Derivation of the Tolman-Oppenheimer-Volkoff equation

[maverick280857]

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

$$\nabla_\mu T^{\mu\nu} = 0$$

to

$$(\rho + p)\frac{d\alpha}{dr} = -\frac{dp}{dr}$$

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 $sin^2 \theta$ 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!

 

[PeterDonis]

when I try to derive it through the condition that the energy-momentum tensor is covariantly conserved, I get terms involving $sin^2 \theta$

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.