On the backward stability of the Schwarzschild black hole 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.

6 Figures

References in corpus (1)

• Analytic description of singularities in Gowdy spacetimes

Cited by in corpus (9)

• Stable Big Bang formation for Einstein's equations: The complete sub-critical regime
• The maximal development of near-FLRW data for the Einstein-scalar field system with spatial topology ${\mathbb{S}}^3$
• Generic blow-up results for the wave equation in the interior of a Schwarzschild black hole
• Polynomial Blow-up Upper Bounds for the Einstein-scalar field System Under Spherical Symmetry
• The non-linear stability of the Schwarzschild family of black holes
• Stable space-like singularity formation for axi-symmetric and polarized near-Schwarzschild black hole interiors
• Curvature blow-up rates in spherically symmetric gravitational collapse to a Schwarzschild black hole
• Cyclic spacetimes through singularity scattering maps. The laws of quiescent bounces
• Kasner-like description of spacelike singularities in spherically symmetric spacetimes with scalar matter