The Lorentz Condition

We are now in a position to see what the Lorentz condition means. For an isolated stationary charge, the scalar potential is given by equation (16.1) and the vector potential $\mbox{\bf A}$ is zero. The Lorentz condition reduces to

$$\frac{1}{c^2} \frac{\partial \phi}{\partial t} = \frac{1}{4 \pi \epsilon_0 r c^2} \frac{dq}{dt} = 0 .$$ (17.31)

From this we see that the Lorentz condition applied to the four-potential for a point charge is equivalent to the statement that the charge on a point particle is conserved, i. e., it doesn't change with time. This is extended to any stationary distribution of charge by the superposition principle.

We thus see that the Lorentz condition is a consequence of charge conservation for the four-potential of any charge distribution in the reference frame in which the charge is stationary. If we can further show that the Lorentz condition is an equation which is equally valid in all reference frames, then we will have demonstrated that it is true for the four-potential produced by moving charged particles as well.

If the Lorentz condition is valid in one reference frame, it is valid in all frames for the special case of a plane electromagnetic wave. This follows from substituting the four-potential for a plane wave into the Lorentz condition, as was done in equation (16.27) in the previous section. In this case the Lorentz condition reduces to $\underline{k} \cdot \underline{a} = 0$. Since the dot product of two four-vectors is a relativistic scalar, the Lorentz condition is equally valid in all frames. Since we believe that charge is indeed conserved in all circumstances, the Lorentz condition must always be satisfied.