The non-isentropic Einstein-Euler system written in a symmetric hyperbolic form

TL;DR

This paper reformulates the non-isentropic Einstein-Euler system into a symmetric hyperbolic form using pressure as a variable, facilitating initial value problem analysis, with regularization near zero pressure via the Makino variable.

Contribution

It introduces a symmetric hyperbolic formulation of the non-isentropic Einstein-Euler system using pressure, including a regularization method for zero-pressure cases.

Findings

The system is well-suited for hyperbolic initial value problems.

Regularization with the Makino variable handles degeneracy at zero pressure.

The reformulation enhances mathematical analysis of relativistic fluid dynamics.

Abstract

We cast the non--isentropic relativistic Euler system into a symmetric hyperbolic form. Such systems are very suited to treat initial value problems of hyperbolic type. We obtain this form by using the pressure $p$ and not the density $\rho$ as a variable. However, the system becomes degenerate when the pressure $p$ approaches zero, and in these cases we regularise the system by replacing the pressure with an appropriate new matter variable, the Makino variable.