Special Relativity and Electromagnetism

12.1 Covariant Formulation

Maxwell’s equations in covariant form using the field tensor $F^{\mu\nu}$:

$\partial_\mu F^{\mu\nu} = \mu_0 J^\nu \quad \text{(inhomogeneous)}$

$\partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0 \quad \text{(homogeneous / Bianchi identity)}$

The electromagnetic field tensor:

$F^{\mu\nu} = \begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{pmatrix}$

The dual tensor: $\tilde{F}^{\mu\nu} = \frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$.

The Lorentz force: $f^\mu = qF^{\mu\nu}u_\nu$ where $u^\nu = \gamma(c, \mathbf{v})$ is the four-velocity.

12.2 Lorentz Transformation of Fields

Under a boost with velocity $v$ along the $x$-axis:

$E'_x = E_x, \quad B'_x = B_x$

$E'_y = \gamma(E_y - vB_z), \quad B'_y = \gamma\!\left(B_y + \frac{v}{c^2}E_z\right)$

$E'_z = \gamma(E_z + vB_y), \quad B'_z = \gamma\!\left(B_z - \frac{v}{c^2}E_y\right)$

Key insight: $\mathbf{E}$ and $\mathbf{B}$ mix under Lorentz transformations. What appears as a pure electric field in one frame becomes a mixture of electric and magnetic fields in another. There is no frame-independent distinction between $\mathbf{E}$ and $\mathbf{B}$.

Invariants: $E^2 - c^2B^2$ and $\mathbf{E}\cdot\mathbf{B}$ are Lorentz invariants. A pure radiation field ($E = cB$, $\mathbf{E}\perp\mathbf{B}$) satisfies both invariants being zero.

12.3 Electromagnetic Field Momentum and Angular Momentum

Field momentum density:

$\mathbf{g} = \frac{\mathbf{S}}{c^2} = \varepsilon_0\mathbf{E} \times \mathbf{B}$

Field angular momentum: $\mathbf{L} = \int \mathbf{r} \times \mathbf{g}\, d^3r$.

Conservation: \frac{d}{dt}\left(\mathbf{p}_{\text{mech} + \mathbf{p}_{\text{field}\right) = 0}}.

For a charge and a magnetic monopole (if they exist), the field angular momentum $\mathbf{L} = -qg\hat{\mathbf{r}}/(4\pi)$ is quantised in units of $\hbar/2$Leading to the Dirac charge quantisation condition $eg = n\hbar/2$.