A Scattering Theory for Linear Waves on the Interior of Reissner–Nordström Black Holes

Abstract

We develop a scattering theory for the linear wave equation g ψ = 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 ψ 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 ω and . 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 Λ, 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. 6 Proof of Theorem 6: Breakdown of T energy scattering for cosmological constants Λ = 0 41 7 Proof of Theorem 7: Breakdown of T energy scattering for the Klein-Gordon equation 46 A Additional lemmata 47 References 51 on the interior of a Reissner-Nordström black hole, from the bifurcate event horizon H = H A ∪ H B ∪ B − to the bifurcate Cauchy horizon CH = CH A ∪CH B ∪B + , as depicted in Fig. 1. The first main result of our paper

The BTZ black hole violates strong cosmic censorship

“…R in,+ (ω, m; z) has simple poles when ω − mΩ + is a negative integer multiple of iκ + and R out,+ (ω, m; z) has simple poles when ω − mΩ + is a positive integer multiple of iκ + . These results are similar to those for 4d black holes [3,33]. R vev,∞ and R source,∞ are entire functions of ω.…”

Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetime

“…From the factors Γ(±2b − ), one easily verifies the asymptotic behavior (124). Note that in both expressions the fraction of the sums converges to − 1 π sin πν ∞ n=−∞ (−1) n a ν n ∞ n=−∞ a ν n (147) in the limit as ω → 0, so that the divergence at ω → 0 cannot be enhanced but possibly cancelled if ν → Z for ω → 0, which, as can be seen from (141), happens in the RN limit, consistent with results by [76].…”

“…The nature of this pole can also be seen explicitly from the expressions given for the scattering coefficients in sec.5.3. 24 This follows from the fact that the coefficients A, B in (6.5) of[76] are real.…”

Strong cosmic censorship for a scalar field in a Born-Infeld–de Sitter black hole

“…On the other hand, it was observed that a remnant perturbation can exponentially decay in a black hole in asymptotically dS space-time [17][18][19][20][21][22][23][24], which implies that the perturbation might have chance to decay fast enough to violate SCC. More precisely, it showed that, for an asymptotically dS black hole, the competition between the the mass inflation and remnant decaying can be characterized by [25][26][27][28][29][30]…”