Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2000-02-15
J.Math.Phys. 41 (2000) 6944-6958
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
Plain TeX, 19 pages + 3 figures, revised version for publication in J.Math.Phys., significantly improved content
Scientific paper
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.
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