Boundary and Corner Terms in the Action for General Relativity

TL;DR

This paper provides a unified, geometric approach to boundary and corner terms in the action for general relativity, accommodating various boundary types and clarifying the role of creases like black hole horizons.

Contribution

It introduces a unified tetrad formalism treatment for boundary and corner terms in general relativity, including null boundaries and creases, simplifying calculations and revealing new insights.

Findings

Unified treatment of spacelike, timelike, and null boundaries.

Identification of corner and crease terms in the action.

Simplified derivation of known boundary contributions.

Abstract

We revisit the action principle for general relativity motivated by the path integral approach to quantum gravity. We consider a spacetime region whose boundary has piecewise $C^2$ components, each of which can be spacelike, timelike or null and consider metric variations in which only the pullback of the metric to the boundary is held fixed. Allowing all such metric variations we present a unified treatment of the spacelike, timelike and null boundary components using Cartan's tetrad formalism. Apart from its computational simplicity, this formalism gives us a simple way of identifying corner terms. We also discuss "creases" which occur when the boundary is the event horizon of a black hole. Our treatment is geometric and intrinsic and we present our results both in the computationally simpler tetrad formalism as well as the more familiar metric formalism. We recover known results from a simpler and more general point of view and find some new ones.