Why is there no aberation of a static electric field, but there is aberation for EM radiation?

Question

So consider the case where there is a particle and a moving telescope:

If the particle is located at position 1, and the telescope is moving horizontally as shown, the EM radiation appears to be coming from position 2, from the telescopes perspective.

Let us now think of the object 1 as being an electron, producing a static electric field. Now, the telescope will detect the electric field to indeed be at position 1, despite travelling at the horizontal velocity.

Why is there such a difference between EM and electric fields.

Answer 1

It is true that the voltage ($V$), or electric potential does obey the wave equation (Lorenz gauge) $$\frac{\partial^2V}{\partial t^2} - c^2\nabla^2V = \eta_0 c^3 \rho$$ And so the potential is retarded, just as you would expect. So, why doesn't the gradient of the voltage exhibit aberration? Well, except the electric field isn't just the gradient of voltage, but also includes the time derivative of the magnetic potential $$\vec{E} = -\vec{\nabla} V - \frac{\partial \vec{A}}{\partial t}$$ and this potential also obeys the same wave equation $$\frac{\partial^2\vec{A}}{\partial t^2} - c^2\nabla^2\vec{A} = \eta_0 c \vec{J}$$ The source is moving (from the telescope's frame of reference) and so produces a magnetic potential as well. The change in this magnetic potential as the source goes by adds to the electric field, so that the total electric field points to position 1.

This only applies to an unaccelerated source, mind you.

Answer 2

I didn't understood your example very well, but I believe I understood the question. There is no difference between EM field and electrostatic field as you distinguishes them, physically both are the same thing except that purely electrostatic field, once the source or the observer started to move, it's electric field will became (formally speaking) weaker, and the difference goes to another component of it, the magnetic field, in other words the result of source/observer movement will change electric into magnetic field if the source is electron, this actually is a result of Einsteins theory of relativity.