The linear stability of the Schwarzschild spacetime in the harmonic gauge: even part
aa r X i v : . [ g r- q c ] S e p THE LINEAR STABILITY OF THE SCHWARZSCHILD SPACETIME INTHE HARMONIC GAUGE: EVEN PART
PEI-KEN HUNG
Abstract.
In this paper we study the even part of the linear stability of the Schwarzschildspacetime as a continuation of [22]. By taking the harmonic gauge, we prove that the energydecays at a rate τ − for the solution of the linearized Einstein equation after subtracting itsspherically symmetric part. We further show that the spherically symmetric part convergesto a linear combination of two special solutions. One is the gauge-fixed mass change solution[19]. The other is the deformation tensor of a stationary one form, which solves the tensorialwave equation. As a key ingredient, we prove that the solutions of the tensorial wave equationconverge to this stationary one form up to a scalar multiplication. Introduction
Einstein’s general theory of relativity describes how the metric on the spacetime, a 4-dimensionalLorenzian manifold, interacts with the matter fields. When there is no matter field, the theoryreduces to the study of the vacuum Einstein equation, which is equivalent to the Ricci flatequation on the metric g : Ric ( g ) = 0 . (1.1)Choquet-Bruhat and Choquet-Bruhat-Geroch [8, 9] formulated equation (1.1) as a Cauchyproblem with the initial data being a triple (Σ , g ij , k ij ). The triple consists of a 3-dimensionalmanifold Σ, a Remannian metric g ij and a symmetric two tensor k ij . If and only if the constraintequations (Gauss and Codazzi equations) are satisfied, there exist a vacuum spacetime ( M , g ),a 4-dimensional manifold equipped with a Lorentzian metric g solving (1.1), and an embedding i : Σ → M with g and k being the pulled back induced metric and second fundamental formrespectively. This can be viewed as the local (in time) existence theorem for (1.1).With the local existence of (1.1), it is of great attraction to study the long-time behaviorof the solution. A major open problem in this direction is the stability conjecture of the Kerrspacetimes ( M M,a , g
M,a ), which is a family of stationary solutions of (1.1). It is believed thatthe Kerr family is stable as solutions of (1.1): for initial data (Σ , g ij , k ij ) close to one from aKerr spacetime, the solution g has “long-time existence” and converges to a member of the Kerrfamily. We refer readers to [12, section 5.6] for a more precise statement of the conjecture.One needs to understand the linearized equation of (1.1) before studying the stability problem.In the Kerr background, the linearized equation is a linear equation on symmetric two tensors h , which reads dds Ric ( g M,a + sh ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = 0 . (1.2)There are two types of special solutions of (1.2). The first is a 4-dimensional vector space whichcomes from the perturbation within the Kerr family. The second, being infinite dimensional, consists of deformation tensors which correspond to infinitesimal diffeomorphisms. As a conse-quence of the Kerr stability conjecture, the solution h is believed to decay to zero up to thesespecial solutions, which is known as the linear stability problem of Kerr.In this paper, we study the linear stability of the Schwarzschild spacetime . To deal withthe infinite dimensional deformation tensors, we impose the harmonic gauge condition:Γ b [ h ] := ∇ a (cid:18) h ab −
12 ( trh ) g ab (cid:19) = 0 . (1.3)The harmonic gauge is the linearization of the harmonic map gauge used in [8, 9]. Lindblad-Rodnianski [28, 29] and Hintz-Vasy [20] also adapted the harmonic map gauge in proving thestability of the Minkowski spacetime. These accomplishments in the harmonic map gauge isthe main motivation to study the harmonic gauge in the Schwarzschild background. Under theharmonic gauge condition (1.3), the linearized equation (1.2) is equivalent to the Lichnerowiczd’Alembertian equation: (cid:3) h ab + 2 R c da b h cd = 0 , (1.4)where R c da b is the (index raised) Remannian curvature tensor of g . The equation (1.4) is a waveequation on symmetric two tensors and hence has well-posed Cauchy problem. In particular, wehave long-time existence of solutions for regular initial data. Moreover, the gauge condition (1.3)is preserved: for any solution h of the Lichnerowicz d’Alembertian equation (1.4), the gauge onefrom Γ b [ h ] satisfies the tensorial wave equation: (cid:3) Γ a [ h ] = 0 . Therefore Γ b [ h ] vanishes identically provided Γ b [ h ] and its normal derivative vanish initially. Weremark that there are sitll infinite dimensional deformation tensors satisfying (1.4). Since adeformation tensor W π is always a solution of (1.2), it solves (1.4) provided (1.3) holds, whichis equivalent to the tensorial wave equation on the potential one form: (cid:3) W a = 0 . (1.5)We investigate the equations (1.3), (1.4) for even symmetric two tensors and the equation(1.5) for even one forms as the odd part was studied in [22]. See subsection 2.5 for the even/odddecomposition. Denote by W ℓ =0 or h ℓ =0 the spherically symmetric part of a one form W or asymmetric two tensor h respectively. We show that solutions of (1.3) and (1.4) or (1.5) decay tozero after subtracting W ℓ =0 or h ℓ =0 . See Theorem 2.4 and 2.5 for the precise statements of thefollowing results. Theorem 1.1.
Let W = W a dx a be a solution of (1.5) with initial data falling off fast enough.Then the energy of W − W ℓ =0 decays at a rate τ − . Theorem 1.2.
Let h = h ab dx a dx b be a solution of (1.3) and (1.4) with initial data falling offfast enough.. Then the energy of h − h ℓ =0 decays at a rate τ − . The equation (1.5) admits a spherically symmetric and stationary solution W ∗ with finiteinitial energy [19]. See (8.1) in section 8 for the explicit form of W ∗ . We show that sphericallysymmetric solutions of (1.5) converge to a multiple of W ∗ . See Theorem 8.1 for the precisestatement of the following result. Theorem 1.3.
Let W = W a dx a be a spherically symmetric solution of (1.5) with initial datafalling off fast enough. Then W converges to cW ∗ for some constant c . CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 3
The equations (1.3) and (1.4) have two spherically symmetric and non-decaying solutions. Onesolution K ∗ is the gauge-fixed mass perturbation [19], which grows linearly. See subsection 8.3for the explicit form of K ∗ . The other is the deformation tensor of W ∗ , which is stationary. Weshow that under certain conditions, spherically symmetric solutions of (1.3) and (1.4) convergeto a linear combination of these two. See Theorem 8.7 for the precise statement of the followingresult. Theorem 1.4.
Let h = h ab dx a dx b be a spherically symmetric solution of (1.3) and (1.4) withinitial data falling off fast enough. Then h converges to c K ∗ + c ( W ∗ π ) for some constants c and c . . Related Work.
There have been many progresses towards the Kerr stability conjecture. The stability of theMinkowski spacetime was proved in the monumental work of Christodoulou and Klainerman [10].See also [26, 6, 28, 29, 20] for various approaches. Recently, the stability of the Schwarzschildspacetime was established by Klainerman and Szeftel [27] for axial symmetric polarized pertur-bations. In the positive cosmological constant setting, the stability of Kerr-de Sitter with smallangular momentum was proved by Hintz and Vasy [21].The study of equation (1.2) on the Schwarzschild background was initiated by Regge andWheeler [32]. The authors performed the even/odd decomposition and derived the Regge-Wheeler equation for the odd solutions of (1.2). For even solutions, there is a similar equa-tion discovered by Zerilli [38]. Bardeen and Press [4] adapted the Newman-Penrose formalismto study equation (1.2). This approach was extended to Kerr spacetimes by Teukolsky [35],showing that the extreme Weyl curvature components satisfy the Teukolsky equations. In theSchwarzschild spacetime, the transformation theory of Wald [36] and Chandrasekhar [7] relatesRegge-Wheeler-Zerilli-Moncrief system to the Teukolsky equations. See also [34] for further re-finement of the transformation theory. These works accumulated to the proof of mode stabilityfor Kerr by Whiting [37].A major progress which goes beyond mode stability is the work of quantitative linear stabilityof Schwarzschild by Dafermos, Holzegel and Rodnianski [11]. The authors proved the bound-edness and decay estimates for the Regge-Wheeler equation and then for Teukolsky equationsthrough transformation theory. The metric perturbation was reconstructed in the double nullgauge. Keller, Wang and the author [23] worked in the mixed Regge-Wheeler/Chandrasekhargauge and shown t − / decay of the metric coefficient based on Regge-Wheeler/Zerilli equa-tions. Johnson further [24, 25] proved t − decay through Regge-Wheeler/Zerilli equations withan insightful chosen generalized wave gauge in which the metric perturbation is related to Regge-Wheeler/Zerilli quantities by pseudo-differential operators. See also [22] for the t − decay of theodd part in harmonic gauge. For the linearized Einstein-Maxwell equations, Giorgi [15, 16, 17, 18]obtained the boundedness and decay estimates for the Teukolsky system in Reissner-Nordstr¨omspacetimes with small charge and proved linear stability under the gravitational-electromagneticperturbations.Recently, there are huge breakthroughs for the linear stability of Kerr spacetimes. Andersson,B¨ackdahl, Blue and Ma [1] established linear stability of Kerr with small angular momentumwith pointwise t − / decay in the outgoing radiation gauge. Hafner, Hintz and Vasy [19] gavea detailed description of the metric perturbation in Kerr spacetimes with small angular momen-tum under the wave gauge of the present paper. The authors identified 7-dimensional stationaryand additional 4-dimensional linear growth solutions of (1.3) and (1.2); these solutions consist of4-dimensional Kerr family perturbations and 7-dimensional deformation tensors. Under weaker PEI-KEN HUNG initial fall-off condition than the present paper, The authors further established t − − decay uptothese 11-dimensional space. Outline.
In section 2 we introduce the Schwarzschild spacetime and relevant notations.Section 3 contains simple lemmas that we use repeatedly. In section 4 we apply the vector fieldmethod to (1.4) near the horizon and null infinity. Then a gauge transformation from harmonicgauge to Regge-Wheeler gauge is performed. The transformation satisfies (1.5) with source termson the right hand side. In section 5 and section 6 we start to analyze equation (1.5) supportedon ℓ ≥
1, which leads to the main results for ℓ ≥ ℓ = 0 is considered insection 8. Acknowledgments
The author is grateful to Simon Brendle for suggesting this problem and for his initial contri-bution. The author also thanks Sergiu Klainerman and Mu-Tao Wang for their encouragement.The author further thanks T¨ubingen University where part of this work was carried out.2.
Schwarzschild spacetime and notations
Schwarzschild coordinate.
In this subsection we introduce the Schwarzschild coordinateand the vector fields that we need later. Let
M > M can be written as g M = − (cid:18) − Mr (cid:19) dt + (cid:18) − Mr (cid:19) − dr + r ( dθ + sin θdφ ) . The range of the ( t, r, θ, φ ) coordinate is t ∈ R , r > M and ( θ, φ ) ∈ S . To see the scaling ofenergies clearly, we define s := r/M . We suppress the dependence of M in g M and denote it by g . Let ∇ be the the Levi-Civita connection of g . We use x a , x b to denote a spacetime coordinate, x A , x B to denote the spherical coordinate on S and x α , x β to denote the quotient coordinateon R t × (2 M, ∞ ) r . Let /g AB dx A dx B := r ( dθ + sin θdφ ) be the induced metric on the orbitspheres and / ∇ be the Levi-Civita connection of /g . For any scalar function ψ , we denote | / ∇ ψ | := /g AB / ∇ A ψ / ∇ B ψ = 1 r (cid:12)(cid:12)(cid:12)(cid:12) ∂ψ∂θ (cid:12)(cid:12)(cid:12)(cid:12) + sin − θ (cid:12)(cid:12)(cid:12)(cid:12) ∂ψ∂φ (cid:12)(cid:12)(cid:12)(cid:12) ! . The ( t, r, θ, φ ) coordinate system has coordinate singularity at r = 2 M . To remove the singularityand to include the future horizon, it is convenient to work with the ( v, R, θ, φ ) coordinate, where(2.1) v := t + r + 2 M log (cid:16) r M − (cid:17) , R := r. The metric takes the form g = − (cid:18) − MR (cid:19) dv + 2 dvdR + R ( dθ + sin θdφ ) , which is smooth upto R = 2 M, v ∈ R . We denote by M the Schwarzschild exterior includingthe future horizon H + := { R = 2 M, v ∈ R } as M := (cid:18) R t × (2 M ∞ ) r ⊔ R v × [2 M, ∞ ) R / ∼ (cid:19) × S . CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 5
Here we identify ( t, r ) and ( v, R ) coordinates through (2.1).We record here the wave operator in various coordinate systems defined above. Let (cid:3) be thed’Alembertion operator with respect to the Schwarzschild metric g as (cid:3) ψ := g ab ∇ a ∇ b ψ. In particular, in the ( t, r, θ, φ ) coordinate we have (cid:3) = − (cid:18) − Mr (cid:19) − ∂ ∂r + (cid:18) − Mr (cid:19) ∂ ∂r + 2 r (cid:18) − Mr (cid:19) ∂∂r + / ∆ , where / ∆ = 1 r (cid:18) ∂ ∂θ + cot θ ∂∂θ + sin − θ ∂ ∂φ (cid:19) is the Laplacian operator for /g . In the { v, R, θ, φ } coordinate, we have(2.2) (cid:3) = (cid:18) − Mr (cid:19) ∂ ∂R + 2 ∂ ∂v∂R + 2 r (cid:18) − Mr (cid:19) ∂∂R + 2 r ∂∂v + / ∆ . More generally, let g ( r ) be a function and define the new variables τ := t + r + 2 M log (cid:16) r M − (cid:17) − g ( r ) , ρ := r. In this coordinate system, the the d’Alembertion operator reads (cid:3) = (cid:18) − Mr (cid:19) ∂ ∂ρ + 2 r (cid:18) − Mr (cid:19) ∂∂ρ + Z ∂∂τ + / ∆ , (2.3)where Z = − dgdr + (cid:18) − Mr (cid:19) (cid:18) dgdr (cid:19) ! ∂∂τ + (cid:18) − (cid:18) − Mr (cid:19) dgdr (cid:19) ∂∂ρ + (cid:18) − (cid:18) − Mr (cid:19) d gdr − r (cid:18) − Mr (cid:19) dgdr + 2 r (cid:19) . As for any mass parameter
M > K , solves (1.2). Explicitly, K = 1 R dv . (2.4)Similarly, for each a , the Kerr metric g M,a is vacuum and infinitesimal angular momentum changesolve (1.2). Denote them as K m = 1 R /ǫ BA / ∇ B Y m (cid:18) ( dv − dR ) dx A + dx A ( dv − dR ) (cid:19) , m = − , , . (2.5)Here Y m are the spherical harmonic functions supported on ℓ = 1 and /ǫ is the Levi-Civitatensor of /g AB . It can be verified directly that K m satisfies the harmonic gauge (1.3): Γ[ K m ] = 0.However, K fails to be in the harmonic gauge and PEI-KEN HUNG Γ[ K ] = 1 R dv. See subsection 8.3 for more discussion.We start to define the vector fields. The isometry group of g M is 4-dimensional, including onetime translation and 3-dimensional rotation Killing vector fields. We fix the notation for theseKilling vector fields as T := ∂∂t , Ω := ∂∂φ , Ω := cos φ ∂∂θ − cot θ sin φ ∂∂φ , Ω := sin φ ∂∂θ + cot θ cos φ ∂∂φ . Define the outgoing and incoming null vectors L and L as L := ∂∂t + (cid:18) − Mr (cid:19) ∂∂r , L := ∂∂t − (cid:18) − Mr (cid:19) ∂∂r . Note that in the ( v, R, θ, φ ) coordinate, L = − (cid:0) − Mr (cid:1) ∂ R which vanishes along H + . To getnon-zero incoming null vector, we define L ′ := (cid:18) − Mr (cid:19) − L. We are ready to define the red-shift vector Y ( σ ) as in [13]. Fix r + rs := 5 M/ η rs ( r ) = (cid:26) r ∈ [2 M, M/ , r ∈ [5 M/ , ∞ ) . (2.6)For any σ >
0, let Y ( σ ) := η rs ( r ) · (cid:16) σ r − M ) T + (2 + σ ( r − M )) L ′ (cid:17) . (2.7)Note that Y ( σ ) is casual in r ≥ M and is supported in r ∈ [2 M, r + rs ]. We fix the notation forthe following collections of vector fields: K := { M T, Ω , Ω , Ω } ,∂ := { T, r Ω , r Ω , r Ω , L ′ } , D := { M T, Ω , Ω , Ω , rL, M L ′ } . (2.8)The collection K consists Killing vectors and we will use it, together with D , to express higherorder estimates of wave equations. See Proposition A.4 in Appendix A. CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 7 foliation of spacetime.
In this subsection we define the spacelike/null hypersurface alongwhich we measure the solutions. Let R null ≥ M to be determined later. It is the maximumamong R Z , R H and R W determined in subsections 4.1, 5.1 and 6.4 respectively and we keep thedependence of R null in estimates until its value is fixed. Let g ( r ) be a piecewise smooth functiondefined in [2 M, ∞ ) which satisfies the following conditions: In [2 M, R null ], g ( r ) is smooth with dgdr (cid:18) − (cid:18) − Mr (cid:19) dgdr (cid:19) < . In [5 M/ , R null ], g ( r ) = r + 2 M log (cid:16) r M − (cid:17) . In ( R null , ∞ ), g ( r ) = 2 (cid:16) r + 2 M log (cid:16) r M − (cid:17)(cid:17) − (cid:18) R null + 2 M log (cid:18) R null M − (cid:19)(cid:19) . Let Σ τ := { t + r + 2 M log( r/ M − − g ( r ) = τ } for any τ ∈ R . The above requirementsensure that Σ τ is spacelike in r ∈ [2 M, R null ], is null in r ∈ ( R null , ∞ ) and intersects with H + transversally. Let n be the unit future vector of Σ τ in [2 M, R null ] and fix n = L in ( R null , ∞ ).For any τ ≥ τ , let D ( τ , τ ) be the region bounded by Σ τ and Σ τ . We introduce notations forΣ τ and D ( τ , τ ) in r ≤ R null or r ≥ R null respectively.Σ ′′ τ :=Σ τ ∩ { M ≤ r ≤ R null } , Σ ′ τ :=Σ τ ∩ { R null ≤ r } ,D ′′ ( τ , τ ) := D ( τ , τ ) ∩ { M ≤ r ≤ R null } ,D ′ ( τ , τ ) := D ( τ , τ ) ∩ { R null ≤ r } . For any τ ∈ R and r ≥ M , we denote the orbit sphere as S ( τ, r ) := Σ τ ∩ { r = r } . See figure 2.2 for the Penrose diagram of M .Let the spacetime volume form and the volume form of unit sphere be denoted by dvol := r sin θdt ∧ dr ∧ dθ ∧ dφ,dvol S := sin θdθ ∧ dφ. Denote by dvol the three form of Σ τ corresponding to n . In other words, − n ♭ ∧ dvol = dvol, where ( n ♭ ) a = g ab n b . PEI-KEN HUNG H + I + Σ ′′ Σ τ Σ ′ r = R null D ′′ (0 , τ ) D ′ (0 , τ ) S ( τ , R null ) M Figure 1.
Penrose diagram2.3. vector bundles L ( − and L ( − . Besides scalar functions, we also work with sphericalone forms and spherical symmetric traceless two tensors. Let L ( − ⊂ T ∗ M be the subbundleof spherical one forms. Locally, a section ξ of L ( −
1) can be written as ξ = ξ A dx A . Similarly, we denote by L ( − ⊂ T ∗ M ⊗ s T ∗ M the subbundle of spherical symmetric tracelesstwo tensors. Locally, a section Ξ of L ( −
2) can be written asΞ = Ξ AB dx A dx B , /g AB Ξ AB = 0 . The connections on L ( −
1) and L ( −
2) induced by the Levi-Civita connection are denoted by ∇ and ∇ respectively. Equipped with the induced metric, L ( −
1) and L ( −
2) are Riemannianvector bundles. We denote by (cid:3) and (cid:3) the d’Alembertion operator for ∇ and ∇ . We omitthe superscript in ∇ and ∇ when it doesn’t cause confusion.Denote by L (0) the trivial line bundle R × M with sections being scalar functions. Define theoperators /D : L ( − −→ L (0) and /D : L ( − −→ L ( −
1) as: /D ξ := / ∇ A ξ A , ( /D Ξ) A := / ∇ B Ξ AB . Their adjoints are
CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 9 ( /D ∗ ψ ) A := − / ∇ A ψ, ( /D ∗ ξ ) AB := −
12 ( / ∇ A ξ B + / ∇ B ξ A − / ∇ C ξ C /g AB ) . We record here the simple commutation relation between /D , /D , /D ∗ , /D ∗ and ∇ , ∇ : /D ∗ ∇ r = ∇ r /D ∗ + r − /D ∗ , /D ∇ r = ∇ r /D + r − /D ,/D ∗ ∇ r = ∇ r /D ∗ + r − /D ∗ , /D ∇ r = ∇ r /D + r − /D , (2.9) /D ∗ ∇ t = ∇ t /D ∗ , /D ∇ t = ∇ t /D ,/D ∗ ∇ t = ∇ t /D ∗ , /D ∇ t = ∇ t /D . (2.10)Let W a be a one form. we decompose W a as W a dx a = W dt + (cid:18) − Mr (cid:19) − W dr + W ,A dx A . Here W and W are scalar functions and W is a section of L ( − W a is. We record here the components of the deformation tensor W π ab = ∇ a W b + ∇ b W a in terms of W , W and W .( W π ) tt =2 ∇ t W − Mr W , ( W π ) rr =2 (cid:18) − Mr (cid:19) − ∇ r W − Mr (cid:18) − Mr (cid:19) − W , ( W π ) tr = ∇ r W + (cid:18) − Mr (cid:19) − ∇ t W − Mr (cid:18) − Mr (cid:19) − W , ( W π ) tA = − /D ∗ W + ∇ t W , ( W π ) rA = − (cid:18) − Mr (cid:19) − /D ∗ W + ∇ r W − r W ,/tr ( W π ) =2 /D W + 4 r W , W ˆ π AB = − /D ∗ W . (2.11)Here /tr W π is the trace of W π AB with respect to /g AB and W ˆ π AB is the traceless part of W π AB .Let Φ be a scalar function, a section of L ( − L ( −
2) or more generally a section of productsof these vector bundles, the stress-energy tensor of Φ is defined as T ab [Φ] := ∇ a Φ · ∇ b Φ −
12 ( ∇ c Φ · ∇ c Φ) g ab . Here the · stands for the contraction using the bundle metric. The stress-energy tensor satisfiesthe energy condition that T ab [Φ] X a Y b ≥ X and Y . Moreover, forany vector field X orthogonal to S ( τ, r ), one has ∇ a ( T ab [Φ] X b ) = ∇ X Φ · (cid:3) Φ + T ab [Φ] ∇ a X b . (2.12) As Φ is a scalar function, (2.12) holds without the requirement X being perpendicular to S ( τ, r ).As Φ is a section of L ( − L ( −
2) or their products, (2.12) follows from the fact that the curvaturetwo forms of L ( −
1) and L ( −
2) are supported on the tangent plane of S ( τ, r ), which can bechecked through direct computation. An alternative way to see this is to consider L ( −
1) and L ( −
2) as pull back bundles from S . We compute here the divergence of the red-shfit current.Let Y ( σ ) be the red-shift vector defined in (2.7). For any constant c , one has ∇ a (cid:16) T ab [Φ] Y b ( σ ) − cr | Φ | Y a ( σ ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) r =2 M = σ |∇ v Φ | + 12 M |∇ R Φ | + 2 M ∇ R Φ · ∇ v Φ + σ | / ∇ Φ | + cσ M | Φ | + ∇ Y Φ · (cid:16) (cid:3) Φ − cr Φ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) r =2 M . (2.13)Here we use the coordinate vector fields ∂ v and ∂ R in the coordinate (2.1). See [13] for theoriginal computation.For a one form J a , the divergence theorem in D ( τ , τ ) implies Z Σ τ J · n dvol = Z Σ τ J · n dvol + Z H + ( τ ,τ ) J · L dvol H + + Z I + ( τ ,τ ) J · L dvol I + + Z D ( τ ,τ ) div J dvol. (2.14)Here H + ( τ , τ ) := H + ∩ { τ ≤ τ ≤ τ } is part of the boundary of D ( τ , τ ) and dvol H + is thethree form corresponding to the normal vector L . I + ( τ , τ ) is the boundary of D ( τ , τ ) at nullinfinity (See figure 2.2) and the integral is understood as Z I + ( τ ,τ ) J · L dvol I + := lim v →∞ Z { τ ≤ τ ≤ τ ,v = v } J · L dvol L , and dvol L is the three form corresponding to L . The vector field method in estimating wave equa-tion is choosing a suitable multiplier X and applying the divergence theorem to J a = T ab [Φ] X b to control the behavior of Φ at later time Σ τ by the previous data along Σ τ . See Appendix Afor a brief overview.Recall that K , D and ∂ are collection of vector fields defined in (2.8). For A = D or A = ∂ and j ∈ N , we define |A j Φ | := X X ,X , ··· ,X j ∈A |∇ X ∇ X · · · ∇ X j Φ | , |A ≤ j Φ | := j X i =1 |A i Φ | . For the Killing vector fields K , we instead use Lie derivative and abuse the notation as CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 11 |K j Φ | := X X ,X , ··· ,X j ∈K |L X L X · · · L X j Φ | , |K ≤ j Φ | := j X i =1 |K i Φ | . Energy norms.
Let Φ be a smooth function, section of L ( − L ( −
2) or their products,we define inductively for any j ∈ N ,Φ (0) := s · Φ , Φ ( j ) := (cid:18) − Mr (cid:19) − M s · ∇ L Φ ( j − . Consider the energy norms: F [Φ]( τ ) := Z Σ ′′ τ | ∂ Φ | + M − s − | Φ | dvol + Z Σ ′ τ |∇ L Φ | + | / ∇ Φ | + M − s − | Φ | dvol , (2.15) B [Φ]( τ ) := Z Σ τ s − (cid:16)(cid:0) − s − (cid:1) | ∂ Φ | + (1 − s − ) |∇ r Φ | + M − s − | Φ | (cid:17) dvol , (2.16) F T [Φ]( τ ) := Z Σ ′′ τ (1 − s − ) |∇ L ′ Φ | + |∇ L Φ | + | / ∇ Φ | + M − s − | Φ | dvol + Z Σ ′ τ |∇ L Φ | + | / ∇ Φ | + M − s − | Φ | dvol , (2.17) ¯ B [Φ]( τ ) := Z Σ τ s − (cid:0) | ∂ Φ | + M − s − | Φ | (cid:1) dvol , (2.18) E p, ( j ) L [Φ]( τ ) := Z Σ ′ τ s p − |∇ L Φ ( j ) | dvol , (2.19) E p, ( j ) / ∇ [Φ]( τ ) := Z Σ ′ τ s p − | / ∇ Φ ( j ) | dvol , (2.20) E p, ( j ) L, / ∇ [Φ]( τ ) := E p, ( j ) L [Φ]( τ ) + E p, ( j ) / ∇ [Φ]( τ ) . (2.21)The F [Φ]( τ ) energy is the ˙ H ∩ rL norm of Φ along Σ τ together with the L norm of ∇ n Φ.Except the zeroth order term, F T [Φ]( τ ) energy is the boundary integral along Σ τ when applyingdivergence theorem (2.14) to the T -current T ab [Φ] T b . Compared to F [Φ], F T [Φ] degenerates at r = 2 M for the term |∇ L ′ Φ | because T becomes a null vector. This degeneracy can be removedby considering the red-shift vector Y ( σ ) defined in (2.7). The B [Φ]( τ ) energy appears in the bulkterm in the Morawetz type estimates. It has degeneracy at the photon sphere r = 3 M becauseof trapped geodesics. See subsection 6.3. The non-degenerate version is denoted by ¯ B [Φ]. The E p, (0) L, / ∇ energy appears in the the r p -estimates. See subsection 6.4.We note that we only consider E p, ( j ) L , E p, ( j ) / ∇ and E p, ( j ) L, / ∇ for truncated Φ which is supported in r ≥ R null . Throughout this paper, we take a cut-off function η null ( r ) (depending only on R null ) as follows. Let φ ( s ) be a cut-off function such that φ ( s ) = 1 for s ≥ φ ( s ) = 0 for s ≤ η null ( r ) = φ (( r − R null ) /M ) . To simply the notation, we use ˜Φ to stand for η null ( r )Φ. Remark 2.1.
For a symmetric two tensor h ab or a one form W a , we measure their size bycomponents with respect to vector fields in ∂ . In particular, the F -energy norm of h ab is definedas F [ h ]( τ ) := X X ,X ∈ ∂ F [ h ( X , X )]( τ ) . Other energy norms for h ab or W a are defined similarly. We define below the norms for the source term. For any p ∈ R and τ ≥ τ , letE p source [ G ]( τ , τ ) := M Z D ( τ ,τ ) s p +1 | G | dvol + M Z Σ ′′ τ | G | dvol + M Z Σ ′′ τ | G | dvol + M Z D ′′ ( τ ,τ ) (cid:18) − Mr (cid:19) | ∂G | + (cid:18) − Mr (cid:19) | ∂ r G | dvol. The decay of Φ which satisfies a wave equation with a source term G will be formulate in terms ofthe following initial norms for Φ and spacetime bound on G . We fix a small constant 0 < δ < / k ∈ N , let(2.23) I ( k ) [Φ] := F [ K ≤ k +2 Φ] + X i + i ≤ k +1 i ≤ k E − δ, ( i ) L [ K i ˜Φ] (0) . For any ¯ δ ≥ δ >
0, let(2.24) I source , ¯ δ [ G ] := sup p ∈ [ δ, − ¯ δ ] sup τ ≥ τ ≥ (cid:16) τ M (cid:17) − p − ¯ δ E p source [ K ≤ G ]( τ , τ ) . We record in Appendix A the decay estimates for wave equation obtained through the vectorfield method.2.5. even/odd and spherical harmonic decomposition.
A spherical one form ξ A is said tobe even if /ǫ AB / ∇ A ξ B = 0 and is said to be odd if / ∇ A ξ A = 0, where /ǫ is the Levi-Civita tensorof /g . This is equivalent to the Hodge decomposition on S . Similarly, a traceless symmetric twotensor Ξ AB is called even (odd) if / ∇ A Ξ AB is even (odd). All scalar functions are said to be even.Recall that x α , x β is the coordinate of the quotient space M / S and x A , x B is the coordinateof S . For a spacetime one from W a or a spacetime symmetric two tensor h ab , we decomposethem as W a dx a = W α dx α + W A dx A and h ab dx a dx b = h αβ dx α dx β + h αA ( dx α dx A + dx A dx α ) + ˆ h AB dx A dx B + 12 /trh/g AB dx A dx B . CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 13
We use ˆ h AB and /trh to denote the traceless and the trace of h AB respectively. We consider W α , h αβ and /trh as scalars, W A and h αA as spherical one forms and ˆ h AB as a spherical symmetrictwo tensor. Then W a or h ab is called even or odd if its components above are all even or odd. Inthe case that W a and h ab are axially symmetric, being even (odd) is the same as being symmetric(anti-symmetric) under the isometry φ
7→ − φ . Therefore, the following lemma holds. Lemma 2.2.
Any symmetric two tensor h ab or one form W a can be decomposed into even andodd parts. h ab = h even ab + h odd ab ,W a = W even a + W odd a . Furthermore, h ab is a solution of (1.2) , (1.3) or (1.4) if and only if both h even ab and h odd ab aresolutions separately. Similarly, W a is a solution of (1.5) if and only if both W even a and W odd a aresolutions. Let Y mℓ ( θ, φ ) , ℓ ≥ , | m | ≤ ℓ be the spherical harmonic functions on the unit sphere. For anysmooth function ψ ( t, r, θ, φ ), we can perform spherical harmonic decomposition for each ( t, r )and express ψ as ψ ( t, r, θ, φ ) = ∞ X ℓ =0 X | m |≤ ℓ ψ mℓ ( t, r ) · Y mℓ ( θ, φ ) . Similarly, for any spherical one form ξ A , we decompose it as ξ A = ∞ X ℓ =1 X | m |≤ ℓ ξ even mℓ ( t, r ) · / ∇ A Y mℓ + ∞ X ℓ =1 X | m |≤ ℓ ξ odd mℓ ( t, r ) · /ǫ BA / ∇ B Y mℓ . For any spherical symmetric traceless two tensor Ξ AB ,Ξ AB = ∞ X ℓ =2 X | m |≤ ℓ Ξ even mℓ ( t, r ) · ˆ / ∇ AB Y mℓ + ∞ X ℓ =2 X | m |≤ ℓ Ξ odd mℓ ( t, r ) ·
12 ( /ǫ CA / ∇ CB Y mℓ + /ǫ CB / ∇ CA Y mℓ ) . In this paper, we focus only on the even tensors ξ and Ξ for which ξ odd mℓ ≡ odd mℓ ≡
0. Weoften work with a fixed mode ψ mℓ ( t, r ) Y mℓ ( θ, φ ). In this situation, / ∆ is equivalent to multiplyingthe number r − ℓ ( ℓ + 1). We define λ = λ ( ℓ ) and Λ = Λ( ℓ ) by ℓ ( ℓ + 1) = 2 λ + 2 = Λ . (2.25)Using this notation, we have a simple lemma below. Lemma 2.3. /D ∗ ∞ X ℓ =0 X | m |≤ ℓ ψ mℓ Y mℓ = ∞ X ℓ =1 X | m |≤ ℓ − ψ mℓ / ∇ A Y mℓ ,/D ∗ ∞ X ℓ =1 X | m |≤ ℓ ξ mℓ / ∇ A Y mℓ = ∞ X ℓ =2 X | m |≤ ℓ − ξ mℓ ˆ / ∇ AB Y mℓ , and /D ∞ X ℓ =1 X | m |≤ ℓ ξ mℓ / ∇ A Y mℓ = ∞ X ℓ =1 X | m |≤ ℓ − Λ r − ξ mℓ Y mℓ ,/D ∞ X ℓ =2 X | m |≤ ℓ Ξ mℓ ˆ / ∇ AB Y mℓ = ∞ X ℓ =2 X | m |≤ ℓ − λr − Ξ mℓ / ∇ A Y mℓ . We denote the projection for a scalar function ψ as ψ ℓ ≥ ℓ := X ℓ ≥ ℓ X | m |≤ ℓ ψ mℓ ( t, r ) Y mℓ ( θ, φ ) ,ψ ℓ := X | m |≤ ℓ ψ mℓ ( t, r ) Y mℓ ( θ, φ ) . The projection of a section in L ( −
1) or L ( −
2) is defined similarly. We say ψ is supported on ℓ ≥ ℓ if ψ = ψ ℓ ≥ ℓ .We adapt the notation A ≤ s B if for any τ ∈ R and any r ≥ M , Z S ( τ,r ) Advol S ≤ Z S ( τ,r ) Bdvol S . For example, if ψ = ψ ℓ ≥ ℓ , we have r − ℓ ( ℓ + 1) | ψ | ≤ s | / ∇ ψ | by Poincar´e inequality. AlsoWe use the notation A . B for two non-negative quantities A and B to indicate that there isa constant C such that A ≤ CB . The constant C may depend on δ , the small constant we fixthroughout this paper, and the defining function of Σ in r ∈ [2 M, M ]. We also use A ≈ B to denote the relation A . B and B . A . If the constant C depends on other quantities, weindicate the dependence using subscript. For instance, the notation . R null emphasizes that theconstant depends on R null .2.6. main theorems for ℓ ≥ . Now we are ready to state the main theorems for ℓ ≥
1. Theresults for ℓ = 0 have more involved assumptions and we postpone them to section 8. The firstone concerns the even solutions of (1.3) and (1.4) supported on ℓ ≥ Theorem 2.4.
Let h ab be an even solution of (1.3) and (1.4) .(1) Further assume that h ab is supported on ℓ ≥ and that Decay ℓ ≥ [ h ] defined below is finite.Then for any p ∈ [ δ, − δ ] and τ ≥ , we have F [ h ]( τ ) + E p, (0) L [ h ]( τ ) + Z ∞ τ ¯ B [ h ]( τ ′ ) + E p − , (0) L, / ∇ [ h ]( τ ′ ) dτ ′ . (cid:16) τM (cid:17) − p +3 δ Decay ℓ ≥ [ h ] . Here
Decay ℓ ≥ [ h ] := I (0) [ s ( M ∂ ) ≤ K ≤ h, s ( M ∂ ) ≤ K ≤ h, s D ≤ ( M ∂ ) ≤ K ≤ h ] + I (1) [ s ( M ∂ ) ≤ K ≤ h ] . (2) Assume that h ab is supported on ℓ = 1 and that Decay ℓ =1 [ h ] defined below is finite. Thenfor any p ∈ [ δ, − δ ] and τ ≥ , we have CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 15 F [ h ]( τ ) + E p, (0) L [ h ]( τ ) + Z ∞ τ F [ h ]( τ ′ ) + E p − , (0) L, / ∇ [ h ]( τ ′ ) dτ ′ . (cid:16) τM (cid:17) − p +3 δ Decay ℓ =1 [ h ] . Here
Decay ℓ =1 [ h ] := I (0) [ s ( M ∂ ) ≤ K ≤ h, D ≤ K ≤ h ] + I (1) [ K ≤ h ] . The second theorem concerns the even solutions of (1.5).
Theorem 2.5.
Let W a be an even solution of (1.5) . Further assume that W a is supported on ℓ ≥ and that Decay ℓ ≥ [ W ] defined below is finite. Then for any p ∈ [ δ, − δ ] and τ ≥ , wehave F [ W ]( τ ) + E p, (0) L [ W ]( τ ) + Z ∞ τ B [ W ]( τ ′ ) + E p − , (0) L, / ∇ [ W ]( τ ′ ) dτ ′ . (cid:16) τM (cid:17) − p +3 δ Decay ℓ ≥ [ W ] . Here
Decay ℓ ≥ [ W ] := I (0) [ s ( M ∂ ) ≤ K ≤ W, D ≤ ( M ∂ ) ≤ K ≤ W ] + I (1) [( M ∂ ) ≤ K ≤ W ] . Basic Lemmas
We prove in this section some useful lemmas that we need. Once we have an estimate forthe solution ψ of a wave equation, for instance Proposition (A.5), we can commute the equationwith Killing vectors in K to control K ψ . However K doesn’t generate the full tangent space of M . Fortunately, as long as r > M , the missing direction can be recovered using (cid:3) ψ , which isthe content of the lemma below. Lemma 3.1.
Let ψ be a smooth function. Then for any k ≥ and ¯ r > M , we have for r ≥ ¯ r , (3.1) (cid:12)(cid:12) ( M ∂ ) ≤ k u (cid:12)(cid:12) . k, ¯ r (cid:0) | u | + | M ∂ K ≤ k − u | + M | ( M ∂ ) ≤ k − (cid:3) u | (cid:1) . Proof. As k = 1, (3.1) holds trivially. Now we prove (3.1) by induction. Assume (3.1) holds for k −
1. By the commutation relation in ∂ , | ( M ∂ ) ≤ k ψ | . | ( M ∂ ) ≤ k − · M T ψ | + M r | ( M ∂ ) ≤ k − · Ω ψ | + | ( M L ′ ) k ψ | + | ( M ∂ ) ≤ k − ψ | . Applying the induction hypothesis to
M T u, we have (cid:12)(cid:12) ( M ∂ ) ≤ k − · M T ψ (cid:12)(cid:12) . | M ∂ K ≤ k − · M T ψ | + M | ( M ∂ ) ≤ k − (cid:3) · M T ψ | + | M T ψ | . | M ∂ K ≤ k − ψ | + M | ( M ∂ ) ≤ k − (cid:3) ψ | . Similarly, M r (cid:12)(cid:12) ( M ∂ ) ≤ k − · Ω ψ (cid:12)(cid:12) . M r (cid:0) | M ∂ K ≤ k − · Ω ψ | + M | ( M ∂ ) ≤ k − (cid:3) · Ω ψ | + | Ω ψ | (cid:1) . | M ∂ K ≤ k − ψ | + M | ( M ∂ ) ≤ k − (cid:3) ψ | . To deal with (
M L ′ ) k ψ , we note that from (2.2),( L ′ ) = (cid:18) − Mr (cid:19) − (cid:3) + 2 T L ′ + 2 r (cid:18) − Mr (cid:19) L ′ − r T − r X i =1 Ω i ! . Therefore as r ≥ ¯ r , | ( M L ′ ) k ψ | . ¯ r M | ( M L ′ ) k − (cid:3) ψ | + | ( M L ′ ) k − M T ψ | + M r | ( M ∂ ) ≤ k − Ω ψ | + | ( M ∂ ) ≤ k − ψ | . The first term appears on the right hand side of (3.1), the second and the third terms are alreadyestimated above and the last term can be controlled by induction hypothesis. Adding the abovefinishes the proof. (cid:3)
In using the divergence theorem (2.14), it’s often to only have ˙ H type norm without thezeroth order term and we rely on the Hardy inequality to recover such base term. Lemma 3.2.
For any positive number q > and any Lipschitz function ψ , we have M q − sup r ≥ R null Z S ( τ,r ) s − q | ψ | dvol S + M − Z Σ ′ τ s − q − | ψ | dvol ≤ M q − Z S ( τ,R null ) s − q | ψ | dvol S + 8 q − Z Σ ′ τ s − q − |∇ L ψ | dvol . (3.2) For any negative number q < , we have M | q | − Z S ( τ,R null ) s − q | ψ | dvol S + M − Z Σ ′ τ s − q − | ψ | dvol ≤ M | q | − lim inf r →∞ Z S ( τ,r ) s − q | ψ | dvol S + 8 | q | − Z Σ ′ τ s − q − |∇ L ψ | dvol . (3.3) Proof.
For any Lipchitz function f ( r ), a < b and q = 0 we have Z ba r − q − f ( r ) dr = q − r − q f ( r ) (cid:12)(cid:12)(cid:12)(cid:12) r = ar = b + q − Z ba r − q f ( r ) f ′ ( r ) dr. As q >
0, by Cauchy-Schwarz,2 q − b − q f ( b ) + Z ba r − q − f ( r ) dr ≤ q − a − q f ( a ) + 4 q − Z ba r − q +1 ( f ′ ( r )) dr. Applying the above inequality with f ( r ) = ψ ( τ, r, θ, φ ), a = R null , b ∈ ( R null , ∞ ) and integratingalong S , we have2 q − sup r ≥ R null Z S ( τ,r ) r − q | ψ | dvol S + Z ∞ R null Z S ( τ,r ) r − q − | ψ | · r dvol S dr ≤ q − Z S ( τ,R null ) r − q | ψ | dvol S + 4 q − Z ∞ R null Z S ( τ,r ) r − q − |∇ L ψ | · (cid:18) − Mr (cid:19) − r dvol S dr. From R null ≥ M , r dvol S dr = (cid:18) − Mr (cid:19) dvol ≥ dvol , (cid:18) − Mr (cid:19) − r dvol S dr = (cid:18) − Mr (cid:19) − dvol ≤ dvol . Then (3.2) follows as 5 / , / <
2. Similar argument yields (3.3). (cid:3)
CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 17
This lemma has several useful corollaries.
Corollary 3.3.
Let ψ be a smooth function and ˜ ψ = η null ( r ) ψ with the cut-off function η null ( r ) defined in (2.22) . For any p < , E p, (0) [ ˜ ψ ]( τ ) ≥ (1 − p )4 sup r ≥ R null Z S ( τ,r ) M s p − | ˜ ψ (0) | dvol S + (1 − p ) Z Σ ′ τ M − s p − | ˜ ψ (0) | dvol. Proof.
This follows by taking q = 1 − p > ψ = ˜ ψ (0) together with ˜ ψ (cid:12)(cid:12)(cid:12)(cid:12) r = R null = 0. (cid:3) Corollary 3.4.
Let ψ be a smooth function and ˜ ψ = η null ( r ) ψ with the cut-off function η null ( r ) defined in (2.22) . Suppose E p, (0) [ ˜ ψ ]( τ ) < ∞ for some p > and τ ∈ R , then Z Σ ′ τ |∇ L ψ | dvol ≥ Z Σ ′ τ M − s − | ψ | dvol . Proof.
By (3.3) with q = −
1, it’s sufficient to checklim inf r →∞ Z S ( τ,r ) s | ψ | dovl S = 0 , which follows from the assumption E p, (0) [ ˜ ψ ]( τ ) < ∞ and Corollary 3.3. (cid:3) Corollary 3.5.
Let ψ be a smooth function and ˜ ψ = η null ( r ) ψ with the cut-off function η null ( r ) defined in (2.22) . We have F [ ψ ]( τ ) . R null B [ K ≤ ψ ]( τ ) + E , (0) L, / ∇ [ ˜ ψ ]( τ ) ,F [ ψ ]( τ ) . R null ¯ B [ ψ ]( τ ) + E , (0) L, / ∇ [ ˜ ψ ]( τ ) , (3.4) Proof.
Clearly as long as r is bounded, the integrand on the left hand side is controlled by theone on the right hand side. Also, from Corollary 3.3 with p = 0, we can control the base termfrom E , (0) [ ˜ ψ ]( τ ). The only term on the left which is not controlled for large r is |∇ L ψ | , whichcan be controlled by M − s − | ψ | and s − |∇ L ( sψ ) | . (cid:3) linearized Gravity under Harmonic Gauge Let h ab be an even tensor in the Schwarzschild spacetime satisfying (1.3) and (1.4). Westart by decomposing h into components and use (1.4) to obtain red-shift and r p estimate nearhorizon and null infinity respectively. The result is Proposition 4.1. Next, in sebsection 4.2 themode h ℓ ≥ is considered. We perform a gauge transformation to the Regge-Wheeler gauge h RW ,which is governed by the Zerilli quantity ψ Z satisfying the Zerilli equation (4.16). The mode h ℓ = 1 equals a deformation tensor − W π and the relationship between h ℓ =1 and W is discussedin subsection 4.3. decomposition of h . The main result of this subsection is:
Proposition 4.1.
Let h ab be an even solution of (1.4) . There exist R H ≥ M and r rs, H ∈ (2 M, r + rs ) such that the following statement holds. Suppose R null ≥ R H then for any p ∈ [ δ, − δ ] and τ ≥ τ , F [ h ]( τ ) + E p, (0) L [˜ h ]( τ ) + M − Z τ τ ¯ B [ h ]( τ ) + E p − , (0) L, / ∇ [˜ h ]( τ ) dτ . R null F [ h ]( τ ) + E p, (0) L [˜ h ]( τ ) + M − Z D ( τ ,τ ) ∩{ r rs, H ≤ r ≤ R nul + M } | ( M ∂ ) ≤ h | dvol. (4.1) Here ˜ h = η null h and η null ( r ) is the cut-off function defined in (2.22) . This will be proved using standard r p and red-shift estimate of Dafermos-Rodnianski to asuitable decomposition of h . From simplicity, we only prove the case h = h ℓ ≥ . The lower modes ℓ = 1 and ℓ = 0 can be proved similarly. See the remark at the end of this subsection.We split the even tensor h ab into seven components as following. There are four scalarfunctions H = 12 h tt + (cid:18) − Mr (cid:19) h rr ! + Mr /trh, H = 12 /trh, H = 12 (cid:18) − Mr (cid:19) − h tt − (cid:18) − Mr (cid:19) h rr ! , H = (cid:18) − Mr (cid:19) h tr + Mr /trh, two spherical 1-forms H = (cid:18) − Mr (cid:19) h rA dx A , H = h tA dx A , and one spherical symmetric traceless 2-tensor H = 1 √ h AB dx A dx B . Proposition 4.2.
Suppose h ab satisfies (1.3) and (1.4) , then H = ( H , . . . , H ) satisfies (4.2) (cid:3) H = A H + B H + U H . Here A is a seven by seven matrix: CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 19 A = 2 r (cid:0) − s − (cid:1) − s − − s − − s − − s − − − s − − s − − − s − − s − / − s − − s − s − − s − s − − s − − s −
00 0 0 − s − / − s − .B is a first order angular operator: B = 2 r − s − ) /D − /D − /D /D ∗ − /D ∗ ( − s − ) /D ∗ √ − s − ) /D √ /D ∗ − s − /D − s ) /D − s − /D ∗ s − /D ∗ −√ s − /D /D ∗ . And U is a first order derivative along L : U = 2 Mr (cid:18) − Mr (cid:19) − − ∇ L · r ∇ L · r
00 0 0 −∇ L · r ∇ L · r . The derivation of this equation can be found in Appendix C. The leading terms in A and B are self-adjoint and we denote them by A main and B main respectively. A main = 2 r − − − − / / .B main = 2 r /D − /D − /D /D ∗ − /D ∗ − /D ∗ √ /D √ /D ∗ /D /D . Denote the remaining terms as A sub = s ( A − A main ) and B sub = s ( B − B main ). They decayfaster in r as | A sub H | . M − s − | H | , | B sub H | . M − s − | / ∇ H | , | U H | . M − s − |∇ L ( s H ) | . (4.3) Lemma 4.3.
Suppose H = H ℓ ≥ , then | / ∇ H | + H · ( A main + B main ) · H ≥ s . Further assume H = H ℓ ≥ , then we have | / ∇ H | + H · ( A main + B main ) · H & s | / ∇ H | . Proof.
Through spherical harmonic decomposition, it’s sufficient to prove the lemma for H sup-ported on one spherical harmonic mode Y mℓ . In this situation, angular operators become multi-plication. From Lemma 2.3, in terms of the basis Y mℓ , r Λ − / / ∇ A Y mℓ and r λ − / Λ − / ˆ / ∇ AB Y mℓ , B main is of the form B main = 2 r − Λ − / − / − / − Λ − / Λ − / Λ − / − (2 λ ) − / − (2 λ ) − / − Λ − / − Λ − / , Here Λ and λ are defined in (2.25). Similarly, | / ∇ H | = s H · O · H , where O = 1 r Λ 0 0 0 0 0 00 Λ 0 0 0 0 00 0 Λ 0 0 0 00 0 0 Λ − − − . The eigenvalues of r ( A main + B main + O ) are { Λ , ( ℓ − ℓ − , ( ℓ + 2)( ℓ + 3) , ℓ ( ℓ − , ( ℓ + 1)( ℓ + 2) } , where Λ has multiplicity 3. Hence A main + B main + O is non-negative definite for ℓ ≥ O as ℓ ≥ (cid:3) Lemma 4.4.
Let H = H ℓ ≥ be a solution of (4.2) . There exists R H ≥ M such that if R null ≥ R H , for any p ∈ [ δ, − δ ] and τ ≥ τ ≥ we have E p, (0) L [ ˜ H ]( τ ) + M − Z τ τ E p − , (0) L, / ∇ [ ˜ H ]( τ ) dτ + M − Z D ( τ ,τ ) s − − δ |∇ L ˜ H | dvol . R null F [ h ]( τ ) + E p, (0) L [ ˜ H ]( τ ) + M − Z D ( τ ,τ ) ∩{ R null ≤ r ≤ R null + M } | ( M ∂ ) ≤ H | dvol. (4.4) CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 21
Here ˜ H = η null H and η null ( r ) is the cut-off function defined in (2.22) .Proof. Consider the current J p H ,a := s p − (cid:18) − Mr (cid:19) − (cid:18) T ab [ s ˜ H ] L b −
12 ( s ˜ H ) · ( A main + B main + 2 M/r ) · ( s ˜ H ) L a (cid:19) . Clearly, Z Σ ′ τ J p H · L dvol ≈ E p, (0) L [ ˜ H ]( τ ) . We will apply the divergence theorem 2.14 to J p H in the region D ( τ , τ ). To make sure thecontribution along null infinity is non-negative , we compute J p H · L = s p − (cid:16) | / ∇ ( s ˜ H ) | + ( s ˜ H ) · ( A main + B main + 2 M/r ) · ( s ˜ H ) (cid:17) , which is non-negative (after integrated along S ) from Lemma 4.3. We start to estimate div J p H .By direct calculation,div J p H = M − s p − (cid:18) p (cid:18) − Mr (cid:19) − Mr (cid:19) (cid:18) − Mr (cid:19) − |∇ L ( s ˜ H ) | + M − s p − (cid:16) − p (cid:17) | / ∇ ( s ˜ H ) | − M − s p − ( s ˜ H ) · (cid:0) r (cid:2) ∇ r , ( A main + B main + 2 M/r ) (cid:3) + p ( A main + B main + 2 M/r ) (cid:1) · ( s ˜ H )+ s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ H ) · (cid:3) ˜ H − s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ H ) · ( A main + B main ) · ˜ H − s p − (cid:18) − Mr (cid:19) − ˜ H · ( A main + B main ) · ∇ L ( s ˜ H ) . (4.5)From [ ∇ r , r ( A main + B main )] = 0, we have r (cid:2) ∇ r , ( A main + B main + 2 M/r ) (cid:3) + p ( A main + B main + 2 M/r )= − (2 − p )( A main + B main ) − (6 − p ) Mr . Therefore by Lemma 4.3, after integrated along S , M − s p − (cid:16) − p (cid:17) | / ∇ ( s ˜ H ) | − M − s p − ( s ˜ H ) · (cid:18) r (cid:2) ∇ r , ( A main + B main + 2 M/r ) (cid:3) + p (( A main + B main + 2 M/r )) (cid:19) · ( s ˜ H )is non-negative and is bounded below by M − s p − | / ∇ ( s ˜ H ) | if H = H ℓ ≥ . For H = H ℓ =2 , we canstill get M − s p − | / ∇ ( s ˜ H ) | through Corollary 3.3 as M − E p − , (0) L [ ˜ H ]( τ ) ≥ (2 − p ) M − Z Σ ′ τ s p − | s ˜ H | svol & M − E p − , (0) / ∇ [ ˜ H ]( τ ) . Since A main and B main are self-adjoint, the last three terms in (4.5) can be combined as s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ H ) · ( (cid:3) − A main − B main ) · ˜ H = s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ H ) · (cid:16) s − ( A sub + B sub ) ˜ H + η null · U H + (cid:3) η null · H + 2 ∇ η null · ∇ H (cid:17) . The A sub and B sub terms decay faster in r and can be estimated by (4.3) and Cauchy-Schwarz: s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ H ) · s − ( A sub + B sub ) ˜ H = s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ H ) · ( A sub + B sub )( s ˜ H ) . ǫ · M − s p − |∇ L ( s ˜ H ) | + 1 ǫ · M − s p − | s ˜ H | + 1 ǫ · M − s p − | / ∇ ( s ˜ H ) | , where ǫ is any positive real. Similarly, s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ H ) · η null ( r ) U H . ǫ · M − s p − |∇ L ( s ˜ H ) | + 1 ǫ · M − s p − |∇ L ( s ˜ H ) | + 1 ǫ · M − s p − |∇ L η null | | s H | . We take ǫ > ǫ · M − s p − |∇ L ( s ˜ H ) | , and then pick R H large enough to absorb the terms ǫ − · M − s p − |∇ L ( s ˜ H ) | , ǫ − · M − s p − | / ∇ ( s ˜ H ) | and ǫ − · M − s p − | s ˜ H | above. Therefore we have for some constant C > Z D ( τ ,τ ) div J p H dvol + CM Z D ( τ ,τ ) s p +1 (cid:0) | (cid:3) η null · H | + |∇ η null · ∇ H | + M − s − |∇ L η null | | H | (cid:1) dvol & M − Z D ( τ ,τ ) s p − |∇ L ( s ˜ H ) | + s p − | / ∇ ( s ˜ H ) | dvol. Applying the divergence theorem (2.14) yields E p, (0) L [ ˜ H ]( τ ) + M − Z τ τ E p − , (0) L, / ∇ [ ˜ H ]( τ ) dτ . R null E p, (0) L [ ˜ H ]( τ ) + M − Z D ( τ ,τ ) ∩{ R null ≤ r ≤ R null + M } | ( M ∂ ) ≤ H | dvol. To add s − − δ |∇ L ˜ H | term, we consider( J H ,δ ) a := s − δ (cid:18) T ab [ ˜ H ] −
12 ˜ H · ( A main + B main ) · ˜ H g ab (cid:19) T b . (4.6)From Lemma 4.3, J H ,δ · n τ and J H ,δ · L are non-negative after integrated along S . Moreover, CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 23 ≤ Z Σ τ J H ,δ · ndvol . F [ H ]( τ ) . The divergence of J H ,δ isdiv J H ,δ = s δ M − s − − δ |∇ L ˜ H | − δ M − s − − δ |∇ L ˜ H | + η null ( r ) s − δ ∇ t H · ( (cid:3) η null ( r ) · H + 2 ∇ η null ( r ) · ∇ H )+ η null ( r ) s − δ ∇ t ˜ H · ( s − A sub + s − B sub + U ) H . The second line will be grouped into M − | ( M ∂ ) ≤ H | · χ [ R null ,R null + M ] . The last line, by Cauchy-Schwarz, can be bounded as (cid:12)(cid:12) η null ( r ) s − δ ∇ t ˜ H · ( s − A sub + s − B sub + U ) H (cid:12)(cid:12) . R null ǫ · M − s − − δ ( |∇ L ˜ H | + |∇ L ˜ H | )+ 1 ǫ · M − s − − δ ( |∇ L ( s ˜ H ) | + | / ∇ ( s ˜ H ) | + M − s − | s ˜ H | )+ 1 ǫ · M − | ( M ∂ ) ≤ H | · χ [ R null ,R null + M ] . By taking ǫ >
C > J H ,δ ≥ C · M − s − − δ |∇ L ˜ H | − C · M − s − − δ ( |∇ L ( s ˜ H ) | + | / ∇ ( s ˜ H ) | + M − s − | s ˜ H | ) − C · M − | ( M ∂ ) ≤ H | · χ [ R null ,R null + M ] . Because for any p ∈ [ δ, − δ ], p − ≥ − δ > − − δ , the second line can be absorbed into theintegrand of E p − , (0) L, / ∇ [ ˜ H ] through further enlarging the value of R H . Then the result follows byapplying divergence theorem (2.14) to J p H + ǫJ H ,δ with ǫ > (cid:3) We now turn to the red-shift estimate near the horizon H + . To apply the red-shift estimate, theonly thing we need to check is that the coefficient of ∇ L ′ H in (cid:3) H is non-negative (actually, nega-tive part bounded by the surface gravity of Schwarzschild is enough.) The above decompositionis actually not regular on the horizon as they don’t generate the full vector space T ∗ M ⊗ s T ∗ M .We instead consider H ′ := (cid:0) − Mr (cid:1) − ( H − H ) and H ′ := (cid:0) − Mr (cid:1) − ( H − H ). They satisfythe equation (cid:3) H ′ = 4 Mr ∇ L ′ H ′ + 2 r /D H ′ + 2 r H + 2 r H + 2 r H ′ , (cid:3) H ′ = 2 Mr ∇ L ′ H ′ − r /D ∗ H + 2 r /D ∗ H + 2 √ r /D H + 2 r (cid:18) − Mr (cid:19) /D ∗ H ′ − r H + 1 r (cid:18) − Mr (cid:19) H ′ . Let H ′ = ( H , H , H , H , H , H ′ , H ′ ) and consider the red-shift current J H ′ ,a := T ab [ H ′ ] · ( Y ( σ ) + η rs T ) b − r | H ′ | · ( Y ( σ ) + η rs T ) a , (4.7)where σ is a positive number to be determined, η rs is the red-shift cut-off function and Y ( σ ) isthe red-shift vector defined in (2.6) and (2.7). From (2.13), on the horizon r = 2 M we havediv J H ′ (cid:12)(cid:12)(cid:12)(cid:12) r =2 M = σ |∇ v H ′ | + 12 M |∇ R H ′ | + 2 M ∇ R H ′ · ∇ v H ′ + σ | / ∇ H ′ | + σ M | H ′ | + ∇ T + Y H ′ · ( (cid:3) H ′ − r H ′ ) . Since the coefficient of ∇ L ′ H in (cid:3) H ′ is non-negative and Y ( σ ) = 2 L ′ on the horizon, we havefrom Cauchy-Schwarz for some constant C > ∇ Y + T H ′ · ( (cid:3) H ′ − r H ′ ) + 14 M |∇ R H ′ | (cid:12)(cid:12)(cid:12)(cid:12) r =2 M ≥ − C (cid:18) M − |∇ v H ′ | + M − | / ∇ H ′ | + M − | H ′ | (cid:19) , By choosing σ >> M − , we obtaindiv J H ′ (cid:12)(cid:12)(cid:12)(cid:12) r =2 M & M − | ( M ∂ ) ≤ H ′ | . Because of continuity, there exists r rs, H ∈ (2 M, r + rs ) such that the above estimate holds in[2 M, r rs, H ]. Therefore through the divergence theorem (2.14) of J H ′ , we obtain M − Z Σ τ | ( M ∂ ) ≤ H ′ | · χ [2 M,r rs, H ] dvol + M − Z D ( τ ,τ ) | ( M ∂ ) ≤ H ′ | · χ [2 M,r rs, H ] dvol . M − Z Σ τ | ( M ∂ ) ≤ H ′ | · χ [2 M,r + rs ] dvol + M − Z D ( τ ,τ ) | ( M ∂ ) ≤ H ′ | · χ [ r rs, H ,r + rs ] dvol. (4.8) proof of Proposition 4.1. From the view of (4.4), we have controlled E p, (0) L [˜ h ]( τ ) and M − R τ τ E p, (0) L [˜ h ]( τ ) dτ .From (4.8), the part of F [ h ]( τ ) and the integral of M − R τ τ ¯ B [ h ]( τ ) dτ near the horizon r ∈ [2 M, r rs, H ] is also bounded by the right hand side of (4.1). Furthermore, from Corollary 3.3, Z τ τ E p − , (0) L, / ∇ [ ˜ H ]( τ ) dτ + Z D ( τ ,τ ) s − − δ |∇ L ˜ H | dvol ≈ Z D ( τ ,τ ) s p − ( |∇ L ˜ H | + | / ∇ ˜ H | + M − s − | ˜ H | ) + s − − δ |∇ L ˜ H | dvol, which bounds the integrand of ¯ B [ h ]( τ ) in r ≥ R null + M . As the integrand of ¯ B [ h ]( τ ) in r ∈ [ r rs, H , R null + M ] can be bounded by the right hand side of (4.1), the only term on the lefthand side of (4.1) which is not controlled yet is Z Σ ′′ τ ∩{ r ≥ r rs, H } | ∂h | + M − s − | h | dvol + Z Σ ′ τ |∇ L h | + | / ∇ h | + M − s − | h | dvol . This can be bounded by F T [ H ] defined in (2.17). We can add this term by considering thecurrent ( J H ,T ) a := (cid:18) T ab [ H ] − r | H | g ab (cid:19) T b . (4.9) CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 25
Clearly, Z Σ τ J H ,T · ndvol ≈ F T [ H ] . We estimate its divergence as | div J H ,T | . |∇ t H | (cid:18) M s | M ∂ H | + 1 M s | H | (cid:19) . M − | ( M ∂ ) ≤ H | · χ [2 M,R null + M ] + (cid:18) M − s − − δ |∇ t H | + M − s − δ ( |∇ L ( s H ) | + | / ∇ ( s H ) | + M − s − | s H | ) (cid:19) · χ [ R null + M, ∞ ) . The second term already appears in the the bulk term of (4.4). Then we apply the divergencetheorem (2.14) to ǫJ H ,T with ǫ > | div J H ,T | can be absorbed into (4.8)and (4.4) in r ∈ [2 M, r rs, H ] and in r ∈ [ R null + M, ∞ ) respectively. Adding this estimate togetherwith (4.8) and (4.4) yields the result. (cid:3) Remark 4.5.
The equation for H ℓ =1 is the same as the ℓ ≥ case except that there is no H .In particular, the leading term of A ℓ =1 and B ℓ =1 are r − − − − / / . r /D − /D − /D /D ∗ − /D ∗ − /D ∗ /D /D . We verify that in this case the eigenvalue of r ( A ℓ =1 + B ℓ =1 + O ℓ =1 ) is { , , , , , } . Inparticular, A ℓ =1 + B ℓ =1 + O ℓ =1 is semi-positive definite. Therefore, (4.4) and (4.8) and thenProposition 4.1 can be proved by the same argument.Similarly, as ℓ = 0 there’s no H and H . B ℓ =0 = 0 and the leading term of A ℓ =0 is r − − − − , which is still semi-positive definite with eigenvalues { , , , } . relation to Regge-Wheeler gauge. Throughout this subsection we assume h = h ℓ ≥ .An even solution of (1.2), denoted by h RW , is said to be in the Regge-Wheeler gauge if it issupported on ℓ ≥ h RW AB = 0 , h RW tA = h RW rA = 0 . For any even solution of (1.2) supported on ℓ ≥
2, denoted by h ab , there exists a unique even vector field W , which is also supported on ℓ ≥
2, such that h + W π = h RW . We decompose W into components as W = W dt + W (cid:18) − Mr (cid:19) − dr + W ,A dx A , where W and W are scalars and W is an even spherical one form. As /D ∗ and /D ∗ are invertiblefor ℓ ≥
2, They are uniquely determined through (2.9), (2.10) and (2.11) as /D ∗ /D ∗ W = /D ∗ h tA + 12 ∇ t ˆ h AB , (4.11) /D ∗ /D ∗ W = (cid:18) − Mr (cid:19) /D ∗ h rA + 12 (cid:18) − Mr (cid:19) ∇ r ˆ h AB , (4.12) /D ∗ W = 12 ˆ h AB . (4.13) Remark 4.6.
For the mode ℓ = 1 , any even solution h ℓ =1 of (1.2) equals − W π for a uniqueeven vector field W supported on ℓ = 1 . This decomposition was done in the work of Zerilli [39] .See also [30] . In this case W satisfies the equation (1.5) provided h ℓ =1 solves (1.3) . However,the relation between h and W is not given by (4.11) , (4.12) and (4.13) . We will treat thiscase separately in subsection 4.3. For the mode ℓ = 0 , any solution h ℓ =0 of (1.2) is a linearcombination of the mass perturbation K (defined in (2.4) ) and a deformation tensor W π . Wewill discuss this case in section 8. Define the operator / ∆ Z as in [24, 25] by / ∆ Z := / ∆ + 2 r − Mr . (4.14)The Zerilli quantity is defined as ψ Z := − r − (cid:18) − Mr (cid:19) / ∆ − / ∆ − Z ∂ r /trh − r − / ∆ − /trh + 4 r − (cid:18) − Mr (cid:19) / ∆ − / ∆ − Z h rr − r − (cid:18) − Mr (cid:19) / ∆ − Z ( /D ∗ ) − h rA + r − ( /D ∗ ) − ( /D ∗ ) − ˆ h AB . (4.15)Zerilli [38] first defined this quantity and derived the so called Zerilli equation, a wave equationwith potential depending on ℓ . Precisely, as h ab satisfies linearized Einstein equation (1.2), r − ψ Z satisfies.(4.16) (cid:3) ( r − ψ Z,ℓ ) − V Z,ℓ ( r − ψ Z,ℓ ) = 0 , CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 27 where V Z,ℓ = − Mr (2 λ + 3)(2 λ + 3 M/r )( λ + 3 M/r ) , and λ ( ℓ ) is defined as (2.25). Moreover, ψ Z is a gauge-invariant quantity as for any vector field W = W ℓ ≥ , ψ Z [ W π ] = 0. Furthermore, h RW can be expressed in terms of ψ Z as g αβ h RW αβ = 0 , ˆ h RW αβ = ˆ ∇ αβ ( rψ Z ) + 12 Mr / ∆ − Z ∇ α r ˆ ⊗ s ∇ β ψ Z , and /trh RW = − r / ∆ ψ Z + 2 (cid:18) − Mr (cid:19) ∂ r ψ Z + 12 Mr (cid:18) − Mr (cid:19) / ∆ − Z ψ Z . These identities was derived in [39]. See also [24, Theorem 7.2.3] for a modern treatment. Inparticular, | ( M ∂ ) ≤ m h RW | . m | s ( M ∂ ) ≤ m +2 r − ψ Z | . (4.17)Suppose ψ Z = 0, then h RW = 0 and h = − W π . In this case (1.3) is equivalent to tensorialwave equation (1.5). As ψ Z = 0, the Zerilli quantity ψ Z becomes the source term in (1.5). In thefollowing we rewrite (1.5) in terms of three components W , W and W . We fix an integer ℓ ≥ ℓ and denote ψ Z,ℓ , W ,ℓ ,etc. simply by ψ Z , W , etc.Define the quantities S W := − (cid:18) − Mr (cid:19) − ∇ t W + 2 ∇ r W + 2 /D W + 4 r W , (4.18) P even := − r ∇ r W + (cid:18) − Mr (cid:19) − · r ∇ t W − W , (4.19)and Q even := /D ∗ W + (cid:18) − Mr (cid:19) ∇ r W + (cid:18) − Mr (cid:19) r W − r /D ∗ S W − r /D ∗ / ∆ Z ( r − ψ Z ) . (4.20)Here S W is the trace of the deformation tensor W π and P even as well as Q even are componentsof the two form dW provided S W and ψ Z are zero. Let S := trh be the trace of h ab and we notethat S = trh RW − S W . Proposition 4.7.
The quantities W , W , W , S, P even and Q even (to be precise, their projectionon the mode ℓ ) satisfy the wave equations: (4.21) (cid:3) W + 2 Mr W = − Mr P even + 2 λr − λM r − M r ( λr + 3 M ) ∇ t ψ Z + 2 (cid:18) − Mr (cid:19) ∇ t ∇ r ψ Z . (cid:3) W − r (cid:18) − Mr (cid:19) W − r (cid:18) − Mr (cid:19) /D W = − Mr S − M ( r − M ) r ( λr + 3 M ) ∇ r ψ Z + 2 ∇ t ∇ t ψ Z − Mr ( λr + 3 M ) (cid:18) λ ( λ − λ + 1) r + 3 λ (4 λ + 1) M r + 9(3 λ − M r + 36 M (cid:19) ψ Z . (4.22)(4.23) (cid:3) W − r (cid:18) − Mr (cid:19) W − r /D ∗ W = 0 . (4.24) (cid:3) S = 0 . (cid:3) P even = − ∇ t S − (cid:18) − Mr (cid:19) ∇ t ∇ r ψ Z − λ (( λ + 3) r − M ) r ( λr + 3 M ) ∇ t ψ Z . (4.25)(4.26) (cid:3) Q even − r (cid:18) − Mr (cid:19) Q even = 0 . The derivation of these equations was done in [5] and we present the computations in AppendixD for completeness. Our plan is to estimate r − ψ Z and S first. Then it gives bound on the righthand side of (4.25), the equation of P even . However, the right hand side of (4.25) has bad weightin r and we can’t directly apply vector field methods. To resolve this issue, we consider thesubstitution:¯ P even := P even + 32 r ∇ t ψ Z − u + r r (2 λ + 3) · ψ Z − u + r S, (4.27) ¯ W := W − r ∇ t ψ Z + ( u + r )2 r ψ Z , (4.28) ¯ W := W − r (cid:18) − Mr (cid:19) ∇ r ψ Z + 3 M r (cid:18) − Mr (cid:19) (cid:18) λ + 3 Mr (cid:19) − ψ Z , (4.29) ¯ W := W − r /D ∗ ψ Z . (4.30)Here u is defined as u := t − r − M log (cid:16) r M − (cid:17) + R null + 2 M log (cid:18) R null M − (cid:19) such that u = τ for r ≥ R null . From (4.16), (4.24), these quantities satisfy the following waveequations. (cid:3) ¯ P even = (cid:18) − Mr (cid:19) − · Mr ∇ L S + Mr S + (cid:18) − Mr (cid:19) − · (2 λ + 3) 2 Mr ∇ L ( r − ψ Z )+(2 λ + 3) · M ( λr + 3 M ) ∇ t ( r − ψ Z ) − λ + 32 (cid:18) ( u + r ) V Z − Mr (cid:19) ( r − ψ Z ) . (4.31) CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 29 (cid:3) ¯ W + 2 Mr ¯ W = − Mr ¯ P even − (cid:16) ur (cid:17) Mr S − M ( λr + 3 M ) · (2 λ + 3) ∇ t ( r − ψ Z ) − Mr (cid:18) − Mr (cid:19) − ∇ L ( r − ψ Z )+ (cid:18)(cid:18) − (2 λ + 3) Mr + r V Z (cid:19) + ur (cid:18) − (2 λ + 2) Mr + r V Z (cid:19)(cid:19) ( r − ψ Z ) . (4.32) (cid:3) ¯ W − r (cid:18) − Mr (cid:19) ¯ W − r (cid:18) − Mr (cid:19) /D ¯ W = − Mr S − λ ( λ + 1)(2 λ + 3) M ( λr + 3 M ) ( r − ψ Z ) . (4.33) (cid:3) ¯ W − (cid:18) − Mr (cid:19) ¯ W − r /D ∗ ¯ W = λ (2 λ + 3) M r ( λr + 3 M ) /D ∗ ( r − ψ Z ) . (4.34)We used the above substitution to eliminate the terms with bad weights in r under the pricethat weight in u appears for r − ψ Z and S . In order to deal with this, we will need the rapiddecay result of Angelopoulos-Aretakis-Gajic [2], recorded in Appendix A as Proposition A.5.Moreover, u is not regular on the horizon H + , so we perform an interpolation. Let b > bM ≥ R null to be a large number. Its value will be determined in subsection 6.5 and we keepthe dependence of b in estimates until the value is fixed. Let η b ( r ) be a cutoff function with η b ( r ) = (cid:26) r ∈ [2 M, bM ] , r ∈ [2 bM, ∞ ) . Define ˆ P even := P even + η b · (cid:18) r ∇ t ψ Z − u + r r (2 λ + 3) · ψ Z − u + r S (cid:19) , (4.35) ˆ W := W + η b · (cid:18) − r ∇ t ψ Z + ( u + r )2 r ψ Z (cid:19) , (4.36) ˆ W := W + η b · − r (cid:18) − Mr (cid:19) ∇ r ψ Z + 3 M r (cid:18) − Mr (cid:19) (cid:18) λ + 3 Mr (cid:19) − ψ Z ! , (4.37) ˆ W := W + η b · (cid:16) − r /D ∗ ψ Z (cid:17) . (4.38)And denote G P := (cid:3) ˆ P even , (4.39) G := (cid:3) ˆ W + 2 Mr ˆ W , (4.40) G := (cid:3) ˆ W − r (cid:18) − Mr (cid:19) − r (cid:18) − Mr (cid:19) /D ˆ W , (4.41) G := (cid:3) ˆ W − (cid:18) − Mr (cid:19) ˆ W − r /D ∗ ˆ W . (4.42)
Lemma 4.8.
Fix an integer m ≥ . As r ∈ [2 M, bM ] , we have | ( M ∂ ) ≤ m G P | . m,b M − (cid:0) | ( M ∂ ) ≤ m +1 S | + | ( M ∂ ) ≤ m +3 ( r − ψ Z ) | (cid:1) . As r ∈ [ bM, bM ] , we have | ( M ∂ ) ≤ m G P | . m,b M − (cid:16) τM (cid:17) (cid:0) | ( M ∂ ) ≤ m +1 S | + | ( M ∂ ) ≤ m +3 ( r − ψ Z ) | (cid:1) . And as r ∈ [2 bM, ∞ ) , | ( M ∂ ) ≤ m G P | . m s − | ( M ∂ ) ≤ m ∇ L S | + M − s − | ( M ∂ ) ≤ m S | + s − |∇ L (( M ∂ ) ≤ m K ≤ r − ψ Z ) | + s − | ( M ∂ ) ≤ m ∇ t ( r − ψ Z ) | + (cid:16) τM (cid:17) M − s − | ( M ∂ ) ≤ m K ≤ r − ψ Z | . Proof.
We will only show the case m = 0 and m ≥ M, bM ], G P = (cid:3) P even and then | (cid:3) P even | ≤ |∇ t S | + (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) − Mr (cid:19) ∇ t ∇ r ψ Z (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) λ (( λ + 3) r − M ) r ( λr + 3 M ) ∇ t ψ Z (cid:12)(cid:12)(cid:12)(cid:12) . b M − (cid:0) | ( M ∂ ) ≤ S | + | ( M ∂ ) ≤ ( r − ψ Z ) | (cid:1) . Similarly, in [2 bM, ∞ ) G P = (cid:3) ¯ P even and from (4.31) | (cid:3) ¯ P even | ≤ (cid:18) − Mr (cid:19) − · Mr |∇ L S | + Mr | S | + (cid:18) − Mr (cid:19) − · (2 λ + 3) 2 Mr |∇ L ( r − ψ Z ) | +(2 λ + 3) · M ( λr + 3 M ) |∇ t r − ψ Z | + 2 λ + 32 (cid:12)(cid:12)(cid:12)(cid:12) ( τ + r ) V Z − Mr (cid:12)(cid:12)(cid:12)(cid:12) | r − ψ Z | . s − |∇ L S | + M − s − | S | + s − |∇ L ( K ≤ r − ψ Z ) | + s − |∇ t ( r − ψ Z ) | + (cid:16) τM (cid:17) M − s − |K ≤ r − ψ Z | . In [ bM, bM ], we compute (cid:3) ˆ P even = (cid:3) P even + (cid:3) η b · ( ¯ P even − P even )+2 ∇ η b · ∇ ( ¯ P even − P even ) + η b (cid:3) ( ¯ P even − P even ) . Since bM ≤ r ≤ bM , from (4.25) we obtain | (cid:3) P even | ≤ |∇ t S | + (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) − Mr (cid:19) ∇ t ∇ r ψ Z (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) λ (( λ + 3) r − M ) r ( λr + 3 M ) ∇ t ψ Z (cid:12)(cid:12)(cid:12)(cid:12) . b M − | ( M ∇ t ) S | + M − | ( M ∂ ) ≤ ( r − ψ Z ) | . Also, from the definition of P even and ¯ P even | ¯ P even − P even | ≤ r |∇ t r − ψ Z | + u + r λ + 3) · | r − ψ Z | + τ + r | S | . b M (cid:16) τM (cid:17) | ( M ∂ ) ≤ r − ψ Z | + M (cid:16) τM (cid:17) | S | . Similarly,
CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 31 | ∂ ( ¯ P even − P even ) | . b (cid:16) τM (cid:17) | ( M ∂ ) ≤ r − ψ Z | + (cid:16) τM (cid:17) | ( M ∂ ) ≤ S | . Even though (cid:3) ( ¯ P even − P even ) involves one more derivative, since ψ Z and S satisfy wave equations,we still have | (cid:3) ( ¯ P even − P even ) | . b M − (cid:16) τM (cid:17) | ( M ∂ ) ≤ r − ψ Z | + M − (cid:16) τM (cid:17) | ( M ∂ ) ≤ S | . Together with | (cid:3) η b | . b M − , | ∂η b | . b M − , the assertion follows by putting these inequalitiestogether. (cid:3) Lemma 4.9.
Fix an integer m ≥ . In r ∈ [2 M, bM ] , | ( M ∂ ) ≤ m G | . m,b M − | ( M ∂ ) ≤ m ˆ P even | + M − | ( M ∂ ) ≤ m +2 ( r − ψ Z ) | . In r ∈ [ bM, bM ] , | G | . m,b M − | ( M ∂ ) ≤ m ˆ P even | + M − (cid:16) τM (cid:17) | ( M ∂ ) ≤ m S | + M − (cid:16) τM (cid:17) | ( M ∂ ) ≤ m +2 ( r − ψ Z ) | . In [2 bM, ∞ ) , | ( M ∂ ) ≤ m G | . m M − s − | ( M ∂ ) ≤ m ˆ P even | + (cid:16) τM (cid:17) M − s − | ( M ∂ ) ≤ m S | + s − | ( M ∂ ) ≤ m ∇ t ( r − ψ Z ) | + s − | ( M ∂ ) ≤ m ∇ L ( r − ψ Z ) | + M − s − (cid:16) τM (cid:17) | ( M ∂ ) ≤ m K ≤ r − ψ Z | . Proof.
Again, we will only show that case m = 0. As r ∈ [2 M, bM ], ˆ W = W and ˆ P even = P even .From (4.21) we have | G | ≤ Mr | ˆ P even | + (cid:12)(cid:12)(cid:12)(cid:12) λr − λM r − M r ( λr + 3 M ) (cid:12)(cid:12)(cid:12)(cid:12) |∇ t ψ Z | + 2 (cid:18) − Mr (cid:19) |∇ t ∇ r ψ Z | . b M − | ˆ P even | + M − | ( M ∂ ) ≤ ( r − ψ Z ) | . As r ∈ [2 bM, ∞ ), ˆ W = ¯ W and ˆ P even = ¯ P even . From (4.32) we have | G | ≤ Mr | ˆ P even | + (cid:16) τr (cid:17) Mr | S | + 3 M ( λr + 3 M ) · (2 λ + 3) |∇ t ( r − ψ Z ) | + 2 Mr (cid:18) − Mr (cid:19) − |∇ L ( r − ψ Z ) | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − (2 λ + 3) Mr + r V Z (cid:19) + ur (cid:18) − (2 λ + 3) Mr + r V Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) | ( r − ψ Z ) | . M − s − | ˆ P even | + (cid:16) τM (cid:17) M − s − | S | + s − |∇ t ( r − ψ Z ) | + s − |∇ L ( r − ψ Z ) | + M − s − (cid:16) τM (cid:17) |K ≤ · r − ψ Z | . In r ∈ [ bM, bM ], we have G = (cid:18) (cid:3) + 2 Mr (cid:19) · W + (cid:3) η b · ( ¯ W − W )+2 ∇ η b · ∇ ( ¯ W − W ) + η b · ( (cid:3) + 2 M/r )( ¯ W − W ) . From (4.21), (cid:18) (cid:3) + 2 Mr (cid:19) · W = − Mr P even + 2 λr − λM r − M r ( λr + 3 M ) ∇ t ψ Z + 2 (cid:18) − Mr (cid:19) ∇ t ∇ r ψ Z = − Mr ˆ P even + 2 λr − λM r − M r ( λr + 3 M ) ∇ t ψ Z + 2 (cid:18) − Mr (cid:19) ∇ t ∇ r ψ Z + η b · Mr (cid:18) r ∇ t ψ Z − u + r r (2 λ + 3) · ψ Z − u + r S (cid:19) . Therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) (cid:3) + 2 Mr (cid:19) W (cid:12)(cid:12)(cid:12)(cid:12) . b M − | ˆ P even | + M − (cid:16) τM (cid:17) | S | + M − (cid:16) τM (cid:17) | ( M ∂ ) ≤ ( r − ψ Z ) | . From the definition of ¯ W , in r ∈ [ bM, bM ] | ¯ W − W | ≤ r |∇ t ψ Z | + ( τ + r )2 r | ψ Z | . b M (cid:16) τM (cid:17) | ( M ∂ ) ≤ ( r − ψ Z ) | . Similarly, in r ∈ [ bM, bM ] M | (cid:3) ( ¯ W − W ) | , |∇ ( ¯ W − W ) | . b (cid:16) τM (cid:17) | ( M ∂ ) ≤ ( r − ψ Z ) | Hence the assertion follows by putting these estimates together. (cid:3)
Similarly, we have the bound on G and G below. Lemma 4.10. In r ∈ [2 M, bM ] , | ( M ∂ ) ≤ m G | . m,b M − | ( M ∂ ) ≤ m S | + M − | ( M ∂ ) ≤ m +2 ( r − ψ Z ) | , | ( M ∂ ) ≤ m G | . m,b M − | ( M ∂ ) ≤ m +2 ( r − ψ Z ) | . In r ∈ [2 bM, ∞ ) , | ( M ∂ ) ≤ m G | . m M − s − | ( M ∂ ) ≤ m S | + M − s − | ( M ∂ ) ≤ m K ≤ · r − ψ Z | , | ( M ∂ ) ≤ m G | . m M − s − | ( M ∂ ) ≤ m S | + M − s − | ( M ∂ ) ≤ m K ≤ · r − ψ Z | . The ℓ = 1 mode. In this subsection, we discuss the case that the even two tensor h ab issupported on ℓ = 1. It is well known that in this mode, any solution of linear gravity (1.2) is ofthe form − W π [39, 30]. We put a minus sign before W π to be consistent with the notation in ℓ ≥
2. If h ab further satisfies the harmonic gauge (1.3), then the vector field W is a solution of(1.5). As in the mode ℓ = 1, there is no spherical symmetric traceless two tensor, equation (4.13)holds trivially and doesn’t determine W . In stead, we rely on the equalities below to express W a in terms of h ab . From (2.11) and the commutation relation (2.9), (2.10), one derives that onany fixed mode ℓ ≥ CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 33 r M ∇ t /trh − r M /D h tA − r M (cid:18) − Mr (cid:19) h tr + r M (cid:18) − Mr (cid:19) ∇ t /D h rA − r M ∇ r (cid:18) − Mr (cid:19) /D h tA = − W − r M (1 − − ) ∇ t /D W , (4.43) r M (cid:18) − Mr (cid:19) ∇ r /trh − r M (cid:18) − Mr (cid:19) /D h rA − r M (cid:18) − Mr (cid:19) h rr = − W − r M (cid:18) − Mr (cid:19) (1 − − ) · ∇ r /D W , (4.44) − r M (cid:18) − Mr (cid:19) ∇ r /D ∗ /trh − r M (cid:18) − Mr (cid:19) /D ∗ /trh + r M (cid:18) − Mr (cid:19) /D ∗ h rr + r M (cid:18) − Mr (cid:19) h rA = − W − (Λ − (cid:18) − r M (cid:18) − Mr (cid:19) ∇ r W + r M (cid:18) Mr (cid:19) W (cid:19) . (4.45)Here Λ is given by (2.25). In particular, as ℓ = 1 and Λ = 2, the above three equalities give theexpression of W a in terms of h ab . 5. Analysis of ˆ W In this and the next sections, we study in solution of the wave equations (4.21), (4.22) and(4.23) with ψ Z satisfying the wave equation (4.16). We note that by putting ψ Z ≡
0, thesethree equations is the even part of the equation (1.5). We further restrict ourselve in the case W = W ℓ ≥ in this and the next sections. The case W ℓ =0 will be discussed in the section 8.We will prove the decay estimate for ˆ W , Proposition 5.1, based on the vector field methodin Appendix A. We start with S and r − ψ Z , which satisfy wave equations without source term,(4.16) and (4.24). The bound on S and r − ψ Z then is translated to the one for G P , the sourceterm in (4.39). After that, we have the bound on G , the source term in (4.40), and then getthe estimate for ˆ W . The main result is the following: Proposition 5.1.
Let W , r − ψ Z , P even and S be solutions of (4.21) , (4.16) , (4.25) and (4.24) respectively and are supported on ℓ ≥ . We further assume ψ Z,ℓ =1 = 0 . Let ˆ W and ˆ P even be defined as in (4.36) and (4.35) . Suppose Decay[ ˆ W ] defined below is finite, then for any p ∈ [ δ, − δ ] and τ ≥ we have, F [ ˆ W ]( τ ) + E p, (0) L [ ˆ W ]( τ ) + M − Z ∞ τ B [ ˆ W ]( τ ′ ) + E p − , (0) L, / ∇ [ ˆ W ]( τ ′ ) dτ ′ . b,R nul (cid:16) τM (cid:17) − p +3 δ · Decay[ ˆ W ] . (5.1) Here
Decay[ ˆ W ] := I (0) [ ˆ W , K ≤ ˆ P even ] + M I (0) [ D ≤ K ≤ S, D ≤ K ≤ S ] + M I (1) [ K ≤ S, K ≤ r − ψ Z ] . (5.2)5.1. Analysis of S and r − ψ Z . We begin with S = trh , which satisfies the wave equation(4.24). From remark A.3 we have Proposition 5.2.
Let S be an solution of (4.24) . Suppose I (1) [ S ] is finite, then for any p ∈ [ δ, − δ ] and τ ≥ , we have (cid:16) F [ S ] + E p, (0) L [ ˜ S ] (cid:17) ( τ ) + M − Z ∞ τ (cid:16) B [ S ] + E p − , (0) L [ ˜ S ] (cid:17) ( τ ′ ) dτ ′ . (cid:16) τM (cid:17) − p +3 δ I (1) [ S ] . (5.3) Here ˜ S = η null S and η null is the cut-off function define in (2.22) . The initial norm I (1) [ · ] isdefined in (2.23) . Lemma 5.3.
There exists R Z large enough such that for any R null ≥ R Z and ℓ ≥ , the potential V Z,ℓ for the Zerilli equation (4.16) is a member of ∈ V ( δ, R null , ℓ, defined in A.2. Moreover,the constants in (A.4) and (A.9) for k = 1 can be chosen independent of ℓ . Therefore Proposition A.4 and Proposition A.5 with k = 1 apply to r − ψ Z,ℓ with constantsbeing uniform in ℓ ≥ Proof.
For R null large enough, V Z,ℓ ∈ V ( δ, R null ,
2) is proved in [24, Theorem 2] and [23, Theorem16]. Here we check V Z,ℓ further satisfies (A.7), (A.8) and (A.9). Recall that V (1) Z,ℓ = V Z,ℓ − r + 14 Mr . Through direct computation, (1 − p/ λ + 2) − r dV (1) Z,ℓ dr + pr V (1) Z,ℓ !! × (cid:18) Mλr (cid:19) =(2 − p )( λ −
1) + s − · Q [ λ − , s − , p ] , where Q [ λ − , s − , p ] is a polynomial in λ − , s − , p . Hence as R Z is large enough, one has forany p ∈ [ δ, − δ ], λ ≥ r ≥ R null (2 − p )( λ −
1) + s − · Q [ λ − , s − , p ] ≥
12 (2 − p )( λ − , which verifies (A.7). For (A.8), one checks thatlim r →∞ r V (1) Z,ℓ = − . Finally, (A.9) clearly holds for each V Z,ℓ . The uniformness comes fromlim ℓ →∞ V Z,ℓ = − Mr . (cid:3) CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 35
Therefore from Proposition A.5 we obtain
Proposition 5.4.
Let ψ Z = ψ Z,ℓ ≥ be a solution of (4.16) . Suppose that I (1) [ r − ψ Z ] is finite,then for any p ∈ [ δ, − δ ] and τ ≥ , we have (cid:16) F [ r − ψ Z ] + E p, (0) L [ r − ˜ ψ Z ] (cid:17) ( τ ) + M − Z ∞ τ (cid:16) B [ r − ψ Z ] + E p − , (0) L [ r − ˜ ψ Z ] (cid:17) ( τ ′ ) dτ ′ . (cid:16) τM (cid:17) − p +3 δ I (1) [ r − ψ Z ] , (5.4) Here ˜ ψ Z = η null ψ Z and η null is the cut-off function define in (2.22) . The initial norm I (1) [ · ] isdefined in (2.23) . Analysis of ˆ P even . We move to equation (4.39) and estimate ˆ P even . To begin we translatethe estimate of S and r − ψ Z above to the one for G P through Lemma 4.8. Lemma 5.5. I source , δ [ G P ] . b,R null M I (0) [ D ≤ K ≤ S, D ≤ K ≤ r − ψ Z ] + M I (1) [ K ≤ S, K ≤ r − ψ Z ] . Here I source , δ [ · ] and I (1) [ · ] are define in (2.24) and (2.23) .Proof. We first deal with the spacetime integrand
M s p +1 | G P | . For r ∈ [2 M, bM ], from Lemma4.8, we have
M s p +1 | G P | . b M − | ( M ∂ ) ≤ S | + M − | ( M ∂ ) ≤ r − ψ Z | , which is bounded by the integrand of M B [ D ≤ S ] and M B [ D ≤ r − ψ Z ]. Note that we lose onederivative here because of the photon sphere. Then from Proposition A.4, Z D ( τ ,τ ) M s p +1 | G P | · χ [2 M,bM ] dvol . b,R null M F [ D ≤ S, D ≤ r − ψ Z ]( τ ) . R null (cid:16) τ M (cid:17) − δ · M I (0) [ D ≤ S, D ≤ r − ψ Z ] . For r ∈ [ bM, bM ], from Lemma 4.8, we have M s p +1 | G P | . b M − (cid:16) τM (cid:17) | ( M ∂ ) ≤ S | + M − (cid:16) τM (cid:17) | ( M ∂ ) ≤ r − ψ Z | , which is bounded by the integrand of (cid:0) τM (cid:1) · M B [ S ] and (cid:0) τM (cid:1) · M B [ D ≤ r − ψ Z ]. Notethat in this region T is strictly timelike and from (3.1) we can replace M B [ D ≤ r − ψ Z ] by M B [ K ≤ r − ψ Z ]. From Proposition 5.2 and 5.4, we have M − Z ∞ τ B [ S ]( τ ) + B [ K ≤ r − ψ Z ]( τ ) dτ . R null (cid:16) τ M (cid:17) − δ I (1) [ S, K ≤ r − ψ Z ] . Therefore Z D ( τ , ∞ ) M s p +1 | G P | · χ [ bM, bM ] dvol . b,R null (cid:16) τ M (cid:17) − δ · M I (1) [ S, K ≤ r − ψ Z ] . For r ∈ [2 bM, ∞ ], from Lemma 4.9 we have M s p +1 | G P | . M s p − |∇ L S | + M − s p − | S | + M s p − |∇ L K ≤ r − ψ Z | + M s p − |∇ t r − ψ Z | + (cid:16) τM (cid:17) M − s p − |K ≤ r − ψ Z | . The terms involving S are bounded by the integrand of M E p − , (0) L, / ∇ [ S ] and the terms involving r − ψ Z are bounded by (cid:16) τM (cid:17) · M E p − , (0) L, / ∇ [ K ≤ r − ψ Z ] . By Proposition 5.2 and Proposition 5.4, we obtain Z D ( τ ,τ ) ∩{ r ≥ bM } M s p +1 | G P | dvol . R null (cid:16) τ M (cid:17) − p + δ · M I (0) [ S ]+ (cid:16) τ M (cid:17) − p +3 δ · M I (1) [ K ≤ r − ψ Z ] . Next we estimate the spacetime integrand (cid:0) − Mr (cid:1) | ∂G | + (cid:0) − Mr (cid:1) | ∂ r G | . Note that bM ≥ R null . In this region M | ∂G P | . M | ∂ D ≤ S | + M | ∂ D ≤ r − ψ Z | ,M | ∂ r G P | . M | ∂ r D ≤ S | + M | ∂ r D ≤ r − ψ Z | , which are bounded by the integrand of M B [ D ≤ S ] and M B [ D ≤ S ]. By Proposition A.4, wehave M Z D ′ ( τ ,τ ) (cid:18) − Mr (cid:19) | ∂G P | + (cid:18) − Mr (cid:19) |∇ r G P | dvol . M Z τ τ B [ D ≤ S, D ≤ r − ψ Z ]( τ ) dτ . R null (cid:16) τ M (cid:17) − δ I (0) [ D ≤ S, D ≤ r − ψ Z ] . Finally, the integrals along Σ ′′ τ is bounded as Z Σ ′′ τ M | G P | dvol . Z Σ ′′ τ | ( M ∂ ) ≤ S | + | ( M ∂ ) ≤ r − ψ Z | dvol . R null (cid:16) τ M (cid:17) − δ · M I (0) [ S, D ≤ r − ψ Z ] . Putting these together, we have for any p ∈ [ δ, − δ ], (cid:16) τ M (cid:17) − p − δ E p source [ G P ]( τ , τ ) . b,R null M I (0) [ D ≤ S, D ≤ r − ψ Z ] + M I (1) [ S, K ≤ r − ψ Z ] . Then the result follows form the definition of I source , δ [ · ] in (2.24). (cid:3) Now we can apply Proposition A.4 to show that ˆ P even decays. CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 37
Proposition 5.6.
Let P even , S and r − ψ Z be solutions of (4.25) , (4.24) and (4.16) respectively.Let ˆ P even be defined as in (4.35) . Suppose Decay[ ˆ P even ] defined below is finite, then for any p ∈ [ δ, − δ ] and τ ≥ we have F [ ˆ P even ]( τ ) + E p, (0) L [ ˆ P even ]( τ ) + M − Z ∞ τ B [ ˆ P even ]( τ ) + E p − , (0) L, / ∇ [ ˆ P even ]( τ ) dτ . b,R null (cid:16) τM (cid:17) − p +3 δ · Decay[ ˆ P even ] . (5.5) Here
Decay[ ˆ P even ] := I (0) [ ˆ P even ] + M I (0) [ D ≤ K ≤ S, D ≤ K ≤ r − ψ Z ] + M I (1) [ K ≤ S, K ≤ r − ψ Z ] , (5.6) and I ( k ) [ · ] is defined in (2.23) . proof of Proposition 5.1. We turn to equation (4.40) and prove the Proposition 5.1.Still, we begin with the source term G . Comparing Lemma 4.8 and Lemma 4.9, we are able tobound G with the estimate of G P and ˆ P even . Lemma 5.7. I source , δ [ G ] . b,R null I (0) [ K ≤ ˆ P even ] + M I (0) [ D ≤ K ≤ S, D ≤ K ≤ r − ψ Z ] + M I (1) [ K ≤ S, K ≤ r − ψ Z ] . Here I source , δ [ · ] and I ( k ) [ · ] are define in (2.24) and (2.23) . Next, we verify the potential of (4.40) is a member of V ( δ, R null , Lemma 5.8.
For any R null ≥ M , − M/r belongs to V ( δ, R null , defined in (A.1) .Proof. The requirements other than the Morawetz one (A.4) can be verified through directcomputation. The choice of Moratwez function f ( r ) is f ( r ) := (cid:18) − Mr (cid:19) (cid:18) Mr + 2 M r (cid:19) ,ω ( r ) = (cid:18) − Mr (cid:19) (cid:18) fr + dfdr (cid:19) . Note that dfdr = Mr + M r + M r > . And we calculate for V ( r ) = − M/r , f (cid:18) − Mr (cid:19) · r + (cid:18) − (cid:3) ω − f (cid:18) − Mr (cid:19) dVdr − Mr f V (cid:19) = r − (cid:18) − s − − s − + 7 s − + 130 s − − s − (cid:19) , which is positive in r ≥ M . Hence (A.4) holds for ℓ = 1. (cid:3) proof of Proposition 5.1. From the view of Lemma 5.7 and Lemma 5.8, Proposition A.4 yieldsthe desired result. (cid:3)
Recall that Q even satisfies the wave equation (4.26). Proposition 5.9.
Let Q even be a solution of (4.26) . Suppose I (0) [ Q even ] is finite, then for any p ∈ [ δ, − δ ] and τ ≥ , we have F [ Q even ]( τ ) + E p, (0) L [ Q even ]( τ ) + M − Z ∞ τ B [ Q even ]( τ ) + E p − , (0) L, / ∇ [ Q even ]( τ ) dτ . R null (cid:16) τM (cid:17) − p + δ · I (0) [ Q even ] . (5.7) Here I ( k ) [ · ] is defined in (2.23) .Proof. This actually a consequence of − M/r ∈ V ( δ, R null ,
1) proved in Lemma 5.8. Note that Q even is a section of L ( −
1) and is automatically supported on ℓ ≥
1. This is also the reason thatwe consider the potential − M/r in stead of r (cid:0) − Mr (cid:1) . The equation (4.26) is equivalentto (4.40) without G in terms application of Proposition A.4. This can be seen through thecommunication relation r /D · (cid:3) = (cid:18) (cid:3) + 1 r (cid:19) · r /D , which implies r /D Q even satisfies the equation ( (cid:3) + 2 M/r ) · r /D Q even = 0. Then the resultfollows from Lemma 5.8 and Proposition A.4 (cid:3) Analysis of ˆ W and ˆ W In this section we study the remaining two quantities ˆ W and ˆ W in the vector field ˆ W . Recallthat ˆ W and ˆ W satisfy a coupled wave equation (4.41) and (4.42). We denote W := ( ˆ W , ˆ W )and G := ( G , G ). The equations (4.41) and (4.42) can be rewritten as (cid:3) W = A W W + G , (6.1)where A W = 1 r (cid:20) (cid:0) − Mr (cid:1) r (cid:0) − Mr (cid:1) /D r /D ∗ (cid:0) − Mr (cid:1) (cid:21) . The main result of this section is
Proposition 6.1.
Let W , W , W , r − ψ Z , and S be solutions of (4.21) , (4.22) , (4.23) , (4.16) ,and (4.24) respectively and are supported on ℓ ≥ . We further assume that ψ Z,ℓ =1 = 0 andthat W is even . Let ˆ W , ˆ W , ˆ W and ˆ P even be defined as in (4.36) , (4.37) , (4.38) and (4.35) .Suppose Decay[ W ] defined below is finite. Then for any p ∈ [ δ, p − δ ] and τ ≥ , we have F [ W ]( τ ) + E p, (0) L [ ˜ W ]( τ ) + M − Z ∞ τ ¯ B [ W ]( τ ′ ) + E p − , (0) L, / ∇ [ ˜ W ]( τ ′ ) dτ ′ . (cid:16) τM (cid:17) − p +3 δ Decay[ W ] . (6.2) Here W = ( ˆ W , ˆ W ) , ˜ W = η null W with η null defined in (2.22) , Decay[ W ] := F [ W ](0) + E − δ, (0) L [ ˜ W ](0) + I (0) [ K ≤ ˆ W , K ≤ ˆ P even ]+ M I (0) [ D ≤ K ≤ S, D ≤ K ≤ r − ψ Z , Q even ] + M I (1) [ K ≤ S, K ≤ r − ψ Z ] , CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 39 and I ( k ) [ · ] is defined in (2.23) . From now on we assume that for all τ ≥ E − δ, (0) [ ˜ W ]( τ ) < ∞ , Z τ E − δ, (0) [ ˜ W ]( τ ′ ) dτ ′ < ∞ . (6.3)The purpose of the assumption is to add a zeroth order term into the boundary term of T -currentin subsection 6.1 through Corollary 3.4. This assumption is actually satisfied for any smoothsolution with E − δ, (0) [ ˜ W ](0) < ∞ . In subsections below, we construct for all p ∈ [ δ, − δ ] acurrent J p W such that for any τ ≥ Z Σ τ J p W · ndvol ≈ F [ W ]( τ ) + E p, (0) [ ˜ W ]( τ ) . (6.4)And for any r ≥ R null + M , Z S ( τ, M ) J p W · L dvol S ≥ , (6.5) Z S ( τ,r ) J p W · L r dvol S ! − . s − (2 − δ ) M − E − δ, (0) L [ ˜ W ]( τ ) , (6.6)where ( · ) − stands for the negative part. For we want to show that Z Σ τ div J p W − Err[ W , G ] − Err [ W , F ] dvol & b,R null M − (cid:0) ¯ B [ W ]( τ ) + E p − , (0) L, / ∇ [ ˜ W ]( τ ) (cid:1) . (6.7)Here Err[ W , G ] is of the form ∇ W · G . The term Err [ W , F ] is a quadratic form involving W andits first order derivatives. The reason we group them together is that they can be rewritten interms of S , r − ψ Z , ˆ P even , Q even and ˆ W , which we already controlled. We record below therelations we need to do so.By rewriting the definition of S W , Q even and P even , (4.18), (4.20) and (4.19), we have ∇ r W + /D W + 2 r W = F , (6.8) /D ∗ W + (cid:18) − Mr (cid:19) / ∇ r W + (cid:18) − Mr (cid:19) · r W = F , (6.9) ∇ t W = F . (6.10)Here F := − S + 12 trh RW + (cid:18) − Mr (cid:19) − ∇ t W ,F := Q even − r /D ∗ S + r /D ∗ trh RW + r /D ∗ / ∆ Z ( r − ψ Z ) ,F := (cid:18) − Mr (cid:19) ∇ r W + (cid:18) − Mr (cid:19) · r W + (cid:18) − Mr (cid:19) · r P even . (6.11) T -current. In this subsection we construct a current J which satisfies the following twoproperties. First, there exists a constant C T > C T F T [ W ]( τ ) ≤ Z Σ τ J · ndvol ≤ C T F T [ W ]( τ ) . (6.12)Second, div J = Err [ W , G ] + Err , [2 M, R T ] [ W ] + Err , [ R T , ∞ ) [ W ] . (6.13)Here Err [ W , G ] := ∇ t W · G + η T · (9 ∇ t W · G + ∇ t W · G ) ,Err , [ R T , ∞ ) [ W ] := (cid:18) − η ′ T (cid:19) (cid:16) − |∇ L ˆ W | − |∇ L ˆ W | (cid:17) + χ [ R T , ∞ ) ( r ) · Mr ∇ t ˆ W /D ∗ ˆ W ,Err , [2 M, R T ] [ W , F ] := (cid:20) − χ [2 M,R T ] ( r ) · Mr + η T ( r ) · (cid:18) r − Mr (cid:19)(cid:21) ∇ t ˆ W /D ˆ W + η T · r (cid:18) − Mr (cid:19) ∇ t ˆ W · ˆ W . The cut-off function η T ( r ) and a large number R T will be defined below. Here Err [ W , G ] is ofthe form ∇ t W · G and will be part of Err[ W , G ]. The term Err , [ R T , ∞ ) [ W ] falls off rapidly in r and can be absorbed by using K p in subsection 6.4. The term Err , [2 M, R T ] [ W , F ] comes fromthe feature that A W is not self-adjoint. We will rely on (6.10) and (6.9) to estimate it.Another difficulty in constructing J is to make J · n positive definite. Let A W , main := 1 r (cid:20) r /D r /D ∗ (cid:21) be the leading term of A W . We focus on a fixed mode ℓ ≥ | m | ≤ ℓ . By using Y mℓ and r Λ − / / ∇ A Y mℓ as basis, we calculate that | / ∇ W | + W · A W , main · W = s W · r − (cid:20) Λ + 2 − √ Λ − √ Λ Λ (cid:21) · W . The worst case is as ℓ = 1 and Λ = 2, the above matrix is only semi-positive definite witheigenvalues { , } . Hence we need to be careful about the lower order term even as s is large.Now we start the construction. We use C T to stand for a constant which may increase fromline to line. Define two self-adjoint operators(6.14) A W , ex := 1 r (cid:20) (cid:0) − Mr (cid:1) r (cid:0) − Mr (cid:1) /D r (cid:0) − Mr (cid:1) /D ∗ (cid:0) − Mr (cid:1) (cid:21) , (6.15) A W , in := 1 r (cid:20) (cid:0) − Mr (cid:1) r /D r /D ∗ (cid:0) − Mr (cid:1) (cid:21) . Let R T ≥ R null be a large number. It will be fixed in subsection 6.4 and we remark that C T will not depend on R T . Let CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 41 (6.16) A W ,T := (cid:26) A W , in r ∈ [2 M, R T ] ,A W , ex r ∈ ( R T , ∞ ) . Consider(6.17) ( J , ) a := (cid:18) T ab [ W ] − W · A W ,T · W g ab (cid:19) T b . Note J , is not continuous across r = R T . But the it has no contribution as we apply divergencetheorem (2.14) since T · ∇ r = 0. The reason to use A W , ex for large r is to make sure J , · n is positive definite for r ≥ R T . To ensure the positivity in r ∈ [2 M, R T ] we need an auxiliarycurrent. Let η T ( r ) be a cut-off function such that η T ( r ) := (cid:26) r ∈ [2 M, R T ] , r ∈ [4 R T , ∞ ) , | dη T /dr | ≤ r − . Define A W , aux := 1 r (cid:20) r /D r /D ∗ (cid:0) − Mr (cid:1) (cid:21) , and ( J , ) a = η T ( r ) (cid:18) T ab [ W ] + T ab [ W ] − W · A W , aux · W g ab (cid:19) T b . (6.18)Let J := J , + J , . It’s easy to check that Z Σ τ J · n dvol ≤ C T F T [ W ]( τ ) , with the constant C T doesn’t depend on R T . We start to verify the positivity of J · n . In[ R T , ∞ ), as n = L , J · n = 12 |∇ L W | + 9 η T ( r )2 |∇ L W | + η T ( r )2 |∇ L W | + 12 (cid:18) − Mr (cid:19) (cid:18) | / ∇ W | + 9 η T ( r ) | / ∇ W | + η T ( r ) | / ∇ W | (cid:19) . + 12 (cid:18) − Mr (cid:19) W · ( A W , ex + η T ( r ) A W , aux ) · W . On a fixed mode ℓ , the positivity of the sumand in the second and the third line (after integratedalong S ) is equivalent to the positivity of the matrix (cid:20) Λ + 2 − s − − √ Λ(1 − s − ) − √ Λ(1 − s − ) Λ − s − (cid:21) + η T (cid:20) − √ Λ − √ Λ Λ − s − (cid:21) . Here Λ = Λ( ℓ ) is defined in (2.25). The first matrix has eigenvalues { Λ + 1 − s − ± (1 − s − ) √
4Λ + 1 } , which is positive for ℓ ≥ ≥
2) and s ≥
10. Moreover, for ℓ ≥ ≥ we finish the verification for r ∈ [ R T , ∞ ). Similarly, the positivity of J · n in r ∈ [2 M, R T ] isequivalent to the positivity of the matrix (cid:20) Λ + 2 − s − − √ Λ − √ Λ Λ − s − (cid:21) + (cid:20) − √ Λ − √ Λ Λ − s − (cid:21) = (cid:20)
10Λ + 2 − s − − √ Λ − √ Λ 2Λ − s − (cid:21) ≥ (cid:20) − − √ Λ − √ Λ 2Λ − (cid:21) . In the last inequality we used s ≥
2. The last matrix has eigenvalues 6Λ − ± p Λ(Λ + 1),which are positive and comparable to Λ for ℓ ≥ ≥
2. We now have checked that for ℓ ≥ Z Σ τ J · ndvol ≥ C T Z Σ ′′ τ (1 − s − ) |∇ L ′ W | + |∇ L W | + | / ∇ W | + M − s − | W | dvol + 1 C T Z Σ ′ τ |∇ L W | + | / ∇ W ℓ ≥ | dvol . By the assumption (6.3) and Corollary 3.4, we can add zeroth order term along Σ ′ τ and finishthe proof of (6.12).We proceed to compute the divergence of J and to verify (6.13). From (6.1), we calculate(6.19) div J , = s (cid:26) ∇ t W · G − M/r · ∇ t ˆ W /D ˆ W r ∈ [2 M, R T ] , ∇ t W · G + 6 M/r · ∇ t ˆ W /D ∗ ˆ W r ∈ ( R T , ∞ ) . and div J , = (cid:18) − η ′ T (cid:19) (cid:16) |∇ L ˆ W | − |∇ L ˆ W | ) + ( |∇ L ˆ W | − |∇ L ˆ W | ) (cid:17) + η T · (cid:16) ∇ t ˆ W · G + ∇ t ˆ W · G (cid:17) + η T · (cid:18) r − Mr (cid:19) ∇ t ˆ W · /D ˆ W + η T · r (cid:18) − Mr (cid:19) ∇ t ˆ W · ˆ W . Therefore in r ∈ [2 M, R T ) , div J = (cid:20) − Mr + (cid:18) r − Mr (cid:19) · η T (cid:21) ∇ t ˆ W /D ˆ W + η T · r (cid:18) − Mr (cid:19) ∇ t ˆ W · ˆ W + ∇ t W · G + η T · (9 ∇ t ˆ W · G + ∇ t ˆ W · G ) . In ( R T , ∞ ), div J ≥ (cid:18) − η ′ T (cid:19) (cid:16) − |∇ L ˆ W | − |∇ L ˆ W | (cid:17) + 6 Mr ∇ t ˆ W /D ∗ ˆ W + (cid:18) r − Mr (cid:19) · η T ∇ t ˆ W /D ˆ W + η T · r (cid:18) − Mr (cid:19) ∇ t ˆ W · ˆ W + ∇ t W · G + η T · (9 ∇ t ˆ W · G + ∇ t ˆ W · G ) . Then (6.13) follows by collecting terms.
CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 43 red-shift current.
In this subsection, we construct the red-shift current J which satisfiesthe following two properties. First, there exists a constant C rs > < ǫ ≤ ǫC rs F [ W ]( τ ) ≤ ǫ Z Σ τ J · n dvol + F T [ W ]( τ ) ≤ C rs F [ W ]( τ ) . (6.20)Second, with the same constant C rs , we havediv J = K + Err [ W , G ] + Err , [ r rs, W ,r + rs ] [ W ] , (6.21)Here K ≥ C rs M − | ( M ∂ ) ≤ W | · χ [2 M,r rs, W ] , Err [ W , G ] = ∇ Y W · G , (cid:12)(cid:12) Err , [ r rs, W ,r + rs ] [ W ] (cid:12)(cid:12) ≤ C rs M − | ( M ∂ ) ≤ W | · χ [ r rs, W ,r + rs ] . The vector field Y and r rs, W ∈ (2 M, r + rs ) will be defined below. The term Err [ W , G ] will be partof Err[ W , G ]. The term Err , [ r rs, W ,r + rs ] [ W ] will be absorbed into K in subsection 6.3.We start the construction of J . We use C rs to stand for a constant that may increase fromline to line. Let σ > Y ( σ ) be the red-shift vector define in(2.7). Consider J ,a = (cid:18) T ab [ W ] − r | W | g ab (cid:19) Y b . Since Y ( σ ) (cid:12)(cid:12) r =2 M = 2 L ′ , we deduce (6.20) with the constant C rs depending on σ .We now compute div J . From (2.13), on the horizon r = 2 M we havediv J (cid:12)(cid:12)(cid:12)(cid:12) r =2 M = σ |∇ v W | + 12 M |∇ R W | + 2 M ∇ R W · ∇ v W + σ | / ∇ W | + σ M | W | + ∇ Y W · ( (cid:3) W − r W ) . By choosing σ large enough depending on A W , one can makediv J (cid:12)(cid:12)(cid:12)(cid:12) r =2 M − ∇ Y W · G ≥ C rs M − | ( M ∂ ) ≤ W | . Through continuity, there exists r − rs, W ∈ (2 M, r + rs ) such thatdiv J − ∇ Y W · G ≥ C rs (cid:0) M − | ∂ W | + M − | W | (cid:1) in [2 M, r − rs, W ] , (cid:12)(cid:12) div J − ∇ Y W · G (cid:12)(cid:12) ≤ C rs (cid:0) M − | ∂ W | + M − | W | (cid:1) in [ r − rs, W , r + rs ] , which yields (6.21). Morawetz current.
In this subsection, we construct a current J which satisfies the fol-lowing two properties. First, there exists a constant C Mor such that(6.22) (cid:12)(cid:12)(cid:12)(cid:12)Z Σ τ J · n dvol (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Mor F T [ W ] . Second, with the same constant C Mor ,(6.23) div J = K + Err [ W , G ] + Err , [2 M,bM ] [ W , F ] + Err , [ bM, ∞ ) [ W ] . Here K ≥ s C Mor M − s − (cid:18) (1 − s − ) |∇ r W | + |∇ t W | + | / ∇ W | + M − s − | W | (cid:19) , Err [ W , G ] := (cid:18) ∇ X W + (cid:18) ω − f (cid:19) W (cid:19) · G ,Err , [ bM, ∞ ) [ W ] := (cid:18) − Mr ∇ X ˆ W · /D ˆ W + 4 Mr ∇ X ˆ W · /D ∗ ˆ W (cid:19) · χ [ bM, ∞ ) . The number b is the same constant we used to subsection 4.2 and its value will be determinedin subsection 6.5. The functions f ( r ) , ¯ f ( r ) , ω ( r ) and the vector field X will be defined below.The term Err [ W , G ] will be part of Err[ W , G ]. The term Err , [ bM, ∞ ) [ W ] will be treated as per-turbation and will be absorbed into K p in subsection 6.4. The term Err , [2 M,bM ] [ W , F ] is moreinvolved and will be defined later in this subsection. Compared to the scalar case, K doesn’tdegenerate at photon sphere because of (6.9) and (6.8).We start the construction of J . We use C Mor to stand for a constant that may increase fromline to line. Let A W ,Mor := 1 r (cid:20) − s − (2 r − M ) /D (2 r − M ) /D ∗ − s − (cid:21) , and f ( r ) := (cid:18) − Mr (cid:19) (cid:18) Mr + 2 M r (cid:19) ,ω ( r ) = (cid:18) − Mr (cid:19) (cid:18) fr + dfdr (cid:19) X := f ( r ) (cid:18) − Mr (cid:19) ∂∂r . Note that dfdr = Mr (1 + 8 s − + 18 s − ) ≥ Mr >
0. We consider the current( J , ) a = T ab [ W ] X b − ∇ a ω | W | + 14 ω ∇ a | W | −
12 ( W · A W ,Mor · W ) X a . (6.24)Through direct computation, CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 45 div J , = s (cid:18) − Mr (cid:19) dfdr |∇ r W | + fr (cid:18) − Mr (cid:19) | / ∇ W | + W · (cid:18) − (cid:3) ω −
12 [
X, A W ,Mor ] − Mr f A W ,Mor (cid:19) · W − Mr ∇ X ˆ W · /D ˆ W + 4 Mr ∇ X ˆ W · /D ∗ ˆ W + M ωr ˆ W · /D ∗ ˆ W + (cid:18) ∇ X W + 12 ω W (cid:19) · G . Here we used A W − A W ,Mor = 1 r (cid:20) − M /D M /D ∗ (cid:21) . The last line comes from the source term G and will be part of Err [ W , G ]. Let Err , [ bM, ∞ ) [ W ] := (cid:18) − Mr ∇ X ˆ W · /D ˆ W + 4 Mr ∇ X ˆ W · /D ∗ ˆ W (cid:19) · χ [ bM, ∞ ) . We will take b large enough and show that the remaining terms are positive definite. To simplifythe computation, we focus on a fixed mode ℓ ≥ | m | ≤ ℓ . Using the basis Y mℓ , r Λ − / / ∇ Y mℓ with Λ given by (2.25), we calculate W · M [ bM, ∞ ) · W = s fr (cid:18) − Mr (cid:19) | / ∇ W | + M ωr ˆ W · /D ∗ ˆ W + W · (cid:18) − (cid:3) ω −
12 [
X, A W ,Mor ] − Mr f A W ,Mor (cid:19) · W , where M [ bM, ∞ ) is the matrix M [ bM, ∞ ) := f (1 − s − ) r (cid:20) Λ 00 Λ − (cid:21) + M ω r (cid:20) −√ Λ −√ Λ 0 (cid:21) + (cid:18) − (cid:3) ω (cid:19) · (cid:20) (cid:21) + f (1 − s − ) r (cid:20) − s − ( − s − ) √ Λ( − s − ) √ Λ 1 − s − (cid:21) − fsr (cid:20) − s − ( − s − ) √ Λ( − s − ) √ Λ 1 − s − (cid:21) We claim that M [ bM, ∞ ) is positive definite for ℓ ≥ s is large enough. The asymptotic of M [ bM, ∞ ) reads, r M [ bM, ∞ ) = (cid:20) (Λ + 2) − (4Λ + 33 / s − ( − s − ) √ Λ( − s − ) √ Λ Λ − (4Λ − / s − (cid:21) + (cid:20) O ( s − )Λ O ( s − ) √ Λ O ( s − ) √ Λ O ( s − )Λ (cid:21) . Here the constant in O ( s − ) term doesn’t depend on ℓ . The determinant has the expansiondet (cid:0) r M [ bM, ∞ ) (cid:1) = Λ(Λ − − (8Λ − − s − + O (Λ s − ) . For ℓ ≥ ≥
6, the above determinant is positive and comparable to Λ for s large enough.For ℓ = 1 and Λ = 2, the above determinant has the asymptotic 25 s − + O ( s − ), which is also positive for s large enough. Therefore, there exists a constant s Mor such that for all s ≥ s Mor and ℓ ≥ W · M [ bM, ∞ ) · W & M − s − Λ | W | ≈ s M − s − | / ∇ W | + M − s − | W | . We from now on require b ≥ s Mor and obtain that up to
Err , [ bM, ∞ ) [ W ] and Err [ W , G ], div J , is positive definite in [ bM, ∞ ).We turn to the region [2 M, bM ]. Since r ≤ bM , we can use (6.8) to replace ∇ r ˆ W = ∇ r W by − /D W − r W + F as (cid:18) − Mr (cid:19) dfdr |∇ r W | − Mr ∇ X W · /D W = (cid:18) − Mr (cid:19) dfdr (cid:12)(cid:12)(cid:12)(cid:12) /D W + 2 r W (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) − Mr (cid:19) Mr f (cid:18) | /D W | + 2 r W · /D W (cid:19) + (cid:18) − Mr (cid:19) dfdr (cid:18) | F | − (cid:18) /D W + 2 r W (cid:19) · F (cid:19) − (cid:18) − Mr (cid:19) Mr f F · /D W . We denote the last line by
Err ′ , [2 M,bM ] [ W , F ]. Similarly, using the substitution (6.9) to replace (cid:0) − Mr (cid:1) ∇ r W by − /D ∗ W − (cid:0) − Mr (cid:1) r W + F , we obtain (cid:18) − Mr (cid:19) dfdr |∇ r W | + 4 Mr ∇ X W · /D ∗ ˆ W = dfdr (cid:12)(cid:12)(cid:12)(cid:12) /D ∗ W + (cid:18) − Mr (cid:19) r W (cid:12)(cid:12)(cid:12)(cid:12) − Mr f (cid:18) | /D ∗ W | + (cid:18) − Mr (cid:19) r W · /D ∗ W (cid:19) + dfdr (cid:18) | F | − (cid:18) /D ∗ W + (cid:18) − Mr (cid:19) r W (cid:19) · F (cid:19) + 4 Mr f F · /D ∗ W . We denote the last line by
Err ′′ , [2 M,bM ] [ W , F ]. Then up to Err ′ , [3 M,bM ] [ W , F ], Err ′′ , [3 M,bM ] [ W , F ]and Err [ W , G ], div J , becomes a quadratic from involving W , W , /D ∗ W and /D W : fr (cid:18) − Mr (cid:19) | / ∇ W | + M ωr ˆ W · /D ∗ ˆ W + W · (cid:18) − (cid:3) ω −
12 [
X, A W ,Mor ] − Mr f A W ,Mor (cid:19) · W + (cid:18) − Mr (cid:19) dfdr (cid:12)(cid:12)(cid:12)(cid:12) /D W + 2 r W (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) − Mr (cid:19) Mr f (cid:18) | /D W | + 2 r W · /D W (cid:19) + dfdr (cid:12)(cid:12)(cid:12)(cid:12) /D ∗ W + (cid:18) − Mr (cid:19) r W (cid:12)(cid:12)(cid:12)(cid:12) − Mr f (cid:18) | /D ∗ W | + (cid:18) − Mr (cid:19) r W · /D ∗ W (cid:19) . On a fixed mode ℓ ≥ | m | ≤ ℓ , after integrated along S , it equals W · M [3 M,bM ] · W , where M [3 M,bM ] is a two by two matrix M [3 M,bM ] = M [ bM, ∞ ) + dfdr r (cid:20) Λ − (1 − s − ) √ Λ − (1 − s − ) √ Λ (1 − s − ) (cid:21) +(1 − s − ) dfdr r (cid:20) − √ Λ − √ Λ Λ (cid:21) + 2 Mr f (cid:20) −
2Λ 00 (1 − s − )Λ (cid:21) . Through computation, det[ r M [2 M,bM ] ] = p ( s )Λ + p ( s )Λ + p ( s ), where CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 47 p ( s ) =1 − s − + 15 s − + 42 s − + 7 s − − s − − s − + 3084 s − + 4032 s − ,p ( s ) = − s − − s − + 561 s − − s − − s − + 6834 s − + 3564 s − − s − − s − − s − ,p ( s ) =3 s − − s − + 252 s − + 354 s − − s − + 4095 s − − s − − s − + 6156 s − + 6804 s − . We check that for s ≥
2, 4 p ( s ) + 2 p ( s ) + p >
0, hence M [2 M,bM ] is positive definite for ℓ = 1.We further check that for s ≥ p ( s ), 4 p ( s ) + p ( s ) and 4 p ( s ) + p ( s ) are positive, hence M [2 M,bM ] is positive definite for ℓ ≥ M [2 M,bM ] is positive definite, we obtain for all r ≥ M ,div J , ≥ C Mor M − (cid:18) s − |∇ r W | + (1 − s − ) s − | / ∇ W | + M − s − | W | (cid:19) +Err [ W , G ] + Err , [ bM, ∞ ) [ W ] + Err ′ , [2 M,bM ) [ W , F ] + Err ′′ , [2 M,bM ) [ W , F ] . Moreover, from the view of (6.9) and (6.8), we can remove the degeneracy of | / ∇ W | at r = 3 M with the help of F and F . Thus we manage to show that for all r ≥ M ,div J , ≥ C Mor M − (cid:18) s − (1 − s − ) |∇ r W | + s − | / ∇ W | + M − s − | W | (cid:19) +Err [ W , G ] + Err , [ bM, ∞ ) [ W ] + Err , [2 M,bM ) [ W , F ] . with Err , [2 M,bM ) [ W , F ] := Err ′ , [2 M,bM ) [ W , F ] + Err ′′ , [2 M,bM ) [ W , F ] + C Mor M − ( | F | + | F | ) χ [2 M,bM ) . We proceed to add |∇ t W | into the bulk. Let ¯ f ( r ) ≥ r . Weconsider ( J , ) a = − ¯ f ∇ a | W | + ∇ a ¯ f | W | . (6.25)We compute div J , = − f ∇ a W · ∇ a W − f W · G + 14 (cid:3) ¯ f · | W | − f W · A W · W =2 ¯ f (cid:0) (1 − s − ) − |∇ t W | − (1 − s − ) |∇ r W | − | / ∇ W | (cid:1) − f W · G + 14 (cid:3) ¯ f · | W | − f W · A W · W By taking ¯ f ( r ) = ǫ , Mr (cid:18) − Mr (cid:19) , we can add |∇ t W | into the divergence by defining J := J , + J , . (6.26)Now (6.23) follows from the above discussion. The requirement (6.22) can be verified by theasymptotics of f ( r ) , ω ( r ) , ¯ f ( r ) and A W ,Mor . We finish this subsection by recording Err , [2 M,bM ) [ W , F ],which is supported in r ∈ [2 M, bM ). Err , [2 M,bM ) [ W , F ] = (cid:26)(cid:18) − Mr (cid:19) dfdr (cid:18) | F | − (cid:18) /D W + 2 r W (cid:19) · F (cid:19) − (cid:18) − Mr (cid:19) Mr f F · /D W + dfdr (cid:18) | F | − (cid:18) /D ∗ W + (cid:18) − Mr (cid:19) r W (cid:19) · F (cid:19) + 4 Mr f F · /D ∗ W + C Mor M − ( | F | + | F | ) (cid:27) · χ [2 M,bM ) . (6.27)6.4. r p -current. In this subsection, we construct a current J p for p ∈ [ δ, − δ ], which satisfiesthe following two properties. First, for any p ∈ [ δ, − δ ] E p, (0) L [ ˜ W ]( τ ) ≤ Z Σ τ J · n dvol ≤ E p, (0) L [ ˜ W ]( τ ) + F [ W ]( τ ) . (6.28)Here ˜ W = η null W and η null is the cut-off function defined in (2.22). Second, there exists aconstant C rp > τ ≥ τ and p ∈ [ δ, − δ ], Z D ( τ ,τ ) div J p dvol = Z D ( τ ,τ ) K p + Err [ W , G ] + Err , [ R null ,R null + M ] dvol. (6.29)Here K p ≥ C rp M − (cid:18) s p − |∇ L ( s ˜ W ) | + s p − | / ∇ ( s ˜ W ) | + s − − δ |∇ L ˜ W | + M − s p − | s ˜ W | (cid:19) ≥ C rp M − (cid:18) s p − |∇ L ˜ W | + s p − | / ∇ ˜ W | + s − − δ |∇ L ˜ W | + M − s p − | ˜ W | (cid:19) , Err p [ W , G ] = s p − η null (cid:18) − Mr (cid:19) − ∇ L ( s ˜ W ) + ǫ , η null s − δ ∇ t ˜ W ! · G ,Err p , [ R null ,R null + M ] = s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ W ) + ǫ , s − δ ∇ t ˜ W ! · ( (cid:3) η null · W + 2 ∇ η null · ∇ W ) . The term Err p [ W , G ] will be part of Err p [ W , G ]. The term Err p , [ R null ,R null + M ] comes from thederivative of the cut-off function η null and will be absorbed into K .Now we start the construction. We use C rp to stand for a constant which may increase fromline to line. Recall that A W , main = 1 r (cid:20) r /D /D ∗ (cid:21) CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 49 is the leading part of A W . Let A W , sub = s ( A W − A W , main ) be the subleading part and ˜ W = η null ( r ) W with η null defined in (2.22). Define( J p , ) a := s p − (cid:18) − Mr (cid:19) − (cid:18) T ab [ s ˜ W ] L b −
12 ( s ˜ W ) · ( A W , main + 2 M/r ) · ( s ˜ W ) | L a (cid:19) . (6.30)We computediv J p , = s M − s p − (cid:18) p (cid:18) − Mr (cid:19) − Mr (cid:19) (cid:18) − Mr (cid:19) − |∇ L ( s ˜ W ) | + M − s p − (cid:16) − p (cid:17) | / ∇ ( s ˜ W ) | − M − s p − ( s ˜ W ) · (cid:0) r (cid:2) ∂ r , ( A W , main + 2 M/r ) (cid:3) + p (( A W , main + 2 M/r )) (cid:1) · ( s ˜ W )+ s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ W ) · ( (cid:3) ˜ W − A W , main ˜ W ) . Since | / ∇ W | + W · A W , main · W is at least non-negative for ℓ ≥ | / ∇ W | for ℓ ≥
2, there exists a constant C rp such that M − s p − (cid:18) p (cid:18) − Mr (cid:19) − Mr (cid:19) (cid:18) − Mr (cid:19) − |∇ L ( s ˜ W ) | + M − s p − (cid:16) − p (cid:17) | / ∇ ( s ˜ W ) | − M − s p − ( s ˜ W ) · (cid:0) r (cid:2) ∇ r , ( A W , main + 2 M/r ) (cid:3) + p (( A W , main + 2 M/r )) (cid:1) · ( s ˜ W ) ≥ s C rp M − s p − (cid:16) |∇ L ( s ˜ W ) | + | / ∇ ( s ˜ W ℓ ≥ ) | + M − | ˜ W | (cid:17) . By Corollary 3.3, we can recover s p − | / ∇ ( s ˜ W ℓ =1 ) | ≈ s s p − M − | ˜ W ℓ =1 | through Hardy inequal-ity. Therefore, after integrated in D ( τ , τ ), the above is bounded from below as1 C rp M − s p − (cid:16) |∇ L ( s ˜ W ) | + | / ∇ ( s ˜ W ) | + M − | ˜ W | (cid:17) , with a larger constant C rp depending on δ . For the remaining terms we compute s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ W ) · ( (cid:3) ˜ W − A W , main ˜ W )= s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ W ) · ( s − A W , sub ˜ W + η null G )+ s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ W ) · ( (cid:3) η null ( r ) · W + 2 ∇ η null ( r ) · W ) . The contribution of s p − (1 − s − ) − ∇ L ( s ˜ W ) · A W , sub ˜ W has higher power in s − and can beabsorbed by ǫ · M − s p − |∇ L ( s ˜ W ) | + 1 ǫ · M − s p − | / ∇ ( s ˜ W ) | + 1 ǫ · M − s p − | s ˜ W | . Hence there exists R W such that for r ≥ R W , we can absorb s p − (1 − s − ) − ∇ L ( s ˜ W ) · A W , sub ˜ W into the positive terms above. We require R null ≥ R W . From now on we fix R null = max { R H , R Z , R W } and drop the dependence of R null in estimates. After integrated along D ( τ , τ ), div J p , isbounded from below by ≥ C rp M − s p − (cid:16) |∇ L ( s ˜ W ) | + | / ∇ ( s ˜ W ) | + M − | ˜ W | (cid:17) + s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ W ) · ( (cid:3) η null ( r ) · W + 2 ∇ η null ( r ) · W )+ s p − η null (cid:18) − Mr (cid:19) − ∇ L ( s ˜ W ) · G . The last line will be part of Err [ W , G ] and the second last line will be part of Err , [ R null ,R null+ M ] [ W ].To add the |∇ L W | term into the divergence, we consider( J , ) a := s − δ (cid:18) T ab [ ˜ W ] T b −
12 ˜ W · A W , main · ˜ W T a (cid:19) , (6.31)and compute div J , = δ M − s − − δ |∇ L ˜ W | − δ M − s − − δ |∇ L ˜ W | + η null ( r ) s − δ ∇ t ψ · ( (cid:3) η null ( r ) · W + 2 ∇ η null ( r ) · ∇ W )+ η null ( r ) s − δ ∇ t W · G + s − − δ ∇ t ˜ W · A W , sub ˜ W . The term involving A W , sub has higher power of s − and can be estimated as s − − δ ∇ t ˜ W · A W , sub ˜ W ≥ − δ M − s − − δ |∇ t ˜ W | − δ − M s − − δ | A W , sub ˜ W | ≥ − δ s − − δ M − ( |∇ L ˜ W | + |∇ L ˜ W | ) − C rp M − s − − δ | / ∇ ˜ W | . Hence div J , ≥ δ M − s − − δ |∇ L ˜ W | − δ M − s − − δ |∇ L ˜ W | + η null ( r ) s − δ ∇ t ψ · ( (cid:3) η null ( r ) · W + 2 ∇ η null ( r ) · ∇ W ) − C rp M − s − δ − | / ∇ ˜ W | + η null ( r ) s − δ ∇ t W · G . Take ǫ , small enough such that1 C rp M − s p − (cid:16) |∇ L ( s ˜ W ) | + | / ∇ ( s ˜ W ) | + M − | ˜ W | (cid:17) ≥ ǫ , · (cid:18) δ M − s − − δ |∇ L ˜ W | + C rp M − s − δ − | / ∇ ˜ W | (cid:19) . Then define J p = J p , + ǫ , J , . (6.32) CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 51
We have, after integrated along D ( τ , τ ), div J p is bounded from below by1 C rp M − s p − (cid:16) |∇ L ( s ˜ W ) | + | / ∇ ( s ˜ W ) | + M − | ˜ W | (cid:17) + 1 C rp M − s − − δ |∇ L ˜ W | + s p − (cid:18) − Mr (cid:19) − ∇ L ( s ˜ W ) + ǫ , s − δ ∇ t ˜ W ! · ( (cid:3) η null ( r ) · W + 2 ∇ η null ( r ) · ∇ W )+ s p − η null ( r ) (cid:18) − Mr (cid:19) − ∇ L ( s ˜ W ) + ǫ , η null ( r ) s − δ ∇ t ˜ W ! · G . We denote the last line by Err [ W , G ] and the second last one by Err , [ R null ,R null + M ] [ W ] andobtain (6.29). The estimate (6.20) holds easily from the form of J p .6.5. Proof of Proposition 6.1.
In this subsection we combine the currents above to construct J p W . We define J p W := C J + ǫ J + J + ǫ J p , (6.33) Err p [ W , G ] := C Err p [ W , G ] + ǫ Err [ W , G ] + Err [ W , G ] + ǫ Err p [ W , G ] , (6.34)with constants C , ǫ , ǫ > ǫ and ǫ small enough such that K + ǫ Err , [ r rs, W ,r + rs ] [ W ] + ǫ Err , [ R null ,R null + M ] [ W ] ≥ K . We take C large to ensure (6.4), (6.5) and (6.6) as follows. We ask C ≥ C Mor C T to get (6.4)from (6.12), (6.20), (6.22) and (6.28). To obtain (6.5), we calculate at r = 2 M , J · L = 12 |∇ L W | + 92 |∇ L ˆ W | + 12 |∇ L ˆ W | ,J · L = f |∇ L W | . Here we used T = (1 − s − ) ∂∂r = L and ω, ¯ f , ∇ L ω, ∇ L ¯ f all vanish at r = 2 M . Together with J · L ≥ J = 0 at r = 2 M , (6.5) follows by requiring C ≥ | f (2 M ) | . To guarantee (6.6),we calculate for any r ≥ M , Z S ( τ,r ) J · L r dvol S & Z S ( τ,r ) (cid:0) |∇ L W | + | / ∇ W ℓ ≥ | (cid:1) r dvol S , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z S ( τ,r ) J · L r dvol S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Z S ( τ,r ) (cid:0) |∇ L W | + | / ∇ W | (cid:1) r dvol S . The only obstruction to make whole thing positive is | / ∇ W ℓ =1 | , which can be controlled as Z S ( τ,r ) | / ∇ W ℓ =1 | r dvol S ≈ Z S ( τ,r ) | W ℓ =1 | dvol S . s − (2 − δ ) M − E − δ, (0) L [ ˜ W ]( τ ) . for any r ≥ R null + M . Here we used Corollary 3.3 with p = 1 − δ . Therefore this term goes to zeroas r goes to infinity by assumption 6.3. Together with J p · L ≥ J = 0 for r ≥ R null + M ,(6.6) follows for C large enough and the value of C is then determined. We can now fix the value of R T , which will be large enough such that the Err , [ R T , ∞ ) [ W ]term in div J can be absorbed into K p . To begin, we require R T M ≥ (cid:18) C C rp ǫ (cid:19) /δ . Then for r ≥ R T and p ∈ [ δ, − δ ], C (cid:12)(cid:12)(cid:12)(cid:12) η ′ T |∇ L W | (cid:12)(cid:12)(cid:12)(cid:12) ≤ C M − s − |∇ L W | ≤ C rp M − s δ − |∇ L W | ≤ ǫ K p . Next, we absorb the term ∇ t ˆ W /D ˆ W into K p for r large. (cid:12)(cid:12)(cid:12)(cid:12) Mr ∇ t ˆ W · /D ∗ ˆ W (cid:12)(cid:12)(cid:12)(cid:12) . M − s − |∇ t ˆ W | + M − s − | / ∇ ˆ W | , which falls off faster than K p for any p ≥ δ . We further require R T to be large enough such thatfor r ≥ R T , C (cid:12)(cid:12)(cid:12)(cid:12) Mr ∇ t ˆ W · /D ∗ ˆ W (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ K p . The value of R T is then determined. We proceed to fix the value of b . Besides b ≥ max { s Mor , R T / } ,We further require that for all r ≥ bM and p ∈ [ δ, − δ ], | Err , [ bM, ∞ ) [ W ] | ≤ ǫ K p . This can be done since | Err , [ bM, ∞ ) [ W ] | is bounded by M − (cid:18) s − |∇ r W | + s − | / ∇ W | + M − s − | W | (cid:19) , which decays faster than K p . The value of b is now determined and from now on we drop thedependence of b in estimates. We have managed to show that div J p W , after integrated along D ( τ , τ ), is bounded from below by Z D ( τ ,τ ) ǫ K + 12 K + ǫ K p dvol + Z D ( τ ,τ ) Err p [ W , G ] + Err , [2 M, R T ] [ W , F ] + Err , [2 M,bM ] [ W , F ] dvol. We turn to estimating
Err , [2 M, R T ] [ W , F ] and Err , [2 M,bM ] [ W , F ] define in subsection 6.1 andsubsection 6.3. Note that ˆ W = W for r ≤ bM . From (6.10), in [2 M, R T ] ⊂ [2 M, bM ], |∇ t ˆ W | = |∇ t W | = | F | . Similarly, from (6.8), | /D ∗ ˆ W | = | /D ∗ W | . |∇ r W | + M − s − | W | + | F | . Thus through Cauchy-Schwarz, one has for any ǫ > CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 53 (cid:12)(cid:12)(cid:12)(cid:12)Z Σ τ Err , [2 M, R T ] [ W , F ] dvol (cid:12)(cid:12)(cid:12)(cid:12) . ǫB [ W ]( τ ) + 1 ǫ Z Σ τ M − ( | F | + | F | ) · χ [2 M, R T ] dvol . From (6.27) and Cauchy-Schwarz, we have (cid:12)(cid:12)(cid:12)(cid:12)Z Σ τ Err , [2 M,bM ] [ W ] dvol (cid:12)(cid:12)(cid:12)(cid:12) . ǫB [ W ] + 1 ǫ Z Σ τ M − ( | F | + | F | ) · χ [2 M,bM ] dvol . Thus, by choosing ǫ > J p W in D ( τ , τ ) by Z D ( τ ,τ ) (cid:18) ǫ K + 12 K + ǫ K p (cid:19) dvol + Z D ( τ ,τ ) Err p [ W , G ] dvol − C Z D ( τ ,τ ) M − ( | F | + | F | + | F | ) · χ [2 M,bM ] dvol. Next we deal with the term Err p [ W , G ].Err p [ W , G ] = C · ( ∇ t W · G + η T · (9 ∇ t W · G + ∇ t W · G )) + ǫ · ( ∇ Y W · G )+ (cid:18) ∇ X W + 12 ω W − f W (cid:19) · G + ǫ · s p − η null ( r ) (cid:18) − Mr (cid:19) − ∇ L ( ˜ s W ) + ǫ , η null ( r ) s − δ ∇ t ˜ W ! · G . From Cauchy-Schwarz, we have for any p ∈ [ δ, − δ ] and ǫ > p [ W , G ] . ǫ (cid:18) ǫ K + 12 K + ǫ K p (cid:19) + 1 ǫ M s p +1 | G | . Again through picking ǫ > J p W in D ( τ , τ ) is boundedfrom below by Z D ( τ ,τ ) (cid:18) ǫ K + 12 K + ǫ K p (cid:19) dvol − C Z D ( τ ,τ ) M s p +1 | G | dvol − C Z D ( τ ,τ ) M − ( | F | + | F | + | F | ) · χ [2 M,bM ] dvol. Proposition 6.2.
Let W be a solution of (6.1) which also satisfies (6.8) , (6.9) and (6.10) .Further assume that F [ W ](0) + E − δ, (0) L [ W ](0) < ∞ , and Z D ( τ ,τ ) M s − δ | G | dvol < ∞ , for any τ ≥ τ ≥ . Then we have for any p ∈ [ δ, − δ ] F [ W ]( τ ) + E p, (0) L [ W ]( τ ) + M − Z τ τ ¯ B [ W ]( τ ) + E p − , (0) L, / ∇ [ W ]( τ ) dτ . F [ W ]( τ ) + E p, (0) L [ W ]( τ ) + Z D ( τ ,τ ) M s p +1 | G | dvol + Z D ( τ ,τ ) M − ( | F | + | F | + | F | ) · χ [2 M,bM ] dvol. (6.35) Proof.
Clearly the integrand of ¯ B [ W ] and E p − , (0) [ W ] is bounded by ǫ K + K + ǫ K p . Byapplying divergence theorem (2.14) to J p W , it’s sufficient to show that the assumption (6.3) holds.By applying the divergence theorem (2.14) to J − δ in D (0 , τ ), we have Z Σ τ J − δ · n dvol + Z D (0 ,τ ) K − δ dvol ≤ Z Σ J − δ · n dvol + Z D (0 ,τ ) − Err − δ [ W , G ] − Err − δ , [ R null ,R null + M ] [ W ] dvol. Here we drop the boundary term along null infinity, which is non-negative. Through Cauchy-Schwarz, we have for any ǫ > − δ [ W , G ] = s − δ η null ( r ) (cid:18) − Mr (cid:19) − ∇ L ( s ˜ W ) + ǫ , η null ( r ) s − δ ∇ t W ! · G . ǫM − (cid:0) s − − δ |∇ L ( s ˜ W ) | + s − − δ |∇ t ˜ W | (cid:1) + 1 ǫ M ( s − δ + s − δ +1 ) | G | . ǫK − δ + 1 ǫ M s − δ | G | . By taking ǫ >
0, we get Z Σ τ J − δ · n dvol + Z D (0 ,τ ) K − δ dvol . Z Σ J − δ · n dvol + Z D (0 ,τ ) M s − δ | G | − Err − δ , [ R null ,R null + M ] [ W ] dvol. The term
Err − δ , [ R null ,R null + M ] [ W ] is supported in [ R null , R null + M ] and can be shown to havebounded integral by using e − Ct T as multiplier with C > E − δ, (0) [ ˜ W ]( τ ) and its integral is finite for any τ ≥
0, which is the assumption (6.3). (cid:3) proof of Proposition 6.1.
From the view of (6.35), it suffices to estimate Z D ( τ ,τ ) M s p +1 | G | dvol + Z D ( τ ,τ ) M − ( | F | + | F | + | F | ) · χ [2 M,bM ] dvol. From the view of Lemma 4.10, the term
M s p +1 | G | can be bounded by the integrand of B [ S, D ≤ r − ψ Z ]and E p − , (0) L, / ∇ [ S, D ≤ r − ψ Z ] in the regions r ∈ [2 M, bM ] and r ∈ [2 bM, ∞ ) respectively. FromProposition A.4, CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 55 Z D ( τ ,τ ) M s p +1 | G | dvol . (cid:16) τ M (cid:17) − p + δ M I (0) [ S, D ≤ r − ψ Z ] . We don’t keep the dependence on b and R null since their values are already fixed. Now we turnto | F i | terms. Through the definition of F , F and F in (6.11) , M − ( | F | + | F | + | F | ) · χ [2 M,bM ] . the intrgrand of (cid:18) M − B [ K ≤ ˆ W , ˆ P even ]+ M B [ D S, D ≤ r − ψ Z , Q even ] (cid:19) . Thus from Propositions 5.1, 5.6, 5.9 and Proposition A.4, we have Z D ( τ ,τ ) M − ( | F | + | F | + | F | ) · χ [2 M,bM ] dvol . (cid:16) τ M (cid:17) − p +3 δ · Decay ′ [ W ] . Here Decay ′ [ W ] = I (0) [ K ≤ ˆ W , K ≤ ˆ P even ] + M I (0) [ D ≤ K ≤ S, D ≤ K ≤ r − ψ Z , Q even ]+ M I (1) [ K ≤ S, K ≤ r − ψ Z ] . This together with (6.35) and (3.4) implies that for any p ∈ [1 , − δ ] and τ ≥ τ ≥
0, we have F [ W ]( τ ) + E p, (0) L [ W ]( τ ) + M − Z τ τ F [ W ]( τ ) + E p − , (0) L, / ∇ [ W ]( τ ) dτ . F [ W ]( τ ) + E p, (0) L [ W ]( τ ) + (cid:16) τ M (cid:17) − p +3 δ · Decay ′ [ W ] . (6.36)Furthermore, as p ∈ [ δ, F [ W ]( τ ) + E p, (0) L [ W ]( τ ) + M − Z τ τ E p − , (0) L, / ∇ [ W ]( τ ) dτ . F [ W ]( τ ) + E p, (0) L [ W ]( τ ) + (cid:16) τ M (cid:17) − p +3 δ · Decay ′ [ W ] . (6.37)The result then follows through an elementary argument we sketch below. By taking ¯ τ k := 2 k M and applying (6.36) with τ = ¯ τ k +1 , τ = ¯ τ k and p = 2 − δ together with the mean value theorem,we get a sequence ˜ τ k ≈ k M such that F [ W ](˜ τ k ) + E − δ, (0) L [ W ](˜ τ k ) . (cid:18) τ k M (cid:19) − (cid:16) F [ W ](¯ τ k ) + E − δ, (0) L [ W ](¯ τ k ) + Decay ′ [ W ] (cid:17) . (cid:18) τ k M (cid:19) − Decay[ W ] . Here we used (6.36) with τ = ¯ τ k , τ = 0 and p = 2 − δ to estimate F [ W ](¯ τ k ) + E − δ, (0) L [ W ](¯ τ k )and Decay[ W ] is defined in Proposition 6.1. Through (6.37) with p = 1 − δ and τ = τ ≥ ˜ τ k = τ , F [ W ]( τ ) + E − δ, (0) L [ W ]( τ ) . (cid:18) τ k M (cid:19) − Decay[ W ] . As each τ ≥ τ k , τ k ] or [0 , M ], we conclude for all τ ≥ F [ W ]( τ ) + E − δ, (0) L [ W ]( τ ) . (cid:16) τM (cid:17) − Decay[ W ] . Through interpolation we have for p ∈ [1 − δ, − δ ], F [ W ]( τ ) + E p, (0) L [ W ]( τ ) . (cid:16) τM (cid:17) − p +3 δ Decay[ W ] . Repeating the argument one more time gives the result. (cid:3) Proof of Theorem 2.4 and 2.5
In this section we prove Theorem 2.4 and 2.5. We first use ψ Z and W to bound h in the region r ∈ [ r rs, H , R null + M ], which leads to the decay of F [ h ]( τ ) from the view of Propositions 4.1, 5.4,5.1 and 6.1. Then the main results follow by expressing the initial norms of r − ψ Z , ˆ P even and soon in terms of h or W . Proposition 7.1.
Let h ab be an even solution of (1.3) and (1.4) supported on ℓ ≥ . Furtherassume that Decay ′ [ h ] defined below is finite. Then for any p ∈ [ δ, − δ ] and τ ≥ τ ≥ wehave F [ h ]( τ ) + E p, (0) L [˜ h ]( τ ) + Z τ τ ¯ B [ h ]( τ ) + E p − , (0) L, / ∇ [˜ h ]( τ ) dτ . F [ h ]( τ ) + E p, (0) L [˜ h ]( τ ) + (cid:16) τ M (cid:17) − δ Decay ′ [ h ] . Here
Decay ′ [ h ] := M − F [ K ≤ W ](0) + M − E − δ, (0) L [ K ≤ W ](0) + M − I (0) [ K ≤ ˆ W , K ≤ ˆ P even ]+ I (0) [ D ≤ K ≤ S, D ≤ K ≤ r − ψ Z , K ≤ Q even ] + I (1) [ K ≤ S, K ≤ r − ψ Z ] , and ˜ h = η null h with η null ( r ) defined in (2.22) and the initial norm I ( k ) [ · ] is defined in (2.23) .Proof. From Proposition 4.1, it suffices to show that M − Z D ( τ ,τ ) ∩{ r rs, H ≤ r ≤ R nul + M } | ( M ∂ ) ≤ h | dvol . (cid:16) τ M (cid:17) − δ Decay ′ [ h ] . From h = h RW − W π , (2.11) and (4.17), we have in r ∈ [ r rs, H , R nul + M ], M − | ( M ∂ ) ≤ h | . M − |D ≤ r − ψ Z | + M − | ( M ∂ ) ≤ W | . From (3.1), (4.21), (4.22) and (4.23), the above is bounded by M − |D ≤ r − ψ Z | + M − | ( M ∂ ) ≤ K ≤ W | + M − | ( M ∂ r ) K ≤ W | + M − |K ≤ W | , which is bounded by the integrand of M − B [ D ≤ r − ψ Z ] + M − ¯ B [ K ≤ W ] + M − B [ K ≤ ˆ W ].Then the result follows from Proposition 5.4, 5.1 and 6.1. (cid:3) CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 57 proof of Theorem 2.4 (1).
From the previous proposition and the dyadic argument at the end ofsection 6, the result holds with Decay ℓ ≥ [ h ] replaced by Decay ′ [ h ] + F [ h ](0) + E − δ, (0) L [˜ h ](0). As F [ h ](0) + E − δ, (0) L [˜ h ](0) is clearly bounded by Decay ℓ ≥ [ h ], it remains to show that Decay ′ [ h ] isalso bounded by Decay ℓ ≥ [ h ]. Fix an integer m ≥
0. From the definition of W , W and W ,(4.11), (4.12) and (4.13), we have | ( M ∂ ) ≤ m W | , | ( M ∂ ) ≤ m W | , | ( M ∂ ) ≤ m W | . s,m | M s ( M ∂ ) ≤ m +1 h | . From the definition of ψ Z , P even and Q even , (4.15), (4.19) and (4.20), we have | ( M ∂ ) ≤ m r − ψ Z | . s,m | s ( M ∂ ) ≤ m +1 h | , | ( M ∂ ) ≤ m P even | . s,m | M s ( M ∂ ) ≤ m +1 h | , | ( M ∂ ) ≤ m Q even | . s,m | s ( M ∂ ) ≤ m +3 h | . Note that P even can be bounded by one instead of two more derivative of h since there’s acancellation in the definition of P even (4.19) from (4.11) and (4.12). Similarly, even though Q even in (4.20) involves three more angular derivative of ψ Z , from (4.15) the angular derivative willactually cancels and Q even can be bounded by three more derivatives of h . Then along Σ , | ( M ∂ ) ≤ m ˆ W | , | ( M ∂ ) ≤ m ˆ W | , | ( M ∂ ) ≤ m ˆ W | , | ( M ∂ ) ≤ m ˆ P even | . s,m | M s ( M ∂ ) ≤ m +2 h | . ThusDecay ′ [ h ] = M − F [ K ≤ W ](0) + M − E − δ, (0) L [ K ≤ W ](0) + M − I (0) [ K ≤ ˆ W , K ≤ ˆ P even ]+ I (0) [ D ≤ K ≤ S, D ≤ K ≤ r − ψ Z , K ≤ Q even ] + I (1) [ K ≤ S, K ≤ r − ψ Z ] . I (0) [ s ( M ∂ ) ≤ K ≤ h, s ( M ∂ ) ≤ K ≤ h, s D ≤ ( M ∂ ) ≤ K ≤ h ]+ I (1) [ s ( M ∂ ) ≤ K ≤ h ]=Decay ℓ ≥ [ h ] . And the proof is finished. (cid:3) proof of Theorem 2.4 (2).
In this proof we assume all quantities are supported on the mode ℓ = 1. Similar to the proof of Theorem 2.4 (1), it suffices to show that Decay ′ [ h ] is bounded byDecay ℓ =1 [ h ]. From the definition of W , W and W , (4.43), (4.44) and (4.45) and noting thethe /D , /D ∗ behaves like r − on fixed mode, we have | ( M ∂ ) ≤ m W | , | ( M ∂ ) ≤ m W | , | ( M ∂ ) ≤ m W | . s,m | M s ( M ∂ ) ≤ m +1 h | . From the definition of P even and Q even , (4.19) and (4.20), we have | ( M ∂ ) ≤ m P even | . s,m | M s ( M ∂ ) ≤ m +2 h | , | ( M ∂ ) ≤ m Q even | . s,m | s ( M ∂ ) ≤ m +2 h | . Hence with ψ Z ≡
0, along Σ we have | ( M ∂ ) ≤ m ˆ P even | . s,m | M s ( M ∂ ) ≤ m +2 h | . Thus,Decay ′ [ h ] = M − F [ K ≤ W ](0) + M − E − δ, (0) L [ K ≤ W ](0) + M − I (0) [ K ≤ W , K ≤ ˆ P even ]+ I (0) [ D ≤ K ≤ S, K ≤ Q even ] + I (1) [ K ≤ S ] . I (0) [ s ( M ∂ ) ≤ K ≤ h, D ≤ K ≤ h ] + I (1) [ K ≤ h ]=Decay ℓ =1 [ h ] . (cid:3) proof of Theorem 2.5. We recall that in this case r − ψ Z is zero, ˆ W i = W i and S = − S W .From Proposition 5.1 and Proposition 6.1, it suffices to show that Decay[ ˆ W ] and Decay[ W ] arebounded by Decay ℓ ≥ [ W ]. In particular, Decay[ ˆ W ] ≤ Decay[ W ] as we used W in controlling W and W . Hence it is enough to deal with Decay[ ˆ W ]. From the definition of S W , P even , ˆ P even and Q even , we have along Σ , | ( M ∂ ) ≤ m S W | . m | M − ( M ∂ ) ≤ m +1 W | , | ( M ∂ ) ≤ m P even | , | ( M ∂ ) ≤ m ˆ P even | . m | s ( M ∂ ) ≤ m +1 W | , | ( M ∂ ) ≤ m Q even | . m | M − s ( M ∂ ) ≤ m +2 W | . Hence Decay[ W ] = F [ W ](0) + E − δ, (0) L [ W ](0) + I (0) [ K ≤ ˆ W , K ≤ ˆ P even ]+ M I (0) [ K ≤ D ≤ S, Q even ] + M I (1) [ K ≤ S ] . I (0) [ s ( M ∂ ) ≤ K ≤ W, D ≤ ( M ∂ ) ≤ K ≤ W ] + I (1) [( M ∂ ) ≤ K ≤ W ]=Decay ℓ ≥ [ W ] . (cid:3) The ℓ = 0 mode In this section we turn to the mode ℓ = 0. We start with equation (1.5) with sphericallysymmetric solutions. Then we discuss the ℓ = 0 mode of equations (1.3) and (1.4) in subsection8.3. To begin, we estimate the trace of the deformation tensor, S W , and P even as before usingvector field method in Appendix A. However, it’s impossible to show W a , a solution of (1.5)supported on ℓ = 0, decays as there exists a stationary solution W ∗ with finite initial energy.The explicit form of W ∗ is given by W ∗ a dx a := 1 r dt + 2 Mr (cid:18) − Mr (cid:19) − dr. (8.1)Therefore we adapt a different approach.Through equation (1.5), one can show that provided P even and S W both vanish, W a is actuallystationary ((8.12) and (8.23)). Then the wave equation, from the view of (2.3), becomes a secondorder ODE in ρ . This ODE has two explicit linear independent solutions but only one is smoothupto the horizon ρ = 2 M , which produces the stationary solution we discussed.With the estimate of P even and S W , we can follow the argument above by introducing errorterms, denoted by O j , that would vanish if P even and S W do. Instead of showing there’s no sin-gular part, we will argue that the coefficient of the singular solution decays through the red-shift CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 59 estimate.Let W a be an spherically symmetric solution of (1.5). It has only two components W a dx a = W dt + (cid:0) − Mr (cid:1) − W dr , and W and W are spherical symmetric functions. Similar to (4.21)and (4.22) The equation (1.5) can be rewritten as (cid:3) W + 2 Mr W = − Mr P even , (8.2) (cid:3) W − r (cid:18) − Mr (cid:19) W = Mr S W . (8.3)Here P even and S W are defined as in (4.19) and in (4.18) with W ≡
0. Similar to (4.25) and(4.24), P even and S W satisfies the wave equations(8.4) (cid:3) P even = ∇ t S W , (8.5) (cid:3) S W = 0 . Recall that b ≥ R null /M is a large number determined in subsection 6.5 and η b ( r ) is a cut-offfunction which equals zero in [2 M, bM ] and equals one in [2 bM, ∞ ). As in section 4, defineˆ P even := P even + η b · (cid:18) u + r S W (cid:19) . (8.6)Now we state the main result for the spherically symmetric solutions of (1.5). Theorem 8.1.
Let W a be an even solution of (1.5) and is supported on ℓ = 0 . Recall that W ∗ is a stationary solution defined in (8.1) . Let S W , P even and ˆ P even be defined as in (4.18) , (4.19) and (4.35) with ψ Z ≡ . Then c ( ∞ ) := − M ( M S W − P even − W ) (cid:12)(cid:12)(cid:12)(cid:12) H + is a constant. Further suppose that Decay ℓ =0 [ W ] defined below is finite. Then ˆ W := W − c ( ∞ ) W ∗ (8.7) converges to zero with the estimate M − Z Σ ′′ τ | ( M ∂ ) ≤ ˆ W | dvol + M − Z D ′′ ( τ, ∞ ) | ( M ∂ ) ≤ ˆ W | dvol . (cid:16) τM (cid:17) − δ Decay ℓ =0 [ W ] . Here
Decay ℓ =0 [ W ] := F [ D ≤ W ](0) + (cid:18) F [ D ≤ ˆ P even ] + E − δ, (0) L [ D ≤ ˆ P even ] (cid:19) (0)+ (cid:18) F [ D ≤ S W ] + E − δ, (0) L [ D ≤ S W ] (cid:19) (0) . Analysis of S W and P even . By applying Proposition A.6 (1) to (8.5), we obtain
Proposition 8.2.
Let m ≥ be a fixed integer. For any p ∈ [ δ, − δ ] and τ ≥ , we have F [ D ≤ m S W ]( τ ) + E p, (0) L [ D ≤ m ˜ S W ]( τ ) + M − Z ∞ τ ¯ B [ D ≤ m S W ]( τ ′ ) + E p − L [ D ≤ m ˜ S W ]( τ ′ ) dτ ′ . m (cid:16) τM (cid:17) − δ + p (cid:16) F [ D ≤ m S W ] + E − δ, (0) L [ D ≤ m ˜ S W ] (cid:17) (0) , provided the right hand side is finite. Let G P := (cid:3) ˆ P even . To apply Proposition A.6 (2) to ˆ P even , we need a bound I source ,ℓ =0 ,δ [ D ≤ m G P ]defined in (A.12). Similar to lemma 4.8, we have Lemma 8.3.
Fix an integer m ≥ . In r ∈ [2 M, bM ] , we have |D ≤ m G P | . m M − ( | ( M ∂ ) ≤ D ≤ m S W | . As r ∈ [ bM, bM ] , we have |D ≤ m G P | . m M − (cid:16) τM (cid:17) | ( M ∂ ) ≤ D ≤ m S W | . And as r ∈ [2 bM, ∞ ) , |D ≤ m G P | . s − |∇ L D ≤ m S W | + M − s − |D ≤ m S W | . The direct consequence isI source ,ℓ =0 ,δ [ D ≤ m G P ] . m M (cid:16) F [ D ≤ m S W ] + E − δ, (0) L [ D ≤ m ˜ S W ] (cid:17) (0) . Therefore, from Proposition A.6 (2), we have
Proposition 8.4.
Let m ≥ be a fixed integer. Further assume that Decay ℓ =0 [ D ≤ m ˆ P even ] defined below is finite. Then for any p ∈ [ δ, − δ ] and τ ≥ , F [ D ≤ m ˆ P even ]( τ ) + E p, (0) L [ D ≤ m ˆ P even ]( τ ) + Z ∞ τ ¯ B [ D ≤ m ˆ P even ]( τ ′ ) + E p, (0) L [ D ≤ m ˆ P even ]( τ ′ ) dτ ′ . m (cid:16) τM (cid:17) − δ + p · Decay ℓ =0 [ D ≤ m ˆ P even ] . Here
Decay ℓ =0 [ D ≤ m ˆ P even ] = (cid:16) F [ D ≤ m ˆ P even ] + E − δ, (0) L [ D ≤ m ˆ P even ] (cid:17) (0) + M (cid:16) F [ D ≤ m S W ] + E − δ, (0) L [ D ≤ m S W ] (cid:17) (0) . Analysis of W . Define W as W := (cid:18) − Mr (cid:19) − ( W − W ) . (8.8)Since on the horizon dt = (cid:0) − Mr (cid:1) − dr , W is smooth upto the horizon. From direct compu-tation, P even and S W can be rewritten as CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 61 P even = r ∇ t W + r ∇ L ′ W − W , (8.9) S W = − ∇ L ′ W + 2 (cid:18) − Mr (cid:19) ∇ r W + 4 r W + 4 r (cid:18) − Mr (cid:19) W . (8.10)and W satisfies the equation(8.11) (cid:3) W − Mr ∇ L ′ W − r (cid:18) − Mr (cid:19) W − r W = 0 . Note that the time derivative of W can be controlled by P even and S W through ∇ t W = r (cid:18) − Mr (cid:19) ∇ r S W − ∇ t P even . (8.12)Here we used the equation for W as well as spherical symmetry of W . We denote(8.13) O := r (cid:18) − Mr (cid:19) ∇ r S W − ∇ t P even . Recall g ( r ) is the function such that Σ τ = { t + r + 2 M log( r/ M − − g ( r ) = τ } and that ρ = r together with τ, θ and φ forms a coordinate. From (2.3), for spherical symmetric functions, (cid:3) = (cid:18) − Mr (cid:19) ∂ ∂ρ + 2 r (cid:18) − Mr (cid:19) ∂∂ρ + Z ∂∂τ , with Z = − dgdr + (cid:18) − Mr (cid:19) (cid:18) dgdr (cid:19) ! ∂∂τ + (cid:18) − (cid:18) − Mr (cid:19) dgdr (cid:19) ∂∂ρ + (cid:18) − (cid:18) − Mr (cid:19) d gdr − r (cid:18) − Mr (cid:19) dgdr + 2 r (cid:19) . Then (8.2) can be rewritten as(8.14) (cid:18) − Mr (cid:19) ∂ ρ M + 2 r (cid:18) − Mr (cid:19) ∂ ρ M + 2 Mr M = O , where O := − Z · O − Mr P even . (8.15)We view (8.14) as a second order ODE with source term O . The homogeneous solutions are W , ( ρ ) = 1 r ,W , ( ρ ) = 1 + 2 Mr log (cid:16) r M − (cid:17) . (8.16)The Wronskian of W , and W , is (cid:12)(cid:12)(cid:12)(cid:12) W , W , W ′ , W ′ , (cid:12)(cid:12)(cid:12)(cid:12) = 1 r (cid:18) − Mr (cid:19) − . Then W can be expressed as(8.17) W ( τ, ρ ) = c ( τ ) W , ( ρ ) + c ( τ ) W , ( ρ ) + O , where c ( τ ) = r (cid:18) − Mr (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) W ( τ, M ) W , (4 M ) ∂ ρ W ( τ, M ) ∂ ρ W , (4 M ) (cid:12)(cid:12)(cid:12)(cid:12) = − M ∂ ρ W ( τ, M ) + 2 M W ( τ, M ) , (8.18) c ( τ ) = r (cid:18) − Mr (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) W , ( τ, M ) W (4 M ) ∂ ρ W , ( τ, M ) ∂ ρ W (4 M ) (cid:12)(cid:12)(cid:12)(cid:12) =2 M · ∂ ρ W ( τ, M ) + 12 W ( τ, M ) . (8.19)And O ( τ, ρ ) = Z ρ M − ( ρ ′ ) W , ( ρ ′ ) O ( τ, ρ ′ ) dρ ′ · W , ( ρ ) + Z ρ M ( ρ ′ ) W , ( ρ ′ ) O ( τ, ρ ′ ) dρ ′ · W , ( ρ ) . (8.20)We will show that c ( τ ) converges to zero. The time derivative of c ( τ ) and c ( τ ) are alreadycontrolled as c ′ ( τ ) = − M ∂ ρ O ( τ, M ) + 2 M O ( τ, M ) =: O c , (8.21)and c ′ ( τ ) =2 M · ∂ ρ O ( τ, M ) + 12 O ( τ, M ) =: O c . (8.22)By (8.9) and ∂ ρ = − L ′ + dgdr ∂ τ , we obtain P even = r∂ τ W + r dgdr ∂ τ W − r∂ ρ W − W . Equivalently, ∂ τ W = 1 r P even − dgdr ∂ τ W + ∂ ρ W + 1 r W = (cid:18) r P even − dgdr O + (cid:18) ∂ ρ + 1 r (cid:19) O (cid:19) + c ( τ ) · r (cid:18) − Mr (cid:19) − = c ( τ ) r (cid:18) − Mr (cid:19) − + O , where O is defined by the last equality and we used (8.19) in the second equality. By takingone more time derivative, we have CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 63 ∂ τ W = O c · r (cid:18) − Mr (cid:19) − + ∂ τ O =: O . By L ′ = dgdr ∂ τ − ∂ ρ and (2.3), (8.11) can be written as0 = (cid:18) − Mr (cid:19) ∂ ρ W + 2 r ∂ ρ W + (cid:18) Z − Mr g ′ (cid:19) ∂ τ W − r (cid:18) − Mr (cid:19) W − r W . Taking one more time derivative, we obtain0 = (cid:20)(cid:18) − Mr (cid:19) ∂ ρ + 2 r ∂ ρ − r (cid:18) − Mr (cid:19)(cid:21) ∂ τ W + (cid:18) Z − Mr g ′ (cid:19) ∂ τ W − r ∂ τ W = (cid:18)(cid:18) − Mr (cid:19) ∂ ρ + 2 r ∂ ρ − r (cid:18) − Mr (cid:19)(cid:19) r (cid:18) − Mr (cid:19) − c ( τ ) + O ! + (cid:18) Z − Mr g ′ (cid:19) O − r O = − r c ( τ ) + O . Here O is defined by the last equality as O := (cid:18)(cid:18) − Mr (cid:19) ∂ ρ + 2 r ∂ ρ − r (cid:18) − Mr (cid:19)(cid:19) O + (cid:18) Z − Mr g ′ (cid:19) O − r O . In particular, c ( τ ) ≡ S W and P even vanish. In general, c ( τ ) decays to zero if O do.Moreover, by using c ( τ ) = r / ∗ O , we have W ( τ, ρ ) = c ( τ ) ρ + O , O := 2 r O W , ( ρ ) + O , and ∂ τ W = (cid:18) r O (cid:19) r (cid:18) − Mr (cid:19) − + O =: O . (8.23)From (8.10) and L ′ = dgdr ∂ τ − ∂ ρ , we have2 (cid:18) − Mr (cid:19) ∂ ρ W + 4 r (cid:18) − Mr (cid:19) W = S W + 2 dgdr ∂ τ W − ∂ ρ W + (cid:18) − (cid:18) − Mr (cid:19) dgdr (cid:19) ∂ τ W − r W = S W + 2 dgdr O − ∂ ρ O + (cid:18) − (cid:18) − Mr (cid:19) dgdr (cid:19) O − r O − c ( τ ) r =: − c ( τ ) r + O . We view (8.24) 2 (cid:18) − Mr (cid:19) ∂ ρ W + 4 r (cid:18) − Mr (cid:19) W = − c ( τ ) r + O as an ODE. A homogeneous solution is r (cid:0) − Mr (cid:1) − and a particular solution for the − c ( τ ) r source term is − c ( τ ) r . Therefore, W ( τ, ρ ) = − c ( τ ) r + d ( τ ) r (cid:18) − Mr (cid:19) − + O . Here O ( τ, ρ ) = Z ρ M
12 ( ρ ′ ) O ( τ, ρ ′ ) dρ ′ · ρ − (cid:18) − Mρ (cid:19) − , and d ( τ ) = 8 M ( W ( τ, M ) + c ( τ )4 M ) . We would show that d ( τ ) decays to zero and c ( τ ) converges to a constant as τ goes to infinity.Let ¯ W := W − c ( τ ) r , (8.25) ¯ W := W + c ( τ ) r , (8.26) W ℓ =0 :=( ¯ W , ¯ W ) . (8.27)We remark that ¯ W = O , (8.28) ¯ W = d ( τ ) r (cid:18) − Mr (cid:19) − + O . (8.29)As d ( τ ) is the coefficient of a singular function, it must be zero if O = 0. In general, we usered-shift to obtain integrated decay estimate of d ( τ ) and d ′ ( τ ).By direct computation, we have (cid:3) ¯ W + 2 Mr ¯ W = O , (8.30) (cid:3) ¯ W − Mr ∇ L ′ ¯ W − r (cid:18) − Mr (cid:19) ¯ W − r ¯ W = O . (8.31)Here O = − Z ( r − · O c ) − Mr P even , (8.32) O = (cid:18) Z − Mr dgdr (cid:19) ( r − · O c ) . (8.33) CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 65
Let σ > η rs ( r ) be the cur-off function defined in (2.6) and Y ( σ ) be thered-shit vector field defined in (2.7). We consider(8.34) ( J ℓ =0 ) a := (cid:18) T ab [ W ℓ =0 ] − r | W ℓ =0 | g ab (cid:19) ( Y b + η rs T b ) . Then from (2.13), on the horizon r = 2 M ,div J ℓ =0 (cid:12)(cid:12)(cid:12)(cid:12) r =2 M = σ |∇ v W ℓ =0 | + 12 M |∇ R W ℓ =0 | + 2 M ∇ v W ℓ =0 · ∇ R W ℓ =0 + σ M | W ℓ =0 | + ∇ T + Y W ℓ =0 · (cid:18) (cid:3) W ℓ =0 − r W ℓ =0 (cid:19) . Note that the coefficient of ∇ L ′ W ℓ =0 in (cid:3) W ℓ =0 is non-negative. Therefore by choosing σ >>M − , there exists a constant C rs >
0, which may increase from line to line, such that on thehorizon div J ℓ =0 (cid:12)(cid:12)(cid:12)(cid:12) r =2 M ≥ C rs M − | ( M ∂ ) ≤ W ℓ =0 | − C rs M ( | O | + | O | ) . By continuity, there exists r rs,ℓ =0 ∈ (2 M, r + rs ) such that in [2 M, r rs,ℓ =0 ] we still havediv J ℓ =0 + C rs M ( | O | + | O | ) ≥ C rs M − | ( M ∂ ) ≤ W ℓ =0 | . Together with (8.29), we have in [2
M, r rs,ℓ =0 ]div J ℓ =0 + C rs M ( | O | + | O | ) + C rs M − | ( M ∂ ) ≤ O | ≥ C rs M − (cid:18) − Mr (cid:19) − | d ′ ( τ ) | + M − (cid:18) − Mr (cid:19) − | d ( τ ) | ! . In r ∈ [ r rs,ℓ =0 , r + rs ], we have | div J ℓ =0 | ≤ C rs (cid:0) M − | ( M ∂ ) ≤ W ℓ =0 | + M | O | + M | O | (cid:1) . Through (8.28) and (8.29), in r ∈ [ r rs,ℓ =0 , r + rs ], | div J ℓ =0 | can be further bounded by C rs (cid:18) M − | d ′ ( τ ) | + M − | d ( τ ) | + M − | ( M ∂ ) ≤ O | + M − | ( M ∂ ) ≤ O | + M | O | + M | O | (cid:19) . Now we fix a number r ∈ (2 M, r rs,ℓ =0 ) close to 2 M such that for any τ ≥ τ Z D ( τ ,τ ) " M − (cid:18) − Mr (cid:19) − | d ′ ( τ ) | + M − (cid:18) − Mr (cid:19) − | d ( τ ) | · χ [ r,r rs,ℓ =0 ] dvol ≥ C rs Z D ( τ ,τ ) (cid:2) M − | d ′ ( τ ) | + M − | d ( τ ) | (cid:3) · χ [ r rs,ℓ =0 ,r + rs ] dvol. This can be done as the left hand side diverges when r goes to 2 M . Then in applying the diver-gence theorem 2.14 to J ℓ =0 , we obtain positive M − | ( M ∂ ) W ℓ =0 | term in r ∈ [2 M, r ], positive | d ( τ ) | | d ′ ( τ ) | terms in r ∈ [ r, r rs,ℓ =0 ] which overcomes the negative ones in r ∈ [ r rs,ℓ =0 , r + rs ] anderror O j terms. Throwing away the positive boundary term along H + ( τ , τ ), we obtain Z Σ τ J ℓ =0 · ndvol + Z τ τ M − | d ′ ( τ ) | + M − | d ( τ ) | dτ + M − Z D ( τ ,τ ) | ( M ∂ ) ≤ W ℓ =0 | · χ [2 M,r ] dvol . Z Σ τ J ℓ =0 · ndvol + M Z D ( τ ,τ ) (cid:0) | O | + | O | (cid:1) · χ [2 M,r + rs ] dvol + M − Z D ( τ ,τ ) (cid:0) | ( M ∂ ) ≤ O | + | ( M ∂ ) ≤ O | (cid:1) · χ [ r,r + rs ] dvol. The boundary term can be estimated from above as Z Σ τ J ℓ =0 · n dvol . M − Z Σ τ | ( M ∂ ) ≤ W ℓ =0 | · χ [2 M,r ] dvol + M − | d ′ ( τ ) | + M − | d ( τ ) | + M − Z Σ τ (cid:0) | ( M ∂ ) ≤ O | + | ( M ∂ ) ≤ O | (cid:1) · χ [ r,r + rs ] dvol . Here we used (8.28), (8.29) in r ∈ [ r, r + rs ]. Similarly, Z Σ τ J ℓ =0 · n dvol + C rs M − Z Σ τ | ( M ∂ ) ≤ O | · χ [ r,r + rs ] dvol . & M − Z Σ τ | ( M ∂ ) ≤ W ℓ =0 | · χ [2 M,r ] dvol + M − | d ′ ( τ ) | + M − | d ( τ ) | . Denote(8.35) E ( τ ) := M − Z Σ τ | ( M ∂ ) ≤ W ℓ =0 | · χ [2 M,r ] dvol + M − | d ′ ( τ ) | + M − | d ( τ ) | . Then we obtain for any τ ≥ τ ≥ E ( τ ) + M − Z τ τ E ( τ ) dτ . E ( τ ) + Err ℓ =0 ( τ , τ ) , where Err ℓ =0 ( τ , τ ) = M Z D ( τ ,τ ) (cid:0) | O | + | O | (cid:1) · χ [2 M,r + rs ] dvol + M − Z D ( τ ,τ ) } (cid:0) | ( M ∂ ) ≤ O | + | ( M ∂ ) ≤ O | (cid:1) · χ [ r,r + rs ] dvol + M − Z Σ τ (cid:0) | ( M ∂ ) ≤ O | + | ( M ∂ ) ≤ O | (cid:1) · χ [ r,r + rs ] dvol + M − Z Σ τ | ( M ∂ ) ≤ O | · χ [ r,r + rs ] dvol . We will show in Appendix B that
Err ℓ =0 ( τ , τ ) is bounded as Lemma 8.5.
Err ℓ =0 ( τ , τ ) . (cid:16) τ M (cid:17) − δ Decay ℓ =0 [ ˆ P even ] . CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 67
Thus, E ( τ ) + Z ∞ τ E ( τ ′ ) dτ ′ . (cid:16) τM (cid:17) − δ · Decay[ E ] , where Decay[ E ] = E (0) + Decay ℓ =0 [ ˆ P even ]. In particular, W ℓ =0 decays to zero near the horizon.Furthermore, we will show that the coefficient c ( τ ) converges as τ goes to infinity. From (8.12)and (cid:0) − Mr (cid:1) ∂ r = ∂ t at r = 2 M , we have0 = M ∂ τ S W − ∂ τ P even − ∂ τ W (cid:12)(cid:12)(cid:12)(cid:12) r =2 M . Also, from the definition of ¯ W ,0 = ∂ τ W − ∂ τ ¯ W − c ′ ( τ )2 M (cid:12)(cid:12)(cid:12)(cid:12) r =2 M . Therefore c ′ ( τ ) = 2 M ∂ τ (cid:0) M S W − P even − ¯ W (cid:1) ( τ, M ) . From E ( τ ) →
0, Proposition (8.2) and (8.4), S W , P even and ¯ W all decay to zero at τ goes toinfinity. In particular, c ( ∞ ) := c (0) + R ∞ c ′ ( τ ) dτ exists and c ( ∞ ) = c (0) − M ( M S W − P even − ¯ W )(0 , M )= − M ( M S W − P even − W )(0 , M ) . proof of theorem 8.1. We first show that Decay[ E ] . Decay ℓ =0 [ W ] and start by analyzing d ( τ ).From the definition of d ( τ ) and c ( τ ), d ( τ ) = − M ∂ ρ W ( τ, M ) + 8 M W − ( τ, M ) + 4 M W ( τ, M ) . Hence M − | d (0) | . M | ( M ∂ ) ≤ W (0 , M ) | . M − Z Σ ′′ | ( M ∂ ) ≤ W | dvol , which is bounded by Decay ℓ =0 [ W ]. We used the spherical symmetry of W to control its supnorm by its H norm. For the term d ′ ( τ ), we have d ′ ( τ ) = 8 M · O + 2 M · O c . sup [ r,R null ] (cid:18) M | ( M ∂ ) ≤ P even | + M | ( M ∂ ) ≤ S W | (cid:19) . Therefore M − | d ′ (0) | . F [ D ≤ P even ](0) + M F [ D ≤ S W ](0) , which is also bounded by Decay ℓ =0 [ W ]. The last term to bound in E (0) is the H norm of W ℓ =0 = ¯ W , ¯ W on Σ . Clearly Decay ℓ =0 [ W ] includes the H norm of W ℓ =0 = W , W on Σ so if suffices to bound their difference. The difference between ¯ W , ¯ W and W , W is c ( τ ) /r .Hence M − Z Σ τ | ( M ∂ ) ≤ ( ¯ W − W ) | · χ [2 M,r ] dvol ≈ M | c ′ ( τ ) | + M − | c ( τ ) | , which can be estimated by the same way as d ( τ ). Thus we concludeDecay[ E ] . Decay ℓ =0 [ W ] . Therefore E ( τ ) + Z ∞ τ E ( τ ′ ) dτ ′ . (cid:16) τM (cid:17) − δ · Decay ℓ =0 [ W ] . Second, we want to show that Theorem 8.1 holds with ˆ W replaced by ¯ W . Note that E ( τ )controls the H norm of ¯ W in r ∈ [2 M, r ]. For the region r ∈ [ r, R null ], we use (8.28) and (8.29).The d ( τ ) term is also controlled in E ( τ ). To bound O and O , we need Lemma 8.6. M − Z D ′′ ( τ, ∞ ) (cid:0) | ( M ∂ ) ≤ O | + | ( M ∂ ) ≤ O | (cid:1) · χ [ r,R null ] dvol + M − Z Σ ′′ τ (cid:0) | ( M ∂ ) ≤ O | + | ( M ∂ ) ≤ O | (cid:1) · χ [ r,R null ] dvol . (cid:16) τM (cid:17) − δ Decay ℓ =0 [ D ≤ ˆ P even ] . This will be proved in Appendix B. Putting these together, we have M − Z Σ ′′ τ | ( M ∂ ) ≤ ¯ W | dvol + M − Z D ′′ ( τ, ∞ ) | ( M ∂ ) ≤ ¯ W | dvol . (cid:16) τM (cid:17) − δ · Decay ℓ =0 [ W ] . To control ˆ W instead of ¯ W , we turn to their differenceˆ W − ¯ W = − ˆ W − ¯ W = c ( τ ) − c ( ∞ ) r =: ¯ c ( τ ) r . Note that ¯ c ( τ ) = 2 M ( M S W − P even − ¯ W )( τ, M ) . And the time derivative of ¯ c can be estimated as¯ c ′ ( τ ) = c ′ ( τ ) = O c . Thus M − Z Σ ′′ τ | ( M ∂ ) ≤ ( ˆ W − ¯ W ) | dvol ≈ M ¯ c ′ ( τ ) + M − ¯ c ( τ ) . sup [ r,R null ] (cid:0) M | ( M ∂ ) ≤ P even | + M | ( M ∂ ) ≤ S W | (cid:1) + (cid:18) M | ¯ W | + M | P even | + M | S W | (cid:19) ( τ, M ) . E ( τ ) + F [ D ≤ P even ]( τ ) + M F [ D ≤ S W ]( τ ) . Similarly,
CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 69 M − Z D ′′ ( τ, ∞ ) | ( M ∂ ) ≤ ( ˆ W − ¯ W ) | dvol . M − Z ∞ τ E ( τ ′ ) + F [ D ≤ P even ]( τ ′ ) + M F [ D ≤ S W ]( τ ′ ) . Thus from the estimate for E ( τ ), Proposition 8.2 and Proposition 8.4, M − Z Σ ′′ τ | ( M ∂ ) ≤ ( ˆ W − ¯ W ) | dvol + M − Z D ′′ ( τ, ∞ ) | ( M ∂ ) ≤ ( ˆ W − ¯ W ) | dvol . (cid:16) τM (cid:17) − δ · Decay ℓ =0 [ W ] . And the proof is finished. (cid:3)
The ℓ = 0 mode of (1.3) and (1.4) . In this subsection we discuss the solution of (1.3)and (1.4) supported on ℓ = 0. Form [39, 30], one can decompose h = h ℓ =0 into the mass change K , defined in (2.4), and a deformation tensor. There exists a unique constant c and a one form W = W ℓ =0 unique upto the time translation T ♭ such that h ℓ =0 = cK + W π, From (2.11), W = r /trh , where W = W · (cid:0) − Mr (cid:1) ∂ r . Hence from (2.11), the constant c isdetermined by cr = (cid:18) − Mr (cid:19) h rr − r (cid:18) − Mr (cid:19) ∇ r /trh − (cid:18) − Mr (cid:19) /trh. (8.37)Suppose c = 0, then W a is a solution of (1.5) supported on ℓ = 0 and Theorem 8.1 applies.Nevertheless, K doesn’t satisfy the harmonic gauge (1.3) asΓ a [ K ] dx a = 1 R dv. To resolve this issue, one can modify K through a deformation tensor W ∗∗ π . An explicit form of W ′ is obtained in [19]: W ∗∗ a dx a := v · W ∗ a dx a − log (cid:16) r M (cid:17) dt + 12 (cid:18) Mr (cid:19) dr. (8.38)Define K ∗ := K + W ∗∗ π. (8.39)From direct calculation K ∗ satisfies (1.3) and hence (1.4). Note that K ∗ grows linearly in τ withleading term being τ · W ∗ π . The remaining one form W ′ = W − cW ∗∗ satisfies (1.3) and decaysto zero if the assumption in Theorem 8.1 holds. In that case we can conclude that h convergesto cK ∗ + c ( ∞ ) W ∗ π with the constant c ( ∞ ) given in Theorem 8.1. Since trK ∗ = 0, the traceof W ′′ π is the same as trh . However, the value of P even , defined in (4.19), is only determinedupto a constant as P even [ T ♭ ] = 1. Theorem 8.7.
Let h = h ℓ =0 be a solution of (1.3) and (1.4) and c be the constant defined in (8.37) . Let W = W ℓ =0 be an even one form such that h = cK ∗ + W π. Suppose that S = trh satisfies (cid:0) F [ D ≤ S ] + E − δ, (0) L [ D ≤ S ] (cid:1) (0) < ∞ . Further assume W can be chosen such that (cid:0) F [ D ≤ ˆ P even ] + E − δ, (0) L [ D ≤ ˆ P even ] (cid:1) (0) < ∞ . Then h converges to cK ∗ + c ( ∞ ) W ∗ π . Appendix A. wave equation in Schwarzschild spacetime In this section we record the estimates for the wave equation equation (cid:3) ψ − V ( r ) ψ = G, (A.1)or(A.2) (cid:3) ψ − V ψ = 0 , where V ( r ) is a smooth function depending only on r .The study of wave equation in black hole spacetimes using vector field method was initiated byDafermos and Rodnianski. In [13, 14] the authors proposed the red-shfit, Morawetz and r p vectorfields as multiplier and proved Proposition A.4 for m = 0. To get the higher order estimates,one can commute the equation with various vector fields. It’s clear that the equation (A.1)commutes with Killing vector fields in K . Moreover, the authors discovered that one can also usethe red-shift vector as a commutator. Even though the red-shift vector doesn’t commute with (cid:3) ,the lower order terms has special structure that allows the red-shift estimate. Schlue [33] furtherobserved that one can use L and rL as commutator in the r p estimate. Later Moschidis [31]applied this technique to a very general class of asymptotically flat spacetimes. These insightsyields the higher order estimate ( m ≥
1) in Proposition A.4. See [12] for more detail.Angelopoulos-Aretakis-Gajic [2] exploited the vector field method by using r L as the com-mutator. There is actually a bad term in the commutation relation and one key observationthey made is that the bad term can be absorbed using the Poincar´e inequality. This leads toProposition A.5. Furthermore, the authors also show that for the case ψ = ψ ℓ =0 is supportedon ℓ = 0, one can extend the r p estimate to p < ψ , which is the content of Proposition A.6.The above mentioned results focus on the case without the potential term V ( r ). Throughstraight forward modifications, the potential term can be included with suitable assumptionsthat we list in the following.A.1. Assumptions.
Fix an integer ℓ ≥
0. The assumptions in below are designed for a solutionof (A.1) supported on ℓ ≥ ℓ . First, we assume that(A.3) ℓ ( ℓ + 1) r + V ( r ) ≥ . This implies for all ψ = ψ ℓ ≥ ℓ , one has | / ∇ ψ | + V | ψ | ≥ s
0. Hence the boundary term of the T -current is non-negative as CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 71 (cid:18) T ab [ ψ ] − V | ψ | (cid:19) T a n b ≥ s . Second, we assume that there exists a constant C Mor > f ( r ) with thefollowing properties: • dfdr ≥ C Mor · Mr • f ( r )(1 − M/r ) ≥ ω = (cid:0) − Mr (cid:1) (cid:16) fr + dfdr (cid:17) , f (cid:18) − Mr (cid:19) ℓ ( ℓ + 1) r + (cid:18) − (cid:3) ω − f (cid:18) − Mr (cid:19) dVdr − Mr f V (cid:19) ≥ C Mor · r (cid:18) − Mr (cid:19) ℓ ( ℓ + 1) + 1 ! . (A.4)The left hand side of (A.4) appears in the divergence term in the Morawetz current, see subsec-tion 6.3. This assumption ensures one can construct a current like J in subsection 6.3.Third, we assume that for all r ≥ R null and p ∈ [ δ, − δ ],(A.5) (cid:16) − p (cid:17) · ℓ ( ℓ + 1) + (cid:18) − r (cid:18) dVdr + pr V (cid:19) + (3 − p ) M r − (cid:19) ≥ . Here R null is the value of where Σ τ changes from spacelike to null and 0 < δ < /
10 is a fixedconstant. The left hand side of (A.5) appears in the divergence term in the r p current, seesubsection 6.4. The assumption ensures one can construct a current like J p in subsection 6.4.Once the potential V ( r ) satisfies (A.3), (A.4) and (A.5), Proposition A.4 applies with m = 0.To get the higher order estimate, we assume that there exists constants C m > m ≥ (cid:12)(cid:12)(cid:12)(cid:12) r m +2 d m Vdr m (cid:12)(cid:12)(cid:12)(cid:12) ≤ C m . (A.6) Definition A.1.
We denote by V ( δ, R null , ℓ ) be the collection of functions V ( r ) which satisfies (A.3) , (A.4) , (A.5) and (A.6) . We continue to list the requirements for the potential. Fix k ≥ ≤ j ≤ k , V ( j ) ( r ) := V ( r ) + 2 Mr − j ( j + 1) r + 6 M j ( j + 1) r . We assume that for all 1 ≤ j ≤ k , r ≥ R null and p ∈ [ δ, − δ ], (cid:16) − p (cid:17) · ℓ ( ℓ + 1) − r (cid:18) dV ( j ) dr + pr V ( j ) (cid:19) ≥ . (A.7)And we require that(A.8) lim inf r →∞ r V ( k ) ≥ − ℓ ( ℓ + 1) , Then we assume that there exists a constant C ( k ) > ≤ j ≤ k (A.9) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ddr · r (cid:19) j V ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( k ) M j r . Definition A.2.
The set V ( δ, R null , ℓ , k ) is the subset of V ( δ, R null , ℓ ) whose member furthersatisfies (A.7) , (A.8) and (A.9) . The three extra assumptions are needed in using r L as commutator k times. Let ψ be asolution of (A.2). Then r − ( r L ) k ( rψ ) behaves like a solution of (A.2) with potential V ( k ) ( r ) inthe r p estimate. These assumptions guarantee the boundary and the bulk terms are non-negative.Applying divergence theorem, one deduces that E p, ( j ) L [ ψ ]( τ ) decays like τ − p · E , ( j ) L [ ψ ](0) forall 0 ≤ j ≤ k . Then using E , ( j ) L . E , ( j +1) L , which can be proved by Hardy inequality Lemma3.2, E p, (0) L [ ψ ]( τ ) decays like τ − k +1) . Remark A.3.
It’s easy to verify that the zero function satisfies all the assumptions for V ( δ, R null , ℓ ) except (A.4) , the Morawetz assumption. For such potential ( V = 0 ), the Morawetzestimate was proved in [13] . Moreover, clearly satisfies (A.7) , (A.8) and (A.9) provided ℓ ≥ k .This is the case studied in [2] . A.2. decay estimate.
Recall that the initial norms of ψ are defined by I ( k ) [ ψ ] := F [ K ≤ k +2 ψ ] + X i + i ≤ k +1 i ≤ k E − δ, ( i ) L [ K i ˜ ψ ] (0) . The spacetime norm of the source term G , for any ¯ δ ≥ δ , is defined byI source , ¯ δ [ G ] := sup p ∈ [ δ, − δ ] sup τ ≥ τ ≥ (cid:16) τ M (cid:17) − p − ¯ δ E p source [ K ≤ G ]( τ , τ ) . Proposition A.4.
Let ψ be a solution of wave equation (A.1) with source term G and m ≥ be a fixed integer. Suppose ψ = ψ ℓ ≥ ℓ and V ∈ V ( δ, R null , ℓ ) . Suppose that I (0) [ D ≤ m ψ ] and I source , ¯ δ [ D ≤ m G ] are finite, then for any p ∈ [ δ, − ¯ δ ] and τ ≥ we have F [ D ≤ m ψ ]( τ ) + E p, (0) L [ D ≤ m ˜ ψ ]( τ ) + Z ∞ τ B [ D ≤ m ψ ]( τ ′ ) + E p − , (0) L, / ∇ [ D ≤ m ˜ ψ ]( τ ′ ) dτ ′ . m,V,R null (cid:16) τM (cid:17) − p +¯ δ (cid:16) I (0) [ D ≤ m ψ ] + I source , ¯ δ [ D ≤ m G ] (cid:17) . (A.10) Proposition A.5.
Let ψ = ψ ℓ ≥ ℓ be a solution of (A.2) . Assume that V ∈ V ( δ, R null , ℓ , k ) .We have for any p ∈ [ δ, − δ ] and any τ ≥ , F [ ψ ]( τ ) + E p, (0) L [ ˜ ψ ]( τ ) + Z ∞ τ B [ ψ ]( τ ′ ) + E p − , (0) L, / ∇ [ ˜ ψ ]( τ ′ ) dτ ′ . V,R null (cid:16) τM (cid:17) − k +1)+ p +(1+2 k ) δ I ( k ) [ ψ ] . (A.11)As k = 1 or 2, this is part of Proposition 7.5 in [2] with simple modification to include thepotential. The authors in [2] work with asymptotically flat space time with suitable behavior CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 73 near null infinity. In the Schwarzschild spacetiem, one can continue commuting with r L assuggested in [2].Since the trapped geodesic can only be approached by a solutions of wave equation with higherand higher frequency. Once we restrict to a fixed mode ℓ = 0 (or any other modes), the loss ofderivative can be avoided. In particular, we define the norm below.I source ,ℓ =0 , ¯ δ [ G ] := sup τ ≥ sup p ∈ [ δ, − ¯ δ ] (cid:16) τM (cid:17) − p − ¯ δ M Z D ( τ, ∞ ) s p +1 | G | dvol. (A.12) Proposition A.6.
Fix m ≥ .(1) Let ψ = ψ ℓ =0 be a solution of wave equation (cid:3) ψ = 0 supported on the mode ℓ = 0 . Supposethat F [ D ≤ m ψ ](0) + E − δ, (0) [ D ≤ m ψ ](0) is finite. Then for any p ∈ [ δ, − δ ] and τ ≥ , F [ D ≤ m ψ ]( τ ) + E − δ, (0) L [ D ≤ m ˜ ψ ]( τ ) + Z ∞ τ ¯ B [ D ≤ m ψ ]( τ ′ ) + E p − , (0) L, / ∇ [ D ≤ m ˜ ψ ]( τ ′ ) dτ ′ . m,V,R null (cid:16) τM (cid:17) − p + δ (cid:16) F [ D ≤ m ψ ](0) + E − δ, (0) [ D ≤ m ψ ](0) (cid:17) . (A.13) (2) Let ψ = ψ ℓ =0 be a solution of wave equation (cid:3) ψ = G supported on the mode ℓ = 0 .Suppose that F [ D ≤ m ψ ](0) + E − δ, (0) [ D ≤ m ψ ](0) and I source ,ℓ =0 , ¯ δ [ D ≤ m G ] are finite. Then for any p ∈ [ δ, − ¯ δ ] and τ ≥ , F [ D ≤ m ψ ]( τ ) + E − δ, (0) L [ D ≤ m ˜ ψ ]( τ ) + Z ∞ τ ¯ B [ D ≤ m ψ ]( τ ′ ) + E p − , (0) L, / ∇ [ D ≤ m ˜ ψ ]( τ ′ ) dτ ′ . m,V,R null (cid:16) τM (cid:17) − p +¯ δ (cid:16) F [ D ≤ m ψ ](0) + E − δ, (0) [ D ≤ m ψ ](0) + I source ,ℓ =0 , ¯ δ [ D ≤ m G ] (cid:17) . (A.14)Proposition A.6 (1), as m = 0, is part of Theorem 1.5 in [2]. Note that our assumption E − δ, (0) L < ∞ implies the Newman-Penrose constant vanishes. The case m ≥ D as commutator and is closely related to Theorem 1.7 in [2]. Proposition A.6 (2) canbe almost derived from Proposition (A.4) except that we have the non-degenerate norm ¯ B andI source ,ℓ =0 , ¯ δ [ D ≤ m G ] because we focus on a fixed mode ℓ = 0. Appendix B. O j terms in section 8 In this appendix, we give estimates on the bounds of the O j terms in in section 8. Let m ≥ O (8.13), we have | ( M ∂ ) ≤ m O | . m M − | ( M ∂ ) ≤ m +1 P even | + | ( M ∂ ) ≤ m +1 S W | . From the definition of O (8.15), we have | ( M ∂ ) ≤ m O | . m M − | ( M ∂ ) ≤ m +2 P even | + M − | ( M ∂ ) ≤ m +2 S W | . The definition of O involves integration and is singular at r = 2 M because of W , ( ρ ).Therefore we from now on restrict ourselve in the region r ∈ [ r, R null ]. Then as r ∈ [ r, R null ], | ( M ∂ ) ≤ m O | . m sup [ r,R null ] (cid:18) | ( M ∂ ) ≤ m +2 P even | + M | ( M ∂ ) ≤ m +2 S W | (cid:19) . The constant in above inequality depends on r and R null and blows up as r goes to 2 M or R null goes to infinity. However , r and R null are already fixed.From the definition of O c and O c , (8.21) and (8.22), we have | ( M ∂ ) ≤ m O c | . m (cid:18) | ( M ∂ ) ≤ m +2 P even | + M | ( M ∂ ) ≤ m +2 S W | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r =4 M | ( M ∂ ) ≤ m O c | . m (cid:18) M − | ( M ∂ ) ≤ m +2 P even | + | ( M ∂ ) ≤ m +2 S W | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r =4 M . Repeating the same argument, we havesup [ r,R null ] | ( M ∂ ) ≤ m O | . m sup [ r,R null ] (cid:18) M − | ( M ∂ ) ≤ m +3 P even | + | ( M ∂ ) ≤ m +3 S W | (cid:19) , sup [ r,R null ] | ( M ∂ ) ≤ m O | . m sup [ r,R null ] (cid:18) M − | ( M ∂ ) ≤ m +4 P even | + M − | ( M ∂ ) ≤ m +4 S W | (cid:19) , sup [ r,R null ] | ( M ∂ ) ≤ m O | . m sup [ r,R null ] (cid:18) M − | ( M ∂ ) ≤ m +5 P even | + M − | ( M ∂ ) ≤ m +5 S W | (cid:19) . and sup [ r,R null ] | ( M ∂ ) ≤ m O | . m sup [ r,R null ] (cid:18) | ( M ∂ ) ≤ m +5 P even | + M | ( M ∂ ) ≤ m +5 S W | (cid:19) , sup [ r,R null ] | ( M ∂ ) ≤ m O | . m sup [ r,R null ] (cid:18) M − | ( M ∂ ) ≤ m +5 P even | + | ( M ∂ ) ≤ m +5 S W | (cid:19) , sup [ r,R null ] | ( M ∂ ) ≤ m O | . m sup [ r,R null ] (cid:18) M − | ( M ∂ ) ≤ m +6 P even | + | ( M ∂ ) ≤ m +6 S W | (cid:19) , sup [ r,R null ] | ( M ∂ ) ≤ m O | . m sup [ r,R null ] (cid:18) | ( M ∂ ) ≤ m +6 P even | + M | ( M ∂ ) ≤ m +6 S W | (cid:19) . The terms O and O are smooth upto the horizon as they depend on O c and P even in(8.32) and (8.33). | ( M ∂ ) ≤ m O | . m (cid:18) M − | ( M ∂ ) ≤ P even | + M − | ( M ∂ ) ≤ S W | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r =4 M + M − | P even | , | ( M ∂ ) ≤ m O | . m (cid:18) M − | ( M ∂ ) ≤ P even | + M − | ( M ∂ ) ≤ S W | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r =4 M , Now we are ready to prove Lemma 8.5 and 8.6. proof of Lemma 8.5 and 8.6.
From the above discussion,
Err ℓ =0 ( τ , τ ) is bounded by CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 75 (cid:18) F [ D ≤ ˆ P even ] + M F [ D ≤ S W ] (cid:19) ( τ ) + (cid:18) F [ D ≤ ˆ P even ] + M F [ D ≤ S W ] (cid:19) ( τ )+ M − Z τ τ (cid:18) ¯ B [ D ≤ ˆ P even ] + M ¯ B [ D ≤ S W ] (cid:19) ( τ ) dτ. Therefore the claim follows Proposition 8.2 and 8.4. The proof of Lemma 8.6 is similar. (cid:3)
Appendix C. Equation for H In this appendix we derive (4.2), the wave equation for H . We follow closely the computationin [5] except that we adapt a different decomposition which is similar to the one in [3].Recall that for a function ψ ( t, r, θ, φ ), we decompose ψ as ψ = ∞ X ℓ =0 X | m |≤ ℓ ψ mℓ ( t, r ) Y mℓ ( θ, φ ) . Similarly, for any even spherical one form ξ A and even spherical symmetric traceless twotensor Ξ AB , we decompose them as ξ A = ∞ X ℓ =1 X | m |≤ ℓ ξ mℓ ( t, r ) / ∇ A Y mℓ ( θ, φ ) , Ξ AB = ∞ X ℓ =2 X | m |≤ ℓ Ξ mℓ ( t, r ) ˆ / ∇ AB Y mℓ ( θ, φ ) . We also define the operator (cid:3) ℓ on the scalar functions depending on { t, r } as (cid:3) ℓ = − (cid:18) − Mr (cid:19) − ∂ ∂t + (cid:18) − Mr (cid:19) ∂ ∂r + 2 r (cid:18) − Mr (cid:19) ∂∂r − ℓ ( ℓ + 1) r . (C.1)Then we have (cid:3) f = ∞ X ℓ =0 X | m |≤ ℓ (cid:0) (cid:3) ℓ f mℓ ( t, r ) (cid:1) · Y mℓ ( θ, φ ) . Similarly, we define (cid:3) ℓ = r · (cid:3) ℓ · r − + 1 r , (C.2) (cid:3) ℓ = r · (cid:3) ℓ · r − + 4 r (C.3)in order to have (cid:3) ξ A = ∞ X ℓ =1 X | m |≤ ℓ (cid:0) (cid:3) ℓ ξ mℓ ( t, r ) (cid:1) · / ∇ A Y mℓ ( θ, φ ) , (cid:3) Ξ AB = ∞ X ℓ =0 X | m |≤ ℓ (cid:0) (cid:3) ℓ Ξ mℓ ( t, r ) (cid:1) · ˆ / ∇ AB Y mℓ ( θ, φ ) . For an even perturbation h ab , the decomposition [5] in in the following: h tt = (cid:18) − Mr (cid:19) ∞ X ℓ =0 X | m |≤ ℓ H ,mℓ · Y mℓ , (C.4) h tr = ∞ X ℓ =0 X | m |≤ ℓ H ,mℓ · Y mℓ , (C.5) h rr = (cid:18) − Mr (cid:19) − ∞ X ℓ =0 X | m |≤ ℓ H ,mℓ · Y mℓ , (C.6) /trh =2 ∞ X ℓ =0 X | m |≤ ℓ K mℓ · Y mℓ . (C.7) h tA = ∞ X ℓ =1 X | m |≤ ℓ h ,mℓ · / ∇ A Y mℓ , (C.8) h rA = ∞ X ℓ =1 X | m |≤ ℓ h ,mℓ · / ∇ A Y mℓ , (C.9) ˆ h AB =2 r ∞ X ℓ =2 X | m |≤ ℓ G mℓ · ˆ / ∇ AB Y mℓ (C.10)From now on we fix ℓ ≥ | m | ≤ ℓ . Let LW ab = (cid:3) h ab + 2 R acbd h cd , (C.11) HG b = ∇ a (cid:18) h ab −
12 (tr h ) g ab (cid:19) . (C.12)Recall that λ = λ ( ℓ ) is defined in (2.25). From [5] (3.1-3.7) or direct computation, LW tt,mℓ = (cid:18) − Mr (cid:19) (cid:3) ℓ H ,mℓ − M r H ,mℓ + 4 Mr ∂ t H ,mℓ + 2 M (3 M − r ) r H ,mℓ + 4 M ( r − M ) r K mℓ . (ME1) LW rr,mℓ = (cid:18) − Mr (cid:19) − (cid:3) ℓ H ,mℓ + 2 Mr (3 M − r )( r − M ) H ,mℓ + 8 λ + 8 r h ,mℓ + 4 M ( r − M ) ∂ t H ,mℓ − (4 r − M r + 18 M ) r ( r − M ) H ,mℓ + 4( r − M ) r ( r − M ) K mℓ . (ME2) CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 77 LW tr,mℓ = (cid:3) ℓ H ,mℓ + 4(1 + λ ) r h ,mℓ + 2 Mr − M r ∂ t H ,mℓ + 2( r − M r + 2 M ) r (2 M − r ) H ,mℓ + 2 Mr − M r ∂ t H ,mℓ . (ME3) LW tA,mℓ = (cid:3) ℓ h ,mℓ + 4 Mr h ,mℓ + 2( M − r ) r ∂ r h ,mℓ + 2 Mr ∂ t h ,mℓ + 2( r − M ) r H ,mℓ . (ME4) LW rA,mℓ = (cid:3) ℓ h ,mℓ + 4 λr G mℓ + 2 M ( r − M ) ∂ t h ,mℓ + 8 M − rr h ,mℓ + 2 r H ,mℓ − r K mℓ + 6 M − rr ∂ r h ,mℓ . (ME5) r /tr LW mℓ = r (cid:18) (cid:3) ℓ K mℓ + 4 Mr H ,mℓ + (8 λ + 8)(2 M − r ) r h ,mℓ + 4( r − M ) r H ,mℓ + 4 r − Mr K mℓ (cid:19) . (ME6) ˆ LW AB,mℓ = 2 r · (cid:3) ℓ G mℓ + 4 G mℓ + 4 r − Mr h ,mℓ . (ME7)Also from [5] (3.8-3.10) or direct computation, the Harmonic gauge condition reads HG t,mℓ = − λ − r h ,mℓ − ∂ t H ,mℓ − M − r ) r H ,mℓ − ∂ t H ,mℓ − ∂ t K mℓ + (cid:18) − Mr (cid:19) ∂ r H ,mℓ . (HG1) HG r,mℓ = − M M r − r H ,mℓ − λ + 2 r h ,mℓ + r M − r ∂ t H ,mℓ + 3 M − r M r − r H ,mℓ − r K mℓ + 12 ∂ r H ,mℓ + 12 ∂ r H ,mℓ − ∂ r K mℓ . (HG2) HG A,mℓ = − λG mℓ + r M − r ∂ t h ,mℓ + 12 H ,mℓ − M − rr h ,mℓ − H ,mℓ + (cid:18) − Mr (cid:19) ∂ r h ,mℓ . (HG3)Now we start to derive the equation for H . From the definition of H , we have H ,mℓ = 12 (cid:18) − Mr (cid:19) ( H ,mℓ + H ,mℓ ) + 2 Mr K mℓ , H ,mℓ = K mℓ , H ,mℓ = 12 ( H ,mℓ − H ,mℓ ) , H ,mℓ = (cid:18) − Mr (cid:19) H ,mℓ + 2 Mr K mℓ . H ,mℓ = (cid:18) − Mr (cid:19) h ,mℓ , H ,mℓ = h ,mℓ . And H ,mℓ = √ r G mℓ . The equation for H can be derived by (cid:3) ℓ H ,mℓ −
12 (ME1) + Mr (ME6) + 12 (cid:18) − Mr (cid:19) (ME2) + 4 M ( r − M ) r (HG2) ! = 2 r (1 − s − ) H ,mℓ − r (1 − s − + 10 s − ) H ,mℓ − r (1 − s − + 20 s − ) H ,mℓ − λ + 4 r (1 − s − ) H ,mℓ . Together with Lemma 2.3, the equation of H follows by summing over all mℓ .The equation for H can be derived by (cid:3) ℓ H ,mℓ − r (ME6)= − r H ,mℓ + 2 r (1 − s − ) H ,mℓ + 2 r (1 − s − ) H ,mℓ + 4 λ + 4 r H ,mℓ . The equation for H can be derived by (cid:3) ℓ H ,mℓ − (cid:18) − Mr (cid:19) − (ME1) − (cid:18) − Mr (cid:19) (ME2) ! = − r H ,mℓ + 2 r (1 − s − ) H ,mℓ + 2 r (1 − s − ) H ,mℓ + 4 λ + 4 r H ,mℓ . The equation for H can be derived by (cid:3) ℓ H ,mℓ − (cid:18)(cid:18) − Mr (cid:19) (ME5) + 2 Mr (HG3) (cid:19) = 2 r (5 / − s − ) H ,mℓ − r H ,mℓ + 2 r H ,mℓ + 2 r (1 − s − ) H ,mℓ − √ λr (1 − s − ) H ,mℓ . The equation for H can be derived by (cid:3) ℓ H ,mℓ − √ r H ,mℓ − √ r H ,mℓ . The equation for H can be derived by (cid:3) ℓ H ,mℓ − (cid:18)(cid:18) − Mr (cid:19) (ME2) + Mr (ME6) + 4 M ( r − M ) r (HG2) (cid:19) = 2 r ( − s − ) H ,mℓ + 2 r (4 s − − s − ) H ,mℓ + 2 r (6 s − − s − ) H ,mℓ + 2 r (1 − s − ) H ,mℓ + 2 r ( s − )(8 λ + 8) H ,mℓ − λ + 4 r (1 − s − ) H ,mℓ − r (1 − s − ) − s − ∇ L ( r ( H ,mℓ − H ,mℓ )) . Finally, the equation for H can be derived by CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 791 (cid:3) ℓ H ,mℓ − (cid:18) (ME4) + 2 Mr (ME3) (cid:19) = 2 r ( − s − ) H ,mℓ + 2 r (1 / − s − ) H ,mℓ + 2 r (2 s − ) H ,mℓ − r ( s − ) H ,mℓ + 2 √ λr s − H ,mℓ − r H ,mℓ + 2 r (1 − s − ) − ( − s − ) r · ∇ L ( H ,mℓ − H ,mℓ ) . Appendix D. Equation for gauge change
Now we start to derive the equation for W , W and W (4.21), (4.22) and (4.23). It isequivalent to [5] except we view W as a section of L ( −
1) and W is defined differently. We willfollow closely the approach in [5]. In this section we always assume ℓ ≥
2. In particular, all /D , /D ∗ , /D and /D ∗ are invertible. Furthermore / ∆ − X | m |≤ ℓ,ℓ ≥ f mℓ Y mℓ = X | m |≤ ℓ,ℓ ≥ (cid:18) − r λ + 2 f mℓ Y mℓ (cid:19) , (D.1) / ∆ − Z X | m |≤ ℓ,ℓ ≥ f mℓ Y mℓ = X | m |≤ ℓ,ℓ ≥ (cid:18) − r λ + 6 M/r f mℓ Y mℓ (cid:19) . (D.2)Together with Lemma 2.3, the mode of the Zerilli quantity ψ is ψ mℓ = − r (cid:18) − Mr (cid:19) ( λ + 1) − ( λ + 3 M/r ) − ∂ r K mℓ + r ( λ + 1) − K mℓ + r (cid:18) − Mr (cid:19) ( λ + 1) − ( λ + 3 M/r ) − H ,mℓ − (cid:18) − Mr (cid:19) ( λ + 3 M/r ) − h ,mℓ + 2 rG mℓ . (D.3)This is (3.34) in [5]. The Zerilli equation for ψ is (cid:3) ℓ ( r − ψ mℓ ) + 2 Mr (2 λ + 3)(2 λ + 3 M/r )( λ + 3 M/r ) · ( r − ψ mℓ ) = 0 . This equation as well as the definition of ψ Z was first derived by Zerilli [38]. It can also beverified directly through0 = − r (6 M + λr )( λ + 1)(3 M + λr ) ∂ t (HG1) − ( r − M ) (6 M + λr )( λ + 1) r (3 M + λr ) ∂ r (HG2) − M − r ) (cid:0) M − M r − λr (cid:1) ( λ + 1) r (3 M + λr ) (HG2)+ 2 (cid:0) − M + 6 M r + λr (cid:1) r (3 M + λr ) (HG3) − (cid:0) M + 2 λM r + λ ( λ + 1) r (cid:1) λ + 1) r (3 M + λr ) (ME6)+ r (6 M + λr )2( λ + 1)(3 M + λr ) (ME1) − λ ( r − M ) λ + 1)(3 M + λr ) (ME2)+ 2( r − M ) r (3 M + λr ) (ME5) + ( r − M )2( λ + 1) r (3 M + λr ) ∂ r (ME6) − r (ME7) . From (4.11),(4.12) and (4.13), W ,mℓ = − h ,mℓ + r ∂ t G mℓ , (D.4) W ,mℓ = − (cid:18) − Mr (cid:19) h ,mℓ + r (cid:18) − Mr (cid:19) ∂ r G mℓ , (D.5) W ,mℓ = − r G mℓ . (D.6)Equivalently, h ,mℓ = − W ,mℓ − ∂ t W ,mℓ ,h ,mℓ = − (cid:18) − Mr (cid:19) − W ,mℓ − ∂ r W ,mℓ + 2 r W ,mℓ ,G mℓ = − r − W ,mℓ . To derive the equation for W , W and W , we first rewrite H ,mℓ , K mℓ , and H ,mℓ in terms of h ,mℓ , h ,mℓ , G mℓ and ψ mℓ . Then we substitute them in (ME4), (ME5) and (ME7) to obtain waveequation for h ,mℓ , h ,mℓ , G mℓ . The next step is to replace h ,mℓ , h ,mℓ , G mℓ by W ,mℓ , W ,mℓ and W ,mℓ . Lemma D.1. H ,mℓ = − r (3 M − r )2 M − r ∂ t G mℓ + 2 M M r − r h ,mℓ + ∂ t h ,mℓ + 3 M + 3 M rλ − λr (2 M − r )(3 M + λr ) ∂ t ψ mℓ − r ∂ t ∂ r G mℓ + ∂ r h ,mℓ + r∂ t ∂ r ψ mℓ . Proof.
This is (3.39) in [5]. Or it can be derive by using (D.3) and0 =
M r (2 M − r )( λ + 1) (HG1) + r M − r )( λ + 1)(3 M + λr ) ∂ t (HG1)+ r (7 M + r ( λ − λ + 1)( λr + 3 M ) ∂ t (HG2) + r λr + 3 M ∂ t (HG3) − r M − r )( λ + 1)( λr + 3 M ) ∂ t (ME1) + r ( − M + r )4( λ + 1)( λr + 3 M ) ∂ t (ME2) − r λ + 2 (ME3) + r λ + 1)( λr + 3 M ) ∂ t (ME6)+ r λ + 2 ∂ r (HG1) + (2 M − r ) r λ + 1)( λr + 3 M ) ∂ t ∂ r (HG2) . (cid:3) Lemma D.2. K mℓ = − λ + 1) G mℓ + 2( r − M ) r h ,mℓ + (6 M + 3 λM r + λ ( λ + 1) r ) r ( λr + 3 M ) ψ mℓ − (2 r − M ) ∂ r G mℓ + (cid:18) − Mr (cid:19) ∂ r ψ mℓ . CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 81
Proof.
This is (3.40) in [5]. Or it can be derive by using (D.3) and0 = − r λ + 1)( λr + 3 M ) ∂ t (HG1) − ( r − M ) ( λ + 1)( λr + 3 M ) (HG2)+ r − Mλr + 3 M (HG3) + r λ + 1)( λr + 3 M ) (ME1)+ r ( r − M ) λ + 1)( λr + 3 M ) (ME2) + r − M λ + 1)( λr + 3 M ) (ME6) − r ( r − M ) λ + 1)( λr + 3 M ) ∂ r (HG2) . (cid:3) Lemma D.3. H ,mℓ = − λG mℓ − r r − M ∂ t G mℓ + 4 r − Mr h ,mℓ + 9 M + 9 λM r + 3 λ M r + λ ( λ + 1) r r ( λr + 3 M ) ψ mℓ + r r − M ∂ t ψ mℓ + 2 M ∂ r G mℓ + 2 (cid:18) − Mr (cid:19) ∂ r h ,mℓ + − M − λM r + λr r ( λr + 3 M ) ∂ r ψ mℓ . Proof.
This is (3.41) in [5]. Or it can be derive by using (D.3) and − r λ + 1)(3 M + λr ) ∂ t ∂ r (HG1) − r (cid:0) M + 3 M (2 λr + r ) + λ r (cid:1) λ + 1)(2 M − r )(3 M + λr ) ∂ t (HG1) − r ( r − M ) λ + 1)(3 M + λr ) ∂ r (HG2) − r (2 M − r ) (cid:0) M + 3(2 λ − M r + ( λ − λr (cid:1) λ + 1)(3 M + λr ) ∂ r (HG2)+ (2 M − r ) (cid:0) M − λ + 3) M r − λ ( λ + 1) r (cid:1) ( λ + 1)(3 M + λr ) (HG2) + r ( r − M )3 M + λr ∂ r (HG3) − (cid:0) − M + (4 λ + 9) M r + λ ( λ + 2) r (cid:1) (3 M + λr ) (HG3) + r (cid:0) M + 3 M (2 λr + r ) + λ r (cid:1) λ + 1)(2 M − r )(3 M + λr ) (ME1) − r ( r − M ) (cid:0) M + (10 λ − M r + ( λ − λr (cid:1) λ + 1)(3 M + λr ) (ME2) + (cid:0) − M + 3 M r + λ ( λ + 2) r (cid:1) λ + 1)(3 M + λr ) (ME6)+ r λ + 1)(3 M + λr ) ∂ r (ME1) + r ( r − M ) λ + 1)(3 M + λr ) ∂ r (ME2)+ 2 r (2 M − r )3 M + λr (ME5) + r (2 M − r )4( λ + 1)(3 M + λr ) ∂ r (ME6) . (cid:3) By substituting H ,mℓ in (ME4), we obtain (cid:3) ℓ h ,mℓ = − r − M ) ∂ t ∂ r G mℓ − (cid:18) − Mr (cid:19) ∂ t G mℓ − Mr ∂ r h ,mℓ + 2 ω ( r − M ) r h ,mℓ + 2 (cid:0) − M − λM r + λr (cid:1) r (3 M + λr ) ∂ r ψ mℓ + (cid:18) − Mr (cid:19) ∂ t ∂ r ψ mℓ . (D.7)By substituting K mℓ and H ,mℓ we have (cid:3) ℓ h ,mℓ = (cid:18) − Mr (cid:19) ∂ r G + 4 r M − r ∂ t G mℓ + 4 r G mℓ + 2 M ( r − M ) ∂ t h ,mℓ − M − r ) r ∂ r h ,mℓ + 4 Mr h ,mℓ − M (cid:0) M + 2 λM r + λ ( λ + 1) r (cid:1) r (3 M + λr ) ψ mℓ − r M − r ∂ t ψ mℓ + 2 M (3 M − ( λ + 3) r ) r (3 M + λr ) ∂ r ψ mℓ . (D.8)By replacing G mℓ and h ,mℓ by W ,mℓ and W ,mℓ in (ME7), we obtain (cid:3) ℓ W ,mℓ = 1 r (cid:18) − Mr (cid:19) W ,mℓ + 2 r W ,ℓ . Summing over all mℓ with ℓ ≥ h ,mℓ , h ,mℓ and G mℓ in (D.7), we obtain (cid:3) ℓ W ,mℓ = 2 Mr ∂ r W ,mℓ − Mr (cid:18) − Mr (cid:19) − ∂ t W ,mℓ + 2 (cid:0) − M − λM r + λr (cid:1) r ( λr + 3 M ) ∂ t ψ mℓ + 2 (cid:18) − Mr (cid:19) ∂ t ∂ r ψ mℓ = − Mr P even ,mℓ − Mr W ,mℓ + 2 (cid:0) − M − λM r + λr (cid:1) r ( λr + 3 M ) ∂ t ψ mℓ + 2 (cid:18) − Mr (cid:19) ∂ t ∂ r ψ mℓ . Here we used the definition of P even , (4.19). Then (4.21) follows from summing over mℓ with ℓ ≥
2. Similarly, from (D.8) we obtain
CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 830 (cid:3) ℓ W ,mℓ = − Mr (cid:18) − Mr (cid:19) − ∂ t W ,mℓ + 2 Mr ∂ r W ,mℓ − λ + 4 r (cid:18) − Mr (cid:19) W ,mℓ + 2 r (cid:18) − Mr (cid:19) W ,mℓ + 6 M (2 M − r ) (cid:0) M + 2 λM r + λ ( λ + 1) r (cid:1) r (3 M + λr ) ψ mℓ − M (2 M − r )(3 M − ( λ + 3) r ) r (3 M + λr ) ∂ r ψ mℓ + 2 ∂ t ψ mℓ = Mr S W,mℓ − λ + 4 r (cid:18) − Mr (cid:19) W ,mℓ + 2 r (cid:18) − Mr (cid:19) W ,mℓ + 6 M (2 M − r ) (cid:0) M + 2 λM r + λ ( λ + 1) r (cid:1) r (3 M + λr ) ψ mℓ − M (2 M − r )(3 M − ( λ + 3) r ) r (3 M + λr ) ∂ r ψ mℓ + 2 ∂ t ψ mℓ . Here we used S W,mℓ = − (cid:18) − Mr (cid:19) − ∂ t W ,mℓ + 2 ∂ r W ,mℓ − λ + 4 r W ,mℓ + 4 r W ,mℓ . By summing over all mℓ with ℓ ≥ S W , P even and Q even . S W − /trh RW satisfies free waveequation since /trh RW = trh RW and trh RW − S W = S . The equation for P even ,mℓ follows the theone for W ,mℓ and W ,mℓ . We compute (cid:3) P even ,mℓ = − (cid:18) − Mr (cid:19) − ∂ t W ,mℓ + 2 ∂ t ∂ r W ,mℓ − λ + 4 r ∂ t W ,mℓ + 4 r ∂ t W ,mℓ + 4 (cid:0) M + λ ( λ + 2) r (cid:1) r (3 M + λr ) ∂ t ψ mℓ − (cid:18) − Mr (cid:19) ∂ t ∂ r ψ mℓ . From the expression of S W,mℓ and( /trh RW ) mℓ = 2 λ + 2 r ψ mℓ + 2 (cid:18) − Mr (cid:19) ∂ r ψ mℓ − Mr ( λr + 3 M ) (cid:18) − Mr (cid:19) ψ mℓ . we compute (cid:3) ℓ P even ,mℓ = ∂ t ( S W,mℓ − ( /trh RW ) mℓ ) − λ (( λ + 3) r − M ) r (3 M + λr ) ∂ t ψ mℓ − (cid:18) − Mr (cid:19) ∂ t ∂ r ψ mℓ . Then after summing over all mℓ with ℓ ≥
2, one has (4.25). From the definition of Q even , wehave Q even ,mℓ = − W ,mℓ + (cid:18) − Mr (cid:19) ∂ r W ,mℓ + r S W,mℓ − r (cid:18) λ + 3 Mr (cid:19) · r ψ mℓ .Q even corresponds to ψ defined in [5, (3.63)]. And the equation for Q even is equivalent to [5,(3.61)]. Alternative, we can also derive it through the equation of W ,mℓ , W ,mℓ , S W,mℓ and ψ mℓ as (cid:3) ℓ Q even ,mℓ − r (cid:18) − Mr (cid:19) Q even ,mℓ = 0 . Then (4.26) follows by summing over all mℓ with ℓ ≥ References [1] Lars Andersson, Thomas B¨ackdahl, Pieter Blue, and Siyuan Ma. Stability for linearized gravity on the Kerrspacetime. arXiv e-prints , page arXiv:1903.03859, Mar 2019.[2] Y. Angelopoulos, S. Aretakis, and D. Gajic. A vector field approach to almost-sharp decay for the waveequation on spherically symmetric, stationary spacetimes.
Ann. PDE , 4(2):Art. 15, 120, 2018.[3] Leor Barack and Carlos O. Lousto. Perturbations of schwarzschild black holes in the lorenz gauge: Formula-tion and numerical implementation.
Phys. Rev. D , 72:104026, Nov 2005.[4] James M. Bardeen and William H. Press. Radiation fields in the Schwarzschild background.
J. MathematicalPhys. , 14:7–19, 1973.[5] Mark V. Berndtson.
Harmonic gauge perturbations of the Schwarzschild metric . PhD thesis, -, Mar 2009.[6] Lydia Bieri. An extension of the stability theorem of the Minkowski space in general relativity.
J. DifferentialGeom. , 86(1):17–70, 2010.[7] S. Chandrasekhar.
The mathematical theory of black holes . Oxford Classic Texts in the Physical Sciences.The Clarendon Press, Oxford University Press, New York, 1998. Reprint of the 1992 edition.[8] Yvonne Choquet-Bruhat. Th´eor`eme d’existence pour certains syst`emes d’´equations aux d´eriv´ees partiellesnon lin´eaires.
Acta Math. , 88:141–225, 1952.[9] Yvonne Choquet-Bruhat and Robert Geroch. Global aspects of the Cauchy problem in general relativity.
Comm. Math. Phys. , 14:329–335, 1969.[10] D. Christodoulou and S. Klainerman. The nonlinear stability of the Minkowski metric in general relativity.In
Nonlinear hyperbolic problems (Bordeaux, 1988) , volume 1402 of
Lecture Notes in Math. , pages 128–145.Springer, Berlin, 1989.[11] Mihalis Dafermos, Gustav Holzegel, and Igor Rodnianski. The linear stability of the Schwarzschild solutionto gravitational perturbations.
Acta Math. , 222(1):1–214, 2019.[12] Mihalis Dafermos and Igor Rodnianski. Lectures on black holes and linear waves. arXiv e-prints , pagearXiv:0811.0354, Nov 2008.[13] Mihalis Dafermos and Igor Rodnianski. The red-shift effect and radiation decay on black hole spacetimes.
Comm. Pure Appl. Math. , 62(7):859–919, 2009.[14] Mihalis Dafermos and Igor Rodnianski. A new physical-space approach to decay for the wave equation withapplications to black hole spacetimes. In
XVIth International Congress on Mathematical Physics , pages421–432. World Sci. Publ., Hackensack, NJ, 2010.[15] Elena Giorgi. Coupled gravitational and electromagnetic perturbations of Reissner-Nordstr¨om spacetimein a polarized setting II - Combined estimates for the system of wave equations. arXiv e-prints , pagearXiv:1804.05986, Apr 2018.[16] Elena Giorgi. Boundedness and decay for the Teukolsky system of spin ± | Q | ≪ M . arXiv e-prints , page arXiv:1811.03526, Nov 2018.[17] Elena Giorgi. Boundedness and decay for the Teukolsky equation of spin ± l = 1 spherical mode. arXiv e-prints , page arXiv:1812.02278, Dec 2018.[18] Elena Giorgi. The linear stability of Reissner-Nordstr¨om spacetime for small charge. arXiv e-prints , pagearXiv:1904.04926, Apr 2019.[19] Dietrich H¨afner, Peter Hintz, and Andr´as Vasy. Linear stability of slowly rotating Kerr black holes. arXive-prints , page arXiv:1906.00860, Jun 2019.[20] Peter Hintz and Andr´as Vasy. A global analysis proof of the stability of Minkowski space and the polyho-mogeneity of the metric. arXiv e-prints , page arXiv:1711.00195, Oct 2017.[21] Peter Hintz and Andr´as Vasy. The global non-linear stability of the Kerr–de Sitter family of black holes. ActaMath. , 220(1):1–206, 2018.[22] Pei-Ken Hung. The linear stability of the Schwarzschild spacetime in the harmonic gauge: odd part. arXive-prints , page arXiv:1803.03881, Mar 2018.[23] Pei-Ken Hung, Jordan Keller, and Mu-Tao Wang. Linear Stability of Schwarzschild Spacetime: Decay ofMetric Coefficients. arXiv e-prints , page arXiv:1702.02843, Feb 2017.[24] Thomas Johnson. On the linear stability of the Schwarzschild solution to gravitational perturbations in thegeneralised wave gauge. arXiv e-prints , page arXiv:1803.04012, Mar 2018.
CHWARZSCHILD LINEAR STABILITY IN HARMONIC GAUGE 85 [25] Thomas Johnson. The linear stability of the Schwarzschild solution to gravitational perturbations in thegeneralised wave gauge. arXiv e-prints , page arXiv:1810.01337, Oct 2018.[26] Sergiu Klainerman and Francesco Nicol`o.
The evolution problem in general relativity , volume 25 of
Progressin Mathematical Physics . Birkh¨auser Boston, Inc., Boston, MA, 2003.[27] Sergiu Klainerman and Jeremie Szeftel. Global Nonlinear Stability of Schwarzschild Spacetime under Polar-ized Perturbations. arXiv e-prints , page arXiv:1711.07597, Nov 2017.[28] Hans Lindblad and Igor Rodnianski. Global existence for the Einstein vacuum equations in wave coordinates.
Comm. Math. Phys. , 256(1):43–110, 2005.[29] Hans Lindblad and Igor Rodnianski. The global stability of Minkowski space-time in harmonic gauge.
Ann.of Math. (2) , 171(3):1401–1477, 2010.[30] Karl Martel and Eric Poisson. Gravitational perturbations of the Schwarzschild spacetime: a practical co-variant and gauge-invariant formalism.
Phys. Rev. D (3) , 71(10):104003, 13, 2005.[31] Georgios Moschidis. The r p -weighted energy method of Dafermos and Rodnianski in general asymptoticallyflat spacetimes and applications. Ann. PDE , 2(1):Art. 6, 194, 2016.[32] Tullio Regge and John A. Wheeler. Stability of a Schwarzschild singularity.
Phys. Rev. (2) , 108:1063–1069,1957.[33] Volker Schlue. Decay of linear waves on higher-dimensional Schwarzschild black holes.
Anal. PDE , 6(3):515–600, 2013.[34] Abhay G. Shah, Bernard F. Whiting, Steffen Aksteiner, Lars Andersson, and Thomas B¨ackdahl. Gauge-invariant perturbations of Schwarzschild spacetime. arXiv e-prints , page arXiv:1611.08291, Nov 2016.[35] S. A. Teukolsky. Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Elec-tromagnetic, and Neutrino-Field Perturbations. apj , 185:635–648, October 1973.[36] Robert Wald. On perturbations of a Kerr black hole.
Journal of Mathematical Physics , 14:1453, October1973.[37] Bernard F. Whiting. Mode stability of the Kerr black hole.
J. Math. Phys. , 30(6):1301–1305, 1989.[38] Frank J. Zerilli. Effective potential for even parity regge-wheeler gravitational perturbation equations.
Phys.Rev.Lett. , 24:737–738, 1970.[39] Frank J. Zerilli. Gravitational field of a particle falling in a Schwarzschild geometry analyzed in tensorharmonics.
Phys. Rev. D (3) , 2:2141–2160, 1970.
Pei-Ken Hung, Massachusetts Institute of Technology, Department of Mathematics, 182 MemorialDrive, Cambridge, MA 02139, USA
E-mail address ::