Can extrinsic curvature of 4D spacetime in some $m$-dimensional spacetime ($m>4$) be used to derive GR equations in the original 4D spacetime?
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asked May 1, 2020 at 18:38
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Almost certainly, but I don't think it is known how many dimensions would be necessary for the most general case. The Nash embedding theorem states that this can always be done for a Riemannian manifold. Spacetime is pseudo-Riemannian, which introduces an additional complication. For spacetime, you typically need six dimensions (I believe 4 spacelike and two timelike), although it is said that for some FLRW cosmologies you can use only 5. However, it is my understanding that these embeddings apply for particular solutions, such as exterior Schwarzschild geometry, which are necessarily simpler than the actual geometry of spacetime (e.g. a FLRW solution assumes perfect homogeneity and isometry, which is not true in practice). In addition, there are possibilities for solutions which are clearly not physical, containing, for example arbitrary wormholes and strange topologies, making the question very obscure.
Some of the key papers
1 A. Friedman, Rev. Mod. Phys. 37 (1965) 201 [2] J. Rosen, Rev. Mod. Phys. 37 (1965) 204 [3] R. Penrose, Rev. Mod. Phys. 37 (1965) 215
answered May 1, 2020 at 19:06
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$\begingroup$ Is more than 1 time dimension neccessary? $\endgroup$ Commented May 1, 2020 at 19:10
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$\begingroup$ Yes, typically two time dimensions are needed. $\endgroup$ Commented May 1, 2020 at 19:11