In Jackson we see equation 6.30 for the vector potential A. Suppose the transverse current is zero since Jackson says the Coulomb gauge is often used in this circumstance.

Consider two observers O and O' moving with constant relativity V with respect to each other. Each observer measures the vector potential A at the point P, (x,y,z,t) for observer O and (x',y',z',t') for observer O'. They are firm believers in Newtonian mechanics and the Galilean transformation used in Newtonian physics.

Say observer O sees the vector potential behave as per equation 6.30. He will use the Galilean transformation to predict what the correct equation should be for his colleague O'. However, O' rejects the equation from O and insists that 6.30 is valid as written in his reference frame.

What is needed so that equation 6.30 is valid for both these observers? Simplify the problem to one dimension in the X direction assuming V = Vx.

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