Abstract
The aim of this chapter is to review and complete the study of geodesics on Gödel-type spacetimes from a variational viewpoint in the last decade (say, from [10] to [2]). In particular, we prove some new results on geodesic connectedness and geodesic completeness for these spacetimes.
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Acknowledgements
The authors of this chapter acknowledge the partial support of the Spanish Grants with FEDER funds MTM2010-18099 (MICINN). Furthermore, R. Bartolo and A.M. Candela acknowledge also the partial support of M.I.U.R. Research Project PRIN2009 “Metodi Variazionali e Topologici nello Studio di Fenomeni Nonlineari” and of the G.N.A.M.P.A. Research Project 2011 “Analisi Geometrica sulle Varietà di Lorentz ed Applicazioni alla Relatività Generale”; J.L. Flores acknowledges also the partial support of the Regional J. Andalucía Grant P09-FQM-4496, with FEDER funds.
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1 Appendix
Taking a connected, finite–dimensional semi–Riemannian manifold \((\mathcal{M},g)\), let \({H}^{1}(I,\mathcal{M})\) be the associated Sobolev space for some auxiliar Riemannian metric on \(\mathcal{M}\). Then, \({H}^{1}(I,\mathcal{M})\) is equipped with a structure of infinite–dimensional manifold modelled on the Hilbert space \({H}^{1}(I, {\mathbb{R}}^{n})\). For any \(z \in {H}^{1}(I,\mathcal{M})\), the tangent space of \({H}^{1}(I,\mathcal{M})\) at z can be written as follows:
If \(\mathcal{M}\) splits globally in the product of two semi–Riemannian manifolds \({\mathcal{M}}_{1}\) and \({\mathcal{M}}_{2}\), i.e. \(\mathcal{M} = {\mathcal{M}}_{1} \times {\mathcal{M}}_{2}\), then
On the other hand, if \(({\mathcal{M}}_{0},\langle \cdot,{\cdot \rangle }_{R})\) is a C 3 complete Riemannian manifold, it can be smoothly and isometrically embedded in a Euclidean space \({\mathbb{R}}^{N}\) (see [24]); moreover such embedding can be chosen closed (see [23]) and this is used in the proof of Lemma 1. Hence, \({H}^{1}(I,{\mathcal{M}}_{0})\) is a closed submanifold of the Hilbert space \({H}^{1}(I, {\mathbb{R}}^{N})\). In this case, we denote by \(d(\cdot,\cdot )\) the distance induced on \({\mathcal{M}}_{0}\) by its Riemannian metric \(\langle \cdot,{\cdot \rangle }_{R}\), i.e.,
Taking z p , \({z}_{q} \in \mathcal{M}\), let us consider
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Bartolo, R., Candela, A.M., Flores, J.L. (2012). Global Geodesic Properties of Gödel-type SpaceTimes. In: Sánchez, M., Ortega, M., Romero, A. (eds) Recent Trends in Lorentzian Geometry. Springer Proceedings in Mathematics & Statistics, vol 26. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4897-6_7
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