On the area of the symmetry orbits in weakly regular Einstein-Euler spacetimes with Gowdy symmetry

TL;DR

This paper proves that in Gowdy-symmetric Einstein-Euler spacetimes, the area of symmetry orbits shrinks to zero in the future under certain conditions, completing the understanding of their global structure.

Contribution

It establishes new bounds and completes the analysis of the global areal foliation for Gowdy-symmetric Einstein-Euler spacetimes, extending prior work.

Findings

Orbit area approaches zero in future contracting spacetimes

Condition for orbit collapse is sharp within spatially homogeneous cases

Results hold when a key geometric invariant is initially non-zero

Abstract

This paper establishes novel bounds for Gowdy-symmetric Einstein-Euler spacetimes and completes the analysis, initiated by LeFloch and Rendall, of the global areal foliation for these spacetimes. We thus consider the initial value problem for the Einstein-Euler equations under the assumption of Gowdy symmetry. We establish that, for the maximal Cauchy development of future contracting initial data, the area of the group orbits approaches zero toward the future. This property holds as one approaches the future boundary of the spacetime, provided a geometry invariant associated with the Gowdy symmetry property is initially non-vanishing. Our condition is sharp within the class of spatially homogeneous spacetimes.