Four-potential

The four-potential is the Lorentz four-vector A^μ = (φ/c, A_x, A_y, A_z) that packages the scalar potential φ and vector potential A of classical electrodynamics into a single Lorentz-covariant object. Under a Lorentz boost, the time component (φ/c) and space components (A) mix exactly the way (ct, x) mix — a fact that turns the apparently separate concepts of "electric potential" and "vector potential" into manifestations of the same underlying object viewed from different inertial frames. A static charge in the lab frame produces a pure φ field (no A); boost into a moving frame and the same physical object now produces both φ' and A'.

The field tensor F^{μν} is constructed from the four-potential as F^{μν} = ∂^μ A^ν − ∂^ν A^μ, where ∂^μ = (∂/c∂t, −∇) is the contravariant four-gradient. This construction makes the homogeneous Maxwell equations ∂_μ F^{μν} = 0 automatic (the antisymmetry of F under permutations and the equality of mixed partials kill the right-hand side identically — the Bianchi identity), so only the inhomogeneous pair ∂_μ F^{μν} = μ₀ J^ν need be enforced as dynamical equations. The four-potential is also subject to gauge freedom: the transformation A_μ → A_μ + ∂_μ Λ for any scalar function Λ leaves F^{μν} unchanged (because ∂_μ ∂_ν Λ is symmetric in μ, ν while F^{μν} is antisymmetric). Common gauges include the Lorenz gauge* ∂_μ A^μ = 0 (manifestly Lorentz-covariant; named for Ludvig Lorenz 1867, NOT Hendrik Lorentz) and the **Coulomb gauge ∇·A** = 0 (frame-dependent but convenient for static problems). The four-potential A^μ is the fundamental dynamical variable of electromagnetism in the Lagrangian formulation, and the EM Lagrangian L = −¼ F_{μν} F^{μν} − A_μ J^μ depends only on derivatives of A^μ — not on A^μ itself in any gauge-invariant combination.