = 0 hold, it would be obviously to see that (cid:0) V r ≤ M − V r ≥ M (cid:1) (cid:12)(cid:12) r =3 M = 0. In particular, we have V r ≤ M − V r ≥ M = ( r − M ) (cid:18) r − M (cid:19) Mr , (3.35)which vanishes at r = 3 M. (cid:3) Step II: The hypergeometric functions associated to the Morawetzpotential.
Denote the lower bound for the Morawetz potential V + 6 U in bothcases by V r ≤ M . = V , and V r ≥ M . = V . (3.36)Notice that (3.35) implies V | r =3 M = V | r =3 M . (3.37)We therefore define the potential in the whole region r ≥ M by V joint . = (cid:40) V , if 2 M ≤ r ≤ M,V , if r ≥ M. (3.38)This is the lower bound for the Morawetz potential in the Zerilli case, and we knowthat V joint ∈ C . In this case, the Hardy type estimate is reduced to finding apositive solution to the ordinary differential equation − ∂ r ( A∂ r φ ) + V φ = 0 , (3.39)on the interval r ∈ [2 M, + ∞ ) with A = A = M ∆ r , V = V joint . (3.40)If φ is a solution to equation (3.39) with A and V being specified in the Zerilli caseby (3.40), we apply the transformation (see Lemma 3.1) u = A φ, x = r − M. (3.41)Then u solves the new ordinary differential equation − ∂ x u + W u = 0 , (3.42)on the interval x ∈ [0 , + ∞ ) with W . = (cid:40) W =
16 15 x − Mx +4 M x ( x +2 M ) , x ≤ M,W = x − Mx +2 M x ( x +2 M ) , x > M. (3.43)We will apply the scheme in Lemma 3.1 to this case, and explore the hypergeo-metric functions associated to (3.42), (3.43) in each of the two region x ≤ M and x > M . To do the calculation explicitly, we may set M = 1 . Index the intervals
ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 17 (a) Positive solution u (b) Positive solution u Figure 2.
Positive hypergeometric functions in Zerilli case x ≤ x > i = 1 , , respectively. Index the hypergeometric (Frobenius)solutions in each interval by j = 1 , . Lemma 3.3 (The hypergeometric differential equation associate to the Zerilli equa-tions) . For x ≤ , there are two Frobenius solutions u , u (defined in (3.44) ) tothe ODE (3.42) , (3.43) and u is positive (see Figure 2(a)).For x > , there are two Frobenius solutions u , u (defined in (3.48) ) to theODE (3.42) , (3.43) and u is positive (see Figure 2(b)).Proof of Lemma 3.3. We find that two linearly independent solutions (adapted to x = 0) to (3.42) in x ≤ u = x α ( x + 2) β F (cid:16) a , b , c ; − x (cid:17) ,u = x α ( x + 2) β x − c F (cid:16) b − c + 1 , a − c + 1 , − c ; − x (cid:17) , (3.44)where we follow Lemma 3.1 to calculate the parameters, α = 12 ± √ , β = 12 ± √ . We take the + sign in α and the − sign in β . Then the a , b , c can be immedi-ately read off in terms of α and β , c = √
153 + 1 ,a + b = 1 + √ − √ ,a b = 143 + √ − √ − √ . The remaining a , b could be solved { a , b } = { √ − √ ± √ } . (3.45)We choose a < b such that a ≈ − . < < b ≈ . < c ≈ . . (3.46)To be clear, we also give the value of α and β , α = 12 + √ , β = 12 − √ , α + β = a + b + 12 . (3.47) For F ( a , b , c ; − x ), its second and third parameters satisfying c > b >
0. Wecan use the integral representation (2.10) to show that F (cid:16) a , b , c ; − x (cid:17) > x > , which says that u is positive.Similarly, let u j , j = 1 , x = 0) to(3.42) in x > u = x α ( x + 2) β F (cid:16) a , b , c ; − x (cid:17) ,u = x α ( x + 2) β x − c F (cid:16) a − c + 1 , b − c + 1 , − c ; − x (cid:17) , (3.48)where the parameters α = 1 ± √ , β = 1 ± . We will follow Lemma 3.1 to chose the parameters. We take the + sign in α andthe − sign in β . Then the a , b , c can be immediately read off in terms of α and β , c = 1 + √ ,a + b = √ − ,a b = 2 − √ . The remaining a , b could be solved { a , b } = { √ − ± √ } . (3.49)We choose a < b such that a ≈ − . < < b ≈ . < c ≈ . . (3.50)We also remark that here we have α = 12 + √ , β = − , α + β = a + b + 12 . (3.51)Those value will be useful in proving Theorem 3.4 in Step III. We find that for F (cid:0) a , b , c ; − x (cid:1) , its second and third parameters satisfying c > b >
0. Againwe can use the integral representation (2.10) to show that F (cid:16) a , b , c ; − x (cid:17) > x > . That is u is positive. (cid:3) Step III: Constructing the positive C solution. We first calculate somequantities which will be useful in constructing the positive C solution to the ordi-nary differential equation (3.42), (3.43). We normalize u (3.44) so that u (1) = 1,and let w = du dx (cid:12)(cid:12) x =1 . (3.52)We have w ≈ . · · · (3.53)For j = 1 , , we normalize u j (3.48) so that u j (1) = 1, and let w j = du j dx (cid:12)(cid:12) x =1 . (3.54)We have w = 0 . · · · , w = − . · · · (3.55) ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 19 (a) Positive solution u (b) u near x = 1 Figure 3.
The positive C solution u Additionally, we observe thatlim x → + ∞ u u . = − Λ = − . · · · (3.56) Theorem 3.4 (Positive hypergeometric function for Zerilli equations) . We define u , normalized to u (1) = 1 , by u = (cid:40) u , x ≤ ωu + (1 − ω ) u , x > where ω is given by ω = w − w w − w = 0 . · · · (3.58) Then u is indeed a positive C solution (see Figure 3(a), 3(b)) to the ordinarydifferential equation (3.42) , (3.43) . Remark 3.5.
In the proof of Theorem 3.4, we had used the numerical value of w (3.53) , w j , j = 1 , and Λ (3.56) to show the positivity of u .Proof for Theorem 3.4. First, we note that, with the above choice (3.57), u is ac-tually C . Hence, u defined in (3.57) is now a C solution to (3.42), (3.43).By construction, we have u > < x ≤
1, since u is positive (see Lemma3.3). One needs to check that for x > u >
0. Notice that u is positive (seeLemma 3.3). We wish to prove that in x >
1, (1 − ω ) u dominates ωu , so that ωu + (1 − ω ) u > W = W in x > − u (cid:48)(cid:48) ( x ) + W u ( x ) = 0 . (3.59)We set z = − x . Then (3.59) has a solution taking the following form¯ u = x α ( x + 2) β F ( z ) , (3.60)where F ( z ) is a solution to the hypergeometric differential equation z (1 − z ) d wd z + ( c − ( a + b + 1) z ) dwdz − a b w = 0 (3.61)with the parameters a , b , c defined in (3.48). Note that, the hypergeometricfunction has possibly regular singularities at z = 0 , , ∞ , namely, x = 0 , − , ∞ . For x > , we only focus on the regular singularity x = ∞ . There are the followingpairs f ( z ) , f ( z ) of Frobenius solutions to (3.61), which is adapted to z = ∞ (seeLemma 3.1), f ( z ) = z − a F (cid:18) a , a − c + 1; a − b + 1; 1 z (cid:19) ,f ( z ) = z − b F (cid:18) b , b − c + 1; b − a + 1; 1 z (cid:19) . (3.62)Submitting (3.62) into (3.60), we know that there are a pair of Frobenius solutions¯ u , ¯ u to (3.59), which are adapted to the singularity x = ∞ ,¯ u = x α ( x + 2) β f , ¯ u = x α ( x + 2) β f . In view of (3.51), we can calculate that the characteristic exponents of singularity x = ∞ are b − a + 12 and a − b + 12 , which could be further specified by the parameters a , b in (3.48). As a result, wehave the asymptotic expansion for ¯ u and ¯ u ¯ u ∼ x b − a → ∞ , ¯ u ∼ x a − b → , as x → ∞ . (3.63)As a remark, the parameters a , b could be chosen in various ways, but the resultingcharacteristic exponents would be always the same. Additionally, we could see that( − b ¯ u is positive. For( − b ¯ u = 2 b x α − b ( x + 2) β F (cid:18) b − c + 1 , b , ; b − a + 1; − x (cid:19) , = 2 b x α − b ( x + 2) β F (cid:18) b − c + 1 , b , ; b − a + 1; − x (cid:19) . (3.64)where we had used the fact that the hypergeometric is symmetric in its first two ar-guments: F ( a, b ; c ; z ) = F ( b, a ; c ; z ). And for F (cid:0) b − c + 1 , b , ; b − a + 1; − x (cid:1) ,the second and third two arguments satisfying b − a + 1 > b > , we can use theintegral representation (2.10) to show that F (cid:18) b − c + 1 , b , ; b − a + 1; − x (cid:19) > x > . The general solution to (3.59), which could be written as linear combinations ofthe Frobenius solutions u , u (adapted to x = 0), is either asymptotically decay-ing as ¯ u or as ¯ u . Due to the observation (3.56), we will construct a combinationof the Frobenius solutions u , u , which will be denoted by G ( x ) below, such that G ( x ) is asymptotically decaying as ¯ u , and hence proportional to ¯ u . Additionally, G ( x ) serves as a positive lower bound for ωu + (1 − ω ) u . In this way, we wouldprove the positivity for ωu + (1 − ω ) u .Recalling (3.56), we have Λ = 5 . · · · (3.65)We define a new function G ( x ) = u + Λ u . (3.66)We wish to prove that G ( x ) > x >
0. Now G ( x ) is a solution of the differentialequation (3.59) with G (1) = 1 + Λ >
0. Moreover, by the construction, we knowthat lim x →∞ G ( x ) u ( x ) = 0 . (3.67) ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 21
With respect to the exponents of the two Frobenius solutions (3.63) adapted tosingularity x = ∞ , we have either G ( x ) ∼ x b − a → ∞ or G ( x ) ∼ x a − b → x → ∞ . The fact of (3.67) further shows that G ( x ) ∼ x a − b → x → ∞ . As a summary, G ( x ) is a solution of the ordinary differential equation (3.59) with G (1) = 1 + Λ , lim x →∞ G ( x ) = 0 . (3.68)On the other hand, G ( x ) could be expressed in terms of linear combinations of theFrobenius solutions ¯ u and ¯ u : Suppose G ( x ) = p ¯ u ( x ) + q ¯ u ( x ) , (3.69)where p, q are some constants. Taking value at x = ∞ yields thatlim x →∞ G ( x ) = p lim x →∞ ¯ u ( x ) + q lim x →∞ ¯ u ( x ) , (3.70)which gives 0 = p · ∞ + q · . Hence p = 0 and G ( x ) = q ¯ u ( x ) . (3.71)We have known that ( − b ¯ u is positive. Hence (3.71) implies that G ( x ) does notchange sign. Besides, we know that G (1) = 1+Λ >
0, therefore G ( x ) > x > . Next, we turn back to ωu + (1 − ω ) u , which could be written as ωu + (1 − ω ) u = ω (cid:18) u + 1 − ωω u (cid:19) . (3.72)Viewing the value of ω in (3.58) and Λ in (3.65), we know that1 − ωω > Λ . (3.73)Additionally, u is positive (see Lemma 3.3). Hence, ωu + (1 − ω ) u > ω ( u + Λ u ) = ωG ( x ) > . (3.74)Therefore, we had proved that the C solution u defined by (3.57) is a positive (seeFigure 3(a)) solution to (3.42). Actually, (3.42) tells that u is also C (see Figure3(b)), since the transformation (3.41) takes V joint to W , and V joint ∈ C , hence W is continuous too.Finally, φ defined as (3.41) would be then a positive C solutions to (3.39). (cid:3) Now we are ready to prove the integrated decay estimate for the Zerilli case.
Proof of Theorem 1.3 for Zerilli equations.
We have constructed a C function φ (3.41), which is a positive solution to (3.39) with V = V joint . Thus the Hardyinequality follows. Hence we complete the proof for the Zerilli case. (cid:3) Remark 3.6.
We can apply the proof of Theorem 1.3 to the spacetime foliation (cid:83) τ Σ τ , hence for τ < τ (cid:90) ττ d t (cid:90) Σ (cid:48) t (cid:0) (1 − µ ) | ∂ r ψ | + | ψ | + | ∂ t ψ | + |∇ / ψ | (cid:1) d µ g (cid:46) (cid:90) Σ τ (cid:88) i ≤ P Tα ( T i ψ ) n α Σ τ , (3.75) where Σ (cid:48) t = Σ t ∩ { r ≤ R } . Non-degenerate energy.
With the integrated decay estimate, we could usethe vector field N to prove the uniform boundedness for the non-degenerated energy. Proof of Theorem 1.2.
For solution ψ of the equation (1.3), taking the vector field ξ = T, the associated energy on t = τ slice is E T ( ψ, τ ) = 12 (cid:90) t = τ (cid:0) | ∂ t ψ | + | ∂ r ∗ ψ | + (1 − µ )( |∇ / ψ | + V g ψ ) (cid:1) r d r ∗ d σ S . In presence of the factor (1 − µ ), the energy E T ( ψ, τ ) is degenerate on the horizon H + . We shall use the globally time-like vector field N and consider the associatedenergy.Away from the horizon Σ eτ = {M| t = τ } ∩ { r > r NH } . Note that N = T for r > r NH . For solutions of Regge-Wheeler or Zerilli equations, they both haveangular frequency (cid:96) ≥
2, which implies that (1.6) holds. This indeed gives thepositivity of energy on Σ eτ : In the case of Regge-Wheeler equation, E N ( ψ, Σ eτ ) ≥ (cid:90) Σ eτ (cid:18) | ∂ t ψ | + | ∂ r ∗ ψ | + (1 − µ ) (cid:18) |∇ / ψ | ψ r (cid:19)(cid:19) r d r ∗ d σ S > . In the case of Zerill equation, (cid:90) Σ eτ |∇ / ψ | + V Zg ψ = (cid:90) Σ eτ (cid:18) λ + 2 − µ (2 λ + 3)(2 λ + µ )( λ + µ ) (cid:19) ψ r ≥ (cid:90) Σ eτ (cid:18) λ − λ + 3 (cid:19) ψ r > . Note that, (1 − µ ) = 1 − Mr ≈ r > r NH .Near the horizon, recalling that Σ iτ . = {M| v = τ + r ∗ NH }∩{ r ≤ r NH } , the energy E N ( ψ, Σ iτ ) ≈ (cid:90) Σ iτ (cid:18) | ∂ u ψ | − µ + (1 − µ )( |∇ / ψ | + V g | ψ | ) (cid:19) d u d σ S . (3.76)The non-degeneracy of E N ( ψ, Σ iτ ) is more apparent if we write this integral in( v, r, θ, φ ) coordinate, which up to some constant is (cid:82) Σ iτ ( | ∂ r ψ | + |∇ / ψ | + V g ψ )d r d σ S . Due to the fact that (cid:96) ≥
2, we have (cid:90) Σ it (cid:18) | ∂ r ψ | + |∇ / ψ | + | ψ | r (cid:19) d r d σ S (cid:46) E N ( ψ, Σ iτ ) . (3.77)Taking ξ = N , we apply the energy identity with the momentum vector P N ( ψ ).Noting that N = T is killing for r > r NH , we have (cid:90) Σ τ P Nα ( ψ ) n α + (cid:90) H + P Nα ( ψ ) n α + (cid:90)(cid:90) { r
0. This is the red-shift effect, which allows us to control thenon-degenerate energy on the horizon.We should combine the conservation of the degenerate energy associated to themultiplier ∂ t with the integrated decay estimate Theorem 1.3 to derive the uniformboundedness of non-degenerate energy.Given data on Σ t for ψ , one may impose data along the horizon to the past of Σ t .This data can be chosen so as to be invariant under the flow along T , i.e. to dependonly on the angular variables and to match ψ where Σ t meets the horizon. Let ˜ ψ denote the solution generated by evolving this data both forward and backward in ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 23 time. Because n is proportional to T along the horizon, P Tα ( ψ ) n α is proportionalto P Tα ( ψ ) T α = (1 / T ψ ) , one finds P Tα ( ψ ) n α = 0 along the horizon, and inparticular the flux through any portion of the horizon is 0. Thus, ˜ ψ is equal to ψ in the future of Σ t and satisfies E T ( ψ, Σ τ ) = E T ( ˜ ψ, τ ) . (3.79)Thus, the last integral of (3.78) (cid:90)(cid:90) { r Corollary 3.7 (Nondegenerate Integrated Decay Estimate) . Let ψ be a solutionof Regge-Wheeler equation (1.4) or Zerilli equations (1.5) , we have for all R > M (cid:90) ττ d t (cid:90) Σ (cid:48) t P Nα ( ψ ) n α Σ τ (cid:46) (cid:90) Σ τ P Nα ( ψ ) n α Σ τ + P Tα ( T ψ ) n α Σ τ , (3.81) where Σ (cid:48) τ = Σ τ ∩ { r < R } . To proceed to higher order case, we use the non-degenerate radial vector fieldˆ Y = (cid:40) (1 − µ ) − ∂ u , if r ≤ r NH ,∂ r , if r > r NH . (3.82)Notice that, near horizon we can also write ˆ Y in ( v, r, ω ) coordinate as ˆ Y = ∂ r , if r ≤ r NH . Corollary 3.8 (Nondegenerate High Order Integrated Decay Estimate) . Let ψ bea solution of Regge-Wheeler equation (1.4) or Zerilli equations (1.5) , we have forall R > M and all integers n ∈ N , (cid:90) ττ d t (cid:90) Σ (cid:48) τ (cid:88) k + l + j ≤ n | N k Ω l ˆ Y j ψ | d µ g (cid:46) (cid:90) Σ τ (cid:88) k + l + j ≤ n P Nα ( T k Ω l ˆ Y j ψ ) n α Σ τ + (cid:88) i ≤ n +1 P Tα ( T i ψ ) n α Σ τ , (3.83) where Σ (cid:48) τ = Σ τ ∩ { r < R } .Proof. First, commuting the equation with T, we still have non-degenerate inte-grated decay estimate for T ψ (cid:90) ττ d t (cid:90) Σ (cid:48) τ (cid:88) k =0 P Nα ( T k ψ ) n α Σ τ (cid:46) (cid:88) k =0 (cid:90) Σ τ P Nα ( T k ψ ) n α Σ τ + P Tα ( T k +1 ψ ) n α Σ τ . (3.84)The elliptic estimate yields the high order integrated decay estimate away from thehorizon. Near horizon, say r < r < r NH , we shall commute the wave operator with Y = (1 − µ ) − ∂ u [15]. This commutator has a good sign, (cid:50) g Y ϕ − Y (cid:50) g ϕ = κY ϕ + f ( Y T ϕ, T ϕ, Y ϕ ) , where f ( Y T ϕ, T ϕ, Y ϕ ) is linearly dependent on Y T ϕ, T ϕ, Y ϕ, and κ > Y and T ,we use the energy identity for N to estimate (cid:90) Σ τ P Nα ( Y ψ ) n α + (cid:90) H + P Nα ( Y ψ ) n α + (cid:90)(cid:90) { r 0, (3.85) gives the estimate (cid:90)(cid:90) { r Corollary 3.9 (High Order Uniform Boundedness) . Let ψ be a solution of theRegge-Wheeler (1.4) or Zerilli equations (1.5) . Then for all n ∈ N , τ > τ , (cid:90) Σ τ (cid:88) k + l + j ≤ n P Nα ( N k Ω l ˆ Y j ψ ) n α Σ τ (cid:46) (cid:90) Σ τ (cid:88) k + l + j ≤ n P Nα ( T i Ω l ˆ Y j ψ ) n α Σ τ + (cid:88) i ≤ n +1 P Tα ( T i ψ ) n α Σ τ . (3.86)4. Decay estimate In this Section we prove quadratic decay of the non-degenerate energy. First ofall, we improve the local integrated decay estimate in Theorem 1.3. We shall usethe r p hierarchy estimate for proving energy decay estimate [32]. ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 25 Σ (cid:48) τ Σ (cid:48) τ H + N τ N τ D ττ I + r = R Figure 4. The spacetime foliation (cid:83) τ Σ (cid:48) τ ∪ N τ Energy decay. Let ψ be a solution of Regge-Wheeler equation (1.4) or Zerilliequations (1.5) and define Ψ . = rψ, (4.1)then (cid:32)L RW Ψ . = ∂ u ∂ v Ψ − η (cid:52) / Ψ + V RW Ψ = 0 , (4.2)in Regge-Wheeler case, and(cid:32)L Z Ψ . = ∂ u ∂ v Ψ − η (cid:52) / Ψ + V Z Ψ = 0 , (4.3)in Zerilli case. Here V RW = η ˆ V RW , ˆ V RW = − Mr , (4.4) V Z = η ˆ V Z , ˆ V Z = − Mr − M (cid:15)r . (4.5)We recall that V Zg = V RWg (1 + ζ ) , where ζ is defined in (3.16) and η = 1 − µ. When there is no confusion, we also refer (4.2) as Regge-Wheeler equation and(4.3) as Zerilli equations. We recall the definition of spacetime foliation (cid:83) τ Σ τ (Section 2.2), where Σ τ = Σ iτ ∪ Σ eτ withΣ iτ = {M| v = τ + r ∗ NH } ∩ { r < r NH } , and Σ eτ = {M| t = τ } ∩ { r ≥ r NH } . For our current purposes, we foliate the spacetime region by (cid:83) τ Σ (cid:48) τ ∪ N τ (Figure4): Fix R > M large enough, define the interior region ∪ τ Σ (cid:48) τ , whereΣ (cid:48) τ = Σ τ ∩ { r ≤ R } . (4.6)In the exterior region { r ≥ R } , let N τ be the outgoing null hypersurface emergingfrom the sphere S ( τ, R ) with constant t = τ and constant r = R , N τ . = { u = τ − R ∗ } ∩ { v ≥ τ + R ∗ } . (4.7)Let us also define a region bounded by the two null hypersurfaces and the time-likehypersurface (Figure 4): D τ τ = { ( u, v ) ∈ M (cid:12)(cid:12) r ( u, v ) ≥ R and τ − R ∗ ≥ u ≥ τ − R ∗ } . (4.8) Lemma 4.1 (Zero Order r p Integrated Decay Estimate) . Let Ψ be a solutionof Regge-Wheeler (4.2) or Zerilli equations (4.3) . Consider the region D τ τ , for < p ≤ , there is the integrated decay estimate (cid:90) N τ r p | ∂ v Ψ | d v d σ S + (cid:90) I + r p (cid:18) |∇ / Ψ | + Ψ r (cid:19) d u d σ S + (cid:90)(cid:90) D τ τ r p − { p | ∂ v Ψ | + 2 − p (cid:18) |∇ / Ψ | + | Ψ | r (cid:19) + 6 pMr | Ψ | r } d u d v d σ S (cid:46) (cid:90) N τ r p | ∂ v Ψ | d v d σ S + (cid:90) { r = R } {| ∂ v Ψ | + |∇ / Ψ | + | Ψ | } d t d σ S . (4.9) Remark 4.2. Comparing with the zero order r p weighted inequality in [32] , we getadditional bound for the spacetime integral (cid:82)(cid:82) D τ τ r p − pMr | Ψ | r d u d v d σ S . This wouldbe important in proving first order r p weighted inequality (for p = 2 ) in Lemma 4.5.Proof. Multiplying η − k r p ∂ v Ψ with 0 < p ≤ , k ≥ R sufficiently large and integrate with respect tothe measure d u d v d σ S in D τ τ , to derive the identity (cid:90) N τ (cid:18) r p η k | ∂ v Ψ | (cid:19) + (cid:90) I + (cid:18) r p η k − |∇ / Ψ | + r p Vη k | Ψ | (cid:19) + (cid:90)(cid:90) D τ τ (cid:0) (2 − p ) r p − η − k +2 + 2 M ( k − r p − η − k +1 (cid:1) |∇ / Ψ | − (cid:90)(cid:90) D τ τ ∂ u (cid:18) r p η k (cid:19) | ∂ v Ψ | + ∂ v (cid:18) r p Vη k (cid:19) | Ψ | = (cid:90) N τ (cid:18) r p η k | ∂ v Ψ | (cid:19) + (cid:90) { r = R } (cid:18) r p η k | ∂ v Ψ | + r p η k − |∇ / Ψ | + r p Vη k | Ψ | (cid:19) , (4.10)where V could be V RW or V Z in those two different cases. For R sufficiently large, − ∂ u ( r p η k ) ≥ p r p − for all 0 < p ≤ k ≤ . Next, we will prove the positivity of the other bulk terms in (4.10) for Regge-Wheeler and Zerilli cases separately. Regge-Wheeler case V = V RW : In the third line of (4.10), the term involving | Ψ | is − (cid:90)(cid:90) D τ τ ∂ v (cid:18) r p V RW η k (cid:19) | Ψ | = (cid:90)(cid:90) D τ τ ∂ v (cid:18) M r p − η k − (cid:19) | Ψ | = (cid:90)(cid:90) D τ τ η − k +1 (cid:0) M ( p − r p − + 12 M (4 − k − p ) r p − (cid:1) | Ψ | . (4.11)(4.11) does not have the good sign. But there is additional positive terms (cid:82)(cid:82) D τ τ M ( k − r p − η − k +1 |∇ / Ψ | in the second line of (4.10). Using the fact that Ψ has angularfrequency (cid:96) ≥ (cid:90) S ( t,r ) M ( k − r p − η − k +1 |∇ / Ψ | ≥ (cid:90) S ( t,r ) M ( k − r p − η − k +3 | Ψ | . (4.12)This additional positive term could be used to absorb the negative terms in (4.11).Therefore, the bulk terms in (4.10) dominate (cid:90)(cid:90) D τ τ (cid:0) M ( p − c k − c ) r p − + 12 M (4 − k − p ) r p − (cid:1) | Ψ | + (cid:90)(cid:90) D τ τ r p − | ∂ v Ψ | + (2 − p ) r p − |∇ / Ψ | + 2 M (1 − c )( k − r p − |∇ / Ψ | , (4.13) ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 27 for some constant c ≤ 1. We take c = , k = 4. Then for R sufficiently large, thebulk terms (4.13) bound (cid:90)(cid:90) D τ τ p r p − | ∂ v Ψ | + 2 − p r p − (cid:18) |∇ / Ψ | + | Ψ | r (cid:19) + 6 pMr r p − | Ψ | r , which is positive. Here we have used the fact that (cid:96) ≥ 2. Therefore, integratingthe identity (4.10) for 0 < p ≤ k = 4 with respect to the measure d u d v d σ S in the region D τ τ , we have (cid:90) N τ r p η | ∂ v Ψ | d v d σ S + (cid:90) I + r p η {|∇ / Ψ | − Mr | Ψ | r } d u d σ S (4.14)+ (cid:90)(cid:90) D τ τ r p − { p | ∂ v Ψ | + 2 − p |∇ / Ψ | + | Ψ | r ) + 6 pMr | Ψ | r } d u d v d σ S (4.15) (cid:46) (cid:90) N τ r p η | ∂ v Ψ | d v d σ S + (cid:90) r = R {| ∂ v Ψ | + |∇ / Ψ | + | Ψ | } d t d σ S . (4.16)Due to the fact that (cid:96) ≥ 2, the integral on the scri I + in (4.14) is actually positive.And η ∼ R large enough, thus we achieve the integrated decay estimate (4.9)for Regge-Wheeler case. Zerilli case V = V Z : In (4.10), the term involving | Ψ | is multiplied by − ∂ v (cid:18) r p V Z η k (cid:19) = ∂ v (cid:18) M r p − η k − (cid:19) + ∂ v (cid:18) M r p − ζη k − (cid:19) > η − k +1 (cid:0) M ( p − r p − + 24 M (4 − k − p ) r p − (cid:1) , (4.17)where we used the fact that (see (3.16). (3.18)), − λ + 3) ≤ ζ ≤ λ , | ζ | ≤ λ , ∂ r ζ > , (4.18)and 0 < p ≤ , k ≤ . As above, those terms in (4.17) do not have the good sign.But the additional positive terms 2 M ( k − r p − η − k +1 |∇ / Ψ | in (4.10) could beused to absorb the negative terms in (4.17). Since (cid:96) ≥ 2, the bulk terms in (4.10)dominate (cid:90)(cid:90) D τ τ (cid:0) M ( p − c k − c ) r p − + 24 M (4 − k − p ) r p − (cid:1) | Ψ | + (cid:90)(cid:90) D τ τ r p − | ∂ v Ψ | + (2 − p ) r p − |∇ / Ψ | + 2 M (1 − c )( k − r p − |∇ / Ψ | , (4.19)for some constant c ≤ 1. We take c = 1 , k = 4, then for R sufficiently large, thebulk terms (4.19) bound (cid:90)(cid:90) D τ τ p r p − | ∂ v Ψ | + 2 − p r p − (cid:18) |∇ / Ψ | + | Ψ | r (cid:19) + 6 pMr r p − | Ψ | r , which is positive. Integrating the identity (4.10) for 0 < p ≤ k = 4, we have (cid:90) N τ r p η | ∂ v Ψ | d v d σ S + (cid:90) I + r p η {|∇ / Ψ | + ˆ V Z | Ψ | } d u d σ S + (cid:90)(cid:90) D τ τ r p − { p | ∂ v Ψ | + 2 − p (cid:18) |∇ / Ψ | + | Ψ | r (cid:19) + 6 pMr | Ψ | r } d u d v d σ S (cid:46) (cid:90) N τ r p η | ∂ v Ψ | d v d σ S + (cid:90) { r = R } {| ∂ v Ψ | + |∇ / Ψ | + | Ψ | } d t d σ S . In the same way, the integral on scri I + : r p η {|∇ / Ψ | + ˆ V Z | Ψ | } is positive since (cid:96) ≥ 2. Thus we achieve the integrated decay estimate (4.9) for Zerilli case. (cid:3) The r p integrated decay estimate and the non-degenerate integrated decay esti-mate lead to the energy decay as follows [32]. Theorem 4.3 (Energy Decay) . Let R > M, and let ψ be a solution of Regge-Wheeler (1.4) or Zerilli equations (1.5) , with the initial data on Σ (cid:48) τ ∪N τ satisfying (cid:90) N τ (cid:88) k ≤ | T k ∂ v ( rψ ) | r d v d σ S + (cid:90) Σ (cid:48) τ ∪N τ (cid:88) k ≤ P Nµ ( T k ψ ) n µ < ∞ , (4.20) then there exist a constant I depending on the initial data (4.20) , such that (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ (cid:46) Iτ . (4.21) Here Σ (cid:48) τ = Σ τ ∩ { r ≤ R } , N τ = {M| u = τ − R ∗ , v ≥ v = τ + R ∗ } . Proof. We will only give the sketch of proof here (refer to [32] for more details).From the non-degenerate integrated decay estimate (Corollary 3.7) and conserva-tion of ∂ t energy (imposing zero data on null infinity to evolve backwards wherenecessary) (cid:90) ττ dt (cid:90) Σ (cid:48) t P Nµ ( ψ ) n µ (cid:46) (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ + (cid:90) Σ τ P Tµ ( T ψ ) n µ (cid:46) (cid:32)(cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ + P Tµ ( T ψ ) n µ (cid:33) . (4.22)Using the spacetime foliation (cid:83) τ Σ (cid:48) τ ∪ N τ , we also have the uniform boundness(Theorem 1.2), for any τ > τ , (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ (cid:46) (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ + P Tµ ( T ψ ) n µ . (4.23)We begin with an inequality, for any τ > τ (cid:90) τ τ d τ (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ (cid:46) (cid:90)(cid:90) D τ τ (cid:18) | ∂ v Ψ | + |∇ / Ψ | + | Ψ | r (cid:19) d u d v d σ S + (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ + P Tµ ( T ψ ) n µ , (4.24)where we had used the non-degenerate integrated decay estimate (4.22). Taking p = 1 in the r p weighted inequality of Lemma 4.1, we can further estimate (4.24)by (cid:90) τ τ d τ (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ (cid:46) (cid:90) N τ r | ∂ v Ψ | d v d σ S + (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ + P Tµ ( T ψ ) n µ . (4.25) ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 29 Next, we take p = 2 in the r p weighted inequality of Lemma 4.1, then there existsa dyadic sequence { τ (cid:48) j } j ∈ N with τ (cid:48) j +1 = 2 τ (cid:48) j and τ (cid:48) = τ (cid:90) N τ (cid:48) j r | ∂ v Ψ | d v d σ S (cid:46) τ (cid:48) j (cid:32)(cid:90) N τ r | ∂ v Ψ | d v d σ S + (cid:90) Σ (cid:48) τ ∪N τ P Tµ ( ψ ) n µ + P Tµ ( T ψ ) n µ (cid:33) . (4.26)Note that, in proving (4.26), there is the boundary term on { r = R } as the righthand side of (4.9), we can use the mean value theorem for the integration in r ∗ ,and the local integrated energy decay (4.22) to bound it by (cid:82) Σ (cid:48) τ ∪N τ P Tµ ( ψ ) n µ + P Tµ ( T ψ ) n µ . Combining (4.25) and (4.26), we have (cid:90) τ (cid:48) j +1 τ (cid:48) j d τ (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ (cid:46) τ (cid:48) j (cid:32)(cid:90) N τ r | ∂ v Ψ | d v d σ S + (cid:90) Σ (cid:48) τ ∪N τ P Tµ ( ψ ) n µ + P Tµ ( T ψ ) n µ (cid:33) + (cid:90) Σ (cid:48) τ (cid:48) j ∪N τ (cid:48) j P Nµ ( ψ ) n µ + P Tµ ( T ψ ) n µ . (4.27)Besides, with the uniform boundness of energy (4.23), we may estimate the lastterm in (4.27) by (cid:90) Σ (cid:48) τ (cid:48) j ∪N τ (cid:48) j (cid:0) P Nµ ( ψ ) n µ + P Tµ ( T ψ ) n µ (cid:1) (cid:46) τ (cid:48) j (cid:90) τ (cid:48) j +1 τ (cid:48) j − d τ (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ + (cid:88) i ≤ P Tµ ( T i ψ ) n µ . (4.28)Again we apply (4.27) on the above term (cid:90) τ (cid:48) j +1 τ (cid:48) j − d τ (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ + (cid:88) i ≤ P Tµ ( T i ψ ) n µ , to derive (cid:90) Σ (cid:48) τ (cid:48) j ∪N τ (cid:48) j (cid:0) P Nµ ( ψ ) n µ + P Tµ ( T ψ ) n µ (cid:1) (cid:46) τ (cid:48) j τ (cid:48) j − (cid:90) N τ (cid:88) k ≤ r | ∂ v T k Ψ | d v d σ S + (cid:90) Σ (cid:48) τ ∪N τ (cid:88) i ≤ P Tµ ( T i ψ ) n µ + 1 τ (cid:48) j (cid:90) Σ (cid:48) τ (cid:48) j − ∪N τ (cid:48) j − P Nµ ( ψ ) n µ + (cid:88) i ≤ P Tµ ( T i ψ ) n µ , (4.29)where by the uniform boundness of energy (4.23), the last term could be furtherbounded by 1 τ (cid:48) j (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ + (cid:88) i ≤ P Tµ ( T i ψ ) n µ . Thus, in view of (4.27) and (4.29), we have (cid:90) τ (cid:48) j +2 τ (cid:48) j d τ (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( ψ ) n µ (cid:46) τ (cid:48) j (cid:90) N τ (cid:88) k ≤ | ∂ v T k Ψ | r d u d v d σ S + (cid:90) Σ (cid:48) τ ∪N τ (cid:88) i ≤ P Nµ ( T i ψ ) n µ . (4.30)Finally performing the pigeon-hole principle, we have (4.21). (cid:3) We further commute the equations with Ω repeatedly, to have the high orderenergy decay. As a result, we have the pointwise decay estimate, Theorem 4.4 (Pointwise Decay) . Let ψ be a solution of Regge-Wheeler (1.4) orZerilli equations (1.5) , with the initial data on Σ (cid:48) τ ∪ N τ satisfying (cid:88) l ≤ (cid:90) N τ (cid:88) k ≤ | T k Ω l ∂ v ( rψ ) | r d v d σ S + (cid:90) Σ (cid:48) τ ∪N τ (cid:88) k ≤ P Nµ ( T k Ω l ψ ) n µ < ∞ , (4.31) then there exist a constant I depending on the initial data (4.31) , such that in thefuture development of initial hypersurface J + (Σ (cid:48) τ ∪ N τ ) r | ψ | ( τ, r ) (cid:46) Iτ . (4.32) Proof. On N τ = {M| u = τ − R ∗ , v ≥ τ + R ∗ } , integrating from infinity, by theCauchy Schwarz inequality, we have for any v ≥ τ + R ∗ ,( r ψ ) ( u, v ) = − (cid:90) ∞ v r ψ∂ v ( r ψ )d v (cid:46) (cid:90) ∞ v | ψ | d v + (cid:90) ∞ v | r∂ v ψ | d v. And then using the Sobolev inequality on the sphere, we have( r ψ ) (cid:46) (cid:90) ∞ v (cid:90) S (cid:88) k ≤ | Ω k ψ | r r d v d σ S + (cid:90) ∞ v (cid:90) S (cid:88) k ≤ | ∂ v Ω k ψ | r d v d σ S (cid:46) (cid:90) N τ (cid:88) k ≤ P Nµ (Ω k ψ ) n µ (cid:46) Iτ . (4.33)On Σ τ ∩ { M < r ≤ r ≤ R } , the Sobolev inequality also yields the pointwisedecay.Near the horizon Σ iτ , which is {M| v = τ + r ∗ NH , u ≤ u ( τ ) } with u ( τ ) < τ − r ∗ to be chosen later, we proceed a similar argument. Integrating from u = τ − r ∗ NH ,we have for any u ≥ u ( τ ), (cid:90) S ψ ( u, v )d σ S = (cid:90) S ψ ( u ( τ ) , v )d σ S + 2 (cid:90) uu ( τ ) (cid:90) S ψ∂ u ψ d u d σ S . The Cauchy Schwarz inequality gives (cid:90) Σ iτ ψ∂ u ψ d u d σ S (cid:46) (cid:90) Σ iτ (1 − µ ) ψ d u d σ S + (cid:90) Σ iτ ( ∂ u ψ ) − µ d u d σ S . Again we apply the Sobolev inequality on the sphere to get the pointwise decayestimate. Besides, applying a pigeon-hole argument in r and replacing ψ in (4.21)by Ω k ψ , we obtain that u ( τ ) could be chosen so that (cid:90) S ψ ( u ( τ ) , v )d σ S (cid:46) τ − . ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 31 In view of the definition of the non-degenerate energy E N ( ψ, Σ iτ ) in (3.76) (andwith the restriction to r ≤ r NH , we have absorbed factors of r into the constant C ), we have ψ ( u, v ) ≤ C (cid:90) Σ iτ (cid:88) k ≤ (cid:18) (1 − µ ) | Ω k ψ | + ( ∂ u Ω k ψ ) − µ (cid:19) r d u d σ S + (cid:90) S ψ ( u ( τ ) , v )d σ S (cid:46) Iτ . (cid:3) We next proceed to high order r p integrated decay estimate. For notationalconvenience, we denote K p − (Ψ) the spacetime integral (4.15) in the r p weightedinequality K p − (Ψ) . = (cid:90)(cid:90) D τ τ (cid:18) p r p − | ∂ v Ψ | + 6 pMr r p − | Ψ | r (cid:19) d u d v d σ S + (cid:90)(cid:90) D τ τ − p r p − (cid:18) |∇ / Ψ | + | Ψ | r (cid:19) d u d v d σ S , (4.34)and S p (Ψ) the energy on I + , S p (Ψ) . = (cid:90) I + r p (cid:18) |∇ / Ψ | + Ψ r (cid:19) d u d σ S . (4.35)In fact, when proceeding to the first order case, we will commute the equation with r∂ v . Define Ψ (1) . = r∂ v Ψ . (4.36)Unavoidably, there will be the leading error term involving angular derivative (cid:52) / Ψappearing during the commuting procedure. And we need to control these errorterms. As we can see from the right hand side of r p weighted inequality, thespacetime integral involving |∇ / Ψ | (4.15) vanishes when p = 2, which implies thatwe will lost the control for the angular derivative in the error terms when p = 2. Toget around this difficulty, we perform the integration by parts twice on the leadingterm (cid:90)(cid:90) D τ τ − r p (cid:52) / Ψ ∂ v Ψ (1) . (4.37)Then instead of (4.37), we get to deal with (cid:90)(cid:90) D τ τ (2 − p ) r p − |∇ / Ψ | , (4.38)as we can see in (4.46), (4.47) and (4.48). Due to the presence of the additionalfactor (2 − p ) , (4.38) which involves the angular derivative vanishes when p = 2.Thus our estimate go through even for p = 2. This idea could be found in [28]. Thisis the main difference from the first order r p weighted inequality in Proposition 5.6of [32]. We will prove the following Lemma. Lemma 4.5 (First Order r p Integrated Decay Estimate) . Let Ψ be a solution toRegge-Wheeler equation (4.2) or Zerilli equations (4.3) , and define D = { r∂ v , (1 − µ ) − ∂ u , r ∇ / } . (4.39) Considering in the region D τ τ , there is the integrated decay estimate for < p ≤ , (cid:88) j ≤ (cid:90) N τ r p | ∂ v D j Ψ | d v d σ S + (cid:90) I + r p (cid:18) | ∂ v D j Ψ | + D j Ψ r (cid:19) d u d σ S + (cid:90)(cid:90) D τ τ (cid:88) j ≤ pr p − (cid:18) | ∂ v D j Ψ | + Mr | D j Ψ | r (cid:19) d u d v d σ S + (cid:90)(cid:90) D τ τ (cid:88) j ≤ (2 − p ) r p − (cid:18) |∇ / D j Ψ | + | D j Ψ | r (cid:19) d u d v d σ S (cid:46) (cid:90) N τ r p (cid:88) j ≤ | ∂ v D j Ψ | + (cid:88) l ≤ | ∂ v Ω l Ψ | d v d σ S + (cid:90) { r = R } (cid:88) i,j ≤ ,l ≤ | ∂ iv ∇ / l D j Ψ | d t d σ S . (4.40) Remark 4.6. In the first order r p weighted energy inequality of [32] (in the proof ofProposition 5.6), the p has only range (0 , . We improve the rang of p to be (0 , in the above Lemma. Based on this, we can further improve the decay estimate fortime derivative as t − , see subsection 4.2. This could be compared with the decayof time derivative t − δ in [32] .Proof. As our proof is the same for both Regge-Wheeler and Zerilli case, we takethe Zerilli case for example. We begin with commuting the equation with r∂ v . Forany smooth function ϕ ∈ C ∞ ( M ), we have the commuting identity,[(cid:32)L Z , r∂ v ] ϕ = η∂ u ∂ v ϕ − η∂ v ϕ − η (cid:18) − Mr (cid:19) (cid:52) / ϕ − η Mr ∂ v ϕ − r∂ v V Z ϕ, (4.41)where (cid:32)L Z is defined as in (4.3). In view of the Zerilli equations (4.3) and commutingidentity(4.41), we have(cid:32)L Z Ψ (1) = − η∂ v Ψ + { η − η (1 − Mr ) }(cid:52) / Ψ − η Mr ∂ v Ψ − ( V Z + r∂ v V Z )Ψ . (4.42)It turns out that the first term on the right hand side of (4.42) has a good sign.Namely, − η∂ v Ψ = − ηr ∂ v Ψ (1) + η r ∂ v Ψ , (4.43)and − ηr ∂ v Ψ (1) has a good sign. Namely, we multiply 2 r p (1 − µ ) − k ∂ v Ψ (1) on bothsides of (4.42), and integrate on the spacetime region D τ τ , to yield that, (cid:90) N τ r p | ∂ v Ψ (1) | d v d σ S + K p − (Ψ (1) ) + S p (Ψ (1) ) (cid:46) (cid:90) N τ r p | ∂ v Ψ (1) | d v d σ S + (cid:90)(cid:90) D τ τ − η − k +1 r p − | ∂ v Ψ (1) | + A + A + A + boundary term on { r = R } , (4.44) ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 33 where A i , i = 1 , · · · , A + A + A . = (cid:90)(cid:90) D τ τ (cid:18) − Mr η − k +1 + 2 η − k +2 (cid:19) r p − ∂ v Ψ ∂ v Ψ (1) + (cid:90)(cid:90) D τ τ (cid:18) η − η (1 − Mr ) (cid:19) η − k r p (cid:52) / Ψ ∂ v Ψ (1) − (cid:90)(cid:90) D τ τ ( V Z + r∂ v V Z ) η − k r p +1 Ψ r ∂ v Ψ (1) . The bulk term in the second line of (4.44) has a good sign. It could be moved tothe right hand side of (4.44), and contributes to integrated decay estimate.Next, we will estimate A , · · · A one by one.For A , an application of Cauchy-Schwarz inequality yields | A | ≤ c (cid:90)(cid:90) D τ τ r p − | ∂ v Ψ | + c (cid:90)(cid:90) D τ τ r p − | ∂ v Ψ (1) | (4.45)for some universal constant c . We choose the constant c to be small enough so thatthe second term on the right hand side of (4.45) can be absorbed by K p − (Ψ (1) )which is on the left hand side of (4.44). And the first term (cid:82)(cid:82) D τ τ r p − | ∂ v Ψ | couldbe controlled by K p − (Ψ), which is of lower order derivative.For A , we rewrite it as A = (cid:90)(cid:90) D τ τ − r p (cid:52) / Ψ ∂ v Ψ (1) + f ( r ) (cid:52) / Ψ ∂ v Ψ (1) , (4.46)with | ∂ jr f | (cid:46) r p − − j , j ∈ N . Integrating by part twice, and using Cauchy-Schwarzinequality, we have | A | (cid:46) (cid:90)(cid:90) D τ τ r p − | ∂ v ΩΨ | + r p − | ∂ v ΩΨ | + (2 − p ) r p − |∇ / Ψ | + (cid:90)(cid:90) D τ τ r p − |∇ / Ψ | + (cid:90) I + r p (cid:0) | ∂ v ΩΨ | + |∇ / Ψ | (cid:1) . (4.47)Here we omit the boundary terms on { r = R } . In (4.47), (cid:82)(cid:82) D τ τ (2 r p − + r p − ) | ∂ v ΩΨ | could be bounded by K p − (ΩΨ), while (cid:90)(cid:90) D τ τ (2 − p ) r p − |∇ / Ψ | ≤ C · K p − (Ψ) , for some C > − p ) (4.48)Note that this holds for all 0 < p ≤ 2. In particular, (4.48) holds when p = 2,while the estimate in the proof of Proposition 5.6 in [32] does not hold for p = 2.Furthermore, noting that p ≤ 2, and then 3 − p > 0, we have (cid:90)(cid:90) D τ τ r p − |∇ / Ψ | (cid:46) (cid:90)(cid:90) D τ τ (3 − p ) r p − |∇ / Ψ | (cid:46) K p − (Ψ) . Finally, the last term in (4.47) could be bounded by S p (Ψ) and S p (ΩΨ).Similar for A , noticing that | ∂ jr ( r p V Z ) | (cid:46) r p − − j , j ∈ N , we integrate by parts,and use Cauchy-Schwarz inequality. Thus | A | (cid:46) (cid:90)(cid:90) D τ τ (cid:0) r p − + r p − (cid:1) | ∂ v Ψ | + r p − Ψ r + (cid:90)(cid:90) D τ τ (2 − p ) r p − Ψ r + (cid:90) I + r p (cid:18) | ∂ v Ψ | + Ψ r (cid:19) . (4.49) In the same way, the bulk integral in the first line of (4.49) could be bounded by K p − (Ψ). And (cid:90)(cid:90) D τ τ (2 − p ) r p − Ψ r ≤ C · K p − (Ψ) , for some C > − p ) . (4.50)Note that this holds for all 0 < p ≤ 2. As in A , the last term in (4.49) could bebounded by S p (Ψ).As a summary, we have (cid:90) N τ r p | ∂ v Ψ (1) | d v d σ S + K p − (Ψ (1) ) + S p (Ψ (1) ) (cid:46) (cid:90) N τ r p | ∂ v Ψ (1) | d v d σ S + K p − (Ψ) + K p − (Ψ) + K p − (ΩΨ)+ S p (Ψ) + S p (ΩΨ) + boundary terms on { r = R } . (4.51)Applying Lemma 4.1 to estimate K p − (Ψ) + K p − (Ψ) + K p − (ΩΨ) + S p (Ψ), we get (cid:88) j ≤ (cid:90) N τ r p | ∂ v (cid:0) ( r∂ v ) j Ψ (cid:1) | d v d σ S + (cid:88) j,l ≤ (cid:90) I + r p (cid:0) | r j − ∂ jv Ω l Ψ | + |∇ / Ψ | (cid:1) + (cid:88) j ≤ (cid:90)(cid:90) D τ τ pr p − (cid:18) | ∂ v ( r∂ v ) j Ψ | + Mr | ( r∂ v ) j Ψ | r (cid:19) d u d v d σ S + (cid:88) j ≤ (cid:90)(cid:90) D τ τ (2 − p ) r p − (cid:18) |∇ / ( r∂ v ) j Ψ | + | ( r∂ v ) j Ψ | r (cid:19) d u d v d σ S (cid:46) (cid:90) N τ r p (cid:88) j ≤ | ∂ v ( r∂ v ) j Ψ | + (cid:88) l ≤ | ∂ v Ω l Ψ | d v d σ S + (cid:88) j,l ≤ (cid:90) { r = R } | ∂ jv ∇ / l Ψ | d t d σ S . (4.52)Next, we commute the equation (4.2) or (4.3) with ∂ u and Ω,[(cid:32)L Z , ∂ u ] = 2 ηr (1 − Mr ) (cid:52) / − ∂ u V Z . (4.53)We know that | ∂ u V Z | (cid:46) Mr . Thus making use of Cauchy Schwarz inequality, wehave the energy inequality, (cid:90) N τ r p | ∂ v ∂ u Ψ | d v d σ S + (cid:90)(cid:90) D τ τ pr p − (cid:18) | ∂ v ∂ u Ψ | + Mr | ∂ u Ψ | r (cid:19) d u d v d σ S + (cid:90)(cid:90) D τ τ (2 − p ) r p − (cid:18) |∇ / ∂ u Ψ | + | ∂ u Ψ | r (cid:19) d u d v d σ S (cid:46) (cid:90) N τ r p | ∂ v ∂ u Ψ | d v d σ S + boundary term on { r = R } . + (cid:90)(cid:90) D τ τ (cid:15)r p − | ∂ v ∂ u Ψ | + 1 (cid:15) r p − (cid:18) |(cid:52) / Ψ | + | Ψ | r M r (cid:19) d u d v d σ S . (4.54) ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 35 We chose (cid:15) to be small enough, then (cid:82)(cid:82) D τ τ (cid:15)r p − | ∂ v ∂ u Ψ | could be absorbed by thesecond line of (4.54). Moreover, since |(cid:52) / Ψ | (cid:46) (cid:80) j ≤ r − | Ω j Ψ | , we have (cid:90)(cid:90) D τ τ r p − (cid:18) |(cid:52) / Ψ | + | Ψ | r M r (cid:19) d u d v d σ S (cid:46) (cid:88) j ≤ K p − (Ω j Ψ) . Combining with Lemma 4.1, we then have (cid:90) N τ r p | ∂ v ∂ u Ψ | d v d σ S + (cid:90)(cid:90) D τ τ pr p − (cid:18) | ∂ v ∂ u Ψ | + Mr | ∂ u Ψ | r (cid:19) d u d v d σ S + (cid:90)(cid:90) D τ τ (2 − p ) r p − (cid:18) |∇ / ∂ u Ψ | + | ∂ u Ψ | r (cid:19) d u d v d σ S (cid:46) (cid:90) N τ r p | ∂ v ∂ u Ψ | + (cid:88) j ≤ | ∂ v Ω j Ψ | d v d σ S +boundary term on { r = R } . (4.55)Finally, we commute the equation with Ω which are killing vector fields, andhence [(cid:32)L Z , Ω] = 0. The statement follows straightforwardly. (cid:3) The general high order r p integrated decay estimate follows by a simple induc-tion. Corollary 4.7 ( r p High Order Integrated Decay Estimate) . Let Ψ be a solutionof the Regge-Wheeler (4.2) or Zerilli equations (4.3) in the region D τ τ , and definethe weighted derivatives D = { r∂ v , (1 − µ ) − ∂ u , r ∇ / } . For < p ≤ , there is theintegrated decay estimate for all n ∈ N , (cid:90) N τ n (cid:88) j =0 r p | ∂ v D j Ψ | d v d σ S + (cid:90)(cid:90) D τ τ n (cid:88) j =0 r p − {| ∂ v D j Ψ | + (2 − p ) (cid:18) |∇ / D j Ψ | + | D j Ψ | r (cid:19) + 6 Mr | D j Ψ | r } d u d v d σ S (cid:46) (cid:90) N τ n (cid:88) j =0 r p | ∂ v D j Ψ | d v d σ S + (cid:90) r = R n (cid:88) j =0 {| ∂ v D j Ψ | + |∇ / D j Ψ | + | D j Ψ | } d t d σ S . (4.56)4.2. Improved decay estimate. Due to the r p first order integrated decay esti-mate (Lemma 4.5), we could improve the decay of first order energy. Corollary 4.8 (Improved Pointwise Decay) . Let ψ be a solution of Regge-Wheeler (1.4) or Zerilli equations (1.5) , with initial data on Σ (cid:48) τ ∪ N τ satisfying (cid:90) N τ (cid:88) k ≤ ,j ≤ r | T k ( r∂ v ) j ψ | + (cid:88) k ≤ ,l ≤ r | T k Ω l ∂ v ( rψ ) | d v d σ S + (cid:90) Σ (cid:48) τ ∪N τ (cid:88) k ≤ ,l ≤ P Nµ ( T k Ω l ψ ) n µ + (cid:88) k ≤ P Nµ ( T k ψ ) n µ < ∞ . (4.57) Then there is a constant I depending on the initial data (4.57) , (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( T ψ ) n µ (cid:46) Iτ , (4.58) where Σ (cid:48) τ = Σ τ ∩ { r ≤ R } , N τ = {M| u = τ − R ∗ , v ≥ v = τ + R ∗ } .Proof. As our proof is the same for both Regge-Wheeler and Zerilli case, we takethe Zerilli case for example. Let ψ (1) . = ∂ v Ψ . (4.59)Recalling that Ψ = rψ , Ψ (1) = rψ (1) in the first order r p weighted inequality (4.52),we make use of the zero order r p weighted estimate in Lemma 4.1, thus for 2 < p ≤ (cid:90) N τ r p | ∂ v ψ (1) | d v d σ S + (cid:90)(cid:90) D τ τ ( p − r p − (cid:18) | ∂ v ψ (1) | + Mr | ψ (1) | r (cid:19) d u d v d σ S + (cid:90)(cid:90) D τ τ (4 − p ) r p − (cid:18) |∇ / ψ (1) | + | ψ (1) | r (cid:19) d u d v d σ S (cid:46) (cid:90) N τ { r p | ∂ v ψ (1) | + (cid:88) j ≤ r p − | ∂ v Ω j Ψ | } d v d σ S + (cid:90) { r = R } (cid:88) j,l ≤ |∇ / j ∂ lv Ψ | d t d σ S . (4.60)We are interested in the quantity ∂ v T Ψ = ∂ v Ψ + ∂ v ∂ u Ψ. In view of the Zerilliequations (4.3), we have | ∂ v T Ψ | (cid:46) | ∂ v ψ (1) | + |(cid:52) / Ψ | + | M | r | Ψ | r . (4.61)For the second term |(cid:52) / Ψ | (cid:46) | ∇ / ΩΨ | r , we repeat the proof of Theorem 4.3 with ΩΨin the place of Ψ, thus (cid:90) N τ r |(cid:52) / Ψ | d v d σ S (cid:46) (cid:90) N τ |∇ / ΩΨ | d v d σ S (cid:46) (cid:90) N τ P Nµ (Ω ψ ) n µ (cid:46) Iτ ., (4.62)where I depends only on the initial data (4.57). Hence, we have (cid:90) N τ r | ∂ v T Ψ | d v d σ S (cid:46) (cid:90) N τ (cid:18) r | ∂ v ψ (1) | + | M | r | Ψ | r (cid:19) d v d σ S + Iτ . (4.63)Notice that, (cid:90)(cid:90) D τ τ r p − M r | Ψ | r d u d v d σ S (cid:46) (cid:90)(cid:90) D τ τ r p − − Mr | Ψ | r d u d v d σ S . (4.64)We shall use (4.60) for 2 < p ≤ < p − ≤ 2, thus (cid:90)(cid:90) D τ τ r p − (cid:18) | ∂ v ψ (1) | + | M | r | Ψ | r (cid:19) d u d v d σ S (cid:46) (cid:90) N τ { r p | ∂ v ψ (1) | + (cid:88) j ≤ r p − | ∂ v Ω j Ψ | } d v d σ S + boundary term at { r = R } . (4.65) ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 37 Next, we would apply the r p hierarchy estimate to improve the energy decay. Taking p = 4 in (4.60) and p = 2 in (4.9) with Ψ being replaed by ΩΨ, we apply the pigeon-hole principle. There exists a sequence { τ } j ∈ N such with τ j +1 = 2 τ j , (cid:90) N τj { r | ∂ v ψ (1) | + (cid:88) j ≤ r | ∂ v Ω j Ψ | } d v d σ S (cid:46) τ j (cid:90) N τ { r | ∂ v ψ (1) | + (cid:88) j ≤ r | ∂ v Ω j Ψ | } d v d σ S + 1 τ j (cid:90) Σ (cid:48) τ ∪N τ (cid:88) k ≤ ,l ≤ P Nµ ( T k Ω l ψ ) n µ + (cid:88) k ≤ P T ( T k ψ ) . (4.66)Furthermore, taking p = 3 in (4.60) and p = 1 in (4.9), we have (cid:90)(cid:90) D τj +1 τj (cid:18) r | ∂ v ψ (1) | + M r | Ψ | r (cid:19) d u d v d σ S (cid:46) (cid:90) N τi { r | ∂ v ψ (1) | + (cid:88) j ≤ r | ∂ v Ω j Ψ | } d v d σ S + (cid:90) Σ (cid:48) τj ∪N τj (cid:88) k ≤ ,l ≤ P Nµ ( T k Ω l ψ ) n µ + (cid:88) k ≤ P T ( T k ψ ) . (4.67)Viewing (4.66) and (4.67), we have (cid:90) τ j +1 τ j (cid:90) N τ (cid:18) r | ∂ v ψ (1) | + M r | Ψ | r (cid:19) d u d v d σ S (cid:46) τ j (cid:90) N τ { r | ∂ v ψ (1) | + (cid:88) j ≤ r | ∂ v Ω j Ψ | } d v d σ S + 1 τ j (cid:90) Σ (cid:48) τ ∪N τ (cid:88) k ≤ ,l ≤ P Nµ ( T k Ω l ψ ) n µ + (cid:88) k ≤ P T ( T k ψ )+ (cid:90) Σ (cid:48) τj ∪N τj (cid:88) k ≤ ,l ≤ P Nµ ( T k Ω l ψ ) n µ + (cid:88) k ≤ P T ( T k ψ ) . (4.68)As in the proof of Theorem 4.3, we make use of the uniform boundness of energy(4.23) to estimate the last term in (4.68). Therefore, in the same way, we have (cid:90) τ j +2 τ j d τ (cid:90) N τ (cid:18) r | ∂ v ψ (1) | + M r | Ψ | r (cid:19) d r ∗ d σ S (cid:46) τ j (cid:90) N τ { r | ∂ v ψ (1) | + (cid:88) j ≤ r | ∂ v Ω j Ψ | } d v d σ S + 1 τ j (cid:90) Σ (cid:48) τ ∪N τ (cid:88) k ≤ ,l ≤ P Nµ ( T k Ω l ψ ) n µ + (cid:88) k ≤ P T ( T k ψ ) . (4.69)We again use the pigeon-hole principle to obtain (cid:90) N τ (cid:18) r | ∂ v ψ (1) | + M r | Ψ | r (cid:19) d r ∗ d σ S (cid:46) τ (cid:90) N τ r | ∂ v ψ (1) | + (cid:88) k ≤ ,l ≤ r | T k Ω l ∂ v Ψ | d v d σ S + 1 τ (cid:90) Σ (cid:48) τ ∪N τ (cid:88) k ≤ ,l ≤ P Nµ ( T k Ω l ψ ) n µ + (cid:88) k ≤ P Tµ ( T k ψ ) n µ . (4.70) Regarding (4.63), we have established (cid:90) N τ r | ∂ v T Ψ | d v d σ S (cid:46) τ (cid:90) N τ r | ∂ v ψ (1) | + (cid:88) k ≤ ,l ≤ r | T k Ω l ∂ v Ψ | d v d σ S + 1 τ (cid:90) Σ (cid:48) τ ∪N τ (cid:88) k ≤ ,l ≤ P Nµ ( T k Ω l ψ ) n µ + (cid:88) k ≤ P Tµ ( T k ψ ) n µ . (4.71)Taking p = 2 and p = 1 in the r p weighted energy inequality (4.9) with Ψ beingreplaced by T Ψ, we repeat the proof of Theorem 4.3, therefore for a dyadic sequence { τ j } j ∈ N , (cid:90) Σ (cid:48) τj +1 ∪N τj +1 P Nµ ( T ψ ) n µ (cid:46) τ j (cid:90) N τj (cid:88) k ≤ r | T k ∂ v Ψ | d v d σ S + (cid:90) Σ (cid:48) τj ∪N τj (cid:88) k ≤ P Nµ ( T k ψ ) n µ . (4.72)As a result of (4.71) and Theorem 4.3, we use the pigeon-hole principle for (4.72),and (4.58) follows. (cid:3) Based on the improved first order energy decay, we can improve the pointwisedecay. Theorem 4.9 (Improved Interior Pointwise Decay) . Let R > M and ψ be asolution of Regge-Wheeler (1.4) or Zerillir equation (1.5) , with initial data on Σ (cid:48) τ ∪N τ satisfying I = (cid:90) N τ (cid:88) k + l ≤ ,l ≤ | T k Ω l ( r∂ v ) j ψ | r d v d σ S + (cid:90) Σ (cid:48) τ ∪N τ (cid:88) k + l ≤ ,l ≤ P Nµ ( T k Ω l ψ ) n µ < ∞ . (4.73) Then we have in the future development of initial hypersurface r | ∂ t ψ | ( τ, r ) (cid:46) Iτ , in J + (Σ (cid:48) τ ∪ N τ ) . (4.74) and the improved interior decay estimate, | ψ | (cid:46) Iτ , for r < R. (4.75) Proof. In Theorem 4.4, we had used the Sobolev inequalities to prove the pointwisedecay estimate: r | ψ | (cid:46) Iτ with I being a constant depending on the initial data(4.31). Similarly, based on the improved first order energy decay (Corollary 4.8),we have r | ∂ t ψ | (cid:46) Iτ , where I is a constant depending on the initial data (4.73).Next we interpolate between ψ and ∂ t ψ to improve the pointwise decay for | ψ | [32]. The basic observation underlying this argument is that for t > t rψ ( t , r ) = rψ ( t , r ) + (cid:90) t t ψ∂ t ψr d t ≤ rψ ( t , r ) + t − (cid:90) t t ψ ( t, r ) r d t + t (cid:90) t t ( ∂ t ψ ) ( t, r ) r d t. (4.76) ORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 39 For r < r < R , rψ ( t, r ) (cid:46) Rψ ( t, R ) + (cid:90) R ∗ r ∗ ψ ( t, r )d r ∗ + (cid:90) R ∗ r ∗ ( ∂ r ∗ ψ ) ( t, r ) r d r ∗ . (4.77)Thus, integrating on t and using the Sobolev inequality on the sphere, we have (cid:90) t t rψ ( t, r )d t (cid:46) (cid:90) t t d t (cid:90) S d σ S (cid:88) l ≤ R (Ω l ψ ) ( t, R )+ (cid:90) t t d t (cid:90) R ∗ r ∗ d r ∗ (cid:90) S d σ S (cid:88) l ≤ r (cid:18) (Ω l ψ ) r + ( ∂ r ∗ Ω l ψ ) (cid:19) . (4.78)That is, (cid:90) t t rψ ( t, r )d t (cid:46) (cid:90) Σ (cid:48) t (cid:88) k,l ≤ P T ( T k Ω l ψ ) . (4.79)Letting ¯ t = 2¯ t , by Theorem 4.3, there exists t (cid:48) ∈ (¯ t , ¯ t ) such that rψ ( t (cid:48) , r ) (cid:46) It (cid:48) . (4.80)Now considering a dyadic sequence { t (cid:48) j } j ∈ N with t (cid:48) j +1 = 2 t (cid:48) j , j ≥ 0, as an applicationof (4.79), we have (cid:90) t (cid:48) j +1 t (cid:48) j r ( ψ ) ( t, r )d t (cid:46) (cid:90) Σ (cid:48) tj (cid:88) k,l ≤ P T ( T k Ω l ψ ) , (cid:90) t (cid:48) j +1 t (cid:48) j r ( ∂ t ψ ) ( t, r )d t (cid:46) (cid:90) Σ (cid:48) tj (cid:88) k,l ≤ P T ( T k Ω l ∂ t ψ ) . (4.81)In regard of Theorem 4.3 and Corollary 4.8, (4.76) yields that rψ ( t (cid:48) j +1 , r ) (cid:46) rψ ( t (cid:48) j , r ) + 1 t (cid:48) j I ( t (cid:48) j ) + t (cid:48) j I ( t (cid:48) j ) . (4.82)By induction on j ∈ N and noting that (4.80) for j = 0, we have rψ ( t (cid:48) j , r ) (cid:46) I ( t (cid:48) j ) , j ∈ N ∪ { } . (4.83)The pigeon-hole principle will give the conclusion.For 2 M ≤ r < r , the same interpolation (4.76) by integration along lines ofconstant radius r < r can be carried out. While (4.77) could be replaced byintegration on v = ( t + r ∗ ), ψ ( t, u ) (cid:46) ψ ( t, u ) + (cid:90) uu (1 − µ ) ψ d u + (cid:90) uu ( ∂ u ψ ) − µ d u. (4.84)where u = ( t − r ∗ ). The above proof could be extended to 2 M ≤ r < r byreplacing P T by P N on the right hand sides of (4.79), (4.81). (cid:3) Remark 4.10. The argument in proving Theorem 4.9 does not hold for R → ∞ . Otherwise, adapted to R → ∞ , (4.77) would be replaced by integrating in v . Tomimic the procedure of deriving (4.79) from (4.78) , we should take p = 0 in thezero order r p inequality (4.9) . But this is impossible, since we require that p > in (4.9) . Combining with the non-degenerate high order integrated decay estimate (Corol-lary 3.8), the weighted high order uniform boundedness (Corollary 3.9), and the r p high order integrated decay estimate (Corollary 4.7), we have the improved highorder pointwise decay estimate by a simple induction. Corollary 4.11 (High Order Decay Estimate) . Let R > M , and define theweighted derivatives D = { r∂ v , (1 − µ ) − ∂ u , r ∇ / } . Let ψ be a solution of Regge-Wheeler (1.4) or Zerilli equations (1.5) , with initial data on Σ (cid:48) τ ∪ N τ satisfying I = (cid:88) i ≤ n (cid:90) N τ (cid:88) j ≤ ,k + l ≤ | D i ( r∂ v ) j Ω l T k ψ | r d v d σ S + (cid:88) k + l ≤ n +9 (cid:90) Σ (cid:48) τ ∪N τ P Nµ ( T k Ω l ψ ) n µ < ∞ . (4.85) for all n ∈ N . 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