# Luca BombelliUniversity of Mississippi

## Statistical Lorentzian geometry and the space of Lorentzian geometries

In this talk, I will describe the construction of a metric on the space
of Lorentzian geometries of finite volume, and arbitrary underlying topology,
based on statistical geometry. When points are randomly scattered in a
Lorentzian manifold, with uniform density according to the volume element,
some information on the topology and metric is encoded in the partial order
that the causal structure induces among those points; one can then define
closeness between Lorentzian geometries by comparing the sets of probabilities
they give for obtaining the same posets. If the density of points is finite,
one gets a pseudo-distance, which only compares the manifolds down to a
finite volume scale, as I will illustrate by a fully worked out example with
two 2-dimensional manifolds of different topology; if the density is allowed
to become infinite, a true metric is obtained on the space of all
Lorentzian geometries satisfying a suitable causality condition.