Linear waves in the interior of extremal black holes I

Understand

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.