Embedding manifolds that are not very flat.

[Spinnor]

I think I read on these forums that a small piece of relatively flat 4D spacetime of General Relativity can be embedded in ten dimensions. What happens if the small piece of spacetime is not very flat, does this change the number of embedding dimensions required?

Thanks for any help!

 

[wofsy]

I think I read on these forums that a small piece of relatively flat 4D spacetime of General Relativity can be embedded in ten dimensions. What happens if the small piece of spacetime is not very flat, does this change the number of embedding dimensions required?

Thanks for any help!

I am not sure what you mean by embed since space time has a metric of siganture -1. Can you elaborate?

 

[Spinnor]

I am not sure what you mean by embed since space time has a metric of siganture -1. Can you elaborate?

Thanks, I'll try. Leave time out. We can embed a 2 dimensional sphere in 3 dimensions. We can embed a 3 dimensional sphere in 4 dimensions. If we have some small piece of some arbitrary 3 dimensional curved manifold my guess is we need a flat space of at most 6 dimensions to properly map the small curved 3 dimensional piece, is that close?

Thanks for any help!

 

[wofsy]

Thanks, I'll try. Leave time out. We can embed a 2 dimensional sphere in 3 dimensions. We can embed a 3 dimensional sphere in 4 dimensions. If we have some small piece of some arbitrary 3 dimensional curved manifold my guess is we need a flat space of at most 6 dimensions to properly map the small curved 3 dimensional piece, is that close?

Thanks for any help!

Space like sections of Minkowski space are flat. They are isometric to flat Euclidean three space. If curvature is added then there is a general isometric embedding theorem of Nash for compact manifolds. For GR there may be constraints that lower the Nash lower bound but I have no idea. My naive sense is that space is only slightly not flat and then only in the near vicinity of large masses.