Linear waves in the interior of extremal black holes I

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.