Boundary term in the gravitational action is the heat content of the null surfaces

Abstract

The Einstein-Hilbert Lagrangian has no well-defined variational derivative with respect to the metric. This issue has to be tackled by adding a suitable surface term to the action, which is a peculiar feature of gravity. We also know that null surfaces in spacetime exhibit (observer-dependent) thermodynamic features. This suggests a possible thermodynamic interpretation of the boundary term when the boundary is a null surface. For timelike/spacelike surfaces it is easy to construct the boundary term but there are some subtleties in the case of the null surface. The correct form of boundary term for null surfaces was obtained recently from first principles. We show that this surface term, as well as its variation, have direct thermodynamic interpretation in terms of a heat density of null surfaces. The implications of the result are discussed.