Charged scalar fields on Reissner--Nordstr\"om spacetimes I: integrated energy estimates

Abstract

This is the first part of a series of papers deriving the precise, late-time behaviour and (in)stability properties of charged scalar fields on near-extremal Reissner--Nordstr\"om spacetimes via energy estimates. In this paper, we establish global, weighted integrated energy decay and energy boundedness estimates for solutions to the charged scalar field equation on (near-)extremal Reissner--Nordstr\"om(--de Sitter) spacetimes. These estimates extend to Reissner--Nordstr\"om spacetimes away from extremality under the assumption of mode stability on the real axis. Together with the companion paper [Gaj26], this paper forms the first global quantitative analysis of the charged scalar field equation on asymptotically flat black hole spacetimes, without a smallness assumption on the scalar field charge. Due to a coupling of the degeneration of the red-shift effect with the presence of superradiance at the linearized level, charged scalar fields on Reissner--Nordstr\"om spacetimes also probe some of the main difficulties encountered when studying the (neutral) wave equation on extremal Kerr spacetimes.