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

TL;DR

This paper develops a unified variational principle for gravity by proposing a boundary counter-term applicable to spacelike, timelike, and null boundaries, ensuring well-posedness of the Einstein-Hilbert action.

Contribution

It introduces a new boundary counter-term that unifies the treatment of all boundary types in gravitational actions, extending previous null boundary results.

Findings

A generalized counter-term for all boundary types.

Elimination of off-the-surface metric variations.

Ensures well-posed variational principle for gravity.

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.