Symmetrization and the Belinfante-Rosenfeld Tensor

Two articles have filed complaints against the canonical stress-energy tensor: it is asymmetric for any field with spin, and for gauge fields it is not even gauge-invariant. One article has supplied the diagnostic — the asymmetry equals minus the divergence of the spin current — and another has certified the legal remedy: currents may be modified by identically conserved improvement terms without changing a single charge. This article performs the surgery. Belinfante and Rosenfeld's construction takes the spin current, reshuffles its three indices into a superpotential $B^{\lambda\mu\nu}$ antisymmetric in its first pair, and adds its divergence to the canonical tensor. The output $\Theta^{\mu\nu}$ is symmetric — exactly, by the spin-asymmetry identity — still conserved, carrying the same four-momentum, and, run on electromagnetism, it converts the diseased canonical object into the gauge-invariant Maxwell tensor whose entries are the familiar $\tfrac12(E^2 + B^2)$ and $\mathbf{E}\times\mathbf{B}$ . Why insist on symmetry? Two independent authorities demand it. Angular momentum: with $\Theta$ symmetric, the total angular-momentum current can be written purely orbitally, $x\wedge\Theta$ , with the spin bookkeeping absorbed into the momentum ledger — one tensor now carries the whole mechanical story. Gravity: Einstein's equations require a symmetric source, and the tensor obtained by varying the action with respect to the metric — the Hilbert tensor — turns out to equal the Belinfante tensor. That equality is the construction's deepest endorsement: the repair invented to fix angular-momentum bookkeeping is precisely what spacetime curvature responds to. The price, spelled out honestly, is local: energy and momentum densities are redistributed pointwise, and only the integrals are guaranteed to match.

Key takeaways

Disease: the canonical $T^{\mu\nu}_{\text{can}}$ is conserved but generally asymmetric (spinful fields) and gauge-dependent (gauge fields).
Remedy: add an improvement $\partial_\lambda B^{\lambda\mu\nu}$ with $B^{\lambda\mu\nu} = -B^{\mu\lambda\nu}$ — conservation and total four-momentum provably survive.
The superpotential is built from the spin current: $B^{\lambda\mu\nu} = \tfrac12(S^{\lambda\mu\nu} + S^{\mu\nu\lambda} - S^{\nu\lambda\mu})$ — the unique index shuffle with the required antisymmetry.
The result $\Theta^{\mu\nu} = T^{\mu\nu}_{\text{can}} + \partial_\lambda B^{\lambda\mu\nu}$ is symmetric, by the spin-asymmetry identity $\partial_\lambda S^{\lambda\mu\nu} = T^{\nu\mu} - T^{\mu\nu}$ .
With $\Theta$ , total angular momentum is purely orbital: $M^{\mu\alpha\beta} = x^\alpha\Theta^{\mu\beta} - x^\beta\Theta^{\mu\alpha}$ — spin absorbed into the momentum ledger.
For electromagnetism, $\Theta^{\mu\nu} = -F^{\mu\lambda}F^\nu{}_\lambda + \tfrac14\eta^{\mu\nu}F^2$ : gauge-invariant, traceless, with $\Theta^{00} = \tfrac12(E^2 + B^2)$ and $\Theta^{0i} = (\mathbf{E}\times\mathbf{B})_i$ .
$\Theta$ equals the Hilbert tensor $\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g_{\mu\nu}}$ : the symmetrized tensor is what sources gravity.
Local densities change pointwise under the improvement; only integrated charges are invariant — never equate $\Theta^{0\nu}$ and $T^{0\nu}_{\text{can}}$ locally.
Key idea: the spin current, repackaged as a superpotential, symmetrizes the stress tensor at zero cost to any conservation law — and gravity confirms the choice.

The core insight

Adding $\partial_\lambda B^{\lambda\mu\nu}$ with $B^{\lambda\mu\nu} = -B^{\mu\lambda\nu}$ leaves conservation and total charges intact; choosing $B$ from the spin current cancels the antisymmetric part of $T^{\mu\nu}_{\text{can}}$ , producing the symmetric, gauge-invariant tensor that sources gravity.

Where this lives today

The Belinfante-Rosenfeld tensor is the default stress-energy of working physics: every general-relativistic source term — neutron-star interiors, binary-merger simulations, cosmological fluids — is a Hilbert/Belinfante tensor, and the gravitational-wave era has made its entries observables (LIGO's waveforms are sensitive to the source's $\Theta^{\mu\nu}$ multipoles; the binary neutron star merger GW170817 constrained the pressure block of nuclear matter). The construction's subtleties have become research fronts in three directions. In hadron physics, the "gravitational form factors" program measures matrix elements of $\Theta^{\mu\nu}$ inside the proton — effectively asking how energy, pressure and shear are distributed in a bound relativistic system, with lattice QCD and Jefferson Lab extractions mapping a pressure profile that must integrate to zero (the von Laue condition of the stress-energy article's history) — and the quark-gluon decomposition of those form factors inherits precisely this article's improvement ambiguities. In condensed matter and hydrodynamics, the choice between canonical and Belinfante tensors has measurable consequences once media break Lorentz invariance or carry intrinsic spin: spin hydrodynamics for quark-gluon plasma (where hyperon polarization at RHIC measures the fluid's spin ledger) and micropolar elasticity both work with asymmetric stress on purpose, using exactly the $T$ -versus- $\Theta$ dictionary to translate. And in quantum gravity and semiclassical gravity, the renormalized $\langle\Theta^{\mu\nu}\rangle$ — Casimir energies, Hawking fluxes, trace anomalies — is defined through the metric-variation route, making (5) not a convenience but the definition on which black-hole thermodynamics runs. Within the chapter: this article closes the stress-energy storyline opened two articles ago; the Lorentz-covariance article now has all ten Poincaré currents in their best-behaved form; and the capstone will quantize a theory whose classical bookkeeping is, thanks to Belinfante and Rosenfeld, in perfect order.

The physical picture

Start from what the asymmetry means, because the cure is designed to its meaning. The spin-current article's identity says $T^{\mu\nu}_{\text{can}}$ is asymmetric exactly where spin flows: the canonical tensor keeps two sets of books — a momentum ledger and a separate spin ledger — and the asymmetry is the reconciliation entry between them. The Belinfante move is an accountant's consolidation: absorb the spin ledger into the momentum ledger. A flow of intrinsic angular momentum through a region is re-booked as a small circulating flow of momentum around that region's edges — physically sensible, since a body absorbing the spin flux does experience torque, i.e. edge-distributed momentum transfer. After consolidation there is one ledger, symmetric, and total angular momentum reads as pure moment-of-momentum, $x\wedge\Theta$ , as if the field were built of spinless constituents.

Figure

The Belinfante repair — one added divergence converts the diseased canonical tensor into the symmetric tensor gravity reads

Why is the re-booking legal? Because the conserved-currents article established the exact size of the allowed freedom: any $\partial_\lambda B^{\lambda\mu\nu}$ with $B$ antisymmetric in $\lambda\mu$ is identically conserved and adds zero total charge. The Belinfante construction is the statement that within this freedom there exists a choice that kills the asymmetry entirely — and that the choice is not clever guesswork but a universal formula in the spin current. The three-derivation core of this article proves the legality, performs the symmetrization, and runs the flagship example. One caution before the machinery: the improvement genuinely moves densities around. $\Theta^{00}(\mathbf{x})$ and $T^{00}_{\text{can}}(\mathbf{x})$ are different functions whose integrals agree; the worked examples exhibit the redistribution with numbers. "Same physics" here means same charges, same conservation laws, same dynamics — not same pointwise attribution.

The construction

The inputs, both inherited: the canonical tensor $T^{\mu\nu}_{\text{can}}$ with $\partial_\mu T^{\mu\nu}_{\text{can}} = 0$ (stress-energy article), and the spin current $S^{\lambda\mu\nu}$ , antisymmetric in its last two indices, obeying the spin-asymmetry identity (spin-current article):

\partial_\lambda S^{\lambda\mu\nu} = T^{\nu\mu}_{\text{can}} - T^{\mu\nu}_{\text{can}} \qquad (1)

where: the right side is (twice) the antisymmetric part of the canonical tensor — the object to be eliminated. Belinfante's superpotential is the unique cyclic reshuffle of $S$ 's indices that is antisymmetric in the first pair:

B^{\lambda\mu\nu} = \tfrac{1}{2}\left(S^{\lambda\mu\nu} + S^{\mu\nu\lambda} - S^{\nu\lambda\mu}\right), \qquad B^{\lambda\mu\nu} = -B^{\mu\lambda\nu} \qquad (2)

where: the antisymmetry check takes one line — swap $\lambda \leftrightarrow \mu$ and use $S$ 's antisymmetry in its last pair on each term. The first-pair antisymmetry is not aesthetic: it is exactly the property that makes $\partial_\lambda B^{\lambda\mu\nu}$ an identically conserved, charge-free improvement, as the first derivation now certifies. The improved tensor is

\Theta^{\mu\nu} = T^{\mu\nu}_{\text{can}} + \partial_\lambda B^{\lambda\mu\nu} \qquad (3)

Derivation AThe improvement preserves conservation and the total four-momentum

We prove the two safety properties: $\Theta$ is conserved whenever $T_{\text{can}}$ is, and both tensors carry the same $P^\nu$ . This is the conserved-currents article's improvement theorem, instantiated for the three-index superpotential — worth one careful pass because the index antisymmetry does all the work and because the $\lambda = 0$ subtlety in the charge computation is a standard exam trap. Strategy: symmetric-antisymmetric contraction for conservation; antisymmetry plus the divergence theorem for the charge.

Step 1. Conservation. Take the divergence of (3):

\partial_\mu\Theta^{\mu\nu} = \underbrace{\partial_\mu T^{\mu\nu}_{\text{can}}}_{= \,0 \text{ on-shell}} + \;\partial_\mu\partial_\lambda B^{\lambda\mu\nu}

The second term vanishes identically: $\partial_\mu\partial_\lambda$ is symmetric under $\mu\leftrightarrow\lambda$ while $B^{\lambda\mu\nu}$ is antisymmetric — a symmetric pair contracted with an antisymmetric pair is zero, no equations of motion required.

Step 2. The added charge. The improvement's contribution to the momentum density is $\partial_\lambda B^{\lambda 0\nu}$ . Split the sum: the $\lambda = 0$ term is $\partial_0 B^{00\nu}$ , and $B^{00\nu} = -B^{00\nu} = 0$ by first-pair antisymmetry — no time-derivative term survives. Only spatial pieces remain:

\int_V \partial_\lambda B^{\lambda 0\nu}\,\mathrm{d}^3x = \int_V \partial_i B^{i0\nu}\,\mathrm{d}^3x

Why this step matters: had a $\partial_0$ term survived, the added charge would be the time derivative of a volume integral — generally nonzero. The first-pair antisymmetry is precisely what forbids it. This is where a wrong index shuffle in (2) would fail.

Step 3. Divergence theorem:

\int_V \partial_i B^{i0\nu}\,\mathrm{d}^3x = \oint_{\partial V} B^{i0\nu}\,\mathrm{d}S_i \;\longrightarrow\; 0

for localized fields (the conserved-currents article's standard decay condition). Hence

P^\nu_\Theta = \int\Theta^{0\nu}\,\mathrm{d}^3x = \int T^{0\nu}_{\text{can}}\,\mathrm{d}^3x = P^\nu

Same four-momentum, exactly. The improvement redistributes; it never creates or destroys. Verified: matches the general improvement theorem of the conserved-currents article specialized to $\Sigma^{\mu\nu} = B^{\lambda\mu\nu}$ -type superpotentials; the $B^{00\nu} = 0$ step per Weinberg Vol. 1 §7.4.

Go deeper · Derivation B · 1 minThe superpotential symmetrizes the tensor

Derivation BThe superpotential symmetrizes the tensor

We prove the headline: $\Theta^{\mu\nu} = \Theta^{\nu\mu}$ , exactly, on-shell. The mechanism is a precise conspiracy between the index shuffle (2) and the spin-asymmetry identity (1): the antisymmetric part of $\partial B$ evaluates to $\partial_\lambda S^{\lambda\mu\nu}$ , which (1) says is minus the antisymmetric part of $T_{\text{can}}$ . Strategy: compute $\Theta^{\mu\nu} - \Theta^{\nu\mu}$ and watch the cancellation assemble itself. The only tool is $S$ 's antisymmetry in its last two indices, $S^{\lambda\mu\nu} = -S^{\lambda\nu\mu}$ .

Step 1. Antisymmetrize the definition (3):

\Theta^{\mu\nu} - \Theta^{\nu\mu} = \big(T^{\mu\nu}_{\text{can}} - T^{\nu\mu}_{\text{can}}\big) + \partial_\lambda\big(B^{\lambda\mu\nu} - B^{\lambda\nu\mu}\big)

Step 2. Evaluate the superpotential's antisymmetric part from (2):

B^{\lambda\mu\nu} - B^{\lambda\nu\mu} = \tfrac12\big(S^{\lambda\mu\nu} + S^{\mu\nu\lambda} - S^{\nu\lambda\mu}\big) - \tfrac12\big(S^{\lambda\nu\mu} + S^{\nu\mu\lambda} - S^{\mu\lambda\nu}\big)

Apply $S^{\lambda\nu\mu} = -S^{\lambda\mu\nu}$ , $S^{\nu\mu\lambda} = -S^{\nu\lambda\mu}$ , $S^{\mu\lambda\nu} = -S^{\mu\nu\lambda}$ (last-pair antisymmetry on each):

= \tfrac12\big(S^{\lambda\mu\nu} + S^{\mu\nu\lambda} - S^{\nu\lambda\mu} + S^{\lambda\mu\nu} + S^{\nu\lambda\mu} - S^{\mu\nu\lambda}\big) = S^{\lambda\mu\nu}

Why the middle terms cancel pairwise: the shuffle (2) was engineered so that under $\mu\leftrightarrow\nu$ the second and third terms trade places with signs — leaving only the original spin current. This is the design secret of the Belinfante combination.

Step 3. Insert and apply the spin-asymmetry identity (1):

\Theta^{\mu\nu} - \Theta^{\nu\mu} = \big(T^{\mu\nu}_{\text{can}} - T^{\nu\mu}_{\text{can}}\big) + \partial_\lambda S^{\lambda\mu\nu} = \big(T^{\mu\nu}_{\text{can}} - T^{\nu\mu}_{\text{can}}\big) + \big(T^{\nu\mu}_{\text{can}} - T^{\mu\nu}_{\text{can}}\big) = 0

Symmetry, exactly, with the asymmetry consumed by the spin current — the two defects of the canonical formalism annihilating each other.

Step 4. Harvest the angular-momentum dividend. With $\Theta$ symmetric and conserved, the purely orbital current $\tilde M^{\mu\alpha\beta} = x^\alpha\Theta^{\mu\beta} - x^\beta\Theta^{\mu\alpha}$ is conserved by itself ( $\partial_\mu\tilde M = \Theta^{\alpha\beta} - \Theta^{\beta\alpha} = 0$ ), and its charges equal the full $J^{\alpha\beta} = L + S$ of the spin-current article (the difference is a surface term built from $B$ ). One symmetric tensor now carries energy, momentum, and angular momentum: $M = x\wedge\Theta$ , spin included. Verified: cancellation recomputed index by index; matches Belinfante (1940) and the standard presentation in Weinberg Vol. 1 §7.4; scalar-field check: $S = 0 \Rightarrow B = 0 \Rightarrow \Theta = T_{\text{can}}$ , already symmetric.

Go deeper · Derivation C · 1 minElectromagnetism healed: from the canonical mess to the Maxwell tensor

Derivation CElectromagnetism healed: from the canonical mess to the Maxwell tensor

We run the construction on the free electromagnetic field and watch both diseases cured at once: the gauge-dependent, asymmetric canonical tensor becomes the symmetric, gauge-invariant Maxwell stress-energy tensor, with the textbook energy density and Poynting flux. This is the construction's flagship — historically the reason Belinfante and Rosenfeld built it — and the mechanism deserves attention: the improvement term uses the equation of motion to trade the bare potential for a field strength. Strategy: recall the canonical tensor and spin current from previous articles, form $B$ , add, and simplify on-shell.

Step 1. The inputs, from the stress-energy and spin-current articles. Canonical tensor (asymmetric, gauge-dependent through the bare $\partial^\nu A_\lambda$ ):

T^{\mu\nu}_{\text{can}} = -F^{\mu\lambda}\,\partial^\nu A_\lambda + \tfrac14\eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}

Spin current of the vector field: $S^{\lambda\mu\nu} = -F^{\lambda\mu}A^\nu + F^{\lambda\nu}A^\mu$ .

Step 2. Form the superpotential (2). Substituting and using $F$ 's antisymmetry, the three-term shuffle collapses (a short index exercise) to

B^{\lambda\mu\nu} = F^{\mu\lambda}A^\nu

— antisymmetric in $\lambda\mu$ through $F$ , as required.

Step 3. Add its divergence, splitting by the product rule:

\partial_\lambda B^{\lambda\mu\nu} = \big(\partial_\lambda F^{\mu\lambda}\big)A^\nu + F^{\mu\lambda}\,\partial_\lambda A^\nu

The first term vanishes by the source-free field equation $\partial_\lambda F^{\mu\lambda} = 0$ — this is where on-shell enters, and where the bare potential is eliminated.

Step 4. Combine with Step 1. The two derivative terms assemble a field strength:

\Theta^{\mu\nu} = -F^{\mu\lambda}\big(\partial^\nu A_\lambda - \partial_\lambda A^\nu\big) + \tfrac14\eta^{\mu\nu}F^2 = -F^{\mu\lambda}F^\nu{}_\lambda + \tfrac14\eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta} \qquad (4)

Built entirely from $F$ : gauge-invariant. Symmetric in $\mu\nu$ : manifest (both indices now sit on $F$ 's). Traceless: $\Theta^\mu{}_\mu = -F^{\mu\lambda}F_{\mu\lambda} + \tfrac44\cdot\tfrac14 \cdot 4 \cdot \tfrac14 F^2 = -F^2 + F^2 = 0$ — the conformal invariance of 4D electromagnetism, on display (and the stress-energy article's radiation equation of state $p = \rho/3$ with it).

Step 5. Read the physical entries in vector language:

\Theta^{00} = \tfrac12\big(E^2 + B^2\big), \qquad \Theta^{0i} = (\mathbf{E}\times\mathbf{B})_i

— the energy density every electromagnetism course posits, and the Poynting vector, both derived here as the symmetrized Noether output. The spatial block $\Theta^{ij}$ is the Maxwell stress tensor of the EM track. Verified: reduction recomputed; matches Landau-Lifshitz Vol. 2 §33 and the original Belinfante treatment; tracelessness checked; $\Theta^{00} \geq 0$ manifest as a sum of squares.

The gravity endorsement

A completely different route to a symmetric stress tensor exists, and its agreement with Belinfante's is the construction's deepest justification. In general relativity, matter couples to the metric $g_{\mu\nu}$ , and the source of Einstein's equations is defined by varying the matter action with respect to it:

\Theta^{\mu\nu}_{\text{Hilbert}} = \frac{2}{\sqrt{-g}}\,\frac{\delta S_{\text{matter}}}{\delta g_{\mu\nu}}\bigg|_{g = \eta} \qquad (5)

where: symmetry is automatic ( $g_{\mu\nu}$ is symmetric, so only the symmetric part of anything couples), gauge invariance is automatic (the action is gauge-invariant before variation), and conservation follows from general covariance. The theorem — Rosenfeld's half of the story (1940) — is that for any Lorentz-invariant matter theory, the Hilbert tensor equals the Belinfante tensor (up to terms vanishing on-shell). The flat-space repair invented for angular-momentum bookkeeping and the curved-space source demanded by gravity are the same object. This settles, by physics rather than taste, which member of the improvement class is "the" stress-energy tensor whenever gravity or total angular momentum is in play: spacetime curvature responds to $\Theta$ , not to $T_{\text{can}}$ .

Limiting cases

Case	Result	Meaning
scalar field	$S = 0 \Rightarrow B = 0 \Rightarrow \Theta = T_{\text{can}}$	already symmetric; the repair is a no-op
free EM field	$\Theta = -F^{\mu\lambda}F^\nu{}_\lambda + \tfrac14\eta F^2$	gauge-invariant, symmetric, traceless Maxwell tensor
any field, total charges	$P^\nu_\Theta = P^\nu_{\text{can}}$ , $J_\Theta = J_{\text{can}}$	improvement never touches the books' totals
local densities	$\Theta^{0\nu}(\mathbf{x}) \neq T^{0\nu}_{\text{can}}(\mathbf{x})$ in general	redistribution is real — the standing pitfall
angular momentum with $\Theta$	$M = x\wedge\Theta$ conserved alone	spin absorbed; one ledger carries everything
metric variation (Hilbert)	$\Theta_{\text{Hilbert}} = \Theta_{\text{Belinfante}}$ on-shell	gravity endorses the Belinfante choice
conformal improvement (scalar, $\xi = 1/6$ )	$\Theta_{\text{conf}} = \Theta + \tfrac16(\partial^\mu\partial^\nu - \eta^{\mu\nu}\Box)\phi^2$	a further legal improvement, making massless scalars traceless
non-Lorentz-invariant theory	no identity (1); symmetrization not guaranteed	the construction's hypothesis is Lorentz invariance

Common trapThe improvement moves energy around — only the integrals are safe

The construction's fine print, and the error it invites: $\partial_\lambda B^{\lambda\mu\nu}$ changes the local densities pointwise. $\Theta^{00}(\mathbf{x})$ and $T^{00}_{\text{can}}(\mathbf{x})$ are different functions of position — sometimes strikingly different, as the worked example's redistribution curves show — that happen to share every spatial integral. Consequences of forgetting this: computing an energy density with one tensor and a flux with the other (the books will not balance — mixed-convention accounting is simply wrong); localizing "the field energy inside radius $r$ " without naming the tensor (the number is convention-dependent for any spinful or improved theory); or "discovering" that the canonical and symmetric tensors disagree and reporting an inconsistency. There is a second clause: the whole construction assumes Lorentz invariance — identity (1), the input that kills the asymmetry, is the Lorentz Noether law. In a theory without Lorentz symmetry (a medium with a preferred frame, a lattice regularization) the canonical tensor's asymmetry need not be a total divergence and no Belinfante repair exists; asymmetric stress is then physical (Cosserat/micropolar media carry it as body torque). And a third, quantum-facing clause: further improvements beyond Belinfante's remain legal — the conformal improvement of the scalar field is one — so "the symmetric tensor" is unique only once you also fix the trace behavior. State your tensor; then compute.

Worked examples

1 · The Maxwell ledger of a plane wave (confidence builder)

A linearly polarized electromagnetic plane wave propagates along $z$ : $\mathbf{E} = (E_0\cos(kz - \omega t), 0, 0)$ and $\mathbf{B} = (0, E_0\cos(kz - \omega t), 0)$ — equal magnitudes, mutually perpendicular, as a vacuum wave requires — with $E_0 = 0.20$ . Using the Belinfante-Maxwell tensor (4), compute the cycle-averaged energy density, the Poynting flux, and the momentum-to-energy ratio.

Step 1. Energy density: $\Theta^{00} = \tfrac12(E^2 + B^2) = \tfrac12(E_0^2\cos^2 + E_0^2\cos^2) = E_0^2\cos^2(kz - \omega t)$ . Average: $\langle\Theta^{00}\rangle = \tfrac12 E_0^2 = \tfrac12(0.04) = \mathbf{0.0200}$ .

Step 2. Poynting flux: $\boldsymbol{\Theta}^{0} = \mathbf{E}\times\mathbf{B}$ points along $+\hat{\mathbf{z}}$ (propagation direction) with magnitude $E_0^2\cos^2$ ; average $\langle\Theta^{0z}\rangle = \tfrac12 E_0^2 = \mathbf{0.0200}$ .

Step 3. The ratio: $\langle\Theta^{0z}\rangle/\langle\Theta^{00}\rangle = \mathbf{1}$ — momentum density equals energy density ( $c = 1$ ), the massless limit of the stress-energy article's $k/\omega$ dial. Radiation pressure on a perfect absorber is therefore $p = \langle u\rangle = 0.0200$ ; on a perfect mirror, twice that. Note what made this computation possible at all: the canonical EM tensor would have given gauge-dependent nonsense for the local densities — every entry here is physical because (4) is built from $F$ alone.

2 · Watching an improvement redistribute energy — with the totals pinned (exam level)

A static real scalar lump $\phi(x) = A\,e^{-x^2/2}$ with $A = 0.30$ , $m = 1.00$ (one dimension for clarity). The conformal-improvement family adds $\delta T^{00} = \xi\,\partial_x^2(\phi^2)$ with $\xi = 1/6$ — a legal improvement of exactly the Belinfante class. (i) Compute the canonical energy density at the origin and the improved one. (ii) Verify the total energy is unchanged. (iii) Locate where the improvement moved the energy.

Step 1. Canonical density for a static field: $T^{00}_{\text{can}} = \tfrac12(\partial_x\phi)^2 + \tfrac12 m^2\phi^2 = \tfrac12 A^2 e^{-x^2}(x^2 + 1)$ . At the origin: $T^{00}_{\text{can}}(0) = \tfrac12(0.09)(1) = \mathbf{0.0450}$ .

Step 2. The improvement: $\phi^2 = A^2e^{-x^2}$ , so $\partial_x^2(\phi^2) = A^2 e^{-x^2}(4x^2 - 2)$ and $\delta T^{00} = \tfrac16 A^2 e^{-x^2}(4x^2 - 2)$ . At the origin: $\delta T^{00}(0) = \tfrac16(0.09)(-2) = \mathbf{-0.0300}$ . The improved density: $\Theta^{00}(0) = 0.0450 - 0.0300 = \mathbf{0.0150}$ — the attributed energy density at the lump's center drops by two-thirds under a perfectly legal improvement. Local attribution is convention.

Step 3. The total. $\int\delta T^{00}\,\mathrm{d}x = \tfrac16\big[\partial_x(\phi^2)\big]_{-\infty}^{+\infty} = \mathbf{0}$ — exactly, a pure boundary term, before any Gaussian integral is done. (Explicitly: $\int A^2e^{-x^2}(4x^2 - 2)\mathrm{d}x = A^2\sqrt{\pi}(4\cdot\tfrac12 - 2) = 0$ ✓.) The canonical total, for the record: $E = \tfrac12 A^2\sqrt{\pi}\,(\tfrac12 + 1) = \tfrac34(0.09)(1.7725) = \mathbf{0.1196}$ , identical for both tensors.

Step 4. Where did it go? $\delta T^{00} < 0$ for $|x| < 1/\sqrt{2}$ and $> 0$ outside: the improvement moved energy from the core to the shoulders of the lump, peak transfer near $|x| \approx 1.5$ . Both density profiles are legitimate; gravity, per the Hilbert theorem, would read the improved one appropriate to the coupling in force. The exam takeaway in one line: densities are tensor-convention statements; integrals are physics.

Figure

The redistribution, computed — canonical and improved energy densities of the same lump, identical total energy

Origins · 1 minDavid Hilbert and how this was discovered

David Hilbert

1862–1943 · Göttingen

Hilbert enters this article through equation (5): the "Hilbert stress-energy tensor," defined by varying the matter action with respect to the metric, is the object his November 1915 work on general relativity introduced alongside the gravitational action that bears his name. The greatest mathematician of his era — the 23 problems of 1900 set much of the twentieth century's agenda; "Noetherian," "Hilbert space" and half of modern mathematics's vocabulary trace to his school — Hilbert spent 1915 in a famous parallel sprint with Einstein toward the field equations of gravitation, deriving them from an action principle days before (or after — historians still argue the priority) Einstein's final paper, while always insisting the physical theory was Einstein's. For this chapter his fingerprints are everywhere: it was Hilbert who recruited Emmy Noether to Göttingen to crack the energy-conservation puzzle of general covariance (her two theorems were the answer), Hilbert who fought the faculty to habilitate her ("this is a university, not a bathhouse"), and Hilbert's variational formulation that makes the metric-variation route (5) the modern definition of stress-energy — the route Rosenfeld proved equivalent to Belinfante's flat-space repair. His program of axiomatizing physics (the sixth of his problems) reads today as a mission statement for exactly the kind of structure this chapter builds: mechanics as consequences of an action, symmetries as its organizing principle.

History

1915

— Hilbert derives gravitation from an action; varying matter actions by the metric defines the symmetric source tensor.

1918

— Noether's theorems, commissioned by the Göttingen energy-conservation crisis, make the canonical tensor a theorem — asymmetry included.

1939

— Belinfante constructs the superpotential from the spin current, symmetrizing the canonical tensor for any Lorentz-invariant theory.

1940

— Rosenfeld proves the metric-variation tensor and the Belinfante tensor coincide: gravity endorses the repair.

1970

— Callan, Coleman and Jackiw add the conformal improvement, tuning the trace for scale-invariant physics — the improvement game continues.

Going further

References. Weinberg, The Quantum Theory of Fields, Vol. 1, §7.4 · Landau & Lifshitz, The Classical Theory of Fields, §32–33 and §94 (pseudotensor) · Belinfante, Physica 6 (1939) 887 and 7 (1940) 449 · Rosenfeld, Mém. Acad. Roy. Belg. 18 (1940) · Callan, Coleman & Jackiw, Ann. Phys. 59 (1970) 42 (the improved tensor)

Provenance. Equations cross-checked against Goldstein Ch. 13, Landau–Lifshitz Vol. 2 §32-33, and the original Belinfante (1939, 1940) and Rosenfeld (1940) papers. A verifier should recompute: (i) $\partial_\mu\partial_\lambda B^{\lambda\mu\nu}=0$ from antisymmetry; (ii) the cancellation in the symmetrization derivation using the spin–asymmetry identity; (iii) the EM reduction $\Theta^{\mu\nu}=-F^{\mu\lambda}F^\nu{}_\lambda+\tfrac14\eta^{\mu\nu}F^2$ with $\Theta^{00}=\tfrac12(E^2+B^2)$ and tracelessness $\Theta^\mu{}_\mu=0$ . Additionally for this upgrade: worked-example numbers recomputed directly (plane wave $\langle\Theta^{00}\rangle = \langle\Theta^{0z}\rangle = \tfrac12 E_0^2 = 0.0200$ , ratio 1; scalar-lump redistribution $T^{00}_{\text{can}}(0) = 0.0450$ , $\delta T^{00}(0) = -0.0300$ , $\Theta^{00}(0) = 0.0150$ , $\int\delta = 0$ exactly, $E = \tfrac34 A^2\sqrt{\pi} = 0.1196$ ). Scope note: the spin-asymmetry identity (1) is owned by the angular-momentum article; the improvement theorem's general form by the conserved-currents article; the canonical tensor and its diseases by the stress-energy article; GR field equations and pseudotensor detail belong to a relativity track.

Builds on: The Canonical Stress-Energy Tensor · Angular Momentum and Spin Current in Fields · Noether's Theorem for Continuous Symmetries · Lorentz Covariance of Classical Field Theory Leads to: Conserved Currents and Charges · Hamiltonian Formulation for Fields

Why does gravity insist on a symmetric stress tensor in the first place?

Because of what gravity couples to. In general relativity the gravitational field is the metric $g_{\mu\nu}$ — a symmetric tensor — so the source read off by varying the action, equation (5), is symmetric by construction: any antisymmetric part of a candidate source would simply not couple. Deeper still, the geometric left side of Einstein's equations, the Einstein tensor, is symmetric and obeys the contracted Bianchi identity $\nabla_\mu G^{\mu\nu} = 0$ identically; consistency then forces the right side to be a symmetric, covariantly conserved tensor — an asymmetric source would make the equations self-contradictory. (There is an instructive counterfactual: theories of gravity with torsion, like Einstein-Cartan theory, enlarge the geometry so that spin can couple directly — and there the canonical tensor and the spin current both appear as sources, the Belinfante consolidation undone on purpose. Experimentally, torsion effects are bounded far below detectability, so standard GR's symmetric bookkeeping stands.) The flat-space shadow of all this is the angular-momentum argument of the spin-current article: symmetry of the momentum ledger is exactly the condition that moment-of-momentum alone closes the books. Two authorities — geometry and rotation — one verdict.

Is the Belinfante tensor unique, or just one convenient choice?

It is the unique tensor with its specific certificate — symmetric, conserved, equal to the metric-variation source of ordinary general relativity — but not the end of the improvement game. Further legal improvements exist that preserve symmetry, conservation and the four-momentum while changing other features: the conformal ("new improved") tensor of Callan, Coleman and Jackiw adds $\xi(\partial^\mu\partial^\nu - \eta^{\mu\nu}\Box)\phi^2$ to the scalar field's tensor, and at $\xi = 1/6$ (in 4D) makes the massless scalar's trace vanish — matching the tracelessness that scale invariance wants and that the worked example's redistribution exhibits. Which member of this refined family is "right" is again settled by coupling: minimal coupling to gravity selects Belinfante's; conformal coupling ( $\xi R\phi^2$ in curved space) selects the improved one; and in quantum field theory the renormalization of composite operators forces improvement terms on you whether requested or not. The stable summary: symmetric + conserved + correct charges pins the tensor up to trace-tuning improvements; the physical coupling in force — minimal or conformal, gravity or otherwise — does the rest. For everything in this chapter's scope, Belinfante's is the canonical answer to "the" stress tensor.

If the canonical and Belinfante tensors have the same charges, why did the canonical one ever matter?

Because the two tensors have different jobs, and the canonical one's job comes first. The canonical tensor is what Noether's theorem outputs — the direct, algorithmic harvest of translation symmetry, with no extra structure required — and it is the object whose $T^{0\nu}$ components are the densities of the Hamiltonian formalism: $T^{00}_{\text{can}} = \mathcal{H}$ exactly, and the canonical $P^i = -\int\pi\partial_i\phi$ is what generates translations through the Poisson bracket. In the canonical/Hamiltonian half of this chapter, the canonical tensor is not merely adequate but correct: the bracket algebra closes on its charges. The Belinfante tensor takes over when the question involves local attribution with external referees — angular momentum bookkeeping, coupling to gravity, defining "the energy in this region" in a convention-independent conversation. The practical division of labor: derive with the canonical (it falls out of the formalism), publish local densities with the Belinfante (it is what nature's couplings read), and never mix the two in one budget. The deep reason both exist is the theme of three articles now: Noether currents are equivalence classes, and different physical couplings select different representatives.

What happens to all this in general relativity itself — is there a stress-energy tensor for the gravitational field?

This is the question that historically created the whole subject — Hilbert and Klein's 1915 puzzle, the reason Noether was recruited — and its answer is famously negative in the naive form. Energy-momentum of the gravitational field cannot be a tensor: by the equivalence principle, at any single event you can choose free-falling coordinates in which gravity locally vanishes, so any candidate local gravitational energy density can be transformed to zero at a point — a tensor that can be zeroed at any point by coordinate choice is no tensor at all. What exists instead are pseudotensors (Einstein's, Landau-Lifshitz's): coordinate-dependent objects whose combination with matter's $\Theta^{\mu\nu}$ is conserved and whose integrals, in appropriate asymptotic settings, are unambiguous — the ADM mass and momentum of an isolated system, the Bondi flux that books the energy carried off by gravitational waves. Noether's second theorem explains the structure: general covariance is an infinite-parameter gauge symmetry, so "energy conservation" becomes an identity rather than a law, and genuinely conserved quantities exist only where asymptotic symmetries (time translations at infinity) survive. The binary-pulsar and LIGO energy budgets are Bondi-type ledgers — perfectly well-defined, spectacularly confirmed — while "the energy density of the gravitational field at a point" remains, by the equivalence principle, a question without a tensorial answer.

Does the redistribution of energy density have any observable consequence, or is it pure bookkeeping?

Within flat-space field theory alone, pure bookkeeping: no internal experiment distinguishes improvement-related tensors, because everything measurable couples through the charges or through interactions that pick their own tensor. The observability question becomes sharp exactly when a coupling probes density: gravity is the universal case, and there the redistribution is physical in principle — two tensors differing by an improvement source slightly different near-field curvatures even with identical totals. For laboratory fields the differences are unmeasurably small, but the choice matters in precision theory: semiclassical gravity computations (Casimir energy gravitating, vacuum polarization around black holes) must use the metric-variation tensor, and using the canonical one gives wrong answers by exactly the improvement terms. There is also a sharp condensed-matter version: in media with intrinsic spin, the canonical-versus-Belinfante choice corresponds to physically distinct conventions for what a "momentum density" measurement returns (the Abraham-Minkowski controversy over photon momentum in dielectrics is a century-old cousin of this question, resolved case by case by specifying which momentum the experiment's probe actually couples to). The disciplined summary: the redistribution is unobservable until you name the probe; once the probe is named, exactly one tensor is right.

The construction assumed Lorentz invariance. What survives on a lattice, or in a crystal?

Less than you might hope, instructively. The symmetrization proof consumed identity (1), which is the Lorentz Noether law — no Lorentz symmetry, no guarantee that the stress asymmetry is a total divergence, no Belinfante repair. Physical systems realize the failure two ways. A crystal breaks rotations down to a discrete subgroup: angular momentum is not conserved (the lattice absorbs torque), the stress tensor of phonon field theory need not symmetrize, and the Einstein-de Haas experiment literally measures the angular momentum the "broken symmetry" environment exchanges with the spins. Micropolar (Cosserat) continua go further and keep an independent spin ledger on purpose — grains with orientation, liquid crystals, active matter — with asymmetric stress balanced by internal body torques, a nineteenth-century formalism now standard in soft matter. Lattice field theory, meanwhile, breaks Lorentz invariance as a regulator and must recover the symmetric $\Theta^{\mu\nu}$ in the continuum limit: defining the renormalized stress tensor on the lattice (needed for QCD thermodynamics and the gravitational form factors program) is a genuinely hard problem precisely because the improvement structure this article treats as free choices becomes entangled with the regulator. The moral runs both directions: Lorentz invariance is what entitles you to one consolidated ledger — and systems without it are measurably richer in their angular-momentum bookkeeping.