electromagnetism - No magnetic field from a static charge - Is there a simple physical argument to show why?

For a charge moving in an electric field $\vec E$, its equation of motion is given by the electric part of the Lorentz force

$$\frac d {dt}\gamma m \vec v = e\vec E$$ This comes from the conservation of relativistic energy in a static electric field. But a magnetic field would still make this conservation law true since the magnetic force is always orthogonal to the velocity of the charge and therefore doesn't change its energy.

Is there a simply physical argument that shows why a static charge doesn't create a magnetic field?