On the area of the symmetry orbits in $T^2$ symmetric spacetimes

TL;DR

This paper proves that in vacuum $T^2$ symmetric spacetimes with non-zero twist, the area of symmetry orbits varies from zero to infinity, confirming the areal coordinate covers the entire spacetime from singularity to infinity.

Contribution

It establishes a global existence result for Einstein equations in $T^2$ symmetric spacetimes, showing the areal coordinate spans from zero to infinity except for flat Kasner cases.

Findings

The area of $T^2$ orbits covers all positive values.

The singularity occurs at the areal coordinate R=0.

The areal coordinate extends to infinity in these spacetimes.

Abstract

We obtain a global existence result for the Einstein equations. We show that in the maximal Cauchy development of vacuum $T^2$ symmetric initial data with nonvanishing twist constant, except for the special case of flat Kasner initial data, the area of the $T^2$ group orbits takes on all positive values. This result shows that the areal time coordinate $R$ which covers these spacetimes runs from zero to infinity, with the singularity occurring at R=0.