Towards black-hole horizons and geodesic focusing in causal sets

TL;DR

This paper explores how to identify black-hole horizons within causal set quantum gravity, introducing local diagnostics and discrete analogs of horizons using ladders and fuzzy ladders in a 1+1D toy model.

Contribution

It develops methods to detect apparent and event horizons in causal sets, advancing the understanding of horizon structure in discrete quantum gravity models.

Findings

A discrete expansion analogue changes sign at the horizon.

Fuzzy ladders enable tracking null geodesics over larger intervals.

Constructed a discrete horizon in a 1+1D black-hole toy model.

Abstract

The event horizon of a black hole is arguably the most dramatic manifestation of the fact that in General Relativity, causal structure is dynamical and spacetimes can be separated into distinct regions by causal boundaries. Causal set quantum gravity is an approach to quantum gravity in which causal relations between spacetime points constitute the basic structure on which the theory is based. This raises the question how a discrete horizon can be identified in a causal set. In our paper, we first construct a local diagnostic to approximate a global concept, namely the event horizon, based on discrete timelike curves. We then turn to the concept of an apparent horizon, which is based on local properties of geodesics, rather than global properties of the entire spacetime. We undertake first steps towards detecting apparent horizons in causal sets, using so-called ladders as tracers of null geodesics. We find that a discrete counterpart of the expansion changes sign across the black-hole horizon, as it should. Finally, we introduce the notion of a fuzzy ladder, which enables us to track null geodesics for larger intervals of the affine parameter. Thereby, we construct a portion of a discrete horizon in a toy-model for a black-hole spacetime in 1+1 dimensions.