Asymptotic Symmetries and Gravitational Charges

Asymptotic symmetries turn boundary conditions into physical conservation laws. The phase space determines which transformations are allowed; quotienting transformations with vanishing generators identifies genuine boundary symmetries; the Einstein constraints and symplectic flux then relate charges on different cuts. In asymptotically flat gravity this route leads from Poincaré and ADM quantities at spatial infinity to BMS charges, Bondi balance, memory, and soft relations at null infinity.

Helpful background. Variational and Symplectic Geometry supplies presymplectic currents; Lorentzian Metrics and Causal Cones and Geodesics and Jacobi Fields supply boundary geometry; Causality and Global Structure gives the volume route; Formulations of General Relativity develops charge construction; Waves, Green Functions, and Geometric Optics supplies radiation.

From boundary conditions to charges

The central definition is a quotient:

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

$G_{\rm allowed}$ preserves the full asymptotic phase space—falloffs, gauge, topology, parity, boundary regularity, and admissible variations. $G_{\rm proper}$ consists of transformations with trivial canonical action and zero normalized charge. The quotient cannot be inferred from how large a coordinate vector appears near infinity.

In covariant phase space, a candidate generator has a surface-charge variation

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

It must be finite, and it must be integrable on phase space if it is to define an ordinary cut charge. At a radiative boundary, a nonintegrable piece can encode symplectic flux; the correct result is then a balance law rather than conservation on every cut. Boundary terms, reference solutions, and presymplectic-potential ambiguities must be fixed consistently.

This logic is developed on Asymptotic Symmetry Groups. The general Noether and presymplectic derivations are treated in Covariant Phase Space, Boundaries, and Charges; here the emphasis is the global structure that selects each asymptotic phase space.

Spatial infinity and null infinity

Poincare Symmetry and ADM Charges begins with asymptotically Euclidean initial data. Standard parity conditions make asymptotic translations, rotations, and boosts differentiable Hamiltonian transformations, giving energy, momentum, angular momentum, and center of mass. Energy–momentum requires less asymptotic moment control than angular momentum and boosts. Twisted parity conditions can retain supertranslations, showing that even the spatial-infinity group depends on the selected phase space.

At null infinity, Bondi gauge and the universal degenerate conformal structure give the Bondi–Metzner–Sachs Group:

$$\mathrm{BMS}_4 =\mathrm{Lorentz}\ltimes C^\infty(S^2).$$

The $\ell=0,1$ spherical harmonics are ordinary translations; the higher modes are pure supertranslations. The Lorentz sector consists of globally smooth conformal Killing fields of the sphere in the standard group. Singular superrotations and smooth $\mathrm{Diff}(S^2)$ extensions require larger phase spaces and are not silently included.

ADM and BMS charges live on different boundaries. Matching them through spatial infinity requires parity, differentiability, and corner conditions. A Penrose diagram in which $\mathscr I^+$ meets $i^0$ supplies orientation, not a matching theorem.

Constraint, flux, and vacuum frame

Bondi Mass Aspect and the Asymptotic Constraint expands the null Einstein equation. In the convention used here,

$$\partial_um =\frac14D_AD_BN^{AB} -\frac18N_{AB}N^{AB} -4\pi G T_{uu}^{(2)}.$$

The sphere average of the divergence vanishes, leaving the Bondi mass-loss law with a nonpositive news-square term and outgoing matter flux. Higher spherical harmonics give supermomentum balance. Detailed waveform flux and detector normalization continue in Gravitational Radiation and Waveforms.

The shear carries both radiative and vacuum-frame information. In a nonradiative interval, higher-mode supertranslations shift $C_{AB}$ while leaving news zero. Supertranslations and Angular-Momentum Ambiguity explains why this gives infinitely many conjugate Lorentz subgroups and why a cut, preferred vacuum, or another prescription is needed for angular momentum. The ambiguity is governed by BMS transformation laws, not an arbitrary freedom to change the answer.

Memory, Soft Relations, and Asymptotic Charges completes the chain. Integrated news produces the shear transition $\Delta C_{AB}$; the constraint rewrites it as a charge balance; after matching past and future symmetries and quantizing the zero mode, the charge conservation law gives a soft Ward identity. Detector memory, infrared-dressed scattering, and loop-level soft statements require their own treatments.

Normalization and independent checks

Every charge formula in the chapter is tested against simple backgrounds. Minkowski space has zero normalized ADM and BMS charges. Isotropic Schwarzschild initial data give

$$E_{\rm ADM}=M,$$

and stationary Bondi data give $M_B=M$. A proper gauge transformation must have vanishing charge. An outgoing news pulse makes the Bondi mass decrease when matter flux is nonnegative. Restoring $G$ checks dimensions: the metric mass coefficient has length, while dividing its sphere average by $G$ gives physical mass.

Algebra supplies another check. Charge brackets must represent the adjusted asymptotic vector-field bracket, including field dependence and only justified central terms. Closure of vector fields alone does not guarantee finite or integrable charges. Conversely, nonzero flux does not destroy the symmetry; it changes a conserved cut charge into a balance law.

Compère and Fiorucci develop both the general surface-charge construction and four-dimensional BMS application in Compère and Fiorucci 2019, §§1.2–1.5 and 3.1–3.3, pp. 10–32, 69–84. Spatial-infinity phase-space choices are compared by Henneaux and Troessaert 2019, §§2–5.

Review the chapter

A vector field preserves the leading flat metric, has finite components at infinity, and closes with nine similar fields under Lie brackets. Is that enough to call the ten fields the asymptotic Poincaré group with ADM charges?

Solution

No. One must specify the full asymptotic phase space and verify preservation of subleading falloffs, derivative conditions, constraints, and parity. The symplectic form and each surface-charge variation must be finite. The transformations with zero generators must be quotiented, while the remaining charge variations must be integrable or assigned justified flux terms. Rotations and boosts require stronger parity and moment control than translations.

Finally, check the algebra using the adjusted bracket when generators depend on fields, normalize Minkowski charges to zero, and recover Schwarzschild ADM mass. Only after these tests does the Poincaré interpretation and its ten charges follow. With different parity conditions, the same leading flat behavior can instead support a larger BMS-type group.

What this chapter enables

After this chapter, you should be able to derive an asymptotic group from boundary data, distinguish proper from improper gauge, state ADM and Bondi normalization assumptions, explain supertranslations and their Lorentz-frame consequence, and trace the limited but precise symmetry–memory–soft correspondence.

The next chapter changes the sign of the cosmological constant and hence the causal character of infinity. Begin with Spacelike Conformal Infinity in de Sitter Geometry.

References

• Compère, Geoffrey, and Adrien Fiorucci, Advanced Lectures on General Relativity, §§1.2–1.5 and 3.1–3.3, pp. 10–32 and 69–84, 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.