Covariant electrodynamics — Faraday tensor

Intuition [Beginner]

Maxwell's equations are four vector equations in the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$. They look like four separate laws governing two different fields. Special relativity reveals this is an illusion: $\mathbf{E}$ and $\mathbf{B}$ are not two fields. They are different parts of a single object, seen from different viewpoints.

A charge sitting at rest feels only an electric force. The same charge, if you run past it, also feels a magnetic force. The field did not change. Your perspective did. What looks like a pure $\mathbf{E}$ field in one frame has $\mathbf{B}$ components in another, the same way a coin seen edge-on looks like a line but seen face-on looks like a circle.

That single object is the electromagnetic field tensor $F$, a $4\times 4$ antisymmetric matrix. Its components are the six numbers $(E_x,E_y,E_z,B_x,B_y,B_z)$. Under a Lorentz boost, the entries mix — the way time and space coordinates mix — and $\mathbf{E}$ rotates into $\mathbf{B}$.

With $F$ in hand, Maxwell's four separate equations collapse into two tensor equations. The payoff: one object, two equations, instead of two objects and four.

Visual [Beginner]

Picture a point charge at rest, surrounded by radial electric field lines pointing outward — the familiar Coulomb-field picture. No magnetic field exists anywhere.

Now run past the charge at speed $v$. The field lines compress in the direction of motion (Lorentz contraction) and the charge becomes a moving current. A moving current produces a $\mathbf{B}$ field circling the line of motion.

The field lines in the two frames are related by a hyperbolic rotation in spacetime — mixing the time coordinate with the spatial coordinate along the direction of motion. The $\mathbf{E}$ and $\mathbf{B}$ components tilt into each other, just as $t$ and $x$ mix under a boost.

Worked example [Beginner]

A point charge $q$ sits at the origin of frame $S$. The electric field is the Coulomb field $\mathbf{E}=(q/4\pi {ϵ}_0)\hat{\mathbf{r}}/r^2$, and $\mathbf{B}=0$.

Now consider frame $S^{\prime }$ moving at velocity $v$ in the $x$-direction relative to $S$. The charge now moves at speed $-v$ in $S^{\prime }$. What fields does an observer in $S^{\prime }$ measure?

The Lorentz transformation of the fields gives, for $v\ll c$:

$${\mathbf{E}}^{\prime }\approx \mathbf{E},{\mathbf{B}}^{\prime }\approx -\frac{\mathbf{v}}{c}\times \mathbf{E}.$$

A magnetic field has appeared. If $\mathbf{v}$ points along $x$ and $\mathbf{E}$ is radial, then ${\mathbf{B}}^{\prime }$ circles the $x$-axis, exactly like the field of a current-carrying wire. The moving charge is a current.

At higher speeds the full result includes a factor of $\gamma =1/\sqrt{1-v^2/c^2}$, and ${\mathbf{E}}^{\prime }$ compresses along the direction of motion. The key point stands: $\mathbf{B}$ is not an independent field. It is the part of $F$ that becomes visible when you change frames.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Throughout this unit we adopt the metric signature $(+,-,-,-)$ and SI units. Greek indices $\mu ,\nu ,\dots$ run over $0,1,2,3$; Latin indices $i,j,\dots$ run over $1,2,3$. The four-position is $x^{\mu }=(ct,x,y,z)$ and the four-derivative is ${\partial }_{\mu }=\frac{1}{c}\frac{\partial }{\partial t},-\nabla$.

The four-potential

The electromagnetic four-potential is

$$A^{\mu }=\left(\frac{\varphi }{c},\mathbf{A}\right),$$

where $\varphi$ is the scalar potential and $\mathbf{A}$ is the vector potential 10.01.01 pending. The corresponding covariant vector is $A_{\mu }=g_{\mu \nu }{A}^{\nu }=(\varphi /c,-\mathbf{A})$.

The Faraday tensor

The electromagnetic field tensor (Faraday tensor) is the rank-2 antisymmetric tensor

$$\boxed{F_{\mu \nu }:={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }.}$$

In matrix form, with $\mu$ as row index and $\nu$ as column index:

$$F_{\mu \nu }=\left(\begin{array}{cccc}0& E_x/c& E_y/c& E_z/c\\ -E_x/c& 0& -B_z& B_y\\ -E_y/c& B_z& 0& -B_x\\ -E_z/c& -B_y& B_x& 0\end{array}\right).$$

The raised-index version $F^{\mu \nu }=g^{\mu \alpha }{g}^{\nu \beta }{F}_{\alpha \beta }$ flips the sign on the electric-field entries:

$$F^{\mu \nu }=\left(\begin{array}{cccc}0& -E_x/c& -E_y/c& -E_z/c\\ E_x/c& 0& -B_z& B_y\\ E_y/c& B_z& 0& -B_x\\ E_z/c& -B_y& B_x& 0\end{array}\right).$$

The six independent components of $F_{\mu \nu }$ are exactly the three components of $\mathbf{E}$ and the three components of $\mathbf{B}$. The four-current is $J^{\mu }=(c\rho ,\mathbf{J})$, where $\rho$ is the charge density and $\mathbf{J}$ is the current density.

Covariant Maxwell equations

Maxwell's four vector equations 10.04.01 pending reduce to two tensor equations. The inhomogeneous pair (sources present) is

$$\boxed{{\partial }_{\mu }{F}^{\mu \nu }={\mu }_0{J}^{\nu }.}$$

The homogeneous pair (no sources — the Bianchi identity) is

$$\boxed{{\partial }_{\lambda }{F}_{\mu \nu }+{\partial }_{\mu }{F}_{\nu \lambda }+{\partial }_{\nu }{F}_{\lambda \mu }=0,\text{equivalently }{\partial }_{[\lambda }{F}_{\mu \nu ]}=0.}$$

The bracket $[\lambda \mu \nu ]$ denotes total antisymmetrisation over the three indices.

Gauge invariance

The Faraday tensor is invariant under the gauge transformation

$$A_{\mu }⟶A_{\mu }+{\partial }_{\mu }\lambda$$

for any smooth function $\lambda (x)$. This follows immediately: $F_{\mu \nu }\to {\partial }_{\mu }(A_{\nu }+{\partial }_{\nu }\lambda )-{\partial }_{\nu }(A_{\mu }+{\partial }_{\mu }\lambda )=F_{\mu \nu }+({\partial }_{\mu }{\partial }_{\nu }-{\partial }_{\nu }{\partial }_{\mu })\lambda =F_{\mu \nu }$, since partial derivatives commute.

Covariant Lorentz force

The equation of motion for a particle of charge $q$ and rest mass $m$ is

$$\frac{d{p}^{\mu }}{d\tau }=q{F}^{\mu \nu }{u}_{\nu },$$

where $p^{\mu }=m{u}^{\mu }$ is the four-momentum, $u^{\mu }=d{x}^{\mu }/d\tau$ is the four-velocity, and $\tau$ is proper time. The spatial components ($\mu =1,2,3$) reproduce the Lorentz force law $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$; the time component ($\mu =0$) reproduces the power equation $dE/dt=q\mathbf{v}\cdot \mathbf{E}$.

The electromagnetic stress-energy tensor

The symmetric rank-2 tensor

$$T^{\mu \nu }=\frac{1}{{\mu }_0}\left(F^{\mu \alpha }{F}^{\nu }{}_{\alpha }-\frac{1}{4}{g}^{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta }\right)$$

encodes the energy density ($T^{00}=\frac{1}{2}({ϵ}_0{E}^2+B^2/{\mu }_0)$), the Poynting vector ($T^{0i}=c(\mathbf{E}\times \mathbf{B}{)}_i/{\mu }_0$), and the Maxwell stress tensor ($T^{ij}$). Energy-momentum conservation is ${\partial }_{\mu }{T}^{\mu \nu }=-F^{\nu \lambda }{J}_{\lambda }$, where the right-hand side is the Lorentz-force density exerted by the field on the sources.

Sign conventions

The sign choices above follow Jackson (3e) and Griffiths (4e): metric $(+,-,-,-)$, $A^{\mu }=(\varphi /c,\mathbf{A})$, and $F_{0i}=E_i/c$. The opposite metric $(-,+,+,+)$ (used by Landau-Lifshitz and much of the particle-physics literature) flips the signs of $F_{0i}$ and $F^{0i}$. State the convention before computing.

Key theorem with proof [Intermediate+]

Theorem (Equivalence of covariant and vector Maxwell equations). The two covariant equations ${\partial }_{\mu }{F}^{\mu \nu }={\mu }_0{J}^{\nu }$ and ${\partial }_{[\lambda }{F}_{\mu \nu ]}=0$, with $F_{\mu \nu }$ and $F^{\mu \nu }$ as defined above, are equivalent to the four standard vector Maxwell equations in SI units.

Proof. We treat the two covariant equations in turn.

Part I: Inhomogeneous equation. Write ${\partial }_{\mu }{F}^{\mu \nu }={\mu }_0{J}^{\nu }$ for each value of $\nu$.

$\nu =0$: ${\partial }_{\mu }{F}^{\mu 0}={\partial }_i{F}^{i0}=\nabla \cdot (\mathbf{E}/c)$, since $F^{00}=0$ and $F^{i0}=-F^{0i}=E_i/c$. The right side is ${\mu }_0{J}^0={\mu }_{0}c\rho$. So $\nabla \cdot \mathbf{E}/c={\mu }_{0}c\rho$, i.e., $\nabla \cdot \mathbf{E}={\mu }_0{c}^2\rho =\rho /{ϵ}_0$. This is Gauss's law.

$\nu =j$ (spatial): ${\partial }_{\mu }{F}^{\mu j}={\partial }_0{F}^{0j}+{\partial }_i{F}^{ij}$. Here $F^{0j}=-E_j/c$, so ${\partial }_0{F}^{0j}=-\frac{1}{c^2}\frac{\partial E_j}{\partial t}$. For the spatial divergence, write $F^{ij}$ in terms of $\mathbf{B}$: for $j=1$, ${\partial }_i{F}^{i1}={\partial }_2{F}^{21}+{\partial }_3{F}^{31}={\partial }_y{B}_z-{\partial }_z{B}_y=(\nabla \times \mathbf{B}{)}_1$. The same pattern holds for $j=2,3$, giving ${\partial }_i{F}^{ij}=(\nabla \times \mathbf{B}{)}_j$. Setting the sum equal to ${\mu }_0{J}^j$:

$$(\nabla \times \mathbf{B}{)}_j-\frac{1}{c^2}\frac{\partial E_j}{\partial t}={\mu }_0{J}_j.$$

Since $1/c^2={\mu }_0{ϵ}_0$, this is $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{ϵ}_0\partial \mathbf{E}/\partial t$. This is the Ampere-Maxwell law.

Part II: Homogeneous equation. Expand ${\partial }_{[\lambda }{F}_{\mu \nu ]}=0$, i.e., ${\partial }_{\lambda }{F}_{\mu \nu }+{\partial }_{\mu }{F}_{\nu \lambda }+{\partial }_{\nu }{F}_{\lambda \mu }=0$.

All indices spatial ($\lambda =1,\mu =2,\nu =3$): ${\partial }_1{F}_{23}+{\partial }_2{F}_{31}+{\partial }_3{F}_{12}=-{\partial }_x{B}_x-{\partial }_y{B}_y-{\partial }_z{B}_z=-\nabla \cdot \mathbf{B}=0$. This is no magnetic monopoles: $\nabla \cdot \mathbf{B}=0$.

One index temporal ($\lambda =0,\mu =1,\nu =2$): ${\partial }_0{F}_{12}+{\partial }_1{F}_{20}+{\partial }_2{F}_{01}=0$. Substituting $F_{12}=-B_z$, $F_{20}=-E_y/c$, $F_{01}=E_x/c$:

$$-\frac{1}{c}\frac{\partial B_z}{\partial t}-\frac{1}{c}{\partial }_x{E}_y+\frac{1}{c}{\partial }_y{E}_x=0.$$

This is $(\nabla \times \mathbf{E}{)}_z=-\partial B_z/\partial t$. Cyclic permutations of the indices give the other two components, yielding $\nabla \times \mathbf{E}=-\partial \mathbf{B}/\partial t$. This is Faraday's law. The remaining index choices ($0,1,3$ and $0,2,3$) reproduce the other components of Faraday's law. $\square$

Corollary (Charge conservation). The antisymmetry $F^{\mu \nu }=-F^{\nu \mu }$ implies ${\partial }_{\nu }{\partial }_{\mu }{F}^{\mu \nu }=0$ identically. Applying ${\partial }_{\nu }$ to the inhomogeneous equation gives ${\partial }_{\nu }{J}^{\nu }=0$, the continuity equation $\frac{\partial \rho }{\partial t}+\nabla \cdot \mathbf{J}=0$.

Exercises [Intermediate+]

Differential-forms formulation [Master]

On Minkowski spacetime $({\mathbb{R}}^{1,3},\eta )$, let $A=A_{\mu }d{x}^{\mu }$ be the electromagnetic potential 1-form. The Faraday tensor is the exterior derivative

$$F=dA.$$

This is a 2-form. The identity $d^2=0$ gives $dF=ddA=0$, which is the homogeneous Maxwell equation $dF=0$ in differential-forms language. This equation is topological: it depends only on $d^2=0$, not on the metric.

The Hodge dual of $F$ (with respect to the Minkowski metric) is

$${\star F}_{\mu \nu }=\frac{1}{2}{ϵ}_{\mu \nu \alpha \beta }{F}^{\alpha \beta },$$

where ${ϵ}_{\mu \nu \alpha \beta }$ is the Levi-Civita symbol with ${ϵ}_{0123}=+1$. The inhomogeneous Maxwell equation becomes

$$d\star F={\mu }_{0}J,$$

where $J=J_{\mu }d{x}^{\mu }$ is the current 3-form (equivalently $J=\star j$ where $j^{\mu }$ is the four-current vector). This equation does depend on the metric, through the Hodge star.

Maxwell's equations in this language are:

$$\boxed{dF=0,d\star F={\mu }_{0}J.}$$

The split is sharp: $dF=0$ is metric-independent and topological (it encodes the statement that $F$ is locally exact, $F=dA$, guaranteed by the Poincare lemma on contractible domains). The equation $d\star F={\mu }_{0}J$ contains the dynamics — the coupling of the field to the metric and to the sources.

Topological vs metric content. The homogeneous equation $dF=0$ has solutions $F=dA$ on any contractible region. Gauge freedom is the statement that $A$ and $A+d\lambda$ give the same $F$. On non-simply-connected topology (e.g., the Aharonov-Bohm configuration, where spacetime has a hole), $F=dA$ holds only locally and the global information is carried by the connection $A$ itself. This distinction — between the local field strength $F$ and the global gauge potential $A$ — is what makes the Aharonov-Bohm effect possible and is the prototype for the topological structure of all gauge theories.

Lagrangian and gauge structure [Master]

The electromagnetic Lagrangian density is

$$\mathcal{L}=-\frac{1}{4{\mu }_0}{F}_{\mu \nu }{F}^{\mu \nu }-J^{\mu }{A}_{\mu }.$$

Varying with respect to $A_{\mu }$ (with appropriate boundary conditions) yields the Euler-Lagrange equation ${\partial }_{\mu }{F}^{\mu \nu }={\mu }_0{J}^{\nu }$, i.e., the inhomogeneous Maxwell equation. The homogeneous equation $dF=0$ is not an equation of motion here — it is the identity $F=dA$.

Noether current and charge conservation. The Lagrangian is invariant under the global $U(1)$ transformation $A_{\mu }\to A_{\mu }$ (constant shift) or, more precisely, the matter sector has a $U(1)$ symmetry $\psi \to e^{i\alpha }\psi$ whose Noether current is $J^{\mu }$. The conservation law ${\partial }_{\mu }{J}^{\mu }=0$ follows either from Noether's theorem applied to the global phase symmetry, or directly from the antisymmetry of $F^{\mu \nu }$ as shown in the corollary above.

$U(1)$ gauge invariance. The Lagrangian is invariant under the local gauge transformation $A_{\mu }\to A_{\mu }+{\partial }_{\mu }\lambda$, $\psi \to e^{i\lambda }\psi$. This local symmetry is the prototype of all gauge symmetries in fundamental physics. The promotion from a global $U(1)$ symmetry to a local one requires the introduction of the gauge field $A_{\mu }$ and the replacement of ordinary derivatives by covariant derivatives $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$ — the minimal coupling prescription.

Connection to Yang-Mills theories. Electromagnetism is the $U(1)$ gauge theory. The non-Abelian generalisation replaces $U(1)$ by $SU(N)$: the gauge field becomes matrix-valued ($A_{\mu }\in \mathfrak{su}(N)$), the field strength acquires a commutator term $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+ig[A_{\mu },A_{\nu }]$, and the Lagrangian becomes $\mathcal{L}=-\frac{1}{4}\mathrm{Tr}(F_{\mu \nu }{F}^{\mu \nu })$. The entire apparatus of covariant derivatives, gauge-covariant field strengths, and Yang-Mills Lagrangians is a direct non-Abelian generalisation of the structure developed in this unit.

Connections to other fields [Master]

The Faraday tensor is the simplest non-degenerate example of several structures that appear throughout theoretical physics.

Riemannian geometry and GR 13.02.01. The Riemann curvature tensor $R^{\mu }{}_{\nu \alpha \beta }$ is a rank-4 tensor with the same antisymmetry-in-pairs structure as $F_{\mu \nu }$ (in fact $R_{\mu \nu \alpha \beta }=-R_{\nu \mu \alpha \beta }=-R_{\mu \nu \beta \alpha }$). The Faraday tensor is a 2-form on spacetime; the Riemann tensor is a 2-form-valued endomorphism of the tangent bundle. The Bianchi identity $dF=0$ is the linearised version of the second Bianchi identity ${\nabla }_{[\lambda }{R}_{\mu \nu ]\alpha \beta }=0$. The Einstein-Maxwell system couples the Einstein equation to the Maxwell equation through the EM stress-energy tensor defined above.

Quantum mechanics and the Dirac equation 12.11.01 pending. The minimal-coupling prescription $p_{\mu }\to p_{\mu }-e{A}_{\mu }$ (or ${\partial }_{\mu }\to D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$ in natural units) is what couples the Dirac field to the electromagnetic field. The gauge-covariant derivative and the $U(1)$ gauge principle established here are the direct input to the Dirac equation and to QED.

Topology and the Aharonov-Bohm effect. In a region where $F=dA=0$ but $A\ne 0$ (a non-contractible loop around a solenoid), the phase $e\oint A\cdot dl$ is physically measurable. This shows that $A$, not $F$, is the fundamental object — a lesson that generalises to the Wilson-loop observables of non-Abelian gauge theories and the Chern-Simons invariants of topological field theories.

Continuum mechanics. The stress-energy tensor $T^{\mu \nu }$ is the electromagnetic instance of the general symmetric rank-2 tensor that appears in fluid dynamics (the perfect-fluid stress-energy tensor $T^{\mu \nu }=(\rho +p)u^{\mu }{u}^{\nu }-p{g}^{\mu \nu }$) and in elasticity theory. The conservation law ${\partial }_{\mu }{T}^{\mu \nu }=0$ (in source-free regions) is the continuum-physics statement of energy-momentum conservation.

Historical and philosophical context [Master]

The unification of electricity and magnetism into a single tensor is one of the great reductions in physics.

Minkowski (1908). Hermann Minkowski, in his lecture "Space and Time" delivered to the 80th Assembly of German Natural Scientists and Physicians, introduced the four-dimensional spacetime formalism and wrote down the electromagnetic field tensor for the first time. Minkowski showed that the Lorentz transformations are rotations in a four-dimensional space with indefinite metric, and that Maxwell's equations take a particularly compact form in this language. His opening words — "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows" — captured the conceptual shift.

Weyl (1918). Hermann Weyl attempted to unify gravity and electromagnetism by allowing the spacetime metric to change by a conformal factor under parallel transport. Einstein objected that this would make the size of objects path-dependent, contradicting experiment. Weyl's idea was rescued by the quantum-mechanical identification of the gauge phase with the electromagnetic vector potential: local $U(1)$ phase invariance, not conformal rescaling, is the correct gauge symmetry. The mathematical apparatus Weyl developed — the gauge connection, the field strength as the curvature of the connection — became the standard language of all gauge theories.

Yang and Mills (1954). Chen-Ning Yang and Robert Mills generalised the $U(1)$ gauge principle to non-Abelian groups ($SU(2)$ in their original paper, motivated by isospin symmetry of the nucleon). The resulting Yang-Mills equations are the non-Abelian generalisation of Maxwell's equations, with self-interacting gauge fields. The Standard Model of particle physics is a Yang-Mills theory with gauge group $SU(3)\times SU(2)\times U(1)$, where the $U(1)$ factor is precisely the electromagnetic gauge group studied in this unit.

Philosophical note. The Faraday tensor illustrates a general pattern in theoretical physics: what appears as multiple distinct phenomena in one frame (here, $\mathbf{E}$ and $\mathbf{B}$) can be revealed as different aspects of a single entity when viewed from a more general perspective (here, the spacetime perspective). The pattern repeats: the electric and magnetic fields unify into $F_{\mu \nu }$; the electric and magnetic charges would unify if magnetic monopoles existed (the Dirac monopole quantisation condition $eg=n\hslash /2$ would make duality between $\mathbf{E}$ and $\mathbf{B}$ exact); and at higher energies the electromagnetic and weak forces unify into the electroweak $SU(2)\times U(1)$ theory.

Bibliography [Master]

• Jackson, J. D. Classical Electrodynamics, 3rd ed. Wiley, 1999. Ch. 11, "Special Theory of Relativity," §11.9–11.11. The standard reference for covariant electrodynamics in SI units with the $(+,-,-,-)$ metric convention.
• Landau, L. D., and E. M. Lifshitz. The Classical Theory of Fields, 4th ed. Course of Theoretical Physics Vol. 2. Pergamon, 1975. §23–26. A terse, coordinate-free presentation emphasising the variational principle and the Lagrangian structure.
• Griffiths, D. J. Introduction to Electrodynamics, 4th ed. Cambridge University Press, 2017. Ch. 12.3. The most accessible presentation of relativistic electrodynamics at the undergraduate level.
• Susskind, L., and A. Friedman. Special Relativity and Classical Field Theory. Basic Books, 2017. Lecture 10. An intuition-first presentation suitable for the Beginner tier.
• Minkowski, H. "Space and Time." Address delivered at the 80th Assembly of German Natural Scientists and Physicians, Cologne, 21 September 1908. Translated in The Principle of Relativity, Dover. The originator paper for the spacetime formalism and the electromagnetic field tensor.
• Weyl, H. Space-Time-Matter. Dover, 1952 (translation of the 4th German edition, 1921). The originator of the gauge principle, initially as a conformal rescaling, later reinterpreted as $U(1)$ phase.
• Yang, C. N., and R. L. Mills. "Conservation of Isotopic Spin and Isotopic Gauge Invariance." Physical Review 96, 191–195 (1954). The originator paper for non-Abelian gauge theory.
• Tong, D. "Lectures on Electromagnetism." §5, "Covariant electrodynamics." University of Cambridge. Available at damtp.cam.ac.uk.