To first order in the angular velocity, $T^{00}=\rho$ and $T^{i0}=\rho v^i$ with $v=(-\Omega y,\Omega x,0)$. The conservation equation $\partial_\mu T^{\mu0}=0$ and time independence give
$$
0=\nabla\mathbin\cdot(\rho\mathbf v)
=\Omega(x\partial_y\rho-y\partial_x\rho).
$$
Thus $\boxed{x\partial_y\rho-y\partial_x\rho=0}$: the density is invariant under rotations about the z-axis.


Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 3 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-309/3/b/i/solution