Lorentzian Manifolds and Causal Sets as Partially Ordered Measure Spaces

2012/12/04 by L. Bombelli, Bombelli, Luca, Johan Noldus +3 · 3 citations
Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Noncommutative and Quantum Gravity Theories

paper · pdf · doi:10.48550/arxiv.1212.0601

openalex publication_date 2012/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Spacetime reconstruction by order and number
• Out of Nowhere: The emergence of spacetime from causal sets
• Gromov's reconstruction theorem and measured Gromov-Hausdorff convergence in Lorentzian geometry

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• The mathematics of causal sets
• On the Axioms of Causal Set Theory
• The causal set approach to quantum gravity
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