Relativistic Thermodynamics, a Lagrangian Field Theory for general flows including rotation

TL;DR

This paper develops a relativistic Lagrangian field theory for thermodynamics and hydrodynamics, including rotation, within the frameworks of Special and General Relativity, extending classical variational principles to more general flows.

Contribution

It introduces a variational principle for relativistic thermodynamics that encompasses general flows with rotation, advancing the coupling of matter and gravity in relativistic theories.

Findings

A relativistic variational principle for general flows is formulated.

The energy-momentum tensor structure is generalized to include rotational flows.

The theory incorporates a conserved mass current analogous to hydrodynamics.

Abstract

The formulation of a dynamical theory of General Relativity, including matter, is viewed as a problem of coupling Einstein's theory of pure gravity, formulated as an action principle, to an independently chosen and well defined field theory of matter. It is well known that this is accomplished in a most natural way when the matter theory is formulated as a relativistic, Lagrangian field theory. Special matter models of this type have been available; here a thermodynamical model that allows for general flows is used. A problem that is of even older date, one that was pursued vigorously by leading scientists of the 19'th century, is that of subjecting hydrodynamics and thermodynamics to an Action Principle. A solution to this problem has been known for some time, but only under the strong restriction to potential flows. A variational principle for general flows has now become available. The present paper lifts this theory to the relativistic context, Special Relativity and General Relativity. The energy momentum tensor has a structure that is more general than that postulated by Tolman, and different from proposed generalizations; it appears to be well suited to represent rotational flows in General Relativity. It incorporates a conserved mass current, the relativistic analogue of the hydrodynamical equation of continuity.