We investigate timelike and null geodesics within the rotating “time machine” spacetime proposed by Ralph, T.C.; et al. Phys. Rev. D 2020, 102, 124013. This is a rotating analogue of Alcubierre’s warp drive spacetime. We obtain geodesics that begin and end in the surrounding flat space region, yet achieve time travel relative to static observers there. This is a global property, as the geodesics remain locally future-pointing, as well as timelike or null.