A Fluid Flow Symmetry: Showing $\pi_b G^b$ is Constant

ergospherical
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I don't know where to start with this problem. If ##\pi_a = (\mu + TS) u_a## then show that \begin{align*}
u^a \nabla_{a} (\pi_b G^b) = 0
\end{align*}where the field ##G^a## is a symmetry generator. [##S## is entropy/baryon, ##T## is temperature, ##u_a## is a one-form field corresponding to a fluid-comoving observer and ##\mu## is chemical potential].
 
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What have you tried? Killing's equation?
 
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How would I use Killing's equation here? The only thing I wrote down so far was ##L_{G} \pi = 0##, the statement of the invariance of ##\pi## under the symmetry transformation corresponding to the Killing vector ##G##, but I don't know how to obtain the target equation.
 
Did you expand out the left hand side? I think Killing’s equation will kill off some terms involving \nabla_a G_b. What remains may vanish due to other properties of, say, u^a that are implicit in your problem statement.
 
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