Lorenz gauge

The Lorenz gauge is the constraint ∇·A + (1/c²) ∂V/∂t = 0 imposed on the scalar and vector potentials V and A. In electrodynamics the fields E and B can be expressed in terms of potentials as E = −∇V − ∂A/∂t, B = ∇×A; but the potentials are not unique — a gauge transformation V' = V − ∂χ/∂t, A' = A + ∇χ for any scalar function χ leaves E and B invariant. The Lorenz gauge is one particular choice of χ that simplifies Maxwell's equations to a symmetric form.

In the Lorenz gauge, Maxwell's four equations collapse to two uncoupled wave equations: □V = ρ/ε₀ and □A = μ₀J, where □ = (1/c²)∂²/∂t² − ∇² is the d'Alembertian operator. Both potentials satisfy the same wave equation — just with different sources — and both propagate at the same speed c. The symmetry is explicit: V and A are related components of a single four-vector potential A^μ = (V/c, A), and the Lorenz-gauge wave equations combine into the manifestly Lorentz-covariant equation □A^μ = μ₀J^μ. This is why the Lorenz gauge is the default choice in relativistic electrodynamics and in any calculation that needs Lorentz-invariance to be obvious.

The name is frequently misspelled *Lorentz* gauge — understandably, because Hendrik Antoon Lorentz was the much better-known contemporary Dutch physicist. But the gauge is named for the Danish physicist Ludvig Valentin Lorenz (1829–1891), who proposed it in 1867, a decade before Hendrik Lorentz's earliest electrodynamics work. The one-letter distinction (z vs. tz) is the only thing separating them in writing, and it is wrong more often than right in textbooks. Both Lorenz's paper and Lorentz's later work are in the long historical arc that produced the modern covariant formulation, but credit for the gauge condition belongs specifically to the older Danish Lorenz.