6.4 Lorenz gauge

The Lorenz gauge condition provides a way to fix the gauge freedom of electromagnetic potentials so that both the scalar and vector potentials satisfy symmetric, decoupled wave equations. This makes it the natural gauge choice for time-dependent and relativistic problems, where you need the equations to treat space and time on equal footing.

Lorenz gauge condition

The Lorenz gauge is a constraint imposed on the scalar potential $\phi$ and vector potential $\mathbf{A}$ to simplify Maxwell's equations when written in terms of potentials. It's named after the Danish physicist Ludvig Lorenz (not Lorentz, though the names are often confused). The condition shows up throughout classical electrodynamics and quantum field theory.

Potentials in Lorenz gauge

The Lorenz gauge condition is:

$\nabla \cdot \mathbf{A} + \frac{1}{c^2} \frac{\partial \phi}{\partial t} = 0$

Here $\nabla \cdot \mathbf{A}$ is the divergence of the vector potential, and $\frac{1}{c^2} \frac{\partial \phi}{\partial t}$ is the time derivative of the scalar potential scaled by $1/c^2$. Why does this help? Without any gauge condition, Maxwell's equations for the potentials contain mixed terms coupling $\phi$ and $\mathbf{A}$. The Lorenz condition eliminates those cross-terms, yielding two independent wave equations.

This condition is also Lorentz invariant: it holds in every inertial frame connected by a Lorentz transformation. That's what makes it the preferred gauge for relativistic electrodynamics.

Invariance under gauge transformations

Recall that a gauge transformation replaces the potentials with new ones that produce the same physical fields $\mathbf{E}$ and $\mathbf{B}$:

$\phi \rightarrow \phi - \frac{\partial \Lambda}{\partial t}, \qquad \mathbf{A} \rightarrow \mathbf{A} + \nabla \Lambda$

If you start in the Lorenz gauge and apply such a transformation, the new potentials will still satisfy the Lorenz condition provided the gauge function $\Lambda$ satisfies the homogeneous wave equation:

$\Box \Lambda = 0$

where $\Box = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2$ is the d'Alembertian. So the Lorenz gauge doesn't fix the potentials uniquely; there's a residual gauge freedom parameterized by solutions to $\Box \Lambda = 0$.

Lorenz gauge vs Coulomb gauge

The Coulomb gauge imposes $\nabla \cdot \mathbf{A} = 0$, which makes the scalar potential satisfy Poisson's equation $\nabla^2 \phi = -\rho/\varepsilon_0$. That's convenient for electrostatics and magnetostatics because $\phi$ is determined instantaneously by the charge distribution.

The trade-off: in the Coulomb gauge the equation for $\mathbf{A}$ still contains $\phi$, so the potentials remain coupled. Worse, the instantaneous Poisson equation for $\phi$ is not manifestly Lorentz invariant.

The Lorenz gauge avoids both problems. The potentials decouple, and the equations are Lorentz covariant. For any time-dependent or relativistic problem, the Lorenz gauge is almost always the better choice.

Wave equations in Lorenz gauge

Once the Lorenz condition is imposed, the coupled Maxwell equations for the potentials reduce to two independent inhomogeneous wave equations driven by the sources.

Wave equation for scalar potential

$\Box \phi = -\frac{\rho}{\varepsilon_0}$

The charge density $\rho$ acts as the source for $\phi$. Because this is a wave equation (not Poisson's equation), changes in $\rho$ propagate outward at speed $c$ rather than appearing instantaneously everywhere.

Wave equation for vector potential

$\Box \mathbf{A} = -\mu_0 \mathbf{J}$

The current density $\mathbf{J}$ sources $\mathbf{A}$. The structure is identical to the scalar case, with $\mu_0 \mathbf{J}$ playing the role of $\rho/\varepsilon_0$.

Decoupling of potentials

Notice that the equation for $\phi$ involves only $\rho$, and the equation for $\mathbf{A}$ involves only $\mathbf{J}$. There is no mixing. This is the central payoff of the Lorenz gauge: you can solve for each potential independently using standard wave-equation techniques (Green's functions, Fourier transforms, etc.). In the Coulomb gauge, by contrast, the equation for $\mathbf{A}$ contains $\phi$ explicitly, forcing you to solve a coupled system.

Retarded potentials

The retarded potentials are the physically causal solutions to the wave equations above. They enforce the requirement that effects propagate forward in time at speed $c$.

Green's function for the wave equation

To solve $\Box \psi = -4\pi f$, you need the Green's function satisfying:

$\Box G(\mathbf{r}, t; \mathbf{r}', t') = -4\pi \delta^3(\mathbf{r} - \mathbf{r}') \delta(t - t')$

The retarded Green's function picks out only contributions from the past:

$G_R(\mathbf{r}, t; \mathbf{r}', t') = \frac{\delta\!\left(t - t' - |\mathbf{r} - \mathbf{r}'|/c\right)}{|\mathbf{r} - \mathbf{r}'|}$

The delta function enforces $t' = t - |\mathbf{r} - \mathbf{r}'|/c$, which is exactly the retarded time: the moment in the past when a signal traveling at $c$ would have had to leave $\mathbf{r}'$ in order to arrive at $\mathbf{r}$ at time $t$.

Retarded scalar potential

Convolving the Green's function with the charge density gives:

$\phi(\mathbf{r}, t) = \frac{1}{4\pi\varepsilon_0} \int \frac{\rho(\mathbf{r}', t_r)}{|\mathbf{r} - \mathbf{r}'|} \, d^3r'$

where $t_r = t - |\mathbf{r} - \mathbf{r}'|/c$ is the retarded time. Each volume element contributes to the potential at $\mathbf{r}$ with a time delay proportional to its distance.

Retarded vector potential

The same structure applies to $\mathbf{A}$:

$\mathbf{A}(\mathbf{r}, t) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}', t_r)}{|\mathbf{r} - \mathbf{r}'|} \, d^3r'$

These two expressions together give you the full retarded potentials in the Lorenz gauge. From them you can compute $\mathbf{E}$ and $\mathbf{B}$ for any source distribution.

Lorenz gauge in electrodynamics

The Lorenz gauge fits naturally into the covariant (four-dimensional) formulation of electrodynamics, where space and time are unified into spacetime.

Covariant formulation

The scalar and vector potentials combine into a single four-potential:

$A^\mu = (\phi/c, \, \mathbf{A}), \qquad \mu = 0, 1, 2, 3$

The electric and magnetic fields are encoded in the electromagnetic field tensor, defined as:

$F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$

This antisymmetric tensor contains all six independent components of $\mathbf{E}$ and $\mathbf{B}$.

Four-potential in Lorenz gauge

The Lorenz condition in covariant notation is simply:

$\partial_\mu A^\mu = 0$

where $\partial_\mu = \left(\frac{1}{c}\frac{\partial}{\partial t}, \, \nabla\right)$ is the four-gradient. This is a single scalar equation, manifestly Lorentz invariant.

With this condition, the wave equations for all four components of $A^\mu$ collapse into one covariant equation:

$\Box A^\mu = -\mu_0 J^\mu$

where $J^\mu = (c\rho, \, \mathbf{J})$ is the four-current density. The elegance here is hard to overstate: one equation, one gauge condition, fully relativistic.

Electromagnetic field tensor

The components of $F^{\mu\nu}$ in terms of the physical fields are:

• $F^{0i} = -F^{i0} = E^i/c$ (electric field components)
• $F^{ij} = -\varepsilon^{ijk} B^k$ (magnetic field components)

where $i, j, k = 1, 2, 3$ and $\varepsilon^{ijk}$ is the Levi-Civita symbol. Maxwell's equations then take the covariant forms:

$\partial_\mu F^{\mu\nu} = \mu_0 J^\nu \qquad \text{(inhomogeneous)}$

$\partial_\mu \tilde{F}^{\mu\nu} = 0 \qquad \text{(homogeneous)}$

where $\tilde{F}^{\mu\nu} = \frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ is the dual field tensor. The first equation contains Gauss's law and the Ampère-Maxwell law; the second contains Faraday's law and $\nabla \cdot \mathbf{B} = 0$.

Lorenz gauge applications

Radiation from moving charges

The retarded potentials in the Lorenz gauge are the starting point for computing radiation fields from accelerating charges. Whether you're analyzing synchrotron radiation (charges in circular orbits) or bremsstrahlung (charges decelerating in a medium), you evaluate the retarded integrals and then differentiate to get the fields. The radiated power follows from the Larmor formula (or its relativistic generalization), both of which are derived within this framework.

Liénard-Wiechert potentials

For a single point charge $q$ moving along a trajectory $\mathbf{w}(t)$, the retarded potential integrals can be evaluated exactly. The result is the Liénard-Wiechert potentials, derived independently by Liénard and Wiechert around 1898-1900:

These potentials account for relativistic effects like the compression of field lines in the forward direction of motion. Differentiating them yields the full electric and magnetic fields of an arbitrarily moving point charge, which split into a "velocity field" (falls off as $1/r^2$) and a "radiation field" (falls off as $1/r$, carrying energy to infinity).

Relativistic transformations of fields

Because the Lorenz gauge preserves Lorentz covariance, it provides a clean framework for transforming fields between inertial frames. The field tensor $F^{\mu\nu}$ transforms as a rank-2 tensor under Lorentz boosts, which mixes $\mathbf{E}$ and $\mathbf{B}$ components depending on the relative velocity between frames. A purely electric field in one frame, for example, acquires magnetic components when viewed from a boosted frame. The covariant formulation in the Lorenz gauge makes these transformations straightforward and systematic.