RADIATION FIELDS ON SCHWARZSCHILD SPACETIME

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1 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME DEA BASKI AD FAG WAG Abstract. In this paper we define the radiation field for the wave equation on the Schwarzschild black hole spacetime. In this context it has two components: the rescaled restriction of the time derivative of a solution to null infinity and to the event horizon. In the process, we establish some regularity properties of solutions of the wave equation on the spacetime. In particular, we prove that the regularity of the solution across the event horizon and across null infinity is determined by the regularity and decay rate of the initial data at the event horizon and at infinity. We also show that the radiation field is unitary with respect to the conserved energy and prove support theorems for each piece of the radiation field.. Introduction In this paper we define the radiation field for the wave equation on the Schwarzschild black hole spacetime. The radiation field is a rescaled restriction of the time derivative of a solution and in this case has two components: one corresponding to null infinity and one corresponding to the event horizon. In the process, we establish some regularity properties of solutions of the wave equation on the spacetime. In particular, we prove that the regularity of the solution across the event horizon and across null infinity is determined by the regularity and decay rate of the initial data at the event horizon and at infinity. We also show that the radiation field is unitary with respect to the conserved energy and prove support theorems for each component of the radiation field. The radiation field for a solution of the wave equation describes the radiation pattern seen by distant observers. On Minkowski space R R n, it is the rescaled restriction of a solution to null infinity. More precisely, one introduces polar coordinates r, ω in the spatial variables as well as the lapse parameter s = t r. The forward radiation field of a solution u of t u = 0 with smooth, compactly supported initial data is given by lim sr n us + r, rω. r Date: May, 03. The authors are grateful to Antônio Sá Barreto for pointing out the simple proof of the + -dimensional support theorem. D.B. was supported by SF postdoctoral fellowship DMS

2 DEA BASKI AD FAG WAG The map taking the initial data to the radiation field of the corresponding solution provides a unitary isomorphism from the space of finite energy initial data to L R s S n ω. The radiation field is a translation representation of the wave group and was initially defined by Friedlander [Fri80], though it is implicit in the work of Lax Phillips e.g., [LP89] and Helgason e.g., [Hel99]. We now recall the structure of the Schwarzschild black hole spacetime. For a more thorough discussion, including many different coordinate systems, we direct the reader to the the book of Hawking and Ellis [HE73] or to the lecture notes of Dafermos and Rodnianski [DR08]. The Schwarzschild spacetime is diffeomorphic to R t M, r S ω with Lorentzian metric given by r M g S = dt r + dr + r dω. r r M Here dω is the round metric on the unit sphere S. We consider the Cauchy problem:. S u = 0, u, t u t=0 = u 0, u where S is the Laplace Beltrami D Alembertian operator for g S : r r M S = t + r + r M r r r M ω + r r. Solutions u of equation. possesses a conserved energy Et: Et = etr dω dr M S where et = M t u + M r u + r r r ωu. Observe that et is positive definite but is ill-behaved at r = M. As there are two spatial ends of the Schwarzschild black hole, there are two ends through which null geodesics i.e., light rays can escape : the event horizon at the r = M end and null infinity at the r = end. There are also trapped null geodesics tangent to the photon sphere r = 3M. In terms of incoming Eddington Finkelstein coordinates, τ = t + r + M logr M, r, ω, the event horizon corresponds to r = M. Similarly, in terms of outgoing Eddington Finkelstein coordinates, τ = t r M logr M, r, ω, null infinity corresponds to r =. Figure depicts the Penrose diagram of the Schwarzschild black hole exterior; E ± corresponds to the event horizons and S ± to null infinity.

3 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME 3 E + S + t = 0 E S Figure. The Penrose diagram of the Schwarzschild exterior. E ± are the event horizons for the black and white holes, while S ± are the future and past null infinities. In this paper we study the behavior of solutions to the Cauchy problem in terms of their radiation fields. We first partially compactify the Schwarzschild spacetime to a manifold with corners and find expansions for solutions of equation. at each boundary hypersurface. Two of the boundary hypersurfaces correspond to null infinity and the event horizon. Currently we do not include a full compactification, as we omit temporal infinity. A full compactification on which solutions are well-behaved is expected to be somewhat complicated, as it must take into account the different expected behaviors of solutions at the event horizon, null infinity, and the photon sphere r = 3M. The main regularity result of this paper is the following: Theorem.. If the initial data φ, ψ has an asymptotic expansion at r = M, i.e., if φ = r M λ/ φ r M, ω, ψ = r M λ/ ψ r M, ω, where φ, ψ C [0, S, then the solution u has an asymptotic expansion in terms of e τ/4m at τ = near the event horizon and the k-th term defined by equation 3.3 in the expansion is C l,δ up to the event horizon, where 0 < λ + k = l + α and δ = min{α ɛ, /} for any ɛ > 0. Similarly, if the initial data has a classical asymptotic expansion at infinity, i.e., if φ = r λ φ/r, ω, ψ = r λ ψ/r, ω, where φ, ψ C [0, /M S, then the solution u has a polyhomogeneous asymptotic expansion in terms of / τ at τ = near null infinity and the k-th term defined by equation 3.5 in the expansion is C l,δ up to null infinity, where l and δ are as above.

4 4 DEA BASKI AD FAG WAG Remark.. The above theorem shows that if the initial data are smooth and have asymptotic expansions at r = M and r =, then the solution has an asymptotic expansion at E 0 and S 0 as pictured in Figure. The regularity at E + respectively, S+ of each term in this expansion is determined by its rate of decay at E 0 respectively, S 0 and hence by the expansion of the initial data. The structure of the partial compactification is discussed in Section. E + τ τ S + b ρ b a ā E 0 S 0 r = M t = 0 r = Figure. The compactification of Schwarzschild spacetime for t 0: E + is the event horizon; S+ is null infinity; E 0 and S 0 are from the blow-up of the spatial ends in the Penrose diagram of the spacetime. One consequence of Theorem. is the existence of the radiation field, i.e., that solutions of equation. may be restricted to the event horizon E + and null infinity S +. As there are two spatial ends of the Schwarzschild black hole exterior, our definition of the radiation field has two components. For smooth initial data φ, ψ compactly supported in M, S, we define the two components of the forward radiation field as follows: R E + φ, ψτ, ω = R S + lim tuτ r M logr M, r, ω, r M φ, ψ τ, ω = lim r t u τ + r + M logr M, r, ω. r The backward radiation fields R E and R S are defined analogously in equation 5.. We show that, under our definition, the radiation field is unitary i.e., norm-preserving: Theorem.3. Given φ, ψ with finite energy, the radiation field of the solution u of equation. is unitary, i.e., 4M RE + φ, ψ + RS L R S + φ, ψ = E0. L R S Related scattering-theoretic work on the unitarity of wave operators has been carried out by Bachelot [Bac94] for the Klein-Gordon equation and Dimock [Dim85] for the wave equation.

5 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME 5 We also prove the following support theorem for the radiation field stated more precisely in Section 5: Theorem.4. Suppose that φ, ψ Cc M, S. If R E + φ, ψ vanishes for τ τ 0 and R E φ, ψ vanishes for τ τ 0, then both φ and ψ are supported in [r 0, S, where r 0 is given implicitly by r + M logr M = τ 0. An analogous statement holds for the component of the radiation field corresponding to null infinity. Remark.5. The smoothness hypothesis above is not essential; we can relax the assumption to C,α for α > 0. In particular, we require only enough smoothness to ensure that the rescaled solution is C,α after being extended by zero across the event horizon or null infinity. In fact, the support hypothesis near the event horizon can be relaxed as well; if the initial data has enough decay there taking λ > in Theorem. should suffice, then the rescaled solution will still have enough smoothness for the uniqueness theorems to apply. ear infinity, however, it is important that we take compactly supported data, as the past and future null infinities do not meet smoothly. Even if this difficulty were overcome, a strong decay condition must be assumed to rule out the counter-examples that exist already in Minkowski space. In particular, for any m, there are smooth functions fz in R n which are not compactly supported, decay like z m, and whose Radon transform and hence radiation field is compactly supported see, e.g., [Hel99]. In the setting of Minkowski space, the Fourier transform of the forward radiation field in the s variable is given in terms of the Fourier transform of the initial data and is related to the Radon transform. In other settings, one may think of the Fourier transform of the radiation field as a distorted Fourier transform. The support theorem can then be seen as a Paley Wiener theorem for a distorted Fourier-type transform on the Schwarzschild spacetime. Similar support theorems in other contexts have been established by Helgason [Hel99] and Sá Barreto [SB08, SB05]. Although we are unable to characterize the range of the radiation field, we do show that our definition of the radiation field captures too much information. In particular, the support theorem above implies the following: Corollary.6. If ψ Cc Similarly, for such a ψ, if R S + M, S and R E + 0, ψ 0, then ψ = 0. 0, ψ 0, then ψ = 0. In other words, for odd, smooth, compactly supported data, knowing that one component of the radiation field vanishes implies that it must vanish on the other component as well. The study of the asymptotic behavior of solutions of the wave equation on the Schwarzschild background has been an active field of research see,

6 6 DEA BASKI AD FAG WAG e.g., [DR09, MMTT0, Luk0] and we refer the reader to the lecture notes of Dafermos and Rodnianski [DR08] for many references. In Section 4 we view the unitarity of the radiation field as a consequence of Price s law [DSS, Tat09, MTT], though we remark that a weaker energy decay result would suffice. Section describes the partial compactification on which we work, while in Section 3, we prove Theorem. via energy estimates. Section 4 is devoted to the proof of Theorem.3 and relies on the pointwise decay of solutions established by the proof of Price s law. Finally, in Section 5, we prove the support theorems... otation. In this section we lay out some notation common to the entire paper. The regularity and asymptotic behavior of solutions near E + and S + may be obtained via energy estimates. As a preparation, for any function v and time-like function T, we give a name to the vector field obtained by contracting the stress-energy tensor of v with respect to a Lorentzian metric g with the gradient of T : F g T, v = T, v g v v, v g + v T. We also record where div g F g T, v = T, v g g v + Q g T, v, Q g T, v = Hess g T dv, dv gt v, v g + v. For a manifold with corners M, we further require the spaces of uniformly degenerate 0- and tangential b- vector fields: V 0 = {X C M, T M : X vanishes at M} V b = {X C M, T M : X is tangent to M} For a given measure dµ, the 0-Sobolev space H0 M, dµ is the space of functions u L dµ so that Xu L dµ for all X V 0. For an integer and a given measure dµ, the b-sobolev space Hb M, dµ consists of functions u L dµ so that X... X k u L dµ for X j V b and k. The mixed 0, b-sobolev space H, 0,b M, dµ consists of those u H 0 M, dµ so that X... X k u H0 M, dµ for X j V b and k. For an introduction to b-sobolev spaces and b-geometry, we refer the reader to the book of Melrose [Mel93].. A partial compactification In this section we describe the partial compactification of Schwarzschild spacetime for t 0 see Figure on which we work. It can be obtained by a suitable blow-up with a logarithmic correction of the Penrose diagram in

7 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME 7 Figure. We describe the smooth structure on this partial compactification by taking explicit local coordinates near the boundary. Each set of local coordinates is valid in the corresponding domain as in Figure. We now describe coordinates giving the smooth structure on the partial compactification and the domains in which they are valid. ear r = M but away from temporal infinity, i.e., for r M < and t + r + M logr M <, we choose coordinates a, b, ω with a = e t+r t+r M, b = e 4M r M. ear the interior of the event horizon E +, i.e., r M < and < t + r + M logr M <, we also can use coordinates τ, ρ, ω with τ = t + r + M logr M, ρ = r M. Here the initial surface {t = 0} for r close to M is equivalent to {a = e M+ab M } for a close to e and b close to 0, which intersect with E 0 smoothly. For r close to, we choose corresponding coordinates as follows: For r large and close to the spatial end S 0, i.e., for r M > 0 and t r M logr M < 0, we choose coordinates ā, b, ω with ā = t + r + M logr M, r b = t + r + M logr M. ear the interior of null infinity, i.e., r M > 0 and < t r M logr M <, we use coordinates τ,, ω with τ = t r M logr M, = /r. Here the initial surface {t = 0} for r large is equivalent to ā = + Mā b log Mā b logā b for ā close to and b close to 0. The initial surface does not intersect S 0 smoothly but instead has a logarithmic correction term. A direct consequence of this lack of smoothness is that for classical initial data which have pure Taylor expansions at r =, the radiation field on S + will have an expansion including logarithmic terms at τ =. We refer the reader to Proposition 3.3 and the surrounding discussion for details. 3. Existence and regularity of the radiation field In this section we establish regularity properties for solutions of equation. on a partial compactification of the Schwarzschild background. We also show that sufficiently regular solutions have asymptotic expansions at the boundary hypersurfaces of this compactification. Taken together, Propositions 3.5 and 3.3 prove Theorem..

8 8 DEA BASKI AD FAG WAG 3.. At the event horizon. For t > 0 we change coordinates to τ = t + r + M logr M, ρ = r M. The coordinates τ, ρ are essentially the incoming Eddington-Finkelstein coordinates. The function τ is a coordinate along the event horizon E +, while ρ is a defining function for E+ as long as τ is bounded away from ±. The metric and its D Alembertian are then: g S = ρ ρ + M dτ + dτ dρ + ρ + M dω, S = τ ρ + ρ ρ + M ρ + ρ + M ω + ρ + M ρ + M ρ + ρ + M τ. We can thus extend g S naturally to a Lorentzian metric in a neighborhood of {ρ = 0} with {ρ = 0} characteristic. Moreover, if Z ij are the rotations of S, then [ S, τ ] = [ S, Z ij ] = 0, 3ρ + M [ S, ρ ρ ] = S ρ + M 3 ω + ρ + M ρ ρ + V, M [ S, ρ ] = ρ + M ρ + ρ + M 3 ω + V, where V, V are vector fields that are tangent to the event horizon: 4M V = ρ + M τ + ρ M ρ + M 3 ρ ρ, V = ρ + M 3 ρ ρ + ρ + M τ. In terms of the coordinates τ, ρ, ω, we choose a time-like function by ρ T = τ ρ, T, T gs = + ρ + M <. We may then compute FT, v, T gs = τ v + τ v + ρ ρv ρ + M + M ρv ρ + M + ρ + 4M ω v ρ + M ρ + M + v, M QT, v = ρ + M ρ v + ρ ρv ρ + M + 4 ρ τ v ωv ρ + M + v. Using Friedlander s argument [Fri80], we now show that solutions of the wave equation with compactly supported smooth initial are smooth across the event horizon E +.

9 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME 9 Proposition 3.. If u 0, u C M, r S are such that suppu 0 suppu M + ɛ, S for some ɛ > 0, then u is smooth down to {ρ = 0} for all τ,. E + S + t = t 0 τ = τ 0 Ω t0 τ = τ 0 t = 0 Figure 3. The domain Ω t0. Proof. By finite speed of propagation, u 0 for τ M + ɛ + M log ɛ. We now fix τ 0 > M + ɛ + M log ɛ large. Let Ω t0 be the domain bounded by {t = 0}, {t = t 0 }, {τ = τ 0 }, and {τ = τ 0 } pictured in Figure 3. Here {t = 0}, {t = t 0 } are space-like with defining function t and {τ = ±τ 0 } are null with defining function τ. Moreover, T, t gs < 0 and T, τ gs < 0 imply that FT, v, t gs 0, FT, v, τ gs 0. Let Σ s = {T = s} with defining function T and Ω s t 0 = Ω t0 {T s}. Define now M u, s = e ct F T, Z I u /, T gs dµ T Σ s Ω t0 I where Z { ρ, τ, Z ij } and dµ T dt = dv gs. ote that each term in sum above is positive, so M controls the first + derivatives of u. Choose c large enough so that div e ct FT, Z I u = e ct c FT, Z I u + T, Z I u gs Z I u + Q gs T, Z I u 0

10 0 DEA BASKI AD FAG WAG for all I. Here c depends on τ 0 and. By Stokes theorem, we then have that M u, s e ct FT, Z I u, T dµ t. {t=0} Ω t0 I Because T is bounded on Ω and {t = 0} Ω t0 is independent of t 0, we have that e ct FT, Z I u, T gs dv gs = M u, s ds C <, Ω t0 I where C depends only on and τ 0. Letting t 0, we obtain e ct FT, Z I u, T gs dv gs C,τ0. Ω I Because and τ 0 are arbitrary, u is smooth up to {ρ = 0} for all τ,. We now consider non-compactly supported data. On the partial compactification depicted in Figure, we use coordinates a and b, defined as follows: a = ρe τ M = e t+r M [0, e ], b = e τ 4M = e t+r 4M r M. These are valid in a neighborhood of E 0, especially near the intersection of E 0 and E +, where a is a defining function for E+ and b is a defining function for E 0. The function r M = ab vanishes on both E 0 and E +. ear the interior of E +, a, b are equivalent to the coordinates τ, ρ as above. ear the initial surface t = 0 and r finite, a, b are equivalent to the coordinates β, t = r M, t. In coordinates a, b, the conformal metric g e = b M g S and its wave operator are: g e = da db b + a ab db + M + ab + M + ab b 4Mb dω, e = a b b a a + We thus have that 4b M + ab M + ab a b a + M + ab b b. 4Mb M + ab b ω M + ab a a gs u = 0 e + γ e ũ = 0, ũ = b Mu, γ e = b e b = b M + ab.

11 Moreover, RADIATIO FIELDS O SCHWARZSCHILD SPACETIME 3. [ ] [ ] e, Z ij = 0, e, b b = b c I Z I, [ ] e, a a = e +b c IZ I, I I where Z {a a, b b, Z ij }and c I, c I are smooth coefficients. In the above and what follows, we use I as a multi-index. In coordinates a, b, ω, we choose time-like functions as follows: ab T = a + log b, T, T ge = a + M + ab < 0, T = a, T, T ab ge = a + M + ab < 0. Here T is asymptotically null when approaching the event horizon. Moreover, we again compute 3. F ge T, v, T ge = + ab M + ab a a v + b bv + ab + M + ab a a v b b v + ab 4M b ω v + a + M + ab M + ab + v, F ge T, v, log b ge = a v ab + + M + ab a a v + 4M b ω v M + ab + v, Q ge T, v = a v b M + ab 8Mb M + ab a v b b v a a v + Θ a a v, b b v, b ω v, v, where Θ v,..., v l is a quadratic form of v,..., v l with smooth coefficients. Proposition 3.. Suppose that u 0, u is in the following Sobolev space β λ H, 0,b [0, β 0 β S dβ dω, β 3 β λ H, 0,b [0, β 0 β S dβ dω, β 3 for some β 0 > 0, > and λ > 0. Then ũ = b Mu is C δ up to {a = 0} for b < β 0 e β 0 +M 4M, where δ = min{λ, }.

12 DEA BASKI AD FAG WAG Proof. The assumption on u 0, u implies that on the Cauchy surface Z I ũ t=0 β λ H, I 0,b [0, β 0 S dβ dω, β 3 for Z {a a, b b, Z ij }. Let Σ s = {T = s}, so that Σ s is space-like with defining function T for s < 0. As s 0, Σ s approaches a characteristic surface. Let Ω be the domain bounded by Σ 0, {t = 0}, {b = 0} and S = {T = log b 0 }. We then define M ũ, s; λ = b λ e ct F ge T, Z I ũ, T ge dµ T, I I Σ s Ω L ũ, s; λ = b λ e ct F ge T, Z I ũ, T ge dµ T S {T <s} Ω where Z {a a, b b, Z ij } and dµ T dt = dv g e, dµ T dt = dv ge. We first choose s 0 close to 0 so that Σ s0 {t = 0} Ω =. Let Ω s0 = Ω {s s 0 } illustrated in Figure 4 so that Ω s0 is bounded by Σ s0, {t = 0}, {b = 0}, and S. Choose c large enough so that in Ω s0 we have div ge b λ e ct F ge T, Z I ũ I = b λ e ct I λ F ge T, Z I ũ, log b ge + Q ge T, Z I ũ c F ge T, Z I ũ, T ge + T, Z I ũ ge e Z I ũ 0. Here c depends only on b 0, s 0, and, and the dependence on s 0 is required to bound a Z I ũ in terms of a a Z I ũ. In order to bound the term of the form T, Z I ũ ge e Z I ũ, we have used that ũ solves e + γ e ũ = 0 and the expressions 3. for the commutators of with Z I. By Stokes theorem, we then have M ũ, s 0 + L ũ, s 0 I {t=0} Ω b λ e ct F ge T Z I ũ, t ge dµ t, where dµ t dt = dv ge. Observe that the right hand side is equivalent to the square of the initial data norm.,

13 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME 3 E + S Σ s0 E 0 Ω s0 r = M t = 0 Figure 4. The domain Ω s0 For s > s 0, we now let Ω s s 0 denote the domain bounded by Σ s0, Σ s, {b = 0}, and S. Again Stokes theorem implies that M ũ, s M ũ, s 0 + L ũ, s L ũ, s 0 = I Ω s s 0 div b λ e ct F ge T, Z I ũ dv ge. Dividing by s s 0, taking a limit, and then using the expressions 3. yields that s M ũ, s + s L ũ, s { M λa + Ca ũ, s λ < Ca M ũ, s, λ with C a constant depending only on b 0 and. ote that the power a / arises from the need to estimate the terms of the form a Z I ũb b Z I ũ. Integrating in s, we find that for all s > s 0, { M C s λ s 0 λ M ũ, s ũ, s 0 λ < C M ũ, s 0 λ, where C = exp e 0 Ca / da. Because >, the Sobolev embedding theorem implies that for a < s 0, we have { C M a ũ ũ, s 0a λ λ < C M ũ, s 0a λ, where C only depends on b 0. Integrating in a then finishes the proof. Corollary 3.3. Suppose λ = k + α with α 0, ] and k 0. Then if u 0, u is in the following space β λ H, 0,b [0, β 0 S dβ dω, β 3 β λ H, 0,b [0, β 0 S dβ dω, β 3

14 4 DEA BASKI AD FAG WAG with > + k, then ũ is C k,δ up to {a = 0} for b < β 0 e β 0 +M 4M, where δ = min{α, }. Proof. otice that [ e, k a] = k k+ a + b [ ] a a, a k = k a, k I +i k+,i k [ ] b b, a k = c I,i Z I i a, [ ] Z ij, a k = 0, where Z {a a, b b, Z ij } and c I,i are smooth coefficients. We set M ũ, s; λ = M i aũ, i s; λ, i k L ũ, s; λ = L i aũ, i s; λ. i k According to the divergence formula in the proof of Proposition 3., we have that s M ũ, s + s L ũ, s αa + Ca M ũ, s α < Ca. M ũ, s α We thus obtain the following in a similar manner to the proof of Proposition 3.: aũ i C M ũ, s 0, a < s 0, i k; { a k+ C M ũ ũ, s 0 a α α < C M ũ, s 0 a α, which finishes the proof. In particular, Schwartz initial data behaves in much the same way that compactly supported data does: Corollary 3.4. If u 0, u is in the following space β H, 0,b [0, β 0 S dβ dω, β 3 β H, 0,b [0, β 0 S, then ũ is smooth up to {a = 0} for b < β 0 e β 0 +M 4M. dβ dω β 3, We now claim that u has a classical asymptotic expansion if the initial data does. Indeed, suppose that u 0, u β λ C [0, β 0 β S,

15 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME 5 and define the k-th term w k by 3.3 w 0 = ũ = b Mu, w k = Π k i= b b λ i ũ for k. otice that b b lifts to β β β t on initial surface t = 0. Hence Z I w k t=0 β λ+k C [0, β 0 β S for Z {a a, b b, Z ij } and multi-index I. Proposition 3.5. If u 0 and u are as above with 0 < λ+k = l +α for some integer l and α 0, ], then w k is C l,δ up to {a = 0} for b < b 0, where δ = min{α ɛ, } with ɛ > 0 arbitrarily small. Proof. otice that because e + γ e ũ = 0, we have e w i = b c I Z I w i + b b + λ i e w i I i = b c I,j Z I,j w j, j=0 I where Z {a a, b b, Z ij } and c I,j are smooth coefficients. We choose 0 < ɛ 0 < ɛ <... < ɛ k = ɛ and let λ i = λ + i ɛ i. By Proposition 3., choosing large and then c = cs 0, b 0,, we may guarantee M w i, s 0 ; λ i I {t=0} Ω for all i k. Assume now that 3.4 M i b λ i e ct F ge T, Z I w i, t ge dµ t C i < w i, s; λ i C i + s λ i, for s > s 0 and i k. This then implies that b λ i+ e ct b Σ s Ω I i c I Z I w i dµ t C i + s λ i. Then for λ i+, we have as in the proof of Proposition 3. s M i w i+, s; λ i+ λ i+ a + Ca M i w i+, s; λ i+ + Ca M i w i+, s; λ i+ + s λ i,

16 6 DEA BASKI AD FAG WAG while if λ i+, we have s M i Hence, w i+, s; λ i+ Ca M i w i+, s; λ i+ + Ca M i w i+, s; λ i+ + s λ+k ɛ k. M i w k, s; λ i+ C i+ + s λ i+ for all s > s 0. ow we know that equation 3.4 holds for i = 0. By induction and the arbitrariness of ɛ, we have thus shown that w k, s; λ + k ɛ C k + s λ+k ɛ M k for all s > s 0, where C k depends on the initial data and ɛ,, and k. We thus have that if λ + k = l + α > 0, then by inserting a l as in the proof of Corollary 3.3, we see that for δ = min{α ɛ, }, wk is C l,δ up to {a = 0}. 3.. At future null infinity. We now adapt the argument from Section 3. to the region near null infinity. For t > 0 we introduce the coordinates τ = t r M logr M, = r. The conformal metric and its wave operator then become: g = g S = M dτ + d τd + dω, = τ + M + 4M + ω. We now calculate the commutators of with various vector fields: [, τ ] = [, Z ij ] = 0, [, ] = 4M 8M ω, [, ] = 3M 6M. Moreover, we have that S u = 0 + γ ũ = 0, ũ = u, γ = = M. In the coordinates τ,, ω, we choose the time-like function for < min { }, M as follows: T = τ, T, T g = + M < 0.

17 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME 7 We then calculate the vector field F in this context: F g T, v, T g = M v + τ v + M v + τ v + M ω v + v, Q g T, v = 3M v + ω. The above computation allows us to prove the following Proposition, whose proof is the same as that of Proposition 3. with the corresponding time-like function: Proposition 3.6. If u 0, u C M, r S satisfy suppu 0 suppu M, ɛ for some ɛ > 0, then ũ is smooth up to { = 0}. We now consider non-compactly supported data. ear S 0 in the partial compactification depicted in Figure, we may use coordinates ā, b given by t + r + M logr M ā = τ = [0, ], b = r τ. Together with the spherical coordinates ω, these are valid coordinates near the intersection of S 0 and S +, where ā is a defining function for E+ and b is a defining function for E 0. ear the interior of E +, we can take coordinates τ, as before. We extend ā, b up to initial surface t = 0, which defines the smooth structure on this partial compactification near S 0. otice that, at t = 0, logr M ā = +. r In the coordinate system ā, b, ω, we have that d b g = dā + ā [ ā Mā b ] d b + dω b b, = ā b b ā ā + Mā b ā ā + 4Mā b ā ā + ω. We also calculate the commutators of with relevant vector fields: [, Z ij ] = 0, [ ], b b = Mā bā ā + 4Mā bā ā, [ ], ā ā = 4Mā b ā ā 8Mā bā ā ω. In this coordinate system we now choose the time like functions T and T as follows: T = ā + log b, T, T g = ā ā Mā b < 0; T = ā, T, T g = ā ā Mā b < 0.

18 8 DEA BASKI AD FAG WAG The vector field F and the function Q are then given by F g T, v, T g = ā Mā b ā āv + + ā ā Mā b āv b bv b bv + + ā ā Mā b ω v + v, Q g T, v = ā 3Mā b āv ω v v + Mā b ā āv. We are now able to prove the following proposition: Proposition 3.7. If the initial data satisfy u 0, u is in the following space [ λ+ H + b 0, [ S d dω, λ+ Hb 0, S d dω,, r 0 with r 0 +M logr 0 M > 0, > and λ > 0, then ũ is C δ up to {ā = 0} for b < b 0, where δ = min{λ, } and b 0 = /r 0 + M logr 0 M. Proof. The proof is the same as the proof of Proposition 3. with the corresponding time-like function. Because t lifts to b b b ā ā, the assumption on the initial data implies that at the Cauchy surface. Z I ũ t=0 λ H + I b [ 0, r 0 S, r 0 d dω for Z {ā ā, b b, Z ij } and multi-index I. As in the previous section, we have the following corollary: Corollary 3.8. Suppose λ = k + α for some integer k and α 0, ]. If the initial data u 0, u is in the following space [ λ+ H + b 0, [ S d dω, λ+ Hb 0, S d dω,, r 0 with r 0 + M logr 0 M > 0 and > + k, then ũ is C k,δ up to {ā = 0} for b < b 0, where δ = min{α, } and b 0 = /r 0 + M logr 0 M. Proof. The proof is the same as the proof of Corollary 3.3 with the corresponding time-like function. This is because satisfies: [, ā] k = k ā k+ + c I,i Z I a, ī I +i k+,i k r 0 where Z {ā ā, b b, Z ij } and c I,i are smooth coefficients. Again as before, Schwartz data behaves in much the same way as compactly supported data:

19 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME 9 Corollary 3.9. If the initial data u 0, u is in the following space [ Hb 0, [ S d dω, Hb 0, S d dω,, r 0 with r 0 + M logr 0 M > 0, then ũ is smooth up to {ā = 0} for b < b 0 = /r 0 + M logr 0 M. The coordinate system given by ā, b, ω valid only on half of null infinity, i.e. τ < 0. To extend Proposition 3.7 to the other half, we may use similar energy estimates in the τ,, ω coordinate system with τ τ 0, τ 0 for arbitrarily large τ 0. In the coordinate system τ,, ω, we choose a pair of time-like functions T, T for τ τ 0, τ 0 as follows: r 0 T = T, T = τ 0 τ, Here T is asymptotically null when approaching null infinity T, T g = τ 0 τ M τ 0 τ 0. and the vector field F is F g T, v, T g = + τ 0 τ M v + τ 0 τ M v + τ v + τ + τ0 τ + M τ 0 τ ω v + v. We use the pair T, T of time-like functions to obtain estimates similar to those in Propositions 3. and 3.7: Proposition 3.0. Suppose r 0 > M is close to M so that τ 0 = r 0 + M logr 0 M > 0. If the intial data is in the following space [ λ+ H + b 0, [ S d dω, λ+ Hb 0, S d dω, r 0 with > and λ > 0, then ũ is C δ up to { = 0} for all τ, τ 0, where δ = min{λ, }. Proof. For τ τ 0, i.e., b b 0, the statement is proved in Proposition 3.7. Let Ω be the domain bounded by {t = 0}, S +, and S = { T = log b 0 } = { T = τ 0 }, S = { T = τ 0 }. r 0

20 0 DEA BASKI AD FAG WAG Let Σ s = { T = s}, which is space-like for s < 0 and approaches null infinity as s 0. We define the quantities M ũ, s = e c T F g T, Z I ũ, T g dµ T, I I Σ s Ω L ũ, s = e c T S { T F g T, Z I ũ, T g dµ T <s} L ũ, s = e c T S { T F g T, Z I ũ, T g <s} I where Z I {ρ ρ, τ, Z ij }. We choose c large enough so that div g e c T F g T, Z I ũ 0. I ote that this is possible because ũ solves the equation + γ ũ = 0. Here c depends on τ 0 and. We now choose s 0 < 0 small enough so that Σ s0 {t = 0} Ω =. By a proof similar to the one in Proposition 3., we have that M ũ, s 0 < C, where C is bounded by the initial data norm and depends on s 0. Moreover, the bound in the proof of Proposition 3.7 implies that s L ũ, s {C s λ λ < C s λ for s > s 0. Here C depends only on the norm of the initial data, τ 0, and s 0. Consider now the domain Ω s s 0 bounded by Σ s0, Σ s, S, and S, for s > s 0. By Stokes theorem, M ũ, s M ũ, s 0 + L ũ, s L ũ, s 0 L ũ, s L ũ, s 0. Dividing by s s 0 and taking a limit then implies that s M ũ, s s L ũ, s. Using the bound above and integrating then shows that, for s > s 0, M ũ, s C + s λ. An application of Sobolev embedding then shows that ũ is C δ up to { = 0} for τ τ 0, τ 0.,,

21 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME Similarly, Corollaries 3.8 and 3.9 may be extended to the entire null infinity. Corollary 3.. Suppose λ = k + α for an integer k and α 0, ]. If the initial data u 0, u is in the following space [ λ+ H + b 0, [ S d dω, λ+ Hb 0, S d dω, r 0 with r 0 + M logr 0 M < 0 and > + k, then ũ is C k,δ up to { = 0} for τ < τ 0 = r 0 + M logr 0 M. Here δ = min{α, }. Proof. For τ τ 0, this reduces to Corollary 3.8. We now define: M ũ, s; λ = M ρũ, ī s; λ, i k L ũ, s; λ = L i ρũ, ī s; λ, i k L ũ, s; λ = L i ρũ, ī s; λ. i k r 0 otice that [ ], k = I +i k+, i k c I,i Z I ī ρ, where Z {a a, b b, Z ij } and c I,i are smooth coefficients. By a proof similar to that of Corollary 3.3, we have: s M ũ, s; λ + s L ũ, s; λ { C s s L ũ, s; λ α, α < C s., α We thus have that k ũ is C δ up to { = 0} for τ < τ 0. Corollary 3.. If the initial data are Schwartz, i.e., u 0, u in the following space [ Hb 0, [ S d dω, Hb 0, S d dω,, r 0 with r 0 + M logr 0 M < 0, then ũ is smooth up to { = 0} for τ < τ 0 = r 0 + M logr 0 M. r 0

22 DEA BASKI AD FAG WAG Suppose now that the initial data has a classical asymptotic expansion at S 0 spatial infinity, i.e., [ u 0, u λ+ C 0, [ S λ+ C 0, S. r 0 r 0 with r 0 + M logr 0 M < 0. Define the k-th term w k as follows: 3.5 w 0 = ũ = u, w k = k i=0 b b λ i ni ũ for k, where n i are integers that only depend on i such that k,i [ Z I w k t=0 λ+k log j C 0, r 0 j= S for Z {ā ā, b b, Z ij }, multi-index I and integers k, I. Here the integers n i and k, I can be calculated according to the lift of ā ā, b b: ā ā = b b = M M t, log + log M M t t t. Proposition 3.3. If λ + k = l + α for some integer l and α 0, ], then w k is C l,δ up to { = 0} for all τ < τ 0 = r 0 + M logr 0 M, where δ = min{, α ɛ} with ɛ > 0 arbitrarily small. Proof. We apply a similar proof to that of Proposition 3.5 for τ < τ 0, i.e. for b < τ 0, and then extend it to τ < τ 0 by a method similar to the method in Proposition The radiation field In this section we define the radiation field, show it extends to an energy space, and then show it is unitary i.e., norm-preserving on this space. Consider now smooth initial data φ, ψ supported in r M + ɛ and r R 0, and let u be the solution of S u = 0 with this initial data. By the results of the last section, u restricts to the event horizon and ru restricts to null infinity. Definition 4.. The forward radiation field of φ, ψ at the event horizon is the restriction of t u to the event horizon. In the coordinates τ, ρ, ω of Section 3., it is given by R E + φ, ψτ, ω = t uτ, ρ, ω ρ=0. The forward radiation field of φ, ψ at null infinity is the restriction of t v to null infinity. In the coordinates τ,, ω of Section 3., it is given by R S + φ, ψ τ, ω = t u τ,, ω =0.

23 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME 3 Finite speed of propagation implies that R E + φ, ψτ, ω vanishes identically for τ M +ɛ+m log ɛ and that R S + φ, ψ τ, ω vanishes identically for τ R 0 + M logr 0 M. The existence of the static Killing field t implies that the energy Et = etr dr dω is conserved, where et = M r For a fixed λ, let us write Et = S + + Let f E + S M t u + M r r M logr M t λ S S =I + II + III. λ t r+m logr M t λ r+m logr M t λ = R E + φ, ψτ, ω and f S + r u + r ωu. etr dr dω etr dr dω etr dr dω = R S + φ, ψ τ, ω. We now compute term I in the τ, ρ, ω coordinates of Section 3.. Recall that near event horizon, τ = t + r + M logr M, ρ = r M. So t lifts to τ and r lifts to ρ + ρ + M ρ τ. Moreover, t is fixed, so we are free to use τ as the variable of integration so we think of r as a function of τ and then r drdω = ρ + Mρdτdω. We then have that I = S τ λ τ u + ρ ρu τ u ρ + M + ρ ρu ρ + M + ρ ωu ρ + M 3 ρ + M dτ dω. Because R E + φ, ψ vanishes for τ τ 0, the convergence of τ u together with its derivatives to R E + φ, ψ is uniform on, λ], and so we may take a limit as t. This implies that lim t I = 4M S λ f E + dτ dω. We then compute term III in the τ,, ω coordinates of Section 3.. ear future null infinity, τ = t r M logr M, = r.

24 4 DEA BASKI AD FAG WAG So t lifts to τ and r lifts to M τ, and the volume form r dr dω becomes 4 d dω. We are again free to use τ as the variable of integration, so that d τ = M d. We then have that III = τ u + M u τ u + M u S τ λ + M ω u d τ dω. We now use the vanishing of R S + φ, ψ for τ τ 0 to conclude that the convergence of τ u and its derivatives to R S + φ, ψ is uniform on, λ]. This allows us to take a limit as t and conclude that lim t III = S λ f S + d τ dω. Term II is positive, so we may now take a limit as λ to conclude that E0 = 4M fe + + fs 4. L R S + L R S + lim lim etr dr dω. λ t S λ t r+m logr M t λ We define the energy space H E of initial data to be the completion of smooth functions compactly supported in M, S with respect to the energy norm φ, ψ H E = S M [ M ψ + M r r + r ωφ ] r dr dω. r φ The above computation then shows that the radiation field extends to a bounded operator H E L R S L R S. We now show that R + is unitary but not necessarily surjective, i.e., that the last term in equation 4. vanishes. We first show this for compactly supported smooth initial data and then extend by density to the full energy space. Versions of this statement were shown by Dimock [Dim85] and Bachelot [Bac94], though they work in a more abstract scattering framework. The main tool we use to establish unitarity is Price s law, which roughly states that waves decay at the order of t 3 on the Schwarzschild exterior. In particular, we use the formulation of Tataru [Tat09] and Metcalfe Tataru Tohaneanu [MTT]. We do not need the full strength of Price s law; a much weaker energy bound would suffice. In the normalized coordinates of Tataru [Tat09] and Metcalfe Tataru Tohaneanu [MTT], one has that solutions of the wave equation with

25 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME 5 smooth, compactly supported data satisfy the following pointwise bounds: u t t r, tu t t r 3. These coordinates correspond to modified Regge Wheeler coordinates and therefore in our setting, t t, r r + M logr M, t τ, r r, near null infinity, near the event horizon. In particular, in our coordinates u is bounded by t τ near null infinity and by τ t + M logr M near the event horizon. We now use the pointwise bounds to bound the integral II above. Let us implicitly insert cutoff functions and treat it as two integrals - one for large r and another for small r. For large r, inserting the pointwise bounds yields an integral bounded by R t λ r 0 S t τ 4 r dr dω t λ + t λ + λ 3. For r near M, the pointwise bounds yield an integral bounded by r0 R λ t S τ t + M logr M 4 r dr dω C r0 t + λ t M 6 λ 6. Here R is the inverse function of R : M,, defined by Rr = r + M logr M. By sending t and then sending λ, we see that the additional contribution of term II tends to 0. This yields that for smooth, compactly supported data, we have E0 = 4M fe + + fs L R S +, L R S i.e., the radiation field is unitary on this subpace. A standard density argument then shows that the radiation field is a unitary operator on the space of initial data with finite energy norm. This finishes the proof of Theorem Support theorems for the radiation field In this section we prove two support theorems for smooth compactly supported initial data. We first show that for such data, the support of the radiation field on the event horizons provides a restriction on the support of the data. We then show a similar theorem for null infinity. In particular, for initial data 0, ψ, ψ Cc M, S, we show that if the radiation field vanishes on E +, then ψ 0. We also show the corresponding statement for S +. One should think of the t as governing the decay we factor out when defining the radiation field.

26 6 DEA BASKI AD FAG WAG First, let us define the two components of the backward radiation field as follows: 5. R S φ, ψτ, ω = R E lim tuτ + r + M logr M, r, ω, r M φ, ψ τ, ω = lim r t u τ r M logr M, r, ω. r 5.. At the event horizon. The main result of this section is the following theorem: Theorem 5.. Suppose φ, ψ Cc M, S. If R E + φ, ψ vanishes for τ τ 0 and R E φ, ψ vanishes for τ τ 0, then both φ and ψ are supported in [r 0, S, where r 0 is given implicitly by r + M logr M = τ 0. Corollary 5.. Suppose ψ Cc M, S. If R E + 0, ψ vanishes for τ τ 0, then ψ is supported in [r 0, S. In particular, if R E + 0, ψ vanishes identically, then ψ 0. Proof of Corollary 5.. Because the initial data is odd, we know that R E + and so we may apply Theorem 5.. 0, ψτ, ω = R E 0, ψ τ, ω, We showed in Section 3. that solutions with compactly supported smooth initial data are smooth up to the event horizons. In fact, the argument above shows they are jointly smooth at the event horizons. Here µ and ν are Kruskal coordinates, given in terms of τ + = t + r + M logr M and τ = t r M logr M as follows: µ = e τ + 4M, ν = e τ 4M. ote that near r = M and therefore in the entire exterior domain, r is a smooth function of µ and ν. In these coordinates, the Schwarzschild metric has the following form: g S = 6M e r/m r dµ dν + r dω. Observe that µ is a defining function for the past event horizon E while ν is a defining function for the future event horizon E +. We summarize the joint smoothness in the following lemma: Lemma 5.3. If φ, ψ Cc M, S and u solves the initial value problem. with data φ, ψ, then u is smooth as a function of µ and ν. Proof of Theorem 5.. Let c 0 = e τ 0/4M. We start by showing that u must vanish to infinite order at ν = 0 for µ [0, c 0 ] and at µ = 0 for ν [0, c 0 ].

27 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME 7 Indeed, consider the D Alembertian S, which is given in these coordinates by S = er 8M r µ r e r ν + 8M r ν r µ + r ω, where r is regarded as a smooth function of µ and ν. Observe that near µ = 0 or ν = 0, r M µν. Because the initial data is compactly supported and smooth, there is some constant a 0 so that uµ, ν, ω vanishes for µ, ν [0, a 0 ]. Moreover, the assumption on the radiation field implies that µ u vanishes when ν = 0 and µ [0, c 0 ], while ν u vanishes for µ = 0 and ν [0, c 0 ]. Moreover, because the initial data is smooth and compactly supported, one may write with a similar expression for uµ, 0, ω u0, ν, ω = ν 0 ν u0, s, ω ds, to conclude that in fact u vanishes for {µ = 0, ν [0, c 0 ]} and {ν = 0, µ [0, c 0 ]}. We now work exclusively at ν = 0 as the µ = 0 case is handled in the same way. Because u satisfies S u = 0, we have that for ν = 0 and µ [0, c 0 ], e r 8M r ν µ u + er 8M r µr ν u + r ωu = 0. Because uµ, 0, ω = 0 for µ [0, c 0 ], r ω u = 0 here as well. Moreover, we note that µ r ν, so the second term vanishes at ν = 0 as well. Recall that we already know u is smooth as a function of µ, ν, ω. In particular, ν µ uµ, 0, ω = 0. By integrating in µ, we may also conclude that ν uµ, 0, ω = 0 for µ [0, c 0 ]. Differentiating the equation 8M r e r S u = 0 in ν yields ν µ u + µ r νu + ν µ r ν u + 8M 8M r e r ω ν u + ν re r ω u = 0. Using that ν u, and u vanish at µ, 0, ω and that µ r vanishes at ν = 0, we conclude that ν µ uµ, 0, ω = 0. Integrating again, we also conclude that νuµ, 0, ω = 0. Continuing inductively, after differentiating the equation k times, we may conclude that ν k+ u vanishes at µ, 0, ω for µ [0, c 0 ]. In particular, u vanishes to infinite order at ν = 0 for µ [0, c 0 ]. We now decompose u into spherical harmonics. We set w j to be an eigenbasis of L S with eigenvalues λ j and write ut, r, ω = j=0 u jt, rw j ω. ote that this decomposition extends to the radiation field as well. In particular, each u j vanishes to infinite order at ν = 0 for µ [0, c 0 ] with a corresponding statement at the other horizon. In terms of the coordinates

28 8 DEA BASKI AD FAG WAG µ and ν, each u j solves the equation 5. ν r µ u j + µ r ν u j 8M λ j re r u j = 0. In particular, we may extend u j by 0 to ν < 0, µ [0, c 0 ] and µ < 0, ν [0, c 0 ]. In other words, we may extend u j to a smooth function for µ, ν, c 0 that vanishes to infinite order at µ + ν = 0 for µ [ c 0, c 0 ]. We now use a unique continuation argument for the + -dimensional wave equation to conclude that each u j must vanish in the region we claim. Indeed, because the extension of u j vanishes to infinite order at µ + ν = 0 for µ [ c 0, c 0 ], finite speed of propagation with respect to µ + ν implies that u vanishes identically for µ and ν in the domain of dependence of {µ + ν = 0, µ [ c 0, c 0 ]}, i.e., in the region {µ c 0, ν c 0 }. In particular, the solution vanishes identically for {µ = ν, µ, ν c 0 }. Because the hypersurface {µ = ν} agrees with the hypersurface {t = 0} away from µ = ν = 0, we may conclude that the spherical components of the initial data are supported in µ c 0, i.e., in r r At null infinity. The main result of this section is the corresponding theorem at null infinity: Theorem 5.4. Suppose φ, ψ Cc M, S. If R S + φ, ψ vanishes for τ τ 0 and R S φ, ψ vanishes for τ τ 0, then both φ and ψ are supported in M, r 0 ] S, where r 0 is given implicitly by r + M logr M = τ 0. Corollary 5.5. Suppose ψ Cc M, S. If R S + 0, ψ vanishes for τ τ 0, then ψ is supported in M, r 0 ] S. In particular, if R S + 0, ψ vanishes identically, then ψ 0. Proof of Corollary 5.5. Because the initial data is odd, one has that R S and we may apply Theorem , ψ τ, ω = R S + 0, ψ τ, ω, Proof of Theorem 5.4. We showed in Section 3. that solutions with compactly supported smooth initial data have expansions at null infinity. If the intial data are in Cc M, S, then the rescaled solution ru is smooth as a function of τ up to = 0. Here = /r and we use τ +, τ to distinguish the coordinates at future and past null infinity if necessary, i.e., τ + = t r M logr M, τ = t + r + M logr M. We start by showing that if the initial data φ, ψ are smooth and compactly supported, and R S + φ, ψ vanishes for τ + τ 0, then v = ru vanishes to infinite order there. A similar argument applies near past null

29 RADIATIO FIELDS O SCHWARZSCHILD SPACETIME 9 infinity as well. First note that by integrating in τ +, we see that v vanishes at = 0 for τ + τ 0. We now use that v satisfies the following equation v + γ v = 0, where γ is a smooth function across = 0 and is the D Alembertian for the conformal metric g S and is given by = τ+ + M + 4M + ω. ote that because v vanishes at = 0 for τ + τ 0, ω v vanishes there as well. Thus at = 0, because v is smooth as a function of, we have that τ+ v =0 = 0, for τ + τ 0. Integrating in τ + shows that v vanishes there as well. Differentiating the equation in yields τ+ v + Ov = 0, where O is the product of and smooth differential operator in and ω. In particular, we have that at = 0, τ+ v = 0 for τ τ 0. Integrating again implies that v = 0 there. Proceeding inductively, after differentiating the equation k times, we may conclude that k+ v = τ+ k+ v = 0 for = 0 and τ + τ 0. In particular, v vanishes to infinite order at = 0 for τ + τ 0. A similar argument shows that if R S φ, ψ vanishes for τ τ 0, then v vanishes to infinite order there as well. We now decompose the solution u and thus also the rescaled solution v into spherical harmonics. Letting w j be an eigenbasis of L S with eigenvalues λ j, we write ut, r, ω = u j t, rw j ω and v τ,, ω = vj τ, w j ω. ote that v j and u j are related by a change of coordinates and the rescaling, i.e., v j = rw j. This decomposition extends to the radiation field as well, so each v j vanishes to infinite order at = 0 for τ + τ 0 and τ τ 0. In τ +, coordinates, each function v j satisfies τ+ v j + M v + 4M v λ jv + γ v = 0. Similarly, in τ, coordinates, each v j satisfies τ v + M v + 4M v λ jv + γ v = 0. We may thus extend v j by 0 to a smooth solution in a neighborhood of = 0 for τ + τ 0. We may perform a similar extension at past null infinity as well. We now use an argument similar to the one in the proof of Theorem 5. to conclude that v and hence u must vanish in the region we claim. Suppose instead that the corresponding initial data for some u j is supported in r r, corresponding to τ. ear = 0 and τ + = τ, v j vanishes to infinite order at τ + + = τ and thus vanishes in a full neighborhood of = 0, τ + = τ. One key difference between the two arguments is that we must proceed incrementally near null infinity because u is not smooth as a function of τ + and τ.

30 30 DEA BASKI AD FAG WAG A similar argument holds for = 0 and τ = τ. Let us say that v j vanishes τ + τ + δ when δ as well as for τ τ δ when δ. In terms of t and, this implies that u j vanishes if r δ τ δ r M logr M t δ τ + r + M logr M. We now use the hyperbolicity of the + -dimensional operator r r M t + r M r + r M r r r λ j r with respect to r to conclude that the initial data for u j vanishes if r + M logr M τ δ, i.e., for r r with r < r. Figure 5 illustrates the process of improving from vanishing for r r to r r. τ + τ 0 τ + δ τ t = 0 r r τ τ δ τ Figure 5. Iteratively improving the support of the initial data near null infinity. Knowing that the solution vanishes identically in the shaded region, we find first that it vanishes in the small white region, then use the hyperbolicity of the one-dimensional operator to improve it to the region outlined by the dashed line. Proceeding in this manner, we find that both φ and ψ must be supported in r r 0. τ 0 References [Bac94] Alain Bachelot. Asymptotic completeness for the Klein-Gordon equation on the Schwarzschild metric. Ann. Inst. H. Poincaré Phys. Théor., 64:4 44, 994.

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