= Horizontal virial balance of a cold magnetized fluid
{title2=$\ddot I_\perp/2=\int[B_z^2/\mu_0-(\partial_z\Phi)^2/(4\pi G)]\,dV$}
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.
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