Einstein's theory of general relativity has solutions of spacetime with closed time curves (CTCs). However, whether CTCs are allowed by other laws in physics is an open question. Some investigations have shown that the stress-energy tensor of vacuum fluctuations in a spacetime with CTCs is divergent on the Cauchy horizon separating the region with CTCs from that without CTCs, indicating that a solution of spacetime with CTCs may be unphysical. Based on this fact, in 1992 Stephen Hawking proposed the Chronology Protection Conjecture: The laws of physics do not allow the appearance of CTCs.
I and my collaborators have written several papers challenging the Chronology Protection Conjecture by presenting counter examples to the conjecture. In a paper published in Physical Review D in 1994, I constructed a spacetime with CTCs but without closed causal geodesics, by placing a spherical mirror between two mouths of a wormhole time machine. The causal structure of the spacetime is shown in the above figure. The null geodetic generators of the Cauchy horizon H all end at the boundary . The initially spacelike (nongeodetic) closed curve C1∪C𝒮∪C2 becomes null and then timelike as it passes through the Cauchy horizon. I showed that the spacetime can be stable against the vacuum fluctuation of matter fields. However, whether such a spacetime is stable against the vacuum fluctuation of gravitational fields is an open question. [For details see L.-X. Li, Phys. Rev. D. 50, R6037 (1994). The work was reviewed by John Gribbin in New Scientist: Time travel: it's all done with smoke and mirrors.]