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Manifold/sub-manifold metric theorem Q- Schwarzschild metric

  1. Feb 17, 2015 #1
    I'm looking at Lecture Notes on General Relativity, Sean M. Carroll, 1997.

    I dont 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.
     
  2. jcsd
  3. Feb 17, 2015 #2

    Orodruin

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    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.
     
  4. Feb 18, 2015 #3
  5. Feb 19, 2015 #4
  6. Feb 19, 2015 #5

    PeterDonis

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    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.
     
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