Lorentz Gauge

Maxwell’s equations in terms of electro dynamic potentials are expressed as
        (∇²A – μ₀ ϵ₀ ∂²A/∂t² = - ∇ (∇.A + μ₀ ϵ₀ ∂V/∂t) = -μ₀J                … (1)
        V²V + ∂ (∇.A)/∂t = - 1ρ /ϵ₀                             … (2)

If we put
        ∇.A = - μ₀ ϵ₀ ∂V/∂t                                … (3)

Eqs. (1) and (2) become
∇2A – μ₀ ϵ₀ ∂²A/∂t² = - μ₀J                                … (4)
∇2V – μ₀ ϵ₀ ∂²A/∂t² = -1/ϵ₀ρ                                … (5)

The condition given by relation (3) is known as Lorentz condition and expresses a relationship between V and A. When the vector and scalar potential satisfy the Lorentz condition, the gauge is known as Lorentz gauge.

Eq. (4) and (5) are written more compactly by defining the d’Alembertian operator as
Then we get
            [ ]² A = - μ2J.}
       [ ]² V = - 1/ϵ₀ ρ.}                                    …. (7)

Lorentz gange treats V and A on an equal footing. The same differential operator occurs in both equations. In special relativity, the d alembertian day somewhat the same role as the Laplacian in classical physics and Eq. (7) may be regarded as a four-dimensional version of Poisson’s equation.