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

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 nonexistence 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 noncompactness 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.

Figure 1

The idea of the perturbative approach is to evaluate $\dot{S}$ along different directions in phase space away from equilibrium and arrive at a contradiction with its strict positivity.

Figure 2

The topological approach relies on phase space compactness and Liouville’s theorem, i.e. the fact that the Hamiltonian flow is volume preserving.

Figure 3

The idea of the proof for the Poincaré recurrence theorem.