Boundary and Corner Terms in the Action for General Relativity

TL;DR

This work develops a unified, geometry-driven treatment of boundary and corner terms in general relativity by employing Cartan's tetrad formalism. By allowing metric variations that fix only the boundary pullback and leveraging differential forms, the authors derive a universal boundary term valid for spacelike, timelike, and null boundaries, and they systematically extract corner terms and creases arising at joins. They also translate the results to the metric formalism, confirming consistency with established results while clarifying reparametrisation issues for null boundaries and the additivity of the action across spacetime splits. The framework clarifies the role of boundary terms in quantum gravity path integrals and offers insights into black-hole horizons, asymptotic structures, and gauge properties of the boundary action.

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.

Figures (5)

• Figure 1: An illustration of how the normal and transverse vectors would be orientated on a patch of $1+1$ Minkowski spacetime whose boundary is a circle.
• Figure 2: An interval in $2+1$ Minkowski spacetime.
• Figure 3: Examples of null-spacelike joins in $2+1$ Minkowski spacetime showing the orientation of the vectors $k^a$, $l^a$, $n^a$ and $m^a$. The subcaptions illustrate whether a given null or spacelike surface is part of the future or past boundary of $\mathcal{M}$.
• Figure 4: Examples of the null-timelike case with a spacelike join in $2+1$ Minkowski spacetime. In each example we show a portion of the null and timelike surfaces $\Sigma_1$ and $\Sigma_2$ respectively. One (Two) dot(s) on a surface indicates that, from the perspective of the diagram, you are seeing the inside (outside) face of the surface with respect the region $\mathcal{M}$ that it bounds. The subcaptions illustrate whether the null surface and join are part of the future or past boundary of $\mathcal{M}$.
• Figure 5: Examples of the null-timelike case with a null join in $2+1$ Minkowski spacetime. In each example we show a portion of the null and timelike surfaces $\Sigma_1$ and $\Sigma_2$ respectively. From the perspective of the diagram the outside faces of the surfaces can be seen, with respect the region $\mathcal{M}$ that they bound. The subcaptions illustrate whether the null surface and join are part of the future or past boundary of $\mathcal{M}$.