Covariant Electrodynamics and the Field Tensor

Maxwell's laws extend steady-current loops to changing fields. We now describe E and B together using Lorentz transformations. Use SI units, flat spacetime, coordinates (ct,x,y,z), and metric eta = diag(1,-1,-1,-1). Greek indices run over 0,x,y,z; repeated indices are summed. The field tensor obeys F^{mu nu} = -F^{nu mu}, so its diagonal is zero. An antisymmetric n by n tensor has n(n-1)/2 independent entries: each off-diagonal pair supplies one value.

With rows and columns ordered (0,x,y,z), set diagonal entries to zero. Use F^{0i} = -Ei/c and F^{i0} = Ei/c, with 1=x, 2=y and 3=z. Set F^{12} = -Bz, F^{23} = -Bx and F^{31} = -By. Fill each reversed-index entry with the opposite sign. These rules assemble F from the three electric and three magnetic components.

Let S' move with velocity v relative to S, with parallel spatial axes. If B=0 at an event in S, then E'_parallel = E_parallel, E'_perp = gamma E_perp, and B' = -gamma(v cross E)/c². Here gamma = 1/sqrt(1-|v|²/c²). Parallel and perpendicular refer to v. A nonzero transverse E produces B'; if v is parallel to E, B' remains zero.

The four-current is J^mu = (c rho, jx, jy, jz). Here rho is charge density and j is current density. The factor c gives all four entries the same units. Under a Lorentz transformation Lambda, J'^mu = Lambda^mu_nu J^nu. Charge-density and current-density components can mix between frames.

In vacuum with charge and current sources, ∂_mu F^{mu nu} = mu0 J^nu. Here ∂_mu differentiates with respect to x^mu, so ∂_0 = (1/c)∂t. Vacuum constants obey 1/c² = mu0 eps0. The nu=0 component gives div E = rho/eps0. The spatial components give curl B - (1/c²)∂t E = mu0 j: Ampere-Maxwell's law.

Lower indices using F_{mu nu} = eta_{mu alpha} eta_{nu beta} F^{alpha beta}. The homogeneous equation is ∂_lambda F_{mu nu} + ∂_mu F_{nu lambda} + ∂_nu F_{lambda mu} = 0. Its spatial-index part gives div B = 0. Its time-space parts give curl E + ∂t B = 0. This pair has no source term, even where electric charges or currents are present.

Transform both indices: F'^{mu nu} = Lambda^mu_alpha Lambda^nu_beta F^{alpha beta}, or F' = Lambda F Lambda^T in matrix form. Read E'i = -c F'^{0i}, B'x = -F'^{23}, B'y = -F'^{31}, and B'z = -F'^{12}. Transform J too when sources are needed. Direct E and B transformation formulas are also valid when their boost and sign conventions match.