Superrotation Charges in Asymptotically Flat Gravity

• Superrotation charges are asymptotic charges in flat gravity, defined through angle-dependent extensions of Lorentz symmetry and incorporating singular generators.
• They are computed using methods like Bondi-Sachs and Penrose completion, with regularization techniques to manage singularities on the sphere.
• Superrotation charges connect to soft graviton theorems, memory effects, and black hole soft hair, revealing deep links between symmetry and infrared physics.

Searching arXiv for recent and foundational papers on superrotation charges and related BMS charge constructions. Superrotation charges are asymptotic or horizon charges associated with angle-dependent extensions of angular momentum and boost symmetries in asymptotically flat gravity. In the four-dimensional null-infinity setting they arise from the $Y^A$ sector of the BMS algebra, where smooth global conformal Killing vectors reproduce the ordinary Lorentz charges, while more general locally defined or meromorphic conformal Killing vectors define the extended superrotation sector. Their study combines Bondi-Sachs methods, Penrose conformal completion, canonical surface-charge constructions, and near-horizon Hamiltonian formalisms, and it repeatedly exposes three structural issues: genuine superrotations are typically singular on $S^2$, classical charges have supertranslation ambiguity unless corrected, and the charge algebra is tightly linked to soft graviton theorems, memory observables, and black-hole soft hair (Akhoury et al., 6 Jul 2025, Chen et al., 2021, Hawking et al., 2016).

1. Symmetry generators and the extended BMS setting

In four-dimensional asymptotically flat gravity, the standard Bondi description writes an asymptotic symmetry generator as

$$\xi^u = \alpha(\theta^A)+\frac12 u D_A Y^A, \qquad \xi^A = Y^A(\theta^B).$$

If $Y^A$ is restricted to the six smooth conformal Killing vectors of $S^2$, one recovers the Lorentz group. Allowing general meromorphic solutions of the sphere conformal Killing equations enlarges the algebra to the infinite-dimensional superrotation sector. In complex coordinates $z,\bar z$, the conformal Killing equations are

$$\partial_z Y^{\bar z}=0, \qquad \partial_{\bar z} Y^z=0,$$

with basis

$$l_m = z^m \partial_z,\qquad \bar l_m = \bar z^m \partial_{\bar z},\qquad m\in\mathbb Z.$$

The smooth subset $m=-1,0,1$ gives the usual Lorentz generators, while higher or lower modes are singular at one or both poles and define superrotations (Akhoury et al., 6 Jul 2025).

This distinction is not merely terminological. Several analyses in the literature work only with the Lorentz subgroup and do not extend to genuinely singular superrotation generators. For example, the first subleading Newman-Penrose dipole charges were identified with the full ordinary Lorentz group—boosts and rotations—rather than with the larger local-conformal superrotation extension, and that analysis explicitly does not pursue the full superrotation algebra (Kol, 2020). A related but distinct line of work constructs supertranslation-invariant dressed Lorentz boosts and rotations and stresses that these are not superrotation charges in the Barnich-Troessaert sense (Javadinezhad et al., 2022).

The generator concept also appears in other asymptotic frameworks. In harmonic gauge, the tower of residual gauge transformations contains vector-type superrotation towers $\xi^R_{(N)}$ with schematic form

$S^2$0

and introduces a “novel” leading superrotation $S^2$1 that is $S^2$2-subleading with respect to Lorentz transformations but leading within that residual-gauge tower (Compère, 2019). In higher even dimensions, the generalized BMS group replaces sphere conformal Killing vectors by $S^2$3 superrotations $S^2$4, so the relevant generator becomes a diffeomorphism of the celestial $S^2$5 rather than a local conformal vector field on $S^2$6 (Chowdhury et al., 2022).

2. Charge definitions at null infinity

The standard Bondi-type superrotation charge is built from the angular momentum aspect. In the black-hole and scattering discussion of asymptotically flat gravity, it is written as

$S^2$7

with $S^2$8 the angular momentum aspect. For global conformal Killing vectors $S^2$9, this reduces to ordinary angular momentum or boost charge; for general $$\xi^u = \alpha(\theta^A)+\frac12 u D_A Y^A, \qquad \xi^A = Y^A(\theta^B).$$0, it gives infinitely many angle-dependent charges (Hawking et al., 2016).

A more elaborate extended-BMS formula is the “classical charge”

$$\xi^u = \alpha(\theta^A)+\frac12 u D_A Y^A, \qquad \xi^A = Y^A(\theta^B).$$1

Here $$\xi^u = \alpha(\theta^A)+\frac12 u D_A Y^A, \qquad \xi^A = Y^A(\theta^B).$$2 is the Bondi mass aspect, $$\xi^u = \alpha(\theta^A)+\frac12 u D_A Y^A, \qquad \xi^A = Y^A(\theta^B).$$3 is the angular momentum aspect, $$\xi^u = \alpha(\theta^A)+\frac12 u D_A Y^A, \qquad \xi^A = Y^A(\theta^B).$$4 is the shear, and $$\xi^u = \alpha(\theta^A)+\frac12 u D_A Y^A, \qquad \xi^A = Y^A(\theta^B).$$5 is the news tensor. For a classical BMS field the last term vanishes, while for an extended BMS field it remains part of the superrotation-sector charge (Chen et al., 2021).

A coordinate-free formulation is provided by Penrose conformal completion and the Geroch-Winicour linkage construction. With $$\xi^u = \alpha(\theta^A)+\frac12 u D_A Y^A, \qquad \xi^A = Y^A(\theta^B).$$6 and $$\xi^u = \alpha(\theta^A)+\frac12 u D_A Y^A, \qquad \xi^A = Y^A(\theta^B).$$7 realized as $$\xi^u = \alpha(\theta^A)+\frac12 u D_A Y^A, \qquad \xi^A = Y^A(\theta^B).$$8, the linkage charge on a cut $$\xi^u = \alpha(\theta^A)+\frac12 u D_A Y^A, \qquad \xi^A = Y^A(\theta^B).$$9 is

$Y^A$0

The geometric asymptotic-symmetry conditions are

$Y^A$1

and the Geroch-Winicour gauge condition

$Y^A$2

removes ambiguity from the conformal factor and the bulk extension of $Y^A$3. In this formalism, the Bondi and Penrose descriptions of superrotation charges agree (Akhoury et al., 6 Jul 2025).

Beyond four dimensions, the six-dimensional generalized-BMS superrotation charge is proposed in hard-plus-soft form,

$Y^A$4

with the full expression

$Y^A$5

Its defining feature is the use of the nonlinear radiative variable $Y^A$6, chosen so that the superrotation action on the radiative data is homogeneous (Chowdhury et al., 2022).

3. Singularities, supertranslation ambiguity, and regularization

The central technical obstruction for genuine superrotation charges is that the generators are singular on the sphere. In the linkage approach, one may still write the formal integral

$Y^A$7

but for superrotations $Y^A$8 is not globally smooth on $Y^A$9. Since the integrand contains $S^2$0, $S^2$1, and higher derivatives integrated over the full sphere, the charge becomes an improper integral and is formally ill-defined (Akhoury et al., 6 Jul 2025).

A substantial result is that the bulk linkage integrand itself remains finite near $S^2$2 despite the $S^2$3 factor, once $S^2$4 is imposed and the superrotation conformal Killing equations are used. The singularity problem is therefore not a failure of conformal completion, but a regularization problem for the sphere integral. The Flanagan-Nichols prescription addresses it by excising small caps around the singular poles, integrating over the punctured sphere, and then taking $S^2$5. In the explicit construction, the superrotation charge is

$S^2$6

with $S^2$7. The finiteness argument relies on harmonic expansions and the cancellation of the would-be divergences after the $S^2$8-integration (Akhoury et al., 6 Jul 2025).

A separate issue is supertranslation ambiguity. For the classical extended-BMS charge, the total flux

$S^2$9

shifts under supertranslations by terms involving $z,\bar z$0 and, in the extended case, also endpoint shear data: $z,\bar z$1 To remove this ambiguity, a corrected invariant charge $z,\bar z$2 is defined by adding explicit shear-potential terms from the Hodge-type decomposition of $z,\bar z$3. The resulting total flux is supertranslation invariant except for the $z,\bar z$4 translation modes, and both classical and invariant extended-BMS charges are boost invariant under the stated decay hypotheses (Chen et al., 2021).

These two subtleties are often conflated, but the literature treats them differently. The singularity of $z,\bar z$5 is handled by regularization of the sphere integral, whereas the supertranslation ambiguity is handled by modifying the charge functional itself. A plausible implication is that “well-defined superrotation charge” can mean either regularized finiteness for singular generators or supertranslation-invariant definition for fluxes; the two problems are related but not identical.

4. Ward identities, soft graviton theorems, and memory

Superrotation charges are tied to soft graviton physics through Ward identities. In the four-dimensional scattering setting, conservation across spatial infinity is written as

$z,\bar z$6

and implies the $z,\bar z$7-matrix identity

$z,\bar z$8

When expressed in terms of hard and soft fluxes at null infinity, this Ward identity reproduces the tree-level subleading soft theorem and, in a phase space admitting gravitational tails, also the logarithmic one-loop soft corrections (Agrawal et al., 2023).

In that formulation, the total superrotation flux is decomposed into hard and soft parts,

$z,\bar z$9

with explicit canonical generators acting on radiative gravitons, matter, the supertranslation Goldstone mode $$\partial_z Y^{\bar z}=0, \qquad \partial_{\bar z} Y^z=0,$$0, and soft news. The paper’s central claim is that the logarithmic soft graviton theorem

$$\partial_z Y^{\bar z}=0, \qquad \partial_{\bar z} Y^z=0,$$1

is encoded in the superrotation Ward identity, provided one works with dressed interacting states rather than free fields. For massive scattering, part of the relevant superrotation charge lies at timelike infinity $$\partial_z Y^{\bar z}=0, \qquad \partial_{\bar z} Y^z=0,$$2 (Agrawal et al., 2023).

The charge-memory-soft relation has also been organized into an infinite tower. In harmonic gauge, subleading residual gauge transformations $$\partial_z Y^{\bar z}=0, \qquad \partial_{\bar z} Y^z=0,$$3 are in one-to-one correspondence with memory observables and with conserved Noether charges $$\partial_z Y^{\bar z}=0, \qquad \partial_{\bar z} Y^z=0,$$4 at spatial infinity. The leading-order mutually commuting supertranslations and the “novel” leading superrotations are both associated with a leading displacement memory effect, while higher members of the tower give subleading memories and degenerate towers of subleading soft graviton theorems (Compère, 2019).

In higher even dimensions, the generalized-BMS superrotation charge has an associated Ward identity

$$\partial_z Y^{\bar z}=0, \qquad \partial_{\bar z} Y^z=0,$$5

or equivalently

$$\partial_z Y^{\bar z}=0, \qquad \partial_{\bar z} Y^z=0,$$6

For six-dimensional massless-scalar scattering, the paper proves that this Ward identity follows from the subleading soft graviton theorem after smearing with $$\partial_z Y^{\bar z}=0, \qquad \partial_{\bar z} Y^z=0,$$7 and using a specific identity for the kernel $$\partial_z Y^{\bar z}=0, \qquad \partial_{\bar z} Y^z=0,$$8 (Chowdhury et al., 2022).

Taken together, these results place superrotation charges at the intersection of asymptotic symmetry, radiative phase space, and infrared structure. They are not merely labels of asymptotic data; they act as canonical generators whose conservation laws become soft graviton theorems, and whose associated observables include memory effects.

5. Black holes, horizon soft hair, and superrotation observables

In black-hole spacetimes, superrotation charges appear both at null infinity and in near-horizon constructions. A central claim is that a supertranslated black hole does not itself carry supertranslation charge, but it does acquire a nontrivial pattern of superrotation charges measured at infinity. The defining asymptotic quantity is again

$$\partial_z Y^{\bar z}=0, \qquad \partial_{\bar z} Y^z=0,$$9

For a linearly supertranslated Schwarzschild black hole, the angular momentum aspect shifts by

$$l_m = z^m \partial_z,\qquad \bar l_m = \bar z^m \partial_{\bar z},\qquad m\in\mathbb Z.$$0

so that

$$l_m = z^m \partial_z,\qquad \bar l_m = \bar z^m \partial_{\bar z},\qquad m\in\mathbb Z.$$1

This is generically nonzero, and it is the sense in which superrotation charges distinguish black holes that are diffeomorphic to Schwarzschild but are related by large BMS transformations and therefore occupy different points in phase space (Hawking et al., 2016).

The same paper derives a linearized horizon supertranslation charge,

$$l_m = z^m \partial_z,\qquad \bar l_m = \bar z^m \partial_{\bar z},\qquad m\in\mathbb Z.$$2

and shows that it generates horizon supertranslations by the Dirac bracket. Although this is not itself a superrotation charge, it clarifies why supertranslation hair is visible through superrotation observables at infinity: the horizon and null-infinity charges are linked by charge conservation on $$l_m = z^m \partial_z,\qquad \bar l_m = \bar z^m \partial_{\bar z},\qquad m\in\mathbb Z.$$3 (Hawking et al., 2016).

A separate near-horizon analysis in asymptotic Rindler geometry defines superrotation charges as surface charges associated with transverse diffeomorphisms $$l_m = z^m \partial_z,\qquad \bar l_m = \bar z^m \partial_{\bar z},\qquad m\in\mathbb Z.$$4. In the canonical Regge-Teitelboim construction, the full surface charge is

$$l_m = z^m \partial_z,\qquad \bar l_m = \bar z^m \partial_{\bar z},\qquad m\in\mathbb Z.$$5

and the superrotation contribution is carried by the $$l_m = z^m \partial_z,\qquad \bar l_m = \bar z^m \partial_{\bar z},\qquad m\in\mathbb Z.$$6 term. In a decomposition

$$l_m = z^m \partial_z,\qquad \bar l_m = \bar z^m \partial_{\bar z},\qquad m\in\mathbb Z.$$7

only the gradient part contributes: $$l_m = z^m \partial_z,\qquad \bar l_m = \bar z^m \partial_{\bar z},\qquad m\in\mathbb Z.$$8 The paper emphasizes that matter crossing the horizon excites these charges, whereas perturbative gravitational waves do not store information in them at first order (Hotta et al., 2016).

In five dimensions, the near-horizon soft-hair construction yields a supertranslation scalar

$$l_m = z^m \partial_z,\qquad \bar l_m = \bar z^m \partial_{\bar z},\qquad m\in\mathbb Z.$$9

and a one-form superrotation charge

$m=-1,0,1$0

For the equal-spin Myers-Perry black hole, the explicit horizon one-form is

$m=-1,0,1$1

with nonvanishing field strength

$m=-1,0,1$2

This shows that the superrotation one-form is not pure gauge, even though the near-horizon algebra is still represented as infinitely many copies of Heisenberg algebras (Mirzaiyan, 2021).

6. Higher-dimensional extensions, neighboring symmetries, and conceptual delimitations

The meaning of “superrotation charge” depends on the asymptotic framework. In six-dimensional generalized BMS, superrotations are the $m=-1,0,1$3 part of

$m=-1,0,1$4

with antipodal matching

$m=-1,0,1$5

The corresponding charge is nonlinear, acts correctly on asymptotically flat geometries through the field $m=-1,0,1$6, and has a Ward identity implied by the subleading soft graviton theorem (Chowdhury et al., 2022).

In the Special Double Null gauge, an even larger algebra appears. Besides supertranslations $m=-1,0,1$7 and superrotations $m=-1,0,1$8, the independent symmetry parameters include $m=-1,0,1$9, $\xi^R_{(N)}$0, $\xi^R_{(N)}$1, and $\xi^R_{(N)}$2, corresponding to leading and subleading hypertranslations and hyperrotations. The refined fall-offs allow the leading hyperrotation and leading hypertranslation generators to appear explicitly in the finite covariant surface charges, and the enlarged algebra $\xi^R_{(N)}$3-bms$\xi^R_{(N)}$4 reduces to standard BMS under appropriate truncation (Krishnan et al., 2023).

These generalizations sharpen an important conceptual boundary. Some analyses treat the ordinary Lorentz charges as the only subleading asymptotic charges of interest: the first subleading Newman-Penrose dipole moments correspond to boosts and rotations and “do not” introduce independent superrotation charges beyond the Lorentz subgroup (Kol, 2020). Other analyses explicitly enlarge the symmetry algebra to singular or local superrotations and then confront regularization and ambiguity directly (Akhoury et al., 6 Jul 2025, Chen et al., 2021). Still others construct dressed Lorentz charges that commute with supertranslations and stress that the resulting observables should not be confused with extended-BMS superrotation charges (Javadinezhad et al., 2022).

A common misconception is therefore that every angle-dependent rotational charge in asymptotically flat gravity is a superrotation charge in the same sense. The literature does not support that identification. Depending on the framework, the charge may be an ordinary Lorentz charge, a corrected extended-BMS charge, a geometric linkage requiring Flanagan-Nichols regularization, a near-horizon one-form charge, or a higher-dimensional $\xi^R_{(N)}$5 charge. What unifies these constructions is that each uses an angular vector generator to define a nontrivial conserved quantity associated with asymptotic or horizon symmetry; what differentiates them is the phase space, the regularity conditions, and the treatment of soft modes.

Superrotation charges therefore occupy a structurally rich position in asymptotically flat gravity. They interpolate between angular momentum and its angle-dependent generalizations, require careful control of singular generators and supertranslation dependence, govern Ward identities for subleading and logarithmic soft graviton theorems, and supply observable soft-hair data for black holes and horizons. The current literature presents not a single universally adopted object, but a family of closely related charge constructions whose agreements and discrepancies are now explicit parts of the subject.