Shock Waves in Plane Symmetric Spacetimes

TL;DR

This paper studies the formation of shock waves in plane symmetric spacetimes governed by Einstein and Euler equations, showing finite-time breakdown of solutions due to shock formation.

Contribution

It introduces a new theorem on the breakdown of solutions for systems of balance laws and links solution breakdown to shock wave formation in Einstein-Euler systems.

Findings

Classical solutions break down in finite time for various equations of state.

Bounded spatial derivatives of energy density and velocity allow solution extension.

Shock wave formation causes the finite-time breakdown of solutions.

Abstract

We consider Einstein's equations coupled to the Euler equations in plane symmetry, with compact spatial slices and constant mean curvature time. We show that for a wide variety of equations of state and a large class of initial data, classical solutions break down in finite time. The key mathematical result is a new theorem on the breakdown of solutions of systems of balance laws. We also show that an extension of the solution is possible if the spatial derivatives of the energy density and the velocity are bounded, indicating that the breakdown is really due to the formation of shock waves.