Boundary Conditions, Energies and Gravitational Heat in General Relativity (a Classical Analysis)

TL;DR

This paper explores the classical geometric relationship between energy variations, boundary conditions, and thermodynamic concepts like heat and free energy in General Relativity, highlighting an analogy with thermodynamic laws.

Contribution

It provides a covariant formulation of energy variation in GR, linking Komar energy to gravitational heat and ADM correction to Helmholtz free energy, within a classical geometric framework.

Findings

Komar energy relates to gravitational heat.

ADM correction acts as Helmholtz free energy.

Thermodynamic analogy applies to various stationary spacetimes.

Abstract

The variation of the energy for a gravitational system is directly defined from the Hamiltonian field equations of General Relativity. When the variation of the energy is written in a covariant form it splits into two (covariant) contributions: one of them is the Komar energy, while the other is the so-called covariant ADM correction term. When specific boundary conditions are analyzed one sees that the Komar energy is related to the gravitational heat while the ADM correction term plays the role of the Helmholtz free energy. These properties allow to establish, inside a classical geometric framework, a formal analogy between gravitation and the laws governing the evolution of a thermodynamic system. The analogy applies to stationary spacetimes admitting multiple causal horizons as well as to AdS Taub-bolt solutions.