The Electromagnetic Field Tensor

Every article in this chapter has been a promissory note written against this one. The kinematics trio built the index machinery and called itself "deliberately a toolkit"; the transformation article derived six field-mixing rules from capacitor plates and called them "the data that the field tensor must explain"; the conservation chapter, three tiers back, deferred the construction of a covariant field object outright. Here the notes are paid. The electric and magnetic fields — six numbers at every event, mixing under boosts by rules that looked conspiratorial when derived plate by plate — are the six independent components of one antisymmetric rank-two tensor, $F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu$ , the four-dimensional curl of the four-potential. The construction takes one definition and delivers everything. Antisymmetry is automatic, and an antisymmetric $4\times4$ array has exactly $4\cdot3/2=6$ slots — precisely enough for $\mathbf{E}$ and $\mathbf{B}$ , none left over. The transformation rules earned with moving plates and velocity addition fall out of one matrix product, $\bar F=\Lambda F\Lambda^{\mathsf T}$ . Charge conservation stops being a separate law and becomes an algebraic identity. And the Lorentz force — force and power, in one line — is the tensor contracted with the four-velocity. This is the summit of the tensor ladder: not a new theory, but the discovery of what Maxwell's theory had been about all along. One object, sliced differently by different observers. The chapter's remaining articles are its corollaries.

Key takeaways

The field tensor $F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu$ packages $\mathbf{E}$ and $\mathbf{B}$ into one antisymmetric object: $4\cdot3/2=6$ slots exactly, and the split into "electric" (time–space block) and "magnetic" (space–space block) is frame-dependent.
In this track's $(+,-,-,-)$ convention: $F^{0i}=-E^i/c$ , $F^{ij}=-\epsilon^{ijk}B_k$ ; lowering indices flips the $E$ entries ( $F_{0i}=+E^i/c$ ) but leaves the $B$ block alone — index altitude is real information.
Gauge invariance is one line: $A^\mu\to A^\mu+\partial^\mu\chi$ shifts $F^{\mu\nu}$ by $\partial^\mu\partial^\nu\chi-\partial^\nu\partial^\mu\chi=0$ .
The tensor law $\bar F=\Lambda F\Lambda^{\mathsf T}$ reproduces, entry for entry, all six mixing rules the parallel-plate route derived physically — two independent routes, one law.
The source equation $\partial_\mu F^{\mu\nu}=\mu_0 J^\nu$ has charge conservation baked in: $\partial_\nu\partial_\mu F^{\mu\nu}\equiv0$ by antisymmetry, so $\partial_\mu J^\mu=0$ rides free.
The covariant Lorentz force $dp^\mu/d\tau=qF^{\mu\nu}u_\nu$ contains $q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$ in its space slots and the power $q\mathbf{E}\cdot\mathbf{v}$ in its time slot.
Key idea: E and B are not two fields but the six components of one antisymmetric spacetime tensor built from the four-potential — and every frame-dependence puzzle of this chapter is that one geometric fact, sliced.

The core insight

The electric and magnetic fields are not independent objects but the six components of a single antisymmetric spacetime tensor $F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu$ ; all of electrodynamics and its frame dependence follow from this one geometric fact.

Where this lives today

The field tensor is the template from which modern physics builds all of its forces. The Standard Model's field strengths, $F^a_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu+gf^{abc}A^b_\mu A^c_\nu$ , are this article's object with one self-interaction term added; gluons and weak bosons are decorated copies of $F^{\mu\nu}$ . On lattice-QCD machines it survives as the plaquette — the smallest closed loop of link variables, measuring $F_{\mu\nu}$ on a spacetime grid — and in general relativity $F_{\mu\nu}$ carries over to curved spacetime unchanged (antisymmetry cancels the connection terms), which is why black-hole magnetospheres and Blandford–Znajek energy extraction are computed with exactly these components in a curved metric. Closer to the laboratory, every accelerator control room implicitly integrates equation (4) — beam-optics codes propagate $qF^{\mu\nu}u_\nu$ through magnet lattices — and heavy-ion physicists reconstruct the strongest electromagnetic fields ever produced ( $B\sim10^{15}$ T in ultraperipheral collisions) as boosted Coulomb tensors of the kind Worked Example 1 built. Within this chapter, the summit opens three roads: the dual tensor, the invariants, and the covariant assembly of the full system — after which electrodynamics fits on a T-shirt: $\partial_\mu F^{\mu\nu}=\mu_0J^\nu$ , plus an identity.

Origins · 2 minHermann Minkowski and how this was discovered

Hermann Minkowski

1864–1909 · Göttingen — whose Grundgleichungen first wrote the field as one object

The tensor's birth certificate is dated 21 December 1907, when Minkowski read "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern" — The Fundamental Equations for Electromagnetic Processes in Moving Bodies — to the Göttingen Academy. Where the famous Cologne lecture the following autumn would sell the philosophy, the Grundgleichungen did the engineering: it introduced the antisymmetric array he called a Traktor (Sommerfeld would rename it the "six-vector"; field tensor came later still), wrote Maxwell's equations as matrix equations, exhibited their covariance by explicit multiplication, and — beyond even Lorentz and Poincaré — derived the covariant electrodynamics of moving media. It was the first time $\mathbf{E}$ and $\mathbf{B}$ appeared as components of one object rather than partners in a system: the first fully covariant field theory ever written down. Minkowski died thirteen months later; Max Born, then his assistant, assembled the unfinished sequel from his notes. The Grundgleichungen's four-dimensional algebra — reorganized by Sommerfeld into this chapter's index calculus — became the template for every field theory since: when Einstein needed a tensor language for gravitation, and when Yang and Mills needed a field strength to generalize, they reached for the object built in that December lecture.

History

1865

— Maxwell's dynamical theory: the fields exist, spread across twenty component equations with E and B as separate 3-vectors

1895

— Lorentz's corresponding states relate fields in moving systems; the mixing rules exist without their container

1905

— Einstein derives the field transformations from the relativity principle; the split of E and B declared frame-relative

1907

— Minkowski's Grundgleichungen: the antisymmetric field array appears, and Maxwell's equations become two matrix statements

1910

— Sommerfeld recasts Minkowski's algebra as four-vector and six-vector calculus — the index grammar of this chapter

1916

— Einstein's general relativity carries F_μν unchanged into curved spacetime; gravitation gets its own tensor field equations

1918

— Weyl's gauge principle: F as the invariant content of a potential with local symmetry — the seed of modern gauge theory

1954

— Yang and Mills generalize the field strength to non-Abelian groups; the Standard Model's forces inherit F's blueprint

The physical picture

Climb the ladder the chapter has already built. A Lorentz scalar is one number every observer shares. A four-vector, like the four-momentum of Relativistic Energy and Four-Momentum, is four numbers whose split into "time part" and "space part" depends on the observer. The field tensor is the next rung: a rank-two antisymmetric object whose split into an "electric part" (the time–space entries) and a "magnetic part" (the space–space entries) depends on the observer in exactly the same way. There is one field; $\mathbf{E}$ and $\mathbf{B}$ are frame-dependent labels for its blocks, no more absolute than "the energy" of a particle. The container has precisely the right shape. A general $4\times4$ tensor has sixteen components; antisymmetry, $F^{\mu\nu}=-F^{\nu\mu}$ , kills the diagonal and pairs the rest, leaving $4\cdot3/2=6$ independent slots — not close to what electromagnetism needs but exactly what it needs: three components of $\mathbf{E}$ , three of $\mathbf{B}$ , no orphan slot begging for a seventh field experiment has never found. Under pure spatial rotations the six split cleanly into two 3-vectors, which is why three centuries of experimenters could believe they were studying two separate fields. Under boosts the blocks bleed into each other — and Transformation of Electric and Magnetic Fields measured the bleeding, rule by rule, before knowing the object that enforced it. The deepest structural fact comes from the construction rather than the counting. $F^{\mu\nu}$ is a four-dimensional curl of the potential, and everything a curl kills, it kills: $\partial^\mu\partial^\nu\chi=\partial^\nu\partial^\mu\chi$ makes $F$ blind to gauge transformations, and — run in reverse, by The Dual Field Tensor — makes the two source-free Maxwell equations automatic identities rather than laws. The field tensor is where the chapter's grammar and its physics finally coincide.

Figure

One antisymmetric container with six slots — the electric block and magnetic block are frame-dependent labels for one object

The math: one definition, then corollaries

The ingredients are banked. The Four-Potential assembled $A^\mu=(V/c,\mathbf{A})$ ; Four-Vectors and the Metric Tensor supplied the four-gradient $\partial^\mu=(\tfrac{1}{c}\partial_t,-\nabla)$ and the rank-two transformation law. The field tensor is their antisymmetrized product — the only rank-two object one potential and one derivative can build with a chance of being gauge-independent:

F^{\mu\nu}\;\equiv\;\partial^\mu A^\nu-\partial^\nu A^\mu. \qquad (1)

where: $F^{\mu\nu}$ is manifestly antisymmetric, and it is a bona fide tensor because it is built from certified four-vectors by legal grammar — no new transformation postulate needed. Physically, (1) is a four-dimensional curl: the construction that extracts a rotation-blind field from a scalar potential in three dimensions here extracts a gauge-blind field from the four-potential. Derivation A opens the box and finds, sitting in the components, the two fields this book has been about since page one — delivering, at last, the covariant field object that Conservation Laws and Spacetime Symmetries (E7) could only promise when it needed one for Noether bookkeeping.

Derivation AOpening the definition — E and B are the six components

The goal is the complete component content of definition (1): all six independent entries, assembled into explicit matrices in both index positions, with the $(+,-,-,-)$ sign bookkeeping done in public — the sign structure here is the classic trap of the subject, and standard references disagree by exactly the flips we are about to track. The strategy is pure evaluation: insert $A^\mu=(V/c,\mathbf{A})$ and $\partial^\mu=(\tfrac{1}{c}\partial_t,-\nabla)$ into (1) and compare with $\mathbf{E}=-\nabla V-\partial_t\mathbf{A}$ and $\mathbf{B}=\nabla\times\mathbf{A}$ , banked in The Four-Potential.

Step 1. Time–space components. With $\mu=0$ , $\nu=i$ (recall $\partial^0=+\tfrac{1}{c}\partial_t$ but $\partial^i=-\partial_i$ — the raised spatial gradient carries the metric's minus sign):

F^{0i}=\partial^0A^i-\partial^iA^0=\frac{1}{c}\,\partial_tA^i+\partial_i\!\left(\frac{V}{c}\right)=\frac{1}{c}\left(\partial_tA^i+\partial_iV\right)=-\frac{E^i}{c}.

Why the sign matters: the bracket is exactly the negative of $E^i=-\partial_iV-\partial_tA^i$ . So the contravariant tensor carries $-E^i/c$ in its top row and, by antisymmetry, $+E^i/c$ down its first column. Neither sign is a convention you can wish away — it is forced by (1), the metric, and the definition of $\mathbf{E}$ .

Step 2. Space–space components. With both indices spatial, both raised gradients flip:

F^{ij}=\partial^iA^j-\partial^jA^i=-\left(\partial_iA^j-\partial_jA^i\right)=-\epsilon^{ijk}B_k,

using $B_k=\epsilon_{kij}\partial_iA^j$ from $\mathbf{B}=\nabla\times\mathbf{A}$ . Concretely: $F^{12}=-B_z$ , $F^{13}=+B_y$ , $F^{23}=-B_x$ — the magnetic field woven antisymmetrically into the spatial block.

Step 3. Assemble both matrices. Rows are $\mu$ , columns $\nu$ :

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},\qquad 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 covariant version comes from lowering both indices: a time–space entry picks up one factor of $(-1)$ , so every electric entry flips sign; a space–space entry picks up two, so the magnetic block is untouched. Remember the asymmetry and half the sign errors of the subject die.

Step 4. Gauge invariance, in one line. Under $A^\mu\to A^\mu+\partial^\mu\chi$ (the freedom catalogued by The Four-Potential),

F^{\mu\nu}\;\to\;F^{\mu\nu}+\partial^\mu\partial^\nu\chi-\partial^\nu\partial^\mu\chi=F^{\mu\nu},

because partial derivatives commute. The fields are the gauge-proof content of the potential — which is why they, and not $A^\mu$ , were discovered first. And the inventory closes: antisymmetry leaves six independent components; Steps 1–2 filled every one. Why the signs are trustworthy: the $\nu=0$ slot of $\partial_\mu F^{\mu\nu}$ evaluates to $\tfrac{1}{c}\nabla\cdot\mathbf{E}$ with these signs, landing Gauss's law with the correct + sign — the audit that catches flipped-E-row conventions. Note also that some printed sources give matrices ( $F^{0i}=+E^i/c$ ) inconsistent with definition (1) in $(+,-,-,-)$ ; the consistent set used throughout is $F^{0i}=-E^i/c$ , $F_{0i}=+E^i/c$ , $F^{ij}=-\epsilon^{ijk}B_k$ .

Read the matrices physically before boosting them. A static charge fills only the electric rim — first row and column; a steady current loop fills only the magnetic core — the inner $3\times3$ block. The claim of relativity, made mechanical by the tensor law of Four-Vectors and the Metric Tensor, is that a boost mixes rim and core:

\bar F^{\mu\nu}=\Lambda^\mu{}_\alpha\,\Lambda^\nu{}_\beta\,F^{\alpha\beta}\qquad\text{(matrix form: }\bar F=\Lambda F\Lambda^{\mathsf T}\text{)}. \qquad (2)

where: $\Lambda$ is the same boost matrix that shuffles coordinates — for a boost at $v=\beta c$ along $x$ , the familiar $\gamma$ , $-\gamma\beta$ block from Lorentz Transformations. Equation (2) is not a new physical input: it is what "tensor" means. The physical input was spent long ago, in the plate-by-plate derivation of the mixing rules — and the next derivation shows that (2) reproduces them wholesale, which is simultaneously a consistency test of two chapters and the moment the rules stop being surprising.

Go deeper · Derivation B · 2 minThe boost of F reproduces the six field-transformation rules

Derivation BThe boost of F reproduces the six field-transformation rules

The goal is every rule of Transformation of Electric and Magnetic Fields — $\bar E_x=E_x$ , $\bar E_y=\gamma(E_y-vB_z)$ , $\bar E_z=\gamma(E_z+vB_y)$ , $\bar B_x=B_x$ , $\bar B_y=\gamma(B_y+\tfrac{v}{c^2}E_z)$ , $\bar B_z=\gamma(B_z-\tfrac{v}{c^2}E_y)$ — obtained this time from the single matrix product (2): no capacitors, no velocity addition, no surface currents. The strategy: execute the mixing pair in full ( $\bar E_y$ from $\bar F^{02}$ , $\bar B_z$ from $\bar F^{12}$ ), dispose of the longitudinal entries with $\gamma^2(1-\beta^2)=1$ , and tabulate the rest. That article derived the rules physically; this one derives them grammatically — two routes with zero shared steps landing on six identical equations, the chapter's strongest internal audit.

Step 1. Longitudinal entries are protected. The $01$ entry contracts both boost rows:

\bar F^{01}=\Lambda^0{}_\alpha\Lambda^1{}_\beta F^{\alpha\beta}=\left(\Lambda^0{}_0\Lambda^1{}_1-\Lambda^0{}_1\Lambda^1{}_0\right)F^{01}=\left(\gamma^2-\gamma^2\beta^2\right)F^{01}=F^{01},

using antisymmetry to pair the cross terms and $\gamma^2(1-\beta^2)=1$ . So $\bar E_x=E_x$ ; and the $23$ entry sees only the identity block of $\Lambda$ , so $\bar B_x=B_x$ . The components along the boost ride through untouched, exactly as the plates and the solenoid found.

Step 2. Execute the transverse electric entry. The index $2$ is inert ( $\Lambda^2{}_\beta=\delta^2{}_\beta$ ), so only $\alpha=0,1$ survive:

\bar F^{02}=\Lambda^0{}_\alpha F^{\alpha2}=\Lambda^0{}_0F^{02}+\Lambda^0{}_1F^{12}=\gamma\left(-\frac{E_y}{c}\right)+(-\gamma\beta)(-B_z)=-\frac{\gamma}{c}\left(E_y-vB_z\right).

Reading off with $\bar F^{02}=-\bar E_y/c$ : $\ \bar E_y=\gamma(E_y-vB_z)$ . Why the signs conspire: the boost row supplies $-\gamma\beta$ and the magnetic entry supplies $F^{12}=-B_z$ ; their product builds the physical $-vB_z$ . Get either sign wrong and the mixing rule flips — how convention errors announce themselves.

Step 3. Execute the transverse magnetic entry, same machinery:

\bar F^{12}=\Lambda^1{}_\alpha F^{\alpha2}=\Lambda^1{}_0F^{02}+\Lambda^1{}_1F^{12}=(-\gamma\beta)\left(-\frac{E_y}{c}\right)+\gamma(-B_z)=-\gamma\left(B_z-\frac{\beta}{c}E_y\right),

and with $\bar F^{12}=-\bar B_z$ : $\ \bar B_z=\gamma\left(B_z-\tfrac{v}{c^2}E_y\right)$ . The $E_y$ – $B_z$ pair transforms into itself.

Step 4. Tabulate the remaining pair by substitution. The $(E_z,B_y)$ sector runs the identical algebra through $\bar F^{03}$ and $\bar F^{13}$ with $F^{13}=+B_y$ supplying the opposite relative sign: $\bar E_z=\gamma(E_z+vB_y)$ , $\bar B_y=\gamma(B_y+\tfrac{v}{c^2}E_z)$ . Collecting all six:

entry executed	rule recovered	plate-route counterpart
$\bar F^{01}=F^{01}$	$\bar E_x=E_x$	plates ⊥ motion: areas don't contract
$\bar F^{23}=F^{23}$	$\bar B_x=B_x$	solenoid: $\gamma n\cdot I/\gamma$ standoff
$\bar F^{02}$ , Step 2	$\bar E_y=\gamma(E_y-vB_z)$	two-observer capacitor, velocity addition
$\bar F^{12}$ , Step 3	$\bar B_z=\gamma(B_z-\tfrac{v}{c^2}E_y)$	current sheets on moving plates
$\bar F^{03}$	$\bar E_z=\gamma(E_z+vB_y)$	apparatus rotated 90° about $x$
$\bar F^{13}$	$\bar B_y=\gamma(B_y+\tfrac{v}{c^2}E_z)$	same rotation, magnetic partner

Line for line, these are equations (1)–(2) of Transformation of Electric and Magnetic Fields. What that article assembled from charge invariance, contraction, and velocity addition, the tensor law delivers as sixteen multiply-adds: $\mathbf{E}$ and $\mathbf{B}$ mix because they are blocks of one matrix and boosts are not block-diagonal. The conspiracy was only ever a slicing. Two independent routes — the plate-by-plate physical derivation and the single matrix product $\Lambda F\Lambda^{\mathsf T}$ — land on six identical equations; Worked Example 1 below confirms the numerical agreement for the capacitor boost.

Figure

A boost is not block-diagonal — it tilts the electric rim into the magnetic core and back

With the object in hand, the field equations compress. The sources couple to $F$ through one covariant statement — the inhomogeneous Maxwell equation,

\partial_\mu F^{\mu\nu}=\mu_0 J^\nu, \qquad (3)

where: $J^\nu=(c\rho,\mathbf{J})$ is the four-current of The Four-Current Density, and the free index $\nu$ makes this four scalar equations bundled as one. Its $\nu=0$ slot reads $\tfrac{1}{c}\nabla\cdot\mathbf{E}=\mu_0c\rho$ — Gauss's law, via $\mu_0\varepsilon_0=1/c^2$ — and its spatial slots carry Ampère–Maxwell; the full unpacking, the companion equation for the source-free pair, and the audit that the covariant system is exactly Maxwell's four belong to Covariant Form of Maxwell's Equations. What belongs here is the one consequence that needs nothing but antisymmetry — the chapter's cheapest profound result.

Go deeper · Derivation C · 1 minCharge conservation is an identity of antisymmetry

Derivation CCharge conservation is an identity of antisymmetry

The goal is $\partial_\mu J^\mu=0$ — local charge conservation — obtained in two lines from the structure of (3), with no reference to what the fields are doing. The strategy is the symmetric-against-antisymmetric annihilation argument, the most reusable trick in tensor calculus; watch it once in slow motion and you own it forever.

Step 1. Take the four-divergence of both sides of (3):

\mu_0\,\partial_\nu J^\nu=\partial_\nu\partial_\mu F^{\mu\nu}.

The right side contracts the derivative pair $\partial_\nu\partial_\mu$ — symmetric under $\mu\leftrightarrow\nu$ , because partial derivatives commute — against $F^{\mu\nu}$ , antisymmetric under the same swap.

Step 2. A symmetric object contracted with an antisymmetric one vanishes identically: relabel the dummy indices $\mu\leftrightarrow\nu$ , use both symmetries, and the expression equals its own negative,

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

Charge conservation is not an extra law layered on electrodynamics — it is a consistency condition the theory enforces on its own sources: a non-conserved current could not source an antisymmetric tensor at all. This is the covariant closure The Four-Current Density promised — the continuity equation it certified as the Lorentz scalar $\partial_\mu J^\mu=0$ is here revealed as automatic. The identity $\partial_\nu\partial_\mu F^{\mu\nu}=0$ holds for arbitrary potential functions; the same symmetric-against-antisymmetric argument, with $u_\mu u_\nu$ in place of $\partial_\mu\partial_\nu$ , underlies the no-work theorem of the force band below.

One more contraction completes the toolkit — the force law. Couple the tensor to a charge's four-velocity (from Proper Time and Four-Velocity, lowered: $u_\nu=\gamma(c,-\mathbf{v})$ ) and postulate the covariant equation of motion

\frac{dp^\mu}{d\tau}=q\,F^{\mu\nu}u_\nu. \qquad (4)

where: this is the only four-force linear in the field and the velocity that grammar permits, and unpacking checks its content. The spatial slots give $dp^i/d\tau=q\gamma\left(E^i+(\mathbf{v}\times\mathbf{B})^i\right)$ ; dividing by $\gamma$ to convert to lab time recovers exactly $d\mathbf{p}/dt=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$ — the Lorentz force law, first met in magnetostatics, now a tensor contraction. The time slot gives $dp^0/d\tau=q\gamma\,\mathbf{E}\cdot\mathbf{v}/c$ , i.e. $dE_{\text{energy}}/dt=q\mathbf{E}\cdot\mathbf{v}$ : the work–energy theorem rides in the same equation, with $\mathbf{B}$ conspicuously absent — and antisymmetry explains why in one line. Contracting (4) with $u_\mu$ gives $qF^{\mu\nu}u_\mu u_\nu=0$ (symmetric against antisymmetric), so the four-force is orthogonal to the four-velocity, the norm $u^\mu u_\mu=c^2$ is preserved, and no electromagnetic force can change a rest mass; magnetic forces do no work as the special case where the surviving electric term also vanishes. Solving (4) — cyclotron orbits, drifts, the machinery of every accelerator — is the business of Relativistic Dynamics of a Charged Particle; building the field's own energy–momentum ledger from $F$ is The Electromagnetic Stress-Energy Tensor's. Both start from (3) and (4) as given.

Limiting cases

case	result	meaning
pure electrostatics ( $\mathbf{B}=0$ )	only the E rim populated; $F_{\mu\nu}F^{\mu\nu}=-2E^2/c^2<0$	invariantly "electric-dominated": no boost erases $\mathbf{E}$
pure magnetostatics ( $\mathbf{E}=0$ )	only the B core; $F_{\mu\nu}F^{\mu\nu}=+2B^2>0$	invariantly "magnetic-dominated": no boost erases $\mathbf{B}$
plane wave	$E=cB$ , $\mathbf{E}\perp\mathbf{B}$ : both invariants vanish	null field — radiation looks like radiation in every frame
$\beta\ll1$ boost	$\bar{\mathbf{E}}\approx\mathbf{E}+\mathbf{v}\times\mathbf{B}$ , $\bar{\mathbf{B}}\approx\mathbf{B}-\mathbf{v}\times\mathbf{E}/c^2$	Galilean-looking mixing; motional-EMF physics recovered
spatial rotations only	E rim and B core rotate separately as 3-vectors	why $\mathbf{E}$ and $\mathbf{B}$ pass as independent fields in any one frame
gauge shift $A^\mu\to A^\mu+\partial^\mu\chi$	$F^{\mu\nu}$ unchanged, exactly	fields are the gauge-proof observables; potentials are scaffolding

Common trapF with indices up and F with indices down differ by real signs — on the electric entries only

The most reliable error generator in covariant electrodynamics: treating $F^{\mu\nu}$ and $F_{\mu\nu}$ as the same array. In $(+,-,-,-)$ , lowering both indices flips each time–space entry once (one spatial metric factor) and each space–space entry twice (net nothing): every electric entry changes sign, no magnetic entry does. With definition (1), $F^{0i}=-E^i/c$ while $F_{0i}=+E^i/c$ — and textbooks compound the hazard by disagreeing among themselves; conventions differ between sources, and slips of exactly this kind reach print. The survival protocol is mechanical. State the signature. Derive the top row from (1) once — four lines. Then run the two structural audits that transcend dialect: the $\nu=0$ slot of (3) must produce Gauss's law with a plus sign, and $\bar F=\Lambda F\Lambda^{\mathsf T}$ must reproduce $\bar E_y=\gamma(E_y-vB_z)$ with a minus. Any sign assignment passing both audits is correct; any failing either is wrong in every convention.

Worked examples

1 · Boost a capacitor by matrix multiplication (confidence builder)

Setup. The first worked example of Transformation of Electric and Magnetic Fields: a capacitor at rest in $S$ with $\mathbf{E}=10^5\,\hat{\mathbf{y}}$ V/m, $\mathbf{B}=0$ , viewed from $\bar S$ at $v=0.8c\,\hat{\mathbf{x}}$ . Redo it with the tensor.

Step 1. Populate the matrix. Only $F^{02}=-F^{20}=-E_y/c=-3.33\times10^{-4}$ V·s/m² is nonzero (with $c=3.00\times10^8$ m/s); the magnetic core is empty. Boost numbers: $\gamma=(1-0.64)^{-1/2}=5/3$ , $\gamma\beta=4/3$ .

Step 2. Multiply $\bar F=\Lambda F\Lambda^{\mathsf T}$ . Antisymmetry leaves two working entries: $\bar F^{02}=\Lambda^0{}_0F^{02}=\gamma F^{02}=-5.56\times10^{-4}$ , and $\bar F^{12}=\Lambda^1{}_0F^{02}=(-\gamma\beta)(-E_y/c)=+4.44\times10^{-4}$ .

Step 3. Read off with the component dictionary: $\bar E_y=-c\bar F^{02}=\gamma E_y=\mathbf{1.67\times10^5}$ V/m and $\bar B_z=-\bar F^{12}=-\gamma\beta E_y/c=\mathbf{-4.44\times10^{-4}}$ T — a magnetic field where the rest frame had none, matching the plate-route example digit for digit.

Step 4. Free audit: $E^2-c^2B^2$ before is $1.00\times10^{10}$ V²/m²; after, $(1.667\times10^5)^2-(3\times10^8)^2(4.44\times10^{-4})^2=1.00\times10^{10}$ . Preserved — the invariant the next two articles will name and exploit.

The lesson: "boost the fields" means one matrix sandwich — all six rules at once, stored in $\Lambda F\Lambda^{\mathsf T}$ .

2 · The covariant force on a beam proton (exam level)

Setup. A proton crosses a beam-cleaning region at $v=0.6c\,\hat{\mathbf{x}}$ ( $\gamma=1.25$ ) through crossed static fields $\mathbf{E}=2.0\times10^6\,\hat{\mathbf{y}}$ V/m, $\mathbf{B}=0.050\,\hat{\mathbf{z}}$ T. Evaluate the covariant force (4), both slots, and audit orthogonality.

Step 1. The lowered four-velocity: $u_\nu=\gamma(c,-\mathbf{v})=(3.75\times10^8,\,-2.25\times10^8,\,0,\,0)$ m/s.

Step 2. Spatial slot of (4): $\mathbf{v}\times\mathbf{B}=-9.0\times10^6\,\hat{\mathbf{y}}$ V/m — the motional term is 4.5× the applied $\mathbf{E}$ and opposes it. Then $dp^y/d\tau=q\gamma(E_y+(\mathbf{v}\times\mathbf{B})_y)=\mathbf{-1.40\times10^{-12}}$ N; per lab time, $dp^y/dt=-1.12\times10^{-12}$ N. The magnetic term wins at this speed (balance would need $v=E/B=4.0\times10^7$ m/s $=0.133c$ ).

Step 3. Time slot: $dp^0/d\tau=q\gamma\,\mathbf{E}\cdot\mathbf{v}/c=0$ , since $\mathbf{E}\perp\mathbf{v}$ . The fields deflect without doing work — energy momentarily constant though a $10^{-12}$ N force acts. The covariant equation carries both books at once.

Step 4. Orthogonality audit: the only nonzero force component is $y$ , and $u_y=0$ , so $u_\mu\,dp^\mu/d\tau=0$ identically — as antisymmetry guarantees for every state of motion, not just this convenient one. Postscript: since $E<cB$ here ( $cB=1.5\times10^7$ V/m), a frame exists where the field is purely magnetic — boost at $0.133c$ and $\mathbf{E}$ vanishes while $B$ drops to $49.6$ mT. Which fields can be transformed away is the classification Lorentz Invariants of the Field delivers; the trajectory the proton actually follows is Relativistic Dynamics of a Charged Particle's opening problem.

The lesson: one contraction yields force and power together, and the invariants already whisper which frame makes the problem trivial.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §12.3.3–12.3.4 · Jackson, Classical Electrodynamics, 3rd ed., §11.9–11.11 · Zangwill, Modern Electrodynamics, §22.3 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 5–6 (Appendix on field transformations) · Landau & Lifshitz, The Classical Theory of Fields, §23–26 The consistent component set for definition (1) in $(+,-,-,-)$ is $F^{0i}=-E^i/c$ , $F_{0i}=+E^i/c$ , $F^{ij}=F_{ij}=-\epsilon^{ijk}B_k$ ; the relation $\mu_0\epsilon_0=1/c^2$ is the standard SI identity. Note that a sign inconsistency appears in some printed sources: matrices with $F^{0i}=+E^i/c$ combined with $F^{ij}=-\epsilon^{ijk}B_k$ fail both the Gauss's-law reduction and the $\Lambda F\Lambda^T$ field-transformation audit. The four key structural checks are: (1) confirm $F^{\mu\nu}$ and $F_{\mu\nu}$ by lowering with $\eta=\mathrm{diag}(+,-,-,-)$ ; (2) reduce the inhomogeneous equation to Gauss and Ampère–Maxwell; (3) verify field-transformation components via $\Lambda F\Lambda^T$ ; (4) check the invariant $F_{\mu\nu}F^{\mu\nu}=2(B^2-E^2/c^2)$ . Worked numbers: capacitor boost $\beta=0.8$ gives $\bar E_y=1.67\times10^5$ V/m, $\bar B_z=-4.44\times10^{-4}$ T, invariant $1.00\times10^{10}$ V²/m² in both frames; proton at $\beta=0.6$ through $2$ MV/m $\times$ $50$ mT crossed fields gives $d\mathbf{p}/dt=-1.12\times10^{-12}\,\hat{\mathbf{y}}$ N, zero power, velocity-selector balance at $0.133c$ , pure-B frame field $49.6$ mT.

Builds on: Four-Vectors and the Metric Tensor · The Four-Potential · The Four-Current Density · Lorentz Transformations · Spacetime and Minkowski Geometry Leads to: Transformation of Electric and Magnetic Fields · The Dual Field Tensor · Lorentz Invariants of the Field · Magnetism as a Relativistic Effect

Is the antisymmetry of the field tensor an assumption, or is it forced?

Forced twice over. Constructively: $F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu$ is antisymmetric by inspection — swap the indices and the terms trade places with a sign. Structurally: the counting must come out to six, since $\mathbf{E}$ and $\mathbf{B}$ have six components between them, and only the antisymmetric part of a rank-two object has six slots; a symmetric tensor carries ten and mixes in a diagonal no field measurement has ever demanded. And dynamically: the charge-conservation identity of Derivation C needs antisymmetry — couple a conserved current to a tensor with any symmetric part and the four-divergence argument fails. Three arguments, one fact: electromagnetism is the theory of a spacetime 2-form.

Different books give different signs for the tensor components. How do I work across references safely?

Three independent binary choices circulate: signature ( $+---$ vs $-+++$ ), definition sign, and units (SI's $E/c$ vs Gaussian's $E$ ) — eight self-consistent dialects. The protocol: never import a component matrix bare; import the book's definition and signature and derive the top row yourself (Derivation A, Step 1 — four lines). Then run the two dialect-proof audits: the $\nu=0$ slot of the source equation must yield Gauss's law as $+\rho/\varepsilon_0$ , and boosting a pure $E_y$ must return $\bar E_y=\gamma(E_y-vB_z)$ . Physical statements no convention can flip: a matrix passing both is right in its dialect, one failing either is wrong in all of them. Printed matrices inconsistent with their stated definition are caught by exactly these audits. Trust the audits, not the table.

Which is fundamental — the field tensor or the four-potential?

Classically the tensor: $F^{\mu\nu}$ is gauge-invariant and locally measurable (a test charge reads it via equation (4)), while infinitely many potentials describe the same $F$ — scaffolding, not structure. Quantum mechanics sharpens and reverses the verdict: a charged particle's phase couples to $A^\mu$ itself, and the Aharonov–Bohm effect detects potentials where $F^{\mu\nu}=0$ locally — though only through gauge-invariant loop integrals $\oint A_\mu dx^\mu$ . The modern synthesis: the fundamental object is the gauge equivalence class, whose local curvature is $F$ and whose global holonomy exceeds it. For everything in this chapter the tensor suffices — and note what the potential buys even classically: $F=\partial A-\partial A$ makes half of Maxwell's equations automatic, as the next article shows.

Where did the other two Maxwell equations go — Faraday's law and the absence of monopoles?

They are hiding as identities. Because $F$ descends from a potential, the cyclic combination $\partial_\lambda F_{\mu\nu}+\partial_\mu F_{\nu\lambda}+\partial_\nu F_{\lambda\mu}$ vanishes automatically — commuting derivatives kill it, no physics required — and unpacked into components that single statement is $\nabla\cdot\mathbf{B}=0$ plus Faraday's law. The covariant formulation thus splits Maxwell's four by logical status: two are dynamics (the source equation, which genuinely constrains the field), two are consequences of the field's descent from a potential. Organizing the identity cleanly — as the vanishing divergence of a second tensor built by dualizing $F$ with the Levi-Civita symbol — is the entire business of The Dual Field Tensor, next in the ladder.

If the no-work contraction is zero, how does a magnetic crane lift a car?

The identity $u_\mu F^{\mu\nu}u_\nu=0$ governs a point charge under equation (4): energy transfer comes exclusively from the electric term, $dE/dt=q\mathbf{E}\cdot\mathbf{v}$ . A crane's load is not a point charge — it is a lattice of magnetized matter. The inhomogeneous field torques and displaces current loops whose rigidity is enforced electrostatically and quantum-mechanically; the microscopic work chain runs through those internal constraint forces, and the joules are paid by the power supply driving the magnet against back-EMF (the Poynting ledger banked in the conservation chapter). The covariant identity survives untouched at the level where it was proven; the lifting is a composite-system effect in which electric fields — in the windings and inside the iron — do every joule of the work.

Why does the field tensor survive unchanged in general relativity when everything else needs covariant derivatives?

Replace $\partial_\mu$ with the covariant derivative and watch: $\nabla_\mu A_\nu-\nabla_\nu A_\mu=\partial_\mu A_\nu-\partial_\nu A_\mu-(\Gamma^\lambda{}_{\mu\nu}-\Gamma^\lambda{}_{\nu\mu})A_\lambda$ , and the connection's symmetry in its lower indices (torsion-free spacetime) kills the bracket. Antisymmetrization is exactly the operation that deletes geometry's correction terms — in form language, $F=dA$ needs no metric or connection at all. So electrodynamics in a curved metric keeps this article's $F_{\mu\nu}$ verbatim, with curvature entering only through index-raising and the covariant divergence in the source equation. That is why black-hole magnetosphere codes carry these components into Kerr geometry with a clear conscience, and why the coming invariants remain well-defined scalars even where spacetime is dynamical.