Consequences of compactness in physics

Question

If we understand spacetime as a $4$-dimensional manifold $M$, from the point of view of physics what are the consquences of a subset of it being compact? My point here is simple: in math we usually think of compactness as some analogue of finiteness because it shares many properties with finite sets, but what are the consequences of this when we deal with physics?

Of course, we need not to go into relativity, we can even think about the usual three space $\mathbb{R}^3$. What are again the consequences of a set $A \subset \mathbb{R}^3$ being compact? Are there any cool things we can get out from this, or we simply use compactness in physics to grant the mathematical properties desired without having any direct impact in the way we understand and interpret those sets?

Answer 1

I think that you should reformulate your question. $A$ can be a single point as well as a closed ball, and I can't see what good physical insights you can make into common properties of these examples. I think that usually we need compactness to think about maxima and minima of continious functions, and boundedness of such function, but it is a pretty mathematical viewpoint.

In fact, we usually deal with very good sets. Like, they have piecewise-smooth boundary etc, which form only a tiny part of all compact sets. Compact sets are just closed bounded sets, and compactness allows really crazy sets. For example, the Сantor set is compact, as well as many other sets that we almost never encounter in Physics.

Answer 2

If your spacetime is compact and if every point has an open neighborhood that locally looks like special relativity, then it contains closed timelike curves (aka time travel). The argument is standard math.

For each point p, there is an open neighborhood that looks like SR and hence contains points q in p's causal past. So the set of causal futures of every point is an open cover of your spacetime (each p has a q that was in its past, so p is in q's causal future). By compactness there is a finite subcover. Since the subcover is finite, we can make it minimal (consider every subcollection of the subcover, there are finitely many, so go through them and pick one of the ones that is of minimal size that is also a cover and use that instead).

So there are points $q_1$, $\dots,$ $q_n$ such that union of their causal futures contain everything. So in particular $q_1$ needs to be in one of those causal futures. If it were in a different $q_k$'s causal future (i.e. $q_k\neq q_1$ for a different $q_k$ and there is a future pointing path from $q_k$ to $q_1$) then the causal future of $q_1$ was redundant (because it was contained in the causal future of $q_k$), contradicting the minimality. But $q_1$ has to be in one of the $q_k$'s causal futures, so it must be in its own. If it is in its own causal future, there is a closed timelike curve (CTC).

How big a deal is this? If the universe must be truly ancient before it repeats is this such a big deal? After all this curve would have to go through the earlier universe and be subject to super high pressures and densities and radiation and maybe inflation and not much information would survive even if there is a geometric line that wiggled throughout that region.

It is even possible that all closed time like curves pass through the same incredible small region, so they are all practically the same so there is no specialness to any of them, to us far from the hot dense early universe they all just look like curves that go back to the hot dense early universe where all the paths near us where very very very very near to each other back then, so no one path feels special.

So you could technically have time travel, but where the modern universe looks like it does now and nothing feels special. So not necessarily a big deal.