Covariant Electromagnetism

Electromagnetism is natively relativistic: Maxwell's equations already respect the Lorentz symmetry, and SR was historically born from making mechanics match them. Written in four-vector form they become two compact, manifestly invariant equations. Convention: $\eta =\mathrm{diag}(+,-,-,-)$, $c=1$, Heaviside–Lorentz units.

Potentials and field tensor

The scalar/vector potentials combine into $A^{\mu }=(\varphi ,\mathbf{A})$. The antisymmetric field-strength tensor

$$F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$$

packs $\mathbf{E}$ and $\mathbf{B}$: $F_{0i}=E_i$, $F_{ij}=-{ϵ}_{ijk}{B}_k$. It is gauge-invariant under $A_{\mu }\to A_{\mu }+{\partial }_{\mu }\chi$.

Maxwell's equations, covariant

The four Maxwell equations reduce to

$${\partial }_{\mu }{F}^{\mu \nu }=j^{\nu },{\partial }_{[\mu }{F}_{\nu \rho ]}=0,$$

with current $j^{\mu }=(\rho ,\mathbf{j})$. Charge conservation ${\partial }_{\mu }{j}^{\mu }=0$ is automatic. In Lorenz gauge ${\partial }_{\mu }{A}^{\mu }=0$ this is $\square A^{\nu }=j^{\nu }$ — the wave equation, with $c$ built in.

Field transformations

Since $F^{\mu \nu }$ is a tensor, $\mathbf{E}$ and $\mathbf{B}$ mix under boosts: a pure electric field in one frame has a magnetic component in another. The invariants are $F_{\mu \nu }{F}^{\mu \nu }\propto B^2-E^2$ and ${ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }{F}_{\rho \sigma }\propto \mathbf{E}\cdot \mathbf{B}$. Magnetism is the relativistic shadow of electrostatics.

Lorentz force

The covariant force on charge $q$ is $f^{\mu }=q{F}^{\mu \nu }{u}_{\nu }$, recovering $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$ and tying into four-force.

This covariance is the template QFT promotes to operators — see bridge to QFT.