Statistical Lorentzian geometry and the closeness of Lorentzian manifolds

I introduce a closeness function between causal Lorentzian geometries of finite volume and arbitrary underlying topology. The construction uses the fact that some information on the manifolds and metrics is encoded in the partial order that the causal structure of each metric induces among points randomly scattered in the corresponding manifold with uniform, finite density according to the volume element. When the density is finite, the closeness function is a pseudodistance, which only compares the manifolds down to the a finite volume scale; this is illustrated by a fully worked out example of two 2-dimensional manifolds of different topology. The introductory and concluding sections include some remarks on the motivation for this definition and its application to quantum geometry, and on the possibility of using it to obtain a distance function which distinguishes all pairs of Lorentzian manifolds. PACS numbers 04.20.Gz, 02.40.-k