Symmetry, Conservation, and Stress-Energy

A continuous symmetry produces a conserved current only through a variational statement with definite boundary conditions. Stress-energy then packages four translation currents, but there is more than one useful construction: the canonical Noether tensor, a Belinfante or other improved tensor, and the Hilbert tensor obtained by varying a background metric answer related but distinct questions.

Throughout this chapter spacetime is four-dimensional Minkowski space with signature $(-,+,+,+)$ and $c=1$. Greek indices run over spacetime components. Curved metrics appear only as nondynamical backgrounds used to define metric response; the Einstein equations, gravitational energy, and asymptotic charges remain outside scope.

Helpful background. Calculus and Linear Algebra for Relativity supplies the local differential manipulations; Classical Fields and Their Actions supplies the variational boundary term; Events, Intervals, and Minkowski Geometry fixes spacetime conventions; the covariant phase-space seam is developed separately in Presymplectic Potentials and Currents.

From symmetry to a local statement

For fields $\phi^a$ with a differentiable local action, the first variation separates into Euler–Lagrange expressions and a boundary term. If a finite-dimensional continuous transformation changes the Lagrangian density only by a divergence, that decomposition gives an off-shell identity,

$$\partial_\mu j^\mu=-E_a(\phi)\,\Delta\phi^a.$$

The field equations $E_a=0$ turn it into local conservation. A time-independent integrated charge requires an additional global condition: convergence of the spatial integral and no unaccounted flux through its boundary. These logical steps are derived carefully in Noether’s First Theorem, following the current construction in Tong, §1.3.1, printed pp. 13–14 and the finite-group theorem in Noether’s original paper, §1, Theorem I.

Three energy-momentum constructions

Translation invariance gives the canonical tensor. It is naturally tied to the chosen fields and Lagrangian, so it may be asymmetric or gauge dependent. Adding the divergence of an antisymmetric superpotential preserves its local divergence identically; the Belinfante choice uses spin current to produce a symmetric tensor on shell. Neither an arbitrary improvement nor the Belinfante construction should be renamed “the Hilbert tensor.”

The Hilbert tensor instead measures metric response:

$$T_{\mu\nu} =-\frac{2}{\sqrt{-g}}\frac{\delta S_{\rm m}}{\delta g^{\mu\nu}}.$$

The inverse-metric definition and its scalar and electromagnetic examples are developed in Tong 2019, §§4.5.2–4.5.4, printed pp. 183–187. For standard minimally covariantized Poincaré-invariant matter theories, the Hilbert and Belinfante tensors agree on shell up to the familiar ambiguities, but the assumptions matter. Pons, §§4.1–4.4 and the Appendix gives a detailed account of their different roles.

Construction	Starting input	What follows directly	What still needs conditions
Canonical	Translation quasi-symmetry of a chosen flat-space Lagrangian	On-shell local divergence equation	Symmetry, gauge independence, and invariance under changes of Lagrangian representative
Superpotential improvement	$T^{\mu\nu}\mapsto T^{\mu\nu}+\partial_\rho B^{\rho\mu\nu}$ with $B^{\rho\mu\nu}=-B^{\mu\rho\nu}$	The divergence is unchanged identically	Equality of charges requires the surface term to vanish or be included
Belinfante	Lorentz component current in a Poincaré-invariant theory	Symmetric tensor on shell, with spin absorbed into orbital form	Boundary falloff and the appropriate field representation
Hilbert	Functional response to a specified background-metric extension	Symmetric tensor off shell	Matter equations for conservation; a covariantization choice for comparison with Belinfante

The rows are related constructions, not interchangeable names. The scalar example makes canonical and Hilbert tensors equal directly; the Maxwell example needs an on-shell improvement; nonminimal curvature couplings can alter the metric response without altering the flat-space equations.

Reading path

The chapter proceeds from theorem to measurement:

• Noether’s First Theorem states the hypotheses, proves the off-shell identity, and separates local conservation from global charge conditions.
• Canonical Energy-Momentum and Improvement derives translation currents and isolates superpotential ambiguities.
• Flat-Background Hilbert Stress-Energy from Metric Variation defines the response tensor and checks its scalar-field energy density and divergence.
• Local Balance Laws and Flux Integrals fixes the outward-flux and hypersurface-orientation signs.
• Angular Momentum, Spin Currents, and Symmetric Stress-Energy explains how intrinsic spin is incorporated by Belinfante improvement.
• Stress-Energy Tensors for Particles, Fields, and Fluids compares what each algebraic form assumes about matter.

Check your understanding

Regional charge

A current satisfies $\partial_\mu j^\mu=0$. For a fixed spatial region $V$, determine the evolution of $Q_V(t)=\int_V j^0\,d^3x$ and state what extra condition makes it a conserved charge.

Solution

Integrating the local equation and applying the divergence theorem gives

$$\frac{dQ_V}{dt} =-\int_V\boldsymbol\nabla\cdot\mathbf j\,d^3x =-\oint_{\partial V}\mathbf j\cdot\mathbf n\,dA.$$

Thus a locally conserved current can carry charge out of a finite region. The regional charge is constant only when the net boundary flux vanishes. For all space, one must additionally require that the integral converges and that the flux at spatial infinity vanishes. Local conservation is the differential input; boundary behavior turns it into a conserved global number.

Improvement and boundary data

Let

$$T'^{\mu\nu}=T^{\mu\nu}+\partial_\rho B^{\rho\mu\nu}, \qquad B^{\rho\mu\nu}=-B^{\mu\rho\nu}.$$

Which statement is automatic: equality of local divergences or equality of integrated four-momenta? State the extra condition needed for the other statement.

Solution

Commuting derivatives and antisymmetry give

$$\partial_\mu T'^{\mu\nu} =\partial_\mu T^{\mu\nu} +\partial_\mu\partial_\rho B^{\rho\mu\nu} =\partial_\mu T^{\mu\nu}$$

identically, without equations of motion. On a constant-time slice, however,

$$P'^\nu-P^\nu =\int_{\partial\Sigma}B^{i0\nu}n_i\,dA.$$

The four-momenta agree only when this surface integral vanishes—for example by compact support, sufficient falloff, or periodic cancellation—or when the boundary contribution is explicitly included in the system. Local divergence equivalence is algebraic; global charge equivalence is boundary-sensitive.

Boundaries of the framework

Noether’s first theorem concerns finite-dimensional continuous quasi-symmetries, not arbitrary gauge functions; the latter lead to differential identities under Noether’s second theorem. Local stress-energy conservation also does not manufacture a coordinate-independent total energy in generic curved spacetime. Those distinctions prevent a flat-space translation charge, a material balance law, and a gravitational charge from being conflated.

Continue to Waves, Optics, and Radiation to apply field equations and local flux balance to propagation, scattering, and radiation.

References

• Emmy Noether, “Invariant Variation Problems,” Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 235–257 (1918), English translation by M. A. Tavel, §1, Theorem I.
• Josep M. Pons, “Noether Symmetries, Energy-Momentum Tensors and Conformal Invariance in Classical Field Theory,” Journal of Mathematical Physics 52, 012904 (2011), doi:10.1063/1.3532941, Open PDF, §§2.2.1 and 4.1–4.4, printed pp. 5–6 and 11–18.
• David Tong, General Relativity, University of Cambridge Part III lecture notes (2019), official PDF, §§4.5.2–4.5.4, printed pp. 183–187.
• David Tong, Quantum Field Theory, University of Cambridge Part III lecture notes (2006/2007), official PDF, §§1.3.1–1.3.3, printed pp. 13–18.