Classical Fields and Electromagnetism

A classical field assigns dynamical data to every spacetime event. An action supplies bulk equations and boundary terms; constraints restrict admissible initial data; gauge freedom identifies redundant descriptions; stress-energy records what the field exchanges with matter. Electromagnetism displays all of these structures in their cleanest relativistic form.

Helpful background. Calculus and Linear Algebra for Relativity supplies variation, vector calculus, and tensor contractions; Events, Intervals, and Minkowski Geometry supplies Lorentzian geometry and index conventions; Differential Equations for Relativity supplies initial-value reasoning; the coordinate-free route uses Exterior Algebra and Differential Forms.

The field-theory pattern

A finite-dimensional trajectory $q(t)$ becomes a field $\phi^a(x)$ with one or more components at every event. The action

$$S[\phi]=\int_\Omega d^4x\, \mathcal L(\phi^a,\partial_\mu\phi^a,x)$$

has the first variation

$$\delta S =\int_\Omega d^4x\,\mathcal E_a\delta\phi^a +\int_{\partial\Omega}d\Sigma_\mu\, \frac{\partial\mathcal L} {\partial(\partial_\mu\phi^a)}\delta\phi^a.$$

The bulk coefficient $\mathcal E_a=0$ is the Euler–Lagrange equation. The second term decides which fixed, free, or mixed boundary variations define a stationary problem. It must be retained until the boundary conditions are specified. Tong 2006–2007, §1.1, printed p. 8, PDF gives the basic field-theory integration by parts.

Electromagnetism adds a second layer. The potential $A_\mu$ is redundant, but its antisymmetric curvature $F_{\mu\nu}$ is gauge-invariant. The homogeneous equation constrains that curvature, the sourced equation determines its response to current, and the antisymmetry of $F$ forces current conservation. Varying the same theory with respect to a background metric yields a stress-energy tensor whose divergence is the negative Lorentz force density.

This chain is the prototype used later in gauge theory and gravitation:

Structure	Electromagnetic realization	Question it answers
Configuration	$A_\mu$	What variables describe the field?
Gauge-invariant field	$F=dA$ locally	What can local measurements depend on?
Equations	$dF=0$ and $d\star F=\star j$	How does the field evolve and respond to sources?
Constraint	Gauss’s law	Which initial data are admissible?
Global datum	Cohomology class or $U(1)$ holonomy	How do local potentials fit together around surfaces and loops?
Metric response	$T_{\mu\nu}$	Where are field energy and momentum stored and transferred?

Conventions that control the signs

This overview collects the chapter-specific consequences of the global conventions. In oriented Cartesian Minkowski coordinates,

$$\eta_{\mu\nu}=\operatorname{diag}(-1,1,1,1), \qquad \epsilon_{0123}=+1, \qquad \epsilon^{0123}=-1.$$

Use rationalized Heaviside–Lorentz units and write

$$A_\mu=(-\Phi,\mathbf A), \qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad J^\mu=(\rho,\mathbf J).$$

The measured fields are defined by

$$F_{0i}=-E_i, \qquad F_{ij}=\epsilon_{ijk}B^k.$$

These choices link the action, equations, and force:

$$\begin{aligned} S[A] &=\int d^4x\left( -\frac14F_{\mu\nu}F^{\mu\nu}+J^\mu A_\mu \right),\\ \partial_\mu F^{\mu\nu} &=-J^\nu,\\ \partial_{[\lambda}F_{\mu\nu]} &=0,\\ \frac{dp^\mu}{d\tau} &=qF^\mu{}_{\nu}u^\nu. \end{aligned}$$

They reproduce

$$\boldsymbol\nabla\mathbin{\cdot}\mathbf E=\rho, \qquad \boldsymbol\nabla\times\mathbf B-\partial_t\mathbf E=\mathbf J,$$

and $d\mathbf p/dt=q(\mathbf E+\mathbf v\times\mathbf B)$. Different sources may reverse several definitions at once or use Gaussian or SI normalization. A consistent simultaneous translation is equivalent; changing one sign or one factor in isolation is not. Tong 2015, §§5.3.1–5.5.3, printed pp. 105–126, PDF develops the tensor, equations, force, and action in a different simultaneous convention set.

Route through the chapter

The six pages form one derivational sequence:

1. Classical Fields and Their Actions derives the field Euler–Lagrange equation while retaining the surface term that distinguishes initial data from wall data.
2. Maxwell’s Equations in Covariant Form builds $F_{\mu\nu}$ from the potential, recovers all four three-vector equations, and separates curl evolution from divergence constraints.
3. Potentials, Gauge Freedom, and Constraints distinguishes local redundancy, residual gauge freedom, boundary data, and compact $U(1)$ holonomy.
4. Electromagnetism with Differential Forms shows which equation is metric-free, how the current becomes a three-form, and why closed need not mean globally exact.
5. Electromagnetic Invariants and Field Classification decides when a boost removes one field, parallelizes both, or reaches the null obstruction.
6. Lorentz Force and Electromagnetic Stress-Energy derives $T_{\mu\nu}$ by metric variation and proves local field-to-matter energy-momentum exchange.

For the coordinate-free bridge, Carroll 1997, Chapter 1, printed pp. 20–24, PDF develops the Hodge form of Maxwell’s equations, while Guillemin and Haine 2019, §§2.6, 4.6, and 5.1, Open PDF supplies local exactness, Stokes’ theorem, and de Rham cohomology.

Check your understanding

Propagating the electric constraint

Suppose initial data obey $\boldsymbol\nabla\mathbin{\cdot}\mathbf E=\rho$. Use Ampère–Maxwell evolution and charge conservation to show why this constraint remains satisfied.

Solution

Take the divergence of

$$\boldsymbol\nabla\times\mathbf B-\partial_t\mathbf E=\mathbf J.$$

Because the divergence of a curl vanishes,

$$\partial_t(\boldsymbol\nabla\mathbin{\cdot}\mathbf E) =-\boldsymbol\nabla\mathbin{\cdot}\mathbf J.$$

Charge conservation gives $\partial_t\rho=-\boldsymbol\nabla\mathbin{\cdot}\mathbf J$, hence

$$\partial_t(\boldsymbol\nabla\mathbin{\cdot}\mathbf E-\rho)=0.$$

The constraint propagates if it holds initially. This does not make Gauss’s law optional: inconsistent initial data remain inconsistent, and a nonconserved prescribed current is incompatible with Maxwell’s equations.

One field viewed four ways

At an event, let $\mathbf E=E_0\widehat{\mathbf x}$ and $\mathbf B=0$. Determine the two quadratic invariants, the field energy density and flux, and the instantaneous force and power on a stationary charge $q$.

Solution

The invariants are

$$I=2(B^2-E^2)=-2E_0^2, \qquad J=4\mathbf E\mathbin{\cdot}\mathbf B=0,$$

so the field is electric-dominated. Its stress-energy components give

$$u=\frac12E_0^2, \qquad \mathbf S=\mathbf E\times\mathbf B=0.$$

For a stationary charge, $\mathbf v=0$, so

$$\frac{d\mathbf p}{dt}=qE_0\widehat{\mathbf x}, \qquad \frac{dE_{\mathrm{particle}}}{dt} =q\mathbf E\mathbin{\cdot}\mathbf v=0.$$

The charge accelerates even though its instantaneous power is zero at that moment. If the field persists, the charge initially acquires velocity in the direction of the electric force $q\mathbf E$, so the field does positive work for either sign of nonzero $q$.

Outcomes and scope

After this chapter, you should be able to vary a local action with its directed surface term, translate between tensor and three-vector Maxwell equations, distinguish gauge equivalence from a physical symmetry, test a global-potential assumption, classify a field invariantly, and verify electromagnetic energy-momentum exchange.

The treatment is classical and emphasizes vacuum electrodynamics with prescribed currents or generic charged matter. Material constitutive laws, detailed radiation theory, quantum electrodynamics, and curved-spacetime propagation are outside the chapter. The next chapter, Symmetry, Conservation, and Stress-Energy, generalizes these conservation structures beyond Maxwell theory.

References

• Sean M. Carroll, “Lecture Notes on General Relativity”, Chapter 1, printed pp. 20–24, arXiv:gr-qc/9712019, 1997.
• Victor Guillemin and Peter J. Haine, Differential Forms, World Scientific, 2019, ISBN 978-981-3272-77-4, DOI 10.1142/11058, Open PDF.
• David Tong, Electromagnetism (PDF), §§5.3.1–5.6.2, printed pp. 105–132, University of Cambridge Mathematical Tripos lecture notes, Lent Term 2015.
• David Tong, Quantum Field Theory (PDF), §1.1, printed p. 8, University of Cambridge Part III Mathematical Tripos lecture notes, Michaelmas Terms 2006 and 2007.