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

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 ${\partial }_{\mu }{T}^{\mu 0}=0$ and time independence giveThus $\boxed{x{\partial }_y\rho -y{\partial }_x\rho =0}$: the density is invariant under rotations about the z-axis.