Variational Principle for Gravity with Null and Non-null boundaries: A Unified Boundary Counter-term

TL;DR

This paper presents a covariant, unified boundary counter-term $\\mathcal{C}_0 = 2\\sqrt{-g} \\Pi^a_{\\,b} \\nabla_a s^b$ that renders the Einstein-Hilbert action variationally well-posed for boundaries of any causal character. By defining $s_a = abla_a \\phi$, an auxiliary vector $t^a$ with $t^a s_a = -1$, and a surface projector $\\Pi^a_{\\,b}$, the authors show the boundary variation splits into a total surface derivative and a normal-derivative term cancelable by $\\mathcal{C}_0$, leaving only metric boundary data. They demonstrate that, for non-null boundaries with $t^a = -s^a/s^2$, $\\mathcal{C}_0$ reduces to the Gibbons-Hawking-York term, while for null boundaries with $t^a = k^a$, it recovers the known null-counter-term; these reductions establish consistency with established prescriptions. Overall, the work provides a single, covariant prescription bridging null and non-null boundaries, improving the robustness and flexibility of gravitational action formulations in varied boundary geometries.

Abstract

It is common knowledge that the Einstein-Hilbert action does not furnish a well-posed variational principle. The usual solution to this problem is to add an extra boundary term to the action, called a counter-term, so that the variational principle becomes well-posed. When the boundary is spacelike or timelike, the Gibbons-Hawking-York counter-term is the most widely used. For null boundaries, we had proposed a counter-term in a previous paper. In this paper, we extend the previous analysis and propose a counter-term that can be used to eliminate variations of the "off-the-surface" derivatives of the metric on any boundary, regardless of its spacelike, timelike or null nature.