Singularity-free orthogonally-transitive cylindrical spacetimes

This talk will deal to a certain extent with the natural continuation of some work in progress that was introduced last year at the Spanish Relativity Meeting in Salamanca 1, 2 and that has recently been published 3. The reason for investigating geodesic completeness of Lorentzian manifolds is twofold. From the mathematical point of view, there is no analogue for the Hopf-Rinow theorem that characterizes geodesic completeness of Riemannian manifolds in terms of metric completeness. Since the Lorentzian metric does not determine a metric structure, just a causal structure, this possibility is hindered. From the physical point of view, the existence of inhomogeneous cosmological models that are causally geodesically complete and, therefore, singularity-free, and whose role in Cosmology is yet to be determined, induces us to try to characterize when such behaviour arises. Of course, there are well-known results, the singularity theorems 4, 5 due to Hawking, Penrose, Tipler . . . , that provide sufficient conditions for a Lorentzian manifold to be incomplete. Furthermore, they discern whether there is a Big Bang, Big Crunch or geodesic imprisonment singularity. On the contrary, sufficient conditions for a spacetime to be causally geodesically complete are not so easily come across in the literature 6, 2, 3. Since such nonsingular cosmological models have arisen within the framework of inhomogeneous cosmologies, I shall devote my talk to orthogonally-transitive G2 cylindrical spacetimes.