Horizontal virial balance of a cold magnetized fluid (${\ddot{I}}_{\perp }/2=\int [B_z^2/{\mu }_0-({\partial }_z\Phi {)}^2/(4\pi G)]dV$)

At an initially resting instant with negligible pressure, sum the horizontal components of the magnetized-fluid tensor virial theorem. The horizontal magnetic trace is $-B_z^2/{\mu }_0$ and the gravitational force-stress trace is $({\partial }_z\Phi {)}^2/(4\pi G)$. Their negatives give the displayed radial second-moment balance. Vacuum field stresses contribute too; being initially at rest does not imply zero acceleration.