# Causal structure of spacetime: intuitive reason for non-compactness?

Apologies in advance if this has been asked and answered somewhere else, I searched but could not find anything.

I just wanted to verify if my intuitive understanding of why space time (if it is to be physically plausible) can not be compact.

According to Hawking and Ellis (and others) space time can not be compact because it would allow for the existence of closed time like curves. I understand (I think) that argument based on limit points, or accumulation points, which define a (sequence) compact space and how this allows such curves to exist.

Now, is it also correct, and much simpler, to assume that *if* space time was compact, this would imply "special" points in space time (i.e. the limit points) which breaks the principle of relativity, somehow...?

I.e. if space time is compact, then there exists an arbitrary number of points which are the accumulation points for sequences of time like curves and (ignoring *completely* the closed-timelike curve problem) this would imply that geodesics would be influenced by other things than just mass/energy (as the causal argument is independent of any solution to Einstein's field equations.)

Or am I very very far off on a tangent indeed..?

## Answers and Replies

Related Special and General Relativity News on Phys.org
I wonder if the following will help:
http://arxiv.org/PS_cache/arxiv/pdf/1005/1005.2591v1.pdf

The author, Paul Kinlaw, recently finished his PhD in math and talked about it where I am. Its a bit mathy (essentially a topology paper) but I think it addresses what you're interested in to some degree. The problems he's interested in were originally motivated in physics but he's a few degrees removed and so has some trouble fielding those types of questions. I straddle math and physics but have yet to really study GR (soon...very soon).

I hope it helps.

Thank you for your response, the article looks very interesting and I'm rolling up my sleeves and digging in to it! If I get to some sort of epiphany I'll post a follow up.

Fredrik
Staff Emeritus
Gold Member
According to Hawking and Ellis (and others) space time can not be compact because it would allow for the existence of closed time like curves.
George Jones posted the proof of this part (or most of it anyway) here.

Thanks Fredrik, I was following Wald's book on GR plus what I could gleam from "The Large Scale Structure of Space Time" on Google books - I think George Jones' reply that you refer to is very clear and helpful - I'm going to wrap my head around this last part in particular (I think it holds a clue to the epiphany I need to get to):
....Consequently, p_1 is in I+(p_1), i.e., there exists a smooth, future-directed timelike curve from p_1 to p_1.
I.e. that a finite subcover and, perhaps on an equally fundamental level, that "M" is Hausdorff, means that points will inevitably lie in their own chronological- and causal -future.

Perhaps this has some relevance to my starting point, i.e. that this also implies that space has some sort of "preferred structure" which would imply non-matter originating effects which could not be encoded in the Einstein tensor and hence are unphysical (leading to compact space times being unphysical) - but that part is still something I'm trying to understand (and I am starting to suspect that it's just plain wrong)