Thermodynamic equilibrium in relativity: four-temperature, Killing vectors and Lie derivatives

TL;DR

This paper reviews the principles of relativistic thermodynamics and statistical mechanics, emphasizing the role of the four-temperature vector as a Killing vector in equilibrium, leading to vanishing Lie derivatives of observables.

Contribution

It clarifies the connection between four-temperature vectors, Killing vectors, and equilibrium conditions in relativistic thermodynamics and mechanics.

Findings

Four-temperature vector  defines a hydrodynamic frame.

Equilibrium requires  to be a Killing vector.

Lie derivatives of all physical observables vanish in equilibrium.

Abstract

The main concepts of general relativistic thermodynamics and general relativistic statistical mechanics are reviewed. The main building block of the proper relativistic extension of the classical thermodynamics laws is the four-temperature vector \beta, which plays a major role in the quantum framework and defines a very convenient hydrodynamic frame. The general relativistic thermodynamic equilibrium condition demands \beta to be a Killing vector field. We show that a remarkable consequence is that all Lie derivatives of all physical observables along the four-temperature flow must then vanish.