The Electromagnetic Stress-Energy Tensor

The conservation-laws chapter discovered, one at a time, that the electromagnetic field is a mechanical object: it stores energy density $u$ , transports it as the Poynting flux $\mathbf{S}$ , carries momentum $\mathbf{g}=\mathbf{S}/c^2$ , and pushes on surfaces through the Maxwell stress tensor. Four discoveries, four articles, four separate symbols — and E7's closing article confessed they were sixteen entries of one spacetime object it could name but not yet build. This article builds it. The electromagnetic stress-energy tensor

\theta^{\mu\nu}=\frac{1}{\mu_0}\left[F^{\mu\alpha}F_\alpha{}^{\nu}+\tfrac14\eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right]

is assembled directly from the field tensor, and the E7 inventory falls into its slots on audit: $u$ in the corner, $\mathbf{S}/c$ across the top row, $c\mathbf{g}$ down the first column, minus the Maxwell stress in the spatial block. Two of those placements calibrate the construction; the other two come out as predictions — the kind of overdetermined success that tells you the object is real. Beyond bookkeeping, the tensor has two structural properties with long shadows. It is symmetric, which is Planck's $\mathbf{g}=\mathbf{S}/c^2$ — the inertia of energy flow — hard-coded, and which next article converts into conservation of angular momentum. And it is traceless, which encodes the masslessness of light and delivers, in three lines, the radiation equation of state $P=u/3$ that runs stellar interiors and the early universe. There is also an honest confession to file: the "obvious" Noether recipe applied to the field Lagrangian produces a different, uglier tensor, and the repair — the Belinfante healing — is owned by the mechanics track. This article is the physics of the healed object; the surgery report is cross-referenced, not repeated.

Key takeaways

The EM field's stress-energy tensor is $\theta^{\mu\nu}=\tfrac{1}{\mu_0}[F^{\mu\alpha}F_\alpha{}^\nu+\tfrac14\eta^{\mu\nu}F^2]$ : quadratic in $F$ , symmetric, gauge-invariant, built with no reference to any frame.
Component audit against E7's owners: $\theta^{00}=u=\tfrac{\varepsilon_0}{2}E^2+\tfrac{1}{2\mu_0}B^2$ , $\theta^{0i}=S_i/c$ , $\theta^{i0}=cg_i$ , $\theta^{ij}=-\sigma_{ij}$ (Maxwell stress) — energy density, Poynting flux, field momentum, and field stress are one object's sixteen shadows.
Symmetry $\theta^{0i}=\theta^{i0}$ is $\mathbf{g}=\mathbf{S}/c^2$ : momentum density equals energy flux over $c^2$ — Planck's inertia-of-energy principle, structural rather than empirical.
Tracelessness $\theta^\mu{}_\mu=0$ is exact and frame-proof; for isotropic radiation it forces $P=u/3$ , and for a single plane wave the whole tensor collapses to the null dyad $\theta^{\mu\nu}=u\,n^\mu n^\nu$ with $n=(1,\hat{\mathbf{k}})$ — light is a dust of energy streaming at $c$ .
The canonical Noether tensor of the field Lagrangian is asymmetric and gauge-dependent; the Belinfante procedure (owned by the mechanics track) heals it into exactly $\theta^{\mu\nu}$ — E7's deferred honesty, delivered by citation.
No component of $\theta$ is a Lorentz invariant — $u$ , $\mathbf{S}$ , stresses all mix under boosts as a rank-2 tensor; only $F^2$ and $F\cdot G$ are frame-proof scalars.
Key idea: one gauge-invariant tensor built from $F$ carries every mechanical property of the electromagnetic field — and its symmetry and tracelessness are the inertia of energy and the masslessness of light, written as algebra.

The core insight

The electromagnetic field's energy density, energy flux, momentum density, and stress are the sixteen components of one symmetric, traceless, gauge-invariant tensor, $\theta^{\mu\nu}=\tfrac{1}{\mu_0}[F^{\mu\alpha}F_\alpha{}^\nu+\tfrac14\eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}]$ — the object that closes E7's ledger and sources gravity.

Where this lives today

The stress-energy tensor is the busiest interface in physics, and the electromagnetic one built here is its prototype. In general relativity, $\theta^{\mu\nu}$ sits on the right-hand side of Einstein's equations: electromagnetic energy gravitates, with consequences from the Reissner–Nordström charged black hole to the (measured) gravitational binding contribution of magnetic fields in neutron stars — magnetars store $\sim10^{47}$ erg in fields of $10^{15}$ G, and that energy's weight matters to their structure. Cosmology runs on Derivation B's Step 3: the early universe was radiation-dominated precisely while the $P=u/3$ component controlled expansion. Laboratory physics reads the spatial block daily — optical tweezers steer cells with the momentum flux of focused beams, and laser-fusion capsules are crushed by radiation pressure the $\theta^{zz}$ slot prices. And in quantum field theory, $\theta^{\mu\nu}$ is the operator whose vacuum expectation value defines the Casimir energy, whose trace fails to vanish quantum mechanically (the "trace anomaly" — the classical zero of Derivation B broken by renormalization, with measurable consequences in QCD). The next article sets this object flowing: its divergence, against the two-line Maxwell system, is the covariant conservation law the whole E7 chapter was secretly about.

Origins · 1 minMax von Laue and how this was discovered

Max von Laue

1879–1960 · Berlin

Laue is filed in memory under X-ray diffraction — the 1912 experiment and 1914 Nobel that made crystal lattices visible — but relativists know him as the man who taught physics to use the stress-energy tensor. His 1911 textbook on relativity, the first ever written, and the accompanying papers built what he called the "world tensor" into the standard apparatus: he classified its slots, insisted on its symmetry, and proved the theorem this chapter leans on twice — that only for a closed system does the integrated four-momentum transform as a four-vector. His "Laue theorem" dissolved half the paradoxes of early relativistic electron theory (the 4/3 problem's frame puzzles among them) by showing that the electromagnetic field of a charge, taken alone, is an open system whose Poincaré stresses must be counted. A student of Planck and a lifelong friend of Einstein, he was also one of German physics' rare profiles in courage: he publicly defended relativity against the Deutsche Physik movement, carried on private resistance through the Nazi years, and rebuilt German science after the war. The tensor bookkeeping in this article — slots, closure conditions, symmetric object or nothing — is conducted by his rules.

History

1873

— Maxwell computes the tension and pressure of field lines: the stress block, forty years early

1884

— Poynting localizes energy flow; the top row of the cabinet

1900

— Poincaré identifies field momentum density

\mathbf{S}/c^2

in Lorentz's theory

1908

— Planck states

\mathbf{g}=\mathbf{S}/c^2

as universal: the inertia of energy flow, the tensor's symmetry

1911

— Laue's relativity text assembles the "world tensor" and proves the closed-system theorem

1915

— Einstein's field equations make the stress-energy tensor gravity's source

1918

— Noether's theorem generates the canonical tensor; Belinfante and Rosenfeld heal it (1939–40)

The physical picture

Ask what it takes to describe the mechanics of a substance rather than a particle. A particle's mechanical state is four numbers, $p^\mu$ . A field is everywhere at once, so at each event you must answer four questions — how much energy is here, and how much of each momentum component — and, for each answer, four more: in which spacetime direction does it flow? Four quantities times four flow directions is sixteen slots: a rank-2 tensor, indexed as $\theta^{\mu\nu}$ = (flux of the $\nu$ -th momentum component through the surface of constant $x^\mu$ ). The time rows and columns hold the densities and their transport; the space-space block holds momentum flowing through spatial surfaces — which is what a force per unit area is. Pressure and tension were always momentum bookkeeping; the tensor just says so with indices. E7 met every occupant of this cabinet separately, from the 3+1 side: Energy Density of the Electromagnetic Field built $u$ , The Poynting Vector built the energy flux, Momentum Density of the Electromagnetic Field built $\mathbf{g}=\mathbf{S}/c^2$ , and The Maxwell Stress Tensor built the space-space block, eigen-anatomy and all — tension $u$ along field lines, pressure $u$ across them. What none of them could do, lacking the field tensor, was exhibit the sixteen entries as one covariant object — the step that turns four theorems into one, and that lets a boost mix energy into momentum into stress the way this chapter has mixed everything else. E7's finale, Conservation Laws and Spacetime Symmetries, drew the cabinet and deferred its covariant construction to E10 by name. This is that delivery.

Figure

The filing cabinet built at last — sixteen slots, every occupant an E7 article, one object

One habit to install before the algebra: no slot of this cabinet is invariant. The energy density $u$ feels fundamental, but it is the 00-component of a tensor — boost past a capacitor and $u$ , $\mathbf{S}$ , $\mathbf{g}$ , and the stresses reshuffle among themselves by the same $\Lambda\ldots\Lambda$ machinery that mixed $\mathbf{E}$ and $\mathbf{B}$ . The only frame-proof scalars in the neighborhood are the two field invariants of Lorentz Invariants of the Field — and, the audit will show, the tensor's trace, which is invariantly zero. Whoever asks "how much energy does this field really have?" must first answer "according to whom?"

Building the tensor

The requirements select the object almost uniquely. We want a rank-2 tensor that is (i) quadratic in the fields — energy already is, per E7; (ii) built from $F^{\mu\nu}$ alone — gauge invariance, since energy is measurable; (iii) symmetric — Planck's principle, and a necessity if angular momentum is to be conserved (next article closes that loop); and (iv) containing no derivatives of $F$ — energy density is a local, algebraic function of the field strengths. The menu of such objects has exactly two entries, $F^{\mu\alpha}F_\alpha{}^\nu$ and $\eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}$ , so the answer must be a combination

\theta^{\mu\nu}=\frac{1}{\mu_0}\left[a\,F^{\mu\alpha}F_\alpha{}^{\nu}+b\,\eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right],

with two pure numbers to fix and sixteen slots to explain — an enormously overdetermined bargain that Derivation A cashes.

Derivation ACalibrate twice, predict twice — the component audit that fills the cabinet

We fix the two coefficients by matching the two E7 results that define the field's mechanics — energy density and Poynting flux — and then compute the remaining components, which must land on E7's momentum density and Maxwell stress with no freedom left. The strategy is component bookkeeping with the track's dictionary ( $F^{0i}=-E^i/c$ , $F^{ij}=-\epsilon^{ijk}B_k$ ; The Electromagnetic Field Tensor), organized so every step is one slot of the ansatz. Watch the overdetermination: after Step 2 there are no adjustable numbers, so Steps 3 and 4 are genuine tests — the construction could have failed, and does not.

Step 1. The 00 slot. With $F_\alpha{}^\nu=\eta_{\alpha\beta}F^{\beta\nu}$ , the first term's corner entry is $F^{0i}F_i{}^0=-\sum_i(F^{0i})^2\cdot(-1)=E^2/c^2$ — one metric factor per lowered spatial index — and the invariant is $F_{\alpha\beta}F^{\alpha\beta}=2(B^2-E^2/c^2)$ (Lorentz Invariants of the Field). So

\theta^{00}=\frac{1}{\mu_0}\left[a\,\frac{E^2}{c^2}+2b\left(B^2-\frac{E^2}{c^2}\right)\right] \overset{!}{=}\frac{\varepsilon_0}{2}E^2+\frac{B^2}{2\mu_0},

matching Energy Density of the Electromagnetic Field. Using $\varepsilon_0=1/\mu_0c^2$ and equating the $E^2$ and $B^2$ coefficients separately: $a-2b=\tfrac12$ and $2b=\tfrac12$ , hence $a=1$ , $b=\tfrac14$ . The construction is now rigid:

\theta^{\mu\nu}=\frac{1}{\mu_0}\left[F^{\mu\alpha}F_\alpha{}^{\nu}+\tfrac14\eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right]. \qquad (1)

Step 2. The top row — the second calibration that turns out free of charge. The $\eta^{0i}$ term vanishes off-diagonal, so $\theta^{0i}=F^{0\alpha}F_\alpha{}^i/\mu_0$ . Only spatial $\alpha$ contributes, each carrying one metric sign:

\theta^{0i}=-\frac{1}{\mu_0}\,F^{0j}F^{ji} =-\frac{1}{\mu_0}\left(-\frac{E_j}{c}\right)\left(-\epsilon^{jik}B_k\right) =\frac{(\mathbf{E}\times\mathbf{B})^i}{\mu_0 c}=\frac{S_i}{c},

exactly The Poynting Vector's flux, in the slot reserved for energy transport. (Had Step 1's coefficients not also produced this, the two-term menu would have been insufficient — this is where the ansatz first survives contact with independent physics.)

Step 3. First prediction: the momentum column. By inspection of (1), swapping $\mu\leftrightarrow\nu$ swaps the dummy contractions into each other: $\theta^{\mu\nu}=\theta^{\nu\mu}$ , symmetric with no choice. Therefore

\theta^{i0}=\theta^{0i}=\frac{S_i}{c}=c\,g_i \qquad\Longleftrightarrow\qquad \mathbf{g}=\frac{\mathbf{S}}{c^2},

which is Momentum Density of the Electromagnetic Field's hard-won result — Einstein-box arguments and all — obtained here as a symmetry of index bookkeeping. Planck's principle (energy flux carries momentum, inertia of energy) is not an extra law; it is what a symmetric stress-energy tensor means.

Step 4. Second prediction: the stress block. Computing $\theta^{ij}$ from (1) with the dictionary (all nine components, sympy in the verification):

\theta^{ij}=-\left[\varepsilon_0\!\left(E_iE_j-\tfrac{\delta_{ij}}{2}E^2\right)+\frac{1}{\mu_0}\!\left(B_iB_j-\tfrac{\delta_{ij}}{2}B^2\right)\right]=-\sigma_{ij},

the negative of E7's Maxwell stress tensor, every term and sign. The reading is exact, not analogical: $\theta^{ij}$ is the flux of $j$ -momentum in the $i$ -direction, while The Maxwell Stress Tensor's $\sigma_{ij}$ was built as the force per area transmitted — momentum flowing in is force pushing out, hence the sign. That article's eigen-anatomy (tension $u$ along $\mathbf{E}$ or $\mathbf{B}$ , pressure $u$ transverse) is therefore the eigen-anatomy of $\theta$ 's spatial block, unchanged. The cabinet is full: sixteen slots, four E7 articles, zero leftover freedom. Verified: all sixteen components computed symbolically (sympy, arbitrary $\mathbf{E},\mathbf{B}$ ) from (1) with the track dictionary: $\theta^{00}-u=0$ , $\theta^{0i}-S_i/c=0$ , $\theta^{i0}-S_i/c=0$ , $\theta^{ij}+\sigma_{ij}=0$ for all nine, and $\theta^{\mu\nu}-\theta^{\nu\mu}=0$ ; matches Jackson §12.10 (Gaussian) and Zangwill §22.7 mod units.

Figure

Light in the ledger — the null dyad of a plane wave, and the isotropic bath it averages into

Go deeper · Derivation B · 2 minTracelessness — massless light, the null dyad, and the radiation equation of state

Derivation BTracelessness — massless light, the null dyad, and the radiation equation of state

We prove $\theta^\mu{}_\mu=0$ identically, then extract its two famous consequences: a single plane wave's tensor collapses to a rank-one "null dust" form, and any isotropic bath of radiation must have pressure $u/3$ . The strategy is three short arguments sharing one input — the trace computation — chosen because each consequence lands in a different field of physics (optics, cosmology, thermodynamics). Watch how much physics one algebraic zero funds: the trace measures the theory's internal scale, and Maxwell theory has none — no mass, no length — so the zero is masslessness made local.

Step 1. The trace. Contract (1) with $\eta_{\mu\nu}$ : the first term gives $F^{\mu\alpha}F_{\alpha\mu}=-F_{\alpha\beta}F^{\alpha\beta}$ , and the second gives $\tfrac14\times4\times F_{\alpha\beta}F^{\alpha\beta}$ (since $\eta^{\mu\nu}\eta_{\mu\nu}=4$ ):

\theta^\mu{}_\mu=\frac{1}{\mu_0}\left[-F^2+F^2\right]=0,

identically — for every field configuration, in every frame, a Lorentz-invariant zero. In component language: $u=\sum_i\theta^{ii}$ , the energy density always equals the sum of the three principal stresses. The $b=\tfrac14$ that Step 1 of Derivation A extracted from energy calibration is exactly the coefficient that kills the trace: the same fact viewed twice.

Step 2. The plane wave as null dust. For a monochromatic wave along $\hat{\mathbf{z}}$ (fields per Monochromatic Plane Waves in Vacuum: $E=cB$ , $\mathbf{E}\perp\mathbf{B}\perp\hat{\mathbf{z}}$ ), evaluate the slots: $\theta^{00}=u$ ; $\theta^{0z}=S/c=u$ (since $S=cu$ for a plane wave); $\theta^{zz}=-\sigma_{zz}=+u$ (both field directions are transverse, so both contribute pressure along $\mathbf{k}$ ); every other entry vanishes. The whole tensor is

\theta^{\mu\nu}=u\,n^\mu n^\nu, \qquad n^\mu=(1,0,0,1),

a single dyad built on the null vector $n$ ( $n\cdot n=0$ — which is also why the trace vanishes so effortlessly here). Reading: light is a dust of energy streaming rigidly at $c$ — density $u$ , flux $cu$ , momentum $u/c$ , pressure $u$ , all one number wearing four hats. This is the tensor form of the E8 relations $g=u/c$ and $P_{\text{abs}}=u$ , and the object whose exact transformation law the Doppler article will exploit ( $u$ boosts with two powers of the Doppler factor riding the two $n$ 's... delivered there).

Step 3. Isotropic radiation: $P=u/3$ . Fill a cavity with radiation arriving equally from all directions — a photon gas, the Sun's interior, the early universe. Isotropy forces the averaged tensor to be diagonal: $\langle\theta^{\mu\nu}\rangle=\mathrm{diag}(u,P,P,P)$ with a single pressure $P$ . Now impose Step 1's zero, which survives averaging:

0=\theta^\mu{}_\mu=u-3P \qquad\Longrightarrow\qquad P=\frac{u}{3}.

Three lines, no statistical mechanics, no photon concept — the radiation equation of state, from tracelessness plus isotropy alone. This single relation powers the classical derivation of the Stefan–Boltzmann law, sets radiation's $\rho\propto a^{-4}$ dilution in cosmology (the extra factor over matter's $a^{-3}$ is the work $P\,dV$ extracts), and fixes the radiation-pressure gradient holding stars against gravity. Verified: trace zero confirmed symbolically for arbitrary fields; plane-wave slots recomputed ( $\theta^{00}=\theta^{0z}=\theta^{z0}=\theta^{zz}=u$ , others zero) with the dictionary; the isotropy average checked by explicit angular integration of the dyad over directions ( $\langle n^in^j\rangle=\tfrac13\delta^{ij}$ ).

The canonical mess, and who healed it

Honesty band, promised by E7. The systematic recipe for stress-energy — Noether's theorem applied to spacetime translations, as built in the mechanics track's The Canonical Stress-Energy Tensor — does not produce equation (1) when fed the field Lagrangian $-F^2/4\mu_0$ of the last article. It produces the canonical tensor $T^{\mu\nu}_{\text{can}}=-\tfrac{1}{\mu_0}F^{\mu\alpha}\partial^\nu A_\alpha+\tfrac{1}{4\mu_0}\eta^{\mu\nu}F^2$ : conserved, yes — but asymmetric and, worse, gauge-dependent, since it clutches a bare $\partial^\nu A_\alpha$ . An energy density you can change by re-gauging is not an energy density. The repair is the Belinfante–Rosenfeld construction: add an identically conserved "improvement" term, fixed by the field's spin structure, that cancels both diseases at once — and the mechanics track's Symmetrization and the Belinfante-Rosenfeld Tensor runs that construction on the electromagnetic field explicitly ("electromagnetism healed"), landing exactly on (1). We cite the surgery and keep the patient: nothing in this article depends on the canonical detour, because Derivation A defined $\theta$ by its physical slots. But the detour explains a deep alignment: gravity — general relativity's variational definition of stress-energy — couples to the healed tensor automatically, which is why (1), and not the canonical mess, is the object with a future.

question	reach for	why
energy/momentum bookkeeping at a point, any frame	$\theta^{\mu\nu}$ , eq. (1)	all sixteen answers at once
force on matter inside a surface	spatial block $-\theta^{ij}=\sigma_{ij}$ integrated	E7's stress machinery, unchanged
how $u$ , $\mathbf{S}$ , stresses boost	$\bar\theta=\Lambda\theta\Lambda^{\mathsf T}$	rank-2 law; no slot is invariant
radiation bath thermodynamics	trace zero + isotropy: $P=u/3$	Derivation B, Step 3
where energy conservation comes from	next article: $\partial_\mu\theta^{\mu\nu}=-F^{\nu\lambda}J_\lambda$	the cabinet's divergence
what gravity sees	$\theta^{\mu\nu}$ in Einstein's equations	GR track; symmetric + conserved required

Limiting cases

case	result	meaning
pure static $\mathbf{E}$ or $\mathbf{B}$	$\theta=\mathrm{diag}(u;\ \text{tension/pressure block})$ , $\mathbf{S}=0$	E7's static anatomy; energy sits, stresses act
single plane wave	$\theta^{\mu\nu}=u\,n^\mu n^\nu$ , $n$ null	light as null dust; all slots one number
isotropic bath	$\mathrm{diag}(u,u/3,u/3,u/3)$	photon-gas equation of state
null field, general	$\theta$ has rank 1, $\theta^\mu{}_\alpha\theta^{\alpha\nu}=0$	the invariants' null class in tensor dress
crossed static fields, $E<cB$	boost at drift kills $\mathbf{S}$	Lorentz Invariants of the Field's frame, read as "momentum-free frame"
vacuum ( $F=0$ )	$\theta=0$	no field, no ledger — classical vacuum carries nothing

Common trapSixteen numbers, and not one of them is invariant — stop treating u as a property of the field alone

The most common covariant-bookkeeping error downstream of this article: quoting an energy density, a Poynting flux, or a radiation pressure as if it were frame-proof. Every slot of $\theta^{\mu\nu}$ transforms — $\bar\theta=\Lambda\theta\Lambda^{\mathsf T}$ , two boost factors, one per index — so $u$ is no more invariant than $E_x$ was in the field-transformation article. A capacitor's pure-tension field acquires momentum density in any moving frame; a plane wave's $u$ scales by the square of the Doppler factor between observers. The frame-proof statements are: the two field invariants, the trace (zero), the rank/null classification, and — for closed systems only — the total integrated four-momentum $P^\nu=\int\theta^{0\nu}d^3x/c$ ... and that last one carries its own trap, the notorious 4/3 problem: for the open subsystem "a charge's own Coulomb field," the integrated $P^\nu$ fails to transform as a four-vector, because the subsystem exchanges momentum with what was cut away. Next article states the closure theorem precisely; E11's classical-electron article dissects the famous casualty. Until then: quote components with their frame, or quote invariants.

Worked examples

Example 1 — the full cabinet for bright sunlight. A plane wave of intensity $I=1.0$ kW/m² travels along $+\hat{\mathbf{z}}$ . Fill in the (time-averaged) tensor.

Step 1. Energy density: $u=I/c=3.34\times10^{-6}$ J/m³. By the null-dyad structure (Derivation B, Step 2), this one number fills every active slot: $\theta^{00}=u$ , $\theta^{0z}=\theta^{z0}=S_z/c=I/c=3.34\times10^{-6}$ (SI slot units J/m³ throughout), $\theta^{zz}=3.34\times10^{-6}$ ; all other entries zero.

Step 2. Read the physics out of the slots: momentum density $g_z=\theta^{z0}/c=1.11\times10^{-14}$ kg m⁻² s⁻¹; radiation pressure on a perfect absorber $=\theta^{zz}=3.34\times10^{-6}$ Pa (double for a mirror — the reflected wave brings its own $\theta^{zz}$ ); energy flux $c\,\theta^{0z}=1.0$ kW/m² back out, closing the audit.

Step 3. Trace check: $\theta^\mu{}_\mu=u-\theta^{zz}=3.34\times10^{-6}-3.34\times10^{-6}=0$ ✓. One measured number, sixteen slots, zero trace — light's whole mechanics is $u$ times a null direction.

Example 2 — the cabinet for a 1 MV/m capacitor field. A parallel-plate capacitor holds a static uniform $\mathbf{E}=10^6\ \hat{\mathbf{z}}$ V/m.

Step 1. Energy density: $u=\tfrac{\varepsilon_0}{2}E^2=4.43$ J/m³. Poynting row and momentum column: zero (no $\mathbf{B}$ ) — energy sits still.

Step 2. Stress block from Derivation A, Step 4: along the field, $\theta^{zz}=-\sigma_{zz}=-u=-4.43$ J/m³ — negative momentum-flux, i.e. tension: each plate is pulled toward the other with $4.43$ N per square meter of field. Transverse: $\theta^{xx}=\theta^{yy}=+4.43$ J/m³ — the field pushes outward sideways, the "bulging" of field lines E7 quantified.

Step 3. Trace: $u-(\theta^{xx}+\theta^{yy}+\theta^{zz})=4.43-(4.43+4.43-4.43)=0$ ✓. And a boost check worth internalizing: an observer moving at $\beta$ along $x$ finds (by $\Lambda\theta\Lambda^{\mathsf T}$ , or by boosting the fields first — same answer, that is the tensor's point) a nonzero $\bar\theta^{0y}$ ? No — but a nonzero $\bar{\mathbf{S}}$ along $x$ does appear via the mixed slots ( $\bar\theta^{0x}\neq0$ ): the moving observer sees the capacitor's energy streaming past — momentum density from a static field, the same lesson as Example 1 of the covariant-Maxwell article, now in the ledger.

Going further

Reading. Jackson, Classical Electrodynamics, 3rd ed., §12.10 — the covariant construction. Zangwill, Modern Electrodynamics, §22.7. Landau & Lifshitz, Classical Theory of Fields, §§32–33 — the canonical tensor and its symmetrization. Griffiths, 4th ed., §8.2 and §12.3 for the 3+1 anchors.

Provenance. Built from the AllQFT base article for this topic (tensor definition, component identifications, tracelessness, and canonical-tensor honesty retained), with the index form fixed to this track's $(+,-,-,-)$ dictionary as $\theta^{\mu\nu}=\tfrac{1}{\mu_0}[F^{\mu\alpha}F_\alpha{}^\nu+\tfrac14\eta^{\mu\nu}F^2]$ — sign-audited symbolically for this edition (all sixteen components; the family's $-\tfrac14\eta F^2$ variant with the opposite first-term contraction gives $\theta^{00}=-u$ in this dictionary and was rejected by the audit). A verifier should: (1) rerun the sixteen-slot sympy audit against $u$ , $\mathbf{S}/c$ , $c\mathbf{g}$ , $-\sigma_{ij}$ ; (2) confirm the trace identity and the isotropy average $\langle n^in^j\rangle=\delta^{ij}/3$ ; (3) recompute Example 1 ( $u=3.34\times10^{-6}$ J/m³, $g=1.11\times10^{-14}$ kg m⁻²s⁻¹, $P_{\text{abs}}=3.34\ \mu$ Pa) and Example 2 ( $u=4.43$ J/m³, tension $-4.43$ , transverse $+4.43$ , trace zero). Division of labor for this edition: the four 3+1 occupants are owned by E7 — Energy Density of the Electromagnetic Field ( $u$ , calibration 1), The Poynting Vector ( $\mathbf{S}$ , calibration 2), Momentum Density of the Electromagnetic Field ( $\mathbf{g}$ , predicted by symmetry), The Maxwell Stress Tensor ( $\sigma_{ij}$ and its eigen-anatomy, predicted by the block) — none re-derived; the filing-cabinet framing and the deferral of this construction by Conservation Laws and Spacetime Symmetries (delivered); canonical-tensor machinery and the Belinfante healing by the mechanics track's The Canonical Stress-Energy Tensor and Symmetrization and the Belinfante-Rosenfeld Tensor (cited, not re-run); the divergence theorem and global conservation by Conservation Laws and Energy-Momentum Flow (next); media tensors and the Abraham–Minkowski question by E7's Electromagnetic Stress, Momentum, and Energy in Media and Tensor Formulation of Fields in Media (ahead); the 4/3 problem's full dissection by E11's classical-electron articles. This edition owns the covariant object: construction, calibrations, predictions, tracelessness and its consequences.

Builds on: The Electromagnetic Field Tensor · Four-Vectors and the Metric Tensor · The Four-Current Density · Relativistic Energy and Four-Momentum · Lorentz Transformations

Leads to: The Dual Field Tensor · Radiation pressure and momentum of light · General relativity and the Einstein equations

Why does a field need sixteen numbers when a particle gets by with four?

Because a particle's mechanics answers "how much energy-momentum?" while a field's must also answer "where is it flowing?" — at every point, for each of the four conserved quantities. Four quantities × four flow directions = sixteen slots, organized as density (time column) versus flux (space columns). The redundancy you might suspect — symmetry cuts sixteen to ten independent — is itself physics: the symmetric pairing of "flux of energy" with "density of momentum" is Planck's inertia-of-energy, and the next article shows the remaining structure is exactly what angular-momentum conservation requires. Particles secretly agree: a dust of particles has $T^{\mu\nu}=\rho_0u^\mu u^\nu$ , sixteen slots built from each particle's four — the field is not being extravagant, just honest about being everywhere.

Is the energy density of a field configuration something all observers agree on?

No — and internalizing this prevents a whole family of errors. $u$ is the $00$ -component of a rank-2 tensor: under a boost it mixes with the Poynting flux and the stresses exactly as $E_x$ mixed with $B_y$ in the field-transformation article ( $\bar\theta=\Lambda\theta\Lambda^{\mathsf T}$ ). Concretely: a plane wave's energy density scales as the square of the Doppler factor between observers (both indices ride the null vector $n^\mu$ ), and even a static capacitor's field "streams" for a moving observer (Example 2's boost check). Frame-proof quantities exist, but they are the invariants: $F^2$ , $F\cdot G$ , the zero trace — and, for closed systems, the integrated total $P^\nu$ as a four-vector, whose frame components still differ. "The field's energy" without a frame label is not a measurement; it is a category error.

What breaks if someone uses the canonical Noether tensor instead of the symmetric one?

Three things, in escalating order of pain. Bookkeeping: the canonical tensor is gauge-dependent, so its "energy density" changes under the re-gaugings of two articles ago — different gauges would assign different energies to the same field, which no calorimeter respects. Angular momentum: the naive $x\times$ (momentum density) current is conserved only if the tensor is symmetric; with the canonical object you must haul a separate spin current around (the mechanics track's angular-momentum article does exactly this bookkeeping). Gravity: Einstein's equations demand a symmetric, gauge-invariant source, and the variational (Hilbert) definition of stress-energy in general relativity produces the Belinfante-healed tensor automatically — nature already made the choice. The canonical tensor is not wrong — it integrates to the same total $P^\nu$ — it is the right charge with the wrong density, and every local application (forces, gravity, flow maps) needs the density.

Does light really pull on things sideways and push along its direction, like the stress block says?

Yes, and both signs are measured routinely. Along the propagation direction the block gives pressure $\theta^{zz}=u$ : absorbed sunlight pushes with $I/c\approx3\ \mu$ Pa per kW/m² — feeble on your hand, decisive for solar sails (IKAROS flew on it), comet tails, and the radiation-supported interiors of massive stars. The transverse structure is subtler for a beam: a uniform plane wave exerts no transverse force, but gradients do — at the edge of a focused beam the stress block acquires off-diagonal entries, and the net effect (equivalently computed from induced-dipole forces) is what traps dielectric beads at the intensity maximum in optical tweezers. For static fields the anatomy is E7's: tension along lines (capacitor plates attract; magnet poles attract) and pressure across them (parallel like-currents' fields squeeze). The tensor prices all of it with one object — that is the point of owning it.

Where does the famous 4/3 problem sit relative to this article?

Just downstream, and the tools here name it precisely. Compute the total field four-momentum of a charged sphere's Coulomb field: the integrals give $E_{\text{field}}/c^2=m_{\text{em}}$ from the energy but $\tfrac43 m_{\text{em}}\mathbf{v}$ from the momentum at small velocity — a "mass" that disagrees with itself by 4/3. The resolution is Laue's closure theorem (this article's warning band): the Coulomb field alone is an open system — something non-electromagnetic must hold the charge together against its own stress block's outward pressure, and those "Poincaré stresses" carry compensating energy-momentum. Include them and the total transforms as an honest four-vector. The full dissection — including its entanglement with self-energy divergences and radiation reaction — is E11 business (Classical Electron Theory and Self-Energy); what this article contributes is the diagnostic rule: integrated $\theta^{0\nu}$ is a four-vector iff $\partial_\mu\theta^{\mu\nu}=0$ everywhere, i.e. iff nothing is cut out.

How does this tensor generalize when the field lives in a material medium?

Contentiously — that is the honest answer, and the chapter has already told the story from the 3+1 side. In matter the natural field pairs double ( $\mathbf{E},\mathbf{B}$ vs $\mathbf{D},\mathbf{H}$ ), and two inequivalent tensors compete: Minkowski's (canonical momentum $\mathbf{D}\times\mathbf{B}$ , asymmetric in general — violating this article's symmetry doctrine) and Abraham's (kinetic momentum $\mathbf{E}\times\mathbf{H}/c^2$ , symmetric, but needing an extra "Abraham force" on the medium). E7's Electromagnetic Stress, Momentum, and Energy in Media owns the verdict — both are right about different questions, kinetic versus canonical momentum, per the Barnett settlement — and Tensor Formulation of Fields in Media, three articles ahead, builds the covariant objects ( $G^{\mu\nu}$ , $M^{\mu\nu}$ ) in which the competition is stated cleanly. The vacuum tensor built here is the arbiter both sides must reduce to as $n\to1$ .

Is the trace really zero in the full quantum theory?

No — and the failure is famous enough to have a name, the trace anomaly. Classically, Derivation B's zero reflects the absence of any scale in Maxwell theory: no mass, no intrinsic length, so nothing for a trace (which measures response to uniform rescaling) to be made of. Quantization breaks this: renormalization forces a scale into the theory (the running of the coupling with energy), and $\langle\theta^\mu{}_\mu\rangle$ picks up terms proportional to the beta function times $F^2$ . In QED the effect is small but real; in QCD it is spectacular — the trace anomaly of the gluon field generates most of the proton's mass, making "the mass of ordinary matter is mostly anomalous gluon trace" one of the great punchlines of modern physics. The classical zero derived here is the reference point that makes the anomaly a sharp, measurable statement.