On the backward stability of the Schwarzschild black hole singularity

TL;DR

This paper proves the local backward-in-time stability of the Schwarzschild black hole singularity under small perturbations without symmetry assumptions, showing that such perturbations evolve into spacetimes with similar singularities.

Contribution

It establishes the first local well-posedness result for the backward evolution of Schwarzschild singularities without symmetry constraints.

Findings

Perturbed spacetimes develop a Schwarzschild-type singularity at a collapsed sphere.

The hypersurface opens up instantly into a smooth spacelike hypersurface.

The analysis relies on precise asymptotics near the Schwarzschild singularity.

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 $\Sigma_0$ 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 $\Sigma_0$ 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 $\Sigma_0$, 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 $\Sigma_0$ `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.