Editing Relativistic Euler equations (section)

===Derivation===
In order to determine the equations of motion, we take advantage of the following spatial projection tensor condition:
<math display="block">
\partial_{\mu}T^{\mu\nu} + u_{\alpha}u^{\nu}\partial_{\mu}T^{\mu\alpha} = 0
</math>

We prove this by looking at <math>\partial_{\mu}T^{\mu\nu} + u_{\alpha}u^{\nu}\partial_{\mu}T^{\mu\alpha}</math> and then multiplying each side by <math>u_{\nu}</math>. Upon doing this, and noting that <math>u^{\mu}u_{\mu} = -1</math>, we have <math>u_{\nu}\partial_{\mu}T^{\mu\nu} - u_{\alpha}\partial_{\mu}T^{\mu\alpha}</math>. Relabeling the indices <math>\alpha</math> as <math>\nu</math> shows that the two completely cancel. This cancellation is the expected result of contracting a temporal tensor with a spatial tensor.

Now, when we note that
<math display="block">
T^{\mu\nu} = wu^{\mu}u^{\nu} + pg^{\mu\nu}
</math>

where we have implicitly defined that <math>w \equiv e+p</math>, we can calculate that
<math display="block">\begin{align}
    \partial_{\mu} T^{\mu\nu}    & = \left(\partial_{\mu} w\right) u^{\mu} u^{\nu}    + w \left(\partial_{\mu} u^{\mu}\right) u^{\nu}    + wu^{\mu} \partial_{\mu} u^{\nu}    + \partial^{\nu}   p \\[1ex]
    \partial_{\mu} T^{\mu\alpha} & = \left(\partial_{\mu} w\right) u^{\mu} u^{\alpha} + w \left(\partial_{\mu} u^{\mu}\right) u^{\alpha} + wu^{\mu} \partial_{\mu} u^{\alpha} + \partial^{\alpha}p
\end{align}</math>

and thus
<math display="block">
u^{\nu}u_{\alpha}\partial_{\mu}T^{\mu\alpha} = (\partial_{\mu}w)u^{\mu}u^{\nu}u^{\alpha}u_{\alpha} + w(\partial_{\mu}u^{\mu})u^{\nu} u^{\alpha}u_{\alpha} + wu^{\mu}u^{\nu} u_{\alpha}\partial_{\mu}u^{\alpha} + u^{\nu}u_{\alpha}\partial^{\alpha}p
</math>

Then, let's note the fact that <math>u^{\alpha}u_{\alpha} = -1</math> and <math>u^{\alpha}\partial_{\nu}u_{\alpha} = 0</math>. Note that the second identity follows from the first. Under these simplifications, we find that
<math display="block">
u^{\nu}u_{\alpha}\partial_{\mu}T^{\mu\alpha} = -(\partial_{\mu}w)u^{\mu}u^{\nu} - w(\partial_{\mu}u^{\mu})u^{\nu} + u^{\nu}u^{\alpha}\partial_{\alpha}p
</math>

and thus by <math>\partial_{\mu}T^{\mu\nu} + u_{\alpha}u^{\nu}\partial_{\mu}T^{\mu\alpha} = 0</math>, we have
<math display="block">
(\partial_{\mu}w)u^{\mu}u^{\nu} + w(\partial_{\mu}u^{\mu}) u^{\nu} + wu^{\mu}\partial_{\mu}u^{\nu} + \partial^{\nu}p -(\partial_{\mu}w)u^{\mu}u^{\nu} - w(\partial_{\mu}u^{\mu})u^{\nu} + u^{\nu}u^{\alpha}\partial_{\alpha}p = 0
</math>

We have two cancellations, and are thus left with
<math display="block">
(e+p)u^{\mu}\partial_{\mu}u^{\nu} = - \partial^{\nu}p - u^{\nu}u^{\alpha}\partial_{\alpha}p
</math>