Wave propagation on microstate geometries

TL;DR

This paper rigorously analyzes wave behavior on supersymmetric microstate geometries, revealing bounded energy solutions and very slow decay rates, which challenge previous conjectures of instability.

Contribution

It provides the first thorough mathematical analysis of the linear wave equation on two and three charge microstate geometries, demonstrating bounded energy and slow decay rates.

Findings

Solutions have uniformly bounded local energy despite ergoregions.

Constructed solutions with non-decaying local energy in three charge geometries.

Decay rates are sub-logarithmic, confirming numerical predictions.

Abstract

Supersymmetric microstate geometries were recently conjectured to be nonlinearly unstable due to numerical and heuristic evidence, based on the existence of very slowly decaying solutions to the linear wave equation on these backgrounds. In this paper, we give a thorough mathematical treatment of the linear wave equation on both two and three charge supersymmetric microstate geometries, finding a number of surprising results. In both cases we prove that solutions to the wave equation have uniformly bounded local energy, despite the fact that three charge microstates possess an ergoregion; these geometries therefore avoid Friedman's "ergosphere instability". In fact, in the three charge case we are able to construct solutions to the wave equation with local energy that neither grows nor decays, although this data must have nontrivial dependence on the Kaluza-Klein coordinate. In the two charge case we construct quasimodes and use these to bound the uniform decay rate, showing that the only possible uniform decay statements on these backgrounds have very slow decay rates. We find that these decay rates are sub-logarithmic, verifying the numerical results of Eperon et al. The same construction can be made in the three charge case, and in both cases the data for the quasimodes can be chosen to have trivial dependence on the Kaluza-Klein coordinates.