Diffeomorphism Noether Currents and Charges

A diffeomorphism generated by a vector field $\xi^a$ has a Noether current even though diffeomorphisms are gauge transformations in the bulk. Diffeomorphism covariance makes that current a sum of field-equation constraints and an exact form. On shell the constraint term vanishes, so the current is locally the exterior derivative of an $(n-2)$-form $Q_\xi$. The integral of $Q_\xi$ alone is not automatically the Hamiltonian charge; boundary variations and integrability still have to be included.

Required background. The Gibbons–Hawking–York Boundary Term supplies the action and its surface variation; Presymplectic Potentials and Currents supplies field-space differential forms; and Killing Fields and Isometry Groups supplies the diffeomorphism generator and exact symmetries.

Helpful background. The Einstein Equation supplies the bulk equations; Hypersurface Normals and Tangential Projectors supplies surface orientation; and Formulations of General Relativity locates Noether charges within this volume.

Current associated with a diffeomorphism

Start from

$$\delta L=E\!\cdot\!\delta\phi+d\Theta(\phi;\delta\phi).$$

For the field variation $\delta_\xi\phi=\mathcal L_\xi\phi$, covariance of the Lagrangian $n$-form gives

$$\delta_\xi L=\mathcal L_\xi L=d(\xi\mathbin{\lrcorner}L).$$

The diffeomorphism Noether current is therefore

$$J_\xi =\Theta(\phi;\mathcal L_\xi\phi) -\xi\mathbin{\lrcorner}L.$$

Taking an exterior derivative and comparing the two variations yields

$$dJ_\xi=-E\!\cdot\!\mathcal L_\xi\phi.$$

Thus $J_\xi$ is closed when the equations of motion hold. Iyer and Wald derive these identities in Iyer and Wald 1994, §4, eqs. (49)–(51), pp. 10–11, PDF.

Constraints plus an exact form

Noether’s second theorem gives the stronger off-shell organization

$$J_\xi=C_\xi+dQ_\xi,$$

where $C_\xi$ is linear in the Euler–Lagrange expressions and vanishes on solutions. In a 3+1 decomposition, its lapse and shift coefficients become the Hamiltonian and momentum constraints. Hence

$$J_\xi\doteq dQ_\xi,$$

where $\doteq$ means equality on shell. Exactness here is local in spacetime; global cohomology can add closed representatives, and $Q_\xi$ itself is defined up to an exact $(n-2)$-form.

For Einstein gravity with $L=(R-2\Lambda)\epsilon/(16\pi G)$, a standard representative is

$$(Q_\xi)_{ab} =-\frac{1}{16\pi G}\epsilon_{abcd}\nabla^c\xi^d.$$

This is the Komar two-form with the normalization arising from the Einstein–Hilbert Lagrangian. It depends on the first derivative of $\xi^a$, not only on the value of the vector at the surface. The explicit Einstein current and charge appear in Iyer and Wald 1994, §5, eqs. (61)–(64), pp. 12–13, PDF.

From Noether form to Hamiltonian variation

Let $\delta g$ satisfy the linearized equations and keep $\xi^a$ field independent. Combining the current variation with the presymplectic current gives

$$\omega(g;\mathcal L_\xi g,\delta g) =d\left(\delta Q_\xi -\xi\mathbin{\lrcorner}\Theta(g;\delta g)\right)$$

on shell, with the site’s ordering $\omega(\delta_1,\delta_2)=\delta_2\Theta(\delta_1)- \delta_1\Theta(\delta_2)$. The order of the two arguments is essential. For a covariant potential and fixed $\xi$, Cartan’s identity gives the short derivation

$$\begin{aligned} \delta J_\xi &=\delta\Theta(\mathcal L_\xi g)-\xi\mathbin{\lrcorner}d\Theta(\delta g),\\ \delta J_\xi-d(\xi\mathbin{\lrcorner}\Theta) &=\delta\Theta(\mathcal L_\xi g)-\mathcal L_\xi\Theta(\delta g)\\ &=\omega(g;\mathcal L_\xi g,\delta g). \end{aligned}$$

Thus, with this ordering, define the generator by $\delta H_\xi=\Omega_\Sigma(\mathcal L_\xi g,\delta g)$. Integrating over $\Sigma$ gives the candidate Hamiltonian variation

$$\delta H_\xi =\int_{\partial\Sigma} \left(\delta Q_\xi -\xi\mathbin{\lrcorner}\Theta\right),$$

after fixing the compatible orientation. This expression shows why $\int Q_\xi$ is generally incomplete: the potential term is required, and the result must be an exact one-form on the allowed solution space before a finite $H_\xi$ exists.

If $\xi^a$ is an exact symmetry, $\mathcal L_\xi g=0$, then the bulk presymplectic pairing vanishes. The boundary integrals on distinct components must balance. This is the mechanism behind relations between a charge at infinity and a horizon contribution, but no black-hole thermodynamic law is assumed on this page. Wald identifies the horizon Noether charge entering the stationary first law in Wald 1993, pp. R3428–R3430.

Ambiguities and boundaries

The shifts $L\mapsto L+d\mu$ and $\Theta\mapsto\Theta+\delta\mu+dY$ induce corresponding shifts in $J_\xi$ and $Q_\xi$. An exact shift of $Q_\xi$ vanishes on a closed integration surface, but corner terms, nonclosed cuts, or field-dependent generators can retain it. Physical comparison therefore requires the same Lagrangian representative, potential prescription, boundary conditions, and generator normalization.

Common pitfalls

On-shell exactness is local. It does not declare every integrated charge zero; the integral is over a codimension-two boundary where $Q_\xi$ need not be globally exact.

Noether charge is not automatically Hamiltonian charge. The $-\xi\mathbin{\lrcorner}\Theta$ term and integrability condition are essential.

A vector normalization is physical boundary data. Rescaling an asymptotic time translation or horizon generator rescales its charge.

Exercise: exact symmetry balance

Let $\xi^a$ be an exact symmetry and let $\delta g$ satisfy the linearized equations. Suppose $\Sigma$ has two closed boundary components $S_1$ and $S_2$. Show the relation between their infinitesimal charge forms.

Solution

Because $\mathcal L_\xi g=0$, the left side of the fundamental identity vanishes, and therefore

$$d\left(\delta Q_\xi -\xi\mathbin{\lrcorner}\Theta\right)=0.$$

Stokes’ theorem on $\Sigma$ gives

$$\int_{S_2}(\delta Q_\xi-\xi\mathbin{\lrcorner}\Theta) -\int_{S_1}(\delta Q_\xi-\xi\mathbin{\lrcorner}\Theta)=0,$$

where the minus sign reflects the induced orientation of the inner component. The two charge variations agree after both surfaces are assigned a common outward-normal convention.

References

• Iyer, Vivek, and Robert M. Wald. “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy.” Physical Review D 50, no. 2 (1994): 846–864. DOI; Open PDF.
• Wald, Robert M. “Black Hole Entropy Is the Noether Charge.” Physical Review D 48, no. 8 (1993): R3427–R3431. DOI.