On the Choquet-Bruhat-York-Friedrich formulation of the Einstein-Euler equations

TL;DR

This paper proves short-time existence for Einstein-Euler and vacuum Einstein equations using a Friedrich-inspired symmetric hyperbolic formulation that treats the Riemann tensor as a fundamental unknown.

Contribution

It introduces a novel formulation of the Einstein-Euler equations based on the Choquet-Bruhat-York approach, ensuring the full Einstein equations are satisfied.

Findings

Short-time existence is established for the Einstein-Euler system.

The formulation preserves the gauge and implies the full Einstein equations.

The Riemann tensor is incorporated as a fundamental unknown.

Abstract

Short-time existence for the Einstein-Euler and the vacuum Einstein equations is proven using a Friedrich inspired formulation due to Choquet-Bruhat and York, where the system is cast into a symmetric hyperbolic form and the Riemann tensor is treated as one of the fundamental unknowns of the problem. The reduced system of Choquet-Bruhat and York, along with the preservation of the gauge, is shown to imply the full Einstein equations.