Higher order linear stability and instability of Reissner–Nordström’s Cauchy horizon

João L Costa,Pedro M Girão

doi:10.1007/s13324-020-00380-5

Abstract

We consider smooth solutions of the wave equation, on a fixed black hole region of a subextremal Reissner–Nordström (asymptotically flat, de Sitter or anti-de Sitter) spacetime, whose restrictions to the event horizon have compact support. We provide criteria, in terms of surface gravities, for the waves to remain in \(C^l\), \(l\geqslant 1\), up to and including the Cauchy horizon. We also provide sufficient conditions for the blow up of solutions in \(C^1\) and \(H^1\).

Highlights

Cauchy horizons are the spacetime boundary of the maximal Cauchy development of initial value problems for the Einstein field equations
Whenever non-empty, their existence and stability puts into question global uniqueness, and challenges the deterministic character of General Relativity
To understand how perturbations of a static charged black hole behave at the Cauchy horizon that lies in its interior, we will study solutions of the wave equation on the black hole region of fixed subextremal ReissnerNordström spacetimes

Summary

Introduction

Cauchy horizons are the spacetime boundary of the maximal Cauchy development of initial value problems for the Einstein field equations. We present criteria for higher order linear stability of the Cauchy horizon, meaning Cl with l 1, in a subextremal Reissner-Nordström spacetime, as well as criteria for linear instability, in both C1 and H1 We will achieve this by considering waves, without symmetry assumptions, whose restrictions to the event horizon have compact support. We will study solutions of the wave equation on a fixed background consisting of the black hole region of a subextremal Reissner-Nordström (asymptotically flat, de Sitter or anti-de Sitter) spacetime. This spacetime has a metric given in a local coordinate system by g.

Basic energy estimates

Existence of a classical solution up to the Cauchy horizon

Solutions with higher regularity

Blow up in C1 and in H1