Continuation Criterion For Solutions To The Einstein Equations

TL;DR

This paper establishes a criterion for extending solutions to the Einstein equations in vacuum spacetime, linking bounded curvature norms and deformation tensors to the indefinite continuation of the spacetime evolution.

Contribution

It introduces a new continuation criterion for Einstein vacuum solutions based on bounded Sobolev norms and deformation tensors, using advanced wave equation techniques.

Findings

Bounded $H^{2}$ Sobolev norm of Riemann curvature ensures indefinite extension.

A spacetime $L^ abla$ bound on the deformation tensor suffices for continuation.

The technique employs Friedlander’s parametrix and Moncrief’s improvements for tensor wave equations.

Abstract

We prove a continuation condition in the context of 3+1 dimensional vacuum Einstein gravity in Constant Mean extrinsic Curvature (CMC) gauge. More precisely, we obtain quantitative criteria under which the physical spacetime can be extended in the future indefinitely as a solution to the Cauchy problem of the Einstein equations given regular initial data. In particular, we show that a gauge-invariant $H^{2}$ Sobolev norm of the spacetime Riemann curvature remains bounded in the future time direction provided the so-called deformation tensor of the unit timelike vector field normal to the chosen CMC hypersurfaces verifies a spacetime $L^\infty$ bound. To this end, we implement a novel technique to obtain this refined estimate by using Friedlander's parametrix for tensor wave equations on curved spacetime and Moncrief's subsequent improvement. We conclude by providing a physical explanation of our result as well as its relation to the issues of determinism and weak cosmic censorship.