Embedding curved spacetime in higher-d flat spacetime

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  • #1
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I have been reading John Baez's introduction to General Relativity (http://math.ucr.edu/home/baez/einstein/einstein.pdf). This part got me thinking:
Our curved spacetime need not be embedded in some higher-dimensional flat spacetime for us to understand its curvature, or the concept of tangent vector. The mathematics of tensor calculus is designed to let us handle these concepts `intrinsically' i.e., working solely within the 4-dimensional spacetime in which we find ourselves.
My question is, in order to completely model our universe's (3+1)-dimensional spacetime as a manifold in a higher-dimension, flat spacetime, how many dimensions are needed? Will a finite number of dimensions suffice, or are an infinite number of flat dimensions required to completely embed all solutions to the Einstein Equation?

If anyone can refer me to texts or publications that treat GR in this manner (as a curved manifold in higher-d flat spacetime) it would be much appreciated.
 

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  • #3
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http://en.wikipedia.org/wiki/Whitney_embedding_theorem

8 dimensions are sufficient in any case. Maybe even less.
That's very interesting. I notice that the Nash embedding theorem (linked from the Whitney embedding theorem Wikipedia page) mentions "isometric embedding", i.e. preserving the length of curves in the manifold, whereas the Whitney theorem does not. Going by this bit of the Nash theorem's explanation:

if M is a given m-dimensional Riemannian manifold (analytic or of class Ck, 3 ≤ k ≤ ∞), then there exists a number n (n = m2+5m+3 will do) and an injective map f : M → Rn (also analytic or of class Ck) such that for every point p of M, the derivative dfp is a linear map from the tangent space TpM to Rn which is compatible with the given inner product on TpM and the standard dot product of Rn
It looks like a 4-d curved spacetime can be isometrically embedded in a flat spacetime of 39 dimensions or less (m=4 -> m2+5m+3 = 16+20+3 = 39). Unwieldy perhaps, but at least it's a finite number of dimensions.
 
  • #4
atyy
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That's very interesting. I notice that the Nash embedding theorem (linked from the Whitney embedding theorem Wikipedia page) mentions "isometric embedding", i.e. preserving the length of curves in the manifold, whereas the Whitney theorem does not.
Does the Nash isometric embedding theorem also work for pseudo-Riemannian manifolds?
 
  • #5
George Jones
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Does the Nash isometric embedding theorem also work for pseudo-Riemannian manifolds?
If the metric for the higher dimensional pseudo-Riemannian manifold is required to restrict down to the metric for 4-dimensional spacetime, then it could take a lot of dimensions.

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.

* Clarke, C. J. S., "On the global isometric embedding of pseudo-Riemannian
manifolds," Proc. Roy. Soc. A314 (1970) 417-428
 
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whats about 10- 11- and 26 dimensional spaces? :)
Or may be these additional dimensions are already a part of a manifold?
 
  • #7
George Jones
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whats about 10- 11- and 26 dimensional spaces? :)
Or may be these additional dimensions are already a part of a manifold?
Right. In the spacetime manifold in which string theory lives, the extra spacetime dimensions are compactified, so they're not easily physically visible.
 
  • #8
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My question is, in order to completely model our universe's (3+1)-dimensional spacetime as a manifold in a higher-dimension, flat spacetime, how many dimensions are needed? Will a finite number of dimensions suffice, or are an infinite number of flat dimensions required to completely embed all solutions to the Einstein Equation?
A d-dimensional (curved) space(time) can be be embedded (immersed) in a flat space of D = (1/2)d(d+1) dimensions. See L. P. Eisenhart, Riemannian Geometry, Princeton, 1997 (org. 1925), Ch. V, Par. 56, p. 188 or T. Levi-Civita, The Absolute Differential Calculus, Dover 1977 (org. 1926), Ch. V, Sec. 21, p. 122.

Notes:
Flat = it is possible to introduce coordinates xm such that ds2 = hmndxm(x)dxn(x) correctly describes distance (ds)2 with hmn fixed (independent of x).
Embedded = When x and dx are inside the sub-space, the ds2 above correctly describes the local distances in the sub-space.
In some cases, a flat space dimension < D suffices. Obviously an already flat space of d dimensions can be embedded in d dimensions.
Eisenhart gives an example of the Schwarzschild spherical space time curved 4-d space being embeddable in a 6 dimensional flat space (a special case).
For d = 4, D = 10 is the general requirement.
This only considers general relativity (gravitation), no other forces.
 
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  • #9
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The dimensionality requirement of (1/2)d(d+1) I mentioned earlier was proven by power series development methods (see references Janet, 1926 and Cartan, 1927 in Eisenhart's book mentioned earlier, also mentioned in the ref. below), and hence are local results (hold in some finite open set around a given point) and require that the original intrinsic space(time) metric is analytic in the coordinates. J. Nash, The Imbedding Problem for Riemannian Manifolds, Ann. of Math., Vol. 63, No. 1, Jan. 1956, pp. 20-63, gives a nice historical introduction to the problem and considers cases when less assumptions are made.
http://abdullahburhanbahce.googlepages.com/theimbeddingproblemforriemmannianman.pdf" [Broken]
 
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  • #10
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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.
Interesting.
What if we work with complex numbers as coordinates and hence completely ignore the difference between spacelike and timelike?
 

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