I took General Relativity at university, years ago and have a question that has recently occurred to me. This might be a dumb one, so I apologise if the answer is a well known negative.
Mathematically, it seems to me that our curved spacetime must be mathematically embedded in a flat space of 5D+. I suspect that spacetime satisfies the requirement of a mathematical manifold, and the Witney embedding theorems would imply the existence of such an embedding. The analogy would be like a curved 2D sheet of paper can be thought of as being embedded in flat 3D Euclidean space.
Furthermore there are the Nash embedding theorems. I am not too familiar with these, but I think they may be a stronger statement that applies to both Riemann manifolds and Pseudo Reimann manifolds. I think the Nash theorem may imply that, given a manifold (or pseudo manifold) with a metric, then a higher dimensional flat space exist which has a metric compatible with the metric of the curved space; inside which the curved space is embedded. So for instance, if spacetime is a 4D pseudo-Reimann manifold of signature (-1, 1, 1, 1), then a flat space of at least 5D exist, which will contain our 4D spacetime, and which will have a metric that is consistent with (-1, 1, 1, 1) when applied locally to our 4D spacetime. This is in analogy to the fact that metric in 3D space of signature (1, 1, 1) clearly is consistent (vi projection) with the metric of (1, 1) on the tangent plane of a point on the curved sheet of paper.
Have I got this totally wrong, or is this a possibility?
Please note: I am asking about the technicality of a compatible metric. I have already seen the link, below, and it does not answer my question