Can soft horizon hair account for black hole entropy and resolve the information paradox? The holographic plate, its capacity, and its quantum completion

Hawking, Perry and Strominger (Phys. Rev. Lett. 116, 231301 (2016)) showed that the infinite number of supertranslation conservation laws in asymptotically flat gravity require black holes to carry zero-energy 'soft' supertranslation hair (and its electromagnetic analog, soft electric hair), described as soft gravitons or photons on the horizon and storing some information about infalling matter on a 'holographic plate' at the future boundary of the horizon. They themselves left open whether this hair, possibly enlarged by superrotations or other soft symmetries, has enough independent degrees of freedom to encode all information crossing the horizon, whether the horizon plate admits a complete quantum description, and whether the Bekenstein-Hawking area-entropy law can be derived microscopically from these horizon soft modes. The subsequent literature (superrotation charges; Kerr horizon Virasoro constructions reproducing the area law under Cardy assumptions; three-dimensional negative results; the island/replica-wormhole Page curves that bypass soft hair) leaves the core question — the storage capacity, complete quantum description, and entropy accounting of black hole soft hair in four-dimensional asymptotically flat gravity — open.

Background

The black hole information paradox originates in Hawking's 1976 argument (S. W. Hawking, 'Breakdown of Predictability in Gravitational Collapse', Phys. Rev. D 14 (1976) 2460): treating gravity semiclassically, the Hawking radiation emitted by an evaporating black hole is thermal and uncorrelated with the state of the collapsing matter, so complete evaporation would map a pure quantum state to a mixed one, violating unitarity. A complete account must at least explain the Bekenstein-Hawking entropy law, which assigns a black hole of horizon area $A$ the entropy $S_{\mathrm{BH}} = A/(4G)$ (with $\hbar=c=k_B=1$), i.e. a number of microstates of order $\exp(A/4G)$.

The setting of the problem is the 'infrared structure' of gauge theory and gravity in asymptotically flat four-dimensional spacetime. Three phenomena are there understood to be equivalent: soft theorems (universal constraints on scattering amplitudes when emitted particles carry zero energy, such as Weinberg's soft photon and soft graviton theorems), asymptotic symmetries, and memory effects (permanent, measurable displacements of detectors after the passage of radiation). The relevant asymptotic symmetry group is the Bondi-Metzler-Sachs (BMS) group of asymptotically flat spacetime; its supertranslation subgroup shifts the retarded time at future null infinity by an arbitrary angle-dependent function $f(z,\bar z)$ on the celestial two-sphere. Large gauge transformations (angle-dependent gauge transformations that approach an arbitrary function at infinity rather than vanishing, as ordinary gauge transformations do) are the electromagnetic analog. The associated conserved charges are carried by zero-energy ('soft') gravitons or photons, and the vacuum is correspondingly infinitely degenerate: acting with a supertranslation charge on a vacuum creates another physically inequivalent vacuum containing one additional soft particle.

Hawking, Perry and Strominger applied this to black holes. Because a horizon hides part of spacetime, future null infinity alone is not a Cauchy surface when massive bodies or black holes are present, and the supertranslation conservation laws acquire a horizon boundary contribution: black holes must carry an infinite set of soft supertranslation charges, with supertranslation hair described by soft gravitons living on the horizon and, in the presence of a Maxwell field, soft electric hair described by soft photons. The quantum state of these horizon soft modes is stored on a two-sphere $\mathcal H^+$ at the future boundary of the horizon, which the authors call the holographic plate; charge conservation yields infinitely many exact relations among the evaporation products of black holes differing only in soft hair. A heuristic argument that hair localized to much less than the Planck length cannot be excited suggests an effective number of soft degrees of freedom scaling with the horizon area in Planck units. The explicit constructions, however, were restricted to spherically symmetric (Vaidya) collapse, massless matter, and leading/linearized order, and used the late-time settling of the geometry (the 'no-hair' behavior, i.e. the theorem that a settled stationary black hole is characterized only by its mass, charge, and angular momentum) in deriving the horizon charges.

The authors state their own open problem plainly: supertranslation hair is expected to be 'too thin' to fully reproduce the area-entropy law; superrotations (asymptotic symmetries associated with angle-dependent, locally conformal transformations of the celestial sphere, conjectured to yield 'thicker' hair) 'have not yet been fully studied or understood'; and 'it is an open question whether or not the current line of investigation, perhaps with additional new ingredients, can characterize all the pixels on the holographic plate', while 'a complete description of the holographic plate and resolution of the information paradox remains an open challenge'. Resolution would require, at a minimum, 'a detailed understanding of the information flow out of black holes as well as a derivation of the Hawking-Bekenstein area-entropy law'.

Later technical context needed to read the subsequent literature: the covariant phase space formalism defines conserved charges for asymptotic symmetries as surface integrals, with an ambiguity fixed by Wald-Zoupas-type counterterms; the Cardy formula gives the entropy of a two-dimensional conformal field theory (CFT) in terms of its Virasoro central charge (the central charge of the Virasoro algebra, the symmetry algebra of two-dimensional conformal transformations), and has been the workhorse of entropy derivations from horizon Virasoro algebras; and 'islands' (regions of the black hole interior bounded by a quantum extremal surface, i.e. a surface extremizing the sum of its area and the entanglement entropy it encloses) are the ingredient in the 2019-2020 replica-wormhole derivations of unitary Page curves (the curve of radiation entropy versus time, which rises then falls back to zero after the Page time if evaporation is unitary) for evaporating black holes in anti-de Sitter (AdS) holography and lower-dimensional models — a resolution route that does not use soft hair.

Problem Statement

For four-dimensional, asymptotically flat black holes, determine whether the soft degrees of freedom associated with horizon asymptotic symmetries can account for black hole entropy and information. The source (Hawking, Perry and Strominger, 2016) leaves this as its stated open challenge: give a complete description of the holographic plate — the full set of independent horizon soft modes ('pixels'), their quantum algebra, and a consistent ultraviolet regularization replacing the heuristic Planck-length cutoff — and decide whether this sector, enlarged as needed by superrotations, near-horizon Virasoro-type symmetries, or any further soft symmetry of the four-dimensional asymptotically flat infrared structure, can (1) microscopically reproduce the Bekenstein-Hawking area-entropy law $S = A/(4G)$ with the exact factor of $1/4$ for generic non-extremal black holes (not only rotating ones, and not merely under unproven assumptions such as the applicability of the Cardy formula to a putative horizon dual conformal field theory); and (2) store, and unitarily return to infinity, all information that crosses the horizon during formation and evaporation, thereby resolving the information paradox in asymptotically flat spacetime — including an explicit understanding of the information flow (an algorithm for the rate at which information is returned) and validity beyond the simplifying assumptions of the original constructions (spherical collapse, massless infalling matter, leading-order and linearized treatment, and late-time settling). Equally admissible as an answer is a rigorous no-go result: a demonstration covering the enumerated soft and horizon-symmetry ingredients — the soft supertranslation (soft-graviton) hair, the soft electric (large-gauge) hair, the superrotations, and the near-horizon Virasoro-type symmetries — treated either one by one or by a single argument covering this whole class, that the horizon soft sector is necessarily too degenerate (too few distinguishable states, or charge constraints that cannot correlate outgoing Hawking radiation with the infalling state) to carry the required information, or that it cannot microscopically reproduce the area-entropy law of requirement (1), with the missing capacity or needed ingredients identified. A refutation of either (1) or (2) over this enumerated class is by itself a complete negative resolution; it need not exclude further soft symmetries outside that class, and it need not supply the holographic-plate description. A positive resolution, by contrast, must supply the holographic-plate description together with both (1) and (2), and may invoke whichever of the above enlargements suffice. A result that establishes one of (1) and (2) while leaving the other undetermined does not resolve the task.

References

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