Boundary Term Contribution to the Volume of a Small Causal Diamond

Abstract

In his calculation of the spacetime volume of a small Alexandrov interval in 4 dimensions Myrheim introduced a term which he referred to as a surface integral [1]. The evaluation of this term has remained opaque and led subsequent authors to evaluate the volume using other techniques [2]. It is the purpose of this work to demystify this integral. We point out that it arises from the difference in the flat spacetime volumes of the curved and flat spacetime intervals. An explicit evaluation using first order degenerate perturbation theory shows that it adds a dimension independent factor to the flat spacetime volume as the lowest order correction. Our analysis admits a simple extension to a more general class of integrals over the same domain. Using a combination of techniques we also find that the next order correction to the volume vanishes.

18 pages, 2 figures

References in corpus (2)

• The Scalar Curvature of a Causal Set
• The Geometry of Large Causal Diamonds and the No Hair Property of Asymptotically de-Sitter Spacetimes

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• The causal set approach to quantum gravity
• Towards a Definition of Locality in a Manifoldlike Causal Set
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• The Complete Barrett-Crane Model and its Causal Structure
• Boundary Terms for Causal Sets
• Geometry of small causal diamonds
• Scalar Field Green Functions on Causal Sets
• Comparison theorems for causal diamonds
• Horizon Molecules in Causal Set Theory
• On the continuum limit of Benincasa-Dowker-Glaser causal set action
• The Geometry of Small Causal Cones
• Aspects of Quantum Fields on Causal Sets