A Local First Law of Gravity

TL;DR

This paper derives a local first law of thermodynamics relating gravitational entropy, matter energy, and work, applicable to various gravity theories and evaluated on stretched light cones in arbitrary spacetimes.

Contribution

It introduces a quasi-local first law of gravity connecting entropy, energy, and work on stretched light cones, extending previous black hole thermodynamics to general spacetimes.

Findings

Derives a local thermodynamic relation in gravity theories.

Applies to Einstein and general theories of gravity.

Evaluates entropy on stretched light cones in arbitrary spacetimes.

Abstract

We find a (quasi-)local first law of thermodynamics, $\Delta E = T \Delta S - W$, connecting gravitational entropy, $S$, with matter energy and work. For Einstein gravity $S$ is the Bekenstein-Hawking entropy, while for general theories of gravity $S$ is the Wald entropy, evaluated on the stretched future light cone of any point in an arbitrary spacetime, not necessarily containing a black hole. The equation can be written as $\rho \Delta V = T \Delta S - p \Delta V$ by regarding the energy-momentum tensor as that of a fluid.