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

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
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