Hawking, Perry and Strominger (JHEP 2017) showed that black holes in classical Einstein gravity carry an infinite head of supertranslation hair and are distinguished by exactly conserved superrotation charges, but left the symmetry status of those charges open: no phase space and bracket was known on which the superrotation charges canonically generate their symmetry, and it was unknown whether the antipodal matching conditions on the mass and angular-momentum aspects at spatial infinity — the foundation of supertranslation and superrotation charge conservation — survive one-loop quantum corrections. The problem asks for (1) a canonical realization of superrotation symmetry at null infinity on an appropriately enlarged phase space, with integrable, covariance-consistent charges and a closed charge algebra whose cocycle structure is determined, and (2) a determination of whether and how the antipodal matching conditions, and the charge conservation laws they imply for scattering including black hole formation and evaporation, are deformed at one loop. Major progress has been made on both fronts — competing phase-space constructions, a no-go result on the standard phase space, centerless charge-algebra results, and Ward-identity derivations of loop-corrected soft theorems — but no consensus canonical construction and no direct one-loop derivation of the deformed matching conditions; the core remains open.

Background

An asymptotically flat spacetime models an isolated gravitating system whose metric approaches Minkowski space at large distances. Near future null infinity I+\mathscr I^+ (the null boundary reached along outgoing light rays) one uses retarded Bondi coordinates (u,r,ΘA)(u,r,\Theta^A) with ΘA\Theta^A angles on the celestial two-sphere. In this gauge the radiative degrees of freedom are the shear CAB(u,Θ)C_{AB}(u,\Theta), describing angle-dependent distortions of the spheres of constant uu and rr, and its retarded-time derivative, the Bondi news NAB=∂uCABN_{AB}=\partial_u C_{AB}, which encodes outgoing gravitational waves. The Einstein constraint equations then introduce the mass aspect m(u,Θ)m(u,\Theta), whose zeroth and first spherical harmonics are the total mass and momentum, and the angular momentum aspect NA(u,Θ)N^A(u,\Theta), whose corresponding modes encode angular momentum and the boost (center-of-mass) charges.

The Bondi–Metzner–Sachs (BMS) group is the asymptotic symmetry group of asymptotically flat gravity: the diffeomorphisms preserving the asymptotic structure but acting nontrivially on the physical phase space. Besides Lorentz transformations it contains supertranslations — angle-dependent translations parametrized by a function f(Θ)f(\Theta) on the sphere. Distinct supertranslated vacua are labelled by the shear at the bottom of I+\mathscr I^+ and are mapped into one another by soft gravitons, i.e. zero-energy gravitons; the permanent shift of this shear between the beginning and end of I+\mathscr I^+ is the gravitational memory effect, a lasting change in the relative separation of freely falling detectors after the passage of radiation. The corresponding black hole labels were named soft hair by Hawking, Perry and Strominger (HPS), who argued that an infinite set of soft charges is carried by black holes and constrains Hawking evaporation, with implications for the information paradox.

Several proposals extend the BMS group further by superrotations: vector fields on the sphere going beyond the six global conformal Killing vectors (vector fields whose flow preserves the sphere's metric up to a position-dependent scale factor; exactly six exist globally, and they generate the Lorentz group). The extended BMS algebra uses (possibly singular) holomorphic conformal Killing vectors of the sphere, while the generalized BMS algebra uses all smooth sphere diffeomorphisms Diff(S2)\mathrm{Diff}(S^2), and the Weyl BMS group adds local Weyl rescalings. Superrotations are subtle: acting with them can create string-like singularities piercing null infinity and can violate peeling — the standard hierarchy of inverse powers of the radius with which the components of the Weyl curvature decay towards null infinity — so they do not preserve the standard phase space of asymptotically flat gravity.

For gravitational scattering to be well posed, HPS impose Lorentz- and CPT-invariant (invariant under the combined charge conjugation, parity and time reversal) antipodal matching conditions at spatial infinity between the top of past null infinity and the bottom of future null infinity: the mass aspect and shear agree, m∣I+−=m∣I−+m|_{\mathscr I^-_+}=m|_{\mathscr I^+_-} and C∣I+−=C∣I−+C|_{\mathscr I^-_+}=C|_{\mathscr I^+_-}, and the angular momentum aspect obeys NA∣I+−=NA∣I−+N^A|_{\mathscr I^-_+}=N^A|_{\mathscr I^+_-} up to the antipodal map on the sphere. These conditions imply an infinite family of exactly conserved charges: supertranslation charges QfQ_f built from the mass aspect and superrotation charges QYQ_Y built from the angular momentum aspect, each equating a number computed from incoming data at past null infinity with one computed from outgoing data at future null infinity. The classical matching of the mass aspect is equivalent to Weinberg's leading soft graviton theorem — the universal limiting form of a graviton scattering amplitude when one external graviton's energy is taken to zero, whose subleading refinements fix the next terms in that zero-energy expansion — and the matching of the angular momentum aspect to the subleading soft theorem.

In the source paper, HPS showed that a Schwarzschild black hole can be supertranslated by throwing in an asymmetric null shock wave; the resulting supertranslated black holes are diffeomorphic to Schwarzschild by a large BMS diffeomorphism but are physically distinct, and they carry superrotation charges which provide a classical diagnostic of the hair. They further constructed, at linearized order around Schwarzschild, a horizon charge Q^fH+\hat Q_f^{\mathcal H^+} which together with a null-infinity piece splits the full linearized supertranslation charge as Q^f+=Q^fI++Q^fH+\hat Q_f^+=\hat Q_f^{\mathscr I^+}+\hat Q_f^{\mathcal H^+}, and showed it generates horizon supertranslations via a Dirac bracket on a linearized horizon phase space. The Dirac bracket is the phase-space implementation of a conserved charge as a symmetry generator; the covariant phase space and symplectic form are the geometric structures from which such brackets are derived — the symplectic form being obtained by integrating the symplectic current, a two-form built from linearized solutions of the equations of motion, over a Cauchy surface — and a charge is integrable when it generates transformations without a flux ambiguity. Upon quantization these brackets become commutators of charge operators, and a commutator can acquire a central extension: an extra contribution, classified up to redefinitions of the charges by a 2-cocycle, an antisymmetric function of the two symmetry parameters that cannot be removed by any redefinition of the generators; whether the quantized charge algebra of asymptotically flat gravity carries such a cocycle is itself a contested question. Charge prescriptions of Wald–Zoupas type are the standard covariant benchmark: they define the charge on a cut of null infinity from the pullback of the symplectic potential (the potential whose exterior derivative yields the symplectic form), with residual ambiguities fixed by requiring the reference structure to be universal across solutions.

Two linked issues were left open and are the core of this problem. First, the canonical realization of superrotations is incomplete: while the supertranslation generators were realized on the standard phase space (as reviewed in the source paper, Sec. 3.1), the superrotation charge QY+Q_Y^+ does not preserve the standard boundary conditions, and HPS state that a larger phase space and associated bracket is needed but has not yet been found. Candidate enlargements in the later literature promote the celestial metric — the degenerate two-metric that each cut of null infinity inherits from the spacetime metric, normally a fixed background structure — to a phase-space variable, or adjoin spin-memory modes (null-infinity observables recording the differential retardation of counter-rotating circularly polarized beams) and gravitational-tail modes (which encode gravitational tails, the late-time power-law decay of the field at null infinity produced by backscatter of radiation off the background curvature). Second, the quantum status of the matching conditions is unknown: the one-loop analyses of Bern, Davies and Nohle show that the subleading soft graviton behavior is modified at loop level with an order-of-limits ambiguity between the soft limit and the loop regularization (the zero-energy limit and the removal of the loop regulator need not commute), and HPS note that the antipodal matching condition on the angular momentum aspect is possibly deformed by a one-loop anomaly, calling the determination of whether the loop corrections deform rather than eliminate the conserved charges an important open problem.

Problem Statement

Establish whether the conserved superrotation charges of asymptotically flat gravity are genuine canonical and quantum-stable symmetries, in two linked parts that together complete, for superrotations, the program that Hawking–Perry–Strominger noted “has been completed for supertranslations but remains underway for superrotations”.

(1) Classical canonical realization. Construct — or prove impossible — a phase space for asymptotically flat gravity at null infinity, enlarging the standard radiative phase space as required, on which the superrotation charges are integrable canonical generators whose Poisson/Dirac brackets realize the superrotation symmetry algebra (extended BMS with conformal Killing vectors of the sphere, generalized BMS with Diff(S2)\mathrm{Diff}(S^2), or Weyl BMS — the answer must state which), with the 2-cocycle and central-extension structure of the quantized algebra determined, with a covariance-consistent charge prescription, and without spoiling asymptotic flatness or the well-posedness of the gravitational scattering problem. The construction must either evade or explicitly resolve the known obstructions: the divergence of the symplectic current found by Flanagan–Prabhu–Shehzad on the standard phase space, the non-integrability of the superrotation charges, and the residual field-dependent 2-cocycle at a cut found in Wald–Zoupas implementations of the extended algebra.

(2) Loop-level fate of the antipodal matching conditions. Determine whether and how the antipodal matching conditions on the mass aspect and on the angular momentum aspect — and hence the conservation of the supertranslation and superrotation charges, including their accounting in scattering processes involving black hole formation and evaporation — are deformed by one-loop quantum corrections and by the order-of-limits ambiguity between the soft limit and loop regularization. In particular, derive the loop-level form of the matching conditions and of the resulting charge conservation laws and charge algebra, consistent with the known one-loop (logarithmic) soft graviton theorems, and reducing in the classical limit to the now-proven classical matching relations including tails.

Part (1) is the classical half and part (2) the quantum half of a single objective: to elevate the superrotation charge structure from a perturbative, linearized statement to an exact symmetry statement about the gravitational S-matrix.

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