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

[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?

 

[micromass]

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.

 

[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.

 

[micromass]

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.

 

[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.

 

[micromass]

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.

 

[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.

 

[micromass]

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?

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.

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).

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.

 

[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.

 

[micromass]

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?

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).

 

[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?

 

[TrickyDicky]

Is there any actual physical necessity for connectedness of all of space-time?

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?

 

[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 which as it says in the Hawking and Ellis book allows us to disregard any possible not connected parts.

 

[HallsofIvy]

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.

 

[micromass]

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?

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?

 

[WannabeNewton]

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.

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.

 

[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 into 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.

books.google.com/books?id=pxA4AAAAIAAJ&pg=PA218&lpg=PA218&dq=connected

There are, none the less, physical arguments for the elimination of a
few classes of manifolds - although these consist essentially of a few
pathologies which are normally permitted mathematically under the
term 'manifold'. Thus one would perhaps not wish to admit a manifold
with boundary as a candidate for the underlying manifold of our universe
(for the boundary would represent physically an 'edge' to
spacetime, while such edges have never been observed); or a
non-Hausdorff manifold (i.e. a manifold in which there are two points which
cannot be separated by disjoint neighborhoods, such behavior would
perhaps violate what we mean physically by 'distinct events'); or a
non-connected manifold (for by no stretch of the imagination could
communication ever be carried out between the separate connected
components, so physically 'our universe' is connected); or a
non-paracompact manifold (i.e. some connected component cannot be
covered by a countable collection of coordinate patches; we shall see the
physical reason for this exclusion shortly). In light of these remarks, we
shall assume hereafter that, unless otherwise stated, 'manifold' means
without boundary, Hausdorff, connected, and paracompact.

www.worldcat.org/title/general-relativity-and-cosmology-varenna-on-lake-como-villa-monastero-20th-june-12th-july-1969/oclc/639816396

Appendix
Hausdorff, connected and paracompact manifolds.

In Sect. 3, we discussed some of the pathological situations which can occur
in the causal structure of a space-time. It is natural to ask whether any similar
anomalies can arise in the underlying manifold itself. There are, roughly speaking,
three classes of pathological manifolds. A non-Hausdorff manifold contains
points which cannot be properly separated by open Sets. A disconnected manifold
can be divided into two or more disjoint manifolds. A nonparacompact
manifold is "too large". One normally assumeS that the manifolds dealt with
in relativity are Hausdorff, connected, and paracompact. Furthermore, it
has not so far been useful to relax any of these assumptions. Since these three
assumptions often appear in the staatements of theorems, however, it is of
interest to know just what situations are thereby being ruled out.
(snip)
A manifold is said to be connected if any two points on the manifold can be
joined by a continuous curve. For example, the manifold consisting of two
disjoint planes is not connected. Physically, a disconnected space-time rrepresents
two or more separate universes: no matter what the metric is, observers
on different components of the space can never communicate with each other.
It is more convenient, therefore, to study each connected component separately,
that is, to impose the condition that models of our universe are connected.

 

[WannabeNewton]

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 which as it says in the Hawking and Ellis book allows us to disregard any possible not connected parts.

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 . 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.

 

[TrickyDicky]

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?

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 which 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 which is true for contractible spaces is necessary.

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 . Here also the author assumes that the underlying manifold of a relativistic space-time is connected.

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

 

[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

 

[PeterDonis]

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.

It's interesting to note that the Geroch quote that robphy gave does not state the assumption that way; it says that our *models* of spacetime must be connected, not that spacetime itself is. In other words, when we model spacetime as a connected manifold, we are not saying that "all of reality" is contained within a single connected manifold; we are just saying that we can only model the connected component that we happen to reside in, because we have no way of testing a model of anything else.

 

[WannabeNewton]

Here are possibly useful passages that can be followed up.

It's interesting to note that the Geroch quote that robphy gave does not state the assumption that way; it says that our *models* of spacetime must be connected, not that spacetime itself is. In other words, when we model spacetime as a connected manifold, we are not saying that "all of reality" is contained within a single connected manifold; we are just saying that we can only model the connected component that we happen to reside in, because we have no way of testing a model of anything else.

Thank you for the passages. I mean it's fine to consider the connected components separately as discussed above (and physically interpret this as 'our universe' as opposed to the other 'disconnected universes') but it just seems that such physical considerations are not strong enough to completely rule out the existence of disconnected space-times. Of course Geroch isn't arguing that such physical considers rule them out (at least from the way I interpreted it) but rather that these are the space-times of physical interest, which is fine of course. I just don't like the idea of defining any and all space-times to be connected when viewed as abstract mathematical objects.

As a side note, I really should read Geroch's "Spacetime Structure from a Global Viewpoint"; Malament constantly refers to it in his own text and based on your quoted passage it seems to be very instructive indeed. Thanks again.

 

[TrickyDicky]

The Einstein field equations cannot in general tell us anything about the global topology of space-time.

Exactly. So something like connectedness has to be introduced either arbitrarily as a purely mathematical convenience (for instance to rule out pathological cases as commented before, in the same vein it is done with assumptions like being haussdorf etc) or also because it is suggested by assumptions of everyday physics like the existence of a magnetic vector potential in my example (to the extent physics can be separated from mathematical physics or plain math). Did you not think it was a valid example?

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, not all authors assume all space-times are simply connected (e.g. Wald and Gannon: http://jmp.aip.org/resource/1/jmapaq/v16/i12/p2364_s1?isAuthorized=no )

I see, but this only holds for asymptotically flat cases which are considered as highly idealized and usually only suited for isolated systems in GR. Cosmological GR (wich is closer to the large scale approach you claim there is no justification for the connectedness assumption) is usually centered on the 3 FRW universes, the three of them are simply connected.

 

[WannabeNewton]

Certainly the topology of FRW universes should be simply connected based on the usual symmetry arguments; I was speaking of arbitrary space-times in the above. I certainly don't disagree with your example either but I'm just saying that using empirical physical considerations to define any and all space-times to be simply connected etc. doesn't seem warranted when in general we just don't know what it is or isn't.

EDIT: For anyone interested, here's the paper by Geroch mentioned above (from Caltech's site): http://www.pma.caltech.edu/~ph236/yr2008/readings/Geroch-SpaceTimeStructure.pdf

I might have to make a separate thread just for the exercises in that paper lol.

 

[robphy]

As a side note, I really should read Geroch's "Spacetime Structure from a Global Viewpoint"; Malament constantly refers to it in his own text and based on your quoted passage it seems to be very instructive indeed. Thanks again.

Before he moved to Irvine, Malament (http://en.wikipedia.org/wiki/David_Malament) was in the Dept of Philosophy at U. Chicago.
(He's not your typical philosopher of science.)
He taught his own courses in relativity for the Conceptual Foundations of Science program there. (I sat in on one.)
His handwritten lecture notes became the basis of his Topics in the Foundations of GR book.
David attended a lot of the Relativity Seminars at Chicago, and the influence of Geroch and Wald (a.k.a. Bob1 and Bob2, in his preface) is evident.

This is an influential paper, especially for approaches to Quantum Gravity that take causality as a primitive structure:
J. Math. Phys. 18, 1399 (1977); http://dx.doi.org/10.1063/1.523436 (6 pages)
"The class of continuous timelike curves determines the topology of spacetime"
http://jmp.aip.org/resource/1/jmapaq/v18/i7/p1399_s1?isAuthorized=no

 

[PAllen]

GR with an assumption of connectedness and without are indistinguishable as to conceivable observations. Since the former is simpler to work with, the latter is only for masochists.

If someone proposed a GR extension in which disconnected components explained some observational anomaly, suddenly this would be very interesting.

Similarly, analyticity is often assumed for no reason other than it makes things easier. If ever someone found a non-analytic solution of GR (for example) that explained something that couldn't be explained with any analytic solution, physicists would no longer assume it.

 

[nitsuj]

What is a black hole in the context of this thread? Is it not an ideal example of a umm.. hole (gap) in spacetime?

what else is a gap? The opposite, something like spacetime expansion?

Oh never mind I see the context is causal system/structure

How else to define a physical system if not by first assuming causal structure, it's physics chaos otherwise lol

 

[TrickyDicky]

GR with an assumption of connectedness and without are indistinguishable as to conceivable observations. Since the former is simpler to work with, the latter is only for masochists.

Indeed.

If someone proposed a GR extension in which disconnected components explained some observational anomaly, suddenly this would be very interesting.

The thing is this is not possible by definition, if such component explained some observation then it would be connected.

Similarly, analyticity is often assumed for no reason other than it makes things easier.

Analiticity is not assumed formally, the way connectedness is. It is implicitly assumed to justify certain gauge fixing that leads to prefer certain solutions over others in a "de facto" way

If ever someone found a non-analytic solution of GR (for example) that explained something that couldn't be explained with any analytic solution, physicists would no longer assume it.

It usually works the other way around, some physicists use the analyticity assumption (usually without even stating it) with the only justification that it leads to some preconceived entity that doesn't really explain any actual observation better than the solution that doesn't assume analyticity.

 

[WannabeNewton]

This is an influential paper, especially for approaches to Quantum Gravity that take causality as a primitive structure:
J. Math. Phys. 18, 1399 (1977); http://dx.doi.org/10.1063/1.523436 (6 pages)
"The class of continuous timelike curves determines the topology of spacetime"
http://jmp.aip.org/resource/1/jmapaq/v18/i7/p1399_s1?isAuthorized=no

Ah yes, he did add a discussion related to this paper as an appendix in his text. Thanks for the link to the original paper. And it definitely is evident from his text that he was influenced by Geroch and Wald; most of the exercises in the first two chapters and the devoted use of the abstract index notation in calculations/proofs (and a general dearth of coordinate based methods) are reminiscent of Wald's text and Geroch's notes. I have to admit that I was surprised when I found out he was a philosopher of science (but my knowledge of that field is quite lacking).

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?

GR with an assumption of connectedness and without are indistinguishable as to conceivable observations. Since the former is simpler to work with, the latter is only for masochists.

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.

 

[nitsuj]

It's one thing to just work with our own connected component and assume this to be the only part of space-time of any physical relevance to us but it's another thing to write off, by definition, the existence of disconnected space-times altogether.

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)

 

[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).