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This question already has an answer here:

can one travel back in time and if not so, what laws prevent time travel to the past. this is quite a challenge to understand.

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marked as duplicate by Martin, Danu, RedGrittyBrick, Kyle Kanos, John Rennie general-relativity Feb 18 '16 at 10:25

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In special relativity the answer is simple: no, you can't (unless the universe is topologically strange).

For general relativity the answer is more complicated: it is possible to construct spacetimes with what are called 'closed timelike loops': curves which are timelike (ie where the curve lies within the lightcone of all points on it) and which are closed: the curve forms a loop. And I am pretty sure you can also arrange things so that the curves are smooth enough to be physically sensible (ie they have no sharp corners).

Such a curve is fairly bad news, any point on such a curve is in its own future (and past). In other words such a spacetime does allow you to travel into your own past.

So the question to answer is: are such spacetimes physically plausible? In particular do they represent stable solutions without silly energy conditions, and could they arise from plausible initial conditions?

Now, my knowledge of this is seriously out of date (as in: more than twenty years out of date), but the situation then was that people strongly suspected that they were not physically plausible, in the sense that you could not arrive at such a solution from physically plausible initial conditions. However, at the time, there was no really convincing proof of this.

Personally, I would be extremely surprised if such solutions are physical. Indeed, if they turn out to be plausible in GR I would regard that as strong evidence that there is something wrong with GR (and I'm saying this as someone who has the field equations on a t-shirt!)

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