A Boundary Term for the Gravitational Action with Null Boundaries

TL;DR

The paper addresses how to formulate a well-posed variational principle for general relativity when the boundary is null. By analyzing the boundary term of the Einstein-Hilbert action, it derives a first-principles null-boundary counter-term, $2\sqrt{-g}(\Theta+\kappa)$ (or $2\sqrt{q}(\Theta+\kappa)$ in adapted coordinates), and identifies the intrinsic data to fix on the null boundary as the 2-metric $q^{ab}$ and the null normal $\ell^a$. It provides two coordinate frameworks—Gaussian Null Coordinates and Null Surface Foliation—to express the boundary term explicitly, and shows that the null boundary term can also be obtained as a limit of non-null boundaries, ensuring consistency. The work discusses the six boundary degrees of freedom associated with $(q^{ab},\ell^a)$ and outlines remaining questions about the precise well-posedness of the variational problem and a rigorous connection to the null initial-value formulation. Overall, the results lay groundwork for a principled treatment of null boundaries in gravitational variational principles with implications for horizon thermodynamics and causal structure analyses.

Abstract

Constructing a well-posed variational principle is a non-trivial issue in general relativity. For spacelike and timelike boundaries, one knows that the addition of the Gibbons-Hawking-York (GHY) counter-term will make the variational principle well-defined. This result, however, does not directly generalize to null boundaries on which the 3-metric becomes degenerate. In this work, we address the following question: What is the counter-term that may be added on a null boundary to make the variational principle well-defined? We propose the boundary integral of $2 \sqrt{-g} \left( Θ+κ\right)$ as an appropriate counter-term for a null boundary. We also conduct a preliminary analysis of the variations of the metric on the null boundary and conclude that isolating the degrees of freedom that may be fixed for a well-posed variational principle requires a deeper investigation.