Magnetic Galilean limit of the pre-Maxwell equations

The magnetic nonrelativistic limit neglects displacement current and uses ${\partial }_t\mathbf{B}=-\nabla \times \mathbf{E}$, $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$, $\nabla \cdot \mathbf{B}=0$. Under a Galilean transformation ${\mathbf{x}}^{\prime }=\mathbf{x}-\mathbf{v}t$, $t^{\prime }=t$, its consistent field transformations are ${\mathbf{B}}^{\prime }=\mathbf{B}$, ${\mathbf{J}}^{\prime }=\mathbf{J}$ and ${\mathbf{E}}^{\prime }=\mathbf{E}+\mathbf{v}\times \mathbf{B}$. Indeed ${\partial }_{t^{\prime }}={\partial }_t+\mathbf{v}\cdot \nabla$ and $\nabla \times (\mathbf{v}\times \mathbf{B})=-(\mathbf{v}\cdot \nabla )\mathbf{B}$, giving covariance of the induction equation. The full charge-current transformation has ${\mathbf{J}}^{\prime }=\mathbf{J}-{\rho }_e\mathbf{v}$; neglecting its convective correction belongs to the magnetic quasineutral approximation, not exact relativistic electrodynamics. With ${\mathbf{u}}^{\prime }=\mathbf{u}-\mathbf{v}$, the combination $\mathbf{E}+\mathbf{u}\times \mathbf{B}$ is invariant. Vanishing electric field in the fluid rest frame therefore gives the ideal magnetohydrodynamic induction equation.