Diffeomorphism Noether Identity and Bianchi Consistency

Diffeomorphism invariance produces a differential identity among Euler–Lagrange expressions. The identity holds off shell: it is true before the metric or matter equations are imposed. The contracted Bianchi identity is the pure-metric instance, while covariant stress-energy conservation is the matter-shell consequence. Treating conservation as an extra independent Einstein equation double-counts the symmetry.

Required background. We use Lie derivatives and Killing geometry, Einstein’s equation, and the completed gravitational action.

Helpful background. The derivation draws on special relativity, tensor fields and integration by parts, and presymplectic potentials.

A scalar-matter derivation

For clarity take an action $S[g,\phi]$ and define its bulk Euler–Lagrange expressions by

$$\delta S =\int_{\mathcal M}\sqrt{-g} \left(\frac12\mathcal E^{ab}\delta g_{ab} +\mathcal E_\phi\delta\phi\right)d^4x +\int_{\partial\mathcal M}\Theta.$$

Let $\xi^a$ have compact support so the boundary term vanishes. Under the active-pullback convention,

$$\delta_\xi g_{ab}=2\nabla_{(a}\xi_{b)}, \qquad \delta_\xi\phi=\xi^a\nabla_a\phi.$$

Diffeomorphism invariance gives

$$0=\int\sqrt{-g} \left(\mathcal E^{ab}\nabla_a\xi_b +\mathcal E_\phi\xi^a\nabla_a\phi\right)d^4x.$$

Integrating the first term by parts and using arbitrary compactly supported $\xi_b$ yields the off-shell identity

$$\nabla_a\mathcal E^{ab} -\mathcal E_\phi\nabla^b\phi=0.$$

For tensor or gauge matter, the second term is replaced by the appropriate formal adjoint of the Lie-derivative generator acting on every matter Euler–Lagrange expression. The principle is unchanged.

Reall derives the link between diffeomorphism invariance, metric variation, and conservation Reall 2013, §§ 7.2 and 10.3, pp. 71–76 and 119–122, PDF.

Pure gravity and the Bianchi identity

For the Einstein–Hilbert action,

$$\mathcal E^{ab}_{\mathrm{grav}} =-\frac{1}{8\pi G}(G^{ab}+\Lambda g^{ab})$$

with the direct-metric variation convention used above. There is no matter Euler–Lagrange term, so the Noether identity becomes

$$\nabla_a(G^{ab}+\Lambda g^{ab})=0.$$

Metric compatibility and constant $\Lambda$ reduce this to $\nabla_aG^{ab}=0$, the contracted differential Bianchi identity. It is valid for every smooth metric; no Einstein equation is required.

This identity explains why four combinations of the ten metric equations are differentially related in four dimensions. It is the covariant origin of constraint propagation, although a well-posed formulation requires additional decomposition and gauge choices.

Matter conservation on shell

For a diffeomorphism-invariant matter action varied with respect to $g_{ab}$,

$$\delta S_{\mathrm m} =\int\sqrt{-g} \left(\frac12T^{ab}\delta g_{ab} +\mathcal E_\phi\delta\phi\right)d^4x+\text{boundary}.$$

The identity gives

$$\nabla_aT^{ab}=\mathcal E_\phi\nabla^b\phi.$$

Thus $\nabla_aT^{ab}=0$ follows when the scalar equation $\mathcal E_\phi=0$ holds. Off shell, the divergence need not vanish. For the canonical scalar one can check directly that

$$\nabla_aT^{ab}=(\Box\phi-V'(\phi))\nabla^b\phi,$$

with exactly this sign: for $S_{\mathrm m}=-\int\sqrt{-g}[\tfrac12(\nabla\phi)^2+V(\phi)]\,d^4x$, integration by parts gives $\mathcal E_\phi=\Box\phi-V'(\phi)$ under the definition above. The direct calculation verifies the Noether identity and prevents an off-shell conservation claim.

Boundaries and large transformations

Compact support made the derivation an identity. If $\xi^a$ acts at a boundary, integration by parts leaves a surface term. Depending on boundary conditions, it can be a flux, a Hamiltonian variation, or an obstruction to differentiability. Discarding it would incorrectly classify a physical symmetry as gauge.

The same distinction explains why Noether’s first and second theorems play different roles. A finite-dimensional rigid symmetry produces a current after the equations of motion are used. A gauge symmetry depending on arbitrary spacetime functions instead produces differential relations among the equations themselves. Boundary conditions can convert selected gauge parameters into genuine symmetry generators, but that conversion is additional structure.

In practical calculations the identity is a diagnostic. If independently derived metric and matter equations violate it, a sign, connection term, or boundary contribution has been lost. Conversely, satisfying the identity does not prove that every equation is correct; it verifies only the symmetry-imposed relation among them.

Carroll relates the Bianchi identity to consistency of Einstein’s equation and covariant matter conservation Carroll 1997, Chapter 4, pp. 109–128, PDF. The equality of bulk divergences is necessary, but conserved global charges additionally require a symmetry and controlled boundary flux.

Exercises

For a canonical scalar with $T_{ab}=\nabla_a\phi\nabla_b\phi-g_{ab}[\tfrac12(\nabla\phi)^2+V(\phi)]$, compute $\nabla^aT_{ab}$.

Solution

Expanding the divergence gives $(\Box\phi)\nabla_b\phi+\nabla_a\phi\nabla^a\nabla_b\phi-\nabla_b[\tfrac12(\nabla\phi)^2]-V'(\phi)\nabla_b\phi$. Covariant derivatives commute on a scalar, so the middle two derivative terms cancel. Hence

$$\nabla^aT_{ab}=(\Box\phi-V'(\phi))\nabla_b\phi,$$

which vanishes on the scalar equation of motion.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013). PDF.