Generic Blow-Up Results for the Wave Equation in the Interior of a Schwarzschild Black Hole

Abstract

We study the behaviour of smooth solutions to the wave equation, g ψ = 0, in the interior of a fixed Schwarzschild black hole. In particular, we obtain a full asymptotic expansion for all solutions towards r = 0 and show that it is characterised by its first two leading terms, the principal logarithmic term and a bounded second order term. Moreover, we characterise an open set of initial data for which the corresponding solutions blow up logarithmically on the entirety of the singular hypersurface {r = 0}. Our method is based on deriving weighted energy estimates in physical space and requires no symmetries of solutions. However, a key ingredient in our argument uses a precise analysis of the spherically symmetric part of the solution and a monotonicity property of spherically symmetric solutions in the interior.

The wave equation near flat Friedmann–Lemaître–Robertson–Walker and Kasner Big Bang singularities

“…Examples are the Schwarzschild singularity or black hole singularities occurring in spherically symmetric solutions to the Einstein-scalar field model [9], where a logarithmic blow up behaviour has been observed for spatially homogeneous waves. Such logarithmic blow up behaviour was recently confirmed [15] for generic linear waves in the Schwarzschild black hole interior. The aforementioned blow up behaviours, however, are in contrast to the behaviour of waves observed near null boundaries, where linear and dynamical waves have been shown in general to extend continuously past the relevant null hypersurfaces [10], [16]- [22], see also [24].…”

Solutions of the wave equation bounded at the Big Bang

“…Interestingly, however, not all (nonconstant) solutions blow up, and in fact one of the difficulties in [2] is to formulate a genericity condition that excludes these special bounded solutions. This scenario, already found in [11], also occurs for solutions of the wave equation in the black hole region of the Schwarzschild spacetime (which can be thought of as a cosmological model approaching a Big Crunch), as discussed in [6]. The results in [10] suggest that a similar situation may occur near compact Cauchy horizons.…”

“…However, we do indeed suspect that this is the case for sufficiently regular solutions. Proving this would probably require the derivation of an accurate enough asymptotic expansion of all linear waves towards the singularity, as was done in [6].…”

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

“…Finally, it is instructive to draw a comparison between the interior of Reissner-Nordström and the interior of Schwarzschild (Q = 0). As opposed to Reissner-Nordström discussed above, the Schwarzschild interior terminates at a singular boundary at which solutions to (1.1) generically blow-up (see [16]). In contrast, the non-singular and, moreover, Killing, Cauchy horizons (see Fig.…”

“…In view of the above, there has been a lot of recent activity analyzing the Cauchy problem on black hole interiors, e.g. [18,17,47,32,16]. However, for certain physical processes it is more natural to consider the scattering problem (see [19] for scattering on the exterior of black holes).…”