Boundary contributions in the causal set action

Fay Dowker

doi:10.1088/1361-6382/abc2fd

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, , of finite volume equals the Einstein Hilbert action of 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.

Highlights

Ni is the number of inclusive order intervals of cardinality i + 1 in C, where the order interval I(a, b) between two causal set elements a and b such that a ≺ b is given by
For each globally hyperbolic Lorentzian spacetime M of dimension d and finite volume, the Poisson process of sprinkling at density ρ := l−d and the causal set action S(d) gives rise to a random variable Sρ(M ) that equals the action evaluated on the random causal set that is the outcome of the sprinkling process
There is some evidence for the conjecture in the literature for the case of flat spacetime: it holds for flat causal intervals in all dimensions [5, 6] and for a null triangle and for a cylinder spacetime in 2 dimensions [5]

Summary

The causal set action

The Benincasa-Dowker-Glaser causal set action [1–4] is a family of actions, SB(dD) G(C) for a finite causal set, {C, }, one action for each natural number d > 1. For each globally hyperbolic Lorentzian spacetime M of dimension d and finite volume, the Poisson process of sprinkling at density ρ := l−d and the causal set action S(d) gives rise to a random variable Sρ(M ) that equals the action evaluated on the random causal set that is the outcome of the sprinkling process.. The causal set scalar d’Alembertian and the scalar curvature analogue have advanced versions gotten by reversing the order in (1.3) and (1.5) so the levels summed over are preceded by a: L1 is the set of elements that are preceded by a and linked to a etc. Running the argument above for this case, when considering the mean of the random discrete action of M we expect only the Einstein Hilbert contribution from points x that lie on the past boundary and not on the future boundary of M.

The Set Up

Metric

The interval

The slab

The triangle

Discussion