Could any curved space be a cut in a higher-dimensional flat space ?

[lalbatros]

For a beginner (as I am since a long time), it is convenient to conceptualize curved spaces as embedded within a familar flat space with more dimensions.

Of course the intrinsic point of view is more elegant and suffice to itself. Nevertheles, I am asking this question: are there some curved spaces that cannot be considered as a surface embedded in some higher dimensional space?

 

[Garth]

For a beginner (as I am since a long time), it is convenient to conceptualize curved spaces as embedded within a familar flat space with more dimensions.
Of course the intrinsic point of view is more elegant and suffice to itself. Nevertheles, I am asking this question: are there some curved spaces that cannot be considered as a surface embedded in some higher dimensional space?

This is certainly an open question!

The problem is trying to imagine all the possible topologically pathological spaces in order to eliminate them all in order to answer in the negative.

In SR we deal with a 4D 'space' of space-time with a Minkowskian metric, which we can imagine as a flat hyper-surface by suppressing one or two of the space dimensions.

In GR we learn that this 4D space-time is curved by the presence of stress-mass-energy-momentum. It is therefore tempting to imagine such curvature, say the familiar Schwarzschild funnel bowling-ball-on-a-rubber-sheet analogy, embedded in some higher, fifth, dimension. The higher dimension is flat by the very virtue of it being a product of our imagination.

However, there is nothing in GR that requires you to believe that this higher dimension actually exists; curvature can be expressed and described intrinsically by the changes in geometry in the hyper-surface itself. All we actually experience are the three dimensions of space and one of time.

Then along come the QM people and invent 10/11/26 dimensions in which to embed their theories!

Whether there are curved hyper-surfaces that cannot be embedded in a higher flat space does not really matter as the geometry of curved spaces can be inclusively and comprehensively described intrinsically within that hyper-surface itself.

Others may know of a specific example that cannot be so embedded.

Garth

 

[George Jones]

are there some curved spaces that cannot be considered as a surface embedded in some higher dimensional space?

I'm not sure what you mean by "curved space." If you mean curved spacetime, the answer to your question is no. Chris Clarke* showed that every 4-dimensional spacetime can be embedded isometically in higher dimensional flat space, and that 90 dimensions suffices - 87 spacelike and 3 timelike. A particular spacetime may be embeddable in a flat space that has dimension less than 90, but 90 guarantees the result for all possible spacetimes.

There is a somewhat related result, the Whitney embedding theorem, for spaces that don't have "metrics": every n-dimensional differential manifold can be embedded in R^(2n).

* Clarke, C. J. S., "On the global isometric embedding of pseudo-Riemannian
manifolds," Proc. Roy. Soc. A314 (1970) 417-428

Regards,
George

 

[Mortimer]

I may add to this the Campbell-Magaard embedding theorem:

The problem of embedding an ND (pseudo-) Riemannian manifold in a Ricci-flat space of one higher dimension was taken up again by Magaard. He essentially proved the theorem in his Ph.D. thesis of 1963.

From: Paul Wesson, "In Defense of Campbell's Theorem as a Frame for New Physics". http://www.arxiv.org/ftp/gr-qc/papers/0507/0507107.pdf

 

[masudr]

the Whitney embedding theorem, for spaces that don't have "metrics": every n-dimensional differential manifold can be embedded in R^(2n).

I was aware of this, but not aware of:

Chris Clarke* showed that every 4-dimensional spacetime can be embedded isometically in higher dimensional flat space, and that 90 dimensions suffices - 87 spacelike and 3 timelike.

Thanks for that, that's very interesting!

 

[masudr]

For a beginner (as I am since a long time), it is convenient to conceptualize curved spaces as embedded within a familar flat space with more dimensions.

I can see where you are coming from, perhaps, but it is hard enough to conceptualise a 4D flat space, let a lone a 4D curved space (for me, anyway). I'll just rely on the power of the symbols we can write down, I feel it's the only way to proceed on the subject. If you can conceptualise the 90 dimensions required for a generally curved space, then well done!

 

[mewmew]

I don't want to junk up this thread but I have a question, we can simulate 3 dimensions on a 2 dimensional surface so if we had some sort of 3 dimensional "cube television" could we simulate 4 dimensions? I have a feeling if we could it wouldn't really be intuitive to understand as our concepts of 3 dimensions seems to be. The question I often think about that is the root of it is why can we only think in 2 and 3 dimensions if we really live in more? Is thinking only in 2 and 3 dimensional euclidean space just a bi-product of learning as opposed to something a-priori? Hopefully that doesn't sound like a stoner hippey question too much.

 

[lalbatros]

Masur: you are right !

it is hard enough to conceptualise a 4D flat space

Embedding is helpful only for lower dimensions, but it helps at the beginning.
It was more or less a way to introduce my question.

If the answer to my question is no: any space can be seen as embedded,
then what is the meaning for the 4-dimensional space of GR?
Is it simpler to take this point of view?
Is it more fundamental, since it goes to 'observables' only?
And what would be the meaning of looking for the embedding-space extra dimensions? Could these dimensions be probed experimentally?

And I need to think further for my next question about non-commutative geometry !

 

[George Jones]

I may add to this the Campbell-Magaard embedding theorem:
From: Paul Wesson, "In Defense of Campbell's Theorem as a Frame for New Physics". http://www.arxiv.org/ftp/gr-qc/papers/0507/0507107.pdf

For the sake of clarification: the Campbell-Magaard theorem is for local embeddings, while Clarke's theorem is for global embeddings.

Reagrds,
George

 

[pervect]

The goal of any scheme is to accurately model the "distances" (sometimes, as in GR, an abstract quantity such as the Lorentz interval rather than a Euclidean distance) between any two points.

The simplest way to do this quantitatively is to use a metric, and not an embedding, IMO. Certain special cases may be handled on an ad-hoc basis more simply with an embedding diagram, and embedding diagrams are used with some limited amount of success to attempt to explain why space-time curvature is equivalent to a force as a pedagogical tool.

There may actually be an experimental way to address the original question, though. We have a proof that any space-time can be embedded in a space of 90 dimensions. So an embedding in such a space-time would be capable of reproducing any configuration of an arbitrary curved space-time. It is plausible, though at this point unproven as far as I know, that a euclidean space of less than 90 dimensions will not be able to model some specific configurations of space-time. To follow this route, we'd have to figure out what these limits on configuration were, and show that our universe obeyed these particular limits via experiment. It's possible that there might be alternative explanations of such limitations on geometry to be due to something other than the universe beeing embedded, but it would make it at least plausible that the universe was actually embedded in some higher dimensional manifold.

This sounds like a very difficult task. I personally wouldn't expect it to be particularly productive, either, though it might be interesting as a "pure math" sort of thing rather than a physics thing. It definitely would be a lot more difficult than, say, just hunkering down and learning relativity via use of a metric.

 

[George Jones]

The simplest way to do this quantitatively is to use a metric, and not an embedding, IMO.

Even though I find Clarke's result to be quite interesting, I agree completely with you. I was just giving a specific answer to a specific question. I have never looked at Clarke's paper, so I'm not sure what motivated Clarke. Interestingly, Clarke coauthored a book, Relativity on Curved Manifolds, that has a very technical discussion of the theory of measurement in GR. And I have the book at hand, since my books finally have almost exactly the same space and time coodinates as I do!

Regards,
George

 

[pervect]

I also find Clarke's result interesting. I was pretty sure that I had read that there was some such result, but I couldn't recall the source, or the number of dimensions required for a general embedding, so I'm very glad you were able to post the specifics.

I was also giving a specific answer to a specific question, the rather broad question of whether it could be fruitful to pursue the idea of what the embedding of space-time is. If there were not any possible experimental predictions from the embedding, this would be a total dead end, but upon thinking about the question it seems to me that there is a possibility of some experimental predictions from such a theory.

While it's not a dead end, I personally don't think that the approach is going to lead anywhere interesting in a physics sense. However, the mathematics involved might be very interesting in a purely abstract sense. The math involved also appears to me to be extremely difficult, something else I wanted to stress, because the Original Poster classified himself as a beginner.