A topological method for geodesic connectedness of space–times: Outer Kerr space–time

For a space–time, the question whether two causally related points can be joined by means of a causal geodesic has a clear physical meaning. More geometrically, geodesic connectedness ~the possibility of joining any two points by a geodesic of any causal type! is a basic property. Some techniques introduced to study geodesic connectedness are appliable to related problems of physical interest, such as the gravitational lensing effect, or the connectedness of two submanifolds by a causal geodesic ~see, e.g., Refs. 1–5!. Different tools have been introduced to study geodesic connectedness of Lorentzian manifolds. The first nontrivial examples were spaceforms, which become specially relevant from the geometrical viewpoint. A complete positive Lorentzian spaceform is geodesically connected if and only if it is not time-orientable. In particular, de Sitter space–time S1 n is not geodesically connected; this happens in spite of the fact that it is globally hyperbolic and, thus, each two causally related points in S1 n can be joined by a causal geodesic. The results about geodesic connectedness of manifolds endowed with an affine connection ~see also Refs. 10–12! are potentially appliable to any manifold endowed with a nondegenerate metric. Geodesics of Lorentzian tori provide interesting examples related to connectedness. But the systematic study of the geodesic connectedness of physically relevant space–times was carried out after the introduction of some variational methods in Lorentzian geometry. These methods permit one to prove the geodesic connectedness of stationary and splitting type manifolds under reasonable conditions ~see the book—Ref. 17 or the survey—Ref. 18!. Moreover, with different improvements, they ensure the connectedness of outer Schwarzschild space–time, intermediate Reissner–Nordström, Gödel type, and other space–times. Recently, the authors have obtained the necessary and sufficient condition for the connectedness of generalized Robertson–Walker space–times. Moreover, topological arguments have been developed which prove the connectedness of multiwarped space–times, under sufficient conditions which are close to necessary conditions ~see also Refs. 24 and 25!. In particular, not only space–times as Schwarzschild black hole are shown to be geodesically connected, but also new proofs of the geodesic connectedness of space–times such as intermediate Reissner–Nördstrom and outer Schwarzschild are obtained. Significantly, geodesic connectedness of outer Kerr space–time has not been studied yet. It is not difficult to prove that the stationary part of Kerr space–time is not geodesically connected; moreover, fast rotating Kerr space–time ~i.e., Kerr space–time with parameters a.m) is not geodesically connected. On the other hand, for values of the parameter a close to 0, the hyper-