A note on incompressibility of relativistic fluids and the instantaneity of their pressures

TL;DR

This paper defines a relativistic incompressibility concept, shows it reduces to classical incompressible Euler equations in the limit, and proves the pressure propagates instantaneously, indicating infinite speed of propagation.

Contribution

It introduces a natural relativistic incompressibility notion and demonstrates the pressure's elliptic nature, implying instantaneity in pressure adjustments.

Findings

Relativistic incompressibility reduces to classical form as c→∞

Pressure satisfies an elliptic equation on hypersurfaces

Pressure propagation is instantaneous, indicating infinite speed

Abstract

We introduce a natural notion of incompressibility for fluids governed by the relativistic Euler equations on a fixed background spacetime, and show that the resulting equations reduce to the incompressible Euler equations in the classical limit as $c\rightarrow \infty$. As our main result, we prove that the fluid pressure of solutions of these incompressible "relativistic" Euler equations satisfies an elliptic equation on each of the hypersurfaces orthogonal to the fluid four-velocity, which indicates infinite speed of propagation.