Entropy theorems in classical mechanics, general relativity, and the gravitational two-body problem

TL;DR

This paper explores entropy in classical mechanics and general relativity, demonstrating why the second law holds in curved spacetime and analyzing the phase space of perturbed black hole solutions, highlighting dissipation signs.

Contribution

It extends entropy theorems to matter fields in curved spacetime and shows why these fail in general relativity, with applications to black hole phase space analysis.

Findings

Entropy theorems are valid for matter fields but fail in general relativity.

The phase space of perturbed Schwarzschild spacetimes is non-compact.

Lack of recurring orbits indicates dissipation and entropy production.

Abstract

In classical Hamiltonian theories, entropy may be understood either as a statistical property of canonical systems, or as a mechanical property, that is, as a monotonic function of the phase space along trajectories. In classical mechanics, there are theorems which have been proposed for proving the non-existence of entropy in the latter sense. We explicate, clarify and extend the proofs of these theorems to some standard matter (scalar and electromagnetic) field theories in curved spacetime, and then we show why these proofs fail in general relativity; due to properties of the gravitational Hamiltonian and phase space measures, the second law of thermodynamics holds. As a concrete application, we focus on the consequences of these results for the gravitational two-body problem, and in particular, we prove the non-compactness of the phase space of perturbed Schwarzschild-Droste spacetimes. We thus identify the lack of recurring orbits in phase space as a distinct sign of dissipation and hence entropy production.