# Geometry of small causal diamonds

## TL;DR

This paper systematically analyzes the geometry of small causal diamonds using three constructions, providing new perturbative results on their properties and revealing a unique area deficit relation in four dimensions.

## Contribution

It introduces a perturbative approach to causal diamond geometry for multiple constructions and uncovers a special four-dimensional relation involving the Bel-Robinson superenergy density.

## Key findings

- Only the light-cone cut construction shows an area deficit proportional to superenergy density in 4D.
- Computed the volume of Alexandrov interval causal diamond in vacuum.
- Provided new geometric formulas useful for quantum gravity and general relativity applications.

## Abstract

The geometry of small causal diamonds is systematically studied, based on three distinct constructions that are common in the literature, namely the geodesic ball, the Alexandrov interval and the lightcone cut. The causal diamond geometry is calculated perturbatively using Riemann normal coordinate expansion up to the leading order in both vacuum and non-vacuum. We provide a collection of results including the area of the codimension-two edge, the maximal hypersurface volume and their isoperimetric ratio for each construction, which will be useful for any applications involving the quantitative properties of causal diamonds. In particular, by solving the dynamical equations of the expansion and the shear on the lightcone, we find that intriguingly only the light-cone cut construction yields an area deficit proportional to the Bel-Robinson superenergy density W in four dimensional spacetime, but such a direct connection fails to hold in any other dimension. We also compute the volume of the Alexandrov interval causal diamond in vacuum, which we believe is important but missing from the literatures. Our work complements and extends the earlier works on the causal diamond geometry by Gibbons and Solodukhin, Jacobson, Senovilla and Speranza and others. Some potential applications of our results in mathematical general relativity and quantum gravity are discussed.