Blowup of the local energy of linear waves at the Reissner-Nordstr\"om-AdS Cauchy horizon

TL;DR

This paper demonstrates that linear wave perturbations on Reissner-Nordström-AdS black holes have infinite local energy at the Cauchy horizon, confirming the linear version of the Strong Cosmic Censorship conjecture in a specific formulation.

Contribution

It establishes that while solutions remain bounded and extend continuously, their local energy diverges at the Cauchy horizon, confirming the linear Strong Cosmic Censorship in the $H^1$ sense.

Findings

Linear perturbations have infinite local energy at the Cauchy horizon.

Solutions are uniformly bounded and extend continuously across the horizon.

Confirms the $H^1$-formulation of the Strong Cosmic Censorship conjecture.

Abstract

We show that generic linear perturbations $\psi$ solving $\Box_g \psi + \frac{\alpha}{l^2} \psi =0$ on Reissner-Nordstr\"om-AdS black holes have infinite local energy at the Cauchy horizon. Combined with the result of arXiv:1812.06142 that such perturbations $\psi$ remain uniformly bounded and extend continuously across the Cauchy horizon, this settles the linear analog of the Strong Cosmic Censorship conjecture for Reissner-Nordstr\"om-AdS: the $C^0$-formulation is false but the $H^1$-formulation is true.