THE ELECTROMAGNETIC FIELD TENSOR

§11.2 left us with a strange-looking fact: under a Lorentz boost, the components of $\mathbf{E}$ and $\mathbf{B}$ mix. A pure electric field in one frame becomes a tilted blend of $\mathbf{E}'$ and $\mathbf{B}'$ in another. Three numbers plus three numbers — six in total — shuffle into each other in a linear, frame-dependent way. That is the unmistakable signature of a tensor: a single multilinear object whose components transform smoothly between frames.

The natural Lorentz-covariant container for six numbers is a rank-2 antisymmetric 4-tensor. A 4×4 matrix has 16 entries; require $F^{\mu\nu} = -F^{\nu\mu}$ and you forbid the four diagonal entries (since $F^{\mu\mu} = -F^{\mu\mu} \Rightarrow F^{\mu\mu} = 0$) and tie each lower-triangular entry to its upper-triangular partner. What's left is exactly $4 \cdot 3 / 2 = 6$ independent numbers. The arithmetic is too clean to be coincidence — and it isn't.

The man who saw this first was , the German mathematician who had taught a young mathematics at ETH Zürich. (Einstein cut his lectures; Minkowski later remembered him as "lazy". The relationship was mutual irritation. Minkowski died at 44 of acute appendicitis in 1909, four years after handing Einstein's 1905 special relativity its mathematical home.) In a 1908 lecture in Cologne he announced what is perhaps the most-quoted sentence in 20th-century physics: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." The "union" was the 4-dimensional pseudo-Euclidean geometry now called Minkowski space, and the field tensor is the object that lives inside it.

What §11.3 does, then, is mechanical. We pack the six numbers $(E_x, E_y, E_z, B_x, B_y, B_z)$ into a single 4×4 antisymmetric tensor $F^{\mu\nu}$, write down the rules that say how its components transform under a Lorentz boost (those rules are forced by the fact that $F^{\mu\nu}$ is a tensor — they are not new physics), and then watch the four scalar Maxwell equations of collapse into two compact tensor expressions on either side of the divergence operator $\partial_\mu$. Two sentences for the entire 1865 Treatise on Electricity and Magnetism, four years after Maxwell wrote the original twenty.

We adopt mostly-minus signature $(+,-,-,-)$ following Griffiths. With Greek indices ranging over $(0,1,2,3) = (ct, x, y, z)$, the prescription that makes the construction work — meaning, that gives the right transformations under boost and the right Maxwell equations on contraction with $\partial_\mu$ — is

$F^{0i} = E_i / c, \qquad F^{ij} = -\varepsilon_{ijk} B_k.$

Spelled out as a 4×4 matrix:

Sixteen entries. Four zeros on the diagonal. Six independent numbers above the diagonal (the three $E_i/c$ in the top row and the three $B_i$ in the lower-right $3\times 3$ block); their negatives appear, antisymmetrically, below the diagonal. The factor of $c$ on the $E$ entries is the only piece of bookkeeping that is annoying the first time and obvious the second: SI $\mathbf{E}$ has units V/m, SI $\mathbf{B}$ has units T = V·s/m², and dividing $E$ by $c$ puts both into the same units. Inside $F^{\mu\nu}$, every entry has units of tesla.

There is a deeper origin for this construction. The four-potential $A^\mu = (\varphi/c, \mathbf{A})$ — which §11.5 builds out — gives both fields by differentiation: $\mathbf{E} = -\nabla\varphi - \partial_t \mathbf{A}$ and $\mathbf{B} = \nabla\times\mathbf{A}$. Writing those two definitions in covariant notation collapses them into

$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu,$

a manifestly antisymmetric object (an exterior derivative). The 4-curl of a 4-vector. Two field-vector definitions, one tensor identity.

Read the matrix as a grid. The top row + first column carry the electric field; the lower-right $3\times 3$ block carries the magnetic field. Mouse over any cell and its antisymmetric partner across the diagonal lights up.

The grid does two things at once. First, it makes the six-number count visible — there are exactly six magenta-or-cyan cells above the diagonal, and the lower triangle is the same six numbers with flipped signs. Second, it reveals the pattern that gets broken if you try to add a magnetic monopole charge: a monopole would source $\nabla\cdot\mathbf{B} \neq 0$, which would mean an extra independent component appears in the structure — i.e. $F^{\mu\nu}$ would no longer fit cleanly into one antisymmetric tensor. (This is the main observation of Dual field tensor.) Without monopoles, six numbers are exactly what we need, and one antisymmetric 4-tensor is exactly what we get. Electromagnetic field tensor

A worked example helps the eye. Switch the preset to "pure E in x". The grid lights up in only two cells — $F^{01} = E_x/c$ and $F^{10} = -E_x/c$ — and everything else is zero. That is the entire content of "a static electric field along the lab x-axis": one number, antisymmetrically duplicated, sitting in two of the sixteen cells. Now switch to "pure B in z". A different pair lights up: $F^{12} = -B_z$ and $F^{21} = +B_z$. Same structural story, different cells. The "general field" preset turns six cells on at once. Six numbers, six cells; the rest is bookkeeping.

Here is where the geometry pays off. Maxwell's four scalar equations, written in 19th-century vector calculus, are two tensor equations once $F^{\mu\nu}$ is in hand:

with the four-current $J^\nu = (c\rho, \mathbf{J})$, and

That's it. Eight scalar equations of vector calculus; two compact tensor identities.

Take the first one and let $\nu = 0$. The contraction is $\partial_0 F^{00} + \partial_1 F^{10} + \partial_2 F^{20} + \partial_3 F^{30}$. The $F^{00}$ piece vanishes (diagonal), and the spatial pieces are $\partial_i F^{i0} = -\partial_i (E_i/c) = -\nabla\cdot\mathbf{E}/c$. The right side is $\mu_0 J^0 = \mu_0 c\rho$. Pulling the minus sign through and using $\mu_0 c^2 = 1/\varepsilon_0$:

$\nabla\cdot\mathbf{E} = \rho/\varepsilon_0.$

Gauss's law. For $\nu = i \in \{1,2,3\}$ the same exercise — now with both temporal and spatial derivatives, and with the $-\varepsilon_{ijk}B_k$ block contributing the curl — recovers $\nabla\times\mathbf{B} - \mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t = \mu_0\mathbf{J}$. That is Ampère-Maxwell.

The dual equation $\partial_\mu {}^{*}F^{\mu\nu} = 0$ does the same job for the source-free pair. $\nu = 0$ gives $\nabla\cdot\mathbf{B} = 0$ (no monopoles); $\nu = i$ gives $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ (Faraday). Eight scalar equations, two tensor expressions, identical content.

The compression is not just notational. It says something about what the equations are. In vector calculus the four laws read like four unrelated facts about how electric and magnetic fields move charges, get sourced, and feed each other. In tensor form they are two statements about a single object: "the divergence of $F$ is the source four-current" and "the divergence of the dual is zero". Frame-invariance, which had to be checked by hand for each of the four scalar equations under a boost, is now manifest — both sides of each tensor equation transform the same way by construction.

The Hodge dual of $F$ is the antisymmetric tensor

${}^{*}F^{\mu\nu} = \tfrac{1}{2}\,\varepsilon^{\mu\nu\rho\sigma}\,F_{\rho\sigma},$

where $\varepsilon^{\mu\nu\rho\sigma}$ is the totally antisymmetric Levi-Civita tensor in 4D. The contraction implements an "exchange" between the top-row + first-column block and the lower-right $3\times 3$ block of $F$. Concretely, ${}^{*}F$ takes the same form as $F$ but with the substitutions $\mathbf{E}/c \rightarrow \mathbf{B}$ and $\mathbf{B} \rightarrow -\mathbf{E}/c$ (signs depend on signature; the structural point is the swap).

If magnetic monopoles existed, the second tensor equation would not be zero on the right-hand side. It would read

$\partial_\mu {}^{*}F^{\mu\nu} = \mu_0\,J_m^\nu,$

with $J_m^\nu = (c\rho_m, \mathbf{J}_m)$ the magnetic charge-current four-vector. Symmetric to the electric source equation in every respect. Paul Dirac, in 1931, showed that monopoles are not just consistent with quantum mechanics — they would force electric charge to be quantised, with quantisation condition $qg = 2\pi\hbar n$ for integer $n$. We have looked for them, hard, for a century. We have not seen one. So the second tensor equation reads zero on the right, the second pair of Maxwell equations are sourceless, and the asymmetry between $F$ and ${}^{*}F$ becomes the most consequential null result in physics. (Forward-pointer: Magnetic monopole in §12.3, where this question gets its own topic.)

The pedagogical voice for the entire section: Maxwell's four equations are two — and the second pair only looks separate because we don't have monopoles.

Two scalar invariants drop out of $F$ by index contraction. Lower both indices via $F_{\mu\nu} = \eta_{\mu\alpha}\eta_{\nu\beta}F^{\alpha\beta}$ (which flips signs on the temporal-spatial blocks but leaves the spatial-spatial block alone), then contract:

These are the same two invariants that fell out of the §11.2 boost calculation — $|\mathbf{E}|^2 - c^2|\mathbf{B}|^2$ and $\mathbf{E}\cdot\mathbf{B}$ — packaged as tensor traces. Sign and prefactor conventions follow Griffiths/Jackson; they depend on the metric signature and on the Lagrangian normalisation. Lorentz invariants of the EM field

The first invariant has a famous use. The kinetic part of the electromagnetic Lagrangian density is

$\mathcal{L}_{\text{EM}} = -\tfrac{1}{4}\,F_{\mu\nu}F^{\mu\nu} = \tfrac{\varepsilon_0}{2}\bigl(|\mathbf{E}|^2 - c^2|\mathbf{B}|^2\bigr),$

which §11.5 plugs into the Euler-Lagrange equations and recovers Maxwell's equations a second way — this time as the equations of motion for the four-potential $A^\mu$. The factor $-1/4$ is not a coincidence; it is what makes the kinetic term canonical. Six numbers in $F$, one scalar in $F_{\mu\nu}F^{\mu\nu}$, the entire dynamics of the classical electromagnetic field in eight characters of LaTeX. The §11.5 reveal is exactly that compression — and you have already seen the object that does the compressing.