A global foliation of Einstein-Euler spacetimes with Gowdy-symmetry on T3

TL;DR

This paper constructs and analyzes low-regularity solutions to Einstein-Euler equations with Gowdy symmetry on T3, revealing the existence of spacetimes with impulsive gravitational waves and shock waves, and establishing a global foliation.

Contribution

It introduces a distributional formulation for Einstein-Euler equations with Gowdy symmetry, enabling the construction of spacetimes with impulsive and shock waves from low-regularity initial data.

Findings

Existence of future development for low-regularity initial data.

Construction of a global foliation based on symmetry orbits.

Presence of impulsive gravitational waves and shock waves in solutions.

Abstract

We investigate the initial value problem for the Einstein-Euler equations of general relativity under the assumption of Gowdy symmetry on T3, and we construct matter spacetimes with low regularity. These spacetimes admit, both, impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularities propagating at light speed) and shock waves in the fluid (i.e., discontinuities propagating at about the sound speed). Given an initial data set, we establish the existence of a future development and we provide a global foliation in terms of a globally and geometrically defined time-function, closely related to the area of the orbits of the symmetry group. The main difficulty lies in the low regularity assumed on the initial data set which requires a distributional formulation of the Einstein-Euler equations.