Spacetime atoms and extrinsic curvature of equi-geodesic surfaces

TL;DR

This paper explores a function $ ho$ representing spacetime atoms of quantum origin, expressing it through the van Vleck scalar and extrinsic curvature, and clarifies its role in counting quantum states rather than atoms.

Contribution

It provides an exact expression for $ ho$ involving the van Vleck scalar and relates it to extrinsic curvature, clarifying its statistical interpretation in quantum spacetime.

Findings

$ ho$ expressed via van Vleck scalar and extrinsic curvature

$ ho$ counts quantum states, not spacetime atoms

Ratio of quantum states compares curved and flat spacetime

Abstract

A recently-introduced function $\rho$ of spacetime event $P$ expressing spacetime as made of 'spacetime atoms' of quantum origin is considered. Using its defining relation, we provide an exact expression for $\rho$ involving the van Vleck biscalar, and show it can be recast in terms of the extrinsic curvature of suitable equi-geodesic surfaces centered at $P$. Moreover, looking at the role $\rho$ plays in the statistical description of spacetime, we point out that this quantity should actually be understood as counting the quantum states of the collection of spacetime atoms rather than counting directly the spacetime atoms themselves (or the degrees of freedom associated to them), and would correspond to the ratio of the number of quantum states 'at $P$' for an assigned spacetime configuration to the number of quantum states for flat spacetime.