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

[WannabeNewton]

If you can simply put it, using words , what is the physical significance of "disconnected spacetime"? could "local spacetime" interact with it & vice versa for example. (supposing not or it'd likely be tested)

No communications whatsoever can ever happen between different "disconnected regions", if that's what you're asking (see my post directly above yours, I edited it to change things around a bit). A causal curve, to clarify, is a null or time-like curve (so these represent the worldlines of light and massive particles respectively).

You can imagine disconnected topological spaces as being "broken up" or, stated more properly, partitioned into different connected subsets (when talking about space-times you may as well imagine this as saying that any two events can be connected by some continuous path). For example the real line with any point removed will be disconnected into two components (the set of reals less than the removed point and the set of reals greater than the removed point).

 

[nitsuj]

No communications whatsoever can ever happen between different "disconnected regions", if that's what you're asking (see my post directly above yours, I edited it to change things around a bit). A causal curve, to clarify, is a null or time-like curve (so these represent the worldlines of light and massive particles respectively).

is this causally the same as a black hole? Don't they do the same thing to a simple metric that isn't made to replicate the horizon.

And thanks for the explanation! Yea I had no clue how to think of "disconnected spacetimes" and worried it was physicist "code" for worm hole or something similar. lol precious causation

 

[WannabeNewton]

I wouldn't say its exactly like the causal nature of a black hole. Ingoing light rays, for example, can still enter Schwarzschild event horizons but with a disconnected space-time, light rays can never go from one connected component to another, regardless of which way (if two events have a continuous path connecting them then they must necessarily be in the same component so there is no way a light ray could go from one component to the other because the associated null geodesic would represent a continuous path from an event in one component to an event in the other but we would have a contradiction because the existence of such a path would imply that these events belonged to the same component).

 

[nitsuj]

Cool thanks WannabeNewton!

 

[WannabeNewton]

Let me also add that ingoing time-like curves (e.g. worldlines of freely falling observers) can also pass through the event horizon of, for example, a Schwarzschild black hole but in the case of a disconnected space-time, time-like curves also cannot connect two events belonging to two different components in either direction (they too count as continuous paths) so it's not just that no one in our component can communicate using light signals with anyone in another component, we also can't actually enter another component (so really, as far as we're concerned, these "disconnected universes" may as well not exist). If I've interpreted Geroch's statement correctly (the one linked by robphy) then this is essentially what he's saying is the reason for taking space-time to be connected in the sense that operationally we can only ever know about our own component so we may as well restrict ourselves to that component (as mentioned before, manifolds are locally connected so these components will necessarily be open in which case they can naturally inherit a smooth structure from the overarching manifold).

 

[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?