I just wanted to check that I am thinking about the coordinate transition correctly. The relativistic generalization of Euler's equation is (from Landau & Lifshitz vol. 6)(adsbygoogle = window.adsbygoogle || []).push({});

## hu^\nu \frac{\partial u_\mu}{\partial x^\nu} - \frac{\partial P}{\partial x^\mu} + u_\mu u^\nu \frac{\partial P}{\partial x^\nu} = 0 ##.

where ## h ## is the enthalpy, ## P ## is the fluid pressure, and ## u^\mu ## are components of the four-velocity. The Greek indices range from 0 to 4. To get this same equation in spherical coordinates (or any new, barred coordinates), I can just bar all of the indices and replace the derivative of ## u_{\overline{\mu}} ## with a covariant derivative, yes? In other words, can I simply write

## hu^{\overline{\nu}} u_{\overline{\mu}; \overline{\nu}} - \frac{\partial P}{\partial x^{\overline{\mu}}} + u_{\overline{\mu}} u^{\overline{\nu}} \frac{\partial P}{\partial x^{\overline{\nu}}} = 0 ##,

where the semicolon represents covariant differentiation?

The reason I am a little worried is that, to get to the relativistic Euler equation in the first place, the four-divergence of the energy-momentum tensor was used. I am wondering whether I have to go back and take the covariant four-divergence of the energy-momentum tensor expressed in barred coordinates.

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# Relativistic Euler equation in spherical coordinates

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