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arXiv · 1504.04079

On the backward stability of the Schwarzschild black hole singularity

Grigorios Fournodavlos
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Abstract

We study the backwards-in-time stability of the Schwarzschild singularity from a dynamical PDE point of view. More precisely, considering a spacelike hypersurface $Σ_0$ in the interior of the black hole region, tangent to the singular hypersurface $\{r=0\}$ at a single sphere, we study the problem of perturbing the Schwarzschild data on $Σ_0$ and solving the Einstein vacuum equations backwards in time. We obtain a local well-posedness result for small perturbations lying in certain weighted Sobolev spaces. No symmetry assumptions are imposed. The perturbed spacetimes all have a singularity at a "collapsed" sphere on $Σ_0$, where the leading asymptotics of the curvature and the metric match those of their Schwarzschild counterparts to a suitably high order. As in the Schwarzschild backward evolution, the pinched initial hypersurface $Σ_0$ `opens up' instantly, becoming a smooth spacelike (cylindrical) hypersurface. This result thus yields classes of examples of non-symmetric vacuum spacetimes, evolving forward-in-time from smooth initial data, which form a Schwarzschild type singularity at a collapsed sphere. We rely on a precise asymptotic analysis of the Schwarzschild geometry near the singularity which turns out to be at the threshold that our energy methods can handle.

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Grigorios Fournodavlos. 2015-04-16. On the backward stability of the Schwarzschild black hole singularity. https://doi.org/10.1007/s00220-016-2681-0

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