Global geometry of T2 symmetric spacetimes with weak regularity

TL;DR

This paper studies the global geometric structure of T2 symmetric spacetimes with weak regularity, establishing a global foliation by symmetry orbits under minimal regularity assumptions, extending understanding beyond smooth solutions.

Contribution

It introduces a framework for weakly regular T2 symmetric spacetimes, proves the existence of a global foliation by symmetry orbit levels, and formulates the Einstein vacuum equations under minimal regularity.

Findings

Existence of a global foliation by symmetry orbit levels

Weak regularity conditions include continuous gradient of R and Sobolev space H1 metric coefficients

All positive values of R are covered except for flat Kasner spacetimes

Abstract

We define the class of weakly regular spacetimes with T2 symmetry, and investigate their global geometry structure. We formulate the initial value problem for the Einstein vacuum equations with weak regularity, and establish the existence of a global foliation by the level sets of the area R of the orbits of symmetry, so that each leaf can be regarded as an initial hypersurface. Except for the flat Kasner spacetimes which are known explicitly, R takes all positive values. Our weak regularity assumptions only require that the gradient of R is continuous while the metric coefficients belong to the Sobolev space H1 (or have even less regularity).