the Lorenz gauge
/ LOR-ents /
For relativistic electromagnetism you want a gauge that treats space and time on the same footing, so that equations look the same in every inertial frame. The Lorenz gauge — named for the Danish physicist Ludvig Lorenz, not the Dutch Hendrik Lorentz — is exactly that choice, and it turns Maxwell's tangle into clean, symmetric wave equations.
The gauge condition is div A + (1/c^2) dV/dt = 0, which in four-vector language is the manifestly covariant d_mu A^mu = 0. Impose it and Maxwell's equations decouple beautifully: the scalar and vector potentials each satisfy their own inhomogeneous wave equation, box V = -rho/epsilon_0 and box A = -mu_0 J, where box is the d'Alembertian d^2/(c^2 dt^2) - laplacian. Gone is the cross-talk between V and A; each potential is sourced by its own charge or current and propagates outward at the speed of light.
The Lorenz gauge is the workhorse of radiation theory because its wave equations are solved directly by the retarded potentials, building light-speed causality in from the start. It is also Lorentz-invariant — the condition holds in every inertial frame, unlike the Coulomb gauge — which is why it is the natural language for the covariant, tensor form of electromagnetism. A residual freedom remains: you may still add any lambda satisfying box lambda = 0 and stay within the gauge.
In the Lorenz gauge, box A = -mu_0 J is solved by the retarded vector potential A(r,t) = (mu_0/4 pi) times the integral of J(r', t_r)/|r - r'|, with t_r = t - |r - r'|/c. The gauge choice is precisely what makes the potentials come out retarded — causal and propagating at c.
div A + (1/c^2) dV/dt = 0 decouples Maxwell into wave equations box V = -rho/epsilon_0, box A = -mu_0 J.
Spell it Lorenz, not Lorentz — the gauge honours Ludvig Lorenz, a different physicist from the Hendrik Lorentz of the Lorentz transformation, though the two are perpetually confused. Unlike the Coulomb gauge the Lorenz condition is Lorentz-invariant, and it still leaves a residual gauge freedom (any lambda with box lambda = 0).