Timelike Killing Fields and Relativistic Statistical Mechanics

TL;DR

This paper develops a relativistic statistical mechanics framework for spacetimes with timelike Killing fields, introducing a new coordinate system and deriving formulas for free energy and thermodynamics, with applications to Kerr spacetime.

Contribution

It introduces a Fermi-Walker-Killing coordinate system and derives an exact relativistic Helmholtz free energy formula for ideal gases in such spacetimes.

Findings

Derived a relativistic Helmholtz free energy formula.

Compared relativistic and Newtonian thermodynamics.

Applied results to Kerr spacetime examples.

Abstract

For spacetimes with timelike Killing fields, we introduce a "Fermi-Walker-Killing" coordinate system and use it to prove a Liouville Theorem for an appropriate volume element of phase space for a statistical mechanical system of particles. We derive an exact relativistic formula for the Helmholtz free energy of an ideal gas and compare it, for a class of spacetimes, to its Newtonian analog, derived both independently and as the Newtonian limit of our formula. We also find the relativistic thermodynamic equation of state. Specific examples are given in Kerr spacetime.