Maxwell Equations in Covariant Form
Scope
These notes build on the calculus tools developed previously and show how Maxwell equations naturally unify into a single geometric statement on spacetime.
The key steps are:
- recognizing that \(\mathbf{E}\) and \(\mathbf{B}\) are observer-dependent projections,
- combining them into the Faraday tensor \(F_{\mu\nu}\),
- writing homogeneous and non-homogeneous Maxwell equations covariantly,
- understanding the geometric role of differential forms in this framework.
From 3D Vector Calculus to Covariant Electromagnetism
Motivation
Classical electromagnetism is usually formulated with a preferred time slicing, where fields are split into spatial vectors \(\mathbf{E}\) and \(\mathbf{B}\). This formulation is not covariant: under Lorentz transformations, time mixes with space, and electric and magnetic fields mix with each other.
This suggests that \(\mathbf{E}\) and \(\mathbf{B}\) are not fundamental objects. Instead, they should arise as components of a single spacetime object.
Our goal is to rewrite Maxwell equations in a form that:
- does not depend on a preferred frame,
- is manifestly Lorentz covariant,
- has a clear geometric meaning.
Homogeneous vs Non-Homogeneous Equations
Maxwell equations naturally split into two groups:
Non-homogeneous (contain sources): \[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}, \qquad \nabla \times \mathbf{B} - \frac{1}{c^2}\partial_t \mathbf{E} = \mu_0\mathbf{J}, \] where \(\rho\) is the charge density, \(\mathbf{J}\) is the current density, and \(\varepsilon_0\), \(\mu_0\) are the vacuum permittivity and permeability. Finally, \(c = 1/\sqrt{\varepsilon_0 \mu_0}\) is the speed of light.
Homogeneous (purely geometric): \[ \nabla \cdot \mathbf{B} = 0, \qquad \nabla \times \mathbf{E} + \partial_t \mathbf{B} = 0. \]
The homogeneous equations do not depend on sources. This suggests they encode a purely geometric structure.
From Vectors to Forms
To make the geometry explicit, we reinterpret fields as differential forms:
- Electric field \(\mathbf{E}\) becomes a 1-form: \[ E = E_i \,\mathrm{d}x^i, \]
- Magnetic field \(\mathbf{B}\) becomes a 2-form: \[ B = \frac{1}{2} B_{ij} \,\mathrm{d}x^i \wedge \mathrm{d}x^j. \]
The magnetic field is naturally a 2-form because it is associated with flux through surfaces, which are 2-dimensional.
Unification into a Single Spacetime Equation
The Key Observation
The homogeneous equations:
- Faraday law: \(\nabla \times \mathbf{E} + \partial_t \mathbf{B} = 0\),
- No magnetic monopoles: \(\nabla \cdot \mathbf{B} = 0\),
are actually different components of a single spacetime equation.
Spacetime Coordinates
Introduce spacetime coordinates with signature \((-,+,+,+)\): \[ x^0 = ct, \qquad x^i \quad (i=1,2,3) \text{ spatial}. \]
The Electromagnetic Field Tensor
Define the Faraday tensor \(F_{\mu\nu}\) as a 2-form on spacetime with components: \[ F_{0i} = -\frac{E_i}{c}, \qquad F_{ij} = B_{ij}=\epsilon_{ijk}B_k, \] and antisymmetry \(F_{\mu\nu} = -F_{\nu\mu}\). This matches the conventions, where the same tensor is fixed by \(F^{0i}=E^i/c\), \(F^{ij}=\epsilon^{ijk}B_k\).
In matrix form: \[ F_{\mu\nu} = \begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & B_z & -B_y \\ E_y/c & -B_z & 0 & B_x \\ E_z/c & B_y & -B_x & 0 \end{pmatrix}. \]
The Homogeneous Maxwell Equation
The homogeneous Maxwell equations become: \[ \mathrm{d}F = 0. \]
This is a single equation on spacetime, encoding all four components:
- the 3 spatial components of Faraday’s law,
- the 1 component of \(\nabla \cdot \mathbf{B} = 0\).
Verification in Components
The equation \(\mathrm{d}F = 0\) expands to: \[ \partial_\alpha F_{\mu\nu} + \partial_\mu F_{\nu\alpha} + \partial_\nu F_{\alpha\mu} = 0. \]
For \((\alpha,\mu,\nu) = (0,i,j)\), using \(\partial_0=\frac1c\partial_t\): \[ \partial_0 F_{ij} + \partial_i F_{j0} + \partial_j F_{0i} = 0, \] substituting \(F_{ij}=B_{ij}\), \(F_{j0}=-F_{0j}=E_j/c\), and \(F_{0i}=-E_i/c\), then multiplying through by \(c\), gives: \[ \partial_t B_{ij} + \partial_i E_j - \partial_j E_i = 0, \] exactly Faraday’s law.
For \((\alpha,\mu,\nu) = (i,j,k)\) (cyclic): \[ \partial_i B_{jk} + \partial_j B_{ki} + \partial_k B_{ij} = 0, \] which is equivalent to \(\nabla \cdot \mathbf{B} = 0\).
Geometric Interpretation
Form Degree and Dimension
- \(F\) is a 2-form on spacetime (6 independent components),
- \(\mathrm{d}F\) is a 3-form on spacetime (4 independent components in 4D).
In 4 dimensions, a 3-form has exactly \[ \binom{4}{3} = 4 \] independent components. These correspond to:
- the 3 components of Faraday’s law,
- the 1 component of \(\nabla \cdot \mathbf{B} = 0\).
Thus, covariance naturally combines them into a single geometric statement.
Closure Condition
The identity \(\mathrm{d}^2 = 0\) guarantees consistency. If \(F\) is itself exact, \(F = \mathrm{d}A\), then: \[ \mathrm{d}F = \mathrm{d}^2 A = 0 \] automatically. This motivates the introduction of the electromagnetic potential \(A_\mu\).
The Non-Homogeneous Equations
Source Term
The non-homogeneous equations involve the current 4-vector: \[ J^\mu = (c\rho, J^x, J^y, J^z). \]
They are written covariantly as: \[ \partial_\mu F^{\mu\nu} = -\mu_0J^\nu, \] or equivalently, using the Hodge dual: \[ \mathrm{d}\star F = -\mu_0\star J. \]
Component Check
For \(\nu = 0\), using \(F^{i0}=-E^i/c\): \[ \partial_i F^{i0} = -\mu_0J^0 = -\mu_0c\rho, \] i.e. \(-\frac1c\nabla\cdot\mathbf{E}=-\mu_0c\rho\), which gives Gauss’s law \(\nabla \cdot \mathbf{E} = \mu_0c^2\rho=\rho/\varepsilon_0\).
For \(\nu = k\) (spatial), using \(\partial_0=\frac1c\partial_t\), \(F^{0k}=E^k/c\), and \(\partial_iF^{ik}=-(\nabla\times\mathbf{B})^k\) (from \(F^{ik}=\epsilon^{ikl}B_l\)): \[ \partial_0 F^{0k} + \partial_i F^{ik} = -\mu_0J^k, \] which expands to: \[ (\nabla \times \mathbf{B})^k - \frac{1}{c^2}\partial_tE^k = \mu_0J^k, \] the Ampère-Maxwell law.
Final Structure
Electromagnetism in covariant form:
Field strength: \[ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu. \]
Homogeneous equations: \[ \mathrm{d}F = 0. \]
Non-homogeneous equations: \[ \partial_\mu F^{\mu\nu} = -\mu_0J^\nu. \]
Conceptual Summary
- The split into \(\mathbf{E}\) and \(\mathbf{B}\) is frame-dependent,
- Vector calculus hides the underlying spacetime geometry,
- Differential forms make the covariant structure explicit,
- Maxwell equations reduce to simple geometric statements on spacetime.
This formulation prepares the ground for:
- understanding gauge freedom (\(A_\mu \to A_\mu + \partial_\mu \chi\)),
- generalizing to curved spacetime (general relativity),
- interpreting electromagnetism as a \(U(1)\) gauge theory.
Exercises
Exercise 1: The Faraday Tensor and the Homogeneous Equation (Essential)
a) Verify that the matrix representation of \(F_{\mu\nu}\) given above is antisymmetric and correctly encodes \(\mathbf{E}\) and \(\mathbf{B}\).
b) Show explicitly that \(\mathrm{d}F=0\) gives the component \(\partial_tB_{12}+\partial_1E_2-\partial_2E_1=0\), and identify this with one component of Faraday’s law.
c) Prove that if \(F=\mathrm{d}A\), then \(\mathrm{d}F=0\) automatically.
Exercise 2: The Non-Homogeneous Equation and Gauge Invariance (Essential)
a) Verify that \(\partial_\mu F^{\mu0}=-\mu_0J^0\) reproduces Gauss’s law when \(F^{i0}=-E^i/c\) and \(J^0=c\rho\).
b) Show that the gauge transformation \(A_\mu\to A_\mu+\partial_\mu\chi\) leaves \(F_{\mu\nu}\) unchanged.
Exercise 3: Electrodynamics from an Action Principle (Essential)
Write the action for free electromagnetism, \[ S=-\frac1{4\mu_0}\int F_{\mu\nu}F^{\mu\nu}\,\mathrm{d}^4x, \] and derive the vacuum Maxwell equations by varying \(A_\mu\).
Exercise 4: Dimension-Counting and the Hodge Dual
a) In 2+1 dimensions, how many independent components does \(F_{\mu\nu}\) have? How many does \(\mathrm{d}F\) have?
b) Explain why the Hodge dual maps a 2-form to another 2-form in 4D specifically, and what role this plays in writing the non-homogeneous equations as \(\mathrm{d}\star F=-\mu_0\star J\).