General Relativity: Manifold/Sub-Manifold Metric Theorem Q-Schwarzschild

[binbagsss]

I'm looking at Lecture Notes on General Relativity, Sean M. Carroll, 1997.

I don't understand eq 7.4 from the theorem 7.2. As I understand, theorem 7.2 is used when you have submanifold that foilate the manifold, and the submanifold must be maximally symmetric.

I know that 2-spheres are maximally symmetric, and foliate a spherically symmetric 3-space .
I don't understand the introduction of 2 coordinates, i.e. applying the therem to 4-d space-time, as the submanifolds only foliate 3-space, I thought the application of the theoremm would be limited to 3-space?

Also does thsi theorem have a name? So i can look it up elsewhere?

Thanks in advance.

 

[Orodruin]

When you ask questions like this, please do not assume that everyone is sitting with a copy of Carrol's lecture notes in front of them. Take the time to write out the equations you are referring to, I guarantee that this will net you more responses. At the very least, provide a link so that people do not have to find it themselves.

 

[binbagsss]

Apologies, link here: http://arxiv.org/pdf/gr-qc/9712019.pdf

 

[binbagsss]

bump

 

[PeterDonis]

I know that 2-spheres are maximally symmetric, and foliate a spherically symmetric 3-space .

If they foliate a single spherically symmetric 3-space, they also foliate a 4-d spacetime composed of spacelike slices each of which is a spherically symmetric 3-space.