Tensor Formulation of Fields in Media

Everything in this chapter so far happened in vacuum, where one tensor sufficed. But the E3 and E5 chapters taught a doubled bookkeeping for matter: alongside $\mathbf{E}$ and $\mathbf{B}$ live $\mathbf{D}$ and $\mathbf{H}$ , absorbing the bound charges and currents of polarization $\mathbf{P}$ and magnetization $\mathbf{M}$ . What happens to that doubled structure when the medium moves? The question is not academic — it decided a century of physics. Fresnel answered a version of it in 1818, predicting that moving water drags light by exactly the fraction $1-1/n^2$ ; Fizeau measured that number in 1851; and its stubborn refusal to fit any honest aether model made it, in Einstein's own telling, one of the two experimental pillars of relativity. This article gives the modern answer: the matter fields assemble into tensors exactly as the vacuum fields did — a polarization-magnetization tensor $M^{\mu\nu}$ packaging $(\mathbf{P},\mathbf{M})$ , an excitation tensor $G^{\mu\nu}$ packaging $(\mathbf{D},\mathbf{H})$ — and macroscopic electrodynamics becomes a two-tensor system whose form no boost can break. The physics lives in the constitutive relations, the material laws $\mathbf{D}=\varepsilon\varepsilon_0\mathbf{E}$ , $\mathbf{H}=\mathbf{B}/\mu\mu_0$ that E3/E5 established in the medium's rest frame. Minkowski's 1908 move — write them covariantly using the medium's four-velocity — produces, for a moving medium, relations in which $\mathbf{D}$ responds to $\mathbf{B}$ and $\mathbf{H}$ to $\mathbf{E}$ : moving matter magnetoelectrically couples. Fresnel's drag coefficient falls out at first order; the Wilson–Wilson experiment (1913) confirmed the crossed couplings with a rotating magnetized cylinder, discriminating relativity's $1-1/\varepsilon\mu$ from the aether's $1-1/\varepsilon$ at the bench. The one fight this article does not re-stage is Abraham–Minkowski: E7 owns the verdict; here we build the objects in which that fight is even stateable.

Key takeaways

Matter's fields tensorize like vacuum's: $M^{\mu\nu}$ packages $(\mathbf{P},\mathbf{M})$ with dictionary $M^{0i}=cP_i$ , $M^{ij}=-\epsilon^{ijk}M_k$ ; the excitation tensor $G^{\mu\nu}=F^{\mu\nu}/\mu_0-M^{\mu\nu}$ packages $(\mathbf{D},\mathbf{H})$ via $G^{0i}=-cD_i$ , $G^{ij}=-\epsilon^{ijk}H_k$ .
Bound sources are one identity: $J_{\text{bound}}^\nu=\partial_\mu M^{\mu\nu}$ reproduces E3/E5's $\rho_b=-\nabla\cdot\mathbf{P}$ and $\mathbf{J}_b=\partial_t\mathbf{P}+\nabla\times\mathbf{M}$ — and conserves itself by antisymmetry.
Macroscopic Maxwell is a two-tensor system: $\partial_\mu G^{\mu\nu}=J_{\text{free}}^\nu$ plus the unchanged Bianchi identity for $F$ — E6's Maxwell's Equations in Matter, covariantly dressed.
Constitutive relations go covariant with the medium's four-velocity $U^\mu$ : $G^{\mu\nu}U_\nu=c^2\varepsilon\varepsilon_0\,F^{\mu\nu}U_\nu$ (and its dual partner) — reducing, for a moving medium, to $\mathbf{D}+\mathbf{v}\times\mathbf{H}/c^2=\varepsilon\varepsilon_0(\mathbf{E}+\mathbf{v}\times\mathbf{B})$ and its twin: motion couples the electric and magnetic responses.
First-order payoff: light in flowing water moves at $u=c/n+v(1-1/n^2)$ — Fresnel's drag coefficient ( $0.437$ for water), Fizeau's 1851 fringe shift ( $\approx0.2$ fringe in his apparatus), pure velocity addition in modern eyes.
Second payoff: the Wilson–Wilson rotating magnetized cylinder develops $V=\tfrac12\omega B(b^2-a^2)(1-1/\varepsilon\mu)$ — the relativistic factor $0.944$ (for $\varepsilon\mu=18$ ) versus the aether's $0.833$ : experiment sided with Minkowski.
Key idea: bound matter adds one antisymmetric tensor to electrodynamics; constitutive laws stated in the medium's rest frame plus Lorentz covariance dictate all the electrodynamics of moving media — drag, magnetoelectricity, and the Röntgen currents of rotating dielectrics included.

The core insight

Macroscopic electrodynamics is the two-tensor system $\partial_\mu G^{\mu\nu}=J_{\text{free}}^\nu$ + Bianchi( $F$ ), with $G^{\mu\nu}=F^{\mu\nu}/\mu_0-M^{\mu\nu}$ absorbing polarization and magnetization; the rest-frame constitutive relations, made covariant with the medium four-velocity $U^\mu$ , then predict every moving-media effect — Fresnel drag and the Wilson–Wilson magnetoelectric coupling among them.

Where this lives today

Moving-media electrodynamics is quietly everywhere. Every magnetohydrodynamic system — fusion plasmas, liquid-metal cooling loops, the Earth's dynamo, astrophysical accretion disks — is system (2)+(3) with a conducting fluid's $U^\mu(x)$ varying point to point; the "ideal MHD" statement $\mathbf{E}+\mathbf{v}\times\mathbf{B}=0$ is Derivation B's first relation in the infinite-conductivity limit. Ring-laser gyroscopes and fiber-optic gyros navigate aircraft by the rotational cousin of Fizeau's phase shift (the Sagnac effect, with the drag coefficient's fingerprints in the medium-filled versions). The magnetoelectric crossing that Wilson–Wilson confirmed has become a materials frontier: engineered magnetoelectric and bianisotropic metamaterials implement, at rest, response tensors of the same mathematical form that motion induces — the covariant classification built here is their design language, and "moving-medium analogues" (light dragged by spinning ruby, slow-light media mimicking horizons) are a live experimental program testing how far the analogy runs. Even the precision frontier reports in: modern Fizeau-type experiments with slow light and structured media still use Fresnel's coefficient, dispersion correction and all, as the benchmark that any claimed exotic propagation must first reproduce.

Origins · 1 minAugustin-Jean Fresnel and how this was discovered

Augustin-Jean Fresnel

1788–1827 · Paris

Fresnel enters this chapter for the second time wearing a different hat. E8 met him as the architect of the wave theory's reflection amplitudes; here he is the author of the single most prophetic wrong-framework calculation in physics. Arago had asked (1818) why prism experiments showed no trace of Earth's motion through the aether; Fresnel replied with partial drag: if a medium of index $n$ carries only the fraction $1-1/n^2$ of the aether within it, all first-order effects of Earth's motion cancel. The coefficient was engineered to save a doomed ontology — condensed aether riding in glass — yet it survived the aether's death untouched, because (as Laue showed in 1907, three lines after velocity addition existed) it is pure relativistic kinematics: $c/n$ composed with $v$ . Fizeau's 1851 measurement of Fresnel's number, and its refusal to fit either full drag or none, became one of Einstein's two named experimental anchors for relativity. A civil engineer who did optics in the evenings, chronically ill and dead at 39, Fresnel had the strange fortune of being right twice: once in the framework that won (wave optics), and once with a number whose true meaning waited ninety years for the framework that could explain it.

History

1818

— Fresnel's letter to Arago: partial aether drag, coefficient

1-1/n^2

, invented to explain null first-order experiments

1851

— Fizeau's flowing-water interferometer measures the coefficient: neither full drag nor none

1888

— Röntgen detects the magnetization of a moving (rotating) dielectric — the P–M crossing observed

1904

— Lorentz derives the dispersion correction to the drag from electron theory

1907

— Laue: the drag coefficient is relativistic velocity addition, in three lines

1908

— Minkowski's Grundgleichungen: the covariant electrodynamics of moving media — this article's system

1913

— Wilson and Wilson's rotating magnetized cylinder confirms the

1-1/\varepsilon\mu

crossing

1914

— Zeeman begins the refined Fizeau runs that verify even the dispersion term

The physical picture

Recall why matter doubled the fields in the first place. A dielectric's molecules shift and align; a magnet's currents circulate; the total microscopic field obeys vacuum Maxwell with every bound charge counted, but nobody wants to track $10^{23}$ dipoles. So E3 and E5 invented the ledger split: let $\mathbf{P}$ and $\mathbf{M}$ summarize the bound response, define $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ and $\mathbf{H}=\mathbf{B}/\mu_0-\mathbf{M}$ to absorb the bound sources, and write field equations sourced by free charge alone. The covariant question is bookkeeping inheritance: the chapter has spent ten articles establishing that $(\mathbf{E},\mathbf{B})$ are one tensor's slots and $(c\rho,\mathbf{J})$ one four-vector's — so the split "total = free + bound" must tensorize too, or it would not survive a boost. It does, and the object doing the work is the pair $(\mathbf{P},\mathbf{M})$ itself. Here is the physical heart of the article, and it is the Transformation of Electric and Magnetic Fields lesson replayed in matter: polarization and magnetization are not separate material properties; they are frame-dependent faces of one bound-charge structure. March past a bar magnet ( $\mathbf{M}$ only, in its rest frame) and you find an electrically polarized object: its circulating bound currents, viewed in motion, develop a charge separation, $\mathbf{P}'\approx-\mathbf{v}\times\mathbf{M}/c^2$ . March past an electret and its bound charges become currents: $\mathbf{M}'\approx\mathbf{v}\times\mathbf{P}$ . This is the magnet-and-conductor asymmetry of Postulates of Special Relativity — the very opening of the relativity arc — now located inside matter: what one observer calls a magnetization another calls a polarization, and only a rank-2 antisymmetric tensor can hold both books at once. The dictionary pattern is forced by the same six-slot logic that built $F^{\mu\nu}$ .

Figure

The two-tensor system — vacuum fields, matter response, and excitation, with the sources split between them

One structural novelty separates this article from every vacuum predecessor: the equations of motion no longer suffice. In vacuum, $\partial_\mu F^{\mu\nu}=\mu_0J^\nu$ plus Bianchi closes — six unknowns, done. In matter, $F$ and $G$ together carry twelve field slots and the field equations connect them to sources without fixing their relationship: the system is open until you add the constitutive relations, the material's own physics. And those relations, unlike the field equations, know a preferred frame — the frame where the medium sits still. That is not a violation of relativity; it is relativity applied honestly: matter, like the CMB of the last article, is stuff, and stuff has a rest frame. The covariant craft is to carry that frame as data — the four-velocity $U^\mu$ — rather than as a coordinate choice, which is exactly Minkowski's 1908 move.

The objects and the system

The dictionaries, fixed by the vacuum limit and the rest-frame definitions (and verified component-by-component for this edition):

M^{0i}=cP_i,\quad M^{ij}=-\epsilon^{ijk}M_k; \qquad G^{0i}=-cD_i,\quad G^{ij}=-\epsilon^{ijk}H_k; \qquad G^{\mu\nu}=\frac{F^{\mu\nu}}{\mu_0}-M^{\mu\nu}. \qquad (1)

where: the $G$ pattern is the $F$ pattern with $(\mathbf{E}/c,\mathbf{B})\to(c\mathbf{D},\mathbf{H})$ — in vacuum ( $M=0$ ), $G=F/\mu_0$ reproduces $\mathbf{D}=\varepsilon_0\mathbf{E}$ , $\mathbf{H}=\mathbf{B}/\mu_0$ identically, the anchor that fixes every sign and factor of $c$ . The macroscopic system is then

\partial_\mu G^{\mu\nu}=J^\nu_{\text{free}}, \qquad \partial_\lambda F_{\mu\nu}+\partial_\mu F_{\nu\lambda}+\partial_\nu F_{\lambda\mu}=0, \qquad (2)

where: the homogeneous line is untouched — Faraday and no-monopoles involve only $\mathbf{E}$ and $\mathbf{B}$ and know nothing of matter ( $F=\partial A$ survives, potentials and gauge theory intact) — while the sourced line now carries only free current. This is E6's Maxwell's Equations in Matter, covariantly packaged; the slot audit runs exactly as in Covariant Form of Maxwell's Equations (the $\nu=0$ slot of (2) gives $\nabla\cdot\mathbf{D}=\rho_f$ , the space slots give $\nabla\times\mathbf{H}-\partial_t\mathbf{D}=\mathbf{J}_f$ — one component checked in Derivation A, the full pattern owned there). Free charge is conserved by the same antisymmetry argument as ever: $\partial_\nu\partial_\mu G^{\mu\nu}\equiv0$ .

Derivation AThe response tensor — bound sources as one identity, and the ledger split certified

We build $M^{\mu\nu}$ from the rest-frame ingredients, verify that the single covariant identity $J^\nu_{\text{bound}}=\partial_\mu M^{\mu\nu}$ reproduces both bound-source formulas of E3/E5, and confirm that the split "total = free + bound" is exactly the difference between the vacuum system and (2). The strategy is definition-chasing with the dictionaries — the same drill as the vacuum audits — and the payoff is consolidation: four separate 3+1 results (two bound-source formulas, their conservation, and the D/H absorption trick) become corollaries of one antisymmetric tensor's existence. Watch conservation come free: it cost E3/E5 a real argument; here antisymmetry pays it.

Step 1. Bound charge. Compute the time slot of $\partial_\mu M^{\mu\nu}$ with dictionary (1): only spatial $\mu$ contribute (antisymmetry kills $M^{00}$ ), and $M^{i0}=-M^{0i}=-cP_i$ :

\partial_\mu M^{\mu0}=\partial_i(-cP_i)=-c\,\nabla\cdot\mathbf{P} \;\Longrightarrow\; J^0_b=c\rho_b\ \text{with}\ \rho_b=-\nabla\cdot\mathbf{P},

exactly Bound Charges: Surface and Volume's density (its charge-counting derivation recalled, not redone — this is bookkeeping consolidation, not new physics).

Step 2. Bound current. The space slots: $\partial_\mu M^{\mu j}=\partial_0M^{0j}+\partial_iM^{ij}=\tfrac1c\partial_t(cP_j)+\partial_i(-\epsilon^{ijk}M_k)$ :

\mathbf{J}_b=\frac{\partial\mathbf{P}}{\partial t}+\nabla\times\mathbf{M},

the polarization current plus Bound Currents' curl — both E-chapters' source formulas from one four-divergence. Conservation rides free: $\partial_\nu J_b^\nu=\partial_\nu\partial_\mu M^{\mu\nu}\equiv0$ by antisymmetry — bound charge can slosh but never leak, a theorem E3 argued physically and the tensor makes algebraic.

Step 3. The split. Write the microscopic (vacuum, all-sources) system with $J_{\text{tot}}=J_{\text{free}}+J_{\text{bound}}$ and substitute Step 1–2's identity:

\partial_\mu\!\left(\frac{F^{\mu\nu}}{\mu_0}\right)=J^\nu_{\text{free}}+\partial_\mu M^{\mu\nu} \;\Longrightarrow\; \partial_\mu\!\left(\frac{F^{\mu\nu}}{\mu_0}-M^{\mu\nu}\right)=J^\nu_{\text{free}},

which is system (2) with $G$ defined by (1): the excitation tensor is not a new postulate but the unique repackaging that moves bound sources to the left-hand side. Audit one slot to anchor the dictionary: $\nu=0$ gives $\nabla\cdot(\varepsilon_0\mathbf{E}+\mathbf{P})=\rho_f$ , i.e. $\nabla\cdot\mathbf{D}=\rho_f$ — The Electric Displacement D recovered (remaining slots: the CMAXW audit pattern, by name).

Step 4. The transformation payoff. Because $M^{\mu\nu}$ is a tensor, $(\mathbf{P},\mathbf{M})$ mix under boosts with exactly the $F$ -pattern algebra: to first order, $\mathbf{P}'=\mathbf{P}-\mathbf{v}\times\mathbf{M}/c^2$ , $\mathbf{M}'=\mathbf{M}+\mathbf{v}\times\mathbf{P}$ . The moving magnet is polarized; the moving electret is magnetized; and the Röntgen experiment (1888) — a rotating dielectric in a field acting as a magnet — stops being a curiosity and becomes a component readout. The chapter's oldest lesson, "one observer's electricity is another's magnetism," holds inside matter, slot for slot. The steps above have been checked symbolically with dictionary (1): both bound-source formulas, the conservation identity, the vacuum limit $G=F/\mu_0$ , and the first-order mixing rules all come out correctly.

Go deeper · Derivation B · 3 minMinkowski's constitutive relations — and Fresnel's drag coefficient from first-order mixing

Derivation BMinkowski's constitutive relations — and Fresnel's drag coefficient from first-order mixing

We promote the rest-frame material laws to covariant statements using the medium four-velocity, extract the 3+1 relations for a moving medium — the crossed couplings — and cash the first-order consequence: light in a moving medium is dragged by the Fresnel coefficient $1-1/n^2$ , Fizeau's fringes computed. The strategy is the standard covariant-completion move (state the law in the preferred frame; write it as a tensor equation that reduces to it there; let covariance do the rest) — the same logic that promoted $\tau$ -derivatives in Proper Time and Four-Velocity. Watch what the four-velocity buys: one vector of data converts two scalars ( $\varepsilon$ , $\mu$ ) into a complete moving-media electrodynamics.

Step 1. Rest-frame laws, tensor dress. In the medium frame, $\mathbf{D}=\varepsilon\varepsilon_0\mathbf{E}$ and $\mathbf{H}=\mathbf{B}/\mu\mu_0$ . Note what these relate: the time columns of $G$ and $F$ (electric slots), and their space blocks (magnetic slots), respectively. Contracting each tensor with the medium's four-velocity $U^\mu$ ( $=(c,\mathbf{0})$ at rest) extracts exactly the electric slots; contracting the duals extracts the magnetic ones. So the pair of laws is equivalent to two covariant statements:

G^{\mu\nu}U_\nu=c^2\varepsilon\varepsilon_0\,F^{\mu\nu}U_\nu, \qquad \tilde G^{\mu\nu}U_\nu=\frac{1}{\mu\mu_0}\,\tilde F^{\mu\nu}U_\nu, \qquad (3)

where: tildes denote duals (the $\epsilon^{\mu\nu\alpha\beta}$ machinery of The Dual Field Tensor), and the factors are fixed by evaluating at rest. Equations (3) are Minkowski's constitutive relations: tensor equations, true in every frame, containing the medium's rest frame as data.

Step 2. Unpack for a medium moving at velocity $\mathbf{v}$ . Insert $U^\mu=\gamma(c,\mathbf{v})$ and read the slots (first order in $v/c$ shown; the exact forms carry $\gamma$ 's):

\mathbf{D}+\frac{\mathbf{v}\times\mathbf{H}}{c^{2}}=\varepsilon\varepsilon_0\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right), \qquad \mathbf{B}-\frac{\mathbf{v}\times\mathbf{E}}{c^{2}}=\mu\mu_0\left(\mathbf{H}-\mathbf{v}\times\mathbf{D}\right).

Why the crossings appear: each side is "the electric (or magnetic) slot as the moving medium sees it," and slots mix under boosts — the constitutive relations are frame-honest versions of "the medium responds to the fields in its own rest frame," with the mixing rules of Transformation of Electric and Magnetic Fields doing the translation. A medium in motion is intrinsically magnetoelectric: $\mathbf{D}$ responds to $\mathbf{B}$ , $\mathbf{H}$ to $\mathbf{E}$ , with coefficients proportional to $v(\varepsilon\mu-1)$ -type combinations that vanish in vacuum ( $\varepsilon\mu=1$ ) — vacuum cannot notice motion, matter must.

Step 3. The drag. Solve system (2)+(3) for a plane wave in a medium flowing at $v$ along the propagation direction: the phase velocity comes out, to first order,

u=\frac{c}{n}+v\left(1-\frac{1}{n^{2}}\right),

Fresnel's 1818 coefficient. The modern two-line route makes its meaning transparent: the wave moves at $c/n$ in the water's frame; add velocities relativistically (Lorentz Transformations' addition law): $u=(c/n+v)/(1+v/nc)\approx c/n+v(1-1/n^2)$ — the "drag" is no drag at all, but the failure of velocities to add linearly. The two routes agreeing (full constitutive solve; velocity addition) is this article's consistency audit, and it explains the aether-era bafflement: the coefficient is not a material property being partially entrained — it is kinematics wearing a material disguise.

Step 4. Fizeau's fringes. Water ( $n=1.333$ , so the coefficient is $1-1/n^2=0.437$ ) flows at $v=7.06$ m/s through paired tubes of length $L=1.487$ m; light traverses with and against the flow and interferes. The travel-time difference $\Delta t\approx4Lvn^2(1-1/n^2)/c^2$ gives a fringe shift $c\,\Delta t/\lambda\approx0.21$ at $\lambda=526$ nm — against Fizeau's observed $0.23\pm0.02$ : a nineteenth-century water pipe measuring, in modern eyes, the second-order structure of the velocity-addition law. (Zeeman's 1914–22 repetitions confirmed even the dispersion correction — the extra $-\tfrac{\omega}{n}\tfrac{dn}{d\omega}$ term from $n(\omega)$ in the moving frame — closing the file.) The slot extraction of Step 1 checks out at rest for both relations; the velocity-addition route and the full constitutive-solve route agree to $6\times10^{-8}$ m/s at Fizeau's parameters; the fringe shift evaluates to $0.207$ at $(L,v,n,\lambda)=(1.487\ \text{m},7.06\ \text{m/s},1.333,526\ \text{nm})$ .

Go deeper · Derivation C · 1 minThe Wilson–Wilson experiment — the crossed couplings put to a bench test

Derivation CThe Wilson–Wilson experiment — the crossed couplings put to a bench test

We work the 1913 Wilson–Wilson configuration: a hollow cylinder of magnetizable dielectric ( $\varepsilon$ , $\mu$ ) rotating at angular velocity $\omega$ in a uniform axial field $B$ , with inner and outer surfaces fitted with electrodes. Relativity's crossed constitutive relations predict a radial polarization and a measurable voltage with the factor $1-1/\varepsilon\mu$ ; the pre-relativistic (Galilean/aether) expectation carries $1-1/\varepsilon$ instead — a clean discrimination precisely because the material is both dielectric and magnetic. The strategy: apply Derivation B's first relation locally (each ring of material moves at $\mathbf{v}=\omega r\,\hat{\boldsymbol\phi}$ ), impose open-circuit conditions, integrate the field for the voltage. Watch the $\mu$ : it enters through the $\mathbf{v}\times\mathbf{H}$ crossing on the left of the constitutive relation — the term Galilean theory does not have.

Step 1. Local fields. With $\mathbf{B}=B\hat{\mathbf{z}}$ axial and $\mathbf{v}=\omega r\,\hat{\boldsymbol\phi}$ , the motional term is radial: $\mathbf{v}\times\mathbf{B}=\omega rB\,\hat{\mathbf{r}}$ . Inside the (linear) medium, $\mathbf{H}=\mathbf{B}/\mu\mu_0$ to leading order, so $\mathbf{v}\times\mathbf{H}=\omega rB/\mu\mu_0\,\hat{\mathbf{r}}$ .

Step 2. Apply the constitutive relation radially:

D_r+\frac{\omega rB}{\mu\mu_0c^{2}}=\varepsilon\varepsilon_0\left(E_r+\omega rB\right) \;\Longrightarrow\; D_r=\varepsilon\varepsilon_0E_r+\varepsilon_0\,\omega rB\left(\varepsilon-\frac{1}{\mu}\right),

using $\mu_0c^2=1/\varepsilon_0$ . The motional source term carries $(\varepsilon-1/\mu)$ — both material constants, the experiment's whole point.

Step 3. Open circuit: no free charge accumulates, $D_r=0$ , so the medium develops a real radial field $E_r=-\omega rB(1-1/\varepsilon\mu)$ , and the electrode voltage is

V=-\int_a^b E_r\,dr=\frac{\omega B\,(b^{2}-a^{2})}{2}\left(1-\frac{1}{\varepsilon\mu}\right).

Step 4. Numbers and the verdict. For a laboratory-scale device — $a=2.0$ cm, $b=3.0$ cm, $B=0.10$ T, $100$ rotations per second ( $\omega=628$ s⁻¹) — a wax-and-steel composite with $\varepsilon=6$ , $\mu=3$ gives $V=\tfrac12(628)(0.10)(5.0\times10^{-4})(0.944)=15$ mV. The discriminator: relativity's factor $1-1/\varepsilon\mu=0.944$ versus the Galilean $1-1/\varepsilon=0.833$ — a 13% difference, decisive at millivolt precision. The Wilsons' data followed the $\varepsilon\mu$ law; moving-media electrodynamics has been Minkowski's ever since. (The honest asterisk, for the record: a rotating frame is not an inertial frame, and the "locally comoving inertial observer" treatment used here — adequate at $\omega r\ll c$ — was itself re-examined and upheld in the modern literature.) The constitutive algebra of Steps 2–3 checks out symbolically; the numbers recompute to $0.944$ vs $0.833$ and $V=14.8$ mV at the stated parameters.

Limiting cases

case	result	meaning
vacuum ( $\varepsilon=\mu=1$ )	$G=F/\mu_0$ , crossings vanish identically	motion undetectable — vacuum has no rest frame
medium at rest	(3) → $\mathbf{D}=\varepsilon\varepsilon_0\mathbf{E}$ , $\mathbf{H}=\mathbf{B}/\mu\mu_0$	E3/E5 recovered as the $U=(c,\mathbf{0})$ slice
$v\ll c$ , $n\to1$	drag coefficient $\to0$	dilute media barely drag — Fresnel's own check
$n(\omega)$ dispersive	drag gains $-\tfrac{\omega}{n}\tfrac{dn}{d\omega}$	Lorentz's correction; Zeeman measured it
perfect conductor limit	constitutive relations replaced by boundary conditions	moving-conductor problems (unipolar induction) route through E6 instead
$\varepsilon\mu\to\infty$	Wilson–Wilson factor $\to1$	the medium's interior field fully motional

Common trapDo not transform D and H with vacuum intuition — and do not let the two-tensor split reopen the momentum war

Two distinct hazards, both born of forgetting which object owns which physics. First: $(\mathbf{D},\mathbf{H})$ are slots of $G$ , and they mix under boosts with $G$ 's transformation law — never by applying $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ frame-by-frame with unmixed $\mathbf{P}$ , and never by copying the $(\mathbf{E},\mathbf{B})$ rules with symbols renamed but constitutive relations assumed intact. The constitutive relations are rest-frame statements; in any other frame they cross (Derivation B), and the crossing is the physics — drop it and Fizeau, Röntgen, and Wilson–Wilson all come out wrong. The reliable protocol: transform the tensors ( $F$ , $G$ , $M$ ), or equivalently carry $U^\mu$ in covariant relations, and only then read slots. Second: the existence of both $F$ and $G$ invites building two stress-energy tensors ( $\theta\sim FG$ à la Minkowski, or the symmetric Abraham repair), and the invitation is a century-old trap — asymmetric tensor, competing momenta, the whole Abraham–Minkowski affair. This chapter's rule: the controversy's verdict (kinetic vs canonical momentum, Barnett's settlement) is owned by E7's Electromagnetic Stress, Momentum, and Energy in Media and is not re-litigated here; this article's tensors are the language of that debate, not a new verdict on it. Build the objects, state the split you use, cite the settlement.

Figure

Two bench tests of moving-media electrodynamics — Fizeau's flowing water and the Wilsons' spinning cylinder

Worked examples

Example 1 — Fizeau's water pipe as a relativity meter. Water ( $n=1.333$ ) flows at $v=7.06$ m/s through tubes of length $L=1.487$ m; monochromatic light ( $\lambda=526$ nm) is split, sent with and against the flow, and recombined.

Step 1. The drag coefficient: $1-1/n^2=1-0.563=0.437$ . Downstream light moves at $u_+=c/n+0.437v$ , upstream at $u_-=c/n-0.437v$ — a velocity difference of $2\times0.437\times7.06=6.2$ m/s riding on $2.25\times10^8$ m/s.

Step 2. Fringe shift: $\Delta=c\Delta t/\lambda$ with $\Delta t\approx4Lvn^2(1-1/n^2)/c^2$ : $\Delta=0.21$ fringes. Fizeau read $0.23\pm0.02$ : an 1851 tabletop resolving a part-per- $10^8$ velocity asymmetry, and — with hindsight — falsifying both full aether drag ( $\Delta$ would be $0.48$ , coefficient 1) and zero drag ( $\Delta=0$ ).

Step 3. The modern audit: velocity addition gives $u_+=(c/n+v)/(1+v/nc)=224{,}900{,}571.73$ m/s; the drag formula gives the same to $6\times10^{-8}$ m/s. Fresnel's "aether dragged by the density ratio" was velocity addition, seventy years early.

Example 2 — a rotating magnetized dielectric as a battery. The Wilson–Wilson cylinder of Derivation C: $a=2.0$ cm, $b=3.0$ cm, $\varepsilon=6$ , $\mu=3$ , axial $B=0.10$ T, spun at 100 Hz.

Step 1. Motional EMF scale: the naive "perfect conductor" value would be $\tfrac12\omega B(b^2-a^2)=15.7$ mV — the upper envelope.

Step 2. Relativistic prediction: multiply by $1-1/\varepsilon\mu=1-1/18=0.944$ : $V=14.8$ mV. Galilean prediction: multiply by $1-1/\varepsilon=0.833$ : $V=13.1$ mV. The 1.7 mV gap is the entire content of the crossed $\mathbf{v}\times\mathbf{H}$ term — a testable 13%.

Step 3. Why both material constants had to be large: with $\mu=1$ the two predictions coincide at $1-1/\varepsilon$ (the crossing term hides inside the vacuum part); the Wilsons' wax-embedded steel spheres were engineered precisely to make $\mu$ matter. Experiment took relativity's column — and every magnetoelectric device analysis since (rotating machines, MHD generators, relativistic beam–plasma media) uses Minkowski's relations as routinely as Ohm's law.

Going further

Reading. Jackson, Classical Electrodynamics, 3rd ed., §11.9–11.10 — the covariant macroscopic formulation and moving-media relations. Landau & Lifshitz, Electrodynamics of Continuous Media, §76 — moving media, with the Wilson–Wilson configuration among the problems. Pauli, Theory of Relativity, §33 — the classic compact treatment. Zangwill, Modern Electrodynamics, Ch. 17 for the rest-frame anchors. The dictionaries are fixed to the $(+,-,-,-)$ conventions and anchored to the vacuum limit: $G^{0i}=-cD_i$ matches $F^{0i}=-E_i/c$ under $\mathbf{E}/c\to c\mathbf{D}$ , component-audited symbolically. Key numerical benchmarks: Fizeau drag coefficient $0.437$ ; fringe shift $\Delta=0.21$ at his parameters; velocity-addition cross-check agrees to $6\times10^{-8}$ m/s; Wilson–Wilson voltage $V=\tfrac12\omega B(b^2-a^2)(1-1/\varepsilon\mu)$ , giving $14.8$ mV relativistic vs Galilean $13.1$ mV at the stated numbers. The 3+1 matter machinery is recalled from E3/E5/E6 — Bound Charges: Surface and Volume, Bound Currents, The Electric Displacement D, The Auxiliary Field H, Maxwell's Equations in Matter — never re-derived here; the field-mixing rules come from Transformation of Electric and Magnetic Fields; velocity addition from Lorentz Transformations; the momentum controversy's verdict from E7's Electromagnetic Stress, Momentum, and Energy in Media (cited, not reopened; this article supplies the tensor language); dispersive response from E8's Absorption and Dispersion; Cherenkov radiation from E11's treatment. This article owns the tensor dictionaries, the bound-current identity, the covariant constitutive relations, and the two experimental audits.

Builds on: The Electromagnetic Field Tensor · The Dual Field Tensor · Maxwell's Equations in Matter · Transformation of Electric and Magnetic Fields · The Four-Current Density

Leads to: Multipole Expansion of Radiation Fields · Magnetohydrodynamics · Cherenkov Radiation

Why doesn't the homogeneous Maxwell pair change inside matter?

Because Faraday's law and the absence of monopoles are statements about $\mathbf{E}$ and $\mathbf{B}$ — the fields that exert forces — and matter, however polarized or magnetized, only ever adds charges and currents, which enter the sourced equations. Structurally: bound matter modifies $J^\nu$ , and the homogeneous pair contains no $J^\nu$ to modify. The deep consequence is that the entire potential-and-gauge edifice survives matter untouched: $F=\partial A$ still solves the homogeneous pair identically, the gauge orbit and Lorenz-slice machinery of this chapter applies verbatim, and only the sourced dynamics gets re-packaged into $G$ . This is also why electromagnetic induction works identically in copper, wax, or vacuum — Faraday's law never consults the medium — while Gauss's and Ampère's laws needed their D-and-H editions.

Is the excitation tensor G physical, or just bookkeeping convenience?

Honest answer: it is exactly as physical as $\mathbf{D}$ and $\mathbf{H}$ always were, which is a subtle status the track has met twice (E3's The Electric Displacement D and E5's honesty doctrine). $G$ is defined by a ledger split — total sources into free plus bound — and the split location is a modeling choice: what counts as "free" in a semiconductor analysis may be "bound" in a plasma one. What is invariant physics: the total-field tensor $F$ , the total current, and the theorem (Derivation A, Step 3) that given a split, the repackaging into $G$ is unique and covariant. The practical rule inherits E5's: measure forces with $F$ ; organize sources with $G$ ; and never assign $G$ 's slots independent reality beyond the split that defined them. The covariant dress changes none of this — it just guarantees the bookkeeping travels correctly between frames.

How could Fresnel get the right coefficient from a wrong theory ninety years early?

Because he was solving the right constraint, not deriving from the right ontology. Arago's data said: no first-order optical effect of Earth's motion in refracting media. Fresnel reverse-engineered the unique drag fraction that cancels all such first-order effects — a functional requirement. Relativity's kinematics satisfies the same requirement automatically (first-order effects of uniform motion cannot exist, by the principle of relativity), so any coefficient engineered to enforce the null result had to coincide with the relativistic one at first order. The lesson generalizes and is worth keeping: experimental null results over-determine formulas even under wrong frameworks — Lorentz's contraction hypothesis, Fresnel's drag, and Poincaré's stresses were all "right answers awaiting the right reason." The dispersion correction is the cautionary coda: where Fresnel's mechanical picture had nothing more to say, Lorentz's electron theory and relativity both predicted the $\omega\,dn/d\omega$ refinement — and Zeeman found it.

Do D and H respond to boosts with the same mixing rules as E and B?

Same pattern, same machinery, different tensor — and the distinction bites. The pairs $(c\mathbf{D},\mathbf{H})$ mix among themselves with the $\Lambda G\Lambda^{\mathsf T}$ algebra exactly as $(\mathbf{E}/c,\mathbf{B})$ do with $\Lambda F\Lambda^{\mathsf T}$ : to first order $\mathbf{D}'=\mathbf{D}+\mathbf{v}\times\mathbf{H}/c^2$ , $\mathbf{H}'=\mathbf{H}-\mathbf{v}\times\mathbf{D}$ . What does not transform simply is the relationship between the two tensors: $\mathbf{D}'=\varepsilon\varepsilon_0\mathbf{E}'$ holds only in the medium's rest frame. So each tensor mixes cleanly; the constitutive bridge between them crosses (Derivation B). Errors come from assuming the bridge is frame-proof — e.g., computing a moving capacitor's $\mathbf{D}$ from the boosted $\mathbf{E}$ with the rest-frame $\varepsilon$ . The safe order of operations: transform tensors first, apply material laws only in the frame where they were measured, or use the covariant relations (3) which enforce this automatically.

Where does the Abraham–Minkowski controversy sit relative to this article's objects?

Downstream of them, and the geography is worth one clear paragraph. This article builds the field tensors of macroscopic electrodynamics; the controversy concerns the stress-energy tensor — how to split total energy-momentum between "field in medium" and "medium." Minkowski's candidate, built naturally from products of $F$ and $G$ , assigns light in glass momentum density $\mathbf{D}\times\mathbf{B}$ (canonical, $n\hbar\omega/c$ per photon) but is asymmetric — colliding with the angular-momentum doctrine of this chapter's conservation articles. Abraham's symmetric repair assigns $\mathbf{E}\times\mathbf{H}/c^2$ (kinetic, $\hbar\omega/nc$ ) at the price of an explicit "Abraham force" on the medium. E7's Electromagnetic Stress, Momentum, and Energy in Media owns the resolution — both are correct answers to different questions, kinetic versus canonical, per the Balazs argument and Barnett's 2010 settlement — and nothing in the covariant dress changes that verdict. What the tensors here add is precision of statement: the two candidates, their symmetries, and their transformation properties can be written in one line each, which is why the modern literature conducts the discussion in exactly this language.

Do the covariant constitutive relations handle dispersive or lossy media?

Not in the simple algebraic form (3), and the honesty matters. Writing $G^{\mu\nu}U_\nu=c^2\varepsilon\varepsilon_0F^{\mu\nu}U_\nu$ with constant $\varepsilon$ assumes the medium responds instantly and losslessly — the same idealization E8's Absorption and Dispersion spent an article dismantling. The honest generalization makes the relation nonlocal along the medium's worldlines: the polarization at an event depends on the field's history in the medium's rest frame, $\varepsilon\to\varepsilon(\omega)$ with $\omega$ measured in that frame — which is precisely where the previous article's Doppler machinery re-enters, since a lab-frame frequency maps to a medium-frame frequency by the Doppler factor. Consequences: the drag coefficient acquires Lorentz's dispersion term (measured by Zeeman); moving absorbing media exhibit genuinely new effects (anomalous drag, amplification in the rotating-body "Zel'dovich" regime); and Kramers–Kronig causality must be imposed in the medium frame. The clean covariant algebra of this article is the non-dispersive skeleton; real materials wear it with frequency-dependent flesh.

Is there a moving-medium effect for light crossing the vacuum near a moving mass — a gravitational Fizeau?

Yes, and the analogy is precise enough to be a working tool. In general relativity, a moving or rotating mass "drags" the spacetime through which light propagates — frame dragging — and to first order the effective equations for light in a weak gravitational field are formally those of a moving optical medium with an index built from the metric (the "Gordon metric" construction inverts this: any moving dielectric mimics a curved spacetime for the light inside it). Fizeau's coefficient reappears in analogue-gravity laboratories: slow-light media with steep dispersion make the drag term enormous, and spinning transparent bodies drag polarization and phase in measured amounts — experiments explicitly framed as tabletop analogues of rotating-black-hole physics. The conceptual chain runs straight through this article: medium four-velocity as data → crossed constitutive response → light dragged by matter's motion; general relativity replaces "constitutive data" by geometry, and the GR track picks up the story from there.