Link btw manifolds and space-time

[Hurkyl]

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 . The discussion about figures 3 and 4, in particular.

 

[MeJennifer]

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"?

 

[coalquay404]

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

 

[vanesch]

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]

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."

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]

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. )

 

[MeJennifer]

No, it cannot. http://en.wikipedia.org/wiki/Manifold#Mathematical_definition

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?

 

[coalquay404]

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.

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.

 

[MeJennifer]

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.

 

[Hurkyl]

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.