Connectivity by geodesics on globally hyperbolic spacetimes with a lightlike Killing vector field

Abstract

Given a globally hyperbolic spacetime endowed with a complete lightlike Killing vector field and a complete Cauchy hypersurface, we characterize the points which can be connected by geodesics. A straightforward consequence is the geodesic connectedness of globally hyperbolic generalized plane waves with a complete Cauchy hypersurface.2000 Mathematics Subject Classification. 53C50, 53C22, 58E10.

Connecting and closed geodesics of a Kropina metric

“…and, as in the proof of Lemma 2.6 in [2], we obtain that the family of curves (x ε ) ε∈(0,1) is bounded in P(S). Moreover, from the second equation in ( 6), using the fact that ∆ ǫ is bounded, as in Lemma 6.2 in [3] we get that the family ( ṫε ) ε∈(0,1) is also bounded in L 2 ([0, 1], R). Now, for each ε, z ε is a critical point of the energy functional I ε of the Lorentzian metric g ε , i.e.…”

“…[13,12,21,14,2,5,22,6,7]). In fact, it has been profitably employed in [3] to study geodesic connectedness of a globally hyperbolic spacetime endowed with a lightlike Killing vector field.…”

Fermat Metrics

“…Some results on the existence of geodesics connecting two points or closed geodesics for a Kropina metric have been recently proved in [86]. We also mention the paper [87], where the existence of geodesics joining two points for the Lorentzian metric (45) is studied. We point out that, since a Kropina metric is not defined on the whole tangent bundle, there always exists couple of point which are not joined by any smooth curve and by no geodesic, see [86].…”

Convex functions and geodesic connectedness of space-times

“…The proof is divided into two cases, depending on whether the timelike convex hypersurface M of E n+1 1 is not or is ruled by parallel null lines. In the latter case, we use the following criterion of Bartolo, Candela, and Flores, the proof of which uses infinite dimensional variational methods [BCF17]. Case 1.…”

“…Geodesic connectedness was studied via an infinite-dimensional variational theory introduced by Benci, Fortunato, Giannoni and Masiello at the end of the 1980s. For Lorentzian manifolds carrying a timelike or null Killing field, geodesic connectedness has only recently become well understood [CFS08,BCF17]. It is also known to hold for globally hyperbolic space-times carrying time-dependent orthogonal splittings satisfying certain conditions, as summarized in Theorem A.2 of in the appendix.…”