Question about pulsating charged sphere

Question

This is in reference to the pulsating sphere question shown here:

A charged sphere with pulsing radius

It was said that:

"Any change in the rate of increase of radius of the sphere does not lead to a change in the electric field at any point farther from the sphere than the maximum radius the sphere can attain while it is pulsating. So the electric field at all points whose distance from the centre of the sphere is larger that the maximum radius that can be attained by the pulsating sphere is constant over time. So beyond the maximum attainable radius there is no electromagnetic radiation."

by the top answer.

Can someone please explain this more? Also, doesn't Guass's law apply to static objects?

I am confused about these aspects because in Feyman's lectures book 1, equation 28.3 (http://www.feynmanlectures.caltech.edu/I_28.html) states that a moving charge's induced electric field at some point is dependent on its acceleration, and velocity. Can someone please explain how this equation is in effect in the pulsating sphere example?

Thanks in advance

Answer 1

There are several ways to understand this. One of my favorites is via symmetry. The pulsating charged sphere and the electric field around it are spherically symmetric. Outside the sphere, the electric field varies as Q/r^2. Because the charge Q on the sphere does not change, the field outside the sphere does not change; and no change means no radiation.

For the region between the maximum and minimum radii, symmetry tells us that any radiation emitted could not have any preferred direction other than radial, and it cannot be chiral, without changing the original symmetry. But electromagnetic radiation can only be transverse, not longitudinal (i.e., in the radial direction). That leaves no degrees of freedom for radiation, and therefore no radiation.