Electromagnetic field tensor

The electromagnetic field tensor F^{μν} is the rank-2 antisymmetric 4×4 tensor that packages the three components of the electric field and three components of the magnetic field into one Lorentz-covariant object. Its components in mostly-minus signature (Griffiths/Jackson convention) are F^{0i} = E_i/c (the time-space components encode E divided by c for unit-matching) and F^{ij} = −ε_{ijk} B_k (the space-space components encode B with the Levi-Civita epsilon). The diagonal F^{μμ} = 0 by antisymmetry, and the six independent components below the diagonal are determined by the antisymmetry F^{μν} = −F^{νμ}: exactly the right number to hold the three E and three B values.

The tensor's defining property is that it transforms as F'^{μν} = Λ^μ_α Λ^ν_β F^{αβ} under Lorentz boosts, which automatically generates the closed-form §11.2 transformation of E and B from a single rule. The two compact equations ∂_μ F^{μν} = μ₀ J^ν and ∂_μ F^{μν} = 0 (where F^{μν} = ½ ε^{μνρσ} F_{ρσ} is the dual tensor) reproduce all four of Maxwell's equations: the first encodes Gauss's law (ν = 0) and Ampère–Maxwell (ν = i); the second encodes Faraday's law and the absence of magnetic monopoles. The tensor formulation also makes Lorentz invariants visible as tensor traces: ½ F^{μν} F_{μν} ∝ c²|B|² − |E|²/c² is the kinetic part of the EM Lagrangian, and ¼ F^{μν} F_{μν} ∝ −E·B*/c is the parity-odd invariant. Hermann Minkowski's 1907–1908 reformulation made this packaging the natural starting point for relativistic electrodynamics.