Special Relativity and Electromagnetism

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.

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.

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 $eg = n\hbar/2$.

• Assuming $\mathbf{E}$ and $\mathbf{B}$ transform independently. They do not; the field tensor transforms as a whole under Lorentz boosts. A pure electric field in one frame becomes a mixture in another.
• Confusing the dual tensor $\tilde{F}^{\mu\nu}$ with $F^{\mu\nu}$. The dual swaps electric and magnetic fields ($\mathbf{E} \to c\mathbf{B}$, $\mathbf{B} \to -\mathbf{E}/c$) and is used in the homogeneous Maxwell equation $\partial_\mu \tilde{F}^{\mu\nu} = 0$.
• Forgetting that $F^{\mu\nu}$ is antisymmetric. This antisymmetry encodes the fact that there are six independent field components (three for $\mathbf{E}$, three for $\mathbf{B}$).
• Misapplying the Lorentz force formula. The relativistic Lorentz force $f^\mu = qF^{\mu\nu}u_\nu$ gives the four-force, not the three-force. The spatial components reduce to $d\mathbf{p}/dt = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})$ in the non-relativistic limit.

Problem 1. Show that $\mathbf{E}\cdot\mathbf{B}$ is a Lorentz invariant.

Solution. $\mathbf{E}\cdot\mathbf{B}$ is proportional to $\frac{1}{4}\tilde{F}^{\mu\nu}F_{\mu\nu}$. Since this is a full contraction of two tensors, it is a scalar and thus invariant. Explicitly: $\tilde{F}^{\mu\nu}F_{\mu\nu} = -4\mathbf{E}\cdot\mathbf{B}/c$. Under any Lorentz transformation, both $F^{\mu\nu}$ and $\tilde{F}^{\mu\nu}$ transform as tensors, so their contraction is invariant. $\blacksquare$

Problem 2. Derive the transformation of the Poynting vector under a Lorentz boost.

Solution. $\mathbf{S} = \mathbf{E} \times \mathbf{B} / \mu_0$ transforms as part of the energy-momentum tensor $T^{\mu\nu}$. The components $T^{0i} = S_i/c$ transform under a boost: $S'_x = S_x$, $S'_y = \gamma(S_y - v u)$, $S'_z = \gamma(S_z + v u)$ where $u$ is the energy density. This shows that energy flux in one frame contributes to energy density in another. $\blacksquare$

• Particle physics: The covariant formulation is essential for quantum electrodynamics (QED), where $F^{\mu\nu}$ couples to the Dirac field via minimal coupling $\partial^\mu \to D^\mu$.
• Plasma physics: Relativistic plasmas require the covariant formulation for correct treatment of high-energy particle motion in strong electromagnetic fields.
• Astrophysics: Pulsar electrodynamics and magnetar fields involve enormous Lorentz factors where the field transformation laws govern radiation emission mechanisms.
• Accelerator physics: The design of particle accelerators requires precise knowledge of how electromagnetic fields appear in the rest frame of relativistic particle bunches.