Retardation from Lorentz Transformation

To obtain the interaction energy in IS we Lorentz transform the coordinates and the 4-vector (cp2,A2)- With the Lorentz transformation A v) — as given by Eq. (3.81) — of the coordinates from IS to IS we obtain the space-like coordinate r from the space-time coordinate (ct,r) in IS as 

Consequently, a distance measurement which yields r — r at time t in IS requires in IS the space and time coordinates (cf, r) of the point of interest as well as those of charge 2, i.e., (cf2, ri)-If we choose r = r, we can evaluate the scalar potential at the position of the first particle. Substituting the distance vector of Eq. (3.210) into Eq. (3.206) we finally obtain 

the positions of both particles at r j and r 2 in IS connect two events in IS at different, but connected times fi and f2- While, by construction, we may choose fi arbitrarily, t2 is completely determined by the time fi2 = fi — 2 needed to transmit the interaction from charge 2 to charge 1 with the speed of light. The scalar potential therefore contains a retardation term, which arises here naturally through the Lorentz transformation. Hence, the correct relativistic treatment of the interaction problem automatically introduces the retardation effect. 

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The Lienard-Wiechert potentials (12) can also be derived from a rotation-free Lorentz transformation (boost) of the four potential of a static charge (13) to the moving frame at retarded time. For a charge moving at constant velocity the potentials can also be expressed in terms of the current position giving [21]... 

It is important to realise that the Lorentz transformation describes accurately both the relativistic effects which are significant because of the high velocity of the electron and also the retardation effects which occur because of the finite time the field takes to reach the field point from the charge (which is a non-relativistic effect). 

Breit constructed a many-electron relativistic theory that takes into aceount sueh a retarded potential in an approximate way. Breit explicitly considered only the electrons of an atom its nucleus (similar to the Dirac theory) created only an external field for the electrons. This ambitious project was only partly successful because the resulting theory turned out to be approximate not only from the point of view of quantum theory (wifli some interactions not taken into account), but also from the point of view of relativity theory (an approximate Lorentz transformation invariance). 

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