Asymptotic Symmetry Groups

An asymptotic symmetry is not simply a coordinate transformation that looks nontrivial near infinity. It is an allowed transformation of a specified asymptotic phase space, modulo transformations whose canonical action and charges are trivial. The boundary conditions therefore come first: they decide which metrics are admitted, which variations are finite, which diffeomorphisms preserve the space, and which of those diffeomorphisms act physically.

Required background. Variational and Symplectic Geometry supplies presymplectic currents and degeneracies. Helpful background. Lorentzian Metrics and Causal Cones and Geodesics and Jacobi Fields supply boundary geometry; Causality and Global Structure supplies conformal infinity; Formulations of General Relativity develops the general charge construction; Waves, Green Functions, and Geometric Optics supplies radiative propagation.

Boundary conditions define the phase space

Fix a set $\mathcal F$ of field configurations obeying equations, gauge choices, topology, falloffs, parity conditions, and boundary regularity appropriate to an asymptotic region. Its tangent vectors $\delta g$ obey the linearized versions of those conditions. A candidate vector field $\xi$ generates an allowed diffeomorphism when

$$\mathcal L_\xi g\in T_g\mathcal F$$

for every relevant $g\in\mathcal F$, possibly up to a compensating gauge transformation that restores the chosen coordinate gauge. This requirement is stronger than preserving only the leading background metric.

The falloffs must make the symplectic structure and intended surface charges finite. They must also be preserved under time evolution. Conditions that are too weak can produce divergent flux or charges; conditions that are too strong can remove radiative modes or genuine boundary symmetries. At spatial infinity, parity conditions control divergent pieces of the symplectic form. At null infinity, the Bondi determinant, radial, and falloff conditions organize the shear, mass aspect, and news.

The result depends on the phase space. Standard Regge–Teitelboim parity conditions at spatial infinity lead naturally to a Poincaré asymptotic symmetry group. Twisted parity conditions can retain the BMS supertranslations there. Neither answer is “the group of the metric at infinity” without those choices. Henneaux and Troessaert work through this dependence in Henneaux and Troessaert 2019, §§2–5.

Quotient by proper gauge transformations

Let $G_{\rm allowed}$ be the transformations preserving the boundary conditions. Some act only as gauge redundancies. Denote by $G_{\rm proper}$ the normal subgroup generated by allowed transformations whose Hamiltonian generators vanish on the phase space, including any necessary boundary normalization. The asymptotic symmetry group is

$$G_{\rm asym}=G_{\rm allowed}/G_{\rm proper}.$$

At the infinitesimal level, vector fields are identified when their difference generates proper gauge. A diffeomorphism that tends to the identity in coordinates can still be improper if it has a finite nonzero boundary charge; a transformation that remains visibly nonzero can be proper if its charge vanishes under the selected conditions. Coordinate size is not the criterion.

In covariant phase space, start from a Lagrangian $n$-form $L$ with

$$\delta L=E\cdot\delta g+d\Theta[g;\delta g].$$

The presymplectic current is

$$\omega[g;\delta_1g,\delta_2g] =\delta_1\Theta[g;\delta_2g]-\delta_2\Theta[g;\delta_1g].$$

For a diffeomorphism generated by $\xi$, the on-shell variation of its charge is a codimension-two integral

$$\delta H_\xi=\int_S k_\xi[g;\delta g].$$

The form $k_\xi$ includes the Noether charge and the contraction of $\Theta$ with $\xi$. Its detailed derivation and ambiguities belong to Diffeomorphism Noether Currents and Charges and Charge Integrability and Ambiguities. Here its role is diagnostic: a finite, integrable, nonzero $H_\xi$ shows that the allowed transformation is not a pure degeneracy.

Finiteness, integrability, and flux

Finiteness of $\delta H_\xi$ is necessary but not sufficient for a charge function $H_\xi[g]$ to exist. Integrability requires the one-form on phase space to be closed:

$$\delta_1\delta_2 H_\xi-\delta_2\delta_1 H_\xi=0,$$

with extra terms when $\xi$ depends on the fields. Radiative boundaries often produce a nonintegrable term proportional to flux. One can separate an integrable charge from a flux term, enlarge the phase space, or change the prescription, but the choice and ambiguity must be stated.

The presymplectic potential itself is ambiguous under $\Theta\mapsto\Theta+dY$, and the Lagrangian can shift by a boundary term. Reference backgrounds and additive constants also affect charges. A sound construction fixes these choices by covariance, boundary conditions, vanishing on a reference solution, agreement with known Hamiltonian expressions, or a specified renormalization prescription. Compère and Fiorucci derive the surface-charge framework and its ambiguities in Compère and Fiorucci 2019, §§1.2–1.5, pp. 10–32.

Flux changes the conservation statement, not necessarily the symmetry. Between cuts $S_1$ and $S_2$ of a boundary region $\Delta\mathscr I$, one expects a balance law of the form

$$H_\xi[S_2]-H_\xi[S_1] =-\int_{\Delta\mathscr I}\mathcal F_\xi,$$

with signs fixed by boundary orientation. A charge is conserved when the corresponding flux vanishes. At future null infinity, gravitational news generally makes Bondi-type charges cut dependent; at spatial infinity, suitable falloffs can eliminate flux through the asymptotic cylinder.

Algebra and central terms

Charges should represent the asymptotic symmetry algebra under the Poisson or covariant bracket:

$$\{H_\xi,H_\eta\}=H_{[\xi,\eta]_*}+K_{\xi,\eta}.$$

The adjusted bracket $[\cdot,\cdot]_*$ accounts for field-dependent generators. If the extra term $K$ is field independent and commutes with all charges, it is a central extension subject to the two-cocycle condition. More general field-dependent extensions need not be central and obey a corresponding generalized consistency condition. Which case occurs depends on dimension, boundary conditions, charge prescription, and normalization. A central term cannot be inferred merely because the vector-field algebra admits one.

Three independent checks control a proposed group and charge:

• on Minkowski data, proper gauge transformations have zero normalized charge;
• the finite-dimensional translation and Lorentz sector reproduces ADM or Bondi four-momentum and angular momentum under compatible conditions; and
• the charge balance has the dimensions and flux sign dictated by $G$ and the boundary orientation.

Closure alone is insufficient. A set of vector fields can close algebraically while failing to preserve subleading falloffs, making the symplectic form divergent, or possessing no integrable generator.

Exercise

Suppose an allowed vector field $\xi$ preserves all falloffs and its charge variation is finite, but there exist tangent variations with $\delta_1\delta_2H_\xi-\delta_2\delta_1H_\xi\neq0$. Is $\xi$ a proper gauge transformation, an asymptotic symmetry with a well-defined charge, or neither conclusion yet justified?

Solution

Preservation of falloffs makes $\xi$ an allowed asymptotic transformation. The nonzero phase-space curl means the displayed variation is not integrable to a single-valued charge $H_\xi$ on that phase space. It therefore has not been shown to be proper gauge—properness would require a vanishing generator—nor to possess an ordinary integrable charge.

One must identify the nonintegrable flux, restrict or enlarge the phase space, or adopt a justified modified charge prescription. Only then can one quotient the transformation as proper or retain it as an improper symmetry with a well-defined generator.

Poincare Symmetry and ADM Charges applies this logic at spatial infinity with explicit falloff and parity assumptions.

References

• Compère, Geoffrey, and Adrien Fiorucci, Advanced Lectures on General Relativity, §§1.2–1.5, pp. 10–32, Springer, 2019.
• Henneaux, Marc, and Cédric Troessaert, “The Asymptotic Structure of Gravity at Spatial Infinity in Four Spacetime Dimensions”, Proceedings of Science CORFU2018, article 154, §§2–5, 2019.