Maxwell Fields on Curved Spacetime

Maxwell theory on a curved spacetime is still a gauge theory of a one-form potential, but its derivatives, volume element, and wave operator are determined by the spacetime metric. The first-order field equations retain their exterior-calculus form; curvature becomes explicit only when they are combined into second-order wave equations. This page treats a prescribed classical geometry and a prescribed conserved current. It does not include electromagnetic backreaction, material constitutive laws, plasma dispersion, or a short-wavelength approximation.

Required background. Classical Fields and Their Actions supplies action variation and gauge symmetry; The Wave Equation and Characteristics supplies hyperbolic propagation and characteristics; Affine Connections and Covariant Derivatives supplies Levi-Civita differentiation and commutators.

Helpful background. Events, Intervals, and Minkowski Geometry reviews causal covectors; Tensor Fields and Natural Operations reviews forms and pullbacks; Timelike Congruences and Spatial Projection prepares the expansion language used for beams later in the chapter.

Field strength, gauge freedom, and the action

Let $A_\mu$ be a one-form and define

$$F_{\mu\nu}=2\nabla_{[\mu}A_{\nu]}=2\partial_{[\mu}A_{\nu]}.$$

The equality follows because the Levi-Civita connection is torsion-free: its symmetric connection coefficients cancel under antisymmetrization. Locally $A_\mu\mapsto A_\mu+\nabla_\mu\chi$ leaves $F_{\mu\nu}$ unchanged. Globally, a closed $F$ need not admit one potential on the whole spacetime, so this local description may require gauge patches.

Use the electromagnetic convention already established in the covariant Maxwell equations: $A_\mu=(-\Phi,\mathbf A)$, $F_{0i}=-E_i$, and $J^\mu=(\rho,\mathbf J)$ in an inertial chart, with rationalized Heaviside–Lorentz normalization. For a fixed metric and current $J^\mu$, take

$$S[A]=- \frac14\int d^4x\,\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu} +\int d^4x\,\sqrt{-g}\,J^\mu A_\mu .$$

After integrating by parts and fixing the variation at the boundary,

$$\delta S =\int d^4x\,\sqrt{-g} \left(\nabla_\mu F^{\mu\nu}+J^\nu\right)\delta A_\nu .$$

Thus the sourced equation is $\nabla_\mu F^{\mu\nu}=-J^\nu$. The source sign belongs to the complete choice of potential, field tensor, and charge convention: it cannot be imported independently from a source using another choice. Gauge invariance of the source coupling, up to a boundary term, requires $\nabla_\mu J^\mu=0$. The source-free component equations and their minimal-coupling interpretation are given explicitly in Reall 2013, §5.1, pp. 49–50, PDF.

Maxwell equations as equations for a two-form

With the inherited orientation $\epsilon_{0123}=+\sqrt{-g}$, define the Hodge dual by

$$(\star F)_{\mu\nu} =\frac12\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}.$$

For the current one-form $j=J_\mu dx^\mu$, the same Hodge convention gives $(\star j)_{\mu\nu\rho}=J^\sigma\epsilon_{\sigma\mu\nu\rho}$. On two-forms in four-dimensional Lorentzian signature, $\star\star F=-F$. The complete Maxwell system is

$$dF=0, \qquad d\star F=\star j,$$

or, in components,

$$\nabla_{[\mu}F_{\nu\rho]}=0, \qquad \nabla_\mu F^{\mu\nu}=-J^\nu.$$

The first equation is an identity when $F=dA$; conversely, the Poincaré lemma produces a potential on a contractible neighborhood. A direct flat-space check fixes the sourced sign: $F^{i0}=-E^i$ gives $\partial_iF^{i0}=-\boldsymbol\nabla\cdot\mathbf E=-\rho$, hence Gauss’s law $\boldsymbol\nabla\cdot\mathbf E=\rho$. In forms, the same check is $(d\star F)_{123}=\boldsymbol\nabla\cdot\mathbf E=(\star j)_{123}$.

Curvature does not appear as an extra algebraic source in the first-order vacuum system. It enters through the metric used by $\star$, index raising, the volume element, and covariant differentiation. Dolan 2018, §§2.1–2.2.2, pp. 3–4, Eqs. (1b) and (4)–(9), PDF gives the corresponding tensor and potential wave equations; its sourced equation is written with the reversed antisymmetric index order $\nabla_bF^{ab}=J^a$.

Where curvature enters wave propagation

Writing the sourced equation in terms of $A^\mu$ gives

$$\Box A^\nu -\nabla^\nu(\nabla_\mu A^\mu) -R^\nu{}_{\mu}A^\mu =-J^\nu.$$

In Lorenz gauge, $\nabla_\mu A^\mu=0$, this becomes

$$\left(\Box\delta^\nu{}_{\mu}-R^\nu{}_{\mu}\right)A^\mu=-J^\nu.$$

The Ricci term comes from commuting covariant derivatives; it is not an effective electric current. Gauge transformations remain available provided the gauge function satisfies $\Box\chi=0$ after Lorenz gauge has been imposed. In four spacetime dimensions the source-free equations for the two-form are conformally invariant: under $g_{\mu\nu}\mapsto\Omega^2g_{\mu\nu}$, the Hodge dual on two-forms is unchanged. The potential wave equation need not look invariant term by term, which is why the field-strength formulation is the cleaner check.

Taking a divergence of $\nabla_\mu F^{\mu\nu}=-J^\nu$ yields current conservation. Indeed, antisymmetry gives

$$\nabla_\nu J^\nu =-\frac12[\nabla_\nu,\nabla_\mu]F^{\mu\nu}=0,$$

because the commutator contracts the antisymmetric $F^{\mu\nu}$ with the symmetric Ricci tensor. In flat spacetime the curvature term vanishes and the Lorenz-gauge equation reduces to the ordinary vector wave equation, independently checking both its sign and its limit.

Regime and physical boundary

Nothing on this page identifies an exact electromagnetic field with a bundle of rays. Vacuum Maxwell theory is nondispersive at the level of its characteristic cone, but diffraction, interference, tails, and finite-frequency structure remain wave phenomena. In a dielectric, plasma, nonlinear electrodynamics model, or quantum effective theory, a constitutive relation or modified action can produce frequency-dependent propagation; then $g^{\mu\nu}k_\mu k_\nu=0$ need not be the relevant dispersion relation. The geometric-optics pages that follow instead assume source-free vacuum propagation over scales on which the wavelength is uniformly small.

Exercise

Show directly that the Lorenz condition is preserved by $A_\mu\mapsto A_\mu+\nabla_\mu\chi$ precisely when $\Box\chi=0$. Then verify that the field strength is unchanged without using a coordinate chart.

Solution

The transformed divergence is

$$\nabla_\mu A'^\mu =\nabla_\mu A^\mu+\nabla_\mu\nabla^\mu\chi =\nabla_\mu A^\mu+\Box\chi.$$

If the original potential is in Lorenz gauge, the transformed one is also in Lorenz gauge exactly when $\Box\chi=0$. As differential forms, $A'=A+d\chi$, so

$$F'=dA'=dA+d^2\chi=F$$

because $d^2=0$. This argument is coordinate-independent.

References

• Dolan, Sam R. “Geometrical Optics for Scalar, Electromagnetic and Gravitational Waves on Curved Spacetime.” International Journal of Modern Physics D 27 (2018): 1843010. DOI. Open PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. PDF.