Closed timelike geodesics in compact spacetimes

Abstract

Let M M be a compact spacetime which admits a regular globally hyperbolic covering, and C {\mathcal C} a nontrivial free timelike homotopy class of closed timelike curves in M . M. We prove that C {\mathcal C} contains a longest curve (which must be a closed timelike geodesic) if and only if the timelike injectivity radius of C {\mathcal C} is finite; i.e., C {\mathcal C} has a bounded length. As a consequence among others, we deduce that for a compact static spacetime there exists a closed timelike geodesic within every nontrivial free timelike homotopy class having a finite timelike injectivity radius.