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 $eg = n\hbar/2$.

flowchart TD
    A[12_Special Relativity And Electromagnetism 12] --> B[Key Concepts]
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Special relativity and electromagnetism are inseparable: Maxwell’s equations are already relativistically correct, and the electric and magnetic fields are merely different aspects of a single electromagnetic field tensor. What one observer calls a pure electric field, a moving observer sees as a mixture of electric and magnetic components. The field tensor $F^{\mu\nu}$ packages all six field components into a single mathematical object that transforms cleanly under Lorentz boosts. The Lorentz force law becomes a compact four-vector equation, and the conservation of energy-momentum extends to include field contributions. The key physical picture is that electricity and magnetism are not separate forces but different faces of the same relativistic coin, unified by the geometry of spacetime.

• 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.

• Electromagnetic Waves — The plane-wave solutions and energy-momentum tensor provide the starting point for the covariant field analysis.

• Special Relativity and Electromagnetism (Ch. 7) — The field tensor and Lorentz transformation laws are developed in detail in the earlier chapter on special relativity.

• Radiation from Accelerating Charges — The synchrotron radiation patterns arise from the field configurations of ultrarelativistic charges treated here.

• Calculus

• Linear Algebra

• Vector Calculus

• Quantum Computing

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

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