Late-time decay of perturbations outside extremal charged black hole

TL;DR

This paper investigates how scalar perturbations decay over time outside extremal charged black holes, revealing decay rates that depend on the wave's origin and spherical harmonic mode, with implications for black hole stability.

Contribution

It provides detailed decay rate analyses for scalar fields in extremal Reissner-Nordstrom spacetime, including differences based on initial data location and mode, extending previous understanding.

Findings

Decay rate for scattered waves: t^{-(2l+3)}

Decay rate for waves from horizon neighborhood: t^{-(2l+2)}

Results align with Schwarzschild decay rates for scattered waves

Abstract

We analyze the late-time decay of scalar perturbations in extremal Reissner-Nordstrom spacetime. We consider individual spherical-harmonic modes $l$ of a test massless scalar field, restricting our attention to initial data of compact support, with generic regular behavior across the horizon. We obtain a decay rate $\propto t^{-(2l+3)}$ (just like in Schwarzschild) for incident waves scattered by the black hole. However, for waves originating at the horizon's neighborhood we obtain a slightly slower decay, $\propto t^{-(2l+2)}$. We discuss relations to previous works.