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I'm asked to show that the ebergy-momentum conservation equation [tex]\nabla_aT^{ab}=0[/tex] is equivalent to [tex](\rho+p)\frac{d\Phi}{dr}=-\frac{dp}{dr}[/tex].

Now, I'm after some advice on how to calculate this. I know all the connection coefficients (they were given in the question). I tried setting up the four equations, one for each value of b, and then attempting to solve these, but each of these equations has 12 terms in, so I figured there must be an easier way to do this!

Does anyone know whether there is another way, or if there are any simplifications I can make; or do I need to plough through the 4 equations?

Thanks