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We have the metric ds^2=-e^{2\Phi}dt^2+e^{2\Lambda}dr^2+r^2d\Omega^2, and the energy momentum tensor takes the form t^{ab}=(\rho+p)u^au^b+pg^{ab} where the 4-velocity is u=e^{-\Phi}\partial_t, and \Phi and \Lambda are functions of r only.
I'm asked to show that the ebergy-momentum conservation equation \nabla_aT^{ab}=0 is equivalent to (\rho+p)\frac{d\Phi}{dr}=-\frac{dp}{dr}.
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
I'm asked to show that the ebergy-momentum conservation equation \nabla_aT^{ab}=0 is equivalent to (\rho+p)\frac{d\Phi}{dr}=-\frac{dp}{dr}.
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