I've been looking a lot at Closed Timelike Curves, and how if a theory allows for these curves it doesn't respect causality. I understand that about the curves themselves (Grandfather Paradox), but can't seem to fathom how a theory would allow for such structures, since they seem to be "geometrically" impossible in a spacetime.
To my understanding, CTC are simply worldlines that loop back on themselves, and are therefore closed. The problem comes in when I actually try to picture a CLOSED worldline: If I start at a point in Minkowski Spacespacetime and draw any closed curve, I end up always having a portion of it be spacelike, and therefore the curve is never fully timelike. Meaning It's impossible to draw a CTC.
So my question is, how can a theory allow for such worldlines, since the fundamental principles behind the geometry of spacetime simply prohibit it?
Answer
In Special Relativity CTCs can't exist (or at least I don't think so) but General Relativity has solutions that include CTCs. The best known is probably Gödel's solution for a rotating universe. The Alcubierre drive could also be used to construct CTCs, as could any FTL mechanism. Also see the Tipler cylinder, and probably many other examples I can't remember.
However, none of these examples of CTCs are realistic. In his paper on the Chronology Protection Conjecture Hawking proved that closed timelike curves cannot be created in a finite system without using exotic matter. The Gödel universe gets round this because it's infinite, while other cunning ideas like the Alcubierre drive require exotic matter.
Now, as far as we know the universe isn't rotating, and exotic matter doesn't exist. So (I guess) most physicists don't believe that time travel is possible, even though Einstein's equation does have solutions that could allow it.
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