2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/33959We 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.General Relativity and Quantum Cosmologytext