A scattering theory for the wave equation on Kerr black hole exteriors
Mihalis Dafermos, Igor Rodnianski, Yakov Shlapentokh-Rothman
aa r X i v : . [ g r- q c ] D ec A scattering theory for the wave equationon Kerr black hole exteriors
Mihalis Dafermos , Igor Rodnianski , and Yakov Shlapentokh-Rothman Princeton University, Department of Mathematics,Fine Hall, Washington Road, Princeton, NJ 08544, United States University of Cambridge, Department of Pure Mathematics and Mathematical Statistics,Wilberforce Road, Cambridge CB3 0WA, United Kingdom Massachusetts Institute of Technology, Department of Mathematics,77 Massachusetts Avenue, Cambridge, MA 02139, United States
December 29, 2014
Abstract
We develop a definitive physical-space scattering theory for the scalar wave equation ◻ g ψ = ∣ a ∣ < M . In particular, we prove results cor-responding to “existence and uniqueness of scattering states” and “asymptotic completeness” and weshow moreover that the resulting “scattering matrix” mapping radiation fields on the past horizon H − and past null infinity I − to radiation fields on H + and I + is a bounded operator. The latter allows us togive a time-domain theory of superradiant reflection. The boundedness of the scattering matrix showsin particular that the maximal amplification of solutions associated to ingoing finite-energy wave packetson past null infinity I − is bounded. On the frequency side, this corresponds to the novel statement thatthe suitably normalised reflection and transmission coefficients are uniformly bounded independently ofthe frequency parameters. We further complement this with a demonstration that superradiant reflec-tion indeed amplifies the energy radiated to future null infinity I + of suitable wave-packets as above.The results make essential use of a refinement of our recent proof [M. Dafermos, I. Rodnianski andY. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: thefull subextremal case ∣ a ∣ < M , arXiv:1402.6034] of boundedness and decay for solutions of the Cauchyproblem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We showin contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerateenergy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shiftinstability when solving the wave equation backwards. Contents T -energy theory and superradiance . . . . . . . . . . . 61.1.2 The N -energy theory and the backwards blue-shift instability . . . . . . . . . . . . . . 61.1.3 The V -energy theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.4 The scattering map S , superradiant reflection R and applications . . . . . . . . . . . 71.1.5 Back to the fixed-frequency theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Related work and further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Detailed overview and statements of the main theorems 8 T -energy theory and its limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 The Schwarzschild a = a ≠ N -energy forward map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.2 A blue-shift instability and the non-existence of an N -energy backwards map . . . . . 122.3.3 The V -energy forward map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.4 The V -energy backwards map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.5 Existence and boundedness of the S -matrix . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.6 A physical space theory of superradiant reflection . . . . . . . . . . . . . . . . . . . . . . 142.3.7 Pseudo-unitarity and non-superradiant unitarity . . . . . . . . . . . . . . . . . . . . . . 152.3.8 Uniqueness of scattering states for ill-posed scattering data . . . . . . . . . . . . . . . . 152.4 Applications to fixed frequency scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.1 Uniform boundedness of the coefficients R and T . . . . . . . . . . . . . . . . . . . . . . 162.4.2 Connection with physical-space theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Nonlinear problems and scattering constructions of dynamical black holes . . . . . . . . . . . . 172.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 a and M and conventions on constants . . . . . . . . . . . . . . . . . . . . . . . 233.4 Useful vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 A foliation by hyperboloidal hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.6 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.7 The non-degenerate boundedness and integrated energy decay statements . . . . . . . . . . . . 263.8 The r p estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 H +≥ and H + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1.2 The energy flux through H +≥ and H + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Null infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.1 The radiation field along I + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.2 The energy flux through I + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 R and T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.2.3 A bound for the Wronskian W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.3 Superradiant estimates for U hor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.3.1 An inhomogeneous ILED in the superradiant regime . . . . . . . . . . . . . . . . . . . . 446.3.2 Properties of the potential V in the superradiant regime . . . . . . . . . . . . . . . . . . 446.3.3 An improved estimate in the superradiant regime . . . . . . . . . . . . . . . . . . . . . . 446.3.4 Proof of Proposition 6.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.4 The large- ℓ limit of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.5 Nonvanishing of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.6 The microlocal r p estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.7 A quantitative estimate on the rate of convergence of the microlocal radiation field . . . . . . 496.8 Relation to the physical space radiation fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ∗ , ˚Σ and Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588.1.2 Scattering data along H +≥ , H + and H + . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.1.3 Scattering data along I + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.2 Definition and boundedness of the forward maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 B − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629.1.2 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639.1.3 Inverting the forward map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679.1.4 A physical-space characterization of B − . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689.2 The backwards map to Σ ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689.3 The backwards map to Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709.4 Aside: Proof of Theorem 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709.5 The scattering matrix S = F + ○ B + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719.6 Aside: A self-contained physical-space treatment of the Schwarzschild case . . . . . . . . . . . 72
10 Further applications 74
11 The backwards blue-shift instability and horizon-singular solutions 80 ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8411.2.4 Positivity on I + and the contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8511.3 Construction of ψ using the degenerate T -scattering theory . . . . . . . . . . . . . . . . . . . . 8611.4 Non-surjectivity of the N -energy forward map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Black holes play a central role in our present general relativistic picture of the universe. At the sametime, however, they are perhaps the example par excellence of a physical object which cannot be observed“directly”. An effective approach to infer both the very presence but also the finer properties of black holes3roceeds through the study of the scattering of waves on their exterior. Hence, a theoretical understandingof scattering theory in this context is of paramount importance.The bulk of the now classical black hole scattering-theory literature concerns only the fixed-frequency study of solutions u ( ω,m,ℓ ) ( r ∗ ) to the radial o.d.e. u ′′ + ω u = V u, (1)where V = V ( ω,m,ℓ ) ( r ∗ ) , resulting from Carter’s remarkable separation [10] of the linear scalar wave equation ◻ g ψ = (M , g a,M ) . One can also consider more complicated systems like theMaxwell equations or the equations of linearised gravity. See Chandrasekhar’s monumental [11] and themonograph [33].Beyond formal fixed-frequency statements concerning ( ) , true scattering results in the “time-domain”,describing actual finite-energy solutions of ( ) and related equations, have only been obtained in variousspecial cases. Let us already mention the pioneering results of Dimock and Kay [26, 28, 27] in the Schwarz-schild a = a ≠
0, on the other hand,despite recent progress on the Cauchy problem, first for the ∣ a ∣ ≪ M case [19, 1, 55] and then, for the fullsubextremal range ∣ a ∣ < M in [24], the most basic questions of scattering theory for ( ) have remained tothis day unanswered. In particular:(a) Can one associate a finite-energy solution of ( ) to every suitable finite-energy past/future asymptoticstate? (Existence of scattering states) (b) Is the above association unique, i.e. do two finite-energy solutions having the same asymptotic statenecessarily coincide? (Uniqueness of scattering states) (c) Do the above solutions parametrised by finite-energy past/future asymptotic states describe the totalityof finite-energy solutions ψ to ( ) ? (Asymptotic completeness) See the classic [49] for a general introduction to the scattering theory framework in physics.At the conceptual level, one of the most interesting new phenomena of black hole scattering which ariseswhen passing from the Schwarzschild a = a ≠ superradiance . Thisalready can be seen at the level of the fixed-frequency o.d.e. ( ) . We review this very quickly for the benefitof the reader familiar with the classical physics literature [11]. For each fixed frequency triple ( ω, m, ℓ ) with ω ∈ R , one can define two complex-valued solutions U hor ( r ∗ ) and U inf ( r ∗ ) of ( ) so that U hor ∼ e − i ( ω − ω + m ) r ∗ as r ∗ → −∞ , U inf ∼ e iωr ∗ as r ∗ → ∞ , corresponding to the asymptotic behaviour of the potential V , which is itself real. Here ω + is related to theKerr parameters a, M by the formula 2 M ω + ( M + √ M − a ) = a . The linear independence of U hor and U inf is the statement of mode stability on the real axis and was proven recently by one of us [53], extending thetransformation theory of [58]. By dimensional considerations, this linear independence at one go answersthe “fixed frequency” analogue of questions (a)–(c) in the affirmative. It follows that since U inf also solves ( ) , we may write T − i ( ω − ω + m ) U hor = R iω U inf + U inf iω , (3)where T = T ( ω, m, ℓ ) and R = R ( ω, m, ℓ ) are known as the transmission and reflexion coefficients. For-mally, these coefficients describe the proportion of “energy” at fixed frequency ( ω, m, ℓ ) transmitted to thehorizon and reflected to infinity, respectively, of purely incoming wave from past infinity. With the precisenormalisation of ( ) , which will be in fact motivated by the considerations of this paper, the energy identityassociated to ( ) yields ∣ R ∣ + ωω − ω + m ∣ T ∣ = . (4) All notations here will be explained in detail in the paper. The reader for which this is unfamiliar can skip directly to thenext paragraph! ω ( ω − ω + m ) − < , (5)the transmission coefficient T is weighted with a negative sign in ( ) allowing thus the reflection coefficient R to have norm strictly greater than 1 ∣ R ( ω, m, ℓ )∣ > . (6)That is to say, there is a nontrivial energy amplification factor at fixed frequency. The first estimates for themaximum reflection coefficient in various frequency regimes go back to pioneering work of Starobinskii [54](see also [56]), but even the statement of the uniform boundedness of R ( ω, m, ℓ ) over all superradiantfrequencies ( ) has remained an open problem.In passing from a fixed-frequency scattering theory to a true time-domain scattering theory, the absenceof an obvious quantitative frequency-independent control of the coefficient R ( ω, m, ℓ ) presents itself as afundamental difficulty. Moreover, an additional difficulty is identifying the correct notion of “energy” withrespect to which solutions should be defined. In particular, one requires a notion of energy which controlssolutions of ( ) not only in the forward but also in the backward direction, i.e. an energy not subject to thelocal red-shift effect associated to the event horizon, which when solving backwards appears as a blue-shift instability.The purpose of this paper is to overcome these difficulties and develop a definitive finite-energy scatteringtheory for ( ) on general subextremal Kerr exteriors (M , g a,M ) with ∣ a ∣ < M , showing in particular: The answer to (a), (b) and (c) is yes.
Existence and uniqueness of scattering states aswell as asymptotic completeness indeed hold for the space of solutions to ( ) and scattering statesdefined by the finiteness of a natural energy flux. We will understand scattering states in the sense of Friedlander [31] (for the Schwarzschild case in thiscontext, see [48]), and our approach to both constructing and estimating the scattering maps can be thoughtof as a combination of what in the traditional literature are known as “stationary” and “time-dependent”methods [39]. We will depend heavily on our recent boundedness results [24] for the Cauchy problem for ( ) , as well as certain decay results of [24], which indeed succeeded in giving a first version of quantitativephysical-space control over superradiance, independent of frequency, and also showed that a suitable class ofsolutions of ( ) can be indeed understood as superpositions of solutions of ( ) over real frequencies ω . Wewill in fact, however, here require a certain refinement of the estimates of [24] so as to apply to a degenerateenergy not subject to the backwards blue-shift instability. This notion of energy lies behind the particularchoice of normalisation of the reflection coefficient R in ( ) . Along the way, we shall in particular providethe missing frequency-independent bound on R over all superradiant frequencies ( ) :sup ( ω,m,ℓ ) ∣ R ( ω, m, ℓ )∣ = S ( a, M ) < ∞ (7)by a finite constant S ( a, M ) depending only on the Kerr parameters, with S ( a, M ) > a ≠ S -matrix S whose boundedness in the operator norm replaces the usual unitarity property. A suitable restrictionof S will be related to a generalisation of the inverse-Fourier transform applied to multiplication by thecoefficients R and T defined by ( ) . Through this, we will give a definitive physical space (i.e. time-domain)interpretation of superradiant reflection, in particular, showing: Superradiant reflection indeed strictly amplifies the energy radiated to infinity of suitably con-structed purely ingoing finite-energy wave packets. The maximum amplification factor, however,is bounded precisely by the constant S ( a, M ) of ( ) . Our results leave open the extremal case g a,M for a = M (see [6]). In particular, it is not known whetherthe limit lim ∣ a ∣→ M S ( a, M ) is finite. We introduce briefly the main theorems of the paper in what follows below. (We will give a more detailedoverview together with precise statements of all theorems listed here in boldface type in Section 2.)5 .1.1 Fron Schwarzschild to Kerr: the T -energy theory and superradiance The first difficulty in constructing a physical-space scattering theory is identifying what constitutes the“correct” class of finite energy solutions and asymptotic states. In the Schwarzschild case, as admissiblesolutions to ( ) it is natural to consider the class of ψ which have finite conserved energy (i.e. finite energycorresponding to the stationary Killing vector field T ) on a Cauchy hypersurface. This in turn suggests acorresponding notion of asymptotic states defined in terms of the completion (with respect to the natural T -energy flux) of the set of Friedlander radiation fields rψ on I + (see [31]), complemented by the analogouscompletion of the set of traces of ψ on the event horizon H + . See Nicolas [48] for a recent formulation ofSchwarzschild scattering theory in precisely these terms. This theory can be constructed entirely in thetime-domain, i.e. using “time-dependent” methods. (We will in fact give our own self-contained version ofthe Schwarzschild theory in Section 9.6.)Turning to the Kerr case, the above conserved energy corresponding to T is clearly unsuitable for ascattering theory, because the inner product it defines is now indefinite, in view of the existence of the well-known ergoregion where T is spacelike. This is the physical-space origin of the phenomenon of superradiancediscussed with respect to ( ) . Recent progress on understanding the Cauchy problem for ( ) on Schwarzschildand Kerr has rested in part on the realisation (see [21, 17, 20]) that a more natural energy quantity forunderstanding forward evolution is that defined by a T -invariant everywhere-timelike vector field N . Eventhough this N -energy is not conserved, it remains, as proven in our recent work [24] (for the full sub-extremalrange of Kerr parameters ∣ a ∣ < M ), uniformly bounded through a suitable spacelike foliation Σ ∗ s of the exteriorregion and controls in fact a spacetime integral quantity. The good divergence properties of the vector field N are related to the celebrated red-shift effect associated to the horizon H + . N -energy theory and the backwards blue-shift instability Despite its success in the context of the Cauchy problem on Kerr, the above N -energy is again unsuitable fordefining a scattering theory, because the helpful red-shift transforms into a lethal blue-shift when trying toassociate admissible solutions to their natural asymptotic states, which requires solving the wave equationbackwards. See the discussion in [14, 50] and also the more recent comments in [48]. The first two results ofour paper are dedicated to making explicit this obstruction. Our Theorems 1 and 2 together show thatwhile one can naturally associate (using our results of [24]) asymptotic states to finite N -energy solutions,this map is not surjective, and thus, one cannot define a one-sided inverse map embodying the existence ofscattering states (cf. (a)). V -energy theory The correct setting for a scattering theory on Kerr would then appear to be an energy quantity defined bya vector field V which (like T in Schwarzschild) is null on the horizon and timelike outside. With the helpof the additional axisymmetric Killing field Φ, one can in fact construct such a vector field V which canbe chosen moreover Killing in a neighbourhood of both H + and I + (though not globally Killing!). Eventhe question of uniform boundedness of solutions assumed to lie only in the energy space defined by V ,however, has not previously been answered. (See however, the very related higher-order weighted estimatesof Andersson–Blue [1] in the very slowly rotating ∣ a ∣ ≪ M case.)The main results of the present paper ( Theorems 3 and 4 ) succeed in constructing a bounded invertiblemap F + associating a unique future asymptotic state to each solution with initially bounded V -energy, withtwo-sided inverse B − satisfying B − ○ F + = Id, F + ○ B − = Id. (8) Let us note that, in contrast to the wave equation ( ) , for the Dirac equation, one still has a coercive L -conservation lawdespite the absence of a globally timelike Killing field. Using this, H¨afner and Nicolas [37] have constructed a scattering theoryfor the Dirac equation on Kerr backgrounds, generalising [47]. This has been extended to Kerr–Newman–de Sitter backgroundsby Daud´e and Nicoleau [25]. In this context, see also H¨afner [36] for scattering results concerning a non-superradiant class ofsolutions of the Klein–Gordon equation for fixed azimuthal mode m . − I + H + H − Σ B The boundedness of the map F + requires a refinement of our previous boundedness results on the Cauchyproblem (see [24]) so as to apply for admissible solutions defined by the finiteness of a suitable V -energy asabove. This will require us to revisit the fixed frequency o.d.e. estimates on ( ) proven in [24]. What willbe the inverse map B − is constructed explicitly via the frequency domain by an appropriate superpositionof solutions to the fixed frequency o.d.e ( ) . Again, to infer the boundedness of B − one needs to exploitquantitative estimates on ( ) adapted from [24], again referring only to the V -energy flux. One may definesimilar maps F − , B + associating solutions to past asymptotic states.In the traditional language of scattering theory, let us note that existence of scattering states (cf. (a))corresponds to the existence of B ∓ , uniqueness of scattering states (cf. (b)) to the injectivity of F ± , andasymptotic completeness (cf. (c)) to the surjectivity of B ∓ . These three statements of course all follow from ( ) . S , superradiant reflection R and applications The asymptotic completeness results allow us in particular to define a scattering map ( S -matrix) S = F + ○ B + taking asymptotic past states to asymptotic future states I − I + H + H − which is moreover bounded in the operator norm with respect to the spaces defined by the flux of the V -energy (see Theorem 5 ).To connect with the usual discussion of superradiant scattering, we may also define a reflection map R anda transmission map T which restricts S to past asymptotic states with no trace on the past event horizon H − and returns only the radiation to future null infinity I + or the future event horizon H + respectively. Itfollows in particular that R and T are also bounded (see Theorem 6 ). On the other hand, we show thatthe operator norm of R satisfies ∥ R ∥ > Theorem 7 ), and thus there exist wave packets correspondingto past asymptotic states supported only on I − such that the energy radiated to I + is strictly greater thanthe energy flux on I − . As discussed above, this gives a physical space interpretation of superradiance (cf. thenumerical [5]). Next, we will show that T ⊕ R is pseudo-unitary in that it preserves an indefinite innerproduct associated to the T -energy ( Theorem 8 ). Upon restricting to “non-superradiant” data along H − and I − the map S becomes unitary in the standard sense ( Theorem 9 ).We finally give a “unique continuation” result that finite V -energy solutions are uniquely characterized bytheir scattering data on any of the “ill-posed” pairs H − ∪H + , I + ∪I − , H − ∪I + or H + ∪I − (see Theorem 10 ).This has the interpretation that for this improper notion of asymptotic states, uniqueness of scattering states(b) holds without existence (a). 7 .1.5 Back to the fixed-frequency theory
We have already noted that our results will require revisiting the estimates proven in [24] for the radialo.d.e. ( ) appearing in Carter’s classical separation of ( ) . In this sense, our work makes contact back with theformal scattering theory literature [33] concerning ( ) at fixed frequency. In particular, our o.d.e. results willyield the uniform boundedness of the reflection and transmission coefficients ( Theorem 11 ), in particular,giving ( ) . This complements the work of Starobinskii and others (see [54, 56]) aimed at numericallyestimating the maximum of these for low fixed values of m , ℓ . Our transmission and reflection maps T and R can in fact be represented as a generalised inverse Fourier transform of multiplication by T and R ( Theorem 12 ). In particular, a posteriori, the boundedness statements of Theorems 6 and 11 are equivalent.This connects the fixed frequency and physical space scattering theories in a very explicit way.
Let us specifically mention here a related recent important advance by Georgescu, G´erard and H¨afner [34]which proves scattering results for fixed-azimuthal mode (i.e. fixed m ) solutions of the Klein-Gordon equationin the very slowly rotating Kerr-de Sitter case ∣ a ∣ ≪ M, Λ. This is in part based on work on the Cauchy prob-lem due to Dyatlov [29]. For additional background on the Cauchy problem on other black hole spacetimes,besides references mentioned previously, we refer the reader to the lecture notes [20].
MD acknowledges support through NSF grant DMS-1405291. IR acknowledges support through NSF grantDMS-1001500 and DMS-1065710. YS acknowledges support through NSF grants DMS-0943787 and DMS-1065710 as well as the hospitality of Princeton University during the period when this research was carriedout.
In this section, we will give a more detailed overview of the main results of this paper. We begin inSection 2.1 with the basic setup for our “time-domain” scattering theory. We shall then briefly turn inSection 2.2 to a discussion first of the Schwarzschild a = T -energy, and then of the problem of superradiance in Kerr for a ≠ ∣ a ∣ < M . We shall relate this backto the fixed-frequency theory in Section 2.4, stating two additional theorems. In Section 2.5, we make abrief comparison with non-linear scattering problems involving black holes, in particular referring to a recentscattering construction of solutions to the Einstein equations themselves which asymptote in time to theKerr family [14]. Finally, we shall give in Section 2.6 a section by section outline of the remainder of thepaper, identifying in particular where each of the main theorems is proven. We begin with the basic setup describing our “time-domain” scattering theory in the Kerr black-hole context.
We will fix subextremal Kerr parameters ∣ a ∣ < M and consider the Kerr metric g a,M defined on a “domainof outer communication” D . See Section 3.2 for an explicit representation of this manifold with stratified8oundary. H − I − I + H + B D
The boundary components H ± correspond to past and future event horizons and meet in the so-calledbifurcation sphere B . (Our convention will be that H ± do not contain B .) Moreover, one can define the two“asymptotic” boundary components future and past null infinity I ± , which, in an auxiliary topology, canindeed be attached to D as boundary. See Section 4.2. We begin by considering smooth solutions ψ of ( ) arising from compactly supported initial data on a suitablehypersurface. We will in fact consider three distinct classes of such data.When we are only interested in future scattering, it is more natural to focus on solutions parametrisedby compactly supported data ( ψ Σ ∗ , ψ ′ Σ ∗ ) on a hypersurfaceΣ ∗ = { t ∗ = } , defined as the level set of a future-horizon penetrating t ∗ -coordinate. See Section 3.2. Here Σ ∗ is understoodas a manifold-with-boundary, so the support of the data can in principle contain the boundary Σ ∗ ∩ H + . Bygeneral theory, such data give rise to a unique smooth solution ψ of ( ) on R ≥ = D + ( Σ ) . We shall call themap from smooth initial data to solution forward evolution : ( ψ , ψ ′ ) ↦ ψ. (9)See Proposition 3.6.1.When we are interested in defining the S -matrix, we need to parameterise solutions ψ by data whichdetermine ψ globally on D . It is in fact natural to distinguish between two cases. Defining˚Σ = { t = } , Σ = ˚Σ ∪ B , where t is the usual Boyer-Lindquist coordinate defined only on the interior of D , we can consider smoothcompactly supported data ( ψ Σ , ψ ′ Σ ) on Σ, or the more restrictive class of smooth compactly supported data ( ψ ˚Σ , ψ ′ ˚Σ ) on ˚Σ. (The latter, thought of as a special case of the former, must vanish in a neighbourhood of B .) We can now associate in either case a global smooth solution ψ on D . See Proposition 3.6.2. We willagain refer to the map ( ) as forward evolution.The significance of considering the restricted data (i.e. data whose support as a subset of ˚Σ is compact)is that the support of the resulting ψ in D is disjoint from an open neighbourhood of B . This will be usefultechnically in defining the backwards map in the frequency domain. It will also facilitate comparison withother results where it has often been this scattering theory that has been implicitly or explicitly considered. The most natural formulation of a scattering theory from the point of view of the present problem describesasymptotic states by an appropriate Hilbert space completion (see below) of the future and past radiationfields on I ± augmented by radiation fields on the horizons.The notion of radiation field along I + is due to Friedlander [31] and in our context is given by thefollowing Proposition: 9 roposition 1. If data ( ψ , ψ ′ ) are smooth of compact support on Σ ∗ , ˚Σ or Σ , then the solution rψ extendsto a smooth function φ defined on I + . We shall infer the above as an essentially trivial consequence of the r p estimates of [18]. See Proposi-tion 3.8.1 and Corollary 4.2.1.The radiation field on the horizon is just the usual restriction of ψ as a smooth function. Let us introducethe notation H +≥ = R ≥ , and H + = H + ∪ B . Since ψ arising from compactly supported data ( ψ Σ ∗ , ψ ′ Σ ∗ ) isonly defined on R ≥ , we may define in this case only ψ ∣ H +≥ ≐ ψ ∣ H +≥ . In the case of solutions arising fromcompactly supported data on ˚Σ and Σ, respectively, ψ is of course defined on all of H + ; nonetheless, we shallrefer to ψ H + ≐ ψ H + in the former case and ψ H + ≐ ψ H + in the latter case. This notation reminds us (cf. theremark at the end of Section 2.1.2 above) that in the former case, the support of ψ H + is disjoint from aneighborhood of B in D , whereas, in the latter case, the support of ψ H + may contain B .To summarise, forward evolution ( ) gives rise to a map on smooth compactly supported initial data ( ψ ∣ Σ ∗ , ˚Σ , or Σ , ψ ′ ∣ Σ ∗ , ˚Σ , or Σ ) ↦ ψ ↦ ( ψ ∣ H +≥ , H + , or H + ≐ ψ ∣ H +≥ , H + , or H + , φ ∣ I + ≐ rψ ∣ I + ) (10)defined by solving the initial value problem for ( ) and restricting to the radiation fields. The forward mapsof our scattering theory will be constructed by completing the above map with respect to suitably definedenergies. The states defining scattering theory are associated to energies which are in turn defined by vector fields.Recall that a general vector field X defines an energy current J X [ ψ ] and an energy flux ∫ S J X [ ψ ] (11)through an arbitrary hypersurface S . (See Section 3.1.)For appropriate vector fields X for which ( ) is nonnegative, the square root of the expression ( ) canin turn be used as a norm to define a space E X Σ ∗ , E X ˚Σ , E X Σ (12)by completion of the set of smooth compactly supported data ( ψ , ψ ′ ) on Σ ∗ , ˚Σ, Σ, respectively. (SeeSection 8.1.) Recall that “compactly supported on ˚Σ” is a more restrictive assumption than “compactlysupported on Σ” and thus E X ˚Σ ⊂ E X Σ .Similarly, the flux ( ) defines asymptotic spaces E X H +≥ ⊕ E X I + E X H + ⊕ E X I + , E X H + ⊕ E X I + , (13)via completion of the space of radiation fields arising from ( ) . Here we have that E X H +≥ embeds (non-uniquely) into E X H + , and also, E X H + ⊂ E X H + . In this picture, the problems (a)–(c) of scattering theory translate into finding bijective maps between ( ) and ( ) induced by the completion of forward evolution ( ) of smooth data, for a suitable choice ofthe vector field X . We will not discuss the construction of wave operators in the spirit of [32, 42] as there isno compelling global “reference dynamics” with which to compare; see [48] for a nice discussion of how toconstruct the latter if desired. T -energy theory and its limitations Before turning to our main theorems, we briefly review the Schwarzschild a = a ≠
0. 10 .2.1 The Schwarzschild a = case In the Schwarzschild case a =
0, the stationary Killing field T is timelike in the interior of D becoming nullon H + ∪ H − and vanishing on B . Thus the energy defined by T degenerates pointwise. Nonetheless, thecompletions E T ˚Σ , E T H + and E T I + define Hilbert spaces and one can obtain a unitary isomorphism E T ˚Σ ≅ E T H + ⊕ E T I + . (14)In our notation, this is the content of the previously known Schwarzschild scattering theory [26, 28, 48].We will give our own self-contained treatment in Section 9.6. One obtains with no additional difficultythe alternative unitary isomorphisms E T Σ ∗ ≅ E T H +≥ ⊕ E T I + and E T Σ ≅ E T H + ⊕ E T I + . a ≠ and the ergoregion Turning to the Kerr case a ≠
0, there is now a non-empty subset S of D known as the ergoregion where T isspacelike. In particular, the energy-fluxes ∫ ˚Σ J T [ ψ ] , ∫ H + J T [ ψ ] defined by T fail to be positive definite. Thisis the physical space origin of the phenomenon of superradiance, discussed in the fixed-frequency theory inthe context of ( ) and ( ) .Part of the conceptual difficulty of formulating a scattering theory in the Kerr case is thus to find thecorrect notion of asymptotic states which replaces those based on E T . At the same time, one must understandwhat property replaces the notion of unitarity in ( ) as a means of quantifying the good properties of thescattering map. We turn now to the statements of the main results of this paper that give a definitiveresolution of this problem. In this section, we will present in detail the main theorems of our paper concerning physical-space (time-domain) scattering theory for the wave equation ( ) on Kerr in the general subextremal case ∣ a ∣ < M . N -energy forward map The first candidate replacement for the (degenerate) Schwarzschild T -energy is the so-called N -energy. Here, N is a globally timelike vector field which is T -invariant outside a neighbourhood of the bifurcation sphere B and moreover such that N = T in a neighbourhood of I + . The energies ( ) associated to this vector fieldare indeed manifestly positive-definite and pointwise non-degenerate.The first main theorem defines asymptotic states for all solutions arising from finite N -energy data onthe hypersurface Σ ∗ , i.e., in the notation ( ) , for all solutions parametrised by E N Σ ∗ . Theorem 1.
Forward evolution ( ) with data on Σ ∗ extends to a bounded map F + ∶ E N Σ ∗ → E N H +≥ ⊕ E T I + . H − I − I + H + ≥ R Σ ∗ See Theorem 8.2.1. (Note that E N I + = E T I + .) For the hard analysis behind the above, the proof reliesin particular on a uniform boundedness statement for the energy ∫ Σ ∗ s J N through a foliation Σ ∗ s defined byfuture-translating Σ ∗ by the flow of T , as well as a weak decay statement, both of which follow from theresults of [24] mentioned previously, here quoted as Theorem 3.7.1.11 .3.2 A blue-shift instability and the non-existence of an N -energy backwards map Satisfactory though the forward theory may be, it turns out that the above N -energy is ill-suited for definingthe asymptotic states of a scattering theory. The fundamental origin of this is the red-shift effect on thehorizon (so favourable for controlling forward evolution!), which for backwards evolution is now seen as ablue-shift. See [14] and Section 3.1.2 of Sbierski [50]. It turns out that one can show explicitly that the mapof Theorem 1 fails to be surjective: Theorem 2.
Already in the Schwarzschild a = case, the map F + of Theorem 1 fails to be surjective. It follows that there does not exist even a one-sided inverse B − satisfying F + ○ B − = Id ; thus, existenceof scattering states (cf. (a)) does not hold in the N -theory. (As we shall see in Section 2.3.4, the above map F + is however injective.)Our proof of the above theorem exploits monotonicity satisfied by the spherical mean under spatial evolu-tion. Though essentially independent of the rest of the paper, the precise statement proven (Theorem 11.1)is deferred to the end (Section 11), so that it can be interpreted both as a non-surjectivity result withrespect to our N -energy scattering theory (Corollary 11.4) and also constructively (Corollary 11.1) usingTheorem 4 of our V -energy scattering theory to be discussed below. Let us already remark, however, thatthe non-surjectivity statement we obtain in Corollary 11.4 is more precise than what we have just statedabove. We elaborate briefly below.First let us note that with the notations of the present paper, the considerations of Section 1.1.6.1 of [14]show that by introducing sufficiently high exponential weights in the spaces defining the scattering data,i.e. considering the spaces E e αv N H + and E e αu T I + , then there indeed exists a bounded one-sided inverse B − ∶ E e αv N H +≥ ⊕ E e αu T I + → E N Σ ∗ (15)such that F + ○ B − = id . Thus, we do have existence of a restricted class of future scattering states.With this setting, our Theorem 11.1 in fact shows (see Corollary 11.4) that e αv above cannot be replacedby ∣ v ∣ p no matter how large p is taken, i.e. the map F + of Theorem 1 is not surjective as a map F − + ( E ∣ v ∣ p N H +≥ ⊕{ }) → E ∣ v ∣ p N H +≥ ⊕ { } . The question of precise characterization of the range of F + remains open. We shallreturn to this issue in Section 2.5. V -energy forward map To define a forward map which one can indeed hope to show is invertible, we must pass to a degenerateenergy class which does not see the red-shift at the horizon.Recall that g M,a admits an additional Killing vector field Φ corresponding to axisymmetry. Although for a ≠
0, the vector field T fails to be globally timelike in the interior of D , the span of T and Φ does form atimelike plane, and the Killing combination K = T + ω + Φ is timelike in a neighbourhood of H + , becomingnull on H + itself. (Note that if a =
0, then K = T , but if a ≠
0, then K is spacelike away from the axisof symmetry near I + .) We define a T -invariant vector field V with the property that V = K near H + and V = T near I + and V is timelike in the interior of D . The energy associated to this vector field is manifestlynon-negative definite, though degenerate analogous to the T -energy in the Schwarzschild case. In the case a ≠
0, there is necessarily a region where V fails to be Killing.Our third main theorem is a degenerate V -energy analogue of Theorem 1 given by Theorem 3.
Forward evolution ( ) extends to bounded maps F + ∶ E V Σ ∗ → E K H +≥ ⊕ E T I + , F + ∶ E V ˚Σ → E K H + ⊕ E T I + , F + ∶ E V Σ → E K H + ⊕ E T I + . See Theorems 8.2.1, 8.2.3 and 8.2.4. The above theorem requires a new version of the boundedness partof Theorem 3.7.1 of [24], depending only on the degenerate energy. This result, which is of independentinterest, is stated as Theorem 7.1 and proven in Section 7. The reader can compare with the higher-orderweighted boundedness result of Andersson and Blue [1] for the ∣ a ∣ ≪ M case, whose degenerate horizonweights are similar to the V -energy. 12et us note that the proof of Theorem 7.1 will require us to revisit the quantitative study of the o.d.e. ( ) at fixed frequency, on which the original results of [24] were based, in particular in the form of Theorem 6.3.1,and a new result, Theorem 6.2.1, which we will prove here by adapting the proof of [24]. In particular, fromthese statements, one can already infer novel results on the fixed-frequency scattering; we defer specificdiscussion of these till Section 2.4. V -energy backwards map Our degenerate-energy class is indeed suitable to construct a bounded inverse of the map of Theorem 3 andthus infer the existence of a satisfactory scattering theory satisfying (a)–(c).
Theorem 4.
There exist bounded maps B − ∶ E K H +≥ ⊕ E T I + → E V Σ ∗ , B − ∶ E K H + ⊕ E T I + → E V ˚Σ , B − ∶ E K H + ⊕ E T I + → E V Σ , (16) which are two-sided inverses to the maps of Theorem 3, i.e. B − ○ F + = Id and F + ○ B − = Id . H − I − I + H + ≥ Σ H − I − I + H + ˚Σ I − I + H + H − Σ B See Theorems 9.2.1, 9.1.1 and 9.3.1. As explained in Section 1.1.3, it is the existence of the map B − whichgives the existence of future scattering states (a), the injectivity of F + which gives the uniqueness of futurescattering spaces (b), and the surjectivity of B − corresponds to asymptotic completeness of future scatteringstates (c). We note that the map of Theorem 1 is in fact the restriction of the first map of Theorem 3. Thusa corollary of the above is that the map F of Theorem 1 is injective. In this sense, for the N -energy theory,one still has uniqueness (b)–but not existence (a)!–of scattering states. Cf. the discussion of the ill-posedproblems of Section 2.3.8.Let us note that in our proof, we construct B − with the help of the frequency domain, again using ouro.d.e. result Theorem 6.2.1, together with a decomposition first given in [24] and which exploits the fact thatthe span of T and Φ is timelike (See Section 9.1.2), to give us the quantitative statement of boundedness.Due to this use of the frequency domain, it is in fact the map B − ∶ E K H + ⊕ E T I + → E V ˚Σ which is most natural toconstruct first.It is perhaps worth explicity noting that even to show the existence of B − , we require appeal to ano.d.e. result which in essence already embodies the totality of the quantitative decay statement for the Cauchyproblem ( ) . This should emphasise how intricately tied in the Kerr case the problem of boundedness is tothe problem of quantitative decay. This is in contrast to many usual problems in scattering theory where“existence of scattering states” (cf. (a)) is a relatively soft result, which can be proven independently of thestructure necessary to obtain asymptotic completeness-type statements. S -matrix We will base our discussion here on the scattering theory associated to ˚Σ or Σ. First, note that applyinga discrete isometry of D which interchanges the future and past of ˚Σ, we infer analogously to Theorems 3and 4 the existence of bounded past forward maps, F − ∶ E V ˚Σ → E K H − ⊕ E T I − , F − ∶ E V Σ → E K H − ⊕ E T I − , and the corresponding bounded two-sided inverses B + ∶ E K H − ⊕ E T I − → E V ˚Σ , B + ∶ E K H − ⊕ E T I − → E V Σ .
13e thus have both existence and uniqueness for past scattering states as well as past asymptotic completeness.The following is then an immediate corollary
Theorem 5.
The composition of S = F + ○ B + defines bounded invertible maps S ∶ E K H − ⊕ E T I − → E K H + ⊕ E T I + , S ∶ E K H − ⊕ E T I − → E K H + ⊕ E T I + . (17) I − I + H + H − The boundedness ∥ S ∥ ≤ C of the map S in the operator norm should be viewed as the quantitativereplacement for the usual unitarity property. Given the scattering map S , we can now give an account of superradiant reflection in physical space, i.e. inthe “time domain”.Recall the standard physical set-up: One wishes to study the scattering of waves with no ingoing contri-bution from the past event horizon H − and we are interested only in the part of the wave reflected to futurenull infinity I + . We thus pass from S to the transmission map T and reflection map R defined by T = π E K H+ ○ S ∣ { }⊕ E T I− , R = π E T I+ ○ S ∣ { }⊕ E T I− (18)where π E T I+ ∶ E K H + ⊕ E T I + → E T I + , π E K H+ ∶ E K H + ⊕ E T I + → E K H + are the natural projections. Note that this map does not depend on whether we consider the domain of S to be either of the choices in ( ) . The map R ∶ E T I − → E T I + takes an asymptotic state corresponding to an incoming wave packet supported solely on past infinity I − (i.e. with no incoming radiation from H − ) and maps it to the part of the asymptotic state which is reflectedto future null infinity I + (i.e. projecting out the part transmitted to the future horizon H + ). Similarly, themap T ∶ E T I − → E K H + takes an asymptotic state corresponding to an incoming wave packet supported solely on past infinity I − and maps it to the part of the asymptotic state which is transmitted to the future event horizon H + .Since S ∣ { }⊕ E T I− = T ⊕ R , the boundedness of S above immediately yields the strictly weaker statement Theorem 6.
The reflection and transmission maps R and T are bounded, i.e. ∥ R ∥ , ∥ T ∥ ≤ C . See the first statement of Theorem 10.1.1. In view of the relation with the fixed-frequency theory to bediscussed in Section 2.4 below, we have sup ( ω,m,ℓ ) ∣ R ( ω, m, ℓ )∣ = ∥ R ∥ , (19)and thus, a posteriori, Theorem 6 gives in particular ( ) . We note however that in the logic of the proof, wewill have essentially already used ( ) in proving the boundedness of both the maps F + and B − .14et us here already mention a further application of the relationship ( ) to our physical-space scatteringtheory. First, note that general soft o.d.e. theory is sufficient to show that the reflection coefficient satisfies ∣ R ( ω, m, ℓ )∣ > ( ) the statement Theorem 7.
For a ≠ , the reflection map R has norm strictly greater than 1, i.e. ∥ R ∥ > . See the second statement of Theorem 10.1.1. The above theorem can be viewed as the definitive physical-space interpretation of the phenomenon of superradiant reflection. To connect with the numerical settingoften studied (e.g. [5, 40]) in which it is difficult to implement past scattering data on I − , we will extract inaddition the following somewhat less natural statement concerning Cauchy data on ˚Σ via a density argument(see Theorem 10.1.2): There exists a smooth solution ψ with the property that its T -energy flux through I + is greater than its T -energy flux through ˚Σ and moreover, the support of the solution on ˚Σ is compact andcan be made arbitrarily close to spatial infinity. Cf. [30]. This addresses in particular some questions raisedin [40].
As we have already discussed, when a ≠ T -energy, provided thisinner product is finite.The T -energy is not finite on the full domain of the scattering matrix S of ( ) . It is, however, finiteif one for instance restricts to past scattering data supported only on I − . Recalling the notation ( ) , onestatement of “pseudo-unitarity” is then captured by the following theorem. Theorem 8.
The map T ⊕ R preserves the T -energy: ∫ H + J Tµ [ T φ ] n µ H + + ∫ I + J Tµ [ R φ ] n µ I + = ∫ I − J Tµ [ φ ] n µ I − . In particular, if the right hand side above is bounded, then the first term on the left hand side, which isunsigned, is integrable. See Theorem 10.2.1.If we restrict to past scattering data on H − ∪ I − that are non-superradiant, i.e. supported in frequencyspace outside the superradiant range, then our scattering map S will indeed be unitary in the usual sense.For this we define Hilbert spaces E T, ♮ H ± ⊕ E T, ♮ I ± by the completion under the inner product ⟨( ψ , φ ) , ( ψ , φ )⟩ = ∫ ∞−∞ ∑ mℓ [ ω ( ω − ω + m ) Re ( ˆ ψ ˆ ψ ) + ω Re ( ˆ φ ˆ φ )] (20)of scattering data whose Fourier transforms are supported in the non-superradiant range {( ω, m, ℓ ) ∶ ω ( ω − ω + m ) > } . We then have
Theorem 9.
The restriction of the first map of ( ) extends to a unitarity isomorphism S ∶ E T, ♮ H − ⊕ E T, ♮ I − → E T, ♮ H − ⊕ E T, ♮ I − with respect to the positive definite inner product (20). See Theorem 10.2.2. Note that the above theorem retrieves in particular the unitarity of the first map of ( ) in the Schwarzschild case a = S restricted to axisymmetric data in the full ∣ a ∣ < M case. Finally, we note that our scattering theory allows us to make the following injectivity statements which canbe understood as statements just of uniqueness of scattering states (cf. (b)) for scattering data determinedon any of the four “ill-posed” pairs of asymptotic boundaries H + ∪ H − , I + ∪ I − , H + ∪ I − and H − ∪ I + .15 heorem 10. The maps F ∶ E V Σ → E K H + ⊕ E K H − , F ∶ E V Σ → E T I + ⊕ E T I − , F ∶ E V Σ → E K H + ⊕ E T I − , F ∶ E V Σ → E K H − ⊕ E T I + are all injective. I − I + H + H − Σ I − I + H + H − Σ I − I + H + H − Σ I − I + H + H − ΣSee Corollary 10.3.1. Together with the previous results, the above implies that finite V -energy solutionsare uniquely determined by their fluxes to any pair of the set { H + , H − , I + , I − } . In contrast, however, tothe forward maps of Theorem 3, it follows already from general local ill-posedness type results for the waveequation (see e.g. the classic textbook [35]) that the above maps F are not surjective. Thus, one does nothave the analogue of “existence of scattering states” (cf. (a)) for scattering states parameterized as above. As we have discussed, the proofs of our theorems of physical space scattering theory required us to revisitour quantitative fixed frequency study of the o.d.e. ( ) conducted in [24]. Thus, along the way, we have infact obtained new results for the fixed-frequency scattering theory initiated by Chandrasekhar [11], as wellas a precise connection of the two through the scattering matrix S . We collect these statements in thissection. R and T We begin with the statement of the uniform boundedness of the transmission and reflection coefficients.
Theorem 11.
The reflection and transmission coefficients as normalised in ( ) are uniformly bounded overall frequencies: sup ( ω,m,ℓ ) ∣ R ( ω, m, ℓ )∣ ≤ C, sup ( ω,m,ℓ ) ∣ T ( ω, m, ℓ )∣ ≤ C. (21)We in fact have a statement for the complete set of coefficients where we also allow for waves normalisedto the past horizon. See Theorem 6.2.2.We will infer the above theorem as an immediate corollary of our o.d.e. estimate Theorem 6.2.1, whichitself is an easy adaptation of an estimate of our previous [24]. We emphasise again that this result requiresin particular appeal to the real-mode stability theorem of [53].To connect with the pioneering heuristic work of Starobinski [54], we may define the following constantdepending only on the Kerr parameters S ( a, M ) ≐ sup ( ω,m,ℓ ) ∣ R ( ω, m, ℓ )∣ , and by Theorem 11, together with the soft statement Corollary 5.3.1 (mentioned already in the context ofTheorem 7), we have 1 < S ( a, M ) < ∞ , < ∣ a ∣ < M. This is of course in sharp distinction to the fixed-frequency theory, for which “existence of scattering states” associated to H + ∪ H − and I + ∪ I − , respectively, corresponds precisely to the existence and linear independence of the pairs U hor , U hor oralternatively U inf , U inf ) described in the beginning of this introduction, on which the whole theory is based. a = S ( , M ) ≤ ℓ estimate givenby Corollary 6.4.1, S ( , M ) = S ( a, M ) , and to understand in particular thelimit lim ∣ a ∣→ M S ( a, M ) . (22) The full scattering map S defined in Section 2.3.5 can be represented as a generalised Fourier transforminvolving the transmission and reflection coefficients T and R defined via ( ) , together with coefficients ˜ T and ˜ R associated to analogously defined solutions U of ( ) normalised to the past event horizon H − . Soas not to define the latter here, for convenience, let us simply state the relations for the physical spacetransmission and reflection maps T and R defined in Section 2.3.6: Theorem 12.
We may represent R [ φ ] = √ π ∫ ∞−∞ ∑ mℓ a I − R e − itω e imφ S mℓ ( aω, cos θ ) dω and T [ φ ] = − √ M πr + ∫ ∞−∞ ∑ mℓ ( ωω − ω + m ) a I − T e − itω e imφ S mℓ ( aω, cos θ ) dω. Here − iωa I − ≐ √ π ∫ ∞−∞ ∫ S ∂ t φ e itω e − imφ S mℓ ( aω, cos θ ) sin θ dt dθ dφ. In particular, ( ) holds. See Theorem 9.5.3 for the full statement concerning S .In fact, a posteriori, in view of Theorem 9.5.3, the statement of Theorem 6.2.2 is equivalent to theboundedness of the map S of Theorem 6. We note in contrast that the boundedness of the maps F + and B + individually (already asserted) does not have an obvious natural interpretation purely in termsof the formal fixed frequency scattering theory. Similarly, the boundedness statement of Theorem 1 (andthe boundedness statement of [24] quoted here as Theorem 3.7.1 which concerns boundedness through aspacelike foliation) cannot be directly interpreted purely in terms of the formal fixed frequency scatteringtheory. These are all distinct manifestations of ways that the phenomenon of “superradiance” allowed by thepresence of an ergoregion can be quantified. As with the question of the finiteness of ( ) , it is a completelyopen question which if any of these boundedness statements survives in the extremal case ∣ a ∣ = M . See [6]. We make a few comments on scattering theory for non-linear generalisations of ( ) . Perhaps the ultimatenonlinear such generalisation is provided by the Einstein vacuum equationsRic ( g ) = all “admissible” solutions by appropriate asymptotic states may turn outto be too ambitious for equations as nonlinear as ( ) . The mere constructing of some, however, in the spiritof the map ( ) , can serve as an important way of obtaining non-trivial examples of solution spacetimeswhich cannot otherwise easily be inferred to exist. A result in that direction has recently been provided by Theorem. [14] Consider asymptotic data on H + ∪ I + for the Einstein vacuum equations (23), decayingtowards Kerr data corresponding to g a,M with ∣ a ∣ ≤ M at a sufficiently fast exponential rate. Then thereexists a vacuum spacetime ( M , g ) attaining the data. ( M , g ) constructed in the above are in fact the first known examples of dynamical vacuumblack holes settling down to Kerr.The above theorem can be thought of as a non-linear analogue of the map ( ) (for energies whichhave additional weights in r however!). In fact, proving the above requires capturing a complicated r p -hierarchy of decay of various components of curvature which in turn allows one to identify a null conditionin the implicit non-linearities in ( ) . (We note in contrast that without additional special structure, theanalogue of the above theorem does not hold even say for the general scalar semilinear equation of the form ◻ g ψ = Q (∇ ψ, ∇ ψ ) .) We refer the reader to [14].In the context of our present paper, let us simply remark that the degenerate V -scattering theory devel-oped here, together with the blow-up result Theorem 11.1 and the upcoming [15], gives further support tothe following conjecture of [14]: Conjecture. [14] Consider asymptotic data on H + ∪ I + as above but which decay to g a,M only at a sufficientlyfast inverse polynomial rate. Then there exists a vacuum spacetime ( M , g ) attaining the data. For genericsuch data, H + is a “weak null singularity” across which the metric extends continuously but with Christoffelsymbols which fail to be locally square integrable. See the discussion of Section 11.3 and [41].
The logic of the paper will depart slightly from the order we have presented the main results above. We thusclose this introduction with a brief section by section outline of the contents of the remainder of the paper,highlighting in bold where each of the main theorems above are actually proven. (More detailed outlineswill be given in the body of the paper at the beginning of each individual section.)In Section 3, we briefly review the structure of the Kerr spacetime, introduce various conventions, andquote some previous results on forward evolution which will be important, in particular, we will state generalwell-posedness results (Propositions 3.6.1–3.6.4), precise versions of our previous boundedness and integratedlocal decay results of [24] (quoted as Theorem 3.7.1 and the higher order Theorem 3.7.2) as well as a general r p weighted estimate (Proposition 3.8.1) which we will derive from [18].In Section 4, we define and establish some basic properties of the radiation fields and energy fluxes along H +≥ (or H + ) and I + for solutions ψ to (2) arising from smooth initial data along Σ ∗ (or Σ) which arecompactly supported. The main result is Proposition 4.2.1, which is the precise statement of Proposition 1 above.In Section 5, we first review Carter’s separation of variables for the wave equation and then define the radial o.d.e. , recalling also some results from its basic asymptotic analysis. This will allow us to define thereflection and transmission coefficients (Definition 5.3.2), deduce fixed-frequency superradiant amplificationin the form of Corollary 5.3.1, and define the so-called microlocal radiation fields (Definition 5.4.1) and fluxes (Definition 5.4.2).In Section 6 we establish various estimates for the radial o.d.e. and give some useful applications. Westart by proving Theorem 6.2.1, an estimate for general solutions to the homogeneous radial o.d.e. The proofof Theorem 6.2.1 will heavily rely on our o.d.e. estimates from [24]. Next, in Section 6.3 we establish animportant estimate for U hor in the superradiant regime (Proposition 6.3.1). We then use related ideas inSection 6.4 to prove Proposition 6.4.1 which states that for fixed ω and m , T vanishes in the large- ℓ limit.In Section 6.5 we show that for each m and ℓ , the reflection coefficient R is not identically 0 as a function of ω . In Section 6.6 we prove Proposition 6.6.1, which is the microlocal version of the r p estimates of [18] (cf.Proposition 3.8.1). The goal of Section 6.7 is to prove Proposition 6.7.1 which establishes uniform estimates,over all frequency parameters, for the rate of convergence of solutions to the radial o.d.e. to their microlocalradiation fields. Finally, in Section 6.8 we prove Propositions 6.8.1 and 6.8.2 which establish that for suitablesolutions ψ to the wave equation, the microlocal radiation fields are essentially the Fourier transform of thephysical space radiation fields defined in Section 4.In Section 7, we prove Theorem 7.1, the statement that the total flux to null infinity I + and the degenerate K -flux to the horizon H + of a solution ψ to the wave equation may be controlled by the V -energy of ψ alongΣ ∗ . The theorem is stated in Section 7.1, after which the reader impatient to proceed to the construction ofour scattering theory may skip to Section 8 below. The proof of Theorem 7.1, which occupies Sections 7.3 is18 modification of the proof of Theorem 3.7.1 quoted from [24]. In a brief aside in Section 7.2, we shall stateTheorem 7.2, which is the full degenerate-energy analogue of our results of [24], quoted as Theorem 3.7.1above. We emphasise that Theorem 7.2 is not in fact necessary for the rest of the paper, and we defer itsproof to Section 9.4, where we can make use of the backwards maps of our scattering theory.In Section 8, we introduce the E V ˚Σ and E N ˚Σ spaces, etc., and define the various “forward” maps andestablish their boundedness. Theorem 8.2.1, the precise version of Theorem 1 , is independent of Section 7,as it relies directly on Theorem 3.7.1 of [24]. Theorem 8.2.2, on the other hand, which together with itscorollaries Theorems 8.2.3 and 8.2.4 embodies the precise version of
Theorem 3 , uses in a fundamental wayTheorem 7.1.In Section 9, we prove first Theorem 9.1.1, then Theorem 9.2.1, then Theorem 9.3.1. This obtains allstatements in
Theorem 4 . As an aside in Section 9.4, we obtain the proof of Theorem 7.2 referred toabove. Next, we construct the “scattering” map S and show that it is a bounded invertible map from dataalong H − ∪ I − to data along H + ∪ I + (Theorem 9.5.2, the precise version of Theorem 5 ). We then proveTheorem 9.5.3 which establishes a formula for the scattering map S explicitly exhibiting the roles of thereflexion and transmission coefficients. This formula will in particular establish the relationship betweenphysical space and fixed frequency scattering theories embodied by Theorem 12 . Finally, as an additionalaside in Section 9.6, we give an alternative self-contained treatment for the Schwarzschild case where it ispossible to exploit purely physical space arguments.In Section 10, we begin by interpreting our scattering results for the reflection operator R . Theorem 10.1.1combines the statements of Theorems 6 and 7 . We also infer the related Theorem 10.1.2. Following this,we study the pseudo-unitarity properties of S and prove the corresponding Theorems 10.2.1 and 10.2.2(cf. Theorems 8 and 9 ). Finally, our “uniqueness of improper scattering states” results are stated asTheorem 10.3.1, giving
Theorem 10 .In Section 11, we prove Theorem 11.1, the statement that solutions ψ of ( ) on Schwarzschild whoseradiation fields on the horizon H + have a precise polynomial tail and whose radiation fields on I + vanish mustnecessarily have infinite N -energy on the hypersurface Σ ∗ . This statement can be understood independentlyof the results concerning our scattering maps, and indeed, Sections 11.1 and 11.2 can be read independentlyof the rest of the paper. In Sections 11.3 and 11.4 we will then return to the scattering framework of ourpaper. We first use the backwards map of our V -scattering theory to infer the existence (See Corollary 11.1)of solutions satisfying the assumptions of Theorem 11.1. Finally, we infer Theorem 2 as Corollary 11.4.
We begin in this section with various preliminaries.After reviewing our notations for energy currents associated to vector fields in Section 3.1, we will definecarefully in Section 3.2 the ambient spacetime D (and related subsets) on which we will consider the Kerrmetric g a,M for subextremal values ∣ a ∣ < M . Our conventions for constants depending only on the Kerrparameters will be reviewed in Section 3.3. These follow our conventions from [24]. Some auxilliary usefulvector fields will be presented in Section 3.4.It will be useful to define a hyperboloidal-type foliation S τ of R and we shall do this in Section 3.5. Theform of the T energy-flux through such a foliation is recorded in Lemma 3.5.1. Section 3.6 states generalwell posedness results (Proposition 3.6.1–3.6.4) for the wave equation ( ) on the Kerr exterior. We shallthen quote our boundedness and integrated decay statement from [24] in Section 3.7, as Theorem 3.7.1 andthe higher order Theorem 3.7.2. The foliation of Section 3.5 will then allow us in Section 3.8 to easilyquote the r p hierarchy of estimates (introduced in [18]) in the form of Proposition 3.8.1 and the higher orderProposition 3.8.2. Given a general Lorentzian manifold ( M , g ) , let Ψ be a sufficiently regular complex function. We define T µν [ Ψ ] ≐ Re ( ∂ µ Ψ ∂ ν Ψ ) − g µν g αβ Re ( ∂ α Ψ ∂ β Ψ ) . X on M , we define the currents J Xµ [ Ψ ] = T µν [ Ψ ] X ν , K X [ Ψ ] = T µν [ Ψ ] ∇ µ X ν = T µν ( X ) π µν , E X [ Ψ ] = − Re (( ◻ g Ψ ) X ν Ψ ,ν ) . Here ( X ) π µν ≐ ∇ µ X ν + ∇ ν X µ is the deformation tensor of X . In particular, K X = X is Killing.Recall the fundamental identity: ∇ µ J Xµ [ Ψ ] = K X [ Ψ ] − E X [ Ψ ] . Then the divergence identity between two homologous spacelike hypersurfaces S − , S + , bounding a region C ,with S + in the future of S − , yields ∫ S + J Xµ [ Ψ ] n µS + + ∫ C ( K X [ Ψ ] − E X [ Ψ ]) = ∫ S − J Xµ [ Ψ ] n µS − , (24)where n S ± denotes the future directed timelike unit normal, and the induced volume forms are to be under-stood. Remark 3.1.1.
In general, in integrals we will either write explicitly a volume form or it is to be understoodthat the integration is with respect to the induced volume form. In the case of a null hypersurface, the volumeelement depends on the choice of a null generator and is defined so that the divergence theorem holds.
We direct the reader unfamiliar with the use of energy currents to the concise introductory book [4].See [12] for a systematic discussion.
In this section we will briefly review the background differentiable structure and various convenient coordinatesystems for the Kerr spacetime. We direct the reader to [19] and [24] for a more thorough discussion of ourconventions and to the books [38] and [46] for a proper introduction to Kerr.As is well known, the Kerr spacetimes (M , g a,M ) are a 2-parameter family of spacetimes which in theparameter range ∣ a ∣ < M may be thought of as the maximal Cauchy development of a Cauchy hypersurfacewith two asymptotically flat ends. The spacetime M possesses a bifurcate Killing horizon separating twoasymptotically flat exterior regions from a black hole and a white hole region (in the case a ≠
0, then M isfurther extendible beyond a smooth Cauchy horizon to a larger spacetime which fails however to be globallyhyperbolic and is thus not uniquely determined by initial data). In this paper, we will work on the subregion D which is the closure of one of the exterior regions M . The boundary of the region D consists of the unionof two null hypersurfaces H + and H − , the future event horizon and the past event horizon , along with B , the bifurcate sphere . Our convention will be that B is not included in H ± and H + ∪ B ∪ H − is a bifurcate nullhypersurface.We proceed to describe explicitly the underlying structure and metric. We start with the smooth manifoldwith boundary R = R ≥ × R × S , (25)parameterized by y ∗ ∈ R ≥ , t ∗ ∈ R and a choice of standard spherical coordinates ( θ ∗ , φ ∗ ) ∈ S . Thiscoordinate system will be known as “Kerr-star coordinates”. Let us denote the coordinate vector field T = ∂ t ∗ and Φ = ∂ φ ∗ and let us denote by Ω , Ω , Ω a basis of standard angular momentum operatorscorresponding to the S factor of ( ) . In particular, the Ω i span the tangent space of S .We define what shall be the future event horizon H + by H + = ∂ R = { y ∗ = } . It will be useful to adoptthe conventions: H +≥ s ≐ H + ∩ { t ∗ ≥ s } , We may take Ω = Φ for instance. + ( s , s ) ≐ H + ∩ { t ∗ ∈ [ s , s ]} , R ≥ s ≐ { t ∗ ≥ s } , ˚ R ≐ int (R) = R ∖ H + , ˚ R ≥ s ≐ R ≥ s ∖ H +≥ s . Next, given a choice of parameters ( a, M ) satisfying ∣ a ∣ < M , we define a new coordinate function r = r ( y ∗ ) on R (with ∞ > C > drdy ∗ ≥ c >
0) so that r ∣ H + = r + ( a, M ) where r ± ≐ M ± √ M − a . It is often convenient toreplace r with yet another rescaled version, r ∗ = r ∗ ( r ) , defined in ˚ R , by dr ∗ dr = r + a ∆ , r ∗ ( M ) = , (26)where ∆ = r − M r + a = ( r − r + )( r − r − ) . (27)Since r − < r + , it follows that ∆ vanishes to first order on H + , and thus the coordinate range ∞ > r > r + covering ˚ R corresponds to the range ∞ > r ∗ > −∞ . It will also be useful to sometimes employ what will bean “approximately null” coordinate system ( ˜ u, ˜ v, θ, φ ) defined by˜ u = ( t − r ∗ ) , ˜ v = ( t + r ∗ ) . Next, we introduce the new coordinates t ( t ∗ , r ) ≐ t ∗ − ¯ t ( r ) , φ ( φ ∗ , r ) ≐ φ ∗ − ¯ φ ( r ) mod 2 π, θ ≐ θ ∗ (28)where ¯ t ( r ) and ¯ φ ( r ) are appropriately chosen smooth functions depending on a and M (see [19] and [24] fordetails) which vanish for sufficiently large r .In these “Boyer-Lindquist coordinates” ( t, r, θ, φ ) , we finally define the Kerr metric by g a,M = − ∆ ρ ( dt − a sin θdφ ) + ρ ∆ dr + ρ dθ + sin θρ ( a dt − ( r + a ) dφ ) , (29)where ρ ≐ r + a cos θ . Though a priori ( ) is only defined in ˚ R , by examining the expression of themetric in Kerr-star coordinates y ∗ and t ∗ (see [19] for the computation), one checks easily that g a,M extendssmoothly to H + making ( R , g a,M ) a smooth Lorentzian manifold-with-boundary. Let us note moreover that T and Φ defined previously can be expressed again as coordinate vector fields T = ∂ t and Φ = ∂ φ , whenceit follows from ( ) that T and Φ are Killing on R . These are the so-called stationary and axisymmetricKilling vector fields.Recall that when a ≠ T is not everywhere timelike. The region S where T is spacelikeis known as the “ergoregion”. Explicitly, we have S = { ∆ − a sin θ < } . (30)Note that S ⊂ { r < M } . Let us also recall that in [19] and [24] we chose the function ¯ t of (28) so that the hypersurfaces t ∗ = s ,denoted by Σ ∗ s , are spacelike with respect to the Kerr metric as just defined. Furthermore, we will have R ≥ s = D + ( Σ ∗ s ) . Let us introduce the notation ˚Σ ≐ { t = } . (31)We have that ˚Σ is also spacelike and a Cauchy hypersurface for ˚ R .Some additional notation from [24]: Note that the definition ∂ r is ambiguous since it depends on thechoice of coordinate system. Thus, we define Z ∗ ≐ ∂ r with respect to coordinates ( t ∗ , r, θ ∗ , φ ∗ ) , (32)21 ≐ ∂ r with respect to coordinates ( t, r, θ, φ ) . (33)Note that Z ∗ is well defined in R and is transversal to H + while Z is only well defined in ˚ R . Finally, we willuse ∇ / to denote the induced covariant derivative on the S factor of R .Though all explicit computations will take place on the manifold-with-boundary R defined above, it isof fundamental importance to understand the existence and properties of a further smooth extension to D = R ∪ H − ∪ B , which will represent precisely the D described at the beginning of this section. We will bebrief in our presentation; we direct the reader to [46] for a very careful and detailed exposition.We begin by attaching H − . Starting with Boyer-Lindquist coordinates ( t, r, θ, φ ) , one defines a newcoordinate system ( ∗ t, ∗ φ, r, ∗ θ ) by ∗ t ( t, r ) ≐ t − ¯ t ( r ) , ∗ φ ( t, r ) ≐ φ − ¯ φ ( r ) mod 2 π, ∗ θ ≐ θ, where ¯ t and ¯ φ are as above. A straightforward computation shows that the metric naturally extends smoothlyso as to be defined also at r = r + in this chart. We may thus use this coordinate chart to extend R to alarger manifold-with-boundary R ∪ H − where H − corresponds to the hypersurface r = r + of this new chart.We shall refer to H − as the past event horizon .One may easily check that the Boyer-Lindquist coordinate defined map ( t, φ ) ↦ ( − t, − φ ) (34)is an isometry of ˚ R which smoothly extends to an isometry of R ∪ H − and furthermore sends H + to H − .Finally, one may even further extend R ∪ H − to a larger Lorentzian manifold ̃ M so that the boundary of R (as a subset) in ̃ M consists of a bifurcate null hypersurface B ∪ H − ∪ H + , with B ⊂ ̃ M a sphere. Our regionof interest D described at the beginning of this section is simply then the manifold-with-stratified boundary D = R ∪ H − ∪ B . We remark that D admits a globally regular coordinate system ( U + , V + , θ, φ ) ∈ [ , −∞) ×[ , ∞) × S so that H + = { U + = , V + ∈ ( , ∞)} , H − = { V + = , U + ∈ (−∞ , )} and B = {( U + , V + ) = ( , )} .Moreover, along B we have g U + U + = g U + θ = g U + φ = g V + V + = g V + θ = g V + φ = . (35)We shall not here require the form of the explicit coordinate transformations defining U + and V + in termsof our previously described charts on R ∪ H − but we remark thatΣ ≐ B ∪ ˚Σ (36)is a smooth manifold-with-boundary (with boundary B ) and interior ˚Σ. Note that smooth functions ( ψ , ψ ′ ) “compactly supported on ˚Σ” extend to smooth compact supported functions on Σ which moreover vanishin a neighbourhood of B . On the other hand, smooth functions compactly supported on Σ do not restrictto compactly supported functions on ˚Σ.It will be convenient to introduce the notation H ± ≐ H ± ∪ B , (37) H +≤ τ ≐ H + ∩ ({ t ∗ ≤ τ }) ∪ B) . (38)These are again smooth hypersurfaces-with-boundary, with boundary B for H ± and boundary B ∪ ( Σ ∗ ∩ H + ) for H +≤ . The reader should in particular again contrast the distinct notions of “compactly supported” on H + and H ± .We have already noted that the vector fields T and Φ are Killing. The event horizon H + is also a Killinghorizon : the Killing field given by the linear combination K ≐ T + ω + Φ , (39)where ω + ≐ a Mr + is the “angular velocity” of the event horizon. The vector field K is null and normal to H + ;thus, H + is in particular a null hypersurface. In integrals associated to energy currents we will denote K by It is globally regular up to the usual degeneration of spherical coordinates. µ H + . It will be useful to recall that the vectorfield K restricted to H + coincides with the smooth extensionof the coordinate vector field ∂ r ∗ of the ( r ∗ , t, θ, φ ) coordinate system.The past event horizon H − is also a Killing horizon with a Killing field also given by K . Note howeverthat the restriction of K to H − coincides with the smooth extensions of the coordinate vector field − ∂ r ∗ ofthe ( r ∗ , t, θ, φ ) coordinate system.Note finally that the vector fields T and Φ and the discrete isometry ( ) both extend smoothly to all of D and K ∣ B = . (40) a and M and conventions on constants In all propositions to follow, unless otherwise stated, ∣ a ∣ < M are fixed parameters and everything refers tothe Kerr metric g M,a on D as described in the previous section. Let us briefly review our conventions from [19] and [24] regarding constants depending on the parameters a and M . Large positive constants will be denoted by B , and small positive constants by b . Both constants B and b depend only on M and a lower bound for M − ∣ a ∣ , and this dependence is always to be understoodeven when not mentioned explicitly. Often these constants will blow up B → ∞ , b − → ∞ in the extremallimit ∣ a ∣ → M .We recall the usual arithmetic properties of b and B : b + b = b, B + B = B, B ⋅ B = B, B − = b, . . . The statement f ∼ g will mean bg ≤ f ≤ Bg.
The statement “for R sufficiently large”, etc., without further qualification, will mean “there exists a constant R ( a, M ) such that for R ≥ R ”.Lastly, if the constant B or b depends on the value of a yet to be fixed parameter, then that dependencewill be explicitly noted. For example, if B depends on a parameter c which has not been fixed, we shalldenote it by B ( c ) . Once the constant c is fixed, we then write B . We recall the following two lemmas proved in [24].
Lemma 3.4.1.
The vector field T + M ar ( r + a ) Φ is a smooth vector field in D , is timelike in ˚ R and null on H ± . Lemma 3.4.2.
There exists a constant ǫ = ǫ ( a, M ) > such that the vector field K (39) is timelike for r ∈ ( r + , r + + ǫ ) . These lemmas allow us to make the following definition.
Definition 3.4.1.
Let ǫ > be from Lemma 3.4.2. Let α ( r ) be a function such that V ≐ T + α ( r ) Φ is asmooth vector field in D , timelike in ˚ R and which satisfies V = K, when r ∈ [ r + , r + + ǫ / ] ,V = T + M ar ( r + a ) Φ , when r ∈ ⎡⎢⎢⎢⎢⎣ r + + ǫ , M ( + √ ) ⎤⎥⎥⎥⎥⎦ ,V = T, when r ≥ M ( + √ ) . emark 3.4.1. This vector field will be useful because it is manifestly T -invariant, it is timelike (hence theassociated energy fluxes J V are positive definite) and because it is Killing for r ≤ r + + ǫ / and r ≥ M ( + √ )/ (hence the error terms K V from the energy identity (24) are supported in r + + ǫ / ≤ r ≤ M ( + √ )/ ). It will be useful to observe the following immediate corollary of Lemmas 3.4.1 and 3.4.2.
Corollary 3.4.1.
For every ǫ > sufficiently small and any r ∈ ( r + , ∞ ) , there exists a vector field ˜ V = ˜ V ( r , ǫ ) of the form T + ˜ α ( r ) Φ for an appropriate function ˜ α ( r ) such that ˜ V is a smooth vector field on D ,is timelike in ˚ R , is Killing in the region r ∈ ( min ( r + , r − ǫ ) , r + ǫ ] , and is equal to V for r sufficiently close to r + and r sufficiently large. We shall apply the above Corollary, for finitely many distinct choices of r , in the context of Section 9.1.2.In order for our non-degenerate energies to have a fixed meaning, it is useful to fix once and for all achoice of a globally defined smooth timelike vector field on D . Definition 3.4.2.
Let N denote any fixed choice of a smooth timelike vector field on D which is invariantunder the flow of T on the complement of a compact set containing the bifurcate sphere B , and satisfies N = T for sufficiently large r . Finally, we note the following easy calculations.
Remark 3.4.2.
Fix an open set U ⊂ D containing the bifurcate sphere B ⊂ U . Then, for s such that Σ ∗ s ∩ U = ∅ we have ∫ Σ ∗ s J Nµ [ ψ ] n µ Σ ∗ s ∼ ∥ ψ ∥ H ( Σ ∗ s ) + ∥ n Σ ∗ s ψ ∥ L ( Σ ∗ s ) ∼ ∫ θ,φ ∗ ∫ ∞ r + (∣ ∂ t ∗ ψ ∣ + ∣ ∂ r ψ ∣ + ∣ ∇ / ψ ∣ g / ) dr dVg / (41) with respect to coordinates ( t ∗ , r, θ ∗ , φ ∗ ) . Remark 3.4.3.
We have ∫ Σ ∗ s J Vµ [ ψ ] n µ Σ ∗ s ∼ ∫ θ,φ ∗ ∫ ∞ r + (∣ ∂ t ∗ ψ ∣ + ( − r + r ) ∣ ∂ r ψ ∣ + ∣ ∇ / ψ ∣ g / ) dr dVg / (42) with respect to coordinates ( t ∗ , r, θ ∗ , φ ∗ ) . It will be convenient to have the following explicit foliation of R by a family of hyperboloidal hypersurfaces. Definition 3.5.1.
For every τ ∈ R we set S τ ≐ { t ∗ = τ r ≤ Mt ∗ − r ∗ + Mr = τ − ( M ) ∗ + r > M. (This hypersurface could be smoothed out, but this does not in fact make a difference.)Some straightforward, if tedious, calculations yield the following lemma. Lemma 3.5.1.
For every τ ∈ R , S τ is a spacelike hypersurface, R = ∪ τ ∈ R S τ , and, for sufficiently large R , ∫ S τ ∩{ r ≥ R } J Tµ [ ψ ] n µS τ ∼ ∫ S τ ∩{ r ≥ R } [∣ ∂ ˜ v ψ ∣ + r − ∣ ∂ ˜ u ψ ∣ + ∣∇/ ψ ∣ ] r sin θ dv dθ dφ. In comparing the way these two integrals are written, we recall our convention that if no volume formis written explicitly (as on the left hand side of the above), the integration is with respect to the inducedvolume form.Later, when r is sufficiently large we will often work in the coordinate system ( τ, r, θ, φ ) associated tothe foliation { S τ } τ ∈ R . We will in fact use this coordinate system to define our notion of null infinity I + inSection 4.2. Note that in view of the vanishing of T on B , one cannot define such a timelike N which is invariant on all of D . .6 Well-posedness Let us briefly recall some basic well-posedness statements.First we consider the case of initial data prescribed on Σ ∗ . Recall that R ≥ = { t ∗ ≥ } = D + ( Σ ∗ ) . In thepropositions below the H s and C k spaces will refer to complex valued functions. Proposition 3.6.1.
Let ( ψ , ψ ′ ) ∈ H s loc ( Σ ∗ ) × H s − ( Σ ∗ ) . Then there exists a unique solution ψ to the waveequation ( ) on R ≥ such that ψ ∈ C τ ∈ [ , ∞ ) ( H s loc ( Σ ∗ τ )) ∩ C τ ∈ [ , ∞ ) ( H s − ( Σ ∗ τ )) ∩ H s loc ( H +≥ ) ,ψ ∣ Σ ∗ = ψ , and n Σ ∗ ψ ∣ Σ ∗ = ψ ′ . Furthermore, the solution map depends continuously on the initial data.Finally, we note that if the initial data ( ψ , ψ ′ ) are smooth, then the solution ψ will be smooth. Next we consider the case of initial data along Σ. Let us define ˜Σ τ to be the image of Σ at time τ of theflow map associated to the vector field N from Definition 3.4.2. Proposition 3.6.2.
Let ( ψ , ψ ′ ) ∈ H s loc ( Σ ) × H s − ( Σ ) . Then there exists a unique solution ψ to the waveequation ( ) in D such that ψ ∈ C τ ∈ ( −∞ , ∞ ) ( H s loc ( ˜Σ τ )) ∩ C τ ∈ ( −∞ , ∞ ) ( H s − ( ˜Σ τ )) ∩ H s loc ( H + ) ∩ H s loc ( H − ) ,ψ ∣ Σ = ψ , and n Σ ψ ∣ Σ = ψ ′ . Furthermore, the solution map depends continuously on the initial data. Finally,we note that if the initial data ( ψ , ψ ′ ) are smooth, then the solution ψ will be smooth. Remark 3.6.1.
In the case when the initial data ( ψ , ψ ′ ) are compactly supported along ˚Σ , then a J K energyestimate immediately implies that bifurcate sphere B lies outside the support of the solution ψ produced byProposition 3.6.2. Also note that if ( ψ , ψ ′ ) are compactly supported on Σ , then ( ψ ∣ Σ ∗ , n Σ ∗ ψ ∣ Σ ∗ ) are compactlysupported on Σ ∗ . It will also be useful to consider the following two mixed characteristic-spacelike initial value problems.For convenience these will both be stated in the smooth category. First we have
Proposition 3.6.3.
Let ψ H +≤ be a smooth function on H +≤ and ( ψ Σ ∗ , ψ ′ Σ ∗ ) be a pair of smooth functionson Σ ∗ such that there exists a smooth function ˜Ψ on D satisfying ˜Ψ ∣ H +≤ = ψ H +≤ , ( ˜Ψ ∣ Σ ∗ , n Σ ∗ ˜Ψ ∣ Σ ∗ ) = ( ψ Σ ∗ , ψ ′ Σ ∗ ) . Then there exists a unique smooth solution ψ to the wave equation (2) in the past of Σ ∗ such that ψ ∣ H +≤ = ψ H +≤ , ( ψ ∣ Σ ∗ , n Σ ∗ ψ ∣ Σ ∗ ) = ( ψ Σ ∗ , ψ ′ Σ ∗ ) . See I − I + H + ≥ Σ ∗ Before giving the next proposition, it is useful to define a function r ( τ, s ) : Let τ > −∞ . Then, for each s > r ( τ, s ) to be the largest solution to s − r ∗ ( τ, s ) + Mr ( τ, s ) ≐ τ − ( M ) ∗ + . { t = s } will intersect the hypersurface S τ along the surface where ( t, r ) = ( s, r ( τ, s )) . Refer to I − I + S τ t = s We have
Proposition 3.6.4.
Let τ < ∞ , let ψ H +≤ τ be a smooth function on H +≤ τ which vanishes in a neighborhood of S τ ∩ H + and let Φ { t = s } ∩ { r ≥ r ( τ,s )} be a smooth compactly supported function on { t = s } ∩ { r ≥ r ( τ, s )} whichvanishes in a neighborhood of { t = s } ∩ { r = r ( τ, s )} . Then there exists a unique smooth solution ψ to thewave equation (2) in the past of H + ≤ τ ∪ ( S τ ∩ { r ≤ r ( τ, s )}) ∪ ({ t = s } ∩ { r ≥ r ( τ, s )}) such that ψ ∣ H +≤ τ = ψ H +≤ τ , ( ψ ∣ S τ ∩ { r ≤ r ( τ,s )} , n S τ ψ ∣ S τ ∩ { r ≤ r ( τ,s )} ) = ( , ) ,rψ ∣ { t = s } ∩ { r ≥ r ( τ,s )} = Φ { t = s } ∩ { r ≥ r ( τ,s )} . In accordance with our conventions (recall Section 3.3), the above propositions refer always to the Kerrmetric with fixed parameters ∣ a ∣ < M . Let us remark that we have defined the differentiable structure in [24]so that we can assert also the smooth dependence of ψ on our parameters a and M ; this, however, shall playno role in the current paper. In this section, we shall recall the precise boundedness and integrated energy decay statements proved in [24].First we recall a few additional notations from [24]:
Definition 3.7.1.
Given s − satisfying r + < M − s − < ∞ , let us define a cutoff function χ ( r ) such that χ = for r ≥ M − s − and χ = for r ≤ ( r + + M − s − )/ . We then set ˜ Z ∗ = χZ + ( − χ ) Z ∗ . Definition 3.7.2.
Given s − and s + satisfying r + < M − s − < M + s + < ∞ we define ζ ( r ) ≐ ( − M / r ) ( − η [ M − s − , M + s + ] ( r )) , (43) where η is the indicator function. The main result of [24] was
Theorem 3.7.1. [24] There exist parameters s − ( a, M ) and s + ( a, M ) satisfying r + < M − s − < M + s + < ∞ such that for all δ > , all sufficiently regular solutions ψ to (2) on R ≥ satisfy the following estimates: ∫ R ≥ ( r − ζ ∣∇/ ψ ∣ + r − − δ ζ ∣ T ψ ∣ + r − − δ ∣ ˜ Z ∗ ψ ∣ + r − − δ ∣ ψ − ψ ∞ ∣ ) ≤ B ( δ ) ∫ Σ ∗ J Nµ [ ψ ] n µ Σ ∗ , (44) ∫ H +≥ ( J Nµ [ ψ ] n µ H + + ∣ ψ − ψ ∞ ∣ ) ≤ B ∫ Σ ∗ J Nµ [ ψ ] n µ Σ ∗ , (45) ∫ Σ ∗ s J Nµ [ ψ ] n µ Σ ∗ s ≤ B ∫ Σ ∗ J Nµ [ ψ ] n µ Σ ∗ , ∀ s ≥ , (46) where πψ ∞ = lim r ′ →∞ ∫ Σ ∗ ∩ { r = r ′ } r − ∣ ψ ∣ .
26e also proved the following higher order version of Theorem 3.7.1:
Theorem 3.7.2. [24] With s ± ( a, M ) as above, then for all δ > , j ≥ , all sufficiently regular solutions ψ to (2) on R ≥ satisfy the following estimates: ∫ R ≥ r − − δ ζ ∑ ≤ i + i + i ≤ j ∣ ∇ / i T i ( ˜ Z ∗ ) i ψ ∣ + r − − δ ∑ ≤ i + i + i ≤ j − (∣ ∇ / i T i ( ˜ Z ∗ ) i + ψ ∣ + ∣ ∇ / i T i ( Z ∗ ) i ψ ∣ ) ≤ B ( δ, j ) ∫ Σ ∗ ∑ ≤ i ≤ j − J Nµ [ N i ψ ] n µ Σ ∗ , (47) ∫ H +≥ ∑ ≤ i ≤ j − J Nµ [ N i ψ ] n µ H + ≤ B ( j ) ∫ Σ ∗ ∑ ≤ i ≤ j − J Nµ [ N i ψ ] n µ Σ ∗ , (48) ∫ Σ ∗ s ∑ ≤ i ≤ j − J Nµ [ N i ψ ] n µ Σ ∗ s ≤ B ( j ) ∫ Σ ∗ ∑ ≤ i ≤ j − J Nµ [ N i ψ ] n µ Σ ∗ , ∀ s ≥ . (49) Remark 3.7.1.
Sufficiently regular may be taken to mean that the initial data lies in H s loc ( Σ ∗ ) for s suitablylarge and that the right hand sides of each inequality are finite. Remark 3.7.2.
Recall that a straightforward elliptic estimate would yield ∫ Σ ∗ s ∑ ≤ i ≤ j − J Nµ [ N i ψ ] n µ Σ ∗ s ∼ ∑ ≤ i ≤ j ∥ ψ ∥ H i ( Σ ∗ s ) + ∥ n Σ ∗ s ψ ∥ H i − ( Σ ∗ s ) . (50) Remark 3.7.3.
In view of the discrete isometry (34), one immediately obtains versions of Theorem 3.7.1and Theorem 3.7.2 for solutions defined in the past of the hypersurface ∗ t = . r p estimates It will be useful to exploit the hierarchy of “ r p estimates” from [18]. For our purposes, it is convenient toapply these estimates in the following form. Proposition 3.8.1.
Let R be sufficiently large. Then for all τ < τ , p ∈ [ , ] , and all ψ sufficiently regularsolutions to (2) on D( τ , τ ) ≐ J + ( S τ ) ∩ J − ( S τ ) , then setting ϕ ≐ ( r + a ) / ψ and keeping Remark 3.1.1in mind, we have ∫ S τ ∩ { r ≥ R } [ r p ∣ ∂ ˜ v ϕ ∣ + r p − ∣ ∇ / ϕ ∣ + r − ∣ ∂ ˜ u ϕ ∣ ] sin θ dv dθ dφ + ∫ D( τ ,τ ) ∩ { r ≥ R } [ pr p − ∣ ∂ ˜ v ϕ ∣ + (( − p ) r p − + r − ) ∣ ∇ / ϕ ∣ + r p − ∣ ϕ ∣ + r − ∣ ∂ ˜ u ϕ ∣ ] sin θ du dv dθ dφ ≤ B ∫ D( τ ,τ ) ∩ { R ≤ r ≤ R + } r p [∣ T ϕ ∣ + ∣ Zϕ ∣ + ∣∇/ ϕ ∣ ] sin θ du dv dθ dφ + B ∫ S τ ∩ { r ≥ R } [ r p ∣ ∂ ˜ v ϕ ∣ + r p − ∣∇/ ϕ ∣ + r − ∣ ∂ ˜ u ϕ ∣ ] sin θ dv dθ dφ. Proof.
One combines the estimates of [18] with an energy estimate, Hardy inequalities, and a Morawetzestimate. This is a special case of a more general computation done in detail in [43] for the general settingof asymptotically flat spacetimes.
Remark 3.8.1.
Note that one may easily check that the boundary terms of the p = estimate relate to thespacetime terms of the p = estimate in such a way as to allow one to combine Theorem 3.7.2 with theiterated pigeon hole argument of [18] in order to conclude for instance that ∫ S τ J Nµ [ ψ ] n µS τ ≤ Bτ − E [ ψ ] ∀ τ > , where E [ ψ ] denotes a weighted second order energy of ψ along Σ ∗ .Note that we will not require such quantitative decays results in this paper.
27e will also need to commute with angular momentum operators Ω ( α ) . We obtain Proposition 3.8.2.
For every multi-index α , let Ω ( α ) denote an arbitrary product of angular momentumoperators as defined in Section 3.2, and set ϕ ( α ) ≐ Ω ( α ) ( r + a ) / ψ . For all sufficiently large R , multi-indices α , τ < τ , and p ∈ [ , ] , we obtain the estimate of Proposition 3.8.1 with ϕ replaced by ϕ ( α ) .Proof. If a =
0, this is of course immediate since then [ Ω ( α ) , ◻ g ] =
0. Otherwise, one proceeds inductivelyin ∣ α ∣ and observes that the the error terms arising from [ Ω ( α ) , ◻ g ] have sufficiently strong r decay so as tobe either absorbed by good bulk terms on the left hand side of the estimate or controlled by the previousstep. In this section, we will define the radiation fields along H + ≥ or H + and I + for solutions ψ to the waveequation (2) arising from smooth initial data along Σ ∗ or Σ which are compactly supported. Since ˚Σ ⊂ Σ,this a fortiori defines the radiation field for solutions with compactly supported data along ˚Σ.The considerations at the horizon are straightforward and will be given in Section 4.1. The finitenessof both the non-degenerate and degenerate radiation fluxes follows as a soft application of Theorem 3.7.1quoted in the previous section.Null infinity will be handled in Section 4.2. We will first have to explicitly define I + as an additionalboundary which can be attached to D (Defintion 4.2.1). The main result is Proposition 4.2.1, which givesthe statement of Proposition 1 of Section 2.1.3. We shall then relate the radiation field as defined to thelimiting energy flux of ψ along I + . Theorem 3.7.1 immediately implies the latter is finite (see Theorem 4.2.1),and according to Proposition 4.2.2 it can by computed from the radiation field. H + ≥ and H + We begin with the radiation field along the horizon.
Definition 4.1.1.
Given a solution ψ to (2) on R ≥ arising from smooth initial data along Σ ∗ which arecompactly supported, the radiation field of ψ along H + ≥ is simply defined to be the restriction of ψ to thehorizon H + ≥ . Similarly, we have
Definition 4.1.2.
Given a solution ψ to (2) on R arising from smooth initial data along Σ which arecompactly supported, the radiation field of ψ along H + is simply defined to be the restriction of ψ to thehorizon H + . Remark 4.1.1.
Note that it follows immediately from Proposition 3.6.2 that the radiation field is smoothalong the horizon.
Remark 4.1.2.
If the initial data for ψ is compactly supported on ˚Σ , then Remark 3.6.1 implies that theradiation field for ψ is supported in H + . Remark 4.1.3.
Of course, given a solution ψ to (2) defined in the past of { ∗ t = } , one may make ananalogous definition for the radiation field along H − ≥ . Similarly, one may define the radiation field along H − for a solution ψ to (2) arising from smooth initial data along Σ . H + ≥ and H + We next define the non-degenerate energy flux along the horizon.
Definition 4.1.3.
Given a solution ψ to (2) on R ≥ arising from smooth initial data along Σ ∗ which arecompactly supported, the non-degenerate N -energy flux of ψ through H + ≥ is defined by ∫ H +≥ J Nµ [ ψ ] n µ H + . emark 4.1.4. Note that Theorem 3.7.1 implies that this energy flux is finite.
Observe that a straightforward computation shows that ∫ H +≥ J Nµ [ ψ ] n µ H + ∼ ∫ H +≥ [∣ Kψ ∣ + ∣∇/ ψ ∣ ] . In particular, all of the derivatives are tangent to the horizon; thus one may think of the non-degenerate fluxas depending only on the radiation field.Finally, we define the degenerate flux along the horizon.
Definition 4.1.4.
Given a solution ψ to (2) on R ≥ arising from smooth initial data along Σ ∗ which arecompactly supported, the degenerate K -energy flux of ψ through H + ≥ is defined by ∫ H +≥ J Kµ [ ψ ] n µ H + . A straightforward computation shows that ∫ H +≥ J Kµ [ ψ ] n µ H + = ∫ H +≥ ∣ Kψ ∣ . Similarly,
Definition 4.1.5.
Given a solution ψ to (2) on R arising from smooth initial data along Σ which arecompactly supported, the degenerate K -energy flux of ψ through H + is defined by ∫ H + J Kµ [ ψ ] n µ H + . A straightforward computation shows that ∫ H + J Kµ [ ψ ] n µ H + = ∫ H + ∣ Kψ ∣ . We first define I + as a suitable additional boundary which can be attached to our spacetime. Definition 4.2.1.
As a differentiable manifold we define I + ≐ R × S and parameterize I + in the standard fashion by coordinates ( τ, θ, φ ) . Next, we extend our background differ-entiable structure R to a manifold with boundary ˜ R ≐ R ∪ I + by declaring that for every sufficiently large R and open set U ⊂ I + , the set U R ≐ {( τ, r, θ, φ ) ∶ r > R and ( τ, θ, φ ) ∈ U } is open (where ( τ, r, θ, φ ) are the coordinates associated to the foliation { S τ } τ ∈ R which we defined in Sec-tion 3.5), identifying I + with the points ( τ, ∞ , θ, φ ) , and then covering the sets U R by a coordinate chart ( τ, s, θ, φ ) ∈ R × [ , ) × S via the map ( τ, s, θ, φ ) ↦ ( τ, Rs − , θ, φ ) . Remark 4.2.1.
Note that for every fixed ( τ, θ, φ ) there exists a unique limit lim r →∞ ( τ, r, θ, φ ) ∈ I + , and, ifwe denote these limits by ( τ, ∞ , θ, φ ) , then the map ( τ, θ, φ ) ↦ ( τ, ∞ , θ, φ ) is a diffeomorphism from R × S to I + . emark 4.2.2. The above “pedestrian” definition of I + is completely equivalent to the usual one involvinga conformal compactification (see [38]). Definition 4.2.2.
Apply the discrete isometry ( t, φ ) ↦ ( − t, − φ ) to the foliation { S τ } to define a new foliation { ˜ S τ } : ˜ S τ ≐ { − ∗ t = τ r ≤ M − ∗ t − r ∗ + Mr = τ + ∗ ( M ) + r > M. Repeating the construction above with respect to this new foliation then defines past null infinity I − . Pro-ceeding in an analogous fashion to Definition 4.2.1, I − may be glued to ˜ R as a suitable boundary. Lastly, it will be useful to introduce the notations I + ≥ s ≐ {( τ, θ, φ ) ∶ τ ≥ s } , I + ≤ s ≐ {( τ, θ, φ ) ∶ τ ≤ s } . I + Recall that given a function ψ , in Section 3.8 we introduced the notation ϕ ≐ ( r + a ) / ψ,ϕ ( α ) ≐ Ω ( α ) ϕ. We now have the following straightforward corollary of Propositions 3.8.1 and 3.8.2.
Proposition 4.2.1.
For all solutions ψ to (2) on R ≥ arising from smooth initial data along Σ ∗ which arecompactly supported, and each ( τ, θ, φ ) ∈ R × S , the function ϕ ( τ, ∞ , r, θ ) ≐ lim r →∞ ϕ ( τ, r, θ, φ ) = lim r →∞ ( r + a ) / ψ ( τ, r, θ, φ ) is well defined, and is in fact a smooth function on I + .Proof. Let r > r . The fundamental theorem of calculus, Cauchy-Schwarz, and a Sobolev inequality on S imply ∣ ϕ ( τ, r , θ, φ ) − ϕ ( τ, r , θ, φ )∣ ≤ B ⎛⎝ ∑ ∣ α ∣≤ ∫ S ∣ ϕ ( α ) ( τ, r , θ, φ ) − ϕ ( α ) ( τ, r , θ, φ )∣ sin θ dθ dφ ⎞⎠ ≤ B ∑ ∣ α ∣≤ ( ∫ S τ ∩ { r ≥ r } [∣ ∂ ˜ v ϕ ( α ) ∣ + r − ∣ ∂ ˜ u ϕ ( α ) ∣] sin θ dr dθ dφ ) ≤ Br − ∑ ∣ α ∣≤ ∫ S τ ∩ { r ≥ r } [ r ∣ ∂ ˜ v ϕ ( α ) ∣ + r − ∣ ∂ ˜ u ϕ ( α ) ∣ ] sin θ dr dθ dφ. In the second inequality we have used the fundamental theorem of calculus along S τ and expressed theresulting derivative in terms of ∂ ˜ v and ∂ ˜ u .Now we conclude the proof of existence of the function ϕ ( τ, ∞ , φ, θ ) by observing that Proposition 3.8.2implies that this last quantity is bounded by B ( τ ) r − .Smoothness of ϕ as a function on I + follows in a straightforward manner by applying the above argumentto ∂ iτ Ω ( α ) ϕ , for i ∈ Z ≥ and ∣ α ∣ ∈ Z ≥ . Remark 4.2.3.
If we combine the proof of Proposition 4.2.1 with Theorem 3.7.1 we may easily concludethat for any τ ∈ R , ( r + a ) / ψ converges to its limit ϕ ∣ r = ∞ in L ∞ R ≥ τ × S . Following [31] and using the previous proposition, we may now define the radiation field along I + . Definition 4.2.3.
Given a solution ψ to (2) on R ≥ arising from smooth initial data along Σ ∗ which arecompactly supported, the radiation field of ψ along I + is defined to be the function ϕ ( τ, ∞ , θ, φ ) . emark 4.2.4. Note that any solution ψ to (2) on D arising from smooth initial data along Σ which arecompactly supported is, a fortiori, a solution to (2) on R ≥ arising from smooth initial data along Σ ∗ whichare compactly supported (cf. Remark 3.6.1). Thus, this definition of the radiation field may be applied tosuch solutions. Remark 4.2.5.
Of course, given a solution ψ to (2) defined in the past of { ∗ t = } , one may analogouslydefine the radiation field along I − . In particular, smooth compactly supported data on Σ give rise to radiationfields along both I + and I − . Remark 4.2.6.
In passing, we observe that the weighted estimates of Proposition 3.8.2 would allow us toeasily show that the radiation field decays along null infinity: ∣ ϕ ( τ, ∞ , θ, φ )∣ ≤ Bτ − / √ E [ ψ ] ∀ τ > , where E is a weighted higher order energy along Σ ∗ . Again, we emphasize that we shall not need to usesuch quantitative decay rates in this paper. I + In this section we will define the energy flux to future null infinity I + for solutions to the wave equation (2)arising from smooth initial data along Σ ∗ which are compactly supported. Recall that Σ ∗ s denotes thehypersurface { t ∗ = s } . We begin with the following lemma: Lemma 4.2.1.
Given a solution ψ to (2) on R ≥ arising from smooth initial data along Σ ∗ which arecompactly supported, then for every τ > , the following limit exists: lim s →∞ ∫ Σ ∗ s ∩ J − ( S τ ) J Tµ [ ψ ] n µ Σ ∗ s . (51) Proof.
First of all, observe that for sufficiently large s , depending on τ , the integration in (51) occurs faroutside the ergoregion (30), so that in particular, T is a timelike Killing vector field in the region underconsideration. With this in mind, a J T energy estimate implies that ∫ S τ J Tµ [ ψ ] n µS τ < ∞ . Consequently, lim s →∞ ∫ S τ ∩ J + ( Σ ∗ s ) J Tµ [ ψ ] n µS τ = . Let s < s both be sufficiently large. Refer to the figure below: I − I + S τ Σ s Σ s It now suffices to observe the following immediate consequence of a J T energy estimate: ∣ ∫ Σ s ∩ J − ( S τ ) J Tµ [ ψ ] n µ Σ s − ∫ Σ s ∩ J − ( S τ ) J Tµ [ ψ ] n µ Σ s ∣ ≤ ∫ S τ ∩ J + ( Σ s ) J Tµ [ ψ ] n µS τ . emark 4.2.7. Observe that this lemma holds for essentially any asymptotically flat spacetime possessinga suitable notion of future null infinity; in particular, we do not appeal to Theorem 3.7.1.
Remark 4.2.8.
We observe that one may easily check that if one considers smooth solutions which satisfy ∫ Σ ∩ { r ≥ R } J Tµ [ ψ ] n µ Σ < ∞ for all sufficiently large R , but are not necessarily compactly supported, then an easymodification of the proof of Lemma 4.2.1 shows that for all τ < τ , the limit lim s →∞ ∫ Σ ∗ s ∩ J − ( S τ ) ∩ J + ( S τ ) J Tµ [ ψ ] n µ Σ ∗ s exists. Lemma 4.2.1 allows us to make the following definitions.
Definition 4.2.4.
Given a solution ψ to (2) on R ≥ arising from smooth initial data along Σ ∗ which arecompactly supported, and τ > −∞ , the energy flux of ψ through I + ≤ τ is defined by ∫ I +≤ τ J Tµ [ ψ ] n µ I + ≐ lim s →∞ ∫ Σ ∗ s ∩ J − ( S τ ) J Tµ [ ψ ] n µ Σ ∗ s . (52) Remark 4.2.9.
There is, of course, great flexibility in the choice of the hypersurfaces S and Σ ∗ , but wewill forgo a systematic treatment of which choice of hypersurfaces leaves the limit (52) unchanged. Since τ < τ implies that J − ( S τ ) ⊂ J − ( S τ ) , it immediately follows that ∫ I +≤ τ J Tµ [ ψ ] n µ I + is an increasingfunction of τ . Thus, we can make the following definition. Definition 4.2.5.
Given a solution ψ to (2) on R ≥ arising from smooth initial data along Σ ∗ which arecompactly supported, the (total) flux of ψ through null infinity I + is defined by ∫ I + J Tµ [ ψ ] n µ I + ≐ lim τ →∞ lim s →∞ ∫ Σ ∗ s ∩ J − ( S τ ) J Tµ [ ψ ] n µ Σ ∗ s ∈ R ≥ ∪ { ∞ } . (53) Remark 4.2.10.
As with Definition 4.2.4, we note that this definition also makes sense for essentially anyspacetime possessing a suitable notion of future null infinity.
Now we observe the following immediate consequence of Definition 4.2.5 and Theorem 3.7.1.
Theorem 4.2.1.
All sufficiently regular solutions ψ to (2) on R ≥ satisfy ∫ I + J Tµ [ ψ ] n µ I + ≤ B ∫ Σ ∗ J Nµ [ ψ ] n µ Σ ∗ . In particular, in the case of smooth compactly supported initial data of Σ ∗ or Σ , the total flux to nullinfinity (53) is finite. Finally, the next proposition establishes the expected connection between the radiation field along nullinfinity with the energy flux to null infinity
Proposition 4.2.2.
Given a solution ψ to (2) on R ≥ arising from smooth initial data along Σ ∗ which arecompactly supported, we have ∫ ( −∞ ,τ ) × S ∣ ∂ τ ϕ ( ∞ , τ, θ, φ )∣ sin θ dτ dθ dφ = ∫ I +≤ τ J Tµ [ ψ ] n µ I + ∀ τ ∈ ( −∞ , ∞ ] . Proof.
First of all, a straightforward computation gives ∫ Σ ∗ s ∩ J − ( S τ ) J Tµ [ ψ ] n µ Σ ∗ s = ∫ Σ ∗ s ∩ J − ( S τ ) ∣ ∂ τ ϕ ∣ sin θ dv dθ dφ + O ( ∫ Σ ∗ s ∩ J − ( S τ ) [∣ ∂ ˜ v ϕ ∣ + ∣ ∇ / ϕ ∣ ] sin θ dv dθ dφ ) as s → ∞ . p ∈ ( , ] ) implies that we can find a(dyadic) sequence { s i } ∞ i = such that lim i →∞ s i = ∞ andlim i →∞ ∫ Σ si ∩ J − ( S τ ) [∣ ∂ ˜ v ϕ ∣ + ∣ ∇ / ϕ ∣ ] sin θ dv dθ dφ = . As in our previous work [24], estimates obtained by exploiting Carter’s separation of the wave equation ( ) will play a fundamental role in our analysis. In this section, we quote a number of results from [53] and [24]concerning the theory of the radial o.d.e ( ) and its relation to ( ) . (In Section 6 to follow, we will thenobtain various refinements of the quantitative o.d.e. estimates of [24] which will be fundamental for ourarguments.)We begin in Section 5.1 by reviewing our relevant formalism based on the Fourier transform of “sufficientlyintegrable” solutions (Definition 5.1.1); the reader should consult [24] for more details.We shall then quote in Section 5.2 some results from [53] concerning the asymptotics of solutions of ( ) ,which in particular allow us to define the special solutions U hor , U inf referred to in the introduction. Westate Proposition 5.2.2, the microlocal version of the energy identity (we will consider more general currentsin Section 6.1 below).The Wronskian W , as well as the reflection R and transmission coefficients T referred to already (togetherwith their dual coefficients ˜ R and ˜ T ), are all defined in Section 5.3, appealing to the real-mode stabilitytheorem of [53]. We then obtain Corollary 5.3.1 which gives that the strict inequality ( ) indeed holds for anysuperradiant frequency and establish a fundamental solution formula for the radial o.d.e. in Proposition 5.3.1.Finally, our separation will allow us to define the “microlocal” radiation fields and fluxes in Section 5.4.(Later, in Section 6.8, these will be related to the radiation fields and degenerate-energy fluxes defined inphysical space.) We begin by recalling the following definition.
Definition 5.1.1.
We say that a smooth function Ψ ∶ ˚ R → C is “sufficiently integrable” if for every j ≥ and r > r + , we have ∑ ≤ j + j + ∣ α ∣≤ j ∫ ∞−∞ ∫ S ∣ ∇ / α ∂ j r ∗ T j Ψ ∣ ∣ r = r sin θ dt dθ dφ < ∞ . Remark 5.1.1.
We note that this definition is in fact weaker than that given in [24].
Remark 5.1.2.
Observe that it follows immediately from Proposition 3.6.2, Theorem 3.7.1 and Remark 3.7.3that any solution to the wave equation arising from smooth compactly supported initial data along Σ issufficiently integrable in the sense of Definition 5.1.1 (cf. Remark 3.6.1). Next, we recall the oblate spheroidal harmonics { S mℓ ( ν, cos θ ) e imφ } mℓ , ν ∈ R , which are the eigenfunctions of the self-adjoint operator P ( ν ) f = − θ ∂∂θ ( sin θ ∂∂θ f ) − ∂ f∂φ θ − ν cos θf The point being that ∫ ∞ ∣ f ( x )∣ x dx < ∞ implies that there exists a sequence { x i } ∞ i = with x i ∈ [ i , i + ] such thatlim i → ∞ f ( x i ) = L ( sin θ dθ dφ ) . We denote the corresponding eigenvalues by λ ( ν ) mℓ ∈ R where m ∈ Z and l ≥ ∣ m ∣ . Thelabeling is uniquely determined by requiring that λ ( ν ) mℓ depends smoothly on ν and setting λ ( ) mℓ = ℓ ( ℓ + ) . These satisfy λ ( ν ) mℓ + ν ≥ ∣ m ∣(∣ m ∣ + ) , (54) λ ( ν ) mℓ + ν ≥ ∣ mν ∣ . (55)Because of the above relations, it is often convenient to work withΛ mℓ ( ν ) ≐ λ mℓ ( ν ) + ν . Let Ψ be sufficiently integrable in the sense of Definition 5.1.1. Then, setting ν = aω , where a is the Kerrparameter, for each ω ∈ R , we decomposeΨ ( t, r, θ, φ ) = √ π ∫ ∞−∞ ∑ mℓ e − iωt Ψ ( aω ) mℓ ( r ) S mℓ ( aω, cos θ ) e imφ dω. The sufficiently integrable assumption implies that for each fixed r , this equality may be interpreted in L t L S . Now define F = ◻ g Ψ . (56)The sufficiently integrable assumption implies that we may define the coefficients ( ρ F ) ( aω ) mℓ ( r ) as above(recall that ρ = r + a cos θ ).Carter’s formal separation [10] of the wave operator yields: Proposition 5.1.1.
Let Ψ be sufficiently integrable in the sense of Definition 5.1.1, and let F be defined by ( ) . Then ∆ ddr ⎛⎝ ∆ d Ψ ( aω ) mℓ dr ⎞⎠ + ( a m + ( r + a ) ω − M raωm − ∆Λ mℓ ) Ψ ( aω ) mℓ = ∆ ( ρ F ) ( aω ) mℓ . (57) Note that the sufficiently integrable assumption allows us to interpret this equality for each r in L ω l mℓ . Remark 5.1.3.
It will turn out to suffice that we study smooth smooth solutions to the o.d.e. (57). See thediscussion in Definition 5.4.1.
Using the definition ( ) of r ∗ and setting u ( aω ) mℓ ( r ) = ( r + a ) / Ψ ( aω ) mℓ ( r ) , (58) H ( aω ) mℓ ( r ) = ∆ ( ρ F ) ( aω ) mℓ ( r )( r + a ) / , (59)we obtain d ( dr ∗ ) u ( aω ) mℓ + ( ω − V ( aω ) mℓ ( r )) u = H ( aω ) mℓ , (60)where V ( aω ) mℓ ( r ) = M ramω − a m + ∆Λ mℓ ( r + a ) + ∆ ( r − M r + a )( r + a ) − r ( r + a ) . (61)We will often refer to (60) as the “radial o.d.e.”As in [24], we shall often suppress the dependence of u , H and V on aω , m , ℓ in our notation. We willalso use the notation ′ = ddr ∗ . (62)Note that r ′ = ∆ r + a . See Proposition B.1 of [52] for a proof that this does indeed uniquely determine { λ ( ν ) mℓ } . .2 Asymptotic analysis of the radial o.d.e. In this section we will collect various facts concerning the asymptotic analysis of the radial o.d.e. (60). Inview of our applications and Remark 5.1.3, all results stated will concern smooth solutions. We will omitproofs as the material is standard (see, e.g., [45]).
Proposition 5.2.1.
Fix parameters ( ω, m, ℓ ) ∈ R × Z × Z ≥∣ m ∣ with ω ≠ and ω ≠ ω + m , and let u be a smoothsolution of the radial o.d.e. (60) u ′′ + ( ω − V ) u = H, where H ( r ) smoothly extends to r = r + and vanishes for large r (of course, by using the relation ddr = r + a ∆ ddr ∗ ,the smoothness condition at r = r + can be translated to a condition on the limits of d k Hd ( r ∗ ) k as r ∗ → −∞ ).Then there exist unique complex numbers a H + , a H − , a I + , and a I − , depending on u , such that u = a I + e iωr ∗ + a I − e − iωr ∗ + O ( r − ) as r → ∞ , (63) u = a H + e − i ( ω − ω + m ) r ∗ + a H − e i ( ω − ω + m ) r ∗ + O ( r − r + ) as r → r + . (64) Here the O ( r − ) and O ( r − r + ) are both preserved upon differentiation in r ∗ . Next, we turn to the “microlocal energy identity”.
Proposition 5.2.2.
Fix parameters ( ω, m, ℓ ) ∈ R × Z × Z ≥∣ m ∣ with ω ≠ and ω ≠ ω + m , and let u be a smoothsolution of the radial o.d.e. (60) with H ( r ∗ ) compactly supported in r ∗ . Then, we have ω ∣ a I + ∣ − ω ∣ a I − ∣ + ω ( ω − ω + m )∣ a H + ∣ − ω ( ω − ω + m )∣ a H − ∣ = ω ∫ ∞−∞ Im ( Hu ) dr ∗ . Proof.
We recall the microlocal energy current from [24]: Q T ≐ ω Im ( u ′ u ) , which satisfies ( Q T ) ′ = ω Im ( Hu ) . (The above is of course the most basic energy current associated to ( ) . We will discuss this and severalother currents in Section 6.1). The proposition then follows immediately from the fundamental theorem ofcalculus and the expansions (63) and (64).It will be useful to introduce the following definitions. Definition 5.2.1.
Let ( ω, m, ℓ ) ∈ R × Z × Z ≥∣ m ∣ . Then we define U hor ( r ∗ , ω, m, l ) to be the unique functionsatisfying1. U ′′ hor + ( ω − V ) U hor = .2. U hor ∼ e − i ( ω − ω + m ) r ∗ near r ∗ = −∞ . ∣ U hor ( −∞ )∣ = . Remark 5.2.1.
Note that this definition makes sense even when ω − ω + m = or ω = ; see, e.g., thediscussion in Appendix C.1 of [52]. Remark 5.2.2.
The physical space interpretation of U hor is that e − itω e imφ S mℓ ( θ ) U hor ( r ∗ ) corresponds toan amplitude normalised solution of the wave equation “frequency localised” to ( ω, m, ℓ ) , with a vanishingenergy flux along H − and a finite energy flux on any compact subset of H + . Definition 5.2.2.
For ω ≠ , define U inf ( r ∗ , ω, m, l ) to be the unique function satisfying1. U ′′ inf + ( ω − V ) U inf = . More precisely, the requirement is that U hor e i ( ω − ω + m ) r ∗ extends to r = r + as a smooth function of r . . U inf ∼ e iωr ∗ near r ∗ = ∞ . ∣ U inf ( ∞ )∣ = . Remark 5.2.3.
The physical space interpretation of U inf is that e − itω e imφ S mℓ ( θ ) U inf ( r ∗ ) corresponds to aan amplitude normalised solution of the wave equation “frequency localised” to ( ω, m, ℓ ) , with a vanishingenergy flux along I − and a finite energy flux on any compact subset of I + . Remark 5.2.4.
When H = , by exploiting the linear independence of the pairs { U hor , U hor } and { U inf , U inf } ,one may easily check that expansions (63) and (64) may be written as the identities u = a I + U inf + a I − U inf ,u = a H + U hor + a H − U hor . Cf. footnote [3]
Proposition 5.2.3.
The constructions of U hor and U inf imply that for each k ≥ , ∣∣ d k d ( r ∗ ) k U hor ∣∣ L ∞ r ∗ ≤ B ( ω, m, ℓ, k ) , and d k d ( r ∗ ) k U hor depends analytically on ω . Similarly, if we additionally assume that ω ≠ , we also have ∣∣ d k d ( r ∗ ) k U inf ∣∣ L ∞ r ∗ ≤ B ( ω, m, ℓ, k ) , and d k d ( r ∗ ) k U inf depends analytically on ω ∈ R ∖ { } . Definition 5.3.1.
For ω ≠ , we define W ( ω, m, ℓ ) to be the Wronskian of U hor and U inf : W ≐ U ′ inf U hor − U inf U ′ hor . Remark 5.3.1.
Note that one may easily check that W does not depend on r ∗ and vanishes if and only if U hor and U inf are linearly dependent. In [53], the following was shown:
Theorem 5.3.1. [53] For all ( ω, m, ℓ ) ∈ R × Z × Z ≥∣ m ∣ with ω ≠ we have W ( ω, m, ℓ ) ≠ , and thus the functions U hor and U inf are linearly independent. The non-vanishing of the Wronskian will allow us to define the reflection and transmission coefficients.First we need the following lemma which follows immediately from Remark 5.3.1 and the non-vanishing ofthe Wronskian.
Lemma 5.3.1.
For ω ≠ and ω ≠ ω + m , there exists a unique set of complex numbers R ( ω, m, ℓ ) , ˜ R ( ω, m, ℓ ) , T ( ω, m, ℓ ) and ˜ T ( ω, m, ℓ ) which satisfy T − i ( ω − ω + m ) U hor = R iω U inf + U inf iω , (65)˜ T iω U inf = ˜ R − i ( ω − ω + m ) U hor + U hor − i ( ω − ω + m ) , (66) More precisely, this means that U inf exhibits a (generally divergent) asymptotic expansion U inf = e iωr ∗ ∑ ∞ i = A i r i as r → ∞ . Definition 5.3.2.
The complex numbers R and ˜ R are called the reflection coefficients, and T and ˜ T arecalled the transmission coefficients. Remark 5.3.2.
If one considers a solution to the wave equation which is “sourced” with a flux along I − equal to and no energy along H − and which is furthermore approximately localised to the frequency ( ω, m, ℓ ) , then R measures the amount of energy “reflected” back to future null infinity I + , and T measuresthe energy “transmitted” to the future event horizon H + . There is a similar interpretation for ˜ R and ˜ T . OurTheorem 9.5.3 will make these interpretations rigorous. Remark 5.3.3.
One often sees the reflection and transmission coefficients R and T defined so that theymeasure the amplitude transmitted to the future event horizon and reflected to future null infinity of a waveof amplitude along I − , see e.g. Section 28 of [11]. However, in the context of scattering theory for finiteenergy solutions, one does not expect to control the radiation fields ψ and φ in L along H ± and I ± , hencean energy normalisation is most natural. Applying Proposition 5.2.2 immediately yields
Corollary 5.3.1.
Fix a frequency triple ( ω, m, ℓ ) which satisfy ω ≠ and ω ≠ ω + m . Then ∣ R ∣ + ωω − ω + m ∣ T ∣ = . In particular, if ω ( ω − ω + m ) < , (67) i.e. the parameters are superradiant, then ∣ R ∣ > . Proof.
For the second statement, it suffices to note that the basic local existence theory for the radialo.d.e. implies that T ≠ H − and I − , for technical reasons they are not always the most convenient way toparameterize solutions to the radial o.d.e. Instead we shall often use the following quantities. Definition 5.3.3.
For ω ≠ and ω ≠ ω + m , we define the complex numbers A I + ( ω, m, ℓ ) , A I − ( ω, m, ℓ ) , A H + ( ω, m, ℓ ) , and A H − ( ω, m, ℓ ) by U hor = A I + e iωr ∗ + A I − e − iωr ∗ + O ( r − ) as r ∗ → ∞ ,U inf = A H + e − i ( ω − ω + m ) r ∗ + A H − e i ( ω − ω + m ) r ∗ + O ( r − r + ) as r → r + . Observe that A I + ( ω, m, ℓ ) , A I − ( ω, m, ℓ ) , A H + ( ω, m, ℓ ) , and A H − ( ω, m, ℓ ) must obey the following con-straints. Lemma 5.3.2. A I + A H + + A I − A H − = , A I + A H + + A I − A H − = , A I + A H − + A I − A H + = , A I − A H + + A H − A I + = . Proof.
We may write U hor = A I + U inf + A I − U inf (68) = A I + ( A H + U hor + A H − U hor ) + A I − ( A H + U hor + A H − U hor )= ( A I + A H + + A I − A H − ) U hor + ( A I + A H − + A I − A H + ) U hor . Similarly, U inf = ( A H + A I + + A I − A H − ) U inf + ( A H + A I − + A I + A H − ) U inf . The lemma follows immediately. 37he following relationships may be easily verified in a similar fashion to Lemma 5.3.2.
Lemma 5.3.3. W = iω A I − , W = i ( ω − ω + m ) A H − , R = − A H + ( A H − ) − , T = − ( ω − ω + m ) ω ( A I − ) − , ˜ R = − A I + ( A I − ) − , ˜ T = − ω ( ω − ω + m ) ( A H − ) − . We close the section with a final remark:
Remark 5.3.4.
By exploiting the underlying analyticity (cf. Corollary 6.5.1), one can in fact define thereflection and transmission coefficients almost everywhere without the mode stability result of [53], quotedhere as Theorem 5.3.1. Given this, we see that Theorem 5.3.1 is equivalent to the statement that the reflectionand transmission coefficients are bounded on any compact set of frequencies, with a bound depending howeveron the set. The fact the reflection and transmission coefficients are uniformly bounded over all frequenciesis the content of Theorem 6.2.2, to be proven in Section 6.2.
We end this section with a final corollary of Theorem 5.3.1 which concerns a fundamental-solution rep-resentation of solutions u of ( ) with vanishing a H − = a I − = Proposition 5.3.1.
Let u be a smooth solution to the radial o.d.e. (60) with a right hand side H such that H ( r ) smoothly extends to r = r + and vanishes for large r , and such that u satisfies a H − = a I − = . Then u isgiven by the following explicit formula: u ( r ∗ ) = W − ⎛⎝ U inf ( r ∗ ) ∫ r ∗ −∞ U hor ( x ∗ ) H ( x ∗ ) dx ∗ + U hor ( r ∗ ) ∫ ∞ r ∗ U inf ( x ∗ ) H ( x ∗ ) dx ∗ ⎞⎠ . Proof.
Given the non-vanishing of the Wronskian (Theorem 5.3.1), this is a trivial computation.
We are now ready to define the microlocal radiation fields. As the name suggests, the definition of themicrolocal radiation fields relies on the Fourier transform; hence, we will only be able to define the microlocalradiation fields for a solution ψ if it is defined on all of ˚ R , not just ˚ R ≥ . Definition 5.4.1.
For all solutions ψ to (2) on ˚ R , which are sufficiently integrable in the sense of Defini-tion 5.1.1, we may apply Carter’s separation to ψ and define the corresponding function u . An easy argument(one can slightly modify the proof of Lemma 5.4.1 of [24]) implies that for almost every ω and every ( m, ℓ ) , u will be a smooth solution to the radial o.d.e. (60) with H = . In particular, we may apply Proposition 5.2.1and easily show that the corresponding a I ± ( ω, m, ℓ ) and a H ± ( ω, m, ℓ ) are measurable functions of ( ω, m, ℓ ) .The microlocal radiation field along I ± associated to ψ is then defined almost everywhere by themeasurable function a I ± ( ω, m, ℓ ) ∶ R × Z × Z ≥∣ m ∣ → C , and the microlocal radiation field along H ± associated to ψ is defined almost everywhere by the measurablefunction a H ± ( ω, m, ℓ ) ∶ R × Z × Z ≥∣ m ∣ → C . We also have the corresponding total fluxes.
Definition 5.4.2.
For all solutions ψ to (2) on ˚ R , which are sufficiently integrable in the sense of Defini-tion 5.1.1, the total microlocal energy flux through I ± associated to ψ is given by ∫ ∞−∞ ∑ mℓ ω ∣ a I ± ∣ dω ∈ R ≥ ∪ { ∞ } , and the total microlocal (degenerate) energy flux through H ± associated to ψ is given by ∫ ∞−∞ ∑ mℓ ( ω − ω + m ) ∣ a H ± ∣ dω ∈ R ≥ ∪ { ∞ } . I + and the degenerate K -energyflux to H + defined previously in Sections 4.2.2 and 4.1.2, respectively. In this section we will produce estimates for the radial o.d.e. (60) and given some useful applications. Theseestimates are refinements of estimates originally proven in [24].In Section 6.1 we review the separated current template from [23] and [24]. These currents form theessential ingredients for all of the o.d.e. estimates of this section.In Section 6.2 we start by proving Theorem 6.2.1 which is a general estimate for solutions to the ra-dial o.d.e. with a vanishing right hand side; the proof of Theorem 6.2.1 will heavily rely on Theorem8.1 from [24]. As a corollary we will obtain the uniform boundedness of all reflexion and transmissioncoefficients (Thereom 6.2.2). This gives in particular
Theorem 11 of Section 2.4.1. We will also obtain aWronskian bound (Proposition 6.2.1) which will be used in Section 7.In Section 6.3 we will prove Proposition 6.3.1 which gives asymptotic control of U hor in the superradiantregime as r → r + independent of the frequency parameters. Proposition 6.3.1 plays an important role inSection 7. The proof of Proposition 6.3.1 will require us to quote a special case of Theorem 8.1 from [24](here given as Theorem 6.3.1).Next, using closely related ideas, in Section 6.4 we will prove Proposition 6.4.1 which states that for fixed ω and m , the large- ℓ limit of T must vanish. As a corollary, we deduce that lim ℓ →∞ ∣ R ∣ = m and ℓ , the reflectioncoefficient R is not identically 0 as a function of ω . Using analyticity of R , one corollary will be that R canonly vanish at isolated points.In Section 6.6 we will interpret the weighted r p hierarchy of estimates of [18] (given previously as Propo-sition 3.8.1 of Section 3.8) directly at the level of the o.d.e. ( ) . The main result is Proposition 6.6.1. Wewill then use this in Section 6.7 to give a quantitative estimate on the rate of convergence of the microlo-cal radiation field (Proposition 6.7.1). Using these results, in Section 6.8, we will succeed in relating themicrolocal radiation fields of Section 5.4 with the physical-space definitions given previously Section 4. In this section we will recall the separated current template from [23] and [24]. All of our o.d.e. estimateswill be based on suitable combinations of these currents.
Proposition 6.1.1.
Fix parameters ( ω, m, ℓ ) ∈ R × Z × Z ≥∣ m ∣ with ω ≠ , and let u be a smooth solution ofthe radial o.d.e. (60) u ′′ + ( ω − V ) u = H. Let h ( r ∗ ) be a C function, y ( r ∗ ) be a C function and z ( r ) a C function of r . Set ˜ V = V − V ∣ r = r + . Thenwe define the Ϙ h current Ϙ h [ u ] ≐ h Re ( u ′ ¯ u ) − h ′ ∣ u ∣ , the ϟ y current ϟ y [ u ] ≐ y (∣ u ′ ∣ + ( ω − V ) ∣ u ∣ ) , the microlocal redshift current Q z red [ u ] ≐ z ∣ u ′ + i ( ω − ω + m ) u ∣ − z ˜ V ∣ u ∣ , (69) the microlocal r p current Q zr p [ u ] ≐ z ∣ u ′ − iωu ∣ − zV ∣ u ∣ , (70) the microlocal T -energy current Q T [ u ] ≐ ω Im ( u ′ u ) (71)39 nd the microlocal K -energy current Q K [ u ] ≐ ( ω − ω + m ) Im ( u ′ u ) . (72) We have ( Ϙ h [ u ]) ′ = h ( ∣ u ′ ∣ + ( V − ω )∣ u ∣ ) − h ′′ ∣ u ∣ + h Re ( u ¯ H ) , (73) ( ϟ y [ u ]) ′ = y ′ (∣ u ′ ∣ + ( ω − V ) ∣ u ∣ ) − yV ′ ∣ u ∣ + y Re ( u ′ H ) , (74) ( Q z red [ u ]) ′ = z ′ ∣ u ′ + i ( ω − ω + m ) u ∣ − ( z ˜ V ) ′ ∣ u ∣ + z Re ( Hu ′ + i ( ω − ω + m ) u ) , (75) ( Q zr p [ u ]) ′ = z ′ ∣ u ′ − iωu ∣ − ( zV ) ′ ∣ u ∣ + z Re ( Hu ′ − iωu ) , (76) ( Q T [ u ]) ′ = ω Im ( Hu ) , ( Q K [ u ]) ′ = ( ω − ω + m ) Im ( Hu ) . (77)The identities above follow by direct computation. Note that we have already used the Q T current (71)in Proposition 5.2.2. Remark 6.1.1.
Note that the microlocal r p current appears for the first time in this paper. The reader mayfind it illuminating to compare (69) with (70). Recall that the microlocal radiation fields a I ± and a H ± were defined in Definition 5.4.1. With this notation,in our previous work [24], estimates for the radial o.d.e. (60), in all frequency ranges, with a non-zero righthand side H and u satisfying a H − = a I − = a H − = a I − =
0. We will thus prove this latter theorem in thepresent section, referring to constructions in our [24]. We will close the section with two corollaries ofTheorem 6.2.1: Theorem 6.2.2 which gives the boundedness of the reflection and transmission coefficientsand Proposition 6.2.1 which gives a uniform bound on the Wronskian.
We will prove here the following variant of Theorem 8.1 of [24] which applies to solutions of the homogeneous o.d.e. ( ) (with H =
0) but allows general asymptotics a H ± ≠ a I ± ≠ Theorem 6.2.1.
There exist parameters s − and s + satisfying r + < M − s − < M + s + < ∞ such that forall −∞ < R ∗− < R ∗+ < ∞ , the following is true. Given ( ω, m, ℓ ) satisfying ω ≠ and ω ≠ ω + m , there exists aparameter r trap ( ω, m, ℓ ) with r trap = or r trap ∈ [ M − s − , M + s + ] , such that for all smooth solutions u to the radial o.d.e. (60) with vanishing right hand side H = , ( ω − ω + m ) ∣ a H − ∣ + ω ∣ a I − ∣ + ∫ R ∗+ R ∗− [ ∣ u ′ ∣ + (( − r trap r − ) ( ω + Λ ) + ) ∣ u ∣ ] dr ∗ (78) ≤ B ( R ∗− , R ∗+ ) [( ω − ω + m ) ∣ a H + ∣ + ω ∣ a I + ∣ ] . Remark 6.2.1.
Recall that the degeneration due to the ( − r trap r − ) term arises because of trapping. Seethe discussion in [24]. Remark 6.2.2.
Note that applying the theorem to u yields the same statement with the roles of a H − and a I − interchanged with a H + and a I + . emark 6.2.3. Let us emphasise the even though we require ω ≠ and ω ≠ ω + m in order to define a H ± and a I ± , the constant B ( R ∗− , R ∗+ ) is according to our conventions in Section 3.3 independent of the frequencyparameters and in particular does not blow up in either of the limits ω → or ω → ω + m .Proof. We recall that in [24] we studied solutions to radial o.d.e. (60) with a non-zero right hand side H and u satisfying a H − = a I − =
0, whereas here H = a H ± , a I ± are in general nontrivial.We begin with the important observation that in (version 2! of) [24] in the proof of Theorem 8.1 weused microlocal currents (see Section 6.1) where the functions f , h , etc. were all bounded as r ∗ → ±∞ . Thecurrents which led to the positive bulk of the microlocal ILED statement produced (1) a term associatedto the inhomogeneity H and (2) boundary terms which were proportional to ( ω − ω + m ) ∣ u ( −∞ )∣ and ω ∣ u ( ∞ )∣ . These boundary terms were eventually controlled with suitable applications of (cut-off versionsof) the Q T and Q K currents. In order for this to work, one key point was that under the assumptions a H − = a I − = ∣ Q T ∣ r = r + ∣ = ∣ ω ( ω − ω + m ) ∣ u ( −∞ )∣∣ , Q Tr = ∞ = ω ∣ u ( ∞ )∣ , (79)Q Kr = r + = ( ω − ω + m ) ∣ u ( −∞ )∣ , ∣ Q Kr = ∞ ∣ = ∣ ω ( ω − ω + m ) ∣ u ( ∞ )∣∣ . (80)Now we consider the case of a solution u to the radial o.d.e. (60) with a vanishing right hand side H but where we make no assumption about the vanishing or non-vanishing of a H ± and a I ± . Since all of themultipliers discussed above are bounded, we immediately observe that we may apply all the currents from(version 2 of) [24] to u .The term associated to the inhomogeneity in the resulting identity, of course, now vanishes since H = K and Q T will yield now various boundaryterms each of which will be proportional to one of ( ω − ω + m ) ∣ a H − ∣ , ( ω − ω + m ) ∣ a H + ∣ , ω ∣ a I + ∣ , or ω ∣ a I − ∣ .Furthermore, the term proportional to ( ω − ω + m ) ∣ a H − ∣ will always enter with the opposite sign of the termproportional to ( ω − ω + m ) ∣ a H + ∣ . An analogous relation holds for the terms proportional to ω ∣ a I + ∣ and ω ∣ a I − ∣ . In particular, we do not have (79) and (80) and we cannot hope to prove an estimate with all ofthe microlocal radiation fields on the left hand side . We are thus forced to always put the boundary termsassociated to one of the pairs ( a H − , a I − ) and ( a H + , a I + ) on the right hand side.Given these observations, the following estimate immediately follows from the proof of Theorem 8.1of [24]: ( ω − ω + m ) ∣ a H − ∣ + ω ∣ a I − ∣ + ∫ R ∗+ R ∗− [ ∣ u ′ ∣ + (( − r trap r − ) ( ω + Λ ) + ) ∣ u ∣ ] dr ∗ (81) ≤ B ( R ∗− , R ∗+ )[( ω − ω + m ) ∣ a H + ∣ + ω ∣ a I + ∣ + { ω low ≤∣ ω ∣≤ ω high } ∩ { Λ ≤ ǫ − ω } ∣ a H − ∣ ] , for a parameter r trap ( ω, m, ℓ ) satisfying r trap = r trap ∈ [ M − s − , M + s + ] , where 1 { ω low ≤∣ ω ∣≤ ω high } ∩ { Λ ≤ ǫ − ω } denotes the indicator function for the set F ♭ ≐ {( ω, m, ℓ ) ∶ ω low ≤ ∣ ω ∣ ≤ ω high and { Λ ≤ ǫ − ω }} . The ω low , ω high and ǫ width are fixed constants which arise during the proof of Theorem 8.1. Thus we haveestablished (78) for frequencies ( ω, m, ℓ ) / ∈ F ♭ .In order to finish the proof we need to show that ( ω, m, ℓ ) ∈ F ♭ implies ∣ a H − ∣ ≤ B [( ω − ω + m ) ∣ a H + ∣ + ω ∣ a I + ∣ ] . This is not so surprising of course, because if we could prove such an estimate we would deduce that u had to vanish! u † to the radial o.d.e. defined by u † = ( a I + W − ( iω )) U hor + ( a H + W − ( i ( ω − ω + m ))) U inf , and let a † H + and a † I + denote the microlocal radiation fields of u † .Observe that Lemma 5.3.3 implies that a † H + = a H + , a † I + = a I + . Thus, applying Theorem 5.3.1 and Remark 5.3.1 to u − u † implies that u = u † . Using the explicit definitionof u † , appealing to Theorem 5.3.1 again and using the compactness of F ♭ , we immediately conclude that ∣ a H − ∣ = ∣ a † H − ∣ ≤ B W − [ ω ∣ a I + ∣ + ( ω − ω + m ) ∣ a H + ∣ ] ≤ B [ ω ∣ a I + ∣ + ( ω − ω + m ) ∣ a H + ∣ ] . R and T Applying Theorem 6.2.1 to the solutions U hor and U inf immediately implies that the reflection and trans-mission coefficients are bounded uniformly in ( ω, m, ℓ ) . Theorem 6.2.2.
The reflection and transmission coefficients are uniformly bounded: ∣ R ∣ + ∣ ˜ R ∣ + ∣ T ∣ + ∣ ˜ T ∣ ≤ B. Proof.
We simply note that by the definition of R and T , there exists a solution u to the radial o.d.e. suchthat a H + = T − i ( ω − ω + m ) , a H − = , a I + = R iw , a I − = iω . Theorem 6.2.1 immediately yields ∣ T ∣ + ∣ R ∣ = ( ω − ω + m ) ∣ a H + ∣ + ω ∣ a I + ∣ ≤ B [( ω − ω + m ) ∣ a H − ∣ + ω ∣ a I − ∣ ] ≤ B. An analogous argument applies for ˜ R and ˜ T .The above in particular already yields Theorem 11 of Section 2.4.1. W We close the section with a uniform bound on the Wronskian which will be useful in Section 7.
Proposition 6.2.1.
For all ( ω, m, ℓ ) ∈ R ∖ { } × Z × Z ≥∣ m ∣ we have ω + ω + m ∣ W ∣ ≤ B. Proof.
We first apply Theorem 6.2.1 with u = U hor . In this case we have a H − = ,a H + = ,a I − = A I + ,a I + = A I − = − W iω . In the last equality we have appealed to Lemma 5.3.3. Theorem 6.2.1 then implies ( ω − ω + m ) + ω ∣ A I + ∣ ≤ B ∣ W ∣ . W implies ( ω − ω + m ) ∣ W ∣ ≤ B. (82)Next we apply Theorem 6.2.1 with u = U inf . In this case we have a H − = A H + ,a H + = A H − = − W i ( ω − ω + m ) ,a I − = A I + = ,a I + = A I − = . Again we have appealed to Lemma 5.3.3. Theorem 6.2.1 then implies ( ω − ω + m ) ∣ A H + ∣ + ω ≤ B ∣ W ∣ . Dividing through by W yields ω ∣ W ∣ ≤ B. (83)Since ω + m = ( ω − ω + m − ω ) ≤ B [( ω − ω + m ) + ω ] , it is clear that (82) and (83) conclude the proof. U hor The frequency range defined below will play an important role in our arguments.
Definition 6.3.1.
For every ǫ > we define the set F ( ǫ ) ♯ by F ( ǫ ) ♯ ≐ {( ω, m, ℓ ) ∈ R × Z × Z ≥∣ m ∣ ∶ amω > and ∣ ω ∣ − ∣ ω + m ∣ < ǫ ∣ m ∣ . } Remark 6.3.1.
Observe that if we set ǫ = , then F ( ǫ ) ♯ would exactly correspond to the superradiant frequen-cies (67). When ǫ > is small, then F ( ǫ ) ♯ contains all frequencies which are “close” to being superradiant.These frequencies will later pose the most serious difficulties in the analysis of Section 7. Remark 6.3.2.
Note that for frequencies in F ( ǫ ) ♯ we have Λ ≥ b ( ǫ ) ( + ω ) . In this section, we shall prove
Proposition 6.3.1.
Let E hor be defined by U hor ( r ∗ ) = e − i ( ω − ω + m ) r ∗ + E hor ( r ∗ ) . Then ( ω, m, ℓ ) ∈ F ( ǫ ) ♯ for a sufficiently small ǫ > and Λ sufficiently large imply ∣ E hor ∣ ≤ B ( ǫ ) ∣ W ∣√ Λ √ r − r + , (84) for sufficiently small r − r + . Remark 6.3.3.
Note that the √ Λ factor above represents a “gain of a derivative” over what one wouldexpect to prove if we were not restricting to ( ω, m, ℓ ) ∈ F ( ǫ ) ♯ . This proposition will be of fundamental importance in Section 7. To prove it, we will need again toreturn to our o.d.e theory for ( ) . We begin with some preliminaries reviewing some additional results andnotation from [24]. The proof proper will be contained in Section 6.3.4.43 .3.1 An inhomogeneous ILED in the superradiant regime The following estimate is a special case of Theorem 8.1 from [24]. Theorem 6.3.1. [24] Let ǫ > be sufficiently small and −∞ < R ∗− < R ∗+ < ∞ , then there exists a constant B ( R ∗− , R ∗+ , ǫ ) such that for all smooth solutions u to the radial o.d.e. (60) with a smooth compactly supportedright hand side H , u satisfying a H − = a I − = , and frequencies ( ω, m, ℓ ) ∈ F ( ǫ ) ♯ with Λ sufficiently large, wehave ( ω − ω + m ) ∣ a H + ∣ + ω ∣ a I + ∣ + ∫ R ∗+ R ∗− [∣ u ′ ∣ + Λ ∣ u ∣ ] dr ∗ (85) ≤ B ( R ∗− , R ∗+ , ǫ ) ∫ ∞−∞ (∣ Hu ′ ∣ + √ Λ ∣ Hu ∣) dr ∗ . Remark 6.3.4.
Note that the integrand on the right hand side of ( ) does not degenerate (cf. ( ) below).This is because the ( ǫ -enlarged) superradiant frequency range F ( ǫ ) ♯ is not trapped. See the discussion in [24]regarding the fortuitous disjointness of the difficulties of superradiance and trapping. V in the superradiant regime We recall the following two propositions proved in [24].
Proposition 6.3.2.
Let ( ω, m, ℓ ) ∈ F ( ǫ ) ♯ for a sufficiently small ǫ > . Then there exists a unique r value r max where the potential V of ( ) achieves its maximum. Furthermore, there exists δ > , independent ofthe frequency parameters, such that ( V − ω ) ∣ r ∈ [ r max − δ,r max + δ ] ≥ b Λ . Furthermore, r max is uniformly bounded away from r + and ∞ . Proposition 6.3.3.
Let ( ω, m, ℓ ) ∈ F ( ǫ ) ♯ for a sufficiently small ǫ > , then there exists δ > , independentof the frequency parameters, such that r ∈ [ r + , r + + δ ] ⇒ dVdr ≥ b ( ǫ ) Λ . We begin by applying Theorem 6.2.1 to U hor and refer to Lemma 5.3.3 concerning the Wronskian. We obtain Corollary 6.3.1.
For all frequencies ( ω, m, ℓ ) ∈ F ( ǫ ) ♯ for a sufficiently small ǫ > and sufficiently large Λ ,and for any constants −∞ < R ∗− < R ∗+ < ∞ , we have ( ω − ω + m ) ⋅ + ω ∣ A I + ∣ + ∫ R ∗+ R ∗− [∣ U ′ hor ∣ + Λ ∣ U hor ∣ ] dr ∗ ≤ B ( R ∗− , R ∗+ , ǫ ) ∣ W ∣ . (86)Proposition 6.3.2 allows us to “gain a derivative” in comparison with Corollary 6.3.1 in the followinglemma. Lemma 6.3.1.
There exists r ∗ > −∞ such that for all frequencies ( ω, m, ℓ ) ∈ F ( ǫ ) ♯ with a sufficiently small ǫ > and Λ sufficiently large, and r ∗ < r ∗ , we have ∫ r ∗ r ∗ [∣ U ′ hor ∣ + Λ ∣ U hor ∣ ] dr ∗ ≤ B ( r ∗ , ǫ ) ∣ W ∣ Λ . (87) Note that we have strengthened the statement of Theorem 8.1 in version 2 of [24] with Theorems 6.3.1 and 6.2.1 in mind. roof. Let u be an arbitrary smooth solution to the homogeneous radial o.d.e. (60), with ( ω, m, ℓ ) ∈ F ( ǫ ) ♯ for a sufficiently small ǫ > h be a smooth positive function supported in [ r max − δ, r max + δ ] which is identically 1 within [ r max − δ / , r max + δ / ] . Then set h ≐ Λ˜ h . Using (73), weobtain Λ ∫ r max + δ / r max − δ / [ ∣ u ′ ∣ + Λ ∣ u ∣ ] dr ∗ ≤ ∫ ∞−∞ ( Ϙ h [ u ]) ′ + B ( ǫ ) Λ ∫ r max + δr max − δ ∣ u ∣ (88) = B ( ǫ ) Λ ∫ r max + δr max − δ ∣ u ∣ ≤ B ( ǫ ) [( ω − ω + m ) ∣ a H + ∣ + ω ∣ a I + ∣ ] . In the last line we used Theorem 6.2.1.In particular, applying the estimate (88) to U hor and then appealing to Corollary 6.3.1 implies ∫ r max + δ / r max − δ / [∣ U ′ hor ∣ + Λ ∣ U hor ∣ ] dr ∗ ≤ B ( ǫ ) ∣ W ∣ Λ . (89)Now let χ be a function which is identically 1 on [ r + , r max − δ / ] and identically 0 on [ r max + δ / , ∞ ) , andthen set ˜ u ≐ χU hor . We have ˜ u ′′ + ( ω − V ) ˜ u = χ ′′ U hor + χ ′ U ′ hor ≐ ˜ H, ˜ a H − = ˜ a I − = . Thus, taking r ∗ sufficiently negative, applying Theorem 6.3.1 to ˜ u yields ∫ r ∗ r ∗ [∣ U ′ hor ∣ + Λ ∣ U hor ∣ ] dr ∗ ≤ B ( r ∗ , ǫ ) ∫ r max + δ / r max − δ / [∣ U ′ hor ∣ + Λ ∣ U hor ∣ ] dr ∗ ≤ B ( r ∗ , ǫ ) ∣ W ∣ Λ . In the last line we used the estimate (89).Now we are ready for the following lemma.
Lemma 6.3.2.
There exists a constant ˜ ǫ > such that for all frequencies ( ω, m, ℓ ) ∈ F ( ǫ ) ♯ with a sufficientlysmall ǫ > and Λ sufficiently large, we have ∫ ( r + + ˜ ǫ ) ∗ −∞ ∣ U ′ hor + i ( ω − ω + m ) U hor ∣ ( r − r + ) − dr ∗ ≤ B ( ǫ ) ∣ W ∣ Λ . Proof.
We consider the microlocal redshift current (69) with z ≐ − Λ˜ V χ ( r ) , where χ is a bump function which is identically 1 for r ∈ [ r + , r + + ˜ ǫ ] and 0 for r ∈ [ ǫ, ∞ ) , for a small positiveconstant ˜ ǫ to be determined. We obtain from ( ) the estimate ∫ ( r + + ˜ ǫ ) ∗ −∞ [ z ′ ∣ U ′ hor + i ( ω − ω + m ) U hor ∣ ] dr ∗ (90) ≤ B ( ǫ, ˜ ǫ ) ∫ ( r + + ǫ ) ∗ ( r + + ˜ ǫ ) ∗ [∣ U ′ hor ∣ + Λ ∣ U hor ∣ ] dr ∗ − Q z red ∣ r = r + . If ˜ ǫ > r ∈ ( r + , r + + ˜ ǫ ] implies that z ′ ≥ b ( r − r + ) − .In particular, after fixing a small choice of ˜ ǫ , we may combine (90) and Lemma 6.3.1 to conclude ∫ ( r + + ˜ ǫ ) ∗ −∞ ∣ U ′ hor + i ( ω − ω + m ) U hor ∣ ( r − r + ) − dr ∗ ≤ B ( ǫ ) ∣ W ∣ Λ − BQ z red ∣ r = r + . (91)We conclude the proof by noting that − Q z red ∣ r = r + = − Λ ≤ . .3.4 Proof of Proposition 6.3.1 Finally, Lemma 6.3.2 easily allows us to prove Proposition 6.3.1.
Proof of Proposition 6.3.1.
Let Λ be sufficiently large. Recall the definition ( ) of E hor . It follows that E ′ hor = U ′ hor + i ( ω − ω + m ) U hor . Assuming r − r + sufficiently small, we then have ∣ E hor ( r ∗ )∣ ≤ ∫ r ∗ −∞ ∣ E ′ hor ∣ ds ∗ = ∫ r ∗ −∞ ∣ U ′ hor + i ( ω − ω + m ) U hor ∣ ds ∗ ≤ B ∫ rr + ∣ U ′ hor + i ( ω − ω + m ) U hor ∣ ( s − r + ) − ds ≤ B √ r − r + √ ∫ rr + ∣ U ′ hor + i ( ω − ω + m ) U hor ∣ ( s − r + ) − ds ≤ B √ r − r + √ ∫ r ∗ −∞ ∣ U ′ hor + i ( ω − ω + m ) U hor ∣ ( s − r + ) − ds ∗ ≤ B ( ǫ ) ∣ W ∣√ Λ √ r − r + . ℓ limit of T It is useful to observe that T must vanish in the large- ℓ limit. Proposition 6.4.1.
For each fixed value of ω and m satisfying ω − ω + m ≠ , we have lim ℓ →∞ T ( ω, m, ℓ ) = , lim ℓ →∞ ˜ T ( ω, m, ℓ ) = . Proof.
We will only consider the case of T as the proof for ˜ T is exactly the same.Fix a pair ω and m such that ω − ω + m ≠
0. Next, pick and fix some value of r ∈ ( r + , ∞) . Then, for allsufficiently large ℓ , there will exist a δ > ( V − ω ) ∣ r ∈ [ r − δ,r + δ ] ≥ b Λ . (92)The basic intuition is that for Λ sufficiently large, this large potential barrier will prevent the transmissionsof waves to H + . To make this rigorous, we observe that an examination of the beginning of proof ofLemma 6.3.1 shows that (92) implies that if ℓ is sufficiently large ∫ r + δ / r − δ / [∣ U ′ hor ∣ + Λ ∣ U hor ∣ ] dr ∗ ≤ B Λ ∫ r + δr − δ [∣ U ′ hor ∣ + Λ ∣ U hor ∣ ] dr ∗ . (93)Keeping in mind that Lemma 5.3.3 implies U hor = − R ( ω − ω + m ) ω T e iωr ∗ − ( ω − ω + m ) ω T e − iωr ∗ + O ( r − ) as r → ∞ , an application of Theorem 6.2.1 implies that ∫ r + δr − δ [∣ U ′ hor ∣ + Λ ∣ U hor ∣ ] dr ∗ ≤ B [ + ( ω − ω + m ) ∣ R ∣ ∣ T ∣ ] . (94)46ombining this with (93) implies ∫ r + δ / r − δ / [∣ U ′ hor ∣ + Λ ∣ U hor ∣ ] dr ∗ ≤ B ( ω, m, r ) Λ [ + ( ω − ω + m ) ∣ R ∣ ∣ T ∣ ] . (95)Intuitively, the estimate (95) shows that that U hor must be small near the large potential barrier.We now want to use an energy estimate to show that if U hor is small near the potential barrier, then T must be small. We thus consider the microlocal K -energy current ( ) from Proposition 6.1.1. Now let χ ( r ) denote a cut-off function which is identically 1 for r ∈ [ r + , r − δ / ] and identically 0 for r ∈ [ r + δ / , ∞ ) .Then, keeping (95) and (77) in mind, ( ω − ω + m ) = ∫ ∞−∞ ( χ Q K ) ′ dr ∗ (96) ≤ B ∫ r + δ / r − δ / [∣ U ′ hor ∣ + ∣ U hor ∣ ] dr ∗ ≤ B Λ [ + ( ω − ω + m ) ∣ R ∣ ∣ T ∣ ] . Now we may multiply (96) through by T , divide through by ( ω − ω + m ) , take ℓ → ∞ and apply Theorem 6.2.2to conclude that lim ℓ →∞ T ( ω, m, ℓ ) = . The following corollary follows easily from Proposition 6.4.1.
Corollary 6.4.1.
For each fixed value of ω and m satisfying ω − ω + m ≠ , we have lim ℓ →∞ R ( ω, m, ℓ ) = , lim ℓ →∞ ˜ R ( ω, m, ℓ ) = . Proof.
This follows immediately from Corollary 5.3.1 and Proposition 6.4.1. R The next proposition shows that for any fixed m and ℓ , the reflection coefficient R cannot be identically 0. Proposition 6.5.1.
For each m and ℓ , there exists ω such that R ( ω, m, ℓ ) ≠ and ˜ R ( ω, m, ℓ ) ≠ .Proof. We will only consider the case of R since ˜ R is treated in a similar fashion.Fix a choice of m and ℓ . Then, for the sake of contradiction, assume that R ( ω, m, ℓ ) is identically 0 in ω . We first consider the case when ω + m ≠
0. Then Corollary 5.3.1 implies ωω − ω + m ∣ T ∣ = . Then we get a contradiction by considering any ω such that ω ( ω − ω + m ) < ω + m = R ( ω, m, ℓ ) implies that for each ω , we can construct a (non-zero!) solution u = u ( r ∗ , ω, m, ℓ ) to the radial o.d.e. suchthat u ∼ e − iωr ∗ as r ∗ → −∞ and u ∼ e − iωr ∗ as r ∗ → ∞ . By direct inspection, one finds that the estimates ofSection 8.7.1 of (version 2! of) [24] go through for such a solution (see Remark 8.7.1 at the end of Section8.7.1), and in particular prove that for ω sufficiently small, u must vanish. This contradiction finishes theproof. Corollary 6.5.1.
The reflection coefficients R and ˜ R cannot vanish on an open set of ω .Proof. Standard o.d.e. theory implies that for each fixed m and ℓ , R and ˜ R are analytic in ω ∈ R ∖ { } , and,because Proposition 6.5.1 implies that they are not identically 0, we conclude that they can only vanish atisolated points in ω . 47 .6 The microlocal r p estimate In this section we will establish an analogue of Proposition 3.8.1 for the function u , using the microlocal r p current ( ) .The following proposition is the microlocal analogue of Proposition 3.8.1. Proposition 6.6.1.
Fix parameters ( ω, m, ℓ ) ∈ R × Z × Z ≥∣ m ∣ with ω ≠ , and let u be a smooth solution ofthe radial o.d.e. (60) u ′′ + ( ω − V ) u = H, such that H ( r ∗ ) is compactly supported and the constant a I − from Proposition 5.2.1 vanishes. Then, for all p ∈ [ , ] and sufficiently large R (independent of ( ω, m, ℓ ) !), ∫ ∞ R + [ r p − ∣ u ′ − iωu ∣ + [( − p ) r p − Λ + r p − ] ∣ u ∣ ] dr ∗ ≤ B ∫ R + R r p ( + ω + Λ ) ∣ u ∣ dr ∗ + B ∫ ∞ R ∣ H ∣ [ r p ∣ u ′ − iωu ∣ + ∣ u ′ ∣] dr ∗ . In the case p = , then we may moreover add the term Λ ∣ a I + ∣ to the left hand side.Proof. We observing that a further asymptotic analysis (see Appendix A of [53]) of u yields u = a I + e iωr ∗ ( + Cr + O ( r − )) as r → ∞ , where C ∈ C is a constant independent of u but depending on ( ω, m, ℓ ) . In particular, we find that u ′ − iωu = O ( r − ) as r → ∞ . Next, let R < ∞ be sufficiently large and let z = χr p where p ∈ [ , ] and χ is a cut-off function which ismonotonically increasing, identically 0 for r ≤ R , and identically 1 for r ≥ R +
1. Keeping in mind that V = Λ r + M [ − ( Λ − amω )] r + O ( r − ) as r → ∞ , we find that Q zr p [ u ]∣ r = ∞ = p ∈ [ , ) ,Q zr p [ u ]∣ r = ∞ = − Λ ∣ a I + ∣ if p = . Furthermore, recalling that by ( ) we haveΛ ≥ ∣ amω ∣ , one may easily check that r sufficiently large and p ∈ [ , ] imply − ( r p V ) ′ ≥ b [( − p ) r p − Λ + r p − ] − B Λ r p − . Thus, applying the fundamental theorem of calculus to the identity ( ) yields ∫ ∞ R + [ pr p − ∣ u ′ − iωu ∣ + [( − p ) r p − Λ + r p − ] ∣ u ∣ ] (97) ≤ B ∫ R + R r p ( + ω + Λ ) ∣ u ∣ dr ∗ + B ∫ ∞ R r p ∣ H ∣ ∣ u ′ − iωu ∣ dr ∗ + B ∫ ∞ R + Λ r p − ∣ u ∣ dr ∗ , where in the case p = ∣ a I + ∣ to the left hand side.It remains to estimate the last term on the right hand side of (97). (Note that for any p ∈ [ , ) we couldtake R sufficiently large depending on p and absorb the troublesome term onto the left hand side. However,this cannot work in the case p = χ be a cut-off which is identically 0 for r ∈ [ r + , R ] and identically1 on [ R + , ∞ ) . Then, taking R sufficiently large and applying the fundamental theorem of calculus to theidentity (74) with y = ˜ χ easily yields ∫ ∞ R + Λ r ∣ u ∣ dr ∗ ≤ B ∫ R + R ( + ω + Λ ) ∣ u ∣ dr ∗ + B ∫ ∞ R ∣ H ∣ ∣ u ′ ∣ dr ∗ , and thus concludes the proof. 48 .7 A quantitative estimate on the rate of convergence of the microlocal radi-ation field The following proposition will be used in Section 6.8 below and also in Section 9.1.2.
Proposition 6.7.1.
Fix parameters ( ω, m, ℓ ) ∈ R × Z × Z ≥∣ m ∣ with ω ≠ , and let u be a smooth solution of theradial o.d.e. (60) with a right hand side H vanishing for sufficiently large r ∗ , such that the constant a I − fromProposition 5.2.1 vanishes. Then there exists a sufficiently large constant R , independent of the frequencyparameters, such that for every ǫ > ∣ ω ( u − e iωr ∗ a I + ) ∣ r = r ∣ ≤ B ( ǫ ) r − + ǫ ∫ R + R ( + Λ ) ∣ u ∣ dr ∗ , ∀ r ≥ R. Remark 6.7.1.
Note that if we allowed the constants B and R to depend on the frequency parameters,standard o.d.e. theory (e.g., see [45]) would allow one to replace − + ǫ with the sharp exponent − . Remark 6.7.2.
As far as the applications of Proposition 6.7.1 are concerned the only thing important aboutthe Λ dependence is that it is polynomial.Proof. Set E ≐ u − e iωr ∗ a I + . Recall that standard o.d.e. theory implies that E = O ( r − ) as r → ∞ (where the implied constant maydepend on ( ω, m, ℓ ) ).Next, we observe that one may find a sufficiently large R < ∞ not depending on the frequency parametersso that r ≥ R implies ∣ V ∣ ≤ B ( Λ r + r ) .A simple computation gives E ′′ + ω E = V E + e iωr ∗ a I + V. Variation of parameters then implies E ( r ) = − ∫ ∞ r ∗ ( e iω ( r ∗ − s ∗ ) − e − iω ( r ∗ − s ∗ ) iω ) ( V ( s ) E ( s ) + e iωs ∗ a I + V ( s )) ds ∗ . In particular, ∣ ωE ( r )∣ ≤ B ⎡ ⎢⎢⎢⎢⎣( ∫ ∞ r ∗ ( + Λ ) ∣ E ( s )∣ s ds ∗ ) + ∣ a I + ∣ ( Λ r + r )⎤ ⎥⎥⎥⎥⎦ . (98)Now we consider the two terms on the right hand side of (98) separately. For the first term, we begin byobserving that e iωr ∗ ( e − iωr ∗ E ) ′ = u ′ − iωu . Keeping this in mind, we have ( ∫ ∞ r ∗ ∣ E ( s )∣ s ds ∗ ) ≤ B ( ǫ ) ∫ ∞ r ∗ ∣ E ( s )∣ s − ǫ ds ∗ (99) = B ( ǫ ) ∫ ∞ r ∗ ∣ e − iωs ∗ E ( s )∣ s − ǫ ds ∗ ≤ B ( ǫ ) ∫ ∞ r ∗ ∣( e − iωs ∗ E ( s )) ′ ∣ s − ǫ ds ∗ ≤ B ( ǫ ) r − + ǫ ∫ ∞ r ∗ s ∣ u ′ − iωu ∣ ds ≤ ( + Λ ) B ( ǫ ) r − + ǫ ∫ R + R ∣ u ∣ dr ∗ . More concretely, we define a function ˜ E by the formula given, note that ( E − ˜ E ) ′′ + ω ( E − ˜ E ) =
0, observe the trivial factthat any solution to g ′′ + ω g = g = O ( r − ) must be identically 0, and deduce that E = ˜ E .
49n the third inequality we used a standard Hardy inequality, and in the final inequality we appealed toProposition 6.6.1.For the second term in (98), we first note that Proposition 6.6.1 with p = ∣ a I + ∣ r ≤ B Λ r − ∫ R + R ∣ u ∣ dr ∗ . (100)For the lower order term we use ∣ a I + ∣ r ≤ B ∫ ∞ r ∗ ∣ a I + ∣ s ds ∗ ≤ B ∫ ∞ r ∗ [ ∣ u ∣ + ∣ E ∣ s ] ds ∗ ≤ B ( + Λ ) r − ∫ R + R ∣ u ∣ dr ∗ . (101)In the last inequality we used the estimates done in (99) and Proposition 6.6.1.Combining (98), (99), (100), and (101) concludes the proof. Definition 5.4.1 is motivated by the following propositions.
Proposition 6.8.1.
For all smooth solutions ψ to (2) on D arising from smooth compactly supported dataalong Σ , let a I + ( ω, m, ℓ ) be the microlocal radiation field along I + . Then ωa I + ∈ L ω l mℓ and ∂ τ ϕ ( τ, ∞ , θ, φ ) = √ π ∫ ∞−∞ ∑ mℓ ωe − iωτ a I + ( ω, m, ℓ ) S mℓ ( aω, cos θ ) e imφ dω. Recall that ϕ ( τ, ∞ , θ, φ ) denotes the radiation field of ψ along future null infinity I + .Proof. First of all, as noted in Remark 5.1.2, ψ is sufficiently integrable in the sense of Definition 5.1.1 andthus the microlocal radiation field a I + is a well defined measurable function.Now, define ψ + ≐ χ ( t ∗ ) ψ,ψ − ≐ ( − χ ( t ∗ )) ψ, where χ ( x ) is a cutoff function which is identically 0 for x ≤ x ≥ t ∗ will not be a problem).We shall denote ( r + a ) / ψ , ( r + a ) / ψ + , and ( r + a ) / ψ − by ϕ , ϕ + , and ϕ − respectively.The following facts are immediate consequences of ψ ’s compact support along Σ and the finite speed ofpropagation.1. ϕ + ∣ I + = ϕ ∣ I + .2. ϕ − ∣ I − = ϕ ∣ I − .3. ψ = ψ + + ψ − .4. ◻ g ψ + vanishes for large r .5. ◻ g ψ − vanishes for large r .Next, we observe the following immediate consequence of Proposition 3.8.1 with p = ψ ’s initial data implies that the norms on the right sides of the estimatesof Theorem 3.7.1 are finite and thus the right hand side of the estimate of Proposition 3.8.1 is uniformlybounded as τ → ∞ ): ∫ ∞−∞ ∫ r ≥ R ∫ π ∫ π ∣( ∂ t + ∂ r ∗ ) (( r + a ) / ψ + )∣ sin θ dt dr dθ dφ < ∞ , (102)50here R is sufficiently large. Applying the discrete isometry ( t, φ ) ↦ ( − t, − φ ) and repeating the aboveargument implies ∫ ∞−∞ ∫ r ≥ R ∫ π ∫ π ∣( ∂ t − ∂ r ∗ ) (( r + a ) / ψ − )∣ sin θ dt dr dθ dφ < ∞ , (103)where R is sufficiently large.Noting that ψ ± are easily seen to be sufficiently integrable in the sense of Definition 5.1.1, we may applyCarter’s separation to ψ + and ψ − and define u + and u − . Now we observe that Plancherel and (102) are easilyseen to imply the existence of a dyadic sequence { r n } such thatlim n →∞ ∫ ∞−∞ ∑ mℓ ∣ u ′+ − iωu + ∣ ∣ r = r n = . In turn, upon passing to a subsequence, this implies that for almost every ω and every ( m, ℓ ) we havelim n →∞ ∣ u ′+ − iωu + ∣ ∣ r = r n = . Finally, combining this with Proposition 5.2.1 clearly implies that u + ∼ e iωr ∗ as r → ∞ . Similarly, we observethat u − ∼ e − iωr ∗ as r → ∞ . Since we clearly have u = u + + u − , we finally conclude that for almost every ω and each ( m, ℓ ) we have u + = a I + e iωr ∗ + O ( r − ) as r → ∞ , (104) u − = a I − e − iωr ∗ + O ( r − ) as r → ∞ . (105)Observe that the Fourier transform in τ of ∂ τ ϕ + is given by ( e − iωr ∗ + O ( ωr )) ωu + as r → ∞ . Furthermore,observe that Theorem 3.7.2 and Plancherel are easily seen to imply that ∫ ∞−∞ ∑ mℓ ∫ R + R ( + Λ + ω ) ∣ u + ∣ dr ∗ dω < ∞ . Thus we may apply Proposition 6.7.1 to conclude that ωe − iωr ∗ u + and ω e iωr ∗ u + converge in L ω l mℓ as r → ∞ to ωa I + and ω a I + respectively. In particular, the Fourier transform in τ of ∂ τ ϕ + converges to ωa I + in L ω l mℓ as r → ∞ . Plancherel then implies that any subsequence { ∂ τ ϕ + } r n is Cauchy in L R × S . Now, recalling that ∂ τ ϕ + ( τ, r, θ, φ ) converges to ∂ τ ϕ + ( τ, ∞ , θ, φ ) in L ∞ R × S (see Remark 4.2.3 and keep in mind that the finitespeed of propagation implies ϕ + is only supported along τ ≥ τ for some τ ∈ R ), we conclude, using theuniqueness of L p limits, that ∂ τ ϕ + ( τ, r, θ, φ ) converges to ∂ τ ϕ + ( τ, ∞ , θ, φ ) in L R × S . Finally, continuity ofthe Fourier transform on L implies that ∂ τ ϕ + ( τ, ∞ , θ, φ ) = √ π ∫ ∞−∞ ∑ mℓ ωe − iωτ a I + ( ω, m, ℓ ) S mℓ ( aω, cos θ ) e imφ dω. To conclude the proof we simply recall that ϕ + ∣ I + = ϕ ∣ I + .Now we turn to the horizon flux. Proposition 6.8.2.
For all solutions ψ to (2) on D arising from smooth compactly supported initial dataalong Σ , let a H + ( ω, m, ℓ ) be the microlocal radiation field along H + . Then ( ω − ω + m ) a H + ∈ L ω l mℓ and Kψ ( t ∗ , r + , θ, φ ) = √ M πr + ∫ ∞−∞ ∑ mℓ ( ω − ω + m ) e − iωt ∗ a H + ( ω, m, ℓ ) S mℓ ( aω, cos θ ) e imφ ∗ dω. (106) Proof.
First of all, as noted in Remark 5.1.2, ψ is sufficiently integrable in the sense of Definition 5.1.1 andthus the microlocal radiation field a H + is a well defined measurable function.We first consider the case where the initial data for ψ are in fact compactly supported along ˚Σ. Wemay then proceed in a completely analogous manner to the proof of Proposition 6.8.1. We note that theargument is in fact simpler since we will be able to rely directly on Theorem 3.7.1 instead of developing ananalogue of Theorem 3.8.1 near the horizon. 51efine ψ + ≐ χ ( t ) ψ,ψ − ≐ ( − χ ( t )) ψ, where χ ( x ) is a cutoff function which is identically 0 for x ≤ x ≥ ψ ’s compact support away from the bifurcate sphere B and the finite speed of propagation.1. ψ + ∣ H + = ψ ∣ H + .2. ψ − ∣ H − = ψ ∣ H − .3. ψ = ψ + + ψ − .4. ◻ g ψ + vanishes for small r − r + .5. ◻ g ψ − vanishes for small r − r + .Recalling that the smooth extension of ∂ r ∗ to H + ∪ H − satisfies ∂ r ∗ ∣ H + = K and ∂ r ∗ ∣ H − = − K , we see thatTheorem 3.7.2 immediately implieslim r → r + ( ∂ r ∗ − K ) ψ + = L t ∗ ,θ ∗ ,φ ∗ ( sin θ ∗ dt ∗ dθ ∗ dφ ∗ ) , (107)lim r → r + ( ∂ r ∗ + K ) ψ − = L ∗ t, ∗ θ, ∗ φ ( sin ∗ θ d ∗ t d ∗ θ d ∗ φ ) . (108)Appealing to Theorems 3.7.1 and 3.7.2, we may apply Carter’s separation to ψ + and ψ − and define u + and u − . Since we clearly have u = u + + u − , Proposition 5.2.1, (107), (108) and a similar argument as we usednear I + (note that the convergence of ψ + to its radiation field along the horizon in both L ∞ R ≥ × S and L R ≥ × S follows immediately from the fundamental theorem of calculus and Theorem 3.7.2) imply that for almostevery ω and for each ( m, ℓ ) , we have u + = a H + e − i ( ω − ω + m ) r ∗ + O ( r − r + ) as r → r + , (109) u − = a H − e i ( ω − ω + m ) r ∗ + O ( r − r + ) as r → r + . (110)Now, we note that Theorem 3.7.1 is easily seen to imply that ψ + ∣ r = s → ψ + ∣ r = r + as s → r + in L t ∗ ,θ ∗ ,φ ∗ .Arguing in a similar fashion as in the proof of Proposition 6.8.1 we conclude that a H + is in L ω l mℓ and ψ ( t ∗ , r + , θ, φ ) = √ M πr + ∫ ∞−∞ ∑ mℓ e − iωt ∗ a H + ( ω, m, ℓ ) S mℓ ( aω, cos θ ) e imφ ∗ dω. (111)Now we consider the case where the support of ψ may contain the bifurcate sphere B . We begin bycommuting (2) with K and conclude that ◻ g ( Kψ ) =
0. Then we recall that Kψ vanishes on the bifurcatesphere in view of (40). Now, let χ ( x ) be a smooth function which is identically 0 for x ∈ ( −∞ , ] andidentically 1 for x ∈ [ , ∞ ) . Set χ ǫ ( x ) ≐ χ ( xǫ ) , and, recalling the coordinate system ( U + , V + , θ, φ ) near thebifurcate sphere which was introduced in Section 3.2, let ( Kψ ) ǫ denote the solution to the wave equationwith the initial data of χ ǫ ( V + ) Kψ . Using that Kψ is smooth and vanishes at the bifurcate sphere, one mayeasily verify that lim ǫ → ∫ ˚Σ J Nµ [ Kψ − ( Kψ ) ǫ ] n µ ˚Σ = . Theorem 3.7.1 then implies that lim ǫ → ∫ H + J Nµ [ Kψ − ( Kψ ) ǫ ] n µ H + = . Since ( Kψ ) ǫ is compactly supported away from the bifurcate sphere, ( Kψ ) ǫ ( t ∗ , r + , θ, φ ) = √ M πr + ∫ ∞−∞ ∑ mℓ e − iωt ∗ a ( K ) ǫ, H + ( ω, m, ℓ ) S mℓ ( aω, cos θ ) e imφ ∗ dω, (112)52here a ( K ) ǫ, H + is the microlocal radiation field along H + for ( Kψ ) ǫ (observe that ( Kψ ) ǫ is easily seen to besufficiently integrable in the sense of Definition 5.1.1).In order to finish the proof, we just need to establish that a ( K ) ǫ, H + → ( ω − ω + m ) a H + in L ω l mℓ as ǫ → ( Kψ ) ǫ to Kψ and Plancherel imply that { a ( K ) ǫ, H + } has an L ω l mℓ limit as ǫ →
0; hence, it suffices to check that ( ω − ω + m ) a ( K ) ǫ, H + converges to ( ω − ω + m ) a H + pointwise almosteverywhere. In order to see this, we let u ( K ) ǫ denote the result of applying Carter’s separation to ( Kψ ) ǫ ,and observe that a ( K ) ǫ, H + is, up to an appropriate normalisation, equal to the Wronskian of u ( K ) ǫ with U hor : a ( K ) ǫ, H + = ( − i ( ω − ω + m )) − (( u ( K ) ǫ ) ′ U hor − u ( K ) ǫ U ′ hor ) . Since Theorem 3.7.1 may be easily used to show that for each ( ω, m, ℓ ) and r ∗ , u ( K ) ǫ ( r ∗ , ω, m, ℓ ) convergesto u ǫ ( r ∗ , ω, m, ℓ ) as ǫ →
0, we conclude that ( ω − ω + m ) a ( K ) ǫ, H + ( ω, m, ℓ ) converges to ( ω − ω + m ) a ( K ) H + as ǫ → This section is dedicated to refining our recent proof from [24] of boundedness for the wave equation so asto apply for finite degenerate V -energy solutions.We will collect all statements which we shall need for the remainder of the paper in Section 7.1. Thekey statement is Theorem 7.1 together with one immediate corollary. (In particular, after digesting thesestatements, the reader impatient to proceed to the scattering theory constructions can skip to Section 8.)In the brief aside of Section 7.2, we shall also state the full degenerate-energy analogue of Theorem 3.7.1in Section 7.2 as Theorem 7.2. We shall not actually require the latter result in the paper and it in factis more convenient to infer it a posteriori with the help of the backwards scattering maps which we shallconstruct in Section 9. Thus, the proof of Theorem 7.2 is in fact deferred till Section 9.4.Section 7.3 gives the proof of Theorem 7.1. We note that the proof will crucially use Proposition 5.3.1,Proposition 6.2.1 and Proposition 6.3.1. The main result which we shall require for later sections is the following.
Theorem 7.1.
For all solutions ψ to (2) on R ≥ arising from smooth initial data on Σ ∗ which are compactlysupported, we have ∫ I + J Tµ [ ψ ] n µ I + + ∫ H +≥ J Kµ [ ψ ] n µ H + ≤ B ∫ Σ ∗ J Vµ [ ψ ] n µ Σ ∗ . (113) Remark 7.1.1.
One can easily formulate and prove higher order versions of Theorem 7.1 but we will notpursue this here.
Given that the restriction of the deformation tensor of V to J − ( Σ ∗ ) ∩ J + ( Σ ) is compactly supportedaway from H + ∪ H − ∪ B , a finite in time energy estimate, i.e. (24) with X = V , immediately implies Corollary 7.1.
For all solutions ψ to (2) on J + ( Σ ) arising from smooth compactly supported initial dataalong Σ , we have ∫ I + J Tµ [ ψ ] n µ I + + ∫ H + J Kµ [ ψ ] n µ H + ≤ B ∫ Σ J Vµ [ ψ ] n µ Σ . (114) We note that we can in fact obtain the full analogue of Theorem 3.7.1 where energy boundedness is givenwith respect to a spacelike foliation, and where integrated local energy decay is proven, both now involvingthe degeneate energy. We will not require this result in the rest of paper and it is in fact convenient to obtainit a posteriori using our scattering theory. 53 heorem 7.2.
For all solutions ψ to (2) on R ≥ arising from smooth initial data on Σ ∗ which are compactlysupported, we have ∫ Σ ∗ s J Vµ [ ψ ] n µ Σ ∗ s ≤ B ∫ Σ ∗ J Vµ [ ψ ] n µ Σ ∗ , ∀ s ≥ , (115) ∫ R ≥ ( r − ζ ∣ ∇ / ψ ∣ + r − − δ ζ ∣ T ψ ∣ + ( r − r + ) r − − δ ∣ ˜ Z ∗ ψ ∣ + r − − δ ∣ ψ ∣ ) ≤ B ( δ ) ∫ Σ ∗ J Vµ [ ψ ] n µ Σ ∗ , (116) where ζ is defined as in the statement of Theorem 3.7.1. The proof is defered till Section 9.4.
Remark 7.2.1.
Note the degeneration of the bulk integral at the horizon. One can easily formulate andprove higher order versions of Theorem 7.2 but we will not pursue this here.
Before we begin the discussion of the proof of Theorem 7.1, let us briefly indicate what would go wrong ifwe simply tried to repeat the proof of Theorem 3.7.1 as given in [24].1. Anytime the redshift estimate of [20] and [21] is applied to ψ , one must put a term ∫ Σ ∗ J Nµ [ ψ ] n µ Σ ∗ onthe right hand side of the resulting estimate.2. In [24], when we proved the integrated energy decay statement for ψ we first proved an estimate for χψ where χ ( t ∗ ) was a cutoff function which was identically 0 in the past of Σ ∗ and identically 1 in thefuture of Σ ∗ . We then studied the inhomogeneous wave equation ◻ g ( χψ ) = g µν ∇ µ χ ∇ ν ψ + ( ◻ g χ ) ψ ≐ F. The resulting estimate in [24] had, in particular, a term on the right hand side proportional to ∫ R ∩ { r ≤ R } ∣ F ∣ , for some constant R > r + . Note that on the horizon, F will contain a term proportional to Z ∗ ψ .Unfortunately, this is exactly the derivative that the J V energy loses control of as r → r + .In order to prove Theorem 7.1 we will first observe that without loss of generality, we can assume thatthe initial data for ψ is supported near the horizon. Applying a J K energy estimate for ψ and Plancherelthen immediately reduce the problem to estimating the microlocal radiation fields for ψ Q along I + in thesuperradiant frequency regime F ( ǫ ) ♯ . Next, using the fundamental solution representation of Proposition 5.3.1we will represent the microlocal radiation fields along I + as an integral in r ∗ of the Fourier transform of F against U hor . Following this, in the most subtle part of the proof, we will crucially exploit the fact thatwe are in a superradiant frequency regime where we can afford to lose a derivative, the fact we only needto estimate the flux to I + , the fact that F is supported near the horizon and the oscillations of U hor in r ∗ (as embodied in Proposition 6.3.1) in order to gain some degeneration in r − r + . Somewhat surprisingly,this step does not use that F = g µν ∇ µ χ ∇ ν ψ + ( ◻ g χ ) ψ ; it treats F as an arbitrary inhomogeneity. Finally,the proof concludes with finite in time energy estimates and Hardy inequalities (of course, the fact that F = g µν ∇ µ χ ∇ ν ψ + ( ◻ g χ ) ψ is used in this step). Proof of Theorem 7.1.
We start with an easy reduction; we may split ψ into ψ and ψ where ψ has initialdata supported near the horizon and ψ has initial data supported away from the horizon. Of course, theestimate (113) for ψ follows from Theorem 3.7.1. Thus, without loss of generality, we will assume that ψ = ψ = ψ has initial data whose support is contained in r ∈ [ r + , M ] .We now define ψ Q ≐ χψ where χ is a function which is identically 1 in the future of Σ ∗ , and identically 0in the past of Σ ∗ . This satisfies ◻ g ψ Q = g µν ∇ µ χ ∇ ν ψ + ( ◻ g χ ) ψ ≐ F. u and H are then defined by applying Carter’s separation to χψ and F respectively. Thissatisfies the radial o.d.e. (60) with a non-zero right hand side H . Let a I + denote the corresponding microlocalradiation field of u .We begin by showing ∫ H +≥ J Kµ [ ψ ] n µ H + + ∫ ∞−∞ ∑ mℓ ω ( ω − ω + m ) ∣ a I + ∣ dω ≤ B ∫ Σ ∗ J Vµ [ ψ ] n µ Σ ∗ . (117)Note that part of the proof of this statement will be that the unsigned quantity ∫ ∞−∞ ∑ mℓ ω ( ω − ω + m ) ∣ a I + ∣ dω is absolutely convergent. (One should think of (117) as corresponding to the formal statement ∫ H +≥ J Kµ [ ψ ] n µ H + + ∫ I + J Kµ [ ψ ] n µ I + ≤ B ∫ Σ ∗ J Vµ [ ψ ] n µ Σ ∗ . However, we will wish to avoid a discussion of the convergence of the un-signed integral ∫ I + J Kµ [ ψ ] n µ I + .)Let s > r > r + . We start with a J K energy estimate in the region bounded by H + ( , s ) , Σ ∗ s ∩ { r ≤ r } , { r = r } ∩ J − ( Σ ∗ s ) , and Σ ∗ . We obtain ∫ H + ( ,s ) J Kµ [ ψ ] n µ H + + ∫ Σ ∗ s ∩ { r ≤ r } J Kµ [ ψ ] n µ Σ ∗ s + ∫ { r = r } ∩ J − ( Σ ∗ s ) ∩ J + ( Σ ∗ ) J Kµ [ ψ ] n µ { r = r } = ∫ Σ ∗ ∩ { r ≤ r } J Kµ [ ψ ] n µ Σ ∗ . (118)It easily follows from Theorem 3.7.2 that for each r , there exists a dyadic sequence { s i } ∞ i = such thatlim i →∞ ∫ Σ ∗ si ∩ { r ≤ r } J Kµ [ ψ ] n µ Σ ∗ si = . We thus obtain ∫ H +≥ J Kµ [ ψ ] n µ H + + ∫ { r = r } ∩ J + ( Σ ∗ ) J Kµ [ ψ ] n µ { r = r } = ∫ Σ ∗ ∩ { r ≤ r } J Kµ [ ψ ] n µ Σ ∗ . (119)Observe that Theorem 3.7.2 allows us to unambiguously assign a value to the unsigned quantity ∫ { r = r } ∩ J + ( Σ ∗ ) J Kµ [ ψ ] n µ { r = r } . Next, recalling that ψ ∣ Σ ∗ is supported with [ r + , r + + M ] , we observe that if r is sufficiently large,then (119) becomes ∫ H +≥ J Kµ [ ψ ] n µ H + + ∫ { r = r } J Kµ [ ψ Q ] n µ { r = r } ≤ B ∫ Σ ∗ J Vµ [ ψ ] n µ Σ ∗ . (120)Now we explicitly compute, apply Plancherel, and integrate by parts: ∫ { r = r } J Kµ [ ψ Q ] n µ { r = r } = ∫ { r = r } (( T ψ Q + ω + Φ ψ Q ) ∂ r ∗ ψ Q ) ( r + a ) sin θ dt dθ dφ (121) = ∫ { r = r } (( T (( r + a ) / ψ Q ) + ω + Φ (( r + a ) / ψ Q )) ∂ r ∗ (( r + a ) / ψ Q )) sin θ dt dθ dφ = ∫ ∞−∞ ∑ mℓ ( ω − ω + m ) Im ( u ′ u ) ∣ r = r dω. Next, we consider the microlocal K -energy current (see Section 6.1): Q K [ u ] ≐ ( ω − ω + m ) Im ( u ′ u ) . This is conserved for r ≥ r for r sufficiently large, i.e. ( Q K ) ′ = . Noting that the proof of Proposition 6.8.1 implies ( u ′ − iωu ) ∣ r = ∞ =
0, we thus obtain ∫ ∞−∞ ∑ mℓ ( ω − ω + m ) Im ( u ′ u ) ∣ r = r dω = ∫ ∞−∞ ∑ mℓ ω ( ω − ω + m ) ∣ a I + ∣ dω. (122)55n particular, the right hand side of (122) is absolutely convergent. Combining (120), (121), and (122)yields (117).Next, we observe that Propositions 6.8.1 and 4.2.2 together imply that ∫ I + J Tµ [ ψ ] n µ I + = ∫ ∞−∞ ∑ mℓ ω ∣ a I + ∣ dω. (123)Now, observing that ( ω, m, ℓ ) /∈ F ( ǫ ) ♯ imply that ω ( ω − ω + m ) ≥ b ( ǫ ) ω , it is clear that in order to finishthe proof we need only show ∫ ∞−∞ ∑ {( m,ℓ ) ∶ ( ω,m,ℓ )∈ F ( ǫ )♯ } ∣ ω + ωm ∣ ∣ a I + ∣ dω ≤ B ( ǫ ) ∫ Σ ∗ J Vµ [ ψ ] n µ Σ ∗ , (124)for some sufficiently small ǫ > ( ) .First, note that Proposition 5.3.1 allows us to write ∣ a I + ∣ = ∣ W ∣ − ∣ ∫ ∞−∞ U hor ( x ∗ ) H ( x ∗ ) dx ∗ ∣ . (125)Keeping in mind that the set {( ω, m, ℓ ) ∈ F ( ǫ ) ♯ ∶ Λ ≤ c } is compact, standard o.d.e. theory implies U hor ( r ∗ ) = e − i ( ω − ω + m ) r ∗ + E hor ( r ∗ ) , (126)where ( ω, m, ℓ ) ∈ F ( ǫ ) ♯ and Λ ≤ c implies ∣ E hor ( r ∗ )∣ ≤ B ( ǫ, c ) ∣ r − r + ∣ , (127)for r − r + sufficiently small. As c → ∞ , however, the dependence of B ( ǫ, c ) may be bad. FortunatelyProposition 6.3.1 shows that if ǫ > ∣ E hor ∣ ≤ B ( ǫ ) ∣ W ∣√ Λ √ r − r + , for sufficiently small r − r + .Applying ( ) , ( ) , ( ) , Proposition 6.2.1 and Proposition 6.3.1, we obtain ∫ ∞−∞ ∑ {( m,ℓ ) ∶ ( ω,m,ℓ )∈ F ( ǫ )♯ } ∣ ω + ωm ∣ ∣ a I + ∣ dω (128) = ∫ ∞−∞ ∑ {( m,ℓ ) ∶ ( ω,m,ℓ )∈ F ( ǫ )♯ } ∣ ω + ωm ∣∣ W ∣ ∣ ∫ ∞−∞ U hor ( r ∗ ) H ( r ∗ ) dr ∗ ∣ dω ≤ B ∫ ∞−∞ ∑ {( m,ℓ ) ∶ ( ω,m,ℓ )∈ F ( ǫ )♯ } ⎡ ⎢⎢⎢⎢⎣ ∣ ω + ωm ∣∣ W ∣ ∣ ∫ ∞−∞ e − i ( ω − ω + m ) r ∗ H ( r ∗ ) dr ∗ ∣ + ∣ ω + ωm ∣∣ W ∣ ∣ ∫ ∞−∞ e − i ( ω − ω + m ) r ∗ E hor ( r ∗ ) H ( r ∗ ) dr ∗ ∣ ⎤ ⎥⎥⎥⎥⎦ dω ≤ B ( ǫ ) ∫ ∞−∞ ∑ {( m,ℓ ) ∶ ( ω,m,ℓ )∈ F ( ǫ )♯ } [∣ ∫ ∞−∞ e − i ( ω − ω + m ) r ∗ H ( r ∗ ) dr ∗ ∣ + ∣ ∫ ∞−∞ √ r − r + ∣ H ( r ∗ )∣ dr ∗ ∣ ] ≐ B ( ǫ ) [ I + II ] . Let us now recall the explicit form of H : H = ∆ ( r + a ) / ∫ ∞−∞ ∫ S e iωt e − imφ S mℓ ( θ, aω ) ( ρ F ) sin θ dt dθ dφ, (129)56 = g µν ∇ µ χ ∇ ν ψ + ( ◻ g χ ) ψ. (130)In particular, directly applying Cauchy-Schwarz, Plancherel, a straightforward Hardy inequality, and afinite in time energy inequality, one may easily check that ∫ ∞−∞ ∑ {( m,ℓ ) ∶ ( ω,m,ℓ )∈ F ( ǫ )♯ } ∣ II ∣ ≤ B ∫ Σ ∗ J Vµ [ ψ ] n µ Σ ∗ . (131)For every γ >
0, the same direct application of Cauchy-Schwarz, Plancherel, a straightforward Hardyinequality, and a finite in time energy inequality to the term I will only give ∫ ∞−∞ ∑ {( m,ℓ ) ∶ ( ω,m,ℓ )∈ F ( ǫ )♯ } ∣ I ∣ ≤ B ( γ ) ∫ Σ ∗ ∩ [ r + ,r + + M ] ( r − r + ) − γ ∣ ˜ Z ∗ ψ ∣ + B ∫ Σ ∗ J Vµ [ ψ ] n µ Σ ∗ . (132)Unfortunately, the first term on the right hand side is (barely) not controlled by ∫ Σ ∗ J Vµ [ ψ ] n µ Σ ∗ .We control the term I as follows (we will not lose anything by allowing the sum in m and ℓ to be overall of Z × Z ≥∣ m ∣ ): ∫ ∞−∞ ∑ mℓ ∣ I ∣ dω = ∫ ∞−∞ ∑ mℓ ∣ ∫ π ∫ π ∫ ∞−∞ ∫ ∞−∞ e − i ( ω − ω + m ) r ∗ e iωt e − imφ S mℓ ( θ, aω ) ∆ ρ ( r + a ) / F sin θ dθ dφ dt dr ∗ ∣ dω ≤ ∫ π ∫ ∞−∞ ∑ m ∣ ∫ ∞−∞ ∫ ∞∞ ∫ π e − iω ( r ∗ − t ) e im ( ω + r ∗ − φ ) ∆ ρ ( r + a ) / F dφ dt dr ∗ ∣ sin θ dω dθ. (133)For each fixed m we have used the orthogonality of the S mℓ in the last inequality.Now we introduce the variables ˜ v ≐ t + r ∗ and ˜ u ≐ t − r ∗ and keep in mind that F is only supported in acompact range of ˜ v . Then (133) becomes ∫ π ∑ m ∫ ∞−∞ ∣ ∫ ∞−∞ ∫ ∞∞ ∫ π e iω ˜ u e im ( ω + ˜ v − ˜ u − φ ) ∆ ρ ( r + a ) / F dφ d ˜ u d ˜ v ∣ sin θ dω dθ (134) ≤ B ∫ π ∫ ∞−∞ ∑ m ∫ ∞−∞ ∣ ∫ ∞−∞ ∫ π e iω ˜ u e im ( ω + ˜ v − ˜ u − φ ) ∆ ρ ( r + a ) / F dφ d ˜ u ∣ sin θ dω d ˜ v dθ. = ( π ) B ∫ π ∫ ∞−∞ ∫ ∞−∞ ∑ m ∣ ∫ π e im ( ω + ˜ v − ˜ u − φ ) ∆ ρ ( r + a ) / F dφ ∣ sin θ d ˜ u d ˜ v dθ = ( π ) B ∫ π ∫ ∞−∞ ∫ ∞−∞ ∑ m ∣ ∫ π e − imφ ∆ ρ ( r + a ) / F dφ ∣ sin θ d ˜ u d ˜ v dθ = ( π ) B ∫ ∞−∞ ∫ ∞−∞ ∫ S ∣ ∆ ρ ( r + a ) / F ∣ sin θ d ˜ u d ˜ v dθ dφ ≤ B ∫ Σ ∗ J Vµ [ ψ ] n µ Σ ∗ . We have used Plancherel in the ω variable and the orthogonality of the e imφ . In the last line we used finite intime energy estimates and the Hardy inequality ∫ ∞ r + f dr ≤ B ∫ ∞ r + ( r − r + ) ( ∂ r f ) dr , which holds for smoothfunctions f which vanish for large r .Putting together ( ) , ( ) , ( ) and ( ) , we have indeed obtained ( ) . The theorem is thusproven. We now turn to our scattering theory proper. 57he first order of business is to carefully set up the relevant spaces described in Section 2.1.4 of theintroduction. This will be accomplished in Section 8.1 below.We will then define in Section 8.2 the various forward maps F + and infer their boundedness. Theboundedness of the map with domain E N Σ ∗ (Theorem 8.2.2) is independent of Section 7. This will give Theorem 1 of Section 2.3.1.The boundedness of the degenerate-energy theory maps with domain E V ˚Σ and E V Σ (Theorems 8.2.3and 8.2.4, respectively) indeed requires the statement of Theorem 7.1 just proven. This will give Theo-rems 3 of Section 2.3.3.
In this section we will define the function spaces for which we will formulate our scattering theory. Σ ∗ , ˚Σ and ΣLet us denote by C ∞ cp ( Σ ∗ ) , C ∞ cp ( ˚Σ ) , C ∞ cp ( Σ ) the vector space of smooth compactly supported pairs offunctions ( ψ , ψ ′ ) defined on Σ ∗ , ˚Σ, Σ, respectively. We will complete these vector spaces with respect toappropriate norms to define the Hilbert spaces of our scattering theory.We start with the non-degenerate N -energy space. We shall only in fact consider this for initial data onΣ ∗ . Definition 8.1.1.
For ( ψ , ψ ′ ) ∈ C ∞ cp ( Σ ∗ ) we set ∣∣( ψ , ψ ′ )∣∣ E N Σ ∗ ≐ √ ∫ Σ ∗ J Nµ [ Ψ ] n µ Σ ∗ , where Ψ is any extension of ψ to R such that n Σ ∗ Ψ = ψ ′ .The above expression gives a norm on the vector space C ∞ cp ( Σ ∗ ) , and we define the space (E N Σ ∗ , ∥ ⋅ ∥ E N Σ ∗ ) to be its completion. Next, we define the degenerate V -energy spaces along Σ ∗ , ˚Σ, and Σ, respectively. Definition 8.1.2.
For ( ψ , ψ ′ ) ∈ C ∞ cp ( Σ ∗ ) , C ∞ cp ( ˚Σ ) , and C ∞ cp ( Σ ) , respectively, we set ∣∣( ψ , ψ ′ )∣∣ E V Σ ∗ ≐ √ ∫ Σ ∗ J Vµ [ Ψ ] n µ Σ ∗ , ∣∣( ψ , ψ ′ )∣∣ E V ˚Σ ≐ √ ∫ ˚Σ J Vµ [ Ψ ] n µ ˚Σ , ∣∣( ψ , ψ ′ )∣∣ E V Σ ≐ √ ∫ Σ J Vµ [ Ψ ] n µ Σ , where Ψ is any extension of ψ to D such that n Σ ∗ Ψ = ψ ′ , n ˚Σ Ψ = , n Σ Ψ = respectively.The above expression give norms on the vector spaces C ∞ cp ( Σ ∗ ) , C ∞ cp ( ˚Σ ) , and C ∞ cp ( Σ ) , respectively, andwe define the spaces (E V Σ ∗ , ∥ ⋅ ∥ E V Σ ∗ ) , (E V ˚Σ , ∥ ⋅ ∥ E V ˚Σ ) , (E V Σ , ∥ ⋅ ∥ E V Σ ) , to be their respective completions. Remark 8.1.1.
Note that the energy density is pointwise degenerate because as r → r + the vector field V becomes null. An explicit calculation gives J Vµ [ ˜ ψ ] n µ Σ ∗ ∼ ∣ ∂ t ∗ ˜ ψ ∣ + ( r − r + ) ∣ Z ∗ ˜ ψ ∣ + ∣ ∇ / ˜ ψ ∣ as r → r + . This does not however affect the positive definitivity of the above norms, which moreover are easily to seento arise from a positive definite inner product. Thus, E N Σ ∗ , E V Σ ∗ , E V ˚Σ , E V Σ are all in fact Hilbert spaces.Note moreover that both E N Σ ∗ and E V Σ ∗ may be identified with subsets of L ( Σ ∗ ) × L ( Σ ∗ ) and, after thisidentification is made, E N Σ ∗ is a proper subset of E V Σ ∗ .Finally, we note that one may easily check that a sufficient condition for a pair of smooth functions ( ψ , ψ ′ ) to lie in E V ˚Σ is that ∥( ψ , ψ ′ )∥ E V ˚Σ < ∞ and lim r → r + ( ψ , ψ ′ ) = lim r →∞ ( ψ , ψ ′ ) = . .1.2 Scattering data along H + ≥ , H + and H + We now carry out similar constructions for data along H + ≥ , H + and H + . Let us denote by C ∞ cp (H + ≥ ) , C ∞ cp (H + ) , C ∞ cp (H + ) the vector space of smooth compactly supported functions ψ defined on H + ≥ , H + , H + , respectively.We start with the case of finite non-degenerate energy data along H + ≥ . Definition 8.1.3.
For ψ ∈ C ∞ cp (H + ≥ ) we set ∣∣ ψ ∣∣ E N H+≥ ≐ √ ∫ H +≥ J Nµ [ ψ ] n µ H + . The above expression gives a norm on the vector space C ∞ cp (H + ≥ ) , and we define the space (E N H +≥ , ∥ ⋅ ∥ E N H+≥ ) to be its completion. Next, we define the K -energy spaces along H + ≥ , H + , and H + , respectively. Definition 8.1.4.
For ψ ∈ C ∞ cp (H + ≥ ) , C ∞ cp (H + ) , and C ∞ cp (H + ) , respectively, we set ∣∣ ψ ∣∣ E K H+≥ ≐ √ ∫ H +≥ J Kµ [ ψ ] n µ H +≥ , ∣∣ ψ ∣∣ E K H+ ≐ √ ∫ H + J Kµ [ ψ ] n µ H + , ∣∣ ψ ∣∣ E K H+ ≐ √ ∫ H + J Kµ [ ψ ] n µ Σ . The above expression give norms on the vector spaces C ∞ cp (H + ≥ ) , C ∞ cp (H + ) , and C ∞ cp (H + ) , respectively, andwe define the spaces (E K H +≥ , ∥ ⋅ ∥ E K H+≥ ) , (E K H + , ∥ ⋅ ∥ E K H+ ) , (E K H + , ∥ ⋅ ∥ E K H+ ) , to be their respective completions. Remark 8.1.2.
Note that the K -based energy densities are pointwise degenerate in that the norms do notcontrol ∂ θ ∗ ψ and ∂ φ ∗ ψ . An explicit calculation gives J Kµ [ ψ ] n µ H + ∼ ∣ Kψ ∣ . Again, this degeneration does not however affect the positive definitivity of the above norms, which moreoverare again easily to seen to arise from a positive definite inner product. Thus, E N H +≥ , E K H +≥ , E K H + , E K H + are allin fact Hilbert spaces. Note moreover that both E N H +≥ and E K H +≥ may be identified with subsets of L ( H + ) and, after this identification is made, E N H +≥ is a proper subset of E K H +≥ . I + Finally, we turn to null infinity. Let us denote by C ∞ cp ( I + ) the vector space of smooth compactly supportedfunctions φ defined on I + .The space of finite energy data along I + is then defined as follows. Definition 8.1.5.
For φ ∈ C ∞ cp ( I + ) we set ∣∣ φ ∣∣ E T I+ ≐ √ ∫ I + ∣ ∂ τ φ ∣ . The above expression gives a norm on the vector space C ∞ cp ( I + ) , and we define the space ( E T I + , ∥ ⋅ ∥ E T I+ ) to be its completion. Remark 8.1.3.
Note that this energy density is pointwise degenerate in that it does not control ∂ θ φ and ∂ φ φ . As before, this does not however affect the positive definitivity of the above norms, which moreover areeasily to seen to arise from a positive definite inner product. Thus, E T I + is in fact a Hilbert space. .2 Definition and boundedness of the forward maps In this section we will define the various forward maps from Cauchy data to scattering data and infer theirboundedness. However, we first need the following corollary of Theorems 3.7.2 and 4.2.1.
Corollary 8.2.1.
For all solutions ψ to (2) on R ≥ arising from initial data in C ∞ cp ( Σ ∗ ) , we have that theradiation fields to H + ≥ and I + lie in the spaces E N H +≥ and E T I + respectively.Proof. Given Theorems 3.7.1 and 4.2.1, the only statement we need to check is that the radiation fields liein the closure of compactly supported smooth functions.In order to prove this, we start by giving a (standard) argument which upgrades Theorem 3.7.2 to thestatement that lim s →∞ ∫ Σ ∗ s ∩ { r ∈ [ r + ,R ]} J Nµ [ ψ ] n µ Σ ∗ s = ∀ R > r + . (135)First we observe that the fundamental theorem of calculus and Theorem 3.7.2 immediately imply thefollowing Lipschitz property: ∣ ∫ Σ ∗ s ∩ { r ∈ [ r + ,R ]} J Nµ [ ψ ] n µ Σ ∗ s − ∫ Σ ∗ s ∩ { r ∈ [ r + ,R ]} J Nµ [ ψ ] n µ Σ ∗ s ∣ ≤ B ( ψ ) ∣ s − s ∣ . (136)Let ǫ >
0. Using (136) we may obtain ∫ Σ ∗ s ∩ { r ∈ [ r + ,R ]} J Nµ [ ψ ] n µ Σ ∗ s ≤ B ( ψ ) ǫ + inf s ′ ∈ [ s − ǫ,s + ǫ ] ∫ Σ ∗ s ′ ∩ { r ∈ [ r + ,R ]} J Nµ [ ψ ] n µ Σ ∗ s ′ . (137)Of course, Theorem 3.7.2 implies thatlim s →∞ inf s ′ ∈ [ s − ǫ,s + ǫ ] ∫ Σ ∗ s ′ ∩ { r ∈ [ r + ,R ]} J Nµ [ ψ ] n µ Σ ∗ s ′ = . Thus (137) implies that lim sup s →∞ ∫ Σ ∗ s ∩ { r ∈ [ r + ,R ]} J Nµ [ ψ ] n µ Σ ∗ s ≤ B ( ψ ) ǫ. Since ǫ was arbitrary, (135) follows.Now, using Theorem 3.7.2 we immediately obtain higher order versions of (135). Sobolev inequalitiesthen imply that lim s →∞ sup Σ ∗ s ∩ { r ∈ [ r + ,R ]} ∣ ψ ∣ = . (138)In particular, we may conclude that the radiation field along the horizon H + lies in the closure of compactlysupported functions.For the radiation field along null infinity, we recall that in the proof of Proposition 6.8.1, we provedthat ∂ τ ϕ ∣ r = r converges as r → ∞ to the ∂ τ derivative of the radiation field in L R ≥ τ × S for some sufficientlynegative τ . For each r , (138) implies that ϕ ∣ r = r lies in the closure of smooth compactly supported functions;completeness thus implies that the radiation field along null infinity also lies in this closure.Similarly, we have the following two corollaries. Corollary 8.2.2.
For all solutions ψ to (2) on R arising from initial data in C ∞ cp ( ˚Σ ) , we have that theradiation fields to H + and I + lie in the spaces E V H + and E T I + respectively.Proof. It follows immediately from a K energy estimate near the bifurcate sphere that the radiation fieldof ψ along H + vanishes for sufficiently negative t ∗ . Since a finite-in-time energy estimate implies that ( ψ ∣ Σ ∗ , n Σ ∗ ψ ∣ Σ ∗ ) ∈ C ∞ cp ( Σ ∗ ) , the rest of the proof may be concluded with an appeal to Corollary 8.2.1. Corollary 8.2.3.
For all solutions ψ to (2) on R arising from initial data in C ∞ cp ( Σ ) , we have that theradiation fields to H + and I + lie in the spaces E V H + and E T I + respectively. roof. The proof is the same as the proof of Corollary 8.2.2.The above three corollaries allow us to make the following definition.
Definition 8.2.1.
We define the “forward maps” F + ∶ C ∞ cp ( Σ ∗ ) → E N H +≥ ⊕ E T I + , F + ∶ C ∞ cp ( ˚Σ ) → E K H + ⊕ E T I + , F + ∶ C ∞ cp ( Σ ) → E K H + ⊕ E T I + , to be the maps ( ψ ∣ Σ ∗ , ˚Σ , or Σ , ψ ′ ∣ Σ ∗ , ˚Σ , or Σ ) ↦ ψ ↦ ( ψ ∣ H +≥ , H + , or H + ≐ ψ ∣ H +≥ , H + , or H + , φ ∣ I + ≐ rψ ∣ I + ) (139) which take smooth initial data in C ∞ cp ( Σ ∗ ) , C ∞ cp ( ˚Σ ) or C ∞ cp ( Σ ) , solve the wave equation to the future andthen compute the radiation fields along H + ≥ , H + or H + , respectively, and I + . Theorem 4.2.1 now implies
Theorem 8.2.1.
The forward map F + uniquely extends by density to a bounded map F + ∶ E N Σ ∗ → E N H +≥ ⊕ E T I + . This gives
Theorem 1 of Section 2.3.1.Similarly, Theorem 7.1 implies the following theorem.
Theorem 8.2.2.
The forward map F + uniquely extends by density to a bounded map F + ∶ E V Σ ∗ → E K H +≥ ⊕ E T I + . Lastly, Corollaries 7.1, 8.2.2 and 8.2.3 now imply the following two theorems.
Theorem 8.2.3.
The forward map F + uniquely extends by density to a bounded map F + ∶ E V ˚Σ → E K H + ⊕ E T I + . Theorem 8.2.4.
The forward map F + uniquely extends by density to a bounded map F + ∶ E V Σ → E K H + ⊕ E T I + . We have obtained thus
Theorem 3 of the Section 2.3.3.
This section represents the heart of the paper. We will here construct bounded maps B − ∶ E K H +≥ ⊕ E T I + → E V Σ ∗ , B − ∶ E K H + ⊕ E T I + → E V ˚Σ , B − ∶ E K H + ⊕ E T I + → E V Σ , (140)which will invert the maps F + from Theorem 8.2.2, 8.2.3 and 8.2.4 and then we shall construct the scatteringmaps S ∶ E V H − ⊕ E T I − → E V H + ⊕ E T I + , S ∶ E V H − ⊕ E T I − → E V H + ⊕ E T I + . (141)It turns out that for technical reasons, it is easiest to first construct the middle map of ( ) and showthat it is a two-sided inverse of the corresponding forward map on E V ˚Σ . This will be the content of Section 9.1where the main result is stated as Theorem 9.1.1. The remaining two backwards maps to Σ ∗ and Σ willthen be easily constructed in Sections 9.2 and 9.3, and these will be shown in Theorems 9.2.1 and 9.3.1 tobe two-sided inverses of the corresponding maps F + . The above three theorems will give Theorem 4 ofSection 2.3.4.The scattering maps ( ) and their boundedness will be deduced as Theorem 9.5.2 in Section 9.5 afterintroducing the past-analogues F − and B + and inferring their boundedness (Theorem 9.5.1). This will61ive Theorem 5 of Section 2.3.5. We shall also represent S in the frequency domain by Theorem 9.5.3,giving the relationship between the fixed-frequency and physical space theories. This will imply in particular Theorem 12 of Section 2.4.2.Finally, this section contains two separate “asides”, Sections 9.4 and 9.6, either of which can be skipped,but both of which could have interest independent of the rest of the paper. In Section 9.4, we will use themaps ( ) to complete the theory of boundedness and integrated decay for the degenerate V -energy bygiving the proof of Theorem 7.2 from Section 7.2. In Section 9.6, we will give an alternative, self-containeddiscussion of the Schwarzschild a = ˚Σ We begin by constructing the map B − ∶ E K H + ⊕ E T I + → E V ˚Σ . B − First, we define what will turn out to be essentially the Fourier transform of our backwards map. We beginby recalling the coefficients A I ± , A H ± and the Wronskian W from Definition 5.3.1 and 5.3.3, as well asTheorem 5.3.1 which states that W never vanishes. Definition 9.1.1.
For all smooth functions a I + ( ω, m, ℓ ) and a H − ( ω, m, ℓ ) which are only supported on acompact set of ( ω, m, ℓ ) , for all ( ω, m, ℓ ) with ω ≠ and ω ≠ ω + m , we define ˆ B − ( a H + , a I + ) ∣ ( r,ω,m,ℓ ) ≐ ( a I + W − ( iω )) U hor + ( a H + W − ( i ( ω − ω + m ))) U inf . The next proposition explains the definition of ˆ B − . Proposition 9.1.1.
For ω ≠ and ω ≠ ω + m , ˆ B − ( a H + , a I + ) is the unique solution u to the radial o.d.e. (60)with vanishing right hand side H = such that there exist complex numbers α ( ω, m, ℓ ) and β ( ω, m, ℓ ) satis-fying u = a H + U hor + α ( ω, m, ℓ ) U hor , (142) u = a I + U inf + β ( ω, m, ℓ ) U inf . (143) Proof.
We start with uniqueness. Suppose that we have two solutions u and ˜ u to the radial o.d.e. (60) witha vanishing right hand side H such that u = a H + U hor + α ( ω, m, ℓ ) U hor , ˜ u = a H + U hor + ˜ α ( ω, m, ℓ ) U hor ,u = a I + U inf + β ( ω, m, ℓ ) U inf , ˜ u = a I + U inf + ˜ β ( ω, m, ℓ ) U inf . Then, for each ( ω, m, ℓ ) with ω ≠ ω ≠ ω + m , u − ˜ u would be a solution the radial o.d.e. (60) with avanishing right hand side such that u − ˜ u ∼ e iωr ∗ as r → ∞ ,u − ˜ u ∼ e − i ( ω − ω + m ) r ∗ as r → r + . These asymptotic conditions imply that u − ˜ u corresponds to a “mode solution” (see Definition 1.1 of [53]),and Theorem 1.6 of [53] proves that there are no non-zero mode solutions.To see that ˆ B − verifies (142) and (143), it suffices to recall the relations U hor = A I + U inf + ( iω ) − W U inf ,U inf = A H + U hor + ( i ( ω − ω + m )) − W U hor . We now introduce a useful function space. 62 efinition 9.1.2.
Let ˇ C ∞ cp denote the set of functions f ∶ R × S → C such that ˆ f ( ω, m, ℓ ) ≐ √ π ∫ ∞−∞ e iωt e − imφ S mℓ ( aω, θ ) f sin θ dt dθ dφ is smooth in ω and vanishes for ( ω, m, ℓ ) outside a compact set of R × Z × Z ≥∣ m ∣ .Next, observing that ˇ C ∞ cp may be naturally identified as a subset of either L ( H + ) or L ( I + ) , we let ˇ C ∞ cp ( H + ) be the result of identifying ˇ C ∞ cp with a subset of E K H + , and let ˇ C ∞ cp ( I + ) be the result of identifying ˇ C ∞ cp with a subset of E T H + . Remark 9.1.1.
One may easily check that ˇ C ∞ cp ( H + ) is dense in E K H + and that ˇ C ∞ cp ( I + ) is dense in E T H + . We now define the map B − on the space ˇ C ∞ cp ( H + ) ⊕ ˇ C ∞ cp ( I + ) . Definition 9.1.3.
For all ( ψ , φ ) ∈ ˇ C ∞ cp ( H + ) ⊕ ˇ C ∞ cp ( I + ) , we define the function B − ( ψ , φ ) ∶ ˚ R → C by B − ( ψ , φ ) ∣ ( t,r,θ,φ ) ≐ ( r + a ) / √ π ∫ ∞−∞ ∑ mℓ e − itω e imφ S mℓ ( aω, θ ) ˆ B − ( ˆ ψ , ˆ φ ) ∣ ( ω,r,m,ℓ ) dω. Note that ˆ B − ( ˆ ψ , ˆ φ ) vanishes for all ( ω, m, ℓ ) outside a compact set; it immediately follows that B − ( ψ , φ ) is a smooth function of ( t, r, θ, φ ) . The following proposition will be used to show that the map ( ψ , φ ) ↦ ( B − ( ψ , φ )∣ ˚Σ , n ˚Σ B − ( ψ , φ )∣ ˚Σ ) isbounded. Proposition 9.1.2.
For all ( ψ , φ ) ∈ ˇ C ∞ cp ( H + ) ⊕ ˇ C ∞ cp ( I + ) , we have ∫ ˚Σ J Vµ [ B − ( ψ , φ )] n µ ˚Σ ≤ B ∫ ∞−∞ ∫ S [ ∣( ∂ τ + ω + ∂ φ ) ψ ∣ + ∣ ∂ τ φ ∣ ] sin θ dτ dθ dφ. Proof.
Set u ≐ ˆ B − ( ˆ ψ , ˆ φ ) ∣ ( ω,r,m,ℓ ) ,ψ ≐ B − ( ψ , φ ) ∣ ( t,r,θ,φ ) . First of all, we observe that ψ ∶ ˚ R → C is a smooth solution to ◻ g ψ =
0, is easily seen to be sufficientlyintegrable in the sense of Definition 5.1.1, and that applying Carter’s separation to ψ yields u .Keeping the explicit formula for ˆ B − in mind, applying Theorem 6.2.1 to u implies that for each −∞ < R ∗− < R ∗+ < ∞ we have ( ω − ω + m ) ∣ a H − ∣ + ω ∣ a I − ∣ + ∫ R ∗+ R ∗− [ ∣ u ′ ∣ + (( − r trap r − ) ( ω + Λ ) + ) ∣ u ∣ ] dr ∗ (144) ≤ B ( R ∗− , R ∗+ ) [( ω − ω + m ) ∣ ˆ ψ ∣ + ω ∣ ˆ φ ∣ ] . The rest of the proof will borrow some ideas from Section 13 of [24]. In order to work around thepresence of the ( − r trap r − ) term in (144), it will be useful to decompose ψ in pieces, each of whichexperience trapping near a specific value of r . We first define the following ranges of ( ω, m, ℓ ) : Definition 9.1.4.
Let ǫ > be a sufficiently small parameter to fixed later. We define F ≐ {( ω, m, ℓ ) ∶ r trap = } , F i ≐ {( ω, m, ℓ ) ∶ r trap ∈ [ M − s − + ǫ ( i − ) , M − s − + ǫi )} ∀ i = , . . . , ⌈ ǫ − ( s + + s − )⌉ . Observe that each value of ( ω, m, ℓ ) lies in exactly one of the F i .63 efinition 9.1.5. We define ψ i by a phase space multiplication of ψ by F i , the indicator function of F i : ψ i ≐ ( r + a ) / √ π ∫ ∞−∞ ∑ mℓ e − iωt F i S mℓ ( aω, cos θ ) e imφ ˆ B − ( ˆ ψ , ˆ φ ) dω. Note that each ψ i is a smooth function from ˚ R to C , satisfies ◻ g ψ i = ψ i is compactly supported in ( ω, m, ℓ ) , Plancherel immediately impliesthat for each r + < r < r < ∞ we have ∫ ∞−∞ ∫ { t = s } ∩ { r ∈ [ r ,r ]} J Vµ [ ψ i ] n µ { t = s } ds < ∞ . (145)In particular, for each r + < r < r < ∞ and i = , . . . , ⌈ ǫ − ( s + + s − )⌉ there exists a constant C i ( r , r ) anda dyadic sequence { s ( i ) n } ∞ n = such that s ( i ) n → ∞ as n → ∞ and ∫ { t = s ( i ) n } ∩ { r ∈ [ r ,r ]} J Nµ [ ψ i ] n µ { t = s ( i ) n } ≤ C i ( r , r ) s ( i ) n . (146)Next, taking ǫ from Definition 9.1.4 sufficiently small (and then fixing ǫ ), for each r i we appeal toCorollary 3.4.1 and construct a T -invariant timelike vector field V i on ˚ R which is Killing in the region r ∈ [ M − s − + ( i − ) ǫ, M − s − + iǫ ) , and is equal to V for r sufficiently close to r + and r sufficiently large.Finally, we are ready for our main estimate. For each r + < r < r < ∞ such that r − r + is sufficientlysmall and r is sufficiently large, we apply the energy identity associated to V i in between the hypersurfaces˚Σ ∩ { r ∈ [ r , r ]} , { r = r } ∩ J + ( ˚Σ ) ∩ J − ({ t = s ( i ) n }) , { r = r } ∩ J + ( ˚Σ ) ∩ J − ({ t = s ( i ) n }) , and { t = s ( i ) n } ∩ { r ∈ [ r , r ]} . We obtain ∫ ˚Σ ∩ { r ∈ [ r ,r ]} J V i µ [ ψ i ] n µ ˚Σ (147) ≤ B ∫ s ( i ) n ∫ { t = s } ∩ supp ( K Vi ) J V i µ [ ψ i ] n µ { t = s } ds + ∫ { t = s ( i ) n } ∩ { r ∈ [ r ,r ]} J V i µ [ ψ i ] n µ { t = s ( i ) n } + B ( ∫ { r = r } + ∫ { r = r } ) ∣ V (( r + a ) / ψ i ) ∂ r ∗ (( r + a ) / ψ i )∣ sin θ dt dθ dφ, where we have used the calculation (121) and that fact that ∣ K V i ∣ ≤ B J V i , where we recall that K V i = ( V i ) π αβ T αβ and ( V i ) π αβ denotes the deformation tensor of V i .Taking n → ∞ and appealing to (146) then yields ∫ ˚Σ ∩ { r ∈ [ r ,r ]} J V i µ [ ψ i ] n µ ˚Σ (148) ≤ B ∫ ∞ ∫ { t = s } ∩ supp ( K Vi ) J V i µ [ ψ i ] n µ { t = s } ds + B ( ∫ { r = r } + ∫ { r = r } ) ∣ V (( r + a ) / ψ i ) ∂ r ∗ (( r + a ) / ψ i )∣ sin θ dt dθ dφ, Next, since ( − r trap r − ) ∣ ˆ ψ i ∣ ∼ ∣ ˆ ψ i ∣ for values of r in the support of K ( V i ) , we observe that applying64lancherel and (144) yields ∫ ∞ ∫ { t = s } ∩ supp ( K Vi ) J V i µ [ ψ i ] n µ { t = s } ds (149) ≤ ∫ ∞−∞ ∫ { t = s } ∩ supp ( K Vi ) ∣ J V i µ [ ψ i ] n µ { t = s } ∣ ds ≤ B ∫ ∞−∞ ∑ ( ω,m,ℓ ) ∈ F i ∫ r ∗ max r ∗ min [∣ u ′ ∣ + (( − r trap r − ) ( ω + Λ ) + ) ∣ u ∣ ] dr ∗ dω ≤ B ∫ ∞−∞ ∑ mℓ [( ω − ω + m ) ∣ ˆ ψ ∣ + ω ∣ ˆ φ ∣ ] dω ≤ B ∫ ∞−∞ ∫ S [∣( ∂ τ + ω + ∂ φ ) ψ ∣ + ∣ ∂ τ φ ∣ ] sin θ dτ dθ dφ. We conclude that ∫ ˚Σ J V i µ [ ψ i ] n µ ˚Σ ≤ (150) B ∫ ∞−∞ ∫ S [∣( ∂ τ + ω + ∂ φ ) ψ ∣ + ∣ ∂ τ φ ∣ ] sin θ dτ dθ dφ + B lim inf r → r + lim inf r →∞ ( ∫ { r = r } + ∫ { r = r } ) ∣ V (( r + a ) / ψ i ) ∂ r ∗ (( r + a ) / ψ i )∣ sin θ dt dθ dφ. It immediately follows from Proposition 5.2.3, the compact support of ˆ ψ i in ( ω, m, ℓ ) , and (144) thatlim inf r → r + ∫ r = r ∣ V (( r + a ) / ψ i ) ∂ r ∗ (( r + a ) / ψ i )∣ sin θ dt dθ dφ (151) = lim inf r → r + ∫ r = r ∣ K (( r + a ) / ψ i ) ∂ r ∗ (( r + a ) / ψ i )∣ sin θ dt dθ dφ ≤ lim inf r → r + B ∫ ∞−∞ ∑ mℓ ∣( ω − ω + m ) uu ′ ∣ dω ≤ B ∫ ∞−∞ ∑ mℓ ( ω − ω + m ) [∣ a H + ∣ + ∣ a H − ∣ ] dω ≤ B ∫ ∞−∞ ∑ mℓ [( ω − ω + m ) ∣ ˆ ψ ∣ + ω ∣ ˆ φ ∣ ] dω ≤ B ∫ ∞−∞ ∫ S [∣( ∂ τ + ω + ∂ φ ) ψ ∣ + ∣ ∂ τ φ ∣ ] sin θ dτ dθ dφ. Remark 9.1.2.
Note that the passing of the limit through the integral and sum that implicitly occurs betweenlines and is justified by Proposition 5.2.3 and the compact support of ˆ ψ i in ( ω, m, ℓ ) . Similarly, it immediately follows from Proposition 6.7.1, Proposition 6.6.1, the compact support of ˆ ψ i ,and (144) thatlim inf r →∞ ∫ r = r ∣ V (( r + a ) / ψ i ) ∂ r ∗ (( r + a ) / ψ i )∣ sin θ dt dθ dφ (152) = lim inf r →∞ ∫ r = r ∣ T (( r + a ) / ψ i ) ∂ r ∗ (( r + a ) / ψ i )∣ sin θ dt dθ dφ ≤ lim inf r →∞ B ∫ ∞−∞ ∑ mℓ ∣ ωuu ′ ∣ ≤ B ∫ ∞−∞ ∑ mℓ ω [∣ a I + ∣ + ∣ a I − ∣ ] dω ≤ B ∫ ∞−∞ ∑ mℓ [( ω − ω + m ) ∣ ˆ ψ ∣ + ω ∣ ˆ φ ∣ ] dω ≤ B ∫ ∞−∞ ∫ S [∣( ∂ τ + ω + ∂ φ ) ψ ∣ + ∣ ∂ τ φ ∣ ] sin θ dτ dθ dφ. ∫ ˚Σ J V i µ [ ψ i ] n µ ˚Σ ≤ B ∫ ∞−∞ ∫ S [∣( ∂ τ + ω + ∂ φ ) ψ ∣ + ∣ ∂ τ φ ∣ ] sin θ dτ dθ dφ. (153)We conclude the proof with the (trivial) observation that ∫ ˚Σ J Vµ [ ψ ] n µ ˚Σ ≤ B ⌈ ǫ − ( s + + s − )⌉ ∑ i = ∫ { t = } J V i µ [ ψ i ] n µ { t = } . The following proposition will be used to show that the range of ( B − ∣ ˚Σ , n ˚Σ B − ∣ ˚Σ ) lies in E V ˚Σ . Proposition 9.1.3.
For all ( ψ , φ ) ∈ ˇ C ∞ cp ( H + ) ⊕ ˇ C ∞ cp ( I + ) , we have lim r → r + sup S ∣ B − ( ψ , φ ) ∣ t = ∣ = , lim r →∞ sup S ∣ B − ( ψ , φ ) ∣ t = ∣ = . Proof.
We start with the limit as r → r + . Since ˆ B − ( ˆ ψ , ˆ φ ) is compactly supported in ( ω, m, ℓ ) , and U hor and W are smooth for ω ∈ R ∖ { } , one may easily establish that for every δ > (∣ ω ∣ , ∣ ω − ω + m ∣) ≥ δ ⇒ ˆ B − ( ˆ ψ , ˆ φ ) = a H + e − i ( ω − ω + m ) r ∗ + a H − e i ( ω − ω + m ) r ∗ + Error , where ∣ Error ∣ ≤ B ( δ, ψ , φ )( r − r + ) . Let χ ( x ) be a cutoff function which is identically 1 in a neighborhood of 0 and identically 0 for ∣ x ∣ > δ > r → r + ∣( r + a ) / √ π B − ∣ ( ,r,θ,φ ) ∣ (154) = lim sup r → r + ∣ ∫ ∞−∞ ∑ mℓ e imφ S mℓ ( aω, θ ) ˆ B − dω ∣ . ≤ lim sup r → r + ∣ ∫ ∞−∞ ∑ mℓ χ ( ωδ − ) χ (( ω − ω + m ) δ − ) e imφ S mℓ ( aω, θ ) ˆ B − dω ∣ + lim sup r → r + ∣ ∫ ∞−∞ ( − χ ( ωδ − )) ( − χ (( ω − ω + m ) δ − )) ∑ mℓ e imφ S mℓ ( aω, θ ) ˆ B − dω ∣ . We estimate the first term simply with Cauchy-Schwarz and (144):lim sup r → r + ∣ ∫ ∞−∞ ∑ mℓ χ ( ωδ − ) χ (( ω − ω + m ) δ − ) e imφ S mℓ ( aω, θ ) ˆ B − dω ∣ (155) ≤ B ( ψ , φ ) lim sup r → r + δ / √ ∫ ∞−∞ ∑ mℓ ∣ ˆ B − ∣ dω ≤ B ( ψ , φ ) δ / √ ∫ ∞−∞ ∑ mℓ [∣ a H + ∣ + ∣ a H − ∣ ] ≤ B ( ψ , φ ) δ / . χ δ ≐ ( − χ ( ωδ − )) ( − χ (( ω − ω + m ) δ − )) . For the second term we use the oscillation in r ∗ :lim sup r → r + ∣ ∫ ∞−∞ ∑ mℓ e imφ ˜ χ δ S mℓ ( aω, θ ) ˆ B − dω ∣ (156) ≤ B ( ψ , φ , δ ) lim sup r → r + ∣ ∫ ∞−∞ ∑ mℓ ˜ χ δ e imφ S mℓ ( a H + e − i ( ω − ω + m ) r ∗ + a H − e i ( ω − ω + m ) r ∗ ) dω ∣ ≤ B ( ψ , φ , δ ) lim sup r → r + δ − ∣ r ∗ ∣ − = . In the second to last line, the decay in r ∗ came from an integration by parts in ω .Since δ may be taken arbitrary small, combining (154), (155), and (156) concludes the proof for the limitwhen r → r + . Moreover, it is easy to see that essentially the same proof works for the limit as r → ∞ .The previous two propositions and Remark 9.1.1 immediately imply the following corollary. Corollary 9.1.1.
The map ( B − ∣ ˚Σ , n ˚Σ B − ∣ ˚Σ ) , which we shall, by a mild abuse of notation, now denote by B − , extends by density to a bounded map B − ∶ E K H + ⊕ E T I + → E V ˚Σ . Proof.
The key point is that a straightforward calculation (remember that V vanishes at the bifurcate sphere!)shows that lim r → r + ( B − ∣ ˚Σ , n ˚Σ B − ∣ ˚Σ ) = r →∞ ( B − ∣ ˚Σ , n ˚Σ B − ∣ ˚Σ ) = ( B − ∣ ˚Σ , n ˚Σ B − ∣ ˚Σ ) lies in E V ˚Σ . Finally, we are ready for the key result of the section.
Theorem 9.1.1.
Let B − and F + be as in Corollary 9.1.1 and Theorem 8.2.3. Then B − and F + are bothbounded isomorphisms and satisfy B − ○ F + = Id and F + ○ B − = Id .Proof. Of course, it suffices to prove the assertions B − ○ F + = Id and F + ○ B − = Id.We start with establishing F + ○ B − = Id. By density (Remark 9.1.1), it suffices to check that ( F + ○ B − ) ∣ ˇ C ∞ cp (H + ) ⊕ ˇ C ∞ cp (I + ) = Id . Let ( ψ , φ ) ∈ ˇ C ∞ cp ( H + ) ⊕ ˇ C ∞ cp ( I + ) . Proposition 9.1.1 implies that there exists functions α ( ω, m, ℓ ) and β ( ω, m, ℓ ) such that ˆ B − ( ˆ ψ , ˆ φ ) = ˆ ψ U hor + αU hor , ˆ B − ( ˆ ψ , ˆ φ ) = ˆ φ U inf + βU inf . Now, using that ( B − ( ψ , φ )∣ ˚Σ , n ˚Σ B − ( ψ , φ )∣ ˚Σ ) lies E V ˚Σ , and Theorem 7.1, one may easily check that the samearguments used in the proofs of Propositions 6.8.1 and 6.8.2 immediately imply ( F + ○ B − ) ( ψ , φ ) = ( ψ , φ ) . We now turn to establishing B − ○ F + = Id. By density, it suffices to study solutions arising from initialdata ( ψ , ψ ′ ) ∈ C ∞ cp ( ˚Σ ) . Let a H + and a I + denote the microlocal radiation fields. Then Proposition 6.8.1and 6.8.2 yield F + ( ψ , ψ ′ ) = ( √ M πr + ∫ ∞−∞ ∑ mℓ e − it ∗ ω e imφ S mℓ a H + dω, √ π ∫ ∞−∞ ∑ mℓ e − iτω e imφ S mℓ a I + dω ) . It immediately follows from Proposition 9.1.1 that ( B − ○ F + ) ( ψ , ψ ′ ) = ( ψ , ψ ′ ) .67 .1.4 A physical-space characterization of B − Before we close the section it will be conceptually clarifying and technically useful to observe that thebackwards map may also be characterized in physical space.
Proposition 9.1.4.
Let ( ψ H + , φ I + ) ∈ C ∞ cp ( H + ) ⊕ C ∞ cp ( I + ) . Pick τ < ∞ such that ψ H + is compactly supportedin H + ( −∞ , τ ) and φ I + is compactly supported in I + τ , and then let Φ I + be any smooth extension of φ I + tothe manifold with boundary ˜ R (see Definition 4.2.1) such that Φ I + vanishes in neighborhood of S τ .Next, using Proposition 3.6.4, for each s > sufficiently large we may uniquely define a smooth solution ψ s to (2) in the past of H + ≤ τ ∪ ( S τ ∩ { r ≤ r ( τ , s )}) ∪ ({ t = s } ∩ { r ≥ r ( τ , s )}) by requiring ψ s ∣ H +≤ τ = ψ H + , ( ψ s ∣ S τ ∩ { r ≤ r ( τ,s )} , n S τ ψ s ∣ S τ ∩ { r ≤ r ( τ,s )} ) = ( , ) ,rψ s ∣ { t = s } ∩ { r ≥ r ( τ,s )} = Φ I + ∣ { t = s } ∩ { r ≥ r ( τ,s )} . Let { s i } ∞ i = be a sequence satisfying s i → ∞ as i → ∞ . It follows that ψ s i ∣ J + ( ˚Σ ) and any finite number ofderivatives form a bounded equicontinuous sequence. In particular, we may extract a smooth limit ψ whichwill be a solution to (2) in the region D − ( S τ ) ∩ J + ( ˚Σ ) . Finally, we have B − ( ψ H + , φ I + ) = ( ψ ∣ ˚Σ , n ˚Σ ψ ˚Σ ) . (157) Proof.
The boundedness and equicontinuity of any finite number of derivatives of { ψ s i } follows immediatelyfrom (higher order) J V energy estimates (it may be useful for the reader to note that the intersection of D − ( S τ ) ∩ J + ( ˚Σ ) and the support of ( V ) π is compact and contained in ∪ s ∈ [ ,τ ] S s ).Next, using Theorem 9.1.1, we note that (157) would follow from F + ( ψ ∣ ˚Σ , n ˚Σ ψ ˚Σ ) = ( ψ H + , φ I + ) . (158)Now, the equality F + ( ψ ∣ ˚Σ , n ˚Σ ψ ˚Σ )∣ H + = ψ H + is a trivial consequence of the definition of the radiation fieldand Proposition 3.6.2.Finally, the equality F + ( ψ ∣ ˚Σ , n ˚Σ ψ ˚Σ )∣ I + = φ I + follows from Proposition 3.8.1 and a straightforward mod-ification of the arguments given in the proof of Proposition 4.2.1. Σ ∗ In this section we will define the backwards map B − on E K H +≥ ⊕ E T I + . Definition 9.2.1.
Let E ∶ C ∞ cp ( H + ≥ ) → C ∞ cp ( H + ) be any map satisfying1. E ( f ) ∣ H +≥ = f .2. ∫ H + J Kµ [ E ( f )] n µ H + ≤ B ∫ H +≥ J Kµ [ f ] n µ H + .Note that such a map is easily constructed.Then we define the backwards map B − ∶ C ∞ cp ( H + ≥ ) ⊕ C ∞ cp ( I + ) → E V Σ ∗ , by B − ( ψ H +≥ , φ I + ) ≐ ( B − ( E ( ψ H +≥ ) , φ I + ) ∣ Σ ∗ , n Σ ∗ B − ( E ( ψ H +≥ ) , φ I + ) ∣ Σ ∗ ) . (159) The reader should keep in mind our standard recycling of the notation concerning the symbol B − . In partic-ular, B − on the right hand side of (159) is as in Definition 9.1.3. The next theorem establishes that the backwards map extends to E K H +≥ ⊕ E T I + and inverts the forward map F + . 68 heorem 9.2.1. The map B − defined above is a bounded map and thus uniquely extends to a map B − ∶ E K H +≥ ⊕ E T I + → E V Σ ∗ . Let F + denote the forward map F + ∶ E V Σ ∗ → E K H + ⊕ E T I + . Then, B − ○ F + = Id and F + ○ B − = Id and thus B − and F + are bounded isomorphisms. Remark 9.2.1.
Observe that one corollary of Theorem 9.2.1 is that B − does not depend on the choice ofextension E .Proof. First of all, we observe that the boundedness of B − and the statement F + ○ B − = Id follow immediatelyfrom Theorem 9.1.1, Proposition 3.6.3 and finite in time energy estimates (cf. the proof of Corollary 7.1).The equality B − ○ F + = Id is a bit more subtle. The key observation is that it suffices to check thison a dense subject and it pays to expend a little effort in creating a convenient one. We thus turn to theconstruction of a useful dense subset. First of all, C ∞ cp ( H + ) ⊕ C ∞ cp ( I + ) is a dense subset of E K H + ⊕ E T I + , and thusTheorem 9.1.1 implies that B − ( C ∞ cp ( H + ) ⊕ C ∞ cp ( I + )) is a dense subset of E V ˚Σ . Now, considering the elementsof B − ( C ∞ cp ( H + ) ⊕ C ∞ cp ( I + )) as Cauchy data along ˚Σ, we may solve the wave equation to the future of ˚Σwith Proposition 3.6.2 and restrict the solutions to Σ ∗ . This defines a subset ˜ C Σ ∗ of E V Σ ∗ . It follows fromProposition 3.6.4 and finite in time energy estimates (cf. the proof of Corollary 7.1) that ˜ C Σ ∗ is in fact adense subset of E V Σ ∗ .We now turn to proving that B − ○ F + ∣ ˜ C Σ ∗ = Id. Let ψ be a solution to (2) in R ≥ whose initial data alongΣ ∗ lie in ˜ C Σ ∗ . We then define a solution ˜ ψ to (2) in R ≥ by applying Proposition 3.6.1 to solve the waveequation with initial data ( B − ○ F + ) ( ψ ∣ Σ ∗ , n Σ ∗ ψ ∣ Σ ∗ ) along Σ ∗ . We need to prove that ψ − ˜ ψ =
0. Now, thekey advantage to considering initial data in ˜ C Σ ∗ is that it immediately follows from Proposition 9.1.4 that ψ ∣ Σ ∗ , n Σ ∗ ψ , ˜ ψ ∣ Σ ∗ and n Σ ∗ ˜ ψ ∣ Σ ∗ are smooth functions and hence that ψ and ˜ ψ extend smoothly to H + ≥ . Since F + ○ B − = Id we conclude in particular that K ( ψ − ˜ ψ ) ∣ H +≥ = ψ † ≐ K ( ψ − ˜ ψ ) . Since the Cauchy data for ψ † along Σ ∗ vanishes at H + , we may easily construct asequence { ψ † i } ∞ i = of solutions to (2) whose initial data along Σ ∗ are smooth and compactly supported awayfrom H + ∩ Σ ∗ and spacelike infinity and which satisfylim i →∞ ∫ Σ ∗ J Vµ [ ˜ ψ − ˜ ψ i ] n µ Σ ∗ = . Since the ψ † i are compactly supported away from H + ∩ Σ ∗ , they may easily be extended as solutions to (2)to all of R by applying Proposition 3.6.3 with vanishing initial data along H + ≤ . We will also denote theextension by ψ † i . Since ψ † i is easily seen to be sufficiently integrable in the sense of Definition 5.1.1, we mayapply Carter’s separation to ψ † i to define u † i and the corresponding microlocal fluxes a † i, H + and a † i, I + . Itfollows immediately from the construction of ψ † i , Corollary 7.1, Proposition 6.8.1 and Proposition 6.8.2 thatlim i →∞ ∫ ∞−∞ ∑ mℓ [∣ a † i, H + ∣ + ∣ a † i, I + ∣ ] = . (160)Proposition 9.1.2 (and an easy density argument) then imply thatlim i →∞ ∫ J Vµ [ ψ † i ] n µ { t = } = . Then, finite in time energy estimates show that ψ † vanishes. Finally, using also (135) from the proof ofCorollary 8.2.1, we conclude that ( ψ − ˜ ψ )∣ R ≥ vanishes.69 .3 The backwards map to Σ With Theorem 9.2.1 proven, we can now revisit scattering to Σ and prove a version of Theorem 9.1.1 where E V ˚Σ is replaced by E V Σ and E K H + is replaced with E K H + . Theorem 9.3.1.
Let F + be the forward map F + ∶ E V Σ → E K H + ⊕ E T I + . Then there exists a backwards map B − ∶ E K H + ⊕ E T I + → E V Σ such that B − is a bounded map, B − ○ F + = Id and F + ○ B − = Id . Thus B − and F + are both boundedisomorphisms.Proof. We begin by introducing the notation H + ≤ ≐ H + ∩ J − ( Σ ∗ ) . Then, we define the function space C ∞ cp ( H + ≤ ∪ Σ ∗ ) to consist of triples ( ψ H +≤ , ψ Σ ∗ , ψ ′ Σ ∗ ) such that ψ H +≤ isa smooth function on H + ≤ , ψ Σ ∗ and ψ ′ Σ ∗ are smooth functions of compact support on Σ ∗ and there exists asmooth function ˜Ψ on D such that ˜Ψ ∣ H +≤ = ψ H +≤ , ˜Ψ ∣ Σ ∗ = ψ Σ ∗ and n Σ ∗ ˜Ψ ∣ Σ ∗ = ψ ′ ∣ Σ ∗ .Proposition 3.6.3 states that to each ( ψ H +≤ , ψ Σ ∗ , ψ ′ Σ ∗ ) ∈ C ∞ cp ( H + ≤ ∪ Σ ∗ ) there exists a unique smoothsolutions ψ to (2) in J − ( Σ ∗ ) . Restricting these solutions to Σ thus defines a map C ∞ cp ( H + ≤ ∪ Σ ∗ ) ↦ C ∞ cp ( Σ ) . (161)Conversely, given any element of ( ψ , ψ ′ ) ∈ C ∞ cp ( Σ ) , Proposition 3.6.2 yields a unique solution to (2)whose Cauchy data along Σ are given by ( ψ , ψ ′ ) . Restricting these solutions to H + ≤ ∪ Σ ∗ defines a map C ∞ cp ( Σ ) ↦ C ∞ cp ( H + ≤ ∪ Σ ∗ ) . (162)It immediately follows that the maps (161) and (162) are inverses of each other and hence that both arebijections.Next, we let E V H +≤ ∪ Σ ∗ denote the completion of C ∞ cp ( H + ≤ ∪ Σ ∗ ) under the norm ∣∣( ψ H +≤ , ψ Σ ∗ , ψ ′ Σ ∗ )∣∣ E V H+≤ ∪ Σ ∗ ≐ √ ∫ H +≤ J Kµ [ ˜Ψ ] n µ H + + ∫ Σ ∗ J Vµ [ ˜Ψ ] n µ Σ ∗ , where ˜Ψ is the smooth extension mentioned in the definition of the space C ∞ cp ( H + ≤ ∪ Σ ∗ ) . Finite in time J V energy estimates and the bijection (162) immediately yield a bounded isomorphism E V Σ ↦ E V H +≤ ∪ Σ ∗ . (163)We conclude the proof by combining (163) with the easily observed fact that Theorem 9.2.1 implies thatforward evolution yields a bounded isomorphism E V H +≤ ∪ Σ ∗ ↦ E K H + ⊕ E T I + . At this point, using the properties of the backwards map B − , we can now complete our study of boundednessand integrated local energy decay for the degenerate V -energy theory by giving the proof of Theorem 7.2 ofSection 7.2. 70 roof of Theorem 7.2. Observe that for any s ≥
0, we could have defined a forward map F ( s ) + ∶ E V Σ ∗ s → E K H +≥ s ⊕ E T I + which, in the case of smooth compactly supported data, computes the radiation field of Cauchydata along Σ ∗ s and, similarly, we could have defined a backwards map B ( s ) − ∶ E K H +≥ s ⊕ E T I + → E V Σ ∗ s . Just as before,we would obtain that F ( s ) + and B ( s ) − are both bounded (with a constant independent of s ) and inverses ofeach other. In particular, since F ( s ) + ( ψ ∣ Σ ∗ s , n Σ ∗ s ψ ∣ Σ ∗ s ) = F + ( ψ ∣ Σ ∗ , n Σ ∗ ψ ∣ Σ ∗ ) , we obtain ∫ Σ ∗ s J Vµ [ ψ ] n µ Σ ∗ s ≤ B ∣∣ F ( s ) + ( ψ ∣ Σ ∗ s , n Σ ∗ s ψ ∣ Σ ∗ s )∣∣ E K H+≥ s ⊕E T I+ (164) ≤ B ∣∣ F + ( ψ ∣ Σ ∗ , n Σ ∗ ψ ∣ Σ ∗ )∣∣ E K H+≥ ⊕ E T I+ ≤ B ∫ Σ ∗ J Vµ [ ψ ] n µ Σ ∗ Next, we observe that during the proof Proposition 9.1.2 an integrated estimate in r is in fact established.Using this, an easy density argument, a finite in time energy estimate and an application of Plancherel easilyshow that for any compact set K ⊂ ( r + , ∞ ) we have ∫ R ≥ ∩ { r ∈ K } ( ζ ∣ ∇ / ψ ∣ + ζ ∣ T ψ ∣ + ∣ ˜ Z ∗ ψ ∣ + ∣ ψ ∣ ) ≤ B ( K ) ∣∣ F + ( ψ ∣ Σ ∗ , n Σ ∗ ψ ∣ Σ ∗ )∣∣ E K H+≥ ⊕ E T I+ (165) ≤ B ( K ) ∫ Σ ∗ J Vµ [ ψ ] n µ Σ ∗ . In order to finish the proof we need to exchange the restriction { r ∈ K } in (165) for the appropriateweights in r and r − r + . For r large, the desired estimate is a trivial consequence of the “large- r estimate” ofProposition 4.6.1 in [24] and the arguments of Section 9.4 in [24]. For r close to the horizon it is possible toapply a degenerate version of the redshift effect [44] to achieve the desired estimate. S = F + ○ B + For notational convenience, we have so far restricted our attention to scattering data along H + and I + .However, in view of the discrete isometry ( ) of D , all of our theorems have exact analogues where H + isreplaced by H − and I + is replaced by I − . In particular, we have the following version of Theorems 9.1.1and 9.3.1. Theorem 9.5.1.
Forward evolution (towards the past) uniquely extends to the bounded maps F − ∶ E V ˚Σ → E K H − ⊕ E T I − , F − ∶ E V Σ → E K H − ⊕ E T I − . There exist bounded maps B + B + ∶ E K H − ⊕ E T I − → E V ˚Σ , B + ∶ E K H − ⊕ E T I − → E V Σ , such that F − ○ B + = Id and B + ○ F − = Id . Combining Theorems 9.1.1 and 9.5.1 allows us to define the maps between scattering data along H − ∪ I − and H + ∪ I + . We immediately obtain the following theorem. Theorem 9.5.2.
We define the scattering map (or S -matrix ) S ∶ E K H − ⊕ E T I − → E K H + ⊕ E T I + , S ∶ E K H − ⊕ E T I − → E K H + ⊕ E T I + , by S ≐ F + ○ B + . (166) The map S is then a bounded isomorphism from E K H − ⊕ E T I − to E K H + ⊕ E T I + and E K H − ⊕ E T I − to E K H + ⊕ E T I + Furthermore, for every ( ψ H − , φ I − ) ∈ E K H − ⊕ E T I − , there exists a unique set of initial data ( ψ , ψ ′ ) ∈ E V ˚Σ such that F − ( ψ , ψ ′ ) = ( ψ H − , φ H − ) and F + ( ψ , ψ ′ ) = S ( ψ H − , φ I − ) . An analogous statement holds for ( ψ H − , φ I − ) ∈ E K H − ⊕ E T I − Theorem 5 of Section 2.3.5.Next, we observe that our scattering map S may be given by an explicit formula involving involving thereflection and transmission coefficients. Theorem 9.5.3.
Let S denote the scattering map from Theorem 9.5.2. Then, for ( ψ H − , φ I − ) lying ineither domain E K H − ⊕ E T I − or E K H − ⊕ E T I − , we have S ( ψ H − , φ I − ) (167) = ⎛⎝ √ M πr + ∫ ∞−∞ ∑ mℓ ( − ( ωω − ω + ) a I − T + a H − ˜ R ) e − itω e imφ S mℓ dω, √ π ∫ ∞−∞ ∑ mℓ ( − ( ω − ω + mω ) a H − ˜ T + a I − R ) e − itω e imφ S mℓ dω ⎞⎠ . Here − i ( ω − ω + m ) a H − ≐ √ M r + π ∫ ∞−∞ ∫ S ( ∂ t + ω + ∂ φ ) ψ H − e it ∗ ω e − imφ S mℓ sin θ dt ∗ dθ dφ, − iωa I − ≐ √ π ∫ ∞−∞ ∫ S ∂ t φ I − e itω e − imφ S mℓ sin θ dt dθ dφ, and we emphasize that in interpreting the formula (167), one must keep in mind that the a I ± ( ω, m, ℓ ) areonly defined as functions such that ωa I ± ∈ L ω l mℓ and that the a H ± ( ω, m, ℓ ) are only defined as functionssuch that ( ω − ω + m ) a H ± ∈ L ω l mℓ .Proof. This follows immediately from the construction of S , Propositions 6.8.1 and 6.8.2 and an easy densityargument.In particular, specialising to the case where ψ H − =
0, this establishes
Theorem 12 of Section 2.4.2.
Remark 9.5.1.
We note that one can define the map S by the expression (167) and prove directly byTheorem 6.2.2 that the map is a bounded isomorphism without relying on Theorem 9.1.1; in fact, the proofis a good deal easier because one need never establish the boundedness of the map B − . However, one wouldstill have to prove the decomposition (166) so as to identify elements of E K H − ⊕ E T I − and E K H + ⊕ E T I + as radiationfields of solutions to the wave equation arising from finite energy Cauchy data. In the Schwarzschild case ( a =
0) there is no superradiance and it is much easier to establish that F isinvertible using the unitarity property. Furthermore, the proofs may all be carried out in physical space,i.e. using “time-dependent methods”. In this section we will give a self-contained treatment of how this canbe done in our set-up (cf. the related [48]).The ease of the Schwarzschild scattering theory is all associated with the following unitarity property. Proposition 9.6.1.
Let a = , and observe that in this case K = T = V . Then the forward maps F + ofTheorems 8.2.2, 8.2.3 and 8.2.4 are unitary: ∣∣ F + ( ψ , ψ ′ )∣∣ E T H+≥ + ∣∣ F + ( ψ , ψ ′ )∣∣ E T I+ = ∣∣( ψ , ψ ′ )∣∣ E T Σ ∗ , (168) ∣∣ F + ( ψ , ψ ′ )∣∣ E T H+ + ∣∣ F + ( ψ , ψ ′ )∣∣ E T I+ = ∣∣( ψ , ψ ′ )∣∣ E T ˚Σ , (169) ∣∣ F + ( ψ , ψ ′ )∣∣ E T H+ + ∣∣ F + ( ψ , ψ ′ )∣∣ E T I+ = ∣∣( ψ , ψ ′ )∣∣ E T Σ . (170)72 roof. We will only prove (168) as the proof of (169) and (170) is exactly the same. By density it sufficesto prove (168) in the case when ( ψ , ψ ′ ) ∈ C ∞ cp ( Σ ∗ ) . Let us assume this for the remainder of the proof. Itfollows now from Proposition 3.8.1 and Theorem 3.7.2 that we can find a dyadic sequence { τ i } ∞ i = such thatlim i →∞ ∫ S τi J Tµ [ F ( ψ , ψ ′ )] n µS τi = . Next, for each τ i a J T energy estimate yields ∫ H + ( ,τ i ) J Tµ [ F + ( ψ , ψ ′ )] n µ H + + ∫ I +≤ τi J Tµ [ F + ( ψ , ψ ′ )] n µ I + + ∫ S τi J Tµ [ F + ( ψ , ψ ′ )] n µS τi = ∣∣( ψ , ψ ′ )∣∣ E T Σ ∗ . We conclude the proof by taking i → ∞ . Remark 9.6.1.
Let us remark that in the Schwarzschild a = case suitable versions of the statements ofTheorems 3.7.1 and 3.7.2 can be obtained without phase space analysis with respect to either time or angularfrequency decompositions. See [20] and [22]. Thus, not only the construction but also all relavent propertiesof F are obtained purely with physical space (i.e. “time-dependent”) methods. Cf. with the Kerr a ≠ casewhere the construction of F is still formulated in the time domain but requires the result of Theorem 3.7.1which is itself based on frequency-analysis. The injectivity of the forward map is an immediate corollary.
Corollary 9.6.1.
Let a = . Then the forward map F + is injective. We now construct the backwards map.
Theorem 9.6.1.
Let a = . Then the forward map F + is a unitary isomorphism (with either domain E V ˚Σ or E V Σ ∗ or E V Σ ) with two-sided unitary inverse B − satisfying B − ○ F + = Id , F + ○ B − = Id .Proof. We consider the case where the domain is E V ˚Σ , the cases E V Σ or E V Σ ∗ are handled in an analogousfashion.First of all, using the physical space construction from the proof of Proposition 9.1.4 we may define thebackwards map on a dense set: B − ∶ C ∞ cp ( H + ≥ ) ⊕ C ∞ cp ( I + ) → E T Σ ∗ . Furthermore, the proof of Proposition 9.1.4 shows that ( ψ H + , φ I + ) ∈ C ∞ cp ( H + ≥ ) ⊕ C ∞ cp ( I + ) implies F + ( B − ( ψ H + , φ I + )) = ( ψ H + , φ I + ) . We thus conclude that the forward map F + has a dense image. Since the unitarity of F + implies thatthe backwards map B − is bounded on its domain, it follows immediately that F + is in fact surjective. Therest of theorem follows immediately. Remark 9.6.2.
It is instructive to compare the above “time-dependent method” construction of B − to thestationary-method construction of Defintion 9.1.1. Of course, one could have defined B − on a dense subsetin the general Kerr case with Proposition 9.1.4, but one would still need to have used the representation ofDefinition 9.1.1 to estimate it so as to take the completion. Applying the discrete isometry t ↦ − t of Schwarzschild yields the analogues of the above statementsfor F − and B + . As before, we then define the scattering map S = F + ○ B + . We immediately obtain thefollowing corollary. Theorem 9.6.2.
Let a = . Then the scattering maps S ∶ E T H − ⊕ E T I − ↦ E T H + ⊕ E T I + are S ∶ E T H − ⊕ E T I − ↦ E T H + ⊕ E T I + unitary isomorphisms. We collect here some further applications of our scattering theory.In Section 10.1, we will construct a physical-space (time-domain) theory of superradiant reflection. The-orem 10.1.1 will give the results of
Theorem 6 and
Theorem 7 of Section 2.3.6. We will also formulateand prove an analogous amplification statement in terms of compactly supported smooth Cauchy data (The-orem 10.1.2).We will then show in Section 10.2 a “pseudo-unitary” property (Theorem 10.2.1) of our scattering map S restricted to past scattering data supported only on I − , as well as a genuine unitarity property of S restrictedto an appopriate Hilbert space of non-superradiant data (Theorem 10.2.2). This will give Theorem 8 and
Theorem 9 of Section 2.3.7.Finally, in Section 10.3 we will establish the injectivity result Theorem 10.3.1, which corresponds to“uniqueness of scattering states” for improperly posed scattering problems (for which there is no existence).This will give
Theorem 10 of Section 2.3.8.
First we define the physical-space reflection and transmission maps referred to already in Section 2.3.6.
Definition 10.1.1.
Define the reflection map R and the transmission map T by R ≐ π E T I+ ○ S ∣ { } ⊕ E T I− , T ≐ π E K H+ ○ S ∣ { } ⊕ E T I− where π E T I+ ∶ E K H + ⊕ E T I + → E T I + , π E K H+ ∶ E K H + ⊕ E T I + → E K H + are the natural projections. We can view S ∣ { } ⊕ E T I− = R ⊕ T . We are now ready for the following theorem.
Theorem 10.1.1.
The operator norms of T and R are bounded ∥ T ∥ ≤ B, ∥ R ∥ ≤ B If a = , then ∥ R ∥ = , whereas if a ≠ then ∥ R ∥ > . Proof.
The maps T and R are compositions of the bounded maps π E T I+ , π E K H+ and S and hence are bounded.Next, it follows immediately from the formula (167) that ∣∣ R ∣∣ = sup ( ω,m,ℓ ) R ( ω, m, ℓ ) . Thus, when a ≠
0, Corollary 5.3.1 shows that ∥ R ∥ >
1, and when a =
0, Corollary 6.4.1 shows that ∥ R ∥ = Theorems 6 and 7 from Section 2.3.6.With a little more work, we can upgrade the above result to the following statement.
Theorem 10.1.2.
Let a ≠ . There exists a smooth solution ψ on D such that the initial data for ψ along ˚Σ is supported away from the bifurcate sphere B (though not necessarily of compact support), ψ has finite V -energy along ˚Σ and we have ∫ H − J Kµ [ ψ ] n µ H − = , ∫ I + J Tµ [ ψ ] n µ I + > ∫ I − J Tµ [ ψ ] n µ I + . lso, for all R < ∞ there exists a solution ψ R to (2) on R such that the initial data for ψ R along ˚Σ arecompactly supported within r ∈ [ R, ∞ ) , and ψ R exhibits superradiance in the sense that ∫ I + J Tµ [ ψ R ] n µ I + > ∫ ˚Σ J Tµ [ ψ R ] n µ ˚Σ . Proof.
We start by letting a I − ( ω, m, ℓ ) be a non-zero smooth function which is compactly supported in theset of ( ω, m, ℓ ) which satisfy ω > , ( ω − ω + m ) < . We define u ≐ T − i ( ω − ω + m ) a I − U hor = R iω a I − U inf + iω a I − U inf ,ψ ≐ ( r + a ) / √ π ∫ ∞−∞ ∑ mℓ e − iωt e imφ S mℓ ( aω, θ ) u dω. Note that Proposition 6.8.1, Theorem 7.1, and Proposition 9.1.2 imply ∫ I − J Tµ [ ψ ] n µ I − = √ ∫ ∞∞ ∑ mℓ ∣ a I − ∣ . Now, Corollary 5.3.1 implies ∣ R ∣ ∣ a I − ∣ ≥ ∣ a I − ∣ + ǫ, for some sufficiently small ǫ > ∫ I + J Tµ [ ψ ] n µ I + = √ ∫ ∞−∞ ∑ mℓ ∣ R ∣ ∣ a I − ∣ dω > √ ∫ ∞−∞ ∑ mℓ ∣ a I − ∣ dω = ∫ I − J Tµ [ ψ ] n µ I − . (171)Finally, applying Proposition 6.8.2, Theorem 7.1, and Proposition 9.1.2 yields ∫ H − J Kµ [ ψ ] n µ H − = . Thus, we may multiply ψ be an appropriate constant to define a solution ψ which will satisfy1. ∫ I + J Tµ [ ψ ] n µ I + > ∫ I − J Tµ [ ψ ] n µ I − = ∫ H − J Kµ [ ψ ] n µ H − = ϕ denote the radiation field for ψ along I − . Let χ ( τ ) be a bump function, ǫ > ψ be the unique solution to (2) whose radiation field vanishes along H − and has the radiation field χ ( τ ǫ − ) ˜ ϕ √ ∫ I − J Tµ [ χ ( τ ǫ − ) ˜ ϕ ] n µ I − along I − . Using the boundedness of the map S from Definition 9.5.2, it is clear that taking ǫ sufficientlysmall (and then fixing ǫ ) will imply that1. ∫ I + J Tµ [ ψ ] n µ I + > ∫ I − J Tµ [ ψ ] n µ I − =
1. 75. ∫ H − J Kµ [ ψ ] n µ H − = ψ along ˚Σ is compactlysupported; thus we may set ψ ≐ ψ .In order to construct ψ R we need to do a little more work. We begin by recalling the estimate (135), theproof of which (being invariant under time reversal) implieslim s →−∞ ∫ { t = s } ∩ [ r + + ǫ ′ ,r + + A ] J Nµ [ ψ ] n µ { t = s } = ∀ < ǫ ′ < A < ∞ . (172)Let χ ( x ) be cut-off which is 0 for x ∈ [ , ] and identically 1 for x ∈ [ , ∞ ) . Letting ǫ ′ be small enough sothat K is timelike for r ∈ [ r + , r + + ǫ ′ ] , applying a J K energy estimate to ( − χ ( r ( ǫ ′ ) − )) ψ ǫ easily implies ∫ { t = s } ∩ [ r + ,r + + ǫ ′ ] J Kµ [ ψ ] n µ { t = s } (173) ≤ ∫ H − J Kµ [ ψ ] n µ H − + B ( ǫ ′ ) ∫ ∞ s ∫ { t = s } ∩ [ r + + ǫ ′ ,r + + ǫ ′ ] [ J Kµ [ ψ ] n µ { t = s } + ∣ ψ ∣ ] ds. Theorem 3.7.1 implies the the second term on the right hand side of this estimate converges to 0 as s → ∞ .Since the first term on the right hand side vanishes, we conclude thatlim sup s →−∞ ∫ { t = s } ∩ [ r + ,r + + ǫ ′ ] J Kµ [ ψ ] n µ { t = s } = . (174)Taking R suitably large, and applying a similar argument in the region r ≥ R , one may easily deduce thatlim sup s →−∞ ∫ { t = s } ∩ { r ∈ [ R / , ∞ )} J Tµ [ ψ ] n µ { t = s } ≤ . (175)Let ǫ ′′ > R sufficiently large and s = s ( ǫ ′′ , R ) sufficiently large and negative so that ∫ { t = s } J Vµ [ ψ − χ ( rR − ) ψ ] n µ { t = s } < ǫ ′′ , (176) ∫ { t = s } J Vµ [ χ ( rR − ) ψ ] n µ { t = s } ≤ + ǫ ′′ , (177)Let ψ be the solution to (2) whose initial data along { t = s } are given by χ ( rR − ) ψ . Now set ψ ( t, r, θ, φ ) ≐ ˜ ψ ( t − s, r, θ, φ ) . It is clear that if we choose ǫ ′′ small enough, then Theorem 7.1 will implythat ∫ I + J Tµ [ ψ ] n µ I + > ∫ { t = } J Tµ [ ψ ] n µ { t = } . Finally, appealing to Theorem 7.1 one more time, we may define ψ R to be the unique solution to (2)whose initial data along { t = } is given by ( − χ ( rS − )) ψ for some sufficiently large S . The next sequence of results expresses the conservation of the J T flux. Since this flux is unsigned along H + we may interpret this as a statement of “pseudo-unitarity”. Proposition 10.2.1.
Let ψ be a solution to (2) whose initial data lines in E N Σ ∗ . Observe that Theorem 3.7.1implies that ∫ H + ∣ J Tµ [ ψ ] n µ H + ∣ ≤ B ∫ H + J Nµ [ ψ ] n µ H + ≤ B ∫ Σ ∗ J Nµ [ ψ ] n µ Σ ∗ . In particular, even though the integrand is unsigned, the integral ∫ H + J Tµ [ ψ ] n µ H + s well defined and finite.We then have ∫ H + J Tµ [ ψ ] n µ H + = ∫ Σ ∗ J Tµ [ ψ ] n µ Σ ∗ − ∫ I + J Tµ [ ψ ] n µ I + . Proof.
By density considerations, we may assume that ψ lies in C ∞ cp ( Σ ∗ ) . As we have already argued a fewtimes before, Proposition 3.8.1 and Theorem 3.7.2 then allow us to find a dyadic sequence { τ i } such that ∫ S τi J Nµ [ ψ ] n µS τi → i → ∞ . For each τ i , a J T energy estimate yields ∫ H + ( ,τ i ) J Tµ [ ψ ] n µ H + + ∫ S τi J Tµ [ ψ ] n µS τi + ∫ I +≤ τi J Tµ [ ψ ] n µ I + = ∫ Σ ∗ J Tµ [ ψ ] n µ Σ ∗ . Now we simply take τ i → ∞ and observe that ∣ J Tµ [ ψ ] n µS τi ∣ ≤ B J Nµ [ ψ ] n µS τi . Remark 10.2.1.
Of course, one may prove a version of Proposition 10.2.1 where the hypersurface Σ ∗ isreplaced by Σ . Theorem 10.2.1.
For any φ ∈ C ∞ cp ( I − ) we have ∫ H + J Tµ [ T φ ] n µ H + + ∫ I + J Tµ [ R φ ] n µ I + = ∫ I − J Tµ [ φ ] n µ I − , (178) ∫ H + ∣ J Tµ [ T φ ] n µ H + ∣ ≤ B ∫ I − J Tµ [ φ ] n µ I − . (179) Then, an easy density argument shows that (178) and (179) hold for arbitrary φ ∈ E T I − . Remark 10.2.2.
As is immediately clear from the proof below, the inequality (179) holds without the non-superradiant assumption.Proof.
The equality (178) follows immediately from Remark 10.2.1 and the fact that φ ∈ C ∞ cp ( I − ) impliesthat ∫ Σ J Nµ [ B + ( , φ )] n µ Σ < ∞ . The inequality (179) follows immediately from Plancherel, Theorem 9.5.3, Theorem 10.1.1 and the factthat combining Theorem 10.1.1 and Corollary 5.3.1 implies that ∣ ωω − ω + T ∣ is uniformly bounded.This gives Theorem 8 of Section 2.3.7.
Remark 10.2.3.
Note that we cannot consider the case of general initial data in E K H − as ψ ∈ E K H − does notimply that ∫ H − J Tµ [ ψ ] n µ H − < ∞ . Finally, we observe that if we restrict the initial data along H − and I − to be non-superradiant, then themap S will be unitary in the standard sense. First we introduce the relevant function spaces. Definition 10.2.1.
We define E T, ♮ I ± to be the Hilbert space consisting of functions f ( τ, θ, φ ) ∶ I ± → C suchthat ˆ f ( ω, m, ℓ ) = √ π ∫ ∞−∞ ∫ S e iωt e − imφ S mℓ f sin θ dt dθ dφ, lies in the closure of functions compactly supported in {( ω, m, ℓ ) ∶ ω ( ω − ω + m ) > } under the inner product ∫ ∞−∞ ∑ mℓ ω Re ( f f ) . efinition 10.2.2. We define E T, ♮ H ± to be the Hilbert space consisting of functions f ( τ, θ, φ ) ∶ H ± → C suchthat ˆ f ( ω, m, ℓ ) = √ π ∫ ∞−∞ ∫ S e iωt e − imφ S mℓ f sin θ dt dθ dφ, lies in the closure of functions compactly supported in {( ω, m, ℓ ) ∶ ω ( ω − ω + m ) > } (180) under the inner product ∫ ∞−∞ ∑ mℓ ω ( ω − ω + m ) Re ( f f ) . The following theorem is an immediate consequence of the microlocal energy identity Proposition 5.2.2.
Theorem 10.2.2.
The restriction of the map S to functions whose Fourier transforms are compactlysupported in (180) extends by density to a map S ∶ E T, ♮ H − ⊕ E T, ♮ I − → E T, ♮ H − ⊕ E T, ♮ I − which is a unitary isomorphismwith respect to the positive definite inner product ⟨( ψ , φ ) , ( ψ , φ )⟩ = ∫ ∞−∞ ∑ mℓ [ ω ( ω − ω + m ) Re ( ˆ ψ ˆ ψ ) + ω Re ( ˆ φ ˆ φ )] . This gives
Theorem 9 of Section 2.3.7. Note that the above reduces again to Theorem 9.6.2 in the case a =
0, where E T, ♮ H ± ⊕ E T, ♮ I ± coincide with E T H ± ⊕ E T I ± . It also yields in particular that S restricted to axisymmetricscattering data is unitary. We turn finally to the “ill-posed case”, where one attempts to pose scattering data on H + ∪ H − , I + ∪ I − , H − ∪ I + or H + ∪ I − .To state our theorems, let us note first that we may define the forward maps F ∶ E V Σ → E T I + ⊕ E T I − , F ∶ E V Σ → E K H + ⊕ E K H − , F ∶ E V Σ → E K H − ⊕ E T I + , F ∶ E V Σ → E K H + ⊕ E T I − , (181)by completion of ( ψ , ψ ′ ) ↦ ψ ↦ ( φ ∣ I + , φ ∣ I − ) or ( ψ ∣ H + , ψ ∣ H − ) or ( ψ ∣ H − , φ ∣ I + ) or ( ψ H + , φ ∣ I − ) , (182)and these are again bounded maps by our previous results. We have the following statement of uniqueness(but not existence!) of “improper” scattering states: Theorem 10.3.1.
The maps F of ( ) are all injective.Proof. We start with the case of the first two maps of ( ) .First of all, the proof is conceptually clearer in the case of smooth compactly supported initial data, andwe thus begin with this case. Consider ( ψ , ψ ′ ) ∈ C ∞ cp ( Σ ) , let ψ be solution of ( ) and assume ( ψ , ψ ′ ) isin the kernel of the first or second map of ( ) . Then, upon an application of Carter’s separation to ψ we have that for almost every ( ω, m, ℓ ) , the resulting u is a smooth solution to the radial o.d.e. (60) suchthat when ω ≠ ∣ a I + ∣ + ∣ a I − ∣ = ∣ a H + ∣ + ∣ a H − ∣ =
0, respectively. It follows immediately from the localexistence theory for these o.d.e.’s that u is identically 0 (see [45]) whenever ω ≠
0, and thus ψ is 0. It followsthat ( ψ , ψ ′ ) = ( , ) .For general ( ψ , ψ ′ ) ∈ E V Σ , let us first consider the case only of the first map of ( ) , i.e., let ( ψ , ψ ′ ) ∈ ker F ∶ E V Σ → E T I + ⊕ E T I − . Let ψ denote the solution of the wave equation ( ) arising from ( ψ , ψ ′ ) . Theo-rem 3.7.1 implies that ψ lies in L ,r L t,r,θ,φ . In particular, we can take the Fourier transform of ψ and definethe Carter separated function u ( r, ω, m, ℓ ) which will lie in L ,r L ω l m,ℓ .Let ǫ >
0, let F ♭ denote an arbitrary compact set of ( ω, m, ℓ ) and let K denote an arbitrary compact setin ( r + , ∞ ) . Now, by regularizing the initial data for ψ , we can produce a solution ψ ǫ to (2) with smoothcompactly supported initial data such that ∫ Σ J Vµ [ ψ − ψ ǫ ] n µ Σ ≤ ǫ.
78t follows immediately from the fact that the forward map is well defined, that ∫ I ± J Tµ [ ψ ǫ ] n µ I ± ≤ Bǫ.
In particular, if we let a ǫ, I ± denote the microlocal radiation fields for ψ ǫ , Propositions 6.8.1 and 4.2.2 implythat ∫ ∞−∞ ∑ mℓ [∣ a ǫ, I + ∣ + ∣ a ǫ, I − ∣ ] dω ≤ Bǫ.
Letting u ǫ denote the result of applying Carter’s separation to ψ ǫ , it now follows immediately from standardo.d.e. theory that ∫ K ∫ ( ω,m,ℓ ) ∈ F ♭ ∣ u ǫ ∣ ≤ B ( K , F ♭ ) ǫ. Finally, an application of Theorem 3.7.1 to the ψ − ψ ǫ followed by an application of Plancherel implies ∫ K ∫ ( ω,m,ℓ ) ∈ F ♭ ∣ u ∣ ≤ B ( K , F ♭ ) ǫ, where u ( r, ω, m, ℓ ) is the result of applying Carter’s separation to ψ . Since ǫ , K , and F ♭ were arbitrary, weconclude that u and hence ψ vanishes.The case where ( ψ , ψ ′ ) lies in the kernel of the second map of ( ) is treated in exactly the same way.We turn now to the case when ( ψ , ψ ′ ) lies in the kernel of the third and fourth map of ( ) . Since ψ isnot necessarily sufficiently integrable, we cannot use Definition 5.4.1 to define the microlocal radiation fields;instead we define a I ± ( ω, m, ℓ ) and a H ± ( ω, m, ℓ ) by applying Carter’s separation to the functions F ± ( ψ , ψ ′ )∣ I ± and F ± ( ψ , ψ ′ )∣ H ± : a I ± ≐ ∫ ∞−∞ ∫ S e itω e − imφ S mℓ F ± ( ψ , ψ ′ )∣ I ± sin θ dt dθ dφ,a H ± ≐ √ M r + ∫ ∞−∞ ∫ S e it ∗ ω e − imφ S mℓ F ± ( ψ , ψ ′ )∣ H ± sin θ dt ∗ dθ ∗ dφ ∗ . Now, we may apply Theorem 9.5.3 (and its complex conjugated version) to conclude that F + ( ψ , ψ ′ )∣ I + = √ π ∫ ∞−∞ ∑ mℓ ( − ( ω − ω + mω ) a H − ˜ T + a I − R ) e − itω e imφ S mℓ dω, (183) F − ( ψ , ψ ′ )∣ I − = √ π ∫ ∞−∞ ∑ mℓ ( − ( ω − ω + mω ) a H + ˜ T + a I + R ) e − itω e imφ S mℓ dω, (184) F + ( ψ , ψ ′ )∣ H + = √ M πr + ∫ ∞−∞ ∑ mℓ ( − ( ωω − ω + ) a I − T + a H − ˜ R ) e − itω e imφ S mℓ dω, (185) F − ( ψ , ψ ′ )∣ H − = √ M πr + ∫ ∞−∞ ∑ mℓ ( − ( ωω − ω + ) a I + T + a H + ˜ R ) e − itω e imφ S mℓ dω. (186)Observe that if ( ψ , ψ ′ ) lies in the kernel of the third map of ( ) , then a H − and a I + will vanish almosteverywhere. Then (183) and (186) imply that a I − R and a H + ˜ R both vanish almost everywhere. However,Corollary 6.5.1 implies that R and ˜ R can only vanish at isolated points in ω . We conclude that a H + and a I − can only be non-zero at isolated points and hence that a H + and a I − vanish almost everywhere. We concludethat F ± ( ψ , ψ ′ ) = ( , ) and thus that ψ vanishes.The case where ( ψ , ψ ′ ) lies in the kernel of the fourth map of ( ) is treated in a similar fashion.We have thus obtained now Theorem 10 of Section 2.3.8.
Remark 10.3.1.
In regard to the first two maps of ( ) , we note that is possible to prove localised versionsof the above via the techniques of “unique continuation”, where ψ is only assumed to vanish on certainportions of H + ∪ H − or portions of I + ∪ H + , but with stronger regularity assumptions and decay at infinity.See [9] for such results in the Schwarzschild case, [2] for such results on general asymptotically flat spacetimesand [3] for such results for (among other things) certain non-linear wave equations on Minkowski space. In this final section, we shall show that any solution of the wave equation ( ) on Schwarzschild assumed tohave a particular choice of radiation field necessarily would have infinite N -energy on the hypersurface Σ ∗ .Our theorem can be stated as follows: Theorem 11.1.
Let a = and let ψ be a smooth spherically symmetric solution of the wave equation in theregion ˚ R ≥ such that1. The initial data for ψ lies in the closure of compactly supported initial data under the norm ∫ Σ ∗ [ J Tµ [ ψ ] + J Tµ [ T ψ ]] n µ Σ ∗ . ∂ ˜ v ψ extends continuously to the function ( t ∗ + ) − p on H + ≥ for some p > .3. ∂ ˜ v ( T ψ ) extends continuously to the function − p ( t ∗ + ) − p − on H + ≥ for the same p as above.4. There exists τ such that τ > τ implies lim r →∞ rψ ∣ τ = τ = . Then ∫ Σ ∗ J Nµ [ ψ ] n µ Σ ∗ = ∞ . We will prove Theorem 11.1 in Sections 11.1–11.2 below. We have stated our theorem in the above formso as to be independent of the existence of the scattering theory maps F + , B − , etc., proven in this paper.Thus, the proof of Theorem 11.1 can be read independently of the rest of our paper. The argument exploitsthe blue-shift factor of the horizon together with a simple monotonicity property of the spherically symmetricwave equation.In combination with the results of our paper, Theorem 11.1 can be reinterpreted in the context of both our N -energy and our T -energy theories. First, applying Theorem 9.2.1, we shall construct solutions ψ satisfyingthe assumptions of Theorem 11.1 such that their induced data lie in E T Σ ∗ and give a short discussion of thesignificance of the existence of such solutions. Finally, in Section 11.4, we shall reinterpret Theorem 11.3 asa statement of the non-surjectivity of the map F + ∶ E N Σ ∗ → E N H + ⊕ E N I + of Theorem 8.2.1. This will thus give Theorem 2 of Section 2.3.2.
Setting a = g Schw = − ( − Mr ) dt + ( − Mr ) − dr + r ( dθ + sin θ dφ ) . (187)To get an explicitly regular expression for the metric near the event horizon H + we introduce the ( v, r, θ, φ ) coordinate system defined by dr ∗ dr ≐ ( − Mr ) − , ˜ v ≐ t + r ∗ . The metric then takes the form g Schw = − ( − Mr ) d ˜ v + d ˜ vdr + r ( dθ + sin θ dφ ) . (188)Note that we have T = ∂ ˜ v in the ( ˜ v, r, θ, φ ) coordinate system. Let us also agree to set Y ≐ ∂ r .80t will also be useful to introduce a ( ˜ t, r, θ, φ ) coordinate system in the following fashion. Let χ ( r ) be acut-off which is identically 0 for r ∈ [ M, M ] and identically 1 for r ∈ [ M, ∞ ) . We then set˜ t ( t, r ) ≐ t + r ∗ − χ ( r ) r ∗ . Note that ∂ ˜ t = T is Killing.Finally, it turns that it often convenient to work in the null coordinate system ( ˜ u, ˜ v, θ, φ ) where ˜ v isdefined as before and ˜ u ≐ t − r ∗ . Then metric then takes the form g Schw = − ( − Mr ) d ˜ ud ˜ v + r ( dθ + sin θ dφ ) . (189) Remark 11.1.1.
These coordinates break down at the horizon, where ˜ u = ∞ . Nevertheless we can stilluse these coordinates in an effective manner near H + as long we remember that the ( − Mr ) − ∂ ˜ u = Y is aregular vector field on H + . If not explicitly noted otherwise, ∂ ˜ v and ∂ ˜ u will also be defined in the ( ˜ u, ˜ v, θ, φ ) coordinate system. Letus agree to set L ≐ ∂ ˜ v .In null coordinates, the wave equation (2) applied to a spherically symmetric function ψ takes the form ∂ ˜ v ( r ∂ ˜ u ψ ) + ∂ ˜ u ( r ∂ ˜ v ψ ) = ⇔ ∂ v, ˜ u ψ + ( ∂ ˜ u r ) ∂ ˜ v ψr + ( ∂ ˜ v r ) ∂ ˜ u ψr = . (190)The equation (190) is equivalent to the following coupled transport equations for r∂ ˜ u ψ and r∂ ˜ v ψ : ∂ ˜ u ( r∂ ˜ v ψ ) = − ( ∂ ˜ v r )( r∂ ˜ u ψ ) r = − ( − Mr ) r∂ ˜ u ψr , (191) ∂ ˜ v ( r∂ ˜ u ψ ) = − ( ∂ ˜ u r )( r∂ ˜ v ψ ) r = ( − Mr ) r∂ ˜ v ψr . (192)Near the event horizon it will be useful to work with transport equations for r ( − Mr ) − ∂ ˜ u ψ and r∂ ˜ v ψ : ( − Mr ) − ∂ ˜ u ( r∂ ˜ v ψ ) = − ( − Mr ) [ r ( − Mr ) − ∂ ˜ u ψ ] r , (193) ∂ ˜ v ( r ( − Mr ) − ∂ ˜ u ψ ) + Mr ( r ( − Mr ) − ∂ ˜ u ψ ) = r∂ ˜ v ψr . (194) Remark 11.1.2.
The fact that Mr ∣ H + = M > represents the positivity of surface gravity and is intimatelytied to the (local) redshift effect. See the discussion in [20]. We are now ready for the proof of Theorem 11.1. We will proceed in four steps.1. Letting ψ be as in Theorem 11.1, we begin by establishing a local energy decay statement with a sharprate: ∫ S τ ∩ { r ≤ R } [( T ψ ) + ( ∂ r ∗ ψ ) ] ≤ B ( R )( + τ ) − p ∀ R > r + .
2. Using the decay from the previous step, for all sufficiently small ǫ > ( + ˜ v ) − p lower bound for ∂ ˜ v ψ along H + to a ( + v ) − p lower bound on the hypersurfaces { r = M + ǫ } . Cf. [13, 16].81. Using the ( ˜ t, r, θ, φ ) coordinate system, define ˜ ψ ( ˜ t, r ) ≐ − ∫ ∞ ˜ t ψ ( ˜ s, r ) d ˜ s . Using the equation (194) andthe previous steps, we will prove that unless ∫ Σ ∗ J Nµ [ ψ ] n µ Σ ∗ = ∞ , then ( − Mr ) − ∂ ˜ u ˜ ψ and ∂ ˜ v ˜ ψ bothare positive along one of the hypersurfaces { r = M + ǫ } .4. Under the assumption that ∫ Σ ∗ J Nµ [ ψ ] n µ Σ ∗ < ∞ we will use some monotonicity hidden in the system (191)and (192), and show that the positivity of ∂ ˜ u ˜ ψ and ∂ ˜ v ˜ ψ along { r = M + ǫ } propagates along outgoingnull curves. Finally, we will see that this positivity of r∂ ˜ u ˜ ψ and r∂ ˜ v ˜ ψ implies that rψ cannot vanishalong I + , yielding a contradiction to the assumption ∫ Σ ∗ J Nµ [ ψ ] n µ Σ ∗ < ∞ . We begin with the following proposition.
Proposition 11.2.1.
Let ψ and p be as in the statement of Theorem 11.1 and satisfy ( . . ) . Then, forall R < ∞ we have ∫ S τ ∩ { r ≤ R } [( T ψ ) + ( ∂ r ∗ ψ ) ] ≤ B ( R )( + τ ) − p . Proof.
We begin by arguing that ∫ S τ J Tµ [ ψ ] n µS τ ≤ B [ ∫ H +≥ τ J Tµ [ ψ ] n µ H + + ∫ I +≥ τ J Tµ [ ψ ] n µ I + ] ≤ B ( + τ ) − p + . (195)Let τ < ∞ . Since ∫ Σ ∗ J Tµ [ ψ ] n µ Σ ∗ < ∞ we may find a sequence of solutions { ψ i } to (2) whose initial datalies in ˜ E Σ ∗ and which satisfy lim i →∞ ∫ S τ J Tµ [ ψ i ] n µS τ = ∫ S τ J Tµ [ ψ ] n µS τ . As we have already observed multipletimes, Theorem 3.7.2 and Proposition 3.8.1 imply that we may find a dyadic sequence { τ ( i ) j } ∞ j = such thatlim j →∞ ∫ S τ ( i ) j J Tµ [ ψ i ] n µS τ ( i ) j =
0. Then we may apply a J T energy estimate to each ψ i and conclude ∫ S τ J Tµ [ ψ i ] n µS τ = ∫ H + ( τ,τ ( i ) j ) J Tµ [ ψ i ] n µ H + + ∫ I + ( τ,τ ( i ) j ) J Tµ [ ψ i ] n µ I + + ∫ S τ ( i ) j J Tµ [ ψ i ] n µS τ ( i ) j . Taking j to infinity and then i to infinity yields ∫ S τ J Tµ [ ψ ] n µS τ = ∫ H +≥ τ J Tµ [ ψ ] n µ H + + ∫ I +≥ τ J Tµ [ ψ ] n µ I + . Finally, using that ∂ ˜ v ψ extends continuously to ( + t ∗ ) − p one may easily show that ∫ H +≥ τ J Tµ [ ψ ] n µ H + ≤ B ∫ ∞ τ ( + t ∗ ) − p dt ∗ , and hence establish (195).Next, we commute with the Killing vector field T and consider the solution T ψ . Repeating the aboveprocedure (using in particular that
T ψ is assumed that have a finite J T energy along Σ ∗ and the assumptionon the limit of ∂ ˜ v ( T ψ ) to H + ≥ ) another J T energy estimate implies ∫ S τ J Tµ [ T ψ ] n µS τ ≤ B [ ∫ H +≥ τ J Tµ [ T ψ ] n µ H + + ∫ I +≥ τ J Tµ [ T ψ ] n µ I + ] ≤ B ( + τ ) − p − . (196)The final ingredient is an integrated local energy decay estimate. Setting X ≐ f ( r ∗ ) ∂ r ∗ for a function f to be fixed later, a straightforward calculation yields the following general formula: ∇ µ J Xµ [ ψ ] = ( f ′ ( − Mr ) − + f r − ) ( T ψ ) + ( f ′ ( − Mr ) − − f r − ) ( ∂ r ∗ ψ ) . (197)We set f ≐ − r − and obtain ∇ µ J Xµ [ ψ ] = r − ( T ψ ) + r − ( ∂ r ∗ ψ ) . (198)82eeping in mind that X ∣ H + = − ( M ) − T and ∣ ∫ S τ J Xµ [ ψ ] n µS τ ∣ ≤ B ∫ S τ J Tµ [ ψ ] n µS τ , combining (198) with (195)and (196) yields the following two estimates: ∫ ∞ τ ∫ S τ ∩ { r ≤ R } [( T ψ ) + ( ∂ r ∗ ψ ) ] ≤ B ( R ) ( + τ ) − p + ∀ R > r + , (199) ∫ ∞ τ ∫ S τ ∩ { r ≤ R } [( T ψ ) + ( ∂ r ∗ T ψ ) ] ≤ B ( R ) ( + τ ) − p − ∀ R > r + . (200)We will now interpolate between these four estimates in a straightforward fashion. For every k ≥
1, using the fact that ∫ k + k ∫ S τ ∩ { r ≤ R } [( T ψ ) + ( ∂ r ∗ ψ ) ] ≤ B ( R ) ( k ) − p + , we may find a τ k ∈ [ k , k + ] such that ∫ S τk ∩ { r ≤ R } [( T ψ ) + ( ∂ r ∗ ψ ) ] ≤ B ( R ) τ − pk . (201)Now consider τ ∈ [ τ k , τ k + ] . The fundamental theorem of calculus and the estimates (199), (200), and (201)imply ∫ S τ ∩ { r ≤ R } [( T ψ ) + ( ∂ r ∗ ψ ) ] (202) ≤ ∫ S τk + ∩ { r ≤ R } [( T ψ ) + ( ∂ r ∗ ψ ) ] + B ∫ τ k + τ k ∫ S s ∩ { r ≤ R } [ τ − k ( T ψ ) + τ − k ( ∂ r ∗ ψ ) + τ k ( T ψ ) + τ k ( ∂ r ∗ T ψ ) ] ≤ B ( R ) [ τ − pk + + τ − pk + τ − pk ] ≤ B ( R ) τ − p . Remark 11.2.1.
If one adds the assumption that ∫ Σ ∗ J Nµ [ ψ ] n µ Σ ∗ < ∞ , then one could establish the localenergy decay statement using only transport equations in the region { r ≤ R } . Corollary 11.2.1.
Of course, the use of the particular foliation { S τ } is not important for the above proposi-tion. By modifying S τ to equal { ˜ v = τ } in the region { r ≤ R } and repeating the above proof, one immediatelyobtains ∫ { ˜ v = τ } ∩ { r ≤ R } [( T ψ ) + ( ∂ r ∗ ψ ) ] ≤ B ( R )( + τ ) − p . Analogously, for ( r , r ) ⊂ ( r + , ∞ ) one may show ∫ { ˜ u = τ } ∩ { r ∈ [ r ,r ]} [( T ψ ) + ( ∂ r ∗ ψ ) ] ≤ B ( r , r )( + τ ) − p . We now turn to the proof of
Proposition 11.2.2.
Let ψ be as in Theorem 11.1. Then, for all ǫ > sufficiently small, b ( + ˜ v ) − p ≤ ( r∂ ˜ v ψ ) ∣ { r ≤ M + ǫ } ≤ B ( + ˜ v ) − p . For example, see [51] for an application of such an interpolation argument to interior decay for the wave equation. roof. Keeping in mind that ( − Mr ) − ∂ ˜ u = Y is equal to ∂ r in ( ˜ v, r, θ, φ ) coordinates, we integrate thetransport equation (191) and obtain ( r∂ ˜ v ψ ) ∣ ( ˜ v,r ) = ( τ, M + ǫ ) = ( r∂ v ψ ) ∣ ( ˜ v,r ) = ( τ, M ) + ∫ { ˜ v = τ } ∩ { r ≤ M + ǫ } ∂ ˜ u ψ dr. (203)Cauchy-Schwarz and Corollary 11.2.1 then yield (( r∂ ˜ v ψ ) ∣ ( ˜ v,r ) = ( τ, M + ǫ ) − ( + τ ) − p ) ≤ ǫ ∫ { ˜ v = τ } ∩ { r ≤ M + ǫ } [( T ψ ) + ( ∂ r ∗ ψ ) ] (204) ≤ Bǫ ( + τ ) − p . ˜ ψ As we have already indicated in the outline, it will be useful to introduce the function˜ ψ ( ˜ t, r ) ≐ ∫ ∞ ˜ t ψ ( ˜ s, r ) d ˜ s. (205)Using the fact that T is Killing, and the fact that ∣ ψ ( τ, r )∣ ≤ B ( r )( + τ ) − p + , one may easily check that˜ ψ is a smooth solution to (2) in ˚ R and that T ˜ ψ = − ψ . The goal of this section is to use the transportequation (194) to show that r ( − Mr ) − ∂ ˜ u ˜ ψ inherits some of r∂ ˜ v ψ ’s positivity.We begin by studying r ( − Mr ) − ∂ ˜ u ψ . Proposition 11.2.3.
Let ψ be as in the statement of Theorem 11.1, ˜ v be a fixed sufficiently large constant,and ( ˜ u, ˜ v ) ∈ { r = M + ǫ } for ˜ v ≥ v and ǫ > sufficiently small. Then ( r ( − Mr ) − ∂ ˜ u ψ ) ∣ ( ˜ u, ˜ v ) ≥ B exp ( − ∫ ˜ v ˜ v Mr d ˜ v ′ ) ∣( r ( − Mr ) − ∂ ˜ u ψ ) ∣ ( ˜ u, ˜ v ) ∣ + b ˜ v − p . Proof.
We may write equation (194) as ∂ ˜ v [ exp ( ∫ ˜ v ˜ v Mr d ˜ v ′ ) r ( − Mr ) − ∂ ˜ u ψ ] = exp ( ∫ ˜ v ˜ v Mr d ˜ v ′ ) ∂ ˜ v ψ. (206)We conclude that ( r ( − Mr ) − ∂ ˜ u ψ ) ∣ ( ˜ u, ˜ v ) (207) = exp ( − ∫ ˜ v ˜ v Mr d ˜ v ′ ) [( r ( − Mr ) − ∂ ˜ u ψ ) ∣ ( ˜ u, ˜ v ) + ∫ ˜ v ˜ v ( exp ( ∫ ˜ v ′ ˜ v Mr d ˜ v ′′ ) ∂ ˜ v ′ ψd ˜ v ′ )] . Next, using Proposition 11.2.2, we observe thatexp ( − ∫ ˜ v ˜ v Mr d ˜ v ′ ) ∫ ˜ v ˜ v ( exp ( ∫ ˜ v ′ ˜ v Mr d ˜ v ′′ ) ∂ ˜ v ′ ψd ˜ v ′ ) (208) ≥ b exp ( − ∫ ˜ v ˜ v Mr d ˜ v ′ ) ∫ ˜ v ˜ v ( exp ( ∫ ˜ v ′ ˜ v Mr d ˜ v ′′ ) ( ˜ v ′ ) − p d ˜ v ′ ) . Using that ∣ ∂ ˜ v r ∣ ≤ Bǫ , a straightforward series of integration by parts yieldsexp ( − ∫ ˜ v ˜ v Mr d ˜ v ′ ) ∫ ˜ v ˜ v ( exp ( ∫ ˜ v ′ ˜ v Mr d ˜ v ′′ ) ( ˜ v ′ ) − p d ˜ v ′ ) (209) ≥ ( r M ) ∣ ( ˜ v, ˜ u ) ˜ v − p − ( r M ) ∣ ( ˜ v , ˜ u ) ( ˜ v ) − p exp ( − ( − Bǫ ) ( ˜ v − ˜ v )) − Bǫ exp ( − ∫ ˜ v ˜ v Mr d ˜ v ′ ) ∫ ˜ v ˜ v ( exp ( ∫ ˜ v ′ ˜ v Mr d ˜ v ′′ ) ( ˜ v ′ ) − p d ˜ v ′ ) − B ( r M ) ∣ ( ˜ v, ˜ u ) ˜ v − p − + b ( r M ) ∣ ( ˜ v , ˜ u ) ( ˜ v ) − p − exp ( − ( − Bǫ ) ( ˜ v − ˜ v )) − B ˜ v − p − .
84e conclude that exp ( − ∫ ˜ v ˜ v Mr d ˜ v ′ ) ∫ ˜ v ˜ v ( exp ( ∫ ˜ v ′ ˜ v Mr d ˜ v ′′ ) ( ˜ v ′ ) − p d ˜ v ′ ) ≥ b ˜ v − p . (210)Combining (210) with (207) finishes the proof. Remark 11.2.2.
If we added the assumption that ( − Mr ) − ∂ ˜ u ψ was uniformly bounded, then for suffi-ciently large ˜ v this proposition would prove that ( − Mr ) − ∂ ˜ u ψ ≥ b ˜ v − p . Cf. [13]. We now have
Proposition 11.2.4.
Let ψ be as in Theorem 11.1, ˜ ψ defined by (205), and ˜ v be a sufficiently large constant.Then ( ˜ u, ˜ v ) ∈ { r = M + ǫ } , ǫ > sufficiently small, and ˜ v ≥ ˜ v imply r ( − Mr ) − ∂ ˜ u ˜ ψ ∣ ( ˜ u, ˜ v ) ≥ − Be − ( − Bǫ ) ˜ v M √ ∫ { ˜ v = ˜ v } ∩ { r ≤ M + ǫ } J Nµ [ ψ ] n µ { ˜ v = ˜ v } + b ˜ v − p + . Proof.
Using that ( ˜ u, ˜ v ) ∈ { r = M + ǫ } implies that ˜ u = ˜ v − ( M + ǫ ) ∗ , applying Proposition 11.2.3 to ψ andintegrating implies that r ( − Mr ) − ∂ ˜ u ˜ ψ ≥ − B ∫ ∞ ˜ v ∣ exp ( − ∫ ˜ v ˜ v Mr d ˜ v ′ ) ( − Mr ) − ∂ ˜ u ψ ∣ ( ˜ v − ( M + ǫ ) ∗ , ˜ v ) ∣ d ˜ v + b ˜ v − p + . (211)Now, we observe that a change of variables yields ∫ ∞ ˜ v ∣ exp ( − ∫ ˜ v ˜ v Mr d ˜ v ′ ) ( − Mr ) − ∂ ˜ u ψ ∣ ( ˜ v − ( M + ǫ ) , ˜ v ) ∣ d ˜ v ≤ ∫ M + Be − ˜ v M M ( r − M ) − Bǫ ∣ Y ψ ∣ ∣ ˜ v = ˜ v dr (212)Cauchy-Schwarz then gives us ∫ M + Be − ˜ v M M ( r − M ) − Bǫ ∣ Y ψ ∣ ∣ ˜ v = ˜ v dr ≤ Be − ( − Bǫ ) ˜ v M √ ∫ { ˜ v = ˜ v } ∩ { r ≤ M + ǫ } J Nµ [ ψ ] n µ { ˜ v = ˜ v } . (213) I + and the contradiction Finally, we will show that if ∂ ˜ v ˜ ψ and ∂ ˜ u ˜ ψ are eventually positive along { r = M + ǫ } , then the null derivativesof ˜ ψ must eventually be positive in a neigbourhood of I + . Proposition 11.2.5.
Let ψ be as in Theorem 11.1 and define ˜ ψ by (205). Additionally, let us assume that ∫ { ˜ v = ˜ v } ∩ { r ≤ M + ǫ } J Nµ [ ψ ] n µ { ˜ v = ˜ v } < ∞ . Then, there exists a constant c such that r∂ ˜ v ˜ ψ ≥ c ˜ v − p + , r∂ ˜ u ˜ ψ ≥ c ˜ u − p + , (214) for all sufficiently large r and ˜ v .Proof. Propositions 11.2.2 and 11.2.4 imply that we may find r > M and ˜ v < ∞ such that ˜ v ≥ ˜ v and ( ˜ u, ˜ v ) ∈ { r ∗ = r ∗ } implies that there exists a constant c > r∂ ˜ v ˜ ψ ≥ c ˜ v − p + , r∂ ˜ u ˜ ψ ≥ c ˜ u − p + . (215)Now, we define A ≐ { s ∈ [ , ∞ ] ∶ ˜ v ≥ ˜ v and r ∗ ( ˜ u, ˜ v ) ∈ [ r ∗ , r ∗ + s ) ⇒ r∂ ˜ v ˜ ψ ≥ c ˜ v − p + and r∂ ˜ u ˜ ψ ≥ c ˜ u − p + } , c is the constant from (215). The proof will be finished if we can prove that A = [ , ∞ ) .It is clear that A is a closed and non-empty subset of [ , ∞ ) , so it suffices to prove that A is open.Suppose that s ∈ A . We show that s + ǫ ∈ A for ǫ > ǫ sufficientlysmall implies that for all r ∗ ∈ [ r ∗ + s , r ∗ + s + ǫ ] and ˜ v ≥ ˜ v we have r∂ ˜ v ˜ ψ ≥ c ˜ v − p + , r∂ ˜ u ˜ ψ ≥ c ˜ u − p + . (216)Given these estimates, we integrate again the transport equations (191) and (192) and now use (216) todetermine that in the region r ∗ ∈ [ r ∗ + s , r ∗ + s + ǫ ] and ˜ v ≥ ˜ v , r∂ ˜ v ˜ ψ is monotonically increasing in − ˜ u and r∂ ˜ u ˜ ψ is monotonically increasing in ˜ v . We conclude that r ∗ ∈ [ r ∗ + s , r ∗ + s + ǫ ] and ˜ v ≥ ˜ v imply ( ) .The next corollary establishes the desired contradiction and thus concludes the proof of Theorem 11.1. Corollary 11.2.2.
Let ψ be as in Theorem 11.1 and define ˜ ψ by (205). Then, for each sufficiently large τ , lim r →∞ rψ ( r, τ ) < . Proof.
Using Proposition 11.2.5 and the facts ∂ ˜ u + ∂ ˜ v = T and T ˜ ψ = − ψ we find that − rψ = rT ˜ ψ = r∂ ˜ u ˜ ψ + r∂ ˜ v ˜ ψ ≥ c ( ˜ u − p + + ˜ v − p + ) . The result follows since lim r →∞ ˜ u ( r, τ ) is bounded above. ψ using the degenerate T -scattering theory We now apply our degenerate scattering theory of Theorem 9.2.1 (see also Section 9.6) to indeed constructsolutions ψ as in the statement of Theorem 11.1. Let ψ H +≥ ∶ H + ≥ → R denote the function ψ H +≥ ( t ∗ ) = ( t ∗ + ) − p + − p + p > Proposition 11.3.1.
Let a = . For ψ H +≥ ⊕ ∈ E N H +≥ ⊕ E T I + ⊂ E T H +≥ ⊕ E T I + above, the solution B − ( ψ H +≥ , ) satisfies all of the hypothesis of Theorem 11.1.Proof. Let us set ψ ≐ B − ( ψ H +≥ , ) . We first note that the spherical symmetry of Schwarzschild, The-orem 9.2.1, and commutations with T and Ω α are easily seen to imply that ψ is a smooth sphericallysymmetric solution in ˚ R and that ∫ Σ ∗ ( J Tµ [ ψ ] + J Tµ [ T ψ ]) n µ Σ ∗ < ∞ . Next, we observe that (191), the fundamental theorem of calculus, Cauchy Schwarz and an easy densityargument show that ∂ ˜ v ψ and ∂ ˜ v ( T ψ ) extend continuously to the functions ( + t ∗ ) − p and − p ( + t ∗ ) − p − respectively along H + ≥ .In order to establish that lim r →∞ rψ ∣ S τ = ∫ Σ ∗ t J Tµ [ ψ ] n µ Σ ∗ t = ∫ H +≥ t J Tµ [ ψ ] n µ H +≥ t ≤ B ( + t ) − p + . (217)Next, for r sufficiently large, the fundamental theorem of calculus implies ∣ ψ ( t, r )∣ ≤ ∫ ∞ r ∣ ∂ r ψ ∣ dr ≤ r − / √ ∫ ∞ r ( ∂ r ψ ) r dr ≤ r − / √ ∫ Σ ∗ t J Tµ [ ψ ] ≤ Br − / ( + t ) − p + / . (218)86ince r is comparable to t along any fixed hypersurface S τ , and p >
2, the estimate (218) immediatelyimplies that lim r →∞ rψ ∣ S τ = . We immediately obtain the following corollary.
Corollary 11.1.
Let a = . For ψ H +≥ ⊕ ∈ E N H +≥ ⊕ E T I + ⊂ E T H +≥ ⊕ E T I + above, then the map B − of Theorem 9.2.1maps B − ( ψ H +≥ , ) ∈ E T Σ ∗ ∖ E N Σ ∗ . More pedestrianly,
Corollary 11.2.
There indeed exist ψ as in Theorem 11.1. We note that by what we have shown in Proposition 11.2.1, ψ has several nice additional properties. Inparticular, we have the following decay result. Corollary 11.3.
Let ψ H +≥ be as in Corollary 11.1. Then, for every R < ∞ we have ∫ S τ ∩ { r ≤ R } [∣ T B − ( ψ H +≥ , )∣ + ∣ ∂ r ∗ B − ( ψ H +≥ , )∣ ] ≤ B ( R )( + τ ) − p ∀ τ ≥ . These strong decay properties lend further support to Conjecture 2.5. N -energy forward map Lastly, we can immediately reinterpret Corollary 11.1 as a non-surjectivity result (cf. the discussion inSection 2.3.2).
Corollary 11.4.
Let a = . The asymptotic state ψ H +≥ ⊕ is not in the image of the map F + ∶ E N Σ ∗ → E N H +≥ ⊕ E T I + . Thus, the map of Theorem 8.2.1 is not surjective, in fact, the image F + (E N Σ ∗ ) has infiniteco-dimension in E N H +≥ ⊕ E T I + and infinite codimension when intersected with E t n ∗ N H +≥ ⊕ for any n ≥ .Proof. If ψ H +≥ ⊕ F + ∶ E N Σ ∗ → E N H +≥ ⊕ E T I + , then Theorem 9.2.1 and Corollary 11.1would immediately yield a contradiction.We have thus obtained the final remaining Theorem 2 of Section 2.3.2.
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