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

TL;DR

This paper investigates the stability and instability of extreme Reissner-Nordstrom black hole spacetimes under linear scalar perturbations, providing detailed decay, non-decay, and blow-up results including energy estimates and conservation laws.

Contribution

It extends previous work by establishing comprehensive decay and blow-up results, along with conservation laws on degenerate horizons, for linear scalar fields on these black hole backgrounds.

Findings

Established energy and pointwise decay estimates.

Proved non-decay and blow-up phenomena at the horizon.

Derived a hierarchy of conservation laws on degenerate horizons.

Abstract

This paper contains the second part of a two-part series on the stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. We continue our study of 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 here obtain definitive energy and pointwise decay, non-decay and blow-up results. Our estimates hold up to and including the horizon. A hierarchy of conservations laws on degenerate horizons is also derived.