Curved space time has 4 dimensions. Mathematically, such a 4-dimensional space embeds in a flat space of no more than 10 dimensions. To what extent do observations or theories suggest answers to the following questions? What is the minimum number of dimensions needed for a flat space that embeds space time? Of those dimensions, how many are space-like? How many are time-like?
This has to be an appearance of the "law of series" :-))) (in the meaning of http://www.scholarpedia.org/article/Law_of_series) - I just answered another question by saying that the answer is book-length, and likely a bit weirder than one'd think .... and as such can hardly be done justice to in this forum's format - which also applies here: the answer is book length
Hi Thomas, i'm pretty sure that the question you're asking depends strongly on the topology of the space time. I mean, the 4-sphere embeds canonically into eclidian 5-space, but, you cannot say the same for, let's say, the 4-torus, right. Also, i guess that you are thinking in a sort of isometric embedding rather than just a topological embedding. In that case, the question not only depends on the topology of the base maniflold, maybe it will depend ond the particular choice of metric. As far as i know, there's a beautifull theorem due to Nash that's the riemannian analogue of Whitney's embedding theorem that gives, for every compact smooth manifld X , a lower bound on the dimension that euclidian flat space should have in order to contain a smooth copy of X. But, i dont know if there's a version of Nash's theorem written for lorentizian manifolds.
The paragraph above was just formal stuff, but your are also asking for a physical point of view. I tend to think that almost all the members of the physics community agree that space-time is an absolute concept. Of course, string theorist are the exception. But recall thar the great discovery of Gauss was that makes sence to talk about intrinsic geometry of manifolds (the original statement was for surfaces, which were the objects of study back then). Mathematically, there's no reason to think space time as a subset of some bigger ambient space. Also, if space time is a sub-manifold of something "bigger", why this ambient space should be euclidian space?
Raul: Thanks. More generally, I am trying to determine the extent to which some of my attempted research (see attachment, if you are interested) may be useful.
Key parts of the research describe something I call 'quantum interaction space.' Some of math seems to have some roots in a perhaps 10-dimensional Minkowski-metric energy-momentum space and in a perhaps 10-dimensional Minkowski-metric space time. (I suppose something similar could be said about usual expressions for plane-waves - but here the numbers would be 4 and 4.)
A key step in some of the work involves apparently assuming (regarding traditional space time, for some purposes) 3 spatial dimensions. So, there is 'no quarrel' with some usual interpretations of space time. My question is more about math than space time itself. Perhaps a characterization of the question could be 'How much curvy-ness?' (This observation seems to point back to thoughts in the first paragraph of your note. Yet, work in the research is more about numbers of harmonic oscillators needed to represent various [interaction-related] phenomena than about dimensions. So, a point from your last sentence above may provide a caution regarding the possible usefulness of thinking that led to my question.)
Appreciated.
Chris, Raul, Robert:
Thanks. Your answers are helping me think that answering this question may not be as important (at least to my attempted research ) as I once thought answering it would have been.
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