Stability and decay-rates for the five-dimensional Schwarzschild metric under biaxial perturbations

TL;DR

This paper proves the nonlinear stability of the five-dimensional Schwarzschild black hole under symmetric vacuum perturbations, demonstrating convergence to a stationary solution with explicit decay rates using vectorfield multiplier techniques.

Contribution

It is the first to establish the existence of non-stationary vacuum black holes that dynamically approach stationary solutions in five dimensions.

Findings

Quantitative polynomial decay rates established

Stability proven for biaxial vacuum perturbations

Techniques applicable to other stability problems

Abstract

In this paper we prove the non-linear asymptotic stability of the five-dimensional Schwarzschild metric under biaxial vacuum perturbations. This is the statement that the evolution of (SU(2) x U(1))-symmetric vacuum perturbations of initial data for the five-dimensional Schwarzschild metric finally converges in a suitable sense to a member of the Schwarzschild family. It constitutes the first result proving the existence of non-stationary vacuum black holes arising from asymptotically flat initial data dynamically approaching a stationary solution. In fact, we show quantitative rates of approach. The proof relies on vectorfield multiplier estimates, which are used in conjunction with a bootstrap argument to establish polynomial decay rates for the radiation on the perturbed spacetime. Despite being applied here in a five-dimensional context, the techniques are quite robust and may admit applications to various four-dimensional stability problems.