There are spacetimes that admit closed timelike curves (for example a cylinder, Gödel's Universe, the Kerr-Solution).
But what does it physically mean to move along a closed timelike curve? If i move along a timelike curve, proper time passes. Thus my carried clock would show a different time, then the time i started with. So i would somehow evolve in time, but come back to an environment in a past state? Or would i also be in the initial state, i started with?
We do not have any experimental access to spacetimes with closed timelike curves. So all statements about physics in such spacetimes is a matter of conjecture. In particular, I will assume the Novikov self consistency conjecture in the remainder of this answer.
Note that just because a spacetime has closed timelike curves does not imply that objects must follow exactly those closed curves. For example, if there is a closed timelike curve then there is also a non-closed timelike curve that converges to the closed curve to within a small distance $\delta$, goes around the curve once never departing more than $\delta$ away during the loop, and then diverges from the closed timelike curve to a distance greater than $\delta$. Such a path would have a different proper time on the converging and diverging legs.
Objects that do follow exactly a closed timelike curve would have to go back completely to their original state. They would necessarily exist only on that loop and would not come in from outside the loop. Such objects would not “age”, including any clock which would necessarily return to their original state. It is also possible that such an object could be its own cause, e.g. a positron produced in a gamma-gamma event which goes through a wormhole, collides with an electron, produces a gamma ray, that interacts with another gamma ray to produce the original positron.
Again, until these are experimentally observed this is all conjecture, but the theory does allow such solutions. Such solutions exist, but they are generally not unique.
