Counting a black hole in Lorentzian product triangulations

TL;DR

This paper develops a method to identify black hole horizons in nonperturbative quantum gravity models using a counting formula based on the expansion rate of null geodesics within Lorentzian dynamical triangulations.

Contribution

It introduces a novel quantum horizon finder using integrated expansion rates in Lorentzian product triangulations, advancing nonperturbative quantum gravity research.

Findings

Derived an expression for null geodesic expansion in causal dynamical triangulations.

Proposed a counting formula for discrete spacetime blocks related to horizon detection.

Introduced Lorentzian dynamical triangulations of product type applicable beyond black-hole geometries.

Abstract

We take a step toward a nonperturbative gravitational path integral for black-hole geometries by deriving an expression for the expansion rate of null geodesic congruences in the approach of causal dynamical triangulations. We propose to use the integrated expansion rate in building a quantum horizon finder in the sum over spacetime geometries. It takes the form of a counting formula for various types of discrete building blocks which differ in how they focus and defocus light rays. In the course of the derivation, we introduce the concept of a Lorentzian dynamical triangulation of product type, whose applicability goes beyond that of describing black-hole configurations.