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I'm trying to work out the logic behind a statement I found in the GR book I'm currently studying. It says that from the conservation equation [tex]\nabla_aT^{ab}=0[/tex], one could deduce the following two equations:

[tex](\varrho+p)\dot{u}^a = \nabla^ap - u^a\dot{p}[/tex]

[tex]\dot{\varrho} + (\varrho+p)\nabla_au^a = 0[/tex]

I've tried calculating [tex]\nabla_aT^{ab}[/tex] from:

[tex]T^{ab}=(\varrho+p)u^au^b - g^{ab}p[/tex]

and the best I could make out of it are the following two lines:

[tex]\nabla_aT^{ab}=(\varrho+p)\nabla_a(u^au^b) + [\nabla_a(\varrho+p)]u^au^b -g^{ab}\nabla_ap =[/tex]

[tex]=(\varrho+p)[u^b\nabla_au^a+u^a\nabla_au^b] + [\nabla_a(\varrho+p)]u^au^b - g^{ab}\nabla_ap[/tex]

Now I'm stuck here, and, what's worse, I'm not even sure these two lines are correct...

Thanks to whoever could provide some light here!

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# Elaborating on stress-energy tensor

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