Proof of entropy principle in Einstein-Maxwell theory

TL;DR

This paper proves that in Einstein-Maxwell theory, the total entropy of a static charged fluid system reaches an extremum if and only if Einstein's and Maxwell's equations are satisfied, confirming the maximum entropy principle's consistency with these equations.

Contribution

It establishes a rigorous link between the extremum of total entropy and the validity of Einstein-Maxwell equations for static charged fluids.

Findings

Total entropy extremum implies Einstein-Maxwell equations.

Einstein-Maxwell equations imply total entropy extremum.

Supports maximum entropy principle in charged gravitational systems.

Abstract

We consider a static self-gravitating charged perfect fluid system in the Einstein-Maxwell theory. Assume Maxwell's equation and the Einstein constraint equation are satisfied, and the temperature of the fluid obeys Tolman's law. Then we prove that the total entropy of the fluid achieves an extremum implies other components of Einstein's equation for any variations of metric and electrical potential with fixed boundary values. Conversely, if Einstein's equation and Maxwell's equations hold, the total entropy achieves an extremum. Our work suggests that the maximum entropy principle is consistent with Einstein's equation when electric field is taken into account.