Transform of the Electrostatic Potential

Transform of the Electrostatic Potential and the 4-Potential

The electrostatic potential is an energy, which like the mass can be related to a frequency via Planck’s constant:

Like the rest mass energy/frequency, this will transform under the Lorentz Transform when observed from a moving reference frame.  Thus in the rest frame of the source charge we have an energy-momentum 4-vector:

We could naively transform to an electromagnetic potential energy-momentum 4-vector in a moving frame as:

  

This is approximately correct if the source and test chargers are stationary (or moving slowly relative to the speed of light).  Later we will derive the relativistically correct expression, the Lienard-Wiechert potentials.  The error is that the connection between the source charge and the test charge is along a null line.  That is, from the point of view of the test charge, the source charge is not only a distance r away, but it is also a distance c delta_t in the past. For now we will work with these expressions.

In the equation above, A is the conventional magnetic vector potential.  Now an important issue is what is beta'?  To answer this question we need to understand where the energy represented by the potential is located.

There are three obvious options here:

       1.  In the electric field between q_s and q_t

       2.  At q_s

       3.  At q_t

The last choice really makes no sense as there would be no chance of getting magnetic field effects (from A).  The first choice appears consistent with conventional electromagnetism, but in this theory, working with potentials, it gives the wrong result!  This is something of fundamental importance we will have to discuss later.  The problem is that the electric field density is centered in the center of charge frame, and thus does not move as fast as the charges themselves.  This gives a factor of two error in the velocity.  What does work is option 2, assuming the potential energy is located at the source charge.  We will make this assumption and show that it gives the correct electromagnetic forces.

Note that the spatial (momentum) components here are the classical magnetic vector potential multiplied by the test charge.  If we divide out the test charge we can get a 4-potential:

The current terms are related to the classical magnetic vector potential:

This form is a bit unconventional as it is in energy units.  The conventional electromagnetic 4-potential is in units of the magnetic vector potential:

The reader should understand that if the roles of source and test charge were exchanged, then from the point of the new test charge the energy of the interaction is located at the new source charge.