Boundary Term Contribution to the Volume of a Small Causal Diamond

TL;DR

This paper clarifies the origin of a surface integral term in the volume calculation of small causal diamonds, showing it arises from differences in curved and flat spacetime volumes and providing explicit first-order corrections.

Contribution

It demystifies the surface integral in the volume of small Alexandrov intervals and computes the first-order correction using perturbation theory.

Findings

The surface integral arises from volume differences between curved and flat spacetime.

First-order correction adds a dimension-independent factor to the flat volume.

Next order correction to the volume vanishes.

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.