The linear stability of Reissner-Nordstr\"om spacetime: the full subextremal range

TL;DR

This paper proves the linear stability of subextremal Reissner-Nordstr"om black holes under gravitational and electromagnetic perturbations using a novel gauge-invariant approach and energy estimates, covering the entire subextremal charge range.

Contribution

It introduces a new gauge-invariant formulation and energy method to establish linear stability for all subextremal Reissner-Nordstr"om spacetimes, extending previous results.

Findings

Bounded energy and Morawetz estimates established.

Decay of perturbations proven in physical space.

Stability results hold for the full subextremal charge range |Q|<M.

Abstract

We prove the linear stability of subextremal Reissner-Nordstr\"om spacetimes as solutions to the Einstein-Maxwell equation. We make use of a novel representation of gauge-invariant quantities which satisfy a symmetric system of coupled wave equations. This system is composed of two of the three equations separately derived in previous works, where the estimates required arbitrary smallness of the charge. Here, the estimates are obtained by defining a combined energy-momentum tensor for the system in terms of the symmetric structure of the right hand sides of the equations. We obtain boundedness of the energy, Morawetz estimates and decay for the full subextremal range |Q|<M, completely in physical space. Such decay estimates, together with the estimates for the gauge-dependent quantities of the perturbations previously obtained, settle the problem of linear stability to gravitational and electromagnetic perturbations of Reissner-Nordstr\"om solution in the full subextremal range |Q|< M.