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

TL;DR

This paper investigates the global existence and properties of $T^2$ symmetric spacetimes with Vlasov matter, demonstrating that the area of symmetry orbits spans all positive values in the maximal development.

Contribution

It proves that in $T^2$ symmetric spacetimes with Vlasov matter, the orbit area covers all positive values, extending understanding of their global structure.

Findings

The orbit area takes all positive values in the maximal Cauchy development.

The work confirms the global hyperbolicity and symmetry properties of the spacetimes.

It advances the understanding of collisionless matter in general relativity.

Abstract

This paper treats the global existence question for a collection of general relativistic collisionless particles, all having the same mass. The spacetimes considered are globally hyperbolic, with Cauchy surface a 3-torus. Furthermore, the spacetimes considered are isometrically invariant under a two-dimensional group action, the orbits of which are spacelike 2-tori. It is known from previous work that the area of the group orbits serves as a global time coordinate. In the present work it is shown that the area takes on all positive values in the maximal Cauchy development.