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PHYS771 Lecture 19: Time Travel

* Let's talk about the more interesting kind of time travel: the backwards kind. Can closed timelike curves (CTCs) exist in Nature? This question has a very long history of being studied by physicists on weekends. It was discovered early on, by Gödel and others, that classical general relativity admits CTC solutions. All of the known solutions, however, have some element that can be objected to as being "unphysical." For example, some solutions involve wormholes, but that requires "exotic matter" having negative mass to keep the wormhole open. They all, so far, involve either non-standard cosmologies or else types of matter or energy that have yet to be experimentally observed. But that's just classical general relativity. Once you put quantum mechanics in the picture, it becomes an even harder question. General relativity is not just a theory of some fields in spacetime, but of spacetime itself, and so once you quantize it, you'd expect there to be fluctuations in the causal structure of spacetime. The question is, why shouldn't that produce CTCs?*

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*Incidentally, there's an interesting metaquestion here: why have physicists found it so hard to create a quantum theory of gravity? The technical answer usually given is that, unlike (say) Maxwell's equations, general relativity is not renormalizable. But I think there's also a simpler answer, one that's much more understandable to a doofus layperson like me. The real heart of the matter is that general relativity is a theory of spacetime itself, and so a quantum theory of gravity is going to have to be talking about superpositions over spacetime and fluctuations of spacetime. One of the things you'd expect such a theory to answer is whether closed timelike curves can exist. So quantum gravity seems "CTC-hard", in the sense that it's at least as hard as determining if CTCs are possible! And even I can see that this can't possibly be a trivial question to settle. Even if CTCs are impossible, presumably they're not going to be proven impossible without some far-reaching new insight. Of course, this is just one instantiation of a general problem: that no one really has a clear idea of what it means to treat spacetime itself quantum-mechanically.

*In the field I come from, it's never our place to ask if some physical object exists or not, it's to assume it exists and see what computations we can do with it. Thus, from now on, we'll assume CTCs exist. What would the consequences be for computational complexity? Perhaps surprisingly, I'll be able to give a clear and specific answer to that.*