The Discrete Geometry of a Small Causal Diamond

Abstract

We study the discrete causal set geometry of a small causal diamond in a curved spacetime using the average abundance of k-element chains or total orders in the underlying causal set C. We begin by obtaining the first order curvature corrections to the flat spacetime expression for the abundance using Riemann normal coordinates. For fixed spacetime dimension this allows us to find a new expression for the discrete scalar curvature of C as well as the time-time component of its Ricci tensor in terms of the abundances of k-chains. We also find a new dimension estimator for C which replaces the flat spacetime Myrheim-Meyer estimator in generic curved spacetimes.

22 pages, 2 figures

References in corpus (8)

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• On Recovering Continuum Topology from a Causal Set
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• Dimensional reduction in manifold-like causal sets
• The Geometry of Small Causal Cones
• Algorithmic Causal Sets and the Wolfram Model
• Path Sums for Propagators in Causal Sets
• In-in correlators and scattering amplitudes on a causal set
• Local Structure of Sprinkled Causal Sets
• Manifold Properties from Causal Sets using Chains
• A Tale of Two Actions: A Variational Principle for Two-Dimensional Causal Sets
• Aspects of Quantum Fields on Causal Sets
• Numerical Evaluation of the Causal Set Propagator in 2D Anti-de Sitter Spacetime
• Endeavours in Discrete Lorentzian Geometry; A thesis in five papers
• Benincasa-Dowker-Glaser causal set actions by quantum counting