On the horizon entropy of a causal set

TL;DR

This paper explores defining horizon entropy within causal set theory, extending previous models to null hypersurfaces, and investigates the behavior of entropy via spacetime mutual information, linking it to horizon area in curved spacetimes.

Contribution

It introduces a new approach to horizon entropy in causal sets, extending horizon molecule definitions to null hypersurfaces and analyzing entropy through spacetime mutual information.

Findings

Extension of horizon molecules to null hypersurfaces is not entropy-local in curved spacetime.

Spacetime mutual information entropy converges to the horizon area.

Null hypersurface entropy definition faces limitations compared to spacelike cases.

Abstract

We discuss how to define a kinematical horizon entropy on a causal set. We extend a recent definition of horizon molecules to a setting with a null hypersurface crossing the horizon. We argue that, as opposed to the spacelike case, this extension fails to yield an entropy local to the hypersurface-horizon intersection in the continuum limit when the causal set approximates a curved spacetime. We then investigate the entropy defined via the Spacetime Mutual Information between two regions of a causal diamond truncated by a causal horizon, and find it does limit to the area of the intersection.