Charge Integrability and Ambiguities

A surface formula first gives an infinitesimal charge one-form on the space of solutions. It defines a finite charge only if that one-form is integrable. This is a field-space question, distinct from finiteness of the surface integral and from conservation under moving the surface. Even after integrability holds, a charge retains a reference constant and can depend on the Lagrangian, presymplectic-potential, corner, and generator prescriptions allowed by the boundary problem.

Required background. The Gibbons–Hawking–York Boundary Term supplies the differentiable action; Presymplectic Potentials and Currents supplies field-space exterior derivatives; and Killing Fields and Isometry Groups supplies symmetry generators.

Helpful background. The Einstein Equation supplies the solution space; Hypersurface Normals and Tangential Projectors supplies surface orientation; and Formulations of General Relativity orients this chapter.

Charge variation as a field-space one-form

For a field-independent vector $\xi^a$ and a closed cut $S$, define the candidate charge variation

$$\not\!\delta H_\xi[\delta\phi;\phi] =\int_S k_\xi[\delta\phi;\phi],$$

where in the Iyer–Wald prescription

$$k_\xi=\delta Q_\xi -\xi\mathbin{\lrcorner}\Theta(\phi;\delta\phi).$$

The slash is a warning: the right side is linear in $\delta\phi$, but it need not equal the variation of a function $H_\xi[\phi]$. Compère formulates this distinction and the field-space closure test in Compère 2019, §1.4.1–§1.4.2, eqs. (1.73)–(1.75), pp. 22–23, PDF.

Integrability condition

For two commuting tangent variations, local integrability requires

$$\delta_1\!\left(\not\!\delta H_\xi[\delta_2]\right) -\delta_2\!\left(\not\!\delta H_\xi[\delta_1]\right)=0.$$

Equivalently,

$$\int_S \left[ \delta_1 k_\xi(\delta_2) -\delta_2 k_\xi(\delta_1) \right]=0.$$

For a fixed generator, fixed integration cut, and the displayed potential prescription, the relation is exact. The second variations of $Q_\xi$ cancel, leaving

$$\begin{aligned} &\delta_1\!\left(\not\!\delta H_\xi[\delta_2]\right) -\delta_2\!\left(\not\!\delta H_\xi[\delta_1]\right)\\ &\qquad=\int_S\xi\mathbin{\lrcorner}\omega(\delta_1,\delta_2). \end{aligned}$$

The contraction converts the spacetime $(n-1)$-form $\omega$ into an $(n-2)$-form that can be integrated on $S$. This is an integrability obstruction on solution space. It is distinct from $\int_{\mathcal B}\omega$ through a time interval, which tests slice independence. Moving cuts, corner modifications, or field-dependent generators require their corresponding additional terms.

Subject to the usual smooth local description of solution space, closure implies a local primitive on a contractible patch. A single-valued primitive on a larger connected region also requires all periods around closed loops to vanish; simple connectedness is a sufficient condition. Field-dependent generators require extra terms from $\delta\xi$ and an adjusted bracket; the fixed-$\xi$ formula cannot be applied unchanged.

When the condition holds, choose a reference solution $\bar\phi$ and a path $\gamma$ to $\phi$:

$$H_\xi[\phi;\bar\phi] =\int_\gamma\not\!\delta H_\xi+N_\xi[\bar\phi].$$

Path independence follows when those periods vanish in the chosen region. The constant $N_\xi[\bar\phi]$ is not fixed by the variation and must be set by a physical normalization, such as zero charge for a reference background.

Independent rectangle check

Suppose solutions depend on parameters $(a,b)$ and

$$\not\!\delta H=A(a,b)\,da+B(a,b)\,db.$$

Integrating around a small rectangle gives

$$\oint\not\!\delta H =\left(\partial_aB-\partial_bA\right)\Delta a\,\Delta b +O(\Delta^3).$$

Thus $\partial_aB=\partial_bA$ is the explicit integrability check. For $A=b$ and $B=0$, the integral along the path that changes $a$ then $b$, minus that along the path that changes $b$ then $a$, is $-\Delta a\Delta b$. No single-valued charge function has that variation.

Ambiguities to track

Five changes are frequently conflated:

1. Lagrangian ambiguity: $L\mapsto L+d\mu$ leaves bulk equations fixed but shifts the potential and Noether representative.
2. Potential ambiguity: $\Theta\mapsto\Theta+dY$ changes the current and surface form by corner terms.
3. Charge-form ambiguity: $Q_\xi\mapsto Q_\xi+dZ_\xi$ is invisible on a closed smooth cut but can matter at corners or punctures.
4. Reference ambiguity: integrating a variation determines $H_\xi$ only up to a constant on each connected component.
5. Generator ambiguity: rescaling or making $\xi$ field dependent changes the quantity being generated and may alter integrability.

Iyer and Wald derive the covariant $L$, $\Theta$, and $Q$ ambiguities in Iyer and Wald 1994, §§3–4, eqs. (40)–(43) and (58)–(60), pp. 8–12, PDF. An allowed ambiguity is not automatically harmless: its surface contribution must vanish, be fixed, or be incorporated into the chosen boundary phase space.

Finiteness, conservation, and integrability

These are separate gates:

Property	Question
Finiteness	Does the surface integral converge?
Integrability	Is the variation an exact one-form on solution space?
Conservation	Is the charge unchanged between homologous cuts?

A charge can be finite and integrable but nonconserved because radiation crosses the boundary. It can be finite and conserved along time yet nonintegrable across the full family of solutions. A sound result states all three properties and the boundary conditions under which they hold.

Common pitfalls

A formula labeled $\delta H$ has not proved that $H$ exists. Antisymmetrize two independent field variations.

Changing the reference is not a local counterterm. It shifts a finite charge by a constant without changing its variation.

Integrability cannot be repaired by silently freezing a varying generator. Field dependence contributes additional terms and must be declared.

Exercise: integrate an exact variation

Let $\not\!\delta H=2a\,da+db$ on a simply connected parameter space. Verify integrability and construct the normalized charge that vanishes at $(a,b)=(0,0)$.

Solution

Here $A=2a$ and $B=1$, so $\partial_aB-\partial_bA=0$. The one-form is exact:

$$\not\!\delta H=d(a^2+b).$$

Path integration therefore gives $H=a^2+b+C$. The prescribed reference normalization sets $C=0$, so $H=a^2+b$. A different reference constant would change the numerical zero but not the Hamiltonian vector field.

References

• Compère, Geoffrey. Advanced Lectures on General Relativity. Lecture Notes in Physics 952. Cham: Springer, 2019. DOI; Open PDF.
• 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.