Macroscopic form of the first law of thermodynamics for an adiabatically evolving non-singular self-gravitating fluid

TL;DR

This paper formulates a global first law of thermodynamics for a non-singular, adiabatically evolving spherical perfect fluid in general relativity, emphasizing the importance of proper volume in pressure work calculations.

Contribution

It introduces the first formulation of a global thermodynamic first law for non-singular fluids within classical general relativity, clarifying the role of proper volume in pressure work.

Findings

Highlights the necessity of using proper volume in thermodynamic work calculations.

Formulates a global first law for adiabatic, non-singular fluids in general relativity.

Clarifies that quantum or semi-classical effects are not considered in this classical framework.

Abstract

We emphasize that the pressure related work appearing in a general relativistic first law of thermodynamics should involve {\em proper volume element} rather than coordinate volume element. This point is highlighted by considering both local energy momentum conservation equation as well as particle number conservation equation. It is also emphasized that we are considering here a {\em non-singular} fluid governed by purely classical general relativity. Therefore, we are not considering here any semi-classical or quantum gravity which apparently suggests thermodynamical properties even for a (singular) black hole. Having made such a clarification, we formulate a global first law of thermodynamics for an adiabatically evolving spherical perfect fluid. It may be verified that such a global first law of thermodynamics, {\em for a non-singular fluid}, has not been formulated earlier.