# Higher Order Linear Stability and Instability of Reissner-Nordstr\"om's Cauchy Horizon

## TL;DR

This paper analyzes the stability of wave solutions near the Cauchy horizon of Reissner-Nordström black holes, establishing criteria for regularity and blow-up based on surface gravities, with implications for black hole interior stability.

## Contribution

It provides new criteria relating surface gravities to wave regularity and blow-up at the Cauchy horizon in Reissner-Nordström spacetimes.

## Key findings

- Criteria for wave regularity up to the Cauchy horizon.
- Conditions for wave blow-up in $C^1$ and $H^1$.
- Applicability to various Reissner-Nordström black hole models.

## Abstract

We consider smooth solutions of the wave equation, on a fixed black hole region of a subextremal Reissner-Nordstr\"om (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 $C^l$, $l\geq 1$, up to and including the Cauchy horizon. We also provide sufficient conditions for the blow up of solutions in $C^1$ and $H^1$.