Some global properties of static spacetimes. (English) Zbl 1044.53028

Caponio, Erasmo; Masiello, Antonio; Piccione, Paolo

Math. Z. 244, No. 3, 457-468 (2003).

Authors’ abstract: “We show that a static Lorentzian manifold satisfying a completeness assumption is geodesically connected. In particular, such condition is satisfied by all compact static manifolds; in the compact case, we also prove the existence of a closed geodesic in every free homotopy class determined by a finite number of deck transformations.”
The main idea behind the proof is the fact, that for a static space-time, one can always find a corresponding positive definite metric being identical to the original Lorentzian metric on the slices \(t =\) constant, if the Killing vector responsible for the spacetime being static is the gradient of \(t\). Therefore, properties of Riemannian geometry can be applied.

Reviewer: Hans-Jürgen Schmidt (Potsdam)

MSC:

53C22	Geodesics in global differential geometry
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
58E10	Variational problems in applications to the theory of geodesics (problems in one independent variable)
83C10	Equations of motion in general relativity and gravitational theory

Keywords:

geodesics; Hopf-Rinow theorem; completeness