Abstract
We develop a scattering theory for the linear wave equation Box _g psi = 0 on the interior of Reissner–Nordström black holes, connecting the fixed frequency picture to the physical space picture. Our main result gives the existence, uniqueness and asymptotic completeness of finite energy scattering states. The past and future scattering states are represented as suitable traces of the solution psi on the bifurcate event and Cauchy horizons. The heart of the proof is to show that after separation of variables one has uniform boundedness of the reflection and transmission coefficients of the resulting radial o.d.e. over all frequencies omega and ell . This is non-trivial because the natural T conservation law is sign-indefinite in the black hole interior. In the physical space picture, our results imply that the Cauchy evolution from the event horizon to the Cauchy horizon is a Hilbert space isomorphism, where the past (resp. future) Hilbert space is defined by the finiteness of the degenerate T energy fluxes on both components of the event (resp. Cauchy) horizon. Finally, we prove that, in contrast to the above, for a generic set of cosmological constants Lambda , there is no analogous finite T energy scattering theory for either the linear wave equation or the Klein–Gordon equation with conformal mass on the (anti-) de Sitter–Reissner–Nordström interior.
Highlights
We develop a scattering theory for the linear wave equation gψ = 0 on the interior of Reissner–Nordstrom black holes, connecting the fixed frequency picture to the physical space picture
We define the Hilbert space of finite T energy ECTH on both components of the Cauchy horizon as the completion of smooth and compactly supported functions Cc∞(CH) the Cauchy horizon CH = CHA ∪CHB ∪B+ with respect to the norm (2.30)
In Theorem 7, stated in Sect. 3.7, we show the non-existence of the T energy scattering map for the Klein–Gordon equation on the interior of Reissner–Nordstrom
Summary
One of the most stunning predictions of general relativity is the formation of black holes, defined by the property that information cannot propagate from their interior region to outside far-away observers. For the Cauchy problem for (1.1) on the interior of both a fixed Kerr and a Reissner–Nordstrom black hole, the works [16,17,24] establish uniform boundedness (in L∞) and continuity up to and including the Cauchy horizon for solutions arising from smooth and compactly supported data on an asymptotically flat spacelike hypersurface. Such data in particular give rise to solutions with polynomial decay along the event horizon. In the last two sections are show our non-existence results: In Sect. 6, we prove Theorem 6 and in Sect. 7, we give the proof of Theorem 7
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