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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!

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# Derivation of the Tolman-Oppenheimer-Volkoff equation

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