The Geometry of Small Causal Diamonds

The geometry of causal diamonds or Alexandrov open sets whose initial and final events $p$ and $q$ respectively have a proper-time separation $\tau$ small compared with the curvature scale is a universal. The corrections from flat space are given as a power series in $\tau$ whose coefficients involve the curvature at the centre of the diamond. We give formulae for the total 4-volume $V$ of the diamond, the area $A$ of the intersection the future light cone of $p$ with the past light cone of $q$ and the 3-volume of the hyper-surface of largest 3-volume bounded by this intersection valid to ${\cal O} (\tau ^4)$. The formula for the 4-volume agrees with a previous result of Myrheim. Remarkably, the iso-perimetric ratio ${3V_3 \over 4 \pi} / ({A \over 4 \pi}) ^{3 \over 2}$ depends only on the energy density at the centre and is bigger than unity if the energy density is positive. These results are also shown to hold in all spacetime dimensions. Formulae are also given, valid to next non-trivial order, for causal domains in two spacetime dimensions. We suggest a number of applications, for instance, the directional dependence of the volume allows one to regard the volumes of causal diamonds as an observable providing a measurement of the Ricci tensor.