Boundary term contribution to the volume of a small causal diamond

In his calculation of the spacetime volume of a small Alexandrov interval in four dimensions, Myrheim introduced a term which he referred to as a surface integral (Myrheim 1978 CERN preprint TH-2538). The evaluation of this term has remained opaque and led subsequent authors to obtain a formula for the volume using other techniques (Gibbons and Solodukhin 2007 Phys. Lett. B 649 317). It is the purpose of this work to explicitly calculate this integral and in the process complete the proof for the volume formula in arbitrary dimensions. We point out that it arises from the difference in the flat spacetime volumes of the curved and flat spacetime intervals. We use first-order degenerate perturbation theory to evaluate this difference and find that it adds a dimension-independent factor to the volume of the flat spacetime interval 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.