Special Relativity and the Covariant Form of Maxwell's Equations

Einstein’s special relativity (1905) was, in a deep sense, already implicit in Maxwell’s equations of electromagnetism (1865). The requirement that the speed of light $c$ be the same in all inertial frames forces a revision of classical kinematics — and, in return, reveals that the six components of E and B are simply different aspects of a single antisymmetric tensor.

Postulates and the Lorentz transformation

Two postulates underlie special relativity:

The laws of physics are the same in all inertial frames.
The speed of light in vacuum is $c$ in all inertial frames.

These force the Lorentz transformation between frames $S$ and $S'$ (moving at velocity $v$ along $x$):

\begin{equation}\label{eq.lorentz} t’ = \gamma!\left(t - \frac{vx}{c^2}\right), \quad x’ = \gamma(x - vt), \quad y’ = y, \quad z’ = z, \end{equation}

with the Lorentz factor $\gamma = (1 - v^2/c^2)^{-1/2}$. The key consequences are time dilation ($\Delta t' = \gamma\,\Delta t_0$) and length contraction ($\Delta x' = \Delta x_0/\gamma$).

Spacetime and four-vectors

Minkowski (1907) realised the correct arena is spacetime $\mathbb{R}^{1,3}$ with the metric

\begin{equation}\label{eq.metric} ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 = \eta_{\mu\nu}\, dx^\mu dx^\nu, \end{equation}

where $\eta_{\mu\nu} = \text{diag}(-1,+1,+1,+1)$ is the Minkowski metric (using the $(-+++)$ signature). Greek indices run $0,1,2,3$.

The four-position is $x^\mu = (ct, x, y, z)$. The Lorentz transformation \eqref{eq.lorentz} is a linear map $x'^\mu = \Lambda^\mu{}_\nu x^\nu$ that preserves $\eta_{\mu\nu}$:

\begin{equation} \Lambda^\alpha{}\mu \,\eta{\alpha\beta}\, \Lambda^\beta{}\nu = \eta{\mu\nu}. \end{equation}

Any set of four quantities transforming the same way as $x^\mu$ is a four-vector. Key examples:

\begin{equation} p^\mu = (E/c,\, \mathbf{p}), \quad j^\mu = (c\rho,\, \mathbf{J}), \quad A^\mu = (\phi/c,\, \mathbf{A}), \end{equation}

where $p^\mu$ is the four-momentum, $j^\mu$ the four-current, and $A^\mu$ the four-potential.

The invariant $p_\mu p^\mu = -m^2c^2$ gives the energy-momentum relation:

\begin{equation} \boxed{E^2 = (pc)^2 + (mc^2)^2.} \end{equation}

The electromagnetic field tensor

The electric and magnetic fields are encoded in the antisymmetric field-strength tensor

\begin{equation}\label{eq.F} F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu, \end{equation}

where $\partial_\mu = (\partial_t/c, \nabla)$. In matrix form:

\begin{equation} 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}. \end{equation}

Under a Lorentz boost along $x$ with velocity $v$, the components mix:

\begin{equation} E’_x = E_x, \quad E’_y = \gamma(E_y - vB_z), \quad E’_z = \gamma(E_z + vB_y), \end{equation} \begin{equation} B’_x = B_x, \quad B’_y = \gamma!\left(B_y + \frac{v}{c^2}E_z\right), \quad B’_z = \gamma!\left(B_z - \frac{v}{c^2}E_y\right). \end{equation}

A purely electric field in one frame has a magnetic component in another — E and B are not independently Lorentz-covariant. They are two faces of $F_{\mu\nu}$.

Maxwell’s equations in covariant form

Maxwell’s four equations reduce to two in covariant notation.

The inhomogeneous equations (Gauss’s law + Ampère’s law) become

\begin{equation}\label{eq.inh} \boxed{\partial_\nu F^{\mu\nu} = \mu_0 j^\mu.} \end{equation}

The homogeneous equations (Gauss’s law for magnetism + Faraday’s law) follow automatically from the definition \eqref{eq.F} via the Bianchi identity:

\begin{equation}\label{eq.hom} \boxed{\partial_{[\lambda} F_{\mu\nu]} = 0,} \end{equation}

or equivalently $\partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0$.

Equation \eqref{eq.inh} encodes:

$\mu = 0$: $\nabla \cdot \mathbf{E} = \rho/\epsilon_0$ (Gauss)
$\mu = i$: $\nabla \times \mathbf{B} - \dot{\mathbf{E}}/c^2 = \mu_0 \mathbf{J}$ (Ampère-Maxwell)

Equation \eqref{eq.hom} encodes:

$\lambda\mu\nu = 123$: $\nabla \cdot \mathbf{B} = 0$
other components: $\nabla \times \mathbf{E} + \dot{\mathbf{B}} = 0$ (Faraday)

In the covariant form, the invariance of Maxwell’s equations under Lorentz transformations is manifest: both sides of \eqref{eq.inh} are four-tensors, so the equation holds in all inertial frames.

The two Lorentz invariants

From $F_{\mu\nu}$ one can form two independent Lorentz scalars:

\begin{equation} \mathcal{F} = \tfrac{1}{2}F_{\mu\nu}F^{\mu\nu} = B^2 - E^2/c^2, \end{equation} \begin{equation} \mathcal{G} = \tfrac{1}{4}F_{\mu\nu}\tilde{F}^{\mu\nu} = \mathbf{E}\cdot\mathbf{B}/c, \end{equation}

where $\tilde{F}^{\mu\nu} = \frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$ is the dual tensor. These are the only independent invariants of the electromagnetic field.

The Lorentz force and equations of motion

The four-force on a charged particle is

\begin{equation} \frac{dp^\mu}{d\tau} = q F^{\mu\nu} u_\nu, \end{equation}

where $u^\mu = dx^\mu/d\tau = \gamma(c, \mathbf{v})$ is the four-velocity. The spatial components reproduce the Lorentz force law $\mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})$; the time component gives the power $dE/dt = q\mathbf{E}\cdot\mathbf{v}$.

From Maxwell to photons

The wave equation follows from \eqref{eq.inh} in vacuum ($j^\mu = 0$) in Lorenz gauge ($\partial_\mu A^\mu = 0$):

\begin{equation} \Box A^\mu = 0, \quad \Box \equiv \partial_\mu \partial^\mu = -\frac{1}{c^2}\partial_t^2 + \nabla^2. \end{equation}

Plane-wave solutions $A^\mu \propto e^{ik_\nu x^\nu}$ require $k_\mu k^\mu = 0$, i.e. $$\omega = c

\mathbf{k}

$— the dispersion relation of **massless photons**. The masslessness of the photon is equivalent to gauge invariance of$A^\mu$$, which is equivalent to the homogeneous equation \eqref{eq.hom}. Special relativity, electromagnetism, and quantum field theory are deeply intertwined from the very start.