The Relativistic Potential from a Moving Charge

In developing a relativistic quantum chemistry we will need to describe a system of interacting particles, all of which have charge, and most of which are assumed to 

We start with the potential set up by a moving charge. Having established that A = (a, (p) is a four-vector, we expect it to transform in analogy with the position four-vector, and the Lorentz transformation of (2.17) should apply if we replace r with A and t with 0/c. More specifically— if S is the stationary frame and S is moving along the x axis with velocity v relative to S—we have the transformation equations 

From this we can deduce the relativistic potential set up by a charge q moving at velocity u. We let S be the observer frame, and place the x axis in the direction of u. In a charge-centered frame S, the potential is the same as from a stationary charge, 

From the Lorentz transformations we get the potential in the stationary frame S  

This signal has traveled some distance before reaching our measuring apparatus, and was therefore emitted at 

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