  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.
