Lorentzian Manifolds and Causal Sets as Partially Ordered Measure Spaces

TL;DR

This paper generalizes the structure of Lorentzian manifolds and causal sets as partially ordered measure spaces, introducing a covariant distance measure to quantify their similarity, bridging classical and quantum gravity frameworks.

Contribution

It introduces a unified framework for Lorentzian manifolds and causal sets as partially ordered measure spaces, including a covariant distance function for comparing their closeness.

Findings

Defined a distance function for partially ordered measure spaces

Showed the distance is valid for compact, separable spaces

Provided a covariant measure of manifoldlikeness and similarity

Abstract

We consider Lorentzian manifolds as examples of partially ordered measure spaces, sets endowed with compatible partial order relations and measures, in this case given by the causal structure and the volume element defined by each Lorentzian metric. This places the structure normally used to describe spacetime in geometrical theories of gravity in a more general context, which includes the locally finite partially ordered sets of the causal set approach to quantum gravity. We then introduce a function characterizing the closeness between any two partially ordered measure spaces and show that, when restricted to compact spaces satisfying a simple separability condition, it is a distance. In particular, this provides a quantitative, covariant way of describing how close two manifolds with Lorentzian metrics are, or how manifoldlike a causal set is.