# Link btw manifolds and space-time

George Jones
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Take a FRW universe with negative cosmological constant. One topology that is consistent with the metric (though not the only one) is S^4. You do not necessarily get CtCs.
But this violates the well-known theorem that every compact spacetime contains closed timelike curves. The simple proof of this theorem is independent of energy conditions and of the cosmological constant; see Theorem 3.3.11 of Naber's Spacetime and Singularities: an Introduction or Proposition 6.4.2 of Hawking and Ellis.

Take a FRW universe with negative cosmological constant. One topology that is consistent with the metric (though not the only one) is S^4. You do not necessarily get CtCs.
You do get closed timelike curves whenever the spacetime is compact. This is not debatable. See, for example, chapter six of Hawking & Ellis

Haelfix
Can you list the assumptions in the proof? I do not have the book handy here. AFAIK ctcs do not occur in closed FRW universes b/c the recurrence time is much larger than the big crunch.

So let me change the topology to S^3 * R^1. Satisfied?

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vanesch
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So let me change the topology to S^3 * R^1. Satisfied?
You guys are beyond me... but isn't this a non-compact manifold ?

I'll reply to Haelfix tomorrow (it's ridiculously late here, and there's rugby on tomorrow), but yes, $S^3\times\mathbb{R}$ is non-compact.

George Jones
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AFAIK ctcs do not occur in closed FRW universes b/c the recurrence time is much larger than the big crunch.

So let me change the topology to S^3 * R^1. Satisfied?
The topology of a (simply connected) closed FRW universe is S^3 x R, which is non-compact, and which doesn't contains CTCs. (Note: the theorem does not say that non-compact spacetimes don't contain closed timelike curves). Even FRW universes that have a cosmological constant/dark energy don't have closed CTCs. For example, observations are consistent with (but don't prove) our universe being a closed S^3 x R universe that expands forever. Galaxies (negelecting peculiar velocities) are given fixed comoving coordinates (in space) on S^3, while the R coordinate, cosmological time, increases. (Proper distances are found by using the time-dependent scale factor.) This is the foliation of spacetime that coalquay404 mentioned.

It is possible to go on journey through space, and, without "turning around", return to the same position in space, i.e., to return the starting S^3 coordinates. But time will have advanced, so the R coordinate will be different.

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George Jones
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Can you list the assumptions in the proof? I do not have the book handy here.
After its proof, Hawking and Ellis makes an interesting point. No compact spacetime is simply connected, so all compact spacetimes can be formed by identifying points of non-compact spacetimes.

Now I don't have my books handy, but I do have a copy of Naber's proof that I made when my books were in storage. Only the concepts of past and future are used, so I suppose orientability is necessary.

Definition: p << q iff there exists a smooth, future-directed timelike curve from p to q.

Lemma: << is transitive. (If p << q << r, connect p to q by a smooth curve, connect q to r by a smooth curve, and "smooth the corner" at q to obtain a smooth curve from p to r.)

Definition: The chronological future of any event p is I+(p) = {q in M | p << q}.

Lemma: For any p, I+(p) is open. (It's the interior of a future lightcone.)

Theorem: Any compact spacetime M contains closed timelike curves.

Proof: {I+(p) | p in M} is an open cover for M since any q in M is in the future of some p (technical details omitted). Since M is compact, this cover must admit a finite subcover {I+(p_1), I+(p_2), ... , I+(p_n)}. We may assume that I+(p_1) is not contained in any I+(p_j) for j >= 2, otherwise I+(p_1) could be removed from the subcover. But then p_1 is not in I+(p_j) for any j>=2 by the transitivity of <<. 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 think the proof is quite beautiful.

Sorry for the late response, I am currently in Thailand and have less access to the internet than normal.

(2) is mistaken. Space-time is a not a Riemannian manifold (instead it is a different particular type of manifold, specifically the type named psuedo-Riemannian), with a metric that has a Lorentzian signature (hence is not positive definite).

I thought you'd seen this earlier along the thread, but nonetheless.. Now that you know the 2nd fibre that the emperor really uses, do you still have any issue with the fabric's consistency?
I don't disagree that space-time is not a Riemann manifold, I was merely paraphrasing the, in my view, incorrect idea that space-time is Riemann and that the (pseudo) metric in some way operates onto this as some sort of algebra.

coalquay404 said:
I wouldn't say you lack intelligence, but you're definitely confused. (2) is incorrect. It should read "Spacetime is a four-dimensional paracompact, connected smooth Hausdorff manifold without boundary, and with an indefinite (or, if you like, pseudo-Riemannian) metric structure ."
Explain to me how a manifold with non positive definite metric can possibly be Hausdorff?

coalquay404 said:
(3) is also completely incorrect.
So what are you saying here, that the shape of the manifold is not determined by the EFE? What then determines the shape according to you?

Pervect said:
I hope you and everyone will agree that that event (the one in Andromeda) is "far away", even though it is connected by a curve of zero Lorentz interval (the null curve, in this case a null geodesic, of the path of the photon) to you.
It is a plain and simple fact of SR and GR that the distance in space-time between an emmited and absorbed photon is 0. You can foliate the two events in 3 dimensions, but in GR there is no prefered foliation. Note that ether theorists have the opinion that space-time is definitively 3D+1.
Sorry Pervect but to say that two events are "far away" from each other when in fact the ds is zero is complete nonsense and counter to the first principles of the theory of relativity.

Einstein demonstrated that both distance and duration are not absolute concepts, they cannot be taken in isolation! What for one observer is "far away" might be "nearby" for another one and what for one observer is "a long time" might be "a short time" for another one.

Again I am not saying that GR is in some way wrong, I am saying that when we want to use a geometric mathematical model we must admit that the mathematics are incomplete. That does not mean we cannot make any calculations but nevertheless it is incomplete.

And we have not even started to consider how it is logical how EFE equations can lead to singularities on a manifold shaped by only smooth deformations.

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Hurkyl
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Explain to me how a manifold with non positive definite metric can possibly be Hausdorff?

The smooth manifold R^2 is Hausdorff. Agreed?

(In the canonical basis) g := dx^2 - dy^2 is a nondegenerate symmetric bilinear form that is not positive definite. Agreed?

Therefore, the pseudo-Riemannian manifold (R^2, g) is Hausdorff pseudo-Riemannian manifold with a metric tensor that is not positive definite.

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I have a reply in progress but since I can't get the LaTeX preview to work, I'll do it offline and upload it in a bit

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Actually, screw the long explanation. MeJennifer, before I continue, you are presumably aware of the differences between topological metricity and tensorial metricity?

(And before you start, yes, I'm perfectly well aware that Hausdorff isn't compatible with pseudometricity in the topological sense.)

Actually, screw the long explanation. MeJennifer, before I continue, you are presumably aware of the differences between topological metricity and tensorial metricity?
Well that was my earlier point. It is just a way to avoid the issue.

If we intepret GR as some algebraic system using tensor algebra then we obviously avoid the whole problem, but then there is no manifold and arguably not even a notion of curvature!

You can’t have your cake and eat it!

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People are getting very confused about two different concepts.

Distinguish the topology of R^4
and
the topology of spacetime in the sense of the global visible universe.

The latter is essentially glued together from the former, and I use the word 'glue' rather nonstandardly here b/c theres several maps going on before this even takes place. The former is purely formal in the sense that we construct a *local* homeomorphism from R^4 --> R^4 to identify our local neighborhood with the usual Lorentzian one. Note, this identification is already topologically restricting, eg we now have a 4 dimensional manifold, and not just any manifold, but one with a special relativity isometry group acting on points. So amongst all possible topologies, we have restricted the set theoretic structure to be both the usual one, as well as endowed a diffeomorphic manifold structure on it.
Sorry but forgive me think this is simply a trick to get around the issue of our current mathematical limitations. Our mathematical knowledge is afteral limited right?

Furthermore, this is actually an imposed limitation on the scope of a non definite positive metric. Feel free to demonstrate, by using physical, not mathematical arguments why this imposed limitation is justified.

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vanesch
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If we intepret GR as some algebraic system using tensor algebra then we obviously avoid the whole problem, but then there is no manifold and arguably not even a notion of curvature!

In order for there to even be a tensor algebra, you need a manifold to start with. If you have your manifold, and if you have such a tensor algebra, which allows for the definition of a symmetric 2-tensor, you have introduced a (pseudo) metric.
In what way is this incompatible ?

The manifold doesn't come from the metric ; the metric is defined over the manifold. It's a 2-tensor.

But even before the 2-tensor was there, there are topological properties to a manifold (by definition of a manifold). These topological properties are not defined by the metric (they only need to be compatible, in that the topology of the manifold needs to allow for the 2-tensor which defines the metric with necessary properties such as smoothness and signature etc...).

Not sure why there is so much resistance to the idea that space-time is non-Hausdorff. I think it is a great feature, but a feature that is obviously beyond our current mathematical understanding.
Many simply don't want to deal with it.

For instance the idea of bifurcating curves opens perspectives for embedding MWI of quantum theory in space-time as mentioned both by Visser and Penrose.

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Hurkyl
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Not sure why there is so much resistance to the idea that space-time is non-Hausdorff.
Nobody was ever resisting the idea that spacetime might be non-Hausdorff. (Of course, GR explicitly asserts that spacetime is Hausdorff)

The "resistance" is to your assertion that manifolds may be non-Hausdorff. Furthermore, that assertion appears to have been caused by confusing the (different!) notions of pseudometric and pseudometric.

By the way, you may be interested in http://philsci-archive.pitt.edu/archive/00002315/02/BranchingUniverses.pdf [Broken]. The discussion about figures 3 and 4, in particular.

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The "resistance" is to your assertion that manifolds may be non-Hausdorff.
A manifold certainly can be non-Hausdorff.

Of course, GR explicitly asserts that spacetime is Hausdorff.
When the theory of general relativity was developed the notion of a manifold being Hausdorff or not did not even exist.

Could you provide some reference for your statement that GR explicitly asserts that. I think it is not explicitly asserted but assumed out of convenience. But there is zero theoretical basis, let alone physical proof, for that restriction.

The prior mentioned Taub-Nut space is an example (see for instance Hawking, Ellis - "The Large Scale Structure of Space-Time" 5.8") where we can have a manifold that is non-Hausdorff and in one case it is not even a non-Hausdorff manifold!

What is wrong in teaching people that we really have no good available math to model non positive definite metrics and that instead we are satisfied with a lot of pseudo definitions that seem to do the job?

By the way, the Hausdorff property is not the only area where our current state of math seem to have difficulties, singularities and geodesic incompleteness also seem to provide difficulties. Should we simple ignore these as well or simply "pseudo-whatever" them and then teach with a straight face that: "all is well"?

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Does anybody else get the idea that this discussion has now become pointless?

vanesch
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I see a difference here between the eventual possibility of considering non-hausdorf manifolds (or other structures) on one hand, and confusing the (locally euclidean) topology of the manifold with the pseudo-metric (which is not a generator for a topology).

Hurkyl
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A manifold certainly can be non-Hausdorff.
No, it cannot. http://en.wikipedia.org/wiki/Manifold#Mathematical_definition

There can certainly be topological spaces that are second countable and locally homeomorphic to Euclidean space, but they aren't manifolds, by definition of manifold.

Could you provide some reference for your statement that GR explicitly asserts that. I think it is not explicitly asserted but assumed out of convenience.
As it's the most convenient reference:

http://en.wikipedia.org/wiki/General_relativity#Overview

What is wrong in teaching people that we really have no good available math to model non positive definite metrics
Why do you think that?

with a lot of pseudo definitions that seem to do the job?
Don't confuse technical words with common English.

By the way, the Hausdorff property is not the only area where our current state of math seem to have difficulties, singularities and geodesic incompleteness also seem to provide difficulties.
Allow me to state a metamathematical tautology:

"Well-behaved spaces behave better than spaces that are not well-behaved." :tongue:

Smooth and separated are some of the "best" qualities a topological space can have. One should not be surprised that the study of spaces that do not have those qualities is more difficult than the study of spaces that do have those qualities.

Hurkyl
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Does anybody else get the idea that this discussion has now become pointless?
Aww, but pointless topology is the most interesting kind!
(Oh, whoops, I guess I should listen to my own advice about confusing technical terms with common English. )

A few references:

Hawking, Ellis - "The Large Scale Structure of Space-Time"
Page 14 for an example of a non-Hausdorff manifold.

Hajicek P. - Causality in non-Hausdorff space-times:
Abstract: Some general properties of completely separable, non-Hausdorff manifolds are studied and the notion of a non-Hausdorff space-time is introduced. It is shown that such a space-time must, under very general conditions, display a kind of causal anomaly.

Wolfram MathWorld on Topological Manifold:
A topological space satisfying some separability (i.e., it is a Hausdorff space) and countability (i.e., it is a paracompact space) conditions such that every point has a neighborhood homeomorphic to an open set in for some . Every smooth manifold is a topological manifold, but not necessarily vice versa. The first nonsmooth topological manifold occurs in four dimensions.

Nonparacompact manifolds are of little use in mathematics, but non-Hausdorff manifolds do occasionally arise in research (Hawking and Ellis 1975). For manifolds, Hausdorff and second countable are equivalent to Hausdorff and paracompact, and both are equivalent to the manifold being embeddable in some large-dimensional Euclidean space.

Visser - "From wormhole to time machine: Remarks on Hawking’s chronology protection conjecture" - Phys. Rev. D 47

Smooth and separated are some of the "best" qualities a topological space can have. One should not be surprised that the study of spaces that do not have those qualities is more difficult than the study of spaces that do have those qualities.
Indeed, and to properly model GR we need to adress these difficulties instead of just ignoring them.

Out of curiosity, how do you, or would you, explain students when they ask you about how our current mathematical abilities deal with singularities, geodesic incompleteness and non-Hausdorff situations in GR?

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Let me jump in here for the final time in this utterly boring thread:

A few references:

Hajicek P. - Causality in non-Hausdorff space-times:
Abstract: Some general properties of completely separable, non-Hausdorff manifolds are studied and the notion of a non-Hausdorff space-time is introduced. It is shown that such a space-time must, under very general conditions, display a kind of causal anomaly.
I was the one who initially pointed you towards Hajicek's relevant work. Had you understood the content of the paper above (as opposed to simply scanning the abstract) you'd understand that it (i) provides no support for your ludicrous claims about indefinite metric structures and (ii) is a study of a pathological case quite unrelated to what we deal with in run of the mill general relativity.

MeJennifer said:
Indeed, and to properly model GR we need to adress these difficulties instead of just ignoring them.

Out of curiosity, how do you, or would you, explain students when they ask you about how our current mathematical abilities deal with singularities, geodesic incompleteness and non-Hausdorff situations in GR?
I tell them to go and read Wald or H&E. Failing that, there are countless acceptable discussions in the literature.

Really, you're beating a dead horse with this one.

is a study of a pathological case quite unrelated to what we deal with in run of the mill general relativity.
So now only "run of the mill relativity" is the true Scotsman of GR?
Sorry, but I hardly find that a scientific argument for discarding non-Hausdorff situations in GR.

I tell them to go and read Wald or H&E. Failing that, there are countless acceptable discussions in the literature.
Well since you think this thread is so boring perhaps you could spice it up by giving me the chapter and pages where Wald addresses the mathematical problems related to singularities, geodesic incompleteness and non-Hausdorff conditions on the Lorentzian manifold.

I don't think he addresses them at all. He does not even discuss Hausdorff in the context of GR, only in appendix A, a review of Topological Spaces.

Regarding mentioning geodesic incompleteness and singularities, he writes in 9.1 on page 216:

"In fact, much more dramatic examples can be given of the failure of geodesic incompleteness to correspond to the intuitive notion of the excision of singular "holes". In a compact spacetime, every sequence of points has an accumulation point, so in a strong intuitive sense, no "holes" can be present."

Is it just me or does he leave out the obvious here? The point that in a non-Hausdorff space the, one accumulation point condition, is anything but guaranteed.

His explanation:

"This failure of geodesic incompleteness to correspond properly to the existence of "holes" is, of course, closely related to the difficulty discussed above of defining a singularity as a "place"."

Unfortunately I am rather unconvinced by this conclusion.

And a bit further he seems to agree that we have more mathematical work to do when he writes:

"Unfortunately, the singularity theorems give virtually no information about the nature of the singularities of which they prove existence."

Taub-NUT spaces remain unmentioned by Wald.

So, in particular I am interested in why you quoted Wald. Did I perhaps miss any sections where these mathematical problems are addressed?

In Hawking, Ellis - "The Large Scale Structure of Space-Time", some of the above mentioned issues are addressed. At least, they discuss the real issues there instead of proclaiming "it is all a closed case" or simply avoiding the issues.
They do discuss non-Hausdorff situations and talk quite extensively about geodesic incompleteness and singularities. Furthermore, as mentioned above, Taub-NUT spaces are discussed as well.

Here is an example, apart from the evident genius of the publication, and Hurkyl you might like this since it involves "cutting a single point from the manifold" with regards to Cauchy surfaces, of why I like Hawking's and Ellis' approach to complexities in GR. Instead of walking away from it they, at least, mention the issues:

"If there were a Cauchy surface for $\mathcal {M}$, one could predict the state of the universe at any time in the past of future if one knew the relevant data on the surface. However one could no know the data unless one was to the future of every point in the surface, which would be impossible in most cases. There does not seem to be any physically compelling reason for believing that the universe admits a Cauchy surface, in fact there are a number of known exact solutions of the Einstein field equations which do not, among them the anti-de Sitter space, plane waves, Taub-NUT space and Reissner-Nordstrom solution, all described in chapter 5."

And, a little further, regarding the Reissner-Nordstrom solution, mind boggling, but also fascinating:

"There could be extra information coming from infinity or from the singularity which would upset any predictions made simply on the basis of data on $\mathcal {S}$"

Anyway, to me, there is a lot of work to be done in mathematics to capture all those, interesting and fascinating, properties of GR.

And I did not even mention imprisoned incompleteness.

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Hurkyl
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Indeed, and to properly model GR we need to adress these difficulties instead of just ignoring them.
Who's ignoring them? As far as I know:

Non-separated points are physically indistinguishable. Thus, the Hausdorff axiom.

The Equivalence principle requires that spacetime is locally like SR -- this gives us "locally Euclidean", "smooth", and "pseudo-Riemannian".

Aside -- there are at least two notions of "singularity" that are relevant here

(1) In a 1-dimensional space shaped like the letter X, one would call the point in the middle a "singularity". (At least, one would when studying schemes. I assume the same term is used in this setting) Similarly, if we have a 1-dim manifold embedded as a V in 2-space, we would say it's singular at the vertex.

(2) The "singularity" in a black hole (and related usage) does not refer to an actual point of the manifold -- instead, it refers to the fact there is a "hole" in space-time. One would say that a field is singular at that hole if it does not converge to a value as you approach that hole.

I suspect that you could "compactify" by filling in the hole, and you could still have a smooth manifold -- but now your fields (like the metric) are not everywhere defined... so for most purposes you would be looking at the subset of your manifold that excludes the filled-in hole. Thus, there really isn't much point to filling in the hole.

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