Recurrence metrics and the physics of closed time-like curves

We investigate vacuum solutions of Einstein's equation for a universe with an S^1 topology of time. Such a universe behaves like a time-machine and has geodesics which coincide with closed time-like curves (CTCs). A system evolving along a CTC experiences the Loschmidt velocity reversion and undergoes a recurrence commensurate with the universal period. We indicate why this universe is free of the causality paradoxes, usually associated with CTCs.