GR: Signature & Dimension of Embedding Space

[dismachaerus]

Hello. I am not familiar with differential geometry and curvature tensors, yet I am having a great deal of questions to ask.
First when we lay a set of coordinates for an n-dimensional plane, let's say 2 coordinates for a surface embedded in a 4D space the vectors we begin with to describe our surface must have 4 coordinates each, right? So, no matter how we arrive at our curvature Riemann or Ricci tensor the space in which our surface is embedded must reflect in our equations somehow... I mean with an increase in the dimensionality of the embeddind space we must get a different set of equations for each 2D surface say. Or is it that the tensor equations are independent of dimensions of surrounding vector space? How is this possible?
With this in mind, what is the signature of the surrounding Euclidean plane in the EFE near a massive body? Is it (-,+,+,+,+) or (-,+,+,+,-) ? Did Einstein use 5D vectors in his equations to arrive at?
In cosmology do we often use higher than 5D embedding spaces to describe the expanding spacetimes, especially for the flat space expanding universes?
And finally, which book is good for me to start learning GR? Is Wheeler's book Gravitation right for a beginner like me or do i need more basics like Spivak's Comprehensive Introduction To Diff. Geometry?

 

[Dale]

Hi dismachaerus, welcome to PF!

Actually, one of the interesting things about Riemannian geometry is that the embedding space is completely irrelevant. All of the notions of curvature and geometry are formulated purely in terms of the lower dimensional curved space and there is no need to reference any higher dimensional flat embedding space.

For example, the surface of a sphere is a 2D curved manifold which you are probably picturing in your head as being embedded in a flat 3D space. But you can do all of your differential geometry on the surface of the sphere using only the 2D curved coordinates.

 

[dextercioby]

It is not true that the 5 as a dimension exists. One of the most complicated (to me) results in modern mathematics is this one: https://en.wikipedia.org/wiki/Whitney_embedding_theorem. This tells something about an 8, not a 5.