On the Einstein-Vlasov system with hyperbolic symmetry

TL;DR

This paper investigates the global structure of spacetimes with collisionless matter under hyperbolic symmetry, demonstrating the existence of constant mean curvature and area radius hypersurfaces, and discusses implications for understanding their global geometry.

Contribution

It extends the analysis of the Einstein-Vlasov system to hyperbolic symmetry, establishing the existence of special hypersurfaces and comparing with spherical and plane symmetry cases.

Findings

Existence of compact hypersurfaces with constant mean curvature

Existence of compact hypersurfaces with constant area radius

Discussion on using these hypersurfaces for global geometric analysis

Abstract

It is shown that a spacetime with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry can be globally covered by compact hypersurfaces on which the mean curvature is constant and by compact hypersurfaces on which the area radius is constant. Results for the related cases of spherical and plane symmetry are reviewed and extended. The prospects of using the global time coordinates obtained in this way to investigate the global geometry of the spacetimes concerned are discussed.