Maxwell's Equations in Covariant Form

The four Maxwell equations are two Lorentz-covariant statements about one antisymmetric tensor. This packaging is more than notation: it fixes how electric and magnetic fields mix between observers, separates evolution from constraints, and makes charge conservation a consistency condition rather than an extra postulate.

Required background. Special Relativity supplies the Minkowski metric, Lorentz covariance, four-vector contractions, and index raising and lowering; Calculus and Linear Algebra supplies partial derivatives, integration by parts, and vector identities. Helpful background. Ordinary Differential Equations supplies initial-value reasoning; Differential Forms and Integration supplies exterior differentiation and the local distinction between closed and exact forms.

The electromagnetic tensor

Use rationalized Heaviside–Lorentz units and the inherited spacetime conventions. In a contractible patch, choose a potential

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

Then

$$\mathbf E=-\boldsymbol\nabla\Phi-\partial_t\mathbf A, \qquad \mathbf B=\boldsymbol\nabla\times\mathbf A,$$

and therefore

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

All six independent components are visible in the matrix

$$(F_{\mu\nu})= \begin{pmatrix} 0&-E_x&-E_y&-E_z\\ E_x&0&B_z&-B_y\\ E_y&-B_z&0&B_x\\ E_z&B_y&-B_x&0 \end{pmatrix}.$$

The current is $J^\mu=(\rho,\mathbf J)$, so the source coupling has the familiar component form

$$J^\mu A_\mu=-\rho\Phi+\mathbf J\mathbin{\cdot}\mathbf A.$$

Writing $F=dA$ is locally justified by the Poincaré lemma when $dF=0$. A single global potential exists only when the cohomology class $[F]$ vanishes. In particular, a nonzero period $\int_C F$ over a closed two-cycle $C$ obstructs global exactness. Such a cycle need not bound a three-chain inside the physical domain, so this is compatible with $dF=0$. See Guillemin and Haine 2019, §2.6, printed pp. 64–65; §4.6, pp. 147–151; and §5.1, pp. 171–173, Open PDF for local exactness, Stokes’ theorem, and de Rham cohomology.

The action and the two tensor equations

With the conventions above, take

$$S[A]=\int_\Omega d^4x\left( -\frac14F_{\mu\nu}F^{\mu\nu}+J^\mu A_\mu \right).$$

Antisymmetry turns the field variation into

$$-\frac12F^{\mu\nu}\delta F_{\mu\nu} =-F^{\mu\nu}\partial_\mu\delta A_\nu.$$

After integration by parts,

$$\delta S =\int_\Omega d^4x\, (\partial_\mu F^{\mu\nu}+J^\nu)\delta A_\nu -\int_{\partial\Omega}d\Sigma_\mu\, F^{\mu\nu}\delta A_\nu.$$

Fixed boundary potential, or another boundary condition that removes the surface term, gives

$$\boxed{\partial_\mu F^{\mu\nu}=-J^\nu.}$$

This action calculation is the site-convention translation of Tong 2015, §5.5.3, printed pp. 125–126, PDF. Tong uses a different simultaneous choice of metric, tensor, and unit conventions, so individual signs should not be imported in isolation.

The definition $F=dA$ also gives

$$\boxed{\partial_{[\lambda}F_{\mu\nu]}=0,}$$

because commuting partial derivatives makes the cyclic sum vanish. The two boxed equations are the inhomogeneous Maxwell equation and the homogeneous Bianchi identity.

Recovering the four three-vector equations

Raising indices gives $F^{0i}=E^i$ and $F^{i0}=-E^i$. The tensor components then reproduce the familiar equations without guesswork:

Tensor component	Calculation	Three-vector equation
$\nu=0$ in $\partial_\mu F^{\mu\nu}=-J^\nu$	$\partial_iF^{i0}=-\boldsymbol\nabla\mathbin{\cdot}\mathbf E=-\rho$	$\boldsymbol\nabla\mathbin{\cdot}\mathbf E=\rho$
$\nu=i$ in $\partial_\mu F^{\mu\nu}=-J^\nu$	$\partial_tE^i-(\boldsymbol\nabla\times\mathbf B)^i=-J^i$	$\boldsymbol\nabla\times\mathbf B-\partial_t\mathbf E=\mathbf J$
Three spatial Bianchi indices	Cyclic derivatives of $F_{ij}$	$\boldsymbol\nabla\mathbin{\cdot}\mathbf B=0$
One temporal Bianchi index	Cyclic derivatives of $F_{0i}$ and $F_{ij}$	$\boldsymbol\nabla\times\mathbf E+\partial_t\mathbf B=0$

The tensor construction and covariant component equations are developed in Tong 2015, §§5.3.2 and 5.4, printed pp. 106–108 and 113–115, PDF, translated consistently to the conventions used here.

Evolution, constraints, and charge conservation

The curl equations give the six first-order time derivatives of the three components of $\mathbf E$ and the three components of $\mathbf B$. The two divergence equations contain no such time derivatives, so they constrain initial data. Their persistence follows from the evolution equations:

$$\partial_t(\boldsymbol\nabla\mathbin{\cdot}\mathbf B)=0,$$

and, using $\partial_t\rho+\boldsymbol\nabla\mathbin{\cdot}\mathbf J=0$,

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

Thus data satisfying both constraints continue to satisfy them in the exact continuum theory. A numerical scheme can still create constraint error, so propagation in the equations is not a substitute for monitoring it in a computation.

Charge conservation itself follows from antisymmetry:

$$\partial_\nu J^\nu =-\partial_\nu\partial_\mu F^{\mu\nu}=0.$$

The same requirement appears from gauge invariance of the source term. Under $A_\mu\mapsto A_\mu+\partial_\mu\chi$,

$$\Delta S_{\mathrm{source}} =\int_{\partial\Omega}d\Sigma_\mu\,J^\mu\chi -\int_\Omega d^4x\,\chi\,\partial_\mu J^\mu.$$

For compactly supported $\chi$, or boundary conditions that remove the flux term, a nonconserved prescribed current is incompatible with gauge invariance and with Maxwell’s equation.

What an observer measures

For an observer with four-velocity $u^\mu$, define the electric field four-vector

$$E_{(u)}^\mu=F^\mu{}_{\nu}u^\nu, \qquad E_{(u)}^\mu u_\mu=0.$$

Its three spatial components in the observer’s rest frame are the force per unit charge on a momentarily comoving test charge. Different observers split the same $F_{\mu\nu}$ into different electric and magnetic fields; they do not transform $\mathbf E$ while leaving $\mathbf B$ untouched.

The covariant Lorentz force is

$$\frac{dp^\mu}{d\tau}=qF^\mu{}_{\nu}u^\nu,$$

whose spatial part is $d\mathbf p/dt=q(\mathbf E+\mathbf v\times\mathbf B)$. See Tong 2015, §5.4.1, printed pp. 115–116, PDF for the covariant force and its component recovery.

Common pitfalls

Changing one sign convention in isolation. The signs of $A_0$, $F_{0i}$, the source action, the inhomogeneous equation, and the Lorentz force form one linked system. Test the full set against Gauss’s law and $q(\mathbf E+\mathbf v\times\mathbf B)$.

Treating Gauss’s laws as optional boundary conditions. They restrict initial data everywhere on a time slice. The curl equations propagate valid constraints but do not repair invalid initial data.

Assuming a global potential without checking topology. The Bianchi identity implies $F=dA$ locally. Global exactness is a stronger statement.

Exercises

Source-free waves

In vacuum, derive a wave equation for $\mathbf E$ from the curl equations, assuming the constraints hold.

Solution

Take the curl of Faraday’s law:

$$\boldsymbol\nabla\times(\boldsymbol\nabla\times\mathbf E) +\partial_t(\boldsymbol\nabla\times\mathbf B)=0.$$

Use $\boldsymbol\nabla\times\mathbf B=\partial_t\mathbf E$ and

$$\boldsymbol\nabla\times(\boldsymbol\nabla\times\mathbf E) =\boldsymbol\nabla(\boldsymbol\nabla\mathbin{\cdot}\mathbf E)-\nabla^2\mathbf E.$$

Since $\boldsymbol\nabla\mathbin{\cdot}\mathbf E=0$ in vacuum,

$$\boxed{(\partial_t^2-\nabla^2)\mathbf E=0.}$$

The same argument gives $(\partial_t^2-\nabla^2)\mathbf B=0$. Omitting the divergence constraint would leave an unwanted longitudinal-gradient term.

A static sign check

Let $\mathbf J=0$, $\partial_t\mathbf A=0$, and $\mathbf E=-\boldsymbol\nabla\Phi$. Show that the covariant sourced equation gives Poisson’s equation with the expected sign.

Solution

The $\nu=0$ component is $\boldsymbol\nabla\mathbin{\cdot}\mathbf E=\rho$. Substituting $\mathbf E=-\boldsymbol\nabla\Phi$ gives

$$\boxed{\nabla^2\Phi=-\rho.}$$

For a positive point charge, the solution away from the origin decreases as $1/r$, and its outward electric field is positive. This checks the linked signs of $A_0$, $F_{0i}$, and the source equation.

Potentials, Gauge Freedom, and Constraints next explains which parts of $A_\mu$ are redundant and how gauge choices expose the physical radiative degrees of freedom.

References

• 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), §§5.3.2, 5.4, 5.4.1, and 5.5.3, printed pp. 106–108, 113–116, and 125–126, University of Cambridge Mathematical Tripos lecture notes, Lent Term 2015.