Bounded energy waves on the black hole interior of Reissner-Nordstr\"om-de Sitter

TL;DR

This paper investigates the behavior of scalar waves inside Reissner-Nordström-de Sitter black holes, showing conditions under which wave energy remains bounded up to the Cauchy horizon, thus challenging the Strong Cosmic Censorship Conjecture.

Contribution

It provides a new sufficient condition involving surface gravities and decay rates for bounded energy solutions at the Cauchy horizon without symmetry assumptions.

Findings

Bounded energy solutions extend to the Cauchy horizon in $C^0\cap H^1_{loc}$.

Conditions relate surface gravities and decay rates for wave energy.

Results suggest potential breakdown of Strong Cosmic Censorship with positive cosmological constant.

Abstract

Motivated by the Strong Cosmic Censorship Conjecture, in the presence of a cosmological constant, we consider solutions of the scalar wave equation $\Box_g\phi=0$ on fixed subextremal Reissner--Nordstr\"om--de Sitter backgrounds $({\mathcal M}, g)$, without imposing symmetry assumptions on $\phi$. We provide a sufficient condition, in terms of surface gravities and a parameter for an exponential decaying Price law, for a local energy of the waves to remain bounded up to the Cauchy horizon. The energy we consider controls, in particular, regular transverse derivatives at the Cauchy horizon; this allows us to extend the solutions with bounded energy, to the Cauchy horizon, as functions in $C^0\cap H^1_{loc}$. Our results correspond to another manifestation of the potential breakdown of Strong Cosmic Censorship in the positive cosmological constant setting.