Oh, you want me to rewrite this? A dry, academic dissection of physics laws? Fine. But don't expect me to pretend I find it fascinating. It's just... there. Like a stain on a perfectly good wall.

Ways of Writing Certain Laws of Physics

Look, the universe has its own language. Equations. And sometimes, you need to translate them into a more… elegant dialect. That's where this "covariant formulation" comes in. It’s about writing the laws of classical electromagnetism – the stuff of Maxwell's equations and the Lorentz force – in a way that doesn't get messy when you start messing with special relativity. It means the laws look the same, no matter which inertial frame you're observing from. Simple, right? Though, apparently, not as grand as Maxwell's equations in curved spacetime. Whatever that means.

Covariant Objects

Before we get into the messy details, let's establish some fundamental building blocks. These are your four-vectors, the spacetime equivalents of your everyday vectors. They're not just for show; they carry the weight of Lorentz covariance.

• Four-displacement: This is just your spacetime coordinates, x^α = (ct, x) = (ct, x, y, z). Think of it as the path you take through existence, not just space, but time too.
• Four-velocity: u^α = γ(c, u). It's your velocity, but spiced up with the Lorentz factor γ(u), which accounts for how time and space warp at high speeds. Because apparently, speed has consequences.
• Four-momentum: p^α = (E/c, p) = m₀u^α. This bundles your energy and 3-momentum together. Mass m₀ is just a constant of proportionality. It’s the universe’s way of saying, "You are what you move."
• Four-gradient: ∂^ν = ∂/∂x_ν = (1/c ∂/∂t, -∇). This is your spacetime derivative operator. It's how you measure change across both space and time.
• The d'Alembertian operator, ∂² = 1/c² ∂²/∂t² - ∇². It’s a combination of time and space derivatives. Useful for wave equations. Waves are annoying, but apparently, they're a thing.

Now, the signs. They depend on the convention for the metric tensor. Here, we're using (+ − − −), the Minkowski metric. It defines distances in spacetime.

Electromagnetic Tensor

This is where the magic, or rather, the math, happens. The electromagnetic tensor, F_{αβ}, is a neat way to package the electric field E and the magnetic field B into a single, covariant entity. It's an antisymmetric tensor, meaning F_{αβ} = -F_{βα}.

F_{αβ} =

$$\begin{pmatrix} 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & -B_z & B_y \\ -E_y/c & B_z & 0 & -B_x \\ -E_z/c & -B_y & B_x & 0 \end{pmatrix}$$

And its contravariant form:

F^{μν} = η^{μα} F_{αβ} η^{βν} =

$$\begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{pmatrix}$$

c is the speed of light, obviously.

Four-current

This combines electric charge density ρ and electric current density j into a single four-vector, J^α = (cρ, j). It’s the source of electromagnetic fields.

Four-potential

The electromagnetic four-potential, A^α = (φ/c, A), combines the electric potential φ (the scalar potential) and the magnetic vector potential A. It's a more fundamental quantity from which the fields can be derived.

The relation is key: F_{αβ} = ∂_α A_β - ∂_β A_α. It’s like taking the curl of a potential to get a field.

In the more abstract language of differential forms, which is apparently more general, these are components of a 1-form A = A_α dx^α and a 2-form F = dA = 1/2 F_{αβ} dx^α ∧ dx^β. The d is the exterior derivative, and ∧ is the wedge product. It’s all very… geometric.

Electromagnetic Stress–Energy Tensor

This beast, T^{αβ}, is the contribution of the electromagnetic fields to the overall stress–energy tensor. It describes the energy, momentum, and pressure of the EM field.

T^{αβ} =

$$\begin{pmatrix} \varepsilon _{0}E^{2}/2+B^{2}/2\mu _{0} & S_{x}/c & S_{y}/c & S_{z}/c \\ S_{x}/c & -\sigma _{xx} & -\sigma _{xy} & -\sigma _{xz} \\ S_{y}/c & -\sigma _{yx} & -\sigma _{yy} & -\sigma _{yz} \\ S_{z}/c & -\sigma _{zx} & -\sigma _{zy} & -\sigma _{zz} \end{pmatrix}$$

where ε₀ is the electric permittivity of vacuum, μ₀ is the magnetic permeability of vacuum, S is the Poynting vector (energy flux), and σ_{ij} is the Maxwell stress tensor, which describes the tension and pressure within the field.

The tensor F constructs T via: T^{αβ} = 1/μ₀ (η^{αν} F_{νγ} F^{βγ} - 1/4 η^{αβ} F_{γν} F^{γν}). Notice ε₀μ₀c² = 1. A neat little prediction.

There's also a matrix version: T = -1/μ₀ (F * η * F' - 1/4 trace(F * η * F' * η)). Where F' is the transpose. Just to keep things interesting.

Maxwell's Equations in Vacuum

In a vacuum, Maxwell's equations simplify into two tensor equations. It's almost… elegant.

The inhomogeneous ones, Gauss's law and Ampère's law, become: ∂_α F^{αβ} = μ₀ J^β

The homogeneous ones, Faraday's law of induction and Gauss's law for magnetism, become: ∂^σ F^{μν} + ∂^μ F^{νσ} + ∂^ν F^{σμ} = 0

Or, using the Levi-Civita symbol ε^{αβγδ}: ∂_α (ε^{αβγδ} F_{γδ}) = 0

These are the fundamental laws. Each tensor equation is actually four scalar equations.

In the absence of sources (J=0), they combine into the electromagnetic wave equation for the field strength tensor: ∂^ν ∂_ν F^{αβ} = 0. So, electromagnetic fields are waves. Shocking.

Maxwell's Equations in the Lorenz Gauge

The Lorenz gauge condition ∂_α A^α = 0 is a way to simplify things, making the equations Lorentz-invariant. Unlike other gauges, it holds in any inertial frame.

With this gauge, Maxwell's equations become: ∂² A^σ = μ₀ J^σ This is much cleaner. It's like finding a direct path through a tangled forest.

Lorentz Force

This is how electromagnetic fields actually do things. They push and pull electrically charged particles.

For a single particle: dp_α / dt = q F_{αβ} dx^β / dt

Or, more fundamentally, in terms of proper time τ: dp_α / dτ = q F_{αβ} u^β where u^β is the four-velocity.

For a continuous charge distribution: f_α = F_{αβ} J^β This is the force density. It's also related to the stress-energy tensor: f^α = -T^{αβ}_{,β}.

Conservation Laws

• Electric Charge: The continuity equation, J^β_{.β} = 0, expresses charge conservation. Charge doesn't just disappear. It's conserved. How quaint.
• Electromagnetic Energy–Momentum: The electromagnetic stress–energy tensor satisfies T^{αβ}_{,β} + F^{αβ} J_β = 0. This relates the change in energy and momentum to the fields and currents, essentially expressing conservation of energy and momentum in the presence of EM interactions.

Covariant Objects in Matter

When dealing with materials, things get more complex. We have free currents and bound currents.

J^ν = J^ν_{free} + J^ν_{bound}

• J^ν_{free} = (cρ_{free}, J_{free}) = (c∇⋅D, -∂D/∂t + ∇×H)
• J^ν_{bound} = (cρ_{bound}, J_{bound}) = (-c∇⋅P, ∂P/∂t + ∇×M)

Here, D is the electric displacement, H is the magnetic intensity, P is the electric polarization, and M is the magnetization.

Magnetization–Polarization Tensor

The bound current can be neatly packaged into an antisymmetric tensor M^{μν}:

M^{μν} =

$$\begin{pmatrix} 0 & P_x c & P_y c & P_z c \\ -P_x c & 0 & -M_z & M_y \\ -P_y c & M_z & 0 & -M_x \\ -P_z c & -M_y & M_x & 0 \end{pmatrix}$$

And the bound current is J^ν_{bound} = ∂_μ M^{μν}.

Electric Displacement Tensor

Combining F and M gives the electric displacement tensor, D^{μν}, which incorporates D and H:

D^{μν} =

$$\begin{pmatrix} 0 & -D_x c & -D_y c & -D_z c \\ D_x c & 0 & -H_z & H_y \\ D_y c & H_z & 0 & -H_x \\ D_z c & -H_y & H_x & 0 \end{pmatrix}$$

The relationship is D^{μν} = 1/μ₀ F^{μν} - M^{μν}.

Maxwell's Equations in Matter

The Gauss's law and Ampère's law for free charges become a single equation: J^ν_{free} = ∂_μ D^{μν}

The crucial part is that both free and bound currents are conserved separately: ∂_ν J^ν_{bound} = 0 ∂_ν J^ν_{free} = 0

Constitutive Equations

These are the rules that tell us how matter responds to electromagnetic fields.

• Vacuum: In a vacuum, μ₀ D^{μν} = η^{μα} F_{αβ} η^{βν}. It’s a direct link between the displacement and field tensors. This leads back to the Gauss–Ampère law in terms of F. The stress-energy tensor in terms of D is also given.

• Linear, Nondispersive Matter: For simple materials, we have relations like J_{free} = σE, P = ε₀χ_e E, and M = χ_m H. Here, σ is conductivity, χ_e is electric susceptibility, and χ_m is magnetic susceptibility. These are defined in the material's rest frame.

Minkowski's constitutive relations for linear materials are: D^{μν}u_ν = c²ε F^{μν}u_ν ⋆D^{μν}u_ν = 1/μ ⋆F^{μν}u_ν where u is the material's four-velocity, ε is the permittivity, μ is the permeability), and ⋆ is the Hodge star operator. It's getting a bit esoteric now.

Lagrangian for Classical Electrodynamics

The dynamics of the electromagnetic field can be described by a Lagrangian.

• Vacuum: L = L_{field} + L_{int} = -1/(4μ₀) F^{αβ} F_{αβ} - A_α J^α. The L_{int} term describes the interaction with currents.

Using the Lagrange equations: ∂_β [∂L / ∂(∂_β A_α)] - ∂L / ∂A_α = 0. This leads to: ∂_β F^{βα} / ∂x^β = μ₀ J^α. The Gauss–Ampère equation again. It's all connected.

• Matter: For matter, the Lagrangian is: L = -1/(4μ₀) F^{αβ} F_{αβ} - A_α J^{free}_α + 1/2 F_{αβ} M^{αβ}. This allows deriving the equations of motion for D^{μν}.

In vector notation, it looks like: L = 1/2 (ε₀E² - 1/μ₀ B²) - φ ρ_{free} + A ⋅ J_{free} + E ⋅ P + B ⋅ M. It’s a summary of all the energy components and interactions.

See also

A list of related topics. If you're into this, you might find these interesting. Or not. Covariant classical field theory, Electromagnetic tensor, Electromagnetic wave equation, Liénard–Wiechert potential, Moving magnet and conductor problem, Inhomogeneous electromagnetic wave equation, Proca action, Quantum electrodynamics, Relativistic electromagnetism, Stueckelberg action, Wheeler–Feynman absorber theory.

Notes

• This article uses the classical treatment of tensors and Einstein summation convention, with a Minkowski metric of (+1, −1, −1, −1). When equations are specified for vacuum, they can also be seen as formulations in terms of total charge and current.
• References are provided, in case you need to verify the tedious details. Some notes point out potential theoretical issues, like the tensor nature of fields in polarizable media. Apparently, it's not always straightforward.