# Stability and instability of the sub-extremal Reissner-Nordstr\"om black hole interior for the Einstein-Maxwell-Klein-Gordon equations in spherical symmetry

## TL;DR

This paper investigates the stability and instability of the interior of sub-extremal Reissner-Nordstr"om black holes under Einstein-Maxwell-Klein-Gordon equations in spherical symmetry, with implications for cosmic censorship.

## Contribution

It extends previous stability and instability results to charged scalar fields, providing new insights into the structure of black hole interiors and cosmic censorship.

## Key findings

- Proves stability of the Cauchy horizon under certain decay conditions.
- Establishes divergence of energy and curvature blow-up at the Cauchy horizon.
- Generalizes previous results to charged scalar fields in spherical symmetry.

## Abstract

We show non-linear stability and instability results in spherical symmetry for the interior of a charged black hole -approaching a sub-extremal Reissner-Nordstr\"om background fast enough at infinity- in presence of a massive and charged scalar field, motivated by the strong cosmic censorship conjecture in that setting : 1. Stability : We prove that spherically symmetric characteristic initial data to the Einstein-Maxwell- Klein-Gordon equations approaching a Reissner-Nordstr\"om background with a sufficiently decaying polynomial decay rate on the event horizon gives rise to a space-time possessing a Cauchy horizon in a neighbourhood of time-like infinity. Moreover if the decay is even stronger, we prove that the spacetime metric admits a continuous extension to the Cauchy horizon. This generalizes the celebrated stability result of Dafermos for Einstein-Maxwell-real-scalar-field in spherical symmetry. 2. Instability : We prove that for the class of space-times considered in the stability part, whose scalar field in addition obeys a polynomial averaged-L^2 (consistent) lower bound on the event horizon, the scalar field obeys an integrated lower bound transversally to the Cauchy horizon. As a consequence we prove that the non-degenerate energy is infinite on any null surface crossing the Cauchy horizon and the curvature of a geodesic vector field blows up at the Cauchy horizon near time-like infinity. This generalizes an instability result due to Luk and Oh for Einstein-Maxwell-real-scalar-field in spherical symmetry. This instability of the black hole interior can also be viewed as a step towards the resolution of the C^2 strong cosmic censorship conjecture for one-ended asymptotically initial data.