This relativistic, time-travel spacetime is everywhere metrically flat, excepting a conical singularity. Observers following timelike geodesics can eventually encounter their past selves, aging in the opposite time sense. The spacetime is not time orientable.

“A conical singularity is unlike more familiar curvature singularities,” said Norton. “No matter how close you come to the singularity in this spacetime, the spacetime geometry remains flat and regular with no indication locally of something pathological.”

A spaceship, traveling in a well-behaved region of spacetime, might then encounter this singularity and find its timelike geodesic to be deflected into its past. It would then return to the earlier, well-behaved region of spacetime and the spaceship occupants would encounter their past selves aging in the opposite time sense.

The author anticipates this model being a useful tool for teaching technical aspects of relativity theory.

“The full geometry of this spacetime model can be recovered using rather simple methods,” said Norton. “In addition to seeing the nature of the conical singularity, the analysis requires that students treat spacetime coordinates with extreme care, since they do not have their usual default physical meanings.”

Source: “A simple Minkowskian time-travel spacetime,” by John D. Norton, American Journal of Physics (2025). The article can be accessed here.