The Lorentz Gauge

The Lorentz gauge, for a variety of reasons, is in my opinion the ``natural'' gauge of electrodynamics. For one thing, it is elegant in four dimensional space-time, and we are gradually working towards the epiphany that we should have formulated all of physics in four dimensional space-time from the beginning, even if we're considering non-relativistic phenomena. Working in it, most problems are relatively tractible if not actually easy. We will therefore consider it first.

Above we derived from MEs and their definitions the two equations of motion for the potentials $\phi$ and $\mbox{\boldmath$A$}$:

$$\nabla^2\phi + \frac{\partial (\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$})}{\partial t} = - \frac{\rho}{\epsilon_0 }$$ (8.45)

$$\nabla^2\mbox{\boldmath$A$}+ - \frac{1}{c^2}\frac{\partial^2 \mbox{\boldmath$A$}}{\partial t^2} = - \mu_0 \mbox{\boldmath$J$} + \mbox{\boldmath$\nabla$}\left (\mb... ...\cdot \mbox{\boldmath$A$}+ \frac{1}{c^2}\frac{\partial \phi}{\partial t}\right)$$ (8.46)

If we can guarantee that we can always find a gauge transformation from a given solution to these equations of motion, $\phi_0,\mbox{\boldmath$A$}_0$, a new one such that new $\phi,\mbox{\boldmath$A$}$ such that the new ones satisfy the constraint (the Lorentz gauge condition):

$$\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$}+ \frac{1}{c^2}\frac{\partial \phi}{\partial t} = 0$$ (8.47)

then the two equations of motion both became the inhomogeneous wave equation for potential waves that propagate at the speed of light into or out of the charge-current source inhomogeneities. This precisely corresponds to our intuition of what should be happening, is elegant, symmetric, and so on. Later we'll see how beautifully symmetric it really is.

We must, however, prove that such a gauge condition actually exists. We propose:

$$\phi = \phi_0 - \frac{\partial \Lambda}{\partial t}$$ (8.48)

$$\mbox{\boldmath$A$} = \mbox{\boldmath$A$}_0 + \mbox{\boldmath$\nabla$}\Lambda$$ (8.49)

and substitute it into the desired gauge condition:

$$\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$}+ \frac{1}{c^2}\frac{\partial \phi}{\partial t} = \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$}_0 + \nabla^2\La... ...tial \phi_0}{\partial t} - \frac{1}{c^2}\frac{\partial^2 \Lambda}{\partial t^2}$$

$$= 0$$ (8.50)

or

$$\nabla^2\Lambda - \frac{1}{c^2} \frac{\partial^2 \Lambda}{\p... ...\frac{\partial \phi_0}{\partial t} = f(\mbox{\boldmath$x$},t)$$ (8.51)

for some computable inhomogeneous sourcevfunction $f(\mbox{\boldmath$x$},t)$.

This equation is solvable for an enormous range of possible $f(\mbox{\boldmath$x$},t)$s (basically, all well-behaved functions will lead to solutions, with issues associated with their support or possible singularities) so it seems at the verly least ``likely'' that such a gauge transformation always exists for reasonable/physical charge-current distributions.

Interestingly, the gauge function $\Lambda$ that permits the Lorentz condition to be satisfied so that $\phi,\mbox{\boldmath$A$}$ satisfy wave equations is itself the solution to a wave equation! It is also interesting to note that there is additional gauge freedom within the Lorentz gauge. For example, if one's original solution $\phi_0,\mbox{\boldmath$A$}_0$ itself satisfied the Lorentz gauge condition, then a gauge transformation to $\phi,\mbox{\boldmath$A$}$ where $\Lambda$ is any free scalar wave:

$$\phi = \phi_0 - \frac{\partial \Lambda}{\partial t}$$ (8.52)

$$\mbox{\boldmath$A$} = \mbox{\boldmath$A$}_0 + \mbox{\boldmath$\nabla$}\Lambda$$ (8.53)

$$\nabla^2\Lambda - \frac{1}{c^2} \frac{\partial^2 \Lambda}{\partial t^2} = 0$$ (8.54)

continues to satisfy the Lorentz gauge condition. Not only are we nearly guaranteed that solutions that satisfy the Lorentz gauge condition exist, we have discovered an infinity of them, connected by a restricted gauge transformation.

In the Lorentz gauge, then, everything is a wave. The scalar and vector potentials, the derived fields, and the scalar gauge fields all satisfy wave equations. The result is independent of coordinates, formulates beautifully in special relativity, and exhibits (as we will see) the causal propagation of the fields or potentials at the speed of light.

The other gauge we must learn is not so pretty. In fact, it is really pretty ugly! However, it is still useful and so we must learn it. At the very least, it has a few important things to teach us as we work out the fields in the gauge.