Lorenzgauge

The Lorenz gauge is a condition used in electromagnetism to fix the gauge freedom of the electromagnetic potentials. It imposes a relation between the scalar potential φ and the vector potential A, and is Lorentz (Lorenz) invariant.

The gauge condition: ∇·A + (1/c^2) ∂φ/∂t = 0, equivalent to ∂μ A^μ = 0 in four-vector notation. Under a gauge transformation A' = A + ∇χ, φ' = φ − ∂χ/∂t, the Lorenz condition becomes □χ = -∇·A - (1/c^2) ∂φ/∂t, so choosing χ appropriately preserves or enforces the condition.

With this gauge, the Maxwell equations reduce to inhomogeneous wave equations for φ and A: ∇^2 φ − (1/c^2) ∂^2 φ/∂t^2 = −ρ/ε0, ∇^2 A − (1/c^2) ∂^2 A/∂t^2 = −μ0 J. The four-divergence constraint is satisfied automatically if the charge continuity equation holds.

The solutions for φ and A in Lorenz gauge are the retarded potentials: φ(r,t) = (1/4πε0) ∫ ρ(r', t − |r−r'|/c)/|r−r'| d^3r', A(r,t) = μ0/(4π) ∫ J(r', t − |r−r'|/c)/|r−r'| d^3r'. It is Lorentz invariant and compatible with special relativity.

Named after Ludvig Lorenz (not to be confused with Hendrik Lorentz), the gauge is widely used in classical and quantum electrodynamics, as it preserves relativistic covariance of the field equations and simplifies treatment of radiation.