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General Relativity, identity isotropic, Ricci tensor

  1. Mar 28, 2017 #1
    1. The problem statement, all variables and given/known data

    Attached

    gollygosh.png
    2. Relevant equations


    3. The attempt at a solution

    So the question says 'some point'. So just a single point of space-time to be isotropic is enough for this identity hold?

    I don't quite understand by what is meant by 'these vectors give preferred directions'. Can someone explain this more please? How do the eigenvectors indicate a preferred direction?

    Also, if it is only isotropic about a single point, then at all other points we do expect there to be preferred directions? So don't we expect something like ##R^u_v## evaluated at the isotropic point would specify no preferred directions, and so indeed ##R^u_v=c\delta^u_v## is needed, however at all the other points space is not isotropic, these preferred directions can be manifest?

    What point is ##R^u_v## being evaluated at?

    Thanks in advance
     
  2. jcsd
  3. Apr 1, 2017 #2
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