Nash embedding for space-time
Mirco Mannucci
Hallo to all.
Does anybody know if there is an analogue of the Nash embedding theorem (i.e. every Riemannian manifold can be realized as a hypersurface of some Euclidean space with the induce metric)
to Lorentz manifolds?
In other words, is it possible to find a R^n with a metric given by a non-positive definite quadratic form such that the manifold sits in it and its Lorentz metric (+ + + -)is just
the one naturally induced by the ambient space?
Thanks.
Mirco Mannucci
Note: perhaps this question is not entirely idle math
curiosity: if the answer is yes, one could think of general relativity from the standpoint of (a higher-dimensional
version of) special relativity (with as many different
“times” as the number of minuses in its signature).
Also it might (?) help our understanding of bundles over
space-time, as one meets all over QFT.
George Jones
Mirco Mannucci wrote:
> Hallo to all.
>
> Does anybody know if there is an analogue of the Nash embedding theorem
> (i.e. every Riemannian manifold can be realized as a hypersurface of
> some Euclidean space with the induce metric) to Lorentz manifolds?
>
> In other words, is it possible to find a R^n with a metric given by a
> non-positive definite quadratic form such that the manifold sits in it
> and its Lorentz metric (+ + + -) is just the one naturally induced by the
> ambient space?
Chris Clarke showed that any spacetime can be embedded isometrically
in some R^n that has constant metric tensor, where n is at most 90
and the number of timelike dimensions is at most 3.
Clarke, C. J. S., "On the global isometric embedding of
pseudo-Riemannian manifolds," Proc. Roy. Soc. A314 (1970) 417-428
Regards,
George
[Moderator's note: presumably in the penultimate paragraph "spacetime"
means something like "4-dimensional Lorentzian manifold"? - jb]
Robert Low
>Chris Clarke showed that any spacetime can be embedded isometrically
>in some R^n that has constant metric tensor, where n is at most 90
>and the number of timelike dimensions is at most 3.
>means something like "4-dimensional Lorentzian manifold"? - jb]
Yes. Clarke gives a general result on embedding a manifold
with a pseudo-Riemannian metric whose diagonal form has
p 1's and q -1's; the result quoted above is for p=1, q=3.
I may be misreading the result sightly, but I thought it said that
any Lorentz four-manifold could be isometrically embedded in
E^{2,89}; so that's 91 dimensional, with number of timelike
dimensions 2. A related result is that if the space-time is
globally hyperbolic, then it can be isometrically embedded in
E^{1,88}.
There is one slight subtlety here: the result guarantees a C^k
embedding for a C^k manifold (k > 2); it doesn't say anything about
C^\infty embeddings for C^\infty manifolds.
Is this stuff relevant to the brane-world lot, or do they
only consider cases where space-time can be embedded with
small codimension?
--
Rob. http://www.mis.coventry.ac.uk/~mtx014/
George Jones
Robert Low wrote:
> I may be misreading the result sightly, but I thought it said that
> any Lorentz four-manifold could be isometrically embedded in
> E^{2,89}; so that's 91 dimensional, with number of timelike
> dimensions 2. A related result is that if the space-time is
> globally hyperbolic, then it can be isometrically embedded in
> E^{1,88}.
I don't have access to Clarke's paper. I got my information from the
book "Tensor Geometry" by Dodson and Poston who say "a spacetime may
need up to 87 spacelike and 3 timelike dimensions."
With reference to a recent discussion in another thread:
The above book is in the Springer Grad Texts in Math series, so it's
definitely a math book, but the authors make an interesting comment
about physics and the Feynman Lectures. The authors mention that
varational techniques gives a relationship between classical and
quantum physics, and say: "This is not the book in which to go
further into this point, however, particularly as it is so lucidly
discussed in Feynman - a work which the reader should in any case read,
mark, learn and inwardly digest."
Regards,
George
squ...@my-deja.com
mtx...@coventry.ac.uk (Robert Low) wrote:
[About embedding space-time into R^91 with signature 89 : 2]
> Is this stuff relevant to the brane-world lot, or do they
> only consider cases where space-time can be embedded with
> small codimension?
Do they really seriously consider the idea 4-dimensional space-time is
a brane? I didn't know that...
Best regards,
squark.
Sent via Deja.com
http://www.deja.com/
zir...@my-deja.com
squ...@my-deja.com wrote:
> Do they really seriously consider the idea 4-dimensional space-time is
> a brane? I didn't know that...
Yes, some people view the Standard Model as
being confined to a 3+1 dimensional subspace
(or 3-brane) in higher dimensions. It is
possible that our whole 3-brane universe
could fit into an extra dimension that is only a
millimeter wide (which is pretty big for an
extra dimension). See this newspaper article:
http://www.zainea-rpg.gr/cosmos.htm
Paul D. Shocklee
> Do they really seriously consider the idea 4-dimensional space-time is
> a brane? I didn't know that...
Oh, yes. It's become a minor industry in the string theory world.
There were some ideas long ago that our spacetime might be a domain wall
of some sort, but one of the first papers to bring D-branes into the
picture was Antoniadis et al, "New Dimensions at a Millimeter to a Fermi
and Superstrings at a TeV", hep-ph/9804398.
Since then, people have been writing papers on brane world gauge groups,
cosmology, black holes, you name it. It seems that these sorts of theories
might make it easier to solve all kinds of problems, from the selection of
the Standard Model gauge group, to supersymmetry breaking, to the hierarchy
problem, to the cosmological constant problem, and more. They also give
rise to new possibilities for near-term experimental verification - large
extra dimensions, stringy physics at TeV scale, new cosmological predictions,
etc.
There are still a number of problems, one of which is, what sort of collection
of branes do we live on? A stack of 3-branes? The intersection of some
higher-dimensional branes? A brane stuck on some kind of orbifold
singularity? Are there other branes close enough to affect us
gravitationally?
One of the coolest possibilities I've seen recently is that we live on a
"brane crystal":
Steven Corley and David Lowe, "Solving the Hierarchy Problem with Brane
Crystals", hep-ph/0101021.
(But maybe I'm biased, because I'm working on string networks. There was a
nice paper out recently on this subject: hep-th/0101080. :))
--
Paul Shocklee
Graduate Student, Department of Physics, Princeton University
Researcher, Science Institute, Dunhaga 3, 107 Reykjavík, Iceland
Phone: +354-525-4429