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Statistical Lorentzian geometry and the closeness of Lorentzian manifolds
I introduce a family of closeness functions between causal Lorentzian
geometries of finite volume and arbitrary underlying topology. 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 illustrated here by a fully worked out example of
two 2-dimensional manifolds of different topology; if the density is allowed to
become infinite, a true distance can be defined on the space of all Lorentzian
geometries. The introductory and concluding sections include some remarks on
the motivation for this definition and its applications to quantum gravity.Comment: Plain TeX, 19 pages + 3 figures, revised version for publication in
J.Math.Phys., significantly improved conten
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