Boundary contributions in the causal set action

TL;DR

This paper investigates the boundary contributions to the causal set action in Lorentzian spacetimes, proposing a conjecture that relates the mean causal set action to the Einstein-Hilbert action plus a boundary volume term, supported by heuristic analysis and examples.

Contribution

It introduces a conjecture linking the causal set action's mean to the Einstein-Hilbert action plus boundary volume, with analysis in 2D and 4D cases.

Findings

Evidence supports the conjecture in specific examples.

The mean causal set action includes a boundary volume term.

Heuristic arguments align with the proposed boundary contribution.

Abstract

Evidence is provided for a conjecture that, in the continuum limit, the mean of the causal set action of a causal set sprinkled into a globally hyperbolic Lorentzian spacetime, M, of finite volume equals the Einstein Hilbert action of M plus the volume of the co-dimension 2 intersection of the future boundary with the past boundary. We give the heuristic argument for this conjecture and analyse some examples in 2 dimensions and one example in 4 dimensions.