Supertranslations and Angular-Momentum Ambiguity

At null infinity there is no unique Poincaré subgroup of the BMS group and therefore no unique Lorentz origin about which to define angular momentum. A supertranslation changes the cuts, shifts the vacuum shear, and conjugates one Lorentz subgroup into another. Angular-momentum charges remain meaningful only after their boundary conditions, charge prescription, and supertranslation frame—or an invariant replacement—are specified.

Required background. Variational and Symplectic Geometry supplies boundary charge ambiguities. Helpful background. Lorentzian Metrics and Causal Cones and Geodesics and Jacobi Fields supply null geometry; Causality and Global Structure supplies cuts of $\mathscr I$; Formulations of General Relativity develops charge construction; Waves, Green Functions, and Geometric Optics supplies radiation.

Cuts and vacuum shear

In Bondi gauge, a cut of $\mathscr I^+$ is a sphere $u=U(x^A)$. A supertranslation with parameter $T(x^A)$ changes the retarded-time labeling and transforms the shear by

$$\delta_TC_{AB} =T\partial_uC_{AB} -2D_AD_BT+q_{AB}D^2T$$

in the convention used in this chapter. In a nonradiative interval $N_{AB}=\partial_uC_{AB}=0$, the inhomogeneous term can carry one vacuum shear into another.

For the electric-parity vacuum sector, write

$$C_{AB} =-2D_AD_BC+q_{AB}D^2C.$$

Then a supertranslation shifts the potential $C$ by $T$, up to the $\ell=0,1$ modes whose trace-free Hessian vanishes. The radiative curvature is zero throughout this family, but the Bondi frames differ. Selecting $C=0$ chooses a vacuum frame; it is not invariant under all BMS transformations.

Ordinary translations already shift special-relativistic angular momentum by orbital terms. Supertranslations generalize the shift with infinitely many angular modes. The issue is therefore not that gravity has “no angular momentum,” but that the boundary structure does not canonically supply the origin and Lorentz subgroup used to decompose total charges.

The angular-momentum aspect

Subleading coefficients of $g_{uA}$ define an angular-momentum aspect $N_A$ after convention-dependent combinations of shear have been fixed. Pairing $N_A$ with a sphere conformal Killing field $Y^A$ gives Lorentz charges, supplemented by shear and mass-aspect terms required for covariance, integrability, and normalization. Schematically,

$$Q_Y[S] =\frac{1}{8\pi G}\int_S Y^A\bigl(N_A+\text{shear terms}\bigr)d\Omega.$$

The phrase “shear terms” cannot be dropped when comparing formulas: changing the presymplectic potential, boundary counterterm, or definition of $N_A$ moves finite pieces between them. A valid expression must reproduce stationary angular momentum, possess the intended flux law, and transform consistently under BMS.

Under a supertranslation, $N_A$ mixes with the mass aspect, derivatives of $T$, shear, and news. Hence a charge associated with a chosen Lorentz field generally changes. For a translation of an isolated system this reduces to the familiar origin dependence. Higher-harmonic $T$ produces additional shifts with no finite-dimensional Poincaré analogue.

The covariant surface-charge construction and its integrability ambiguities are presented in Compère and Fiorucci 2019, §§1.2–1.5 and 3.1–3.3, pp. 10–32, 69–84. The formula used in any calculation must declare the same phase space and conformal frame as that source or translate them explicitly.

Ways to control the ambiguity

Several strategies answer different questions:

1. Choose preferred cuts. In stationary regimes, good cuts or shear-free cuts can select a Poincaré subgroup. Existence and uniqueness require conditions; radiation can prevent a global selection.
2. Fix a vacuum frame. Set the electric vacuum shear potential to a reference value in an early or late nonradiative era. The choice is physically tied to that era and changes across memory.
3. Separate intrinsic and orbital pieces. Construct spin-like quantities insensitive to ordinary translations, often using mass and momentum. General supertranslations still require care.
4. Use super-angular momentum. Enlarge the charge set so transformation under supertranslations is part of the structure rather than treated as an error. This introduces its own regularity and flux issues.
5. Compare balance laws. A charge difference between specified cuts plus its flux can be more robust than either cut charge in isolation.

These approaches are not automatically equivalent. A definition optimized for stationary black holes may differ by finite shear terms from one optimized for radiative scattering. Claims of “the” angular momentum at $\mathscr I^+$ should therefore name the prescription.

Radiation and memory

If news is present between cuts $S_1$ and $S_2$, their shear difference contains radiative memory:

$$\Delta C_{AB}=\int_{u_1}^{u_2}N_{AB}\,du.$$

Even when both eras are nonradiative, they may be related by a supertranslation. A frame chosen to set the early vacuum shear to zero will generally not set the late shear to zero. The radiative process has changed the relative vacuum frame.

This does not make memory a gauge artifact. Detector displacement is expressed by a gauge-invariant relative observable once the asymptotic-to-finite-radius relation is established. The BMS description identifies the boundary transition underlying that observable. The gravitational-radiation volume develops the detailed detector response and nonlinear flux, building on the cut and charge geometry explained here.

Angular-momentum flux is correspondingly delicate. News carries angular momentum, but separating radiative flux from changes induced by moving cuts depends on the prescription. Balance equations should be checked on stationary limits, under ordinary translations, and against the chosen symplectic flux.

What is and is not ambiguous

The ambiguity is structured, not arbitrary. Once one fixes the Bondi phase space, conformal frame, cut, Lorentz generator, and charge prescription, the charge is calculable. Transformations between choices obey BMS laws. The uncertainty lies in selecting a canonical choice from the universal asymptotic geometry alone.

The Bondi four-momentum is less affected because the four translation modes form a distinguished finite-dimensional subspace of supertranslations under the global Lorentz group. The infinite family of Lorentz subgroups, however, is conjugated by pure supertranslations. Thus energy–momentum can be organized covariantly while an origin-independent full angular-momentum tensor remains unavailable without added structure.

Extended BMS proposals alter the issue again. Singular superrotations or smooth sphere diffeomorphisms enlarge the angular generators, phase space, and possible charges. Results obtained there do not automatically apply to the globally smooth standard BMS group.

Exercise

In a nonradiative interval let $C_{AB}=0$. Apply a supertranslation $T$ that is a single spherical harmonic. For which angular momenta $\ell$ does the transformed shear remain zero, and what does this imply for selecting a vacuum frame?

Solution

The transformed shear is

$$C'_{AB}=-2D_AD_BT+q_{AB}D^2T.$$

For $\ell=0$, $T$ is constant and the expression vanishes. For $\ell=1$, $D_AD_BT=-q_{AB}T$ and $D^2T=-2T$, so it also vanishes. For $\ell\geq2$, the trace-free Hessian is generally nonzero.

Thus ordinary spacetime translations preserve the zero-shear vacuum condition, while pure supertranslations move to another vacuum shear. Choosing $C_{AB}=0$ selects a family of Poincaré frames related by the ordinary translations, not a frame invariant under the full BMS group.

Memory, Soft Relations, and Asymptotic Charges next states how vacuum shear change and BMS balance connect to radiative memory and soft limits.

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.