Superselection Sectors of Gravitational Subregions

TL;DR

This paper investigates how to define gravitational subregions in general relativity by imposing conditions that lead to boundaries of extremal area, aiding in understanding graviton entanglement.

Contribution

It introduces a condition on the symplectic form that characterizes gravitational subregions with extremal boundary surfaces and identifies the phase space variables for such subsystems.

Findings

Boundaries of gravitational subregions are extremal surfaces.

The phase space center variables relate to the conformal class of the boundary metric.

The condition discards local boundary deformations to nearby extremal surfaces.

Abstract

Motivated by the problem of defining the entanglement entropy of the graviton, we study the division of the phase space of general relativity across subregions. Our key requirement is demanding that the separation into subregions is imaginary---i.e., that entangling surfaces are not physical. This translates into a certain condition on the symplectic form. We find that gravitational subregions that satisfy this condition are bounded by surfaces of extremal area. We characterise the 'centre variables' of the phase space of the graviton in such subsystems, which can be taken to be the conformal class of the induced metric in the boundary, subject to a constraint involving the traceless part of the extrinsic curvature. We argue that this condition works to discard local deformations of the boundary surface to infinitesimally nearby extremal surfaces, that are otherwise available for generic codimension-2 extremal surfaces of dimension $\geq$ 2.