Relativistic Entropy Inequality

TL;DR

This paper applies the entropy principle to relativistic fluid dynamics equations, deriving a trace condition for the energy system and establishing a relativistic mass-momentum-energy framework using the Liu-Müller sum.

Contribution

It introduces a novel application of the entropy principle to relativistic fluid equations, deriving a trace condition and formulating a relativistic energy system.

Findings

Derived the relativistic mass-momentum-energy system for fluids.

Established the necessity of taking a trace in the energy part of the system.

Utilized the Liu-Müller sum to deduce the Gibbs relation and entropy inequality.

Abstract

In this paper we apply the entropy principle to the relativistic version of the differential equations describing a standard fluid flow, that is, the equations for mass, momentum, and a system for the energy matrix. These are the second order equations which have been introduced in [3]. Since the principle also says that the entropy equation is a scalar equation, this implies, as we show, that one has to take a trace in the energy part of the system. Thus one arrives at the relativistic mass-momentum-energy system for the fluid. In the procedure we use the well-known Liu-M\"uller sum [10] in order to deduce the Gibbs relation and the residual entropy inequality.