Weakly regular Einstein-Euler spacetimes with Gowdy symmetry. The global areal foliation

TL;DR

This paper proves the existence of a global areal foliation in weakly regular Gowdy-symmetric Einstein-Euler spacetimes with shock and impulsive gravitational waves, using a novel adapted random choice scheme.

Contribution

It establishes the global foliation for weakly regular Gowdy-symmetric Einstein-Euler spacetimes with bounded variation initial data, including the contracting case with mass density constraints.

Findings

Existence of a global foliation by spacelike hypersurfaces.

The area function approaches infinity in expanding spacetimes and zero in contracting ones.

The method uses an adapted random choice scheme for Einstein equations with weak regularity.

Abstract

We consider weakly regular Gowdy-symmetric spacetimes on T3 satisfying the Einstein-Euler equations of general relativity, and we solve the initial value problem when the initial data set has bounded variation, only, so that the corresponding spacetime may contain impulsive gravitational waves as well as shock waves. By analyzing, both, future expanding and future contracting spacetimes, we establish the existence of a global foliation by spacelike hypersurfaces so that the time function coincides with the area of the surfaces of symmetry and asymptotically approaches infinity in the expanding case and zero in the contracting case. More precisely, the latter property in the contracting case holds provided the mass density does not exceed a certain threshold, which is a natural assumption since certain exceptional data with sufficiently large mass density are known to give rise to a Cauchy horizon, on which the area function attains a positive value. An earlier result by LeFloch and Rendall assumed a different class of weak regularity and did not determine the range of the area function in the contracting case. Our method of proof is based on a version of the random choice scheme adapted to the Einstein equations for the symmetry and regularity class under consideration. We also analyze the Einstein constraint equations under weak regularity.