Stability and Instability of Extreme Reissner-Nordstr\"om Black Hole Spacetimes for Linear Scalar Perturbations I

TL;DR

This paper investigates the stability and instability of extreme Reissner-Nordstrom black hole spacetimes under linear scalar perturbations, highlighting the challenges posed by the degenerate redshift on the horizon.

Contribution

It provides new analytical results on boundedness, decay, and non-decay of solutions to the wave equation in degenerate horizon settings, advancing understanding of black hole stability.

Findings

Boundedness and decay of scalar perturbations up to the horizon

Non-decay phenomena at the horizon due to degeneracy

New analytical techniques for degenerate horizons

Abstract

We study the problem of stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. Specifically, we consider solutions to the linear wave equation on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface crossing the future event horizon. We obtain boundedness, decay and non-decay results. Our estimates hold up to and including the horizon. The fundamental new aspect of this problem is the degeneracy of the redshift on the event horizon. Several new analytical features of degenerate horizons are also presented.