Linear waves in the interior of extremal black holes I

TL;DR

This paper demonstrates that solutions to the linear wave equation inside extremal Reissner-Nordström black holes can be extended beyond the Cauchy horizon with finite local energy, contrasting with subextremal cases.

Contribution

It provides the first analysis showing finite energy extension of linear waves beyond the Cauchy horizon in extremal black holes.

Findings

Solutions extend continuously beyond the Cauchy horizon.

Local energy remains finite in the extremal case.

Contrasts with blow-up results in subextremal black holes.

Abstract

We consider solutions to the linear wave equation in the interior region of extremal Reissner-Nordstr\"om black holes. We show that, under suitable assumptions on the initial data, the solutions can be extended continuously beyond the Cauchy horizon and moreover, that their local energy is finite. This result is in contrast with previously established results for subextremal Reissner-Nordstr\"om black holes, where the local energy was shown to generically blow up at the Cauchy horizon.