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Derivation of the Tolman-Oppenheimer-Volkoff equation

  1. May 11, 2014 #1
    Hi,

    I am working through Section 5.8 of Sean Carroll's book on GR. Does someone know where I can find the bridging steps that take me from

    [tex]\nabla_\mu T^{\mu\nu} = 0[/tex]

    to

    [tex](\rho + p)\frac{d\alpha}{dr} = -\frac{dp}{dr}[/tex]

    This is equation 5.153, and when I try to derive it through the condition that the energy-momentum tensor is covariantly conserved, I get terms involving [itex]sin^2 \theta[/itex] which make no sense because the solution is spherically symmetric.

    I couldn't find the bridging steps that lead to equation 5.153 anywhere, and I tried using the Bianchi identity to get something but that doesn't help for some reason. Is there some clever mathematical manipulation that I'm missing?

    Thanks in advance!
     
  2. jcsd
  3. May 11, 2014 #2

    PeterDonis

    User Avatar
    2016 Award

    Staff: Mentor

    Can you post more details about the derivation you've tried? That will make it a lot easier to give feedback about what you may be missing.
     
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