The entropy of a hole in spacetime

TL;DR

This paper computes the gravitational entropy of spherical Rindler space, a time-dependent generalization of Rindler space defined by radially accelerating observers disjoint from a spherical hole of radius $R_0$. Through near-horizon reasoning and a replica-trick calculation, the authors show the entropy is $S = \mathcal{A}/(4G)$, where $\mathcal{A}$ is the horizon area of the hole. The result supports the view that spacetime connectivity is rooted in entanglement between interior and exterior quantum-gravity degrees of freedom and remains robust in time-dependent settings. When a small negative cosmological constant is added (AdS), the entropy corresponds to entanglement between ultraviolet and infrared sectors of a dual theory, reinforcing the holographic interpretation of geometric entropy as entanglement entropy.

Abstract

We compute the gravitational entropy of 'spherical Rindler space', a time-dependent, spherically symmetric generalization of ordinary Rindler space, defined with reference to a family of observers traveling along non-parallel, accelerated trajectories. All these observers are causally disconnected from a spherical region H (a 'hole') located at the origin of Minkowski space. The entropy evaluates to S = A/4G, where A is the area of the spherical acceleration horizon, which coincides with the boundary of H. We propose that S is the entropy of entanglement between quantum gravitational degrees of freedom supporting the interior and the exterior of the sphere H.

Figures (1)

• Figure 1: Spherical Rindler Space