Closed timelike geodesics in compact spacetimes

Let $M$ be a compact spacetime which admits a regular globally hyperbolic covering, and ${\mathcal C}$ a nontrivial free timelike homotopy class of closed timelike curves in $M.$ We prove that ${\mathcal C}$ contains a longest curve (which must be a closed timelike geodesic) if and only if the timelike injectivity radius of ${\mathcal C}$ is finite; i.e., ${\mathcal C}$ 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.