# Why connected space-times?

1. Jul 11, 2013

### WannabeNewton

Hi guys! So this question comes from the definition of space-times in both Hawking and Ellis "Large Scale Structure of Space-time" and Malament "Topics in the Foundations of General Relativity and Newtonian Gravitation Theory". My question in particular revolves around the fact that both texts assume a priori that space-time manifolds are connected (Hawking and Ellis p.56 and Malament p.103).

For background: here connected is being used in the topological sense i.e. a topological space $X$ is connected if the only clopen subsets of $X$ are $X$ and $\varnothing$ (well one can also define a connected space as a space which cannot be written as the union of two disjoint non-empty open sets but one can easily show that this definition is equivalent to the previous one). For topological manifolds, connectedness can be shown to be equivalent to path connectedness (every topological manifold is locally path connected and for locally path connected spaces, connectedness is equivalent to path connectedness). By path connectedness, one means that for any two points $p,q\in M$ there exists a continuous map $f:I\rightarrow M$ such that $f(0) = p, f(1) = q$. So we are taking space-times to be such that any two points can be connected by a continuous path (the 0th homology group vanishes). A general topological space can be partitioned into its connected components i.e. the maximal connected subsets of the space (connectedness of subsets is with respect to the subspace topology).

I personally can't imagine simply using local physical experiments to justify a global feature like connectedness/path connectedness of space-time. Malament does not elucidate on why we take space-time to be connected and Hawking states "The manifold $M$ is taken to be connected since we would have no knowledge of any disconnected component." What does he mean by this? Why would we not have knowledge of the connected components of space-time? Is he alluding to what I said before regarding how local physical experiments would be inadequate in determining if space-time was connected or not (since this is a global property)? But even if we couldn't actually determine it, why would that allow us to take space-time to be connected over the possibility of it having non-trivial connected components?

2. Jul 11, 2013

### micromass

Staff Emeritus
Somehow, I feel the same about Hausdorff or second countable. They are global properties and I don't think there is any way to justify them physically.

Thing is, if there were multiple components, then we have no way of knowing about these components. And what happens in these components can not affect us in any way. So instead of studying the entire universe, we can just study what happens locally in our component. So in that sense, everything we study is in a sense local. It makes no sense to study other components that ours since we have no way to verify that our theory is correct.

3. Jul 11, 2013

### WannabeNewton

But I don't see how that justifies making the entire manifold connected. If, according to Hawking and whoever else, our local physical experiments can only determine properties locally within our own connected component, then why assume space-time is connected at all? Why couldn't there be non-trivial connected components? Topological manifolds are automatically locally connected and locally path connected so why isn't that enough, in the spirit of the above? Hausdorff I can be at ease with to some extent because non-Hausdorff manifolds tend to be quite pathological (if space-time had the topology of the line with infinitely many origins then I would be rather surprised-uniqueness of limits of convergent sequences seems to be something natural as far as the physical world is concerned) but disconnected topological manifolds aren't pathological in that sense.

With regards to second countable/separable, I personally don't know of any physical justification for it; the reasons to take the manifolds to be second countable usually seem to be purely of mathematical convenience because we can then use partitions of unity to prove various existence theorems (it is also very hard to find a manifold that isn't second countable so there's also that). On that note, proposition 1.7.1 in Malament (p.45) states that a connected manifold admits a derivative operator $\nabla_{a}$ if and only if it is second countable. So again the connectedness of space-time seems to be of a mathematical convenience, I just can't convince myself of any physical justification for it.

Last edited: Jul 11, 2013
4. Jul 11, 2013

### micromass

Staff Emeritus
I think it's exactly that: a mathematical convenience. Do you know where exactly the connectedness is used.

What I think is that if you take the manifold to be disconnected, then you will have a number of theorems saying things like "On a given components, we have..." And since we only really care about our own component, we might as well put that in our definition.

5. Jul 11, 2013

### WannabeNewton

Well referring back to the same proposition in Malament, he subsequently states the following: "The restriction to connected manifolds here is harmless since, clearly, a manifold admits a derivative operator iff each of its components does." I don't quite understand what he means by this however, particularly with regards to the second direction of the statement. On the one hand, since a manifold is locally connected, the connected components of the manifold must necessarily be open hence they will be connected submanifolds so we can restrict the proposition referenced in post #3 (existence of $\nabla_{a}$) to the connected components and worry about things locally within the various connected components but I don't get why that implies there necessarily exists a derivative operator on the entire manifold if it wasn't connected (his statement about it being "harmless") and I still think that restricting space-time to be connected just for mathematical convenience is more extreme than doing so for second countable for the reasons stated above.

6. Jul 11, 2013

### micromass

Staff Emeritus
By the proposition in #3, we can find a derivative operator on each connected component. And we can then glue these different operators to an operator on the entire manifold.

That is, if we are given a smooth tensor field on the global manifold. Then we can restrict to the connected components and still have smooth tensor fields. We then take the derivative with respect to the derivative operator on the component. The derivative of the global tensor field is then locally equal to the derivatives on the components.

Restricting to connected manifolds might be a bit extreme. But I think that we need to realize that, in that way, GR doesn't talk about the global universe, but only about our component. What happens in other components is not something we can experimentally verify anyway.

7. Jul 11, 2013

### WannabeNewton

This may be a bit off topic but is there some kind of smooth gluing lemma (analogous to the topological gluing lemma) which allows us to smoothly glue together the derivative operators that are guaranteed existence on each connected component of the disconnected manifold?

And if we do have such a smooth gluing lemma at our disposal, why would we ever need connectedness of entire space-times in GR anyways? I've only ever seen it used in conjunction with second countable for existence theorems such as the one for $\nabla_{a}$ referenced above so if we can always relegate a $\nabla_{a}$ to each connected component and then smoothly glue them together then why would we even need connectedness of the entire space-time? I can't seem to think of a deeper physical reason for wanting connectedness of the entire space-time (nor any deeper physical justification for such). If we can never determine, using local physical experiments, if we are in a non-trivial connected component of space-time or a trivial one (i.e. space-time is connected) then it only seems minimal to assume the former, given the above discussion.

8. Jul 11, 2013

### micromass

Staff Emeritus
Yes. For scalar fields, there is a very analogous version to the topological glueing lemma. This is covered in (for example) Lemma 2.1 of Lee, smooth manifolds. There are of course analogous versions for other vector fields, tensor fields, etc.

In modern mathematical language, we abstract the glueing lemma to sheafs. So a sheaf is abstractly a collection of objects that satisfies the glueing lemma. You can find the precise definition on wikipedia, it's not very difficult. We can translate all of differential geometry in terms of sheafs. However, in some sense, sheafs and vector bundles are analogous objects. So any vector bundle gives rise to a sheaf (indeed, consider a vector bundle $p:E\rightarrow M$, then we can consider the local sections $\mathcal{E}(U) = \{\sigma: U\rightarrow E~\vert~p\circ \sigma = 1\}$. And there is also a converse to this. The reason sheafs are not used in differential geometry, is because vector bundles are used.

Anyway, a derivative operator can now best be seen as a sheaf-morphism. A glueing lemma for the derivative operator can now be best seen by applying sheaf theoretic theorems. Of course, we can do it concretely too, but seeing the abstract version is enlightening, I think.

I don't know. For the theorem you mention, connectedness is not needed. Maybe there is another reason?
Does he cover "uniqueness" for the derivative operator? I would think connectedness is rather important there.

9. Jul 11, 2013

### WannabeNewton

Can we actually do it concretely? It seems the usual smooth gluing lemmas apply to sections of tensor bundles, vector bundles etc. whereas a connection is a map on sections so can you actually do it without the language of sheafs and sheaf gluing lemmas?

As for uniqueness, he doesn't seem to use connectedness anywhere for that. The uniqueness statement is as follows: let $\nabla_{a}$ and $\nabla^{'}_{a}$ be two derivative operators on $M$. Then there exists a smooth symmetric tensor field $C^{a}_{bc}$ on $M$ that satisfies $$(\nabla_{m}^{'} - \nabla_{m})\alpha^{a_1...a_r}_{b_1...b_s} = \alpha^{a_1...a_r}_{nb_2...b_s}C^{n}_{mb_1} + ...+\alpha^{a_1...a_r}_{b_1...b_{s-1}b_n}C^{n}_{mb_s}\\ - \alpha^{na_2...a_r}_{b_1...b_s}C^{a_1}_{mn}-...- \alpha^{a_1a_2...a_{r-1}a_n}_{b_1...b_s}C^{a_r}_{mn}$$ for all smooth tensor fields $\alpha^{a_1...a_r}_{b_1...b_s}$ on $M$. Conversely if $\nabla_{a}$ is a derivative operator on $M$ and $\nabla_{a}^{'}$ is defined by the above then it is also a derivative operator on $M$.

Aside from computation, the only extra lemma he makes use of is that given a derivative operator $\nabla_{a}$ on any smooth n-manifold $M$, and $\xi_{b}$ a covector in $T_p^* M$, there exists a smooth scalar field $\alpha$ such that $\xi_{b} = (\nabla_{b}\alpha)|_{p}$. He proves this using nothing more than local coordinates.

10. Jul 11, 2013

### micromass

Staff Emeritus
I'm sorry for not using the right notations, but I hope it'll be clear. So let's say we work with the following set of components $(M_i)_{i\in I}$. We can find a derivative operator on each component $\nabla^i$.

Now, take a global tensor field $T$, then we can restrict to $T\vert_{M_i}$ and take derivatives $\nabla^i T\vert_{M_i}$. We glue this together to form a big derivative operator. So the derivative operator on the global manifold îs defined by $(\nabla T)_p = (\nabla^i T\vert_{M_i})_p$, where $i$ is chosen so that $p\in M_i$. This derivative operator satisfies of course that $(\nabla T)_{M_i} = \nabla^i T\vert_{M_i}$. Now you need to check that it's indeed a derivative operator (starting with checking that the derivative yields an actual tensor field).

11. Jul 11, 2013

### WannabeNewton

Ok I'll look that over but it seems again like we don't really need connectedness of the entirety of space-time for the existence theorems that show up in GR (in particular the one for $\nabla_{a}$) if we can just use smooth gluing lemmas over the connected components of space-time. Maybe it's the minimalist in me speaking but without concrete physical justification (and not "we can't determine it using local experiments so we might as well assume it's true" arguments) it just seems unneeded to assume connectedness of the entire space-time. Is there any actual physical necessity for connectedness of all of space-time?

12. Jul 11, 2013

### TrickyDicky

This is an interesting question I have often thought about. I tend to think that indeed there are good physical arguments that compell us to not only require connectedness but simply-connectedness and contractibility for spacetime. But I'm not at all certain that my arguments are convincing or flawed.

Are you only admitting physical arguments strictly derived from GR or also from electrodynamics or quantum theory? Only abstract principles that would impose connectedness or more specific physical examples?

13. Jul 11, 2013

### TrickyDicky

In any case if one takes the background-independence of the theory in a strict sense I wonder if there are grounds to even talk about a "global spacetime", as opposed to just taking into account the "local spacetime" features wich as it says in the Hawking and Ellis book allows us to disregard any possible not connected parts.

14. Jul 11, 2013

### HallsofIvy

Staff Emeritus
There is no fundamental physical reason why space-time should be connected. But if space-time does have different unconnected components, it is impossible for anything in one component to affect anything in another. Conceptually, all we can know about, all we can experiment on, is our own component.

15. Jul 11, 2013

### micromass

Staff Emeritus
Contractibility is a bit of a large assumption isn't it? Not even n-spheres are contractible. Can you give us the arguments you're thinking of?

16. Jul 11, 2013

### WannabeNewton

I certainly agree Halls my good man which is why it confuses me as to why we make the (rather large assumption) that all of space-time is connected; I just don't see a physical justification for it, given what you said. If we cannot determine such a specific global property, then why make assumptions about said global property at all?

I'm sure there are crucial mathematical conveniences that come out of making the connectedness of all of space-time assumption, but I have yet to see them myself so if anyone knows of any (not counting existence theorems-their nature has been discussed above) that would be quite helpful indeed. Perhaps it becomes useful in global causality and/or the Cauchy problem in one way or another.

Last edited: Jul 11, 2013
17. Jul 11, 2013

### robphy

As others have said earlier in this thread...
imposing connectedness is a mathematical convenience
suggested by classical physics motivations.

(If you don't impose it, the proverbial student in the back of the room
will always try to chime in to suggest a counterexample if connectedness wasn't assumed.

Now, the quantum-minded student might find a better reason
to impose or to not-impose connectedness...
)

Here are possibly useful passages that can be followed up.

18. Jul 11, 2013

### WannabeNewton

But global space-time features are analyzed in great detail theoretically, especially in texts such as Malament and Hawking and Ellis e.g. global causal structure, the topology of space-time etc. See here: http://faculty.washington.edu/manchak/manchak.handbook.pdf [Broken]. Here also the author assumes that the underlying manifold of a relativistic space-time is connected.

It might be interesting to note that Wald doesn't seem to assume that all space-times are connected (he always says if $M$ is connected then such and such). Perhaps it's a conventional thing? In such a case I doubt, as Halls said, there would be any physical justification for assuming all of space-time is connected.

Last edited by a moderator: May 6, 2017
19. Jul 11, 2013

### TrickyDicky

Ok, ignoring for a moment the issues related to the GR principle of "no prior geometry", that to me implies also "no prior global topology", that would make the OP a bit pointless I'll concentrate in certain electrodyamics fact that seems to demand a certain topology (actually the Euclidean topology wich is contractible and therefore simply-connected and of course connected). Of course I guess this can be discarded on the basis that it's really a local example, but such are usually linked to what are considered universal laws in physics if that has any meaning at all.
For instance in justifying the existence of a magnetic vector potential using the Gauss law for magnetism, the often unstated assumption that the second de Rham cohomology vanishes wich is true for contractible spaces is necessary.
Then I guess they don't pay much attention to "background independence" or at least don't give it the same emphasis as authors dealing with quantum gravity, which is ok since this feature of GR is usually not profoundly adressed when the subject is limited to classical GR

Last edited by a moderator: May 6, 2017
20. Jul 11, 2013

### WannabeNewton

The Einstein field equations cannot in general tell us anything about the global topology of space-time. As for the dynamical nature of space-time topology, the concept of topology changing space-times in general relativity certainly isn't ruled out completely but there are a lot of problems associated with topology changing space-times in general relativity; the concept is not intrinsic to general relativity in the way "no prior geometry" is. The concept of topological dynamics is more prevalent in QG than in GR. See here: http://arxiv.org/pdf/gr-qc/9406053v1.pdf

As for simply connectedness, you cannot say anything unequivocal about the fundamental groups of asymptotically flat space-times in particular because of the topological censorship conjecture (if it holds true): http://arxiv.org/pdf/gr-qc/9305017v2.pdf. Also, keep in mind that when topological dynamics are ignored, there are different standard topologies one tends to endow on space-times in general relativity (e.g. the natural smooth structure topology or the path topology). Simply connectedness must of course be looked at with respect to a specific topology; Wald doesn't seem to assume that arbitrary space-times are simply connected under the manifold (smooth structure) topology; I'm still rifling through Hawking and Ellis to see what is assumed in that text. Also see here: http://nestor2.coventry.ac.uk/~mtx014/pubs/pathconn.pdf

Last edited: Jul 11, 2013