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Obviously it is not needed to embed any solution of Einstein's equations in a higher dimensional flat space, although it can be done.

If we had a theory that space-time was a curved 4D manifold in a flat 5D spacetime, how would we test this theory?

This would imply, I believe: $g_{\mu\nu}(x) \equiv \partial_\mu X^N(x) \partial_\nu X^N(x)$ for N=1..5.

What effects could we test for? Are there some solutions of Einsteins equations that could only be embedded in 6D and hence the prediction would be these can't exist?

What other effects might we observe?

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  • $\begingroup$ The Whitney embedding theorem and its extensions say that not every compact orientable 4-D manifold can be embedded in $\mathbb{R}^5$; for at least some 4D manifolds, $\mathbb{R}^7$ is required. But (a) spacetime is not usually assumed to be compact, and (b) this is a topological property of manifolds, and requiring that the 4-D metric is induced by the embedding would introduce additional constraints. $\endgroup$
    – Michael Seifert
    Commented Oct 25, 2018 at 14:09
  • $\begingroup$ @MichaelSeifert: I think it's even worse for non-compact manifolds, isn't it? $\endgroup$
    – user107153
    Commented Oct 25, 2018 at 14:15
  • $\begingroup$ @tfb: I'm not sure about non-compact manifolds; any further references would be most appreciated. $\endgroup$
    – Michael Seifert
    Commented Oct 25, 2018 at 16:25
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    $\begingroup$ Also, note that once you have a smooth embedding of the manifold into $\mathbb{R}^N$ for which the distances between points are not increased (i.e., a nonexpanding map, the Nash embedding theorem guarantees the existence of a map that's arbitrarily close to isometric. So the metric condition isn't as big of an impediment as I originally thought. $\endgroup$
    – Michael Seifert
    Commented Oct 25, 2018 at 16:27
  • $\begingroup$ Also you only need to consider Einsteinian Manifolds. But apart from that interlude, i was really interested about physical experiments. $\endgroup$
    – user84158
    Commented Oct 27, 2018 at 21:30

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