I am reading a paper in which a perfect fluid in the Kerr geometry is studied. The stress-energy tensor is

\begin{equation} T_{\mu \nu} = (\epsilon + P) u_\mu u_\nu + P g_{\mu \nu}, \end{equation}

where $P$ is the fluid pressure and $\epsilon$ the fluid proper energy density. The four-velocity is $u^\mu = u^t (1,0,0,\Omega)$ (circular motion).

They say that in this case, the energy-momentum conservation equation, $\nabla_\nu T^\nu_\mu = 0$, takes the form

\begin{equation} \frac{\nabla_\mu P}{\epsilon + P} = - \nabla_\mu \ln(-u_t) + \frac{\Omega \nabla_\mu \mathcal{l}}{1-\Omega \mathcal{l}}, \end{equation}

with $l \equiv -u_\phi/u_t$, and $\Omega \equiv u^\phi / u^t $.

So, my questions are

  1. Which are the intermediate steps to reach the second form for the energy-momentum conservation equation?

  2. Why the derivatives of the scalars $P$ and $l$ are kept in a covariant-form?

Thank you very much!


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