Fluids
Gravitation box 22.2, p. 564:
Conservative form of Euler fluid equations in 3D:
$\rho_{,t} + (\rho v^j)_{,j} = 0$
$(\rho v^i)_{,t} + (\rho v^i v^j + \delta^{ij} P)_{,j} = 0$
Conservative form of Euler fluid equations in 4D:
$(\rho + P) {u^\mu}_{;\nu} u^\nu + (g^{\mu\nu} + u^\mu u^\nu) P_{;\nu} = 0$
...in flat space:
$(\rho + P) {u^\mu}_{,\nu} u^\nu + (\eta^{\mu\nu} + u^\mu u^\nu) P_{,\nu} = 0$
...splitting apart space and time:
$(\rho + P) ({u^t}_{,t} u^t + {u^t}_{,j} u^j)
+ (\eta^{t t} + u^t u^t) P_{,t}
+ (\eta^{t j} + u^t u^j) P_{,j} = 0$
$(\rho + P) ({u^i}_{,t} u^t + {u^i}_{,j} u^j)
+ (\eta^{i t} + u^i u^t) P_{,t}
+ (\eta^{i j} + u^i u^j) P_{,j} = 0$
$(\rho + P) ({u^t}_{,t} u^t + {u^t}_{,j} u^j)
+ (-1 + u^t u^t) P_{,t}
+ u^t u^j P_{,j} = 0$
$(\rho + P) ({u^i}_{,t} u^t + {u^i}_{,j} u^j)
+ u^i u^t P_{,t}
+ (\delta^{i j} + u^i u^j) P_{,j} = 0$
...low velocities: $0 \approx u^i$, so $-1 = u^\mu u_\mu = -(u^t)^2 + (u^i)^2 \approx -(u^t)^2$ means $u^t \approx \pm 1$. I'll use $u^t = 1$
$ u^j P_{,j} = 0$
$(\rho + P) ({u^i}_{,t} + {u^i}_{,j} u^j) + u^i P_{,t} + (\delta^{i j} + u^i u^j) P_{,j} = 0$