Electrodynamics

Relativistic electrodynamics

Electric and magnetic fields are components of the electromagnetic tensor; Lorentz transformations mix E and B while preserving Maxwell’s structure.

Electricity and magnetism are not wholly separate: relatively moving observers measure different combinations of the same tensor Fμν. Covariant notation packages Maxwell’s four equations into two tensor equations.

∂_μF^{μν} = μ₀J^ν, ∂_[α F_{βγ]} = 0

Definition: Quantities and model

With x^μ = (ct,r), the four-current is J^μ = (cρ,J). Tensor signs depend on metric convention; the covariant equations are equivalent to SI Maxwell and imply ∂_μJ^μ = 0.

Adjust parameters and rotate the view to inspect field structure; this is illustrative, not a general Maxwell solver.

Interpretation and consequences

Under a boost with velocity v, E parallel to v is unchanged; transverse components mix with B (γ = 1/√(1−v²/c²)).

Example: Quantitative example

If E = 0 and B = 1 T in the lab, an observer moving perpendicular to B at v = 0.6c measures transverse electric field γvB = 0.75c T (in SI, V/m), since γ = 1.25.

Solution

Substitute into the stated relation, keep SI units consistent, and check the result dimensionally.

Two Lorentz scalars characterize the field: $F_{\mu\nu}F^{\mu\nu}$ and $F_{\mu\nu}\,{}^*F^{\mu\nu}$ , corresponding to combinations of $E^2-c^2B^2$ and $E\cdot B$ (up to convention-dependent factors). If the invariants permit an electric- or magnetic-dominated rest frame, one can choose a frame where the other field vanishes. If both are nonzero, no inertial frame can eliminate either field completely.

Covariance is more than compact notation: it ensures that every inertial observer describes the same dynamics. The four-divergence of the electromagnetic stress-energy tensor balances the Lorentz force on matter; field energy and momentum can exchange with sources, while the closed total system is conserved. When coupling field equations to particle motion, coordinates, four-current, and field tensor must all transform under the same boost.

Metric and sign conventions for $F^{\mu\nu}$ change intermediate signs, but not physical predictions when definitions are applied consistently.

Quick check

Which relation is correct in the idealized situation described?

What should be checked first when applying a field formula?

References

John David Jackson (1998). Classical Electrodynamics