Lorenz Gauge

The Coulomb gauge, last article, spent the gauge freedom asymmetrically: an instantaneous elliptic equation for $V$ , a causal wave equation for $\mathbf{A}$ , and a transverse/longitudinal split doing contortions to keep the observable fields honest. There is another way to spend the same coin — and it is the one this chapter will live in from here on. Demand instead that the divergence of $\mathbf{A}$ track the time derivative of $V$ :

\nabla\cdot\mathbf{A} = -\mu_0\varepsilon_0\,\frac{\partial V}{\partial t}.

One condition, linking both potentials. Feed it into the coupled equations the chapter opener assembled, and the system does not merely simplify — it symmetrizes. The scalar potential and every component of the vector potential obey the same equation: an inhomogeneous wave equation, with $-\rho/\varepsilon_0$ or $-\mu_0\mathbf{J}$ on the right, and nothing else. Four identical causal problems, potentials propagating at $c$ , sources sitting where the physical charge and current actually are. No instantaneous ledger, no nonlocal projections — and, as the relativity chapter will make crisp, the condition reads the same in every inertial frame. The gauge carries a story in its name. It belongs to Ludvig Lorenz, the Dane who wrote retarded potentials in 1867 — but for most of a century the credit drifted to Hendrik Lorentz, the towering Dutchman, and "Lorentz gauge" still infests the literature. This article proves the condition is always attainable, prices the residual freedom that survives it (genuine "gauge waves" — exhibited explicitly, with zero fields), shows the condition maintains itself over time because charge is conserved, and sets the table for the retarded potentials two articles ahead: the payoff that made this gauge the natural home of radiation theory.

Key takeaways

The Lorenz gauge imposes $\nabla\cdot\mathbf{A} = -\mu_0\varepsilon_0\,\partial V/\partial t$ — one scalar condition tying both potentials together, rather than constraining $\mathbf{A}$ alone.
Its effect on the coupled potential system is surgical: the cross terms vanish and the equations decouple into four identical inhomogeneous wave equations, $\Box^2 V = -\rho/\varepsilon_0$ and $\Box^2\mathbf{A} = -\mu_0\mathbf{J}$ , with $\Box^2 \equiv \nabla^2 - \mu_0\varepsilon_0\,\partial_t^2$ .
The gauge is always attainable: the repairing gauge function solves $\Box^2\lambda = -(\nabla\cdot\mathbf{A} + \mu_0\varepsilon_0\,\partial_t V)$ , a wave equation with known source, which always has solutions.
It is not unique: residual gauge functions with $\Box^2\lambda = 0$ — gauge waves — survive, generating potential pairs that carry zero field. Decay plus initial data pin them away.
The condition, once imposed, stays imposed: the Lorenz quantity obeys a source-free wave equation whose source would be $\mu_0(\partial_t\rho + \nabla\cdot\mathbf{J})$ — charge conservation is what keeps the gauge from leaking.
Unlike the Coulomb condition, the Lorenz condition is Lorentz-invariant ( $\partial_\mu A^\mu = 0$ ): all inertial observers can share it — the reason covariant QED and radiation theory are built here.
Key idea: the Lorenz condition $\nabla\cdot\mathbf{A} = -\mu_0\varepsilon_0\,\partial_t V$ decouples the potential equations into symmetric, causal wave equations sourced by the physical $\rho$ and $\mathbf{J}$ — the gauge in which radiation theory is written.

The core insight

The Lorenz condition $\nabla\cdot\mathbf{A} = -\mu_0\varepsilon_0\,\partial V/\partial t$ decouples the potential equations into symmetric inhomogeneous wave equations, $\Box^2 V = -\rho/\varepsilon_0$ and $\Box^2\mathbf{A} = -\mu_0\mathbf{J}$ , always attainable and self-maintaining for conserved sources.

Where this lives today

The Lorenz gauge is the default coordinate system of relativistic field theory. Covariant QED perturbation theory is built on Lorenz-class gauges — the photon propagators of Feynman and Landau gauge differ only in how the gauge-fixing term weights the $\partial_\mu A^\mu$ constraint — and every cross-section at the LHC flows through them. The gauge's classical structure survives quantization with one famous complication: four potential components describe two physical polarizations, and the Gupta–Bleuler condition (the quantum shadow of this article's residual gauge waves) polices the surplus states out of every physical amplitude. The deepest modern echo is gravitational. Einstein's equations, linearized, are a tensor version of the potential system (1), and the harmonic (de Donder) gauge — $\partial_\mu \bar h^{\mu\nu} = 0$ — is the Lorenz condition's exact analog: it collapses the linearized field equations into wave equations, predicts gravitational waves propagating at $c$ , and its constraint-propagation theorem is Derivation B with indices. Numerical-relativity codes that generate LIGO waveform templates impose generalized harmonic gauges and monitor precisely the "gauge leak" this article's maintenance theorem controls. When the 2017 neutron-star merger showed gravity and light arriving together after 130 million years, the shared wave operator of equations (3) and their gravitational twins was the mathematics being tested — to one part in $10^{15}$ .

Origins · 1 minLudvig Valentin Lorenz and how this was discovered

Ludvig Valentin Lorenz

1829–1891 · Copenhagen — who wrote retarded electrodynamics in 1867 and lost the naming rights by one letter

Lorenz worked in Copenhagen, outside the Maxwellian mainstream, publishing in Danish and German — and in 1867, two years after Maxwell's dynamical theory and entirely independently of it, he constructed electrodynamics from retarded potentials: source integrals evaluated at the earlier time $t - r/c$ , satisfying what we now call the Lorenz condition, with light emerging as an electromagnetic disturbance. He had, in modern terms, the Lorenz-gauge potential theory of this chapter — before Maxwell's own theory was widely believed, and decades before covariance explained why his condition was the natural one. Maxwell himself pushed back (he distrusted the retarded-action viewpoint), and Lorenz's formulation was largely absorbed unattributed. The condition drifted under the name of Hendrik Lorentz, who systematized it in the 1890s–1900s; only in the late twentieth century did textbooks (Jackson's third edition prominently) restore the credit. Lorenz also derived the Lorenz–Mie scattering theory's precursors and the Lorenz–Lorentz relation — that one, delightfully, shared fairly with Lorentz, who found it independently. A careful physicist, twice nearly erased by a more famous near-namesake.

History

1858

— Riemann presents a retarded-potential electrodynamics to the Göttingen Academy (published posthumously 1867)

1867

— Ludvig Lorenz publishes "On the identity of the vibrations of light with electrical currents": retarded potentials and the condition, independent of Maxwell

1892

— Hendrik Lorentz begins the electron-theory program whose authority later attaches his name to the condition

1904

— Lorentz's encyclopedia article systematizes gauge freedom and the "Lorentz" condition; the misnomer sets

1929

— Fermi introduces the gauge-fixing term into quantized electrodynamics; Lorenz-class gauges enter QED

1950

— Gupta and Bleuler tame the quantum surplus states of the covariant gauge

2001

— Jackson and Okun's historical audit "Historical roots of gauge invariance" documents the misattribution; "Lorenz gauge" regains ground

2017

— GW170817: gravitational and electromagnetic waves from a neutron-star merger arrive within seconds — the shared wave-equation structure, tested across 130 million light-years

The physical picture

The chapter opener left the potentials tangled. Written without any gauge choice, their equations of motion shackle $V$ and $\mathbf{A}$ together — the divergence of one feeding the dynamics of the other, a $\nabla(\nabla\cdot\mathbf{A} + \mu_0\varepsilon_0\,\partial_t V)$ block sitting athwart what would otherwise be a clean wave equation. Look at that block once more, because the Lorenz gauge is nothing but the decision to read it as an instruction: the offending object is a gradient of one scalar combination of the potentials, and gauge freedom lets you set one scalar combination of the potentials to anything you like. Set this one to zero. The tangle falls apart. What remains is almost embarrassingly symmetric. $V$ obeys a wave equation sourced by $\rho$ ; each Cartesian component of $\mathbf{A}$ obeys the same wave equation sourced by the matching component of $\mathbf{J}$ . Four copies of one scalar problem — the inhomogeneous wave equation, whose anatomy is the next article's subject and whose solution (the retarded integrals) is the one after. The Coulomb gauge split the world into an instantaneous ledger and a causal remainder; the Lorenz gauge refuses the split. Everything propagates, at $c$ , from the physical sources. When charge wiggles, $V$ and $\mathbf{A}$ hear about it a light-travel-time later — the potentials themselves are causal, not just the fields, and the awkward acquittals of the last article never need to be filed.

Figure

The Lorenz condition as a detangler — the coupled potential system collapses into four identical causal wave equations

There is a price, and honesty requires posting it now. First, the Lorenz slice through the space of potentials is thicker than the Coulomb slice: a genuine family of gauge functions — solutions of the source-free wave equation — preserves the condition, so "Lorenz gauge" names a family of descriptions, not a single one. Worked example 1 exhibits one of these gauge waves explicitly: a full-blooded pair of oscillating potentials whose fields vanish identically. Second, $V$ and $\mathbf{A}$ are now four components doing the work of two physical polarizations, and in the quantum theory the surplus must be policed (the Gupta–Bleuler construction); the Coulomb gauge never carries passengers. For classical radiation problems the trade is overwhelmingly worth it, which is why every article from here to the end of the chapter is written in this gauge.

The condition and the collapse

The chapter opener assembled the general coupled system — recalled here, not re-derived:

\nabla^2 V + \frac{\partial}{\partial t}(\nabla\cdot\mathbf{A}) = -\frac{\rho}{\varepsilon_0}, \qquad \Box^2\mathbf{A} - \nabla\!\left(\nabla\cdot\mathbf{A} + \mu_0\varepsilon_0\frac{\partial V}{\partial t}\right) = -\mu_0\mathbf{J}, \qquad (1)

where: $\Box^2 \equiv \nabla^2 - \mu_0\varepsilon_0\,\partial_t^2$ is the d'Alembertian in Griffiths's notation ( $\mu_0\varepsilon_0 = 1/c^2$ ). Stare at the structure: the vector equation is one gradient away from a clean wave equation, and the scalar equation is one time-derivative-of-a-divergence away from one. Both defects involve the same scalar combination. Now impose the Lorenz condition:

\nabla\cdot\mathbf{A} = -\mu_0\varepsilon_0\frac{\partial V}{\partial t} \qquad\Longleftrightarrow\qquad \nabla\cdot\mathbf{A} + \mu_0\varepsilon_0\frac{\partial V}{\partial t} = 0. \qquad (2)

where: the right-hand form displays it as "the tangle, set to zero." In the vector equation of (1) the gradient block dies on the spot. In the scalar equation, use (2) to replace $\nabla\cdot\mathbf{A}$ by $-\mu_0\varepsilon_0\,\partial_t V$ , and the mixed term becomes the missing piece of the wave operator:

\boxed{\ \Box^2 V = -\frac{\rho}{\varepsilon_0}, \qquad \Box^2\mathbf{A} = -\mu_0\mathbf{J}.\ } \qquad (3)

where: the sources are the physical charge and current densities, evaluated where matter actually is — compare the Coulomb gauge's $\mathbf{J}_\perp$ , which fills all space. Equations (3) are the chapter's load-bearing result: four identical scalar wave problems whose operator anatomy is dissected next article and whose retarded solutions, the article after, become the foundation of every radiation formula in physics. In the static limit $\Box^2 \to \nabla^2$ and (3) collapse onto the Poisson equations of electro- and magnetostatics — the whole earlier course is the $\partial_t \to 0$ shadow of this pair. Two theorems must now be paid for, because (2) was imposed by fiat: that the condition can always be reached (else the gauge is a wish, not a tool), and that it can be kept (else it leaks as the system evolves, and (3) silently fail at later times).

Derivation AAttainability — the Lorenz gauge is always reachable, and what freedom survives

The goal is to show that from any starting potentials $(V, \mathbf{A})$ — however inconvenient — a gauge function $\lambda$ exists whose transformation lands the pair on the Lorenz slice, and then to measure exactly how much gauge freedom remains afterward. The strategy is the standard repair argument of the gauge-transformations article, run with the Lorenz target: demand the condition of the transformed pair and read off the equation $\lambda$ must satisfy. The punch line is that this equation is itself an inhomogeneous wave equation — the very species the gauge is designed to produce — so its solvability is guaranteed by the same machinery the chapter is building. Watch the logic's shape: it is not circular, but it is self-referential in a way worth savoring.

Step 1. Transform and demand. Under $\mathbf{A}' = \mathbf{A} + \nabla\lambda$ , $V' = V - \partial_t\lambda$ , the Lorenz combination of the new pair is

\nabla\cdot\mathbf{A}' + \mu_0\varepsilon_0\frac{\partial V'}{\partial t} = \nabla\cdot\mathbf{A} + \nabla^2\lambda + \mu_0\varepsilon_0\frac{\partial V}{\partial t} - \mu_0\varepsilon_0\frac{\partial^2\lambda}{\partial t^2}.

Why both shifts contribute: the divergence picks up $\nabla^2\lambda$ from the vector shift, and the time derivative picks up $-\mu_0\varepsilon_0\,\partial_t^2\lambda$ from the scalar shift — together they assemble the d'Alembertian of $\lambda$ , which is the structural reason the repair equation comes out hyperbolic rather than elliptic.

Step 2. Read off the repair equation. Setting the combination to zero and writing $s(\mathbf{r},t) \equiv \nabla\cdot\mathbf{A} + \mu_0\varepsilon_0\,\partial_t V$ for the starting pair's "Lorenz defect":

\Box^2\lambda = -s(\mathbf{r},t).

Why this always has solutions: it is an inhomogeneous wave equation with a known, fixed source. The next two articles construct its solutions explicitly (the retarded integral over $s$ ); existence needs nothing about $s$ beyond smoothness and reasonable behavior at infinity. Contrast the Coulomb repair, which was a Poisson problem — elliptic, instantaneous; the Lorenz repair is hyperbolic, and even the act of reaching the gauge propagates causally.

Step 3. Price the residual freedom. Suppose the condition already holds and apply a further transformation with gauge function $\lambda_0$ . By Step 1's algebra, the new Lorenz combination is $0 + \Box^2\lambda_0$ , so the condition survives if and only if

\Box^2\lambda_0 = 0

— the source-free wave equation. Its solutions are genuine waves: propagating profiles of pure gauge, each generating a potential pair with identically zero fields (worked example 1 builds one). The Lorenz slice therefore meets each gauge orbit not in a point but in a family parametrized by free gauge waves.

Step 4. Pin the residue. The gauge waves are eliminated by data, not by more conditions: demanding that the potentials vanish before the sources ever acted (quiescent past), or specifying $\lambda_0$ 's initial value and initial time-derivative as zero, forces $\lambda_0 \equiv 0$ by the uniqueness theorem for the wave equation (next article). In radiation practice the retarded solutions of (3) are the quiescent-past representatives, which is why the chapter can speak of "the" Lorenz-gauge potentials without ambiguity. Verified: matches Griffiths §10.1.2 (Eq. 10.17) and Jackson Eqs. 6.14–6.16; the repair applied to the exotic zero- $V$ point-charge description of the gauge-transformations article lands exactly on the standard Coulomb pair — worked example 2 runs the computation.

Figure

A gauge wave in the Lorenz slice — V and A oscillate in lockstep, and their two contributions to E cancel at every point and instant

Go deeper · Derivation B · 2 minPersistence — charge conservation keeps the gauge from leaking

Derivation BPersistence — charge conservation keeps the gauge from leaking

The goal is the maintenance theorem: if the Lorenz condition holds at one instant (together with its first time derivative) and the potentials evolve by the decoupled equations (3), the condition holds forever — provided, and only provided, the sources conserve charge. This matters practically: in an initial-value setting (numerical electromagnetics, and the exact analog in numerical relativity) one imposes the gauge at $t=0$ and needs a guarantee it will not drift. The strategy is to derive an evolution equation for the Lorenz defect $L \equiv \nabla\cdot\mathbf{A} + \mu_0\varepsilon_0\,\partial_t V$ itself, by combining the divergence of the vector equation with the time derivative of the scalar equation.

Step 1. Divergence of the vector equation. Since $\Box^2$ commutes with $\nabla\cdot$ on smooth fields, taking the divergence of $\Box^2\mathbf{A} = -\mu_0\mathbf{J}$ gives

\Box^2(\nabla\cdot\mathbf{A}) = -\mu_0\,\nabla\cdot\mathbf{J}.

Step 2. Time derivative of the scalar equation. Applying $\mu_0\varepsilon_0\,\partial_t$ to $\Box^2 V = -\rho/\varepsilon_0$ :

\Box^2\!\left(\mu_0\varepsilon_0\frac{\partial V}{\partial t}\right) = -\mu_0\frac{\partial\rho}{\partial t}.

Why these two moves: $L$ is exactly the sum of the arguments on the left-hand sides; building $\Box^2 L$ requires precisely one divergence and one time derivative, and the wave operator's constant coefficients let both slide through.

Step 3. Add, and meet the continuity equation. Summing,

\Box^2 L = -\mu_0\left(\nabla\cdot\mathbf{J} + \frac{\partial\rho}{\partial t}\right) = 0,

where the vanishing is the continuity equation — recalled from the conservation-laws chapter, and guaranteed for any legal source by the consistency identity of the gauge-transformations article. Why conservation is load-bearing and not decorative: for a hypothetical charge-non-conserving source, $L$ would obey a driven wave equation and the gauge would leak at the speed of light from wherever conservation failed — the decoupled equations (3) would remain solvable, but their solutions would no longer be potentials for Maxwell's equations. The Lorenz gauge is consistent exactly because electrodynamics conserves charge.

Step 4. Close with uniqueness. $L$ obeys the source-free wave equation. If $L = 0$ and $\partial_t L = 0$ on the initial slice, the uniqueness theorem for the wave equation (next article) forces $L \equiv 0$ for all time. The gauge, once imposed, maintains itself — the evolution equations (3) are the complete story, with the condition riding along as a conserved constraint. Verified: this is the standard constraint-propagation argument, identical in structure to the maintenance of the harmonic-gauge condition in general relativity (where its failure modes are a live numerical-relativity concern); consistent with Jackson §6.3 and Zangwill §15.3.

Covariance — the statement

Package the potentials as the four-potential $A^\mu = (V/c, \mathbf{A})$ — a preview of the relativity chapter, where the packaging is derived rather than declared. The Lorenz condition then reads

\partial_\mu A^\mu = \frac{1}{c^2}\frac{\partial V}{\partial t} + \nabla\cdot\mathbf{A} = 0, \qquad (4)

where: $\partial_\mu A^\mu$ is a four-divergence — a Lorentz scalar. That is the whole point: a scalar that vanishes in one inertial frame vanishes in all of them. Every boosted observer can adopt the same gauge, and the decoupled equations (3) — whose operator $\Box^2$ is itself invariant — keep their form in every frame. Contrast the Coulomb condition $\nabla\cdot\mathbf{A} = 0$ : three-dimensional divergence, not a four-scalar, true in at most one frame at a time. The full covariant treatment (why $A^\mu$ is a four-vector, how the fields sit inside $F^{\mu\nu}$ ) belongs to The Four-Potential in the relativity chapter; here the statement suffices to explain the gauge's dominance in relativistic and radiation contexts.

Limiting cases

case	result	meaning
statics	condition → $\nabla\cdot\mathbf{A}=0$ ; equations → Poisson	Lorenz and Coulomb gauges coincide; all of electro/magnetostatics is the $\partial_t\to0$ shadow of (3)
source-free space	$\Box^2V=0$ , $\Box^2\mathbf{A}=0$	potentials are free waves at $c$ ; the wave chapter's machinery applies verbatim to the potentials
slowly varying sources	$\Box^2\to\nabla^2$ + corrections of order $(L/c\tau)^2$	quasistatics: retardation corrections enter at second order — the estimate is next article's business
covariant form	$\partial_\mu A^\mu = 0$	frame-invariant condition; the only standard gauge all inertial observers can share
within-gauge freedom	$\Box^2\lambda_0=0$ gauge waves	the slice is a family; quiescent-past (retarded) representatives make it canonical

Common trapLorenz gauge does not mean the potentials are physical — and Lorenz is not Lorentz

Two distinct traps. First: because the Lorenz-gauge potentials propagate causally from physical sources, they feel more real than the Coulomb gauge's instantaneous ledger — but they are exactly as unobservable. Worked example 1 constructs oscillating potentials, legal in this gauge, whose $\mathbf{E}$ and $\mathbf{B}$ vanish identically: no experiment can detect a gauge wave, which is the sharpest possible reminder that only gauge-invariant quantities leave a calculation. Second, the name: the condition is Lorenz's (Ludvig, Copenhagen, 1867), not Lorentz's (Hendrik, Leiden). The misattribution matters beyond etiquette — cite "Lorentz gauge" in a paper and you propagate a century-old citation error; write "Lorenz gauge" and expect at least one referee to "correct" you back. Know the story and hold the line.

Worked examples

1 · A gauge wave — real potentials, zero fields (confidence builder)

Setup. Exhibit the residual freedom of Derivation A, Step 3, concretely. Claim: with $\omega = ck$ , the pair

V(z,t) = V_0\cos(kz - \omega t), \qquad \mathbf{A}(z,t) = \frac{V_0}{c}\cos(kz - \omega t)\,\hat{\mathbf{z}}

satisfies the Lorenz condition and carries identically zero electromagnetic field. Take $V_0 = 10$ V at $f = 100$ MHz.

Step 1. Check the condition. $\nabla\cdot\mathbf{A} = -\tfrac{V_0 k}{c}\sin(kz-\omega t)$ and $\mu_0\varepsilon_0\,\partial_t V = \tfrac{\omega V_0}{c^2}\sin(kz-\omega t)$ . Their sum vanishes precisely when $k/c = \omega/c^2$ , i.e. $\omega = ck$ . ✓ The Lorenz condition enforces the vacuum dispersion relation on this profile — gauge waves travel at $c$ like everything else in this gauge.

Step 2. Compute the fields. $\mathbf{B} = \nabla\times\mathbf{A} = 0$ ( $\mathbf{A}$ is $\hat{\mathbf{z}}$ -directed and varies only along $z$ ). And

E_z = -\partial_z V - \partial_t A_z = V_0 k\sin(kz-\omega t) - \frac{V_0\omega}{c}\sin(kz-\omega t) = V_0\sin(kz-\omega t)\left(k - \frac{\omega}{c}\right) = 0.

Both fields vanish identically — a 10-volt potential wave, oscillating at 100 MHz through all space, carrying nothing.

Step 3. Identify the gauge function. The pair is generated from $(V,\mathbf{A}) = (0,\mathbf{0})$ by $\lambda = \tfrac{V_0}{\omega}\sin(kz-\omega t)$ : indeed $-\partial_t\lambda = V_0\cos(kz-\omega t)$ ✓ and $\nabla\lambda = \tfrac{V_0 k}{\omega}\cos(kz-\omega t)\hat{\mathbf{z}} = \tfrac{V_0}{c}\cos\,\hat{\mathbf{z}}$ ✓, and $\Box^2\lambda = 0$ ✓. Amplitudes: $A_0 = V_0/c = 3.34\times10^{-8}$ V·s/m.

The lesson: "satisfies the Lorenz condition" does not mean "physical." The gauge slice contains pure-gauge traffic; quiescent-past boundary conditions are what clear it.

2 · Repairing the exotic point charge into the Lorenz gauge (exam level)

Setup. The gauge-transformations article described a static charge $q$ by $V' = 0$ , $\mathbf{A}' = -\tfrac{1}{4\pi\varepsilon_0}\tfrac{qt}{r^2}\hat{\mathbf{r}}$ . This pair is not in Lorenz gauge. Repair it, using Derivation A's recipe, and identify where it lands.

Step 1. Measure the defect. $s = \nabla\cdot\mathbf{A}' + \mu_0\varepsilon_0\,\partial_t V' = \nabla\cdot\mathbf{A}'$ . Using $\nabla\cdot(\hat{\mathbf{r}}/r^2) = 4\pi\delta^3(\mathbf{r})$ — the delta-function identity of the vector-calculus chapter:

s = -\frac{qt}{\varepsilon_0}\,\delta^3(\mathbf{r}).

The defect is a point source at the origin with linearly growing strength.

Step 2. Solve the repair equation $\Box^2\lambda = -s = +\tfrac{qt}{\varepsilon_0}\delta^3(\mathbf{r})$ . Borrowing the retarded point-source solution (constructed honestly two articles ahead; here it is a legitimate ansatz to verify): a source $f(t)\delta^3(\mathbf{r})$ drives $\lambda = -f(t - r/c)/(4\pi r)$ for $\Box^2\lambda = f\,\delta^3$ ... applied with $f(t) = qt/\varepsilon_0$ :

\lambda(\mathbf{r},t) = -\frac{q\,(t - r/c)}{4\pi\varepsilon_0\,r}.

Step 3. Transform. The scalar potential: $V'' = V' - \partial_t\lambda = 0 + \tfrac{q}{4\pi\varepsilon_0 r}$ . The vector potential: $\nabla\lambda = \tfrac{q}{4\pi\varepsilon_0}\left[\tfrac{t - r/c}{r^2} + \tfrac{1}{cr}\right]\hat{\mathbf{r}}$ , and adding $\mathbf{A}'$ :

\mathbf{A}'' = \frac{q}{4\pi\varepsilon_0}\left[\frac{t - r/c}{r^2} + \frac{1}{cr} - \frac{t}{r^2}\right]\hat{\mathbf{r}} = \frac{q}{4\pi\varepsilon_0}\left[-\frac{1}{cr} + \frac{1}{cr}\right]\hat{\mathbf{r}} = \mathbf{0}.

Step 4. Identify the destination: $(V'', \mathbf{A}'') = \left(\tfrac{q}{4\pi\varepsilon_0 r},\, \mathbf{0}\right)$ — the standard electrostatic description, which satisfies the Lorenz condition trivially (nothing depends on time). The repair machinery, fed a deliberately perverse description of the simplest system in electrodynamics, marches it back to the textbook page. Note the fingerprint of the method: the $r/c$ retardation inside $\lambda$ — pure gauge here, since the physics is static — cancelled exactly against the $1/cr$ term. The retarded structure is baked into the gauge's plumbing even when the answer is static.

The lesson: the Lorenz repair is an algorithm, not an art: measure the defect $s$ , solve one wave equation, transform. And different descriptions of one physics really do reconnect — Derivation B of the gauge-transformations article, live.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §10.1.2–10.1.3 · Jackson, Classical Electrodynamics, 3rd ed., §6.3–6.5 · Zangwill, Modern Electrodynamics, §15.3 · Jackson & Okun, "Historical roots of gauge invariance," Rev. Mod. Phys. 73, 663 (2001) · Nevels & Shin, "Lorenz, Lorentz, and the gauge," IEEE Ant. Prop. Mag. 43, 70 (2001)

Provenance. Follows Griffiths §10.1.2–10.1.3 (Eqs. 10.12–10.17) and Jackson §6.3. A verifier should recompute: (1) the reduction of the coupled system (1) to the decoupled wave equations (3) under the Lorenz condition; (2) the attainability equation $\Box^2\lambda = -(\nabla\cdot\mathbf{A}+\mu_0\varepsilon_0\partial_t V)$ and the residual condition $\Box^2\lambda_0=0$ ; (3) the covariant form $\partial_\mu A^\mu=0$ . Constants are standard SI vacuum values. Addendum for this edition: per the chapter's division of labor, the base's "Lorentz invariance of the Lorenz condition" derivation is held at statement level (equation (4)) with the tensor machinery deferred to the relativity chapter (The Four-Potential); the base's "from decoupled wave equations to retarded potentials" derivation is owned by Retarded Potentials (EMAG.RAD.RETRD.0607); the base's self-consistency argument is upgraded to the constraint-propagation theorem (Derivation B). The attainability derivation was rescoped here from the gauge-transformations base, per that article's provenance note. The coupled system (1) is quoted from the chapter opener (Scalar and Vector Potentials), not re-assembled. Worked examples (the explicit gauge wave; the Lorenz repair of the exotic point-charge description landing on the static pair) are new to this edition, verified symbolically.

Builds on: Scalar and Vector Potentials · Gauge Transformations · Coulomb Gauge Leads to: Inhomogeneous Wave Equations for Potentials · Retarded Potentials · Liénard–Wiechert Potentials · Jefimenko's Equations

Why is it Lorenz gauge and not Lorentz gauge — and does one letter really matter?

The condition appears in Ludvig Lorenz's 1867 paper — retarded potentials, decades ahead of its time — while Hendrik Lorentz, born 1853, systematized it in the 1890s–1900s and inherited the name by fame. The 2001 historical audits (Jackson & Okun; Nevels & Shin) settled the record, and Jackson's own textbook switched to "Lorenz." It matters the way any citation matters: credit is the currency of science, and this is among its most instructive failures — a physicist erased by one letter for a century. It also matters pedagogically: the condition is Lorenz's, the invariance group under which it is a scalar is Lorentz's, and keeping the names straight keeps the two distinct ideas straight. The coincidence is genuinely accidental; the two also independently found the Lorenz–Lorentz refraction relation, compounding the confusion.

Is the Lorenz condition a law of physics or a choice?

A choice — but a choice the structure of electrodynamics is unusually friendly to. Nothing forces it: Maxwell's equations hold in every gauge, and the Coulomb, temporal, or axial gauges describe identical physics. What elevates the Lorenz condition is a conspiracy of conveniences: it is reachable from anywhere (Derivation A), self-maintaining under evolution precisely because charge is conserved (Derivation B), frame-invariant (equation 4), and it hands the potentials the same causal wave dynamics the fields themselves obey. In field-theory language it is the gauge-fixing choice that keeps Lorentz covariance manifest, which is why QED's covariant formulation and general relativity's harmonic gauge are both built on its template. A choice, then — the way choosing inertial coordinates is a choice.

The potentials now propagate causally. Does that make them more physical than in the Coulomb gauge?

No — and worked example 1 is the proof by construction: a 10-volt gauge wave oscillating through all space at 100 MHz with identically zero fields. Causal-looking potentials are still coordinates. What is true is that the Lorenz gauge's potentials are a more faithful bookkeeping of the physics' causal structure: changes in $V$ and $\mathbf{A}$ here really do trace back to source activity on the past light cone (once gauge waves are cleared by quiescent-past conditions), so intuition trained on them misleads less often than intuition trained on the Coulomb gauge's instantaneous ledger. That is a statement about pedagogy and convenience, not ontology. The gauge-invariant content — fields, forces, fluxes, loop integrals — is identical in both gauges, always.

Why does radiation theory prefer the Lorenz gauge over the Coulomb gauge?

Compactness of sources, mostly. The Lorenz-gauge wave equations are driven by the physical $\rho$ and $\mathbf{J}$ — confined to the antenna, the atom, the accelerating charge — so their retarded solutions are integrals over a finite region, and the far-field expansion of those integrals (the multipole ladder later in this chapter) is clean. The Coulomb gauge's radiating object $\mathbf{A}$ is instead driven by $\mathbf{J}_\perp$ , which fills all space even for a point source, making retarded integrals awkward. Add covariance (radiation problems love boosted frames) and the symmetric treatment of $V$ and $\mathbf{A}$ (they combine into one four-potential with one Green function), and the verdict is lopsided. The Coulomb gauge wins the complementary territory: Hamiltonians, bound states, photon counting.

If four potential components obey wave equations, are there four kinds of electromagnetic waves?

No — two of the four are fakes, and the gauge structure is exactly the machinery that exposes them. The four solutions of $\Box^2 V = 0$ , $\Box^2\mathbf{A}=0$ are constrained by the Lorenz condition (one relation) and still admit gauge waves (one more function's worth of freedom, Derivation A Step 3), leaving $4-1-1 = 2$ physical transverse polarizations — the same count the Coulomb gauge displays explicitly and the wave chapter derived for the fields. In covariant QED the fake pair returns as "scalar" and "longitudinal" photon states, and the Gupta–Bleuler condition — the quantized Lorenz constraint — ensures they appear only in unmeasurable combinations. The counting is the classical shadow of the photon's two polarization states, gauge after gauge.

Do I have to re-impose the Lorenz condition at every time step when solving numerically?

In exact arithmetic, no: Derivation B guarantees the condition propagates itself if imposed (with its time derivative) initially and the sources conserve charge. In floating-point practice, discretization error breaks the guarantee slowly — the numerical Lorenz defect $L$ obeys a driven wave equation with truncation-error sources, and it can grow. Production codes therefore monitor $L$ as a diagnostic and either project it away periodically or add constraint-damping terms that turn $L=0$ into an attractor. The same engineering appears, scaled up, in numerical relativity: generalized-harmonic evolutions add damping precisely because the harmonic (Lorenz-analog) constraint would otherwise drift in long binary-black-hole runs. The theorem tells you what to monitor; the numerics tell you how hard.

Could I impose the Lorenz condition on V only, or use different gauges for V and A?

The question dissolves once the condition is read correctly: it is a single scalar equation relating the two potentials, not a pair of separate demands. There is one gauge function $\lambda$ , so there is one condition to spend it on — you cannot independently gauge-fix $V$ and $\mathbf{A}$ , because one transformation moves both in lockstep (the choreography of the gauge-transformations article). What you can do is choose conditions that reference only one potential (Coulomb: $\nabla\cdot\mathbf{A}=0$ ; temporal: $V=0$ ) — but the choice still spends the whole budget, and the other potential's equation feels the consequences. The Lorenz condition's genius is spending the budget on the combination that appears in the coupled equations' tangle, which is why it alone symmetrizes them.