Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2008-08-24
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
Scientific paper
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.
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