On the existence of closed timelike geodesics in compact spacetimes

In [19], Tipler has shown that a compact spacetime having a regular globally hyperbolic covering space with compact Cauchy surfaces necessarily contains a closed timelike geodesic. The restriction to compact spacetimes with just regular globally hyperbolic coverings (i.e., the Cauchy surfaces are not required to be compact) is still an open question. Here, we shall answer this question negatively by providing examples of compact flat Lorentz space forms without closed timelike geodesics, and shall give some criterion for the existence of such geodesics. More generally, we will show that in a compact spacetime having a regular globally hyperbolic covering, each free timelike homotopy class determined by a central deck transformation must contain a closed timelike geodesic. Whether or not a compact flat spacetime contains closed nonspacelike geodesics is, as far as we know, an open question. We shall answer this question affirmatively. We shall also introduce the notion of timelike injectivity radius for a spacetime relative to a free timelike homotopy class and shall show that it is finite whenever the corresponding deck transformation is central. Received: 9 November 1999; in final form: 19 September 2000 / Published online: 25 June 2001