Covariant Phase Space, Boundaries, and Charges

Covariant phase space begins with the boundary term in a Lagrangian variation and ends, when the necessary conditions hold, with Hamiltonians for spacetime symmetries. The route is boundary sensitive: a locally closed current can carry flux, a diffeomorphism can cease to be pure gauge at a boundary, and a finite surface variation can fail to integrate to a charge. This chapter keeps those logical gates separate and fixes the sign convention used throughout.

Helpful background. The Einstein Equation supplies the solution space; Hypersurface Normals and Tangential Projectors supplies orientations and cuts; The Gibbons–Hawking–York Boundary Term supplies the boundary variational problem; Presymplectic Potentials and Currents supplies field-space forms; Killing Fields and Isometry Groups supplies symmetry generators; and Formulations of General Relativity orients the volume.

The variational chain

For dynamical fields $\phi$ and a covariant Lagrangian $n$-form,

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

This chapter adopts the Iyer–Wald ordering

$$\omega(\phi;\delta_1\phi,\delta_2\phi) =\delta_2\Theta(\phi;\delta_1\phi) -\delta_1\Theta(\phi;\delta_2\phi).$$

For an allowed, field-independent vector $\xi^a$, the Einstein Noether construction leads to

$$J_\xi=\Theta(\mathcal L_\xi\phi) -\xi\mathbin{\lrcorner}L, \qquad J_\xi\doteq dQ_\xi,$$

and the candidate Hamiltonian variation on a cut $S$ is

$$\not\!\delta H_\xi =\int_S (\delta Q_\xi-\xi\mathbin{\lrcorner}\Theta).$$

Iyer and Wald derive this chain in Iyer and Wald 1994, §§3–6, pp. 6–16, PDF. With the displayed ordering, the corresponding bulk identity is $\not\!\delta H_\xi=\Omega_\Sigma(\mathcal L_\xi\phi,\delta\phi)$ when $S=\partial\Sigma$ and the stated on-shell hypotheses hold. Reversing the arguments reverses this sign; it does not change the surface-charge normalization. Each arrow adds hypotheses: equations and linearized equations for current closure, boundary conditions for a phase space, falloff for finiteness, field-space closure for integrability, and normalization for a finite charge.

A coherent route through the chapter

The six pages form the following sequence:

1. Covariant Presymplectic Current derives $\Theta$, $\omega$, its closure, and its exact-form ambiguities.
2. Diffeomorphism Noether Currents and Charges decomposes $J_\xi$ into constraints and $dQ_\xi$ and derives the Einstein charge form.
3. Gauge Degeneracies and Boundary Symmetries tests when $\mathcal L_\xi g$ is a degenerate direction and when it changes a boundary charge.
4. Symplectic Flux and Boundary Conditions relates hypersurface independence to lateral flux and source–response conditions.
5. Charge Integrability and Ambiguities checks whether a surface variation is exact on solution space and records its remaining choices.
6. Komar, ADM, and Covariant Charges Compared identifies the stationary and asymptotic hypotheses under which familiar formulas agree.

The ordering prevents a common shortcut: writing down a Komar-like surface integral before defining the allowed perturbations, flux, and normalization that give it a Hamiltonian meaning.

Four independent boundary questions

Property	Required check	Failure means
Differentiability	Boundary variation is fixed or canceled	No well-defined variational principle
Conservation	Net symplectic flux vanishes or is included	Bulk form depends on the slice
Integrability	Antisymmetrized field variations vanish	No finite Hamiltonian function on that phase space
Gauge reduction	Pairing vanishes for every tangent perturbation	The transformation is boundary-active, not pure gauge

Finiteness is a fifth requirement: every displayed surface integral must converge. These properties constrain one another but none substitutes for all the others. Compère states the charge variation, integrability, and conservation criteria in Compère 2019, §§1.4.1–1.4.3, pp. 22–23, PDF.

Boundaries determine the physical question

The same bulk Einstein Lagrangian supports different phase spaces at spatial infinity, null infinity, a horizon, or a finite timelike wall. The boundary conditions select allowed perturbations and symmetries; counterterms select a compatible potential representative; corner data control exact-form shifts; and flux determines whether the bulk is closed or open. Consequently there is no boundary-independent list of gravitational charges.

This chapter develops the machinery, not the classification of every asymptotic symmetry algebra or the thermodynamic laws of particular spacetimes. Field-dependent generators, anomalous boundaries, and nonintegrable radiative charges require extensions of the fixed-generator formulas stated here.

Review the chapter

For an exact symmetry $\xi^a$ and a linearized solution $\delta g$, state the surface variation that appears on two boundary components, then list the two additional checks needed before calling it a conserved finite charge.

Solution

The exact-symmetry identity gives the closed surface form

$$k_\xi(\delta g) =\delta Q_\xi -\xi\mathbin{\lrcorner}\Theta(g;\delta g), \qquad dk_\xi=0.$$

Its integrals on homologous cuts agree when there is no intervening singularity or unaccounted boundary flux. One must still check that the integral is finite under the declared falloff and that the field-space one-form is integrable. After those checks, a reference solution fixes the additive constant. If $\xi^a$ is field dependent, further variation terms are required before any of these conclusions are used.

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.