Relativistic Formulation

Electromagnetism was relativistic before relativity was discovered. When Einstein formulated special relativity in 1905, he did not need to modify Maxwell's equations. They were already perfectly consistent with Lorentz invariance. It was Newtonian mechanics that needed fixing. In this lesson, we'll see that when electromagnetism is written in the language of spacetime tensors, Maxwell's four equations collapse into just two, the potentials form a natural 4-vector, and the entire theory fits into a Lagrangian of stunning simplicity.

This is more than an exercise in elegant notation. The relativistic formulation reveals that $\mathbf{E}$ and $\mathbf{B}$ are not separate entities. They are components of a single object, the electromagnetic field strength tensor, mixed together by Lorentz transformations. What one observer calls a purely electric field, another observer moving at a different velocity calls a mixture of electric and magnetic fields. The distinction between electricity and magnetism is frame-dependent.

The 4-Potential

In the previous lesson, we introduced the scalar potential $\phi$ and the vector potential $\mathbf{A}$. These four quantities, one scalar and three vector components, combine naturally into a spacetime 4-vector:

That $(\phi/c, \mathbf{A})$ forms a 4-vector is not a choice. It is forced by the requirement that the theory be Lorentz invariant. Under a Lorentz boost with velocity $v$ along the $x$-axis:

$$\phi' = \gamma(\phi - vA_x), \qquad A_x' = \gamma\left(A_x - \frac{v\phi}{c^2}\right)$$

$A_y$ and $A_z$ are unchanged. The scalar and vector potentials mix under boosts, just as time and space do. This is the first hint that electricity and magnetism are two aspects of the same thing.

The gauge transformation also takes a compact 4-dimensional form:

$$A^\mu \to A^\mu + \partial^\mu \chi$$

where $\partial^\mu = (\partial/\partial(ct), -\nabla)$ and $\chi$ is an arbitrary scalar function. The Lorenz gauge condition becomes:

$$\partial_\mu A^\mu = 0$$

This is manifestly Lorentz invariant, a single equation in spacetime, rather than a relation mixing spatial and temporal derivatives.

The Field Strength Tensor

The electric and magnetic fields are obtained from the potentials by differentiation. In 4-dimensional notation, they combine into a single antisymmetric tensor:

This is remarkable. The six independent components of the antisymmetric tensor $F^{\mu\nu}$ are precisely the three components of $\mathbf{E}$ and the three components of $\mathbf{B}$. They are unified into a single geometric object that lives in spacetime.

Under a Lorentz transformation $\Lambda$, the tensor transforms as:

$$F'^{\mu\nu} = \Lambda^\mu{}_\alpha \Lambda^\nu{}_\beta F^{\alpha\beta}$$

This mixing of components under boosts is why electric and magnetic fields transform into each other. A pure electric field in one frame becomes a combination of $\mathbf{E}$ and $\mathbf{B}$ in another.

Maxwell's Equations in Covariant Form

The greatest payoff of the tensor formulation is the compression of Maxwell's four equations into two. We also need the 4-current density:

$$J^\mu = (c\rho, \mathbf{J})$$

which combines the charge density $\rho$ and current density $\mathbf{J}$ into a single 4-vector. Charge conservation becomes the simple 4-divergence condition $\partial_\mu J^\mu = 0$.

Let's unpack the first equation. Setting $\nu = 0$:

$$\partial_\mu F^{\mu 0} = \mu_0 J^0 = \mu_0 c\rho$$

Writing out the sum: $\partial_1 F^{10} + \partial_2 F^{20} + \partial_3 F^{30} = \mu_0 c\rho$. Reading off the components of $F^{\mu\nu}$, this gives:

$$\frac{\partial(E_x/c)}{\partial x} + \frac{\partial(E_y/c)}{\partial y} + \frac{\partial(E_z/c)}{\partial z} = \mu_0 c\rho$$

which simplifies to $\nabla \cdot \mathbf{E} = \rho/\varepsilon_0$, Gauss's law. Setting $\nu = 1, 2, 3$ recovers the three components of the Ampere-Maxwell law.

The homogeneous equation $\partial_{[\alpha} F_{\beta\gamma]} = 0$ (the antisymmetrized derivative) is actually an identity. It follows automatically from $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ because partial derivatives commute. This is the Bianchi identity, and it encodes both $\nabla \cdot \mathbf{B} = 0$ and Faraday's law.

An equivalent and often more convenient form uses the dual tensor $\tilde{F}^{\mu\nu} = \frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$:

$$\partial_\mu \tilde{F}^{\mu\nu} = 0$$

Lorentz Invariants

From $F^{\mu\nu}$, we can construct scalar quantities that are the same in all reference frames: Lorentz invariants. There are exactly two independent ones:

$$F_{\mu\nu}F^{\mu\nu} = 2\left(B^2 - \frac{E^2}{c^2}\right)$$ $$\tilde{F}_{\mu\nu}F^{\mu\nu} = -\frac{4}{c}\mathbf{E} \cdot \mathbf{B}$$

For electromagnetic waves, both invariants are zero: $E = cB$ and $\mathbf{E} \perp \mathbf{B}$. These properties are Lorentz invariant. Electromagnetic waves look like electromagnetic waves in every inertial frame.

The Lorentz Force as 4-Force

The Lorentz force on a charged particle, $\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$, takes a compact covariant form. The equation of motion for a particle with charge $q$ and 4-velocity $u^\mu$ is:

$$\frac{dp^\mu}{d\tau} = qF^{\mu\nu}u_\nu$$

where $p^\mu = mu^\mu$ is the 4-momentum and $\tau$ is the proper time. The spatial components give the ordinary Lorentz force, and the time component gives the rate at which the field does work on the particle:

$$\frac{dE}{dt} = q\mathbf{E} \cdot \mathbf{v}$$

Note that the magnetic force does no work (it is always perpendicular to $\mathbf{v}$). Only the electric field can change a particle's energy.

The Electromagnetic Lagrangian

The entire theory of classical electromagnetism, Maxwell's equations, the Lorentz force, energy-momentum conservation, follows from a single Lagrangian density.

Let's see how. The action is $S = \int \mathcal{L} \, d^4x$. Varying with respect to $A_\nu$ and setting $\delta S = 0$ gives the Euler-Lagrange equation:

$$\partial_\mu \frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\nu)} = \frac{\partial \mathcal{L}}{\partial A_\nu}$$

The left side gives $-\frac{1}{\mu_0}\partial_\mu F^{\mu\nu}$ and the right side gives $-J^\nu$, yielding:

$$\partial_\mu F^{\mu\nu} = \mu_0 J^\nu$$

This is precisely the inhomogeneous Maxwell equation! The homogeneous equation is automatic from the definition $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$. The entire content of Maxwell's equations emerges from a single variational principle.

The Lagrangian $\mathcal{L} = -\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}$ is essentially the simplest possible Lorentz-invariant, gauge-invariant scalar that can be built from the field strength. Higher-order terms like $(F_{\mu\nu}F^{\mu\nu})^2$ are allowed by symmetry but are suppressed at low energies. They correspond to quantum corrections (the Euler-Heisenberg effective Lagrangian).

The Energy-Momentum Tensor

The energy and momentum of the electromagnetic field are encoded in the stress-energy tensor:

$$T^{\mu\nu} = \frac{1}{\mu_0}\left(F^{\mu\alpha}F^\nu{}_\alpha - \frac{1}{4}\eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right)$$

The components have physical meanings:

• $T^{00} = \frac{1}{2}\left(\varepsilon_0 E^2 + \frac{B^2}{\mu_0}\right)$ is the energy density
• $T^{0i}/c = \varepsilon_0(\mathbf{E} \times \mathbf{B})_i$ is the momentum density (the Poynting vector divided by $c^2$)
• $T^{ij}$ is the Maxwell stress tensor (the flow of momentum)

Conservation of energy and momentum is expressed compactly as $\partial_\mu T^{\mu\nu} = -F^{\nu\alpha}J_\alpha$, where the right side is the rate at which the field transfers energy-momentum to matter.

Why Electromagnetism Naturally Lives in Spacetime

It is no accident that electromagnetism is naturally relativistic. Historically, the inconsistency between Maxwell's equations and Galilean relativity was the puzzle that led Einstein to special relativity. Maxwell's equations predict a universal speed of light $c$. They don't say "the speed of light relative to any particular observer" but just "the speed of light." This demanded a new understanding of space and time.

In hindsight, the relativistic structure was always there. The fact that $\mu_0\varepsilon_0 = 1/c^2$ is not a coincidence. It's the signature of a theory built on Lorentz symmetry. The fact that four equations collapse into two tensor equations is not just notation. It reflects the underlying unity of spacetime. And the Lagrangian $-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}$ is the unique simplest theory consistent with Lorentz and gauge invariance.

Key Insights

• The potentials $(\phi/c, \mathbf{A})$ form a Lorentz 4-vector $A^\mu$
• The fields $\mathbf{E}$ and $\mathbf{B}$ are components of the antisymmetric tensor $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$
• Maxwell's four equations compress to two: $\partial_\mu F^{\mu\nu} = \mu_0 J^\nu$ and $\partial_{[\alpha}F_{\beta\gamma]} = 0$
• The Lorentz invariants $B^2 - E^2/c^2$ and $\mathbf{E} \cdot \mathbf{B}$ classify electromagnetic fields
• The Lorentz force law becomes $dp^\mu/d\tau = qF^{\mu\nu}u_\nu$
• The entire theory follows from the Lagrangian $\mathcal{L} = -\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu} - J_\mu A^\mu$
• Electromagnetism was the first, and remains the clearest, example of a relativistic gauge field theory

Looking Ahead

With electromagnetism now in its natural relativistic language, we're ready to dive deep into special relativity itself. In Phase 4, we'll develop the full framework of Lorentz transformations, spacetime geometry, 4-vectors, and relativistic mechanics. We'll see how the concepts we've just used (4-vectors, tensors, invariants) apply to all of physics, and how the simple postulate that the laws of physics are the same in all inertial frames reshapes our understanding of space, time, energy, and mass.