Closed timelike geodesics in compact spacetimes

Abstract

Abstract. Let M be a compact spacetime which admits a regular globally hyperbolic covering, and C a nontrivial free timelike homotopy class of closed timelike curves in M. We prove that C contains a longest curve (which must be a closed timelike geodesic) if and only if the timelike injectivity radius of C is finite; i.e., C has a bounded length. As a consequence among others, we deduce that for a compact static spacetime there exists a closed timelike geodesic within every nontrivial free timelike homotopy class having a finite timelike injectivity radius.

Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field

“…An earlier result by Tipler (see [34]) gives the existence of one periodic timelike geodesic in compact Lorentzian manifolds that admit a regular covering which has a compact Cauchy surface. Recently, this result has been extended by Guediri [14,15,18] and Sánchez [31] to the case that the Cauchy surface in the covering is not necessarily compact, but assuming certain hypotheses on the group of deck transformations. The existence of a periodic timelike geodesic has been established also by Galloway in [11], where he proves the existence of a longest periodic timelike curve, which is necessarily a geodesic, in each stable free timelike homotopy class.…”

“…As to the stationary case, there exist in literature some previous results when the spacetime admits a standard stationary expression and, as a consequence, it is never compact (see [4,24]). As suggested by the authors in [7,18,31], an interesting open question would be to determine if existence results for periodic geodesics also hold for any stationary compact Lorentzian manifold. Moreover, in general there does not exist a globally hyperbolic covering for this class of manifolds (see [31, pag.…”

“…Moreover, in general there does not exist a globally hyperbolic covering for this class of manifolds (see [31, pag. 23], observe also that, unlike the static case, compact stationary Lorentzian manifolds may be simply connected) and a different approach from that of [7,14,15,18,31,34] is needed.…”

Spinning Strings, Black Holes and Stable Closed Timelike Geodesics

“…As a topological problem, the existence of CTGs was proved by Tipler [4] in a class of four-dimensional compact Lorentz manifolds with covering space containing a compact Cauchy surface. In a compact pseudo-Riemaniann manifold with Lorentzian signature (Lorentzian manifold) Galloway [5] found sufficient conditions to have CTGs, see also [6].…”

“…As a topological problem, the existence of CTGs was proved by Tipler [4] in a class of four-dimensional compact Lorentz manifolds with covering space containing a compact Cauchy surface. In a compact pseudo-Riemaniann manifold with Lorentzian signature (Lorentzian manifold) Galloway [5] found sufficient conditions to have CTGs, see also [6].To the best of our knowledge there are four solution to the Einstein equations generated by matter with positive mass density that contain CTGs: a) Soares [7] found a class of cosmological models, solutions of EinsteinMaxwell equations, with a subclass where the timelike paths of matter are closed. For these models the existence of CTGs is demonstrated and explicit examples are given.…”