Is the gravitational action additive?

TL;DR

The paper revisits whether the gravitational action $I$ is additive under spacetime composition via boundary identifications, showing that joint terms can yield a finite residue and that universal additivity cannot be guaranteed. It introduces generalized additivity, $I[\widetilde{\mathcal{M}}]=I[\mathcal{M}]-I[\mathcal{T}]$, to restore an additive framework even when a regional region $\mathcal{T}$ (possibly of finite 4-volume) is involved, and analyzes residues arising from timelike versus spacelike identifications. For timelike boundaries, explicit residues $\mathcal{R}[S]$ can be nonzero (e.g., $\mathcal{R}[J]=A[J]/4$ in a limiting case), while spacelike identifications can yield zero residues, with the Euclidean sector linking residues to horizon entropy. The study discusses implications for quantum gravity superposition, the limits of fixing a boundary functional $C$, and extensions to modified gravity theories, highlighting that generalized additivity provides a robust, broadly applicable additive framework.

Abstract

The gravitational action is not always additive in the usual sense. We provide a general prescription for the change in action that results when different portions of the boundary of a spacetime are topologically identified. We discuss possible implications for the superposition law of quantum gravity. We present a definition of `generalized additivity' which does hold for arbitrary spacetime composition.