The Gravitational Hamiltonian, Action, Entropy, and Surface Terms

TL;DR

This paper derives the gravitational Hamiltonian from the Einstein-Hilbert action, emphasizing surface terms, and explores their relation to energy, entropy, and horizons in various spacetime contexts.

Contribution

It provides a comprehensive derivation of the gravitational Hamiltonian including all surface terms and discusses their implications for energy and entropy in spacetimes with horizons.

Findings

Surface term defines total energy even in non-asymptotically flat spacetimes.

In asymptotically flat spacetimes, surface term matches ADM energy.

Acceleration horizons may not have entropy proportional to horizon area.

Abstract

We give a general derivation of the gravitational hamiltonian starting from the Einstein-Hilbert action, keeping track of all surface terms. The surface term that arises in the hamiltonian can be taken as the definition of the `total energy', even for spacetimes that are not asymptotically flat. (In the asymptotically flat case, it agrees with the usual ADM energy.) We also discuss the relation between the euclidean action and the hamiltonian when there are horizons of infinite area (e.g. acceleration horizons) as well as the usual finite area black hole horizons. Acceleration horizons seem to be more analogous to extreme than nonextreme black holes, since we find evidence that their horizon area is not related to the total entropy.