Connected Space-Times: Why is it Assumed in General Relativity?

[robphy]

That's pretty cool that you got to sit in on Malament's lectures. Did you also ever get to sit in on Geroch's lectures, or Wald's when you were at UChicago?

Yes, I took Wald's GR class. When Geroch taught GR, I sat in on that... and it made more sense to me. (There was no textbook for the course.) At that time, I re-read the GR-from-A-to-B book, the sections in relativity sections in Mathematical Physics, as well as Wald's text. Later, when Malament offered his course in the Philosophy department, I officially registered for that (not for a grade). Each course taught me lots of new stuff and new ways of thinking ("geometrical viewpoint")... different from the more classical presentations I had in college. Because of these numerous viewpoints, time-permitting, I've taken or sat in on every relativity course I could, there and elsewhere. (Nowadays, I even try to watch some of the courses at Perimeter.)

 

[WannabeNewton]

Yes, I took Wald's GR class. When Geroch taught GR, I sat in on that... and it made more sense to me. (There was no textbook for the course.) At that time, I re-read the GR-from-A-to-B book, the sections in relativity sections in Mathematical Physics, as well as Wald's text. Later, when Malament offered his course in the Philosophy department, I officially registered for that (not for a grade). Each course taught me lots of new stuff and new ways of thinking ("geometrical viewpoint")... different from the more classical presentations I had in college. Because of these numerous viewpoints, time-permitting, I've taken or sat in on every relativity course I could, there and elsewhere. (Nowadays, I even try to watch some of the courses at Perimeter.)

Ah you're so lucky! I'm so jealous xP; that's really awesome though that you got to experience their lectures. I have to stick to learning from the textbooks and notes written by people like Malament, Geroch, and Wald for now

 

[PAllen]

Well it seems Geroch's point was that if a space-time indeed had non-trivial connected components, then it must also have non-trivial path components so there cannot exist any two events belonging to two different path components such that there exists a continuous path between them hence no causal curve could ever connect two events in two different path components. This is how I interpreted his statement that no communication can take place between any two observers in different connected components (the "disconnected universes"), since one can easily show that path components are necessarily contained in connected components, and in fact any connected component is a disjoint union of path components. And then he seems to say that because of this, we may as well assume that the space-times of physical interest are connected as we can only ever know about the physical properties of our own connected component.

Is that in accord with what you said? In other words, we take the operational definition of space-time to be some connected component of a possibly non-trivial set of disconnected components of a larger manifold simply because no causal curve could ever go from one component to the other so we can never know the existence of the other components anyways. Is that more or less along what you said? Thanks for the response.

Yes. And my point about when it would be interesting (IMO) to consider manifolds with disconnected 'pieces' is if there is some theory of a type of influence that does not require a path; such that in some way observations in A alone could be different than in A as part of A U B, where A and B are disconnected.

 

[WannabeNewton]

Yes. And my point about when it would be interesting (IMO) to consider manifolds with disconnected 'pieces' is if there is some theory of a type of influence that does not require a path; such that in some way observations in A alone could be different than in A as part of A U B, where A and B are disconnected.

That would be quite interesting indeed. On that note, do disconnected space-times show up at all in quantum gravity/quantum cosmologies?