The soft-hair program derives the Bekenstein–Hawking entropy of a rotating black hole by Cardy counting of a putative two-dimensional conformal field theory (CFT) living on the event horizon, whose Virasoro algebra of horizon diffeomorphisms carries central charge cL=cR=12Jc_L=c_R=12J for Kerr angular momentum JJ. Whether this account is complete remains open: it is not known whether the horizon CFT states encode all information about the matter that formed and fell into the black hole, by what mechanism the outgoing Hawking radiation would carry that information so that evaporation is unitary, why the derived entropy is independent of the elementary-particle spectrum, and how a horizon imprint can coexist with the quantum no-cloning theorem.

Background

Black holes in general relativity carry a thermodynamic entropy given by the Bekenstein–Hawking formula S=A/(4Gℏ)S=A/(4G\hbar), where AA is the event-horizon area. Since this formula was discovered, the outstanding question has been which microstates it counts, in Boltzmann's sense of entropy as a count of microscopic configurations.

The soft-hair program pursues an answer within general relativity itself. Asymptotically flat spacetimes possess an infinite-dimensional group of asymptotic symmetries, the Bondi–Metzner–Sachs (BMS) group, whose supertranslation generators (angle-dependent shifts of the retarded time label at null infinity) are associated with 'soft' (zero-energy) gravitons. Hawking, Perry and Strominger (2016) showed that black holes carry 'soft hair': degeneracies labeled by supertranslation charges on the event horizon, describable as soft gravitons or photons living on the horizon, whose quantum state is recorded on a 'holographic plate' at the future boundary of the horizon.

Haco, Hawking, Perry and Strominger (2018) turned this into an entropy derivation. For a generic Kerr black hole (mass MM, angular momentum JJ), they constructed an explicit pair of Virasoro algebras (the symmetry algebra of a two-dimensional conformal field theory) of diffeomorphisms acting nontrivially on the horizon. The associated covariant surface charges are integrable and associative only after adding a Wald–Zoupas counterterm (a boundary term fixing charge ambiguities on null surfaces), and then carry central charges cL=cR=12Jc_L=c_R=12J. Assuming a quantum Hilbert space on which these charges generate the symmetries, and assuming the applicability of the Cardy formula — the universal high-energy density-of-states formula for a two-dimensional CFT with central charges cL,cRc_L,c_R at temperatures TL,TRT_L,T_R — the counting reproduces exactly S=A/4S=A/4. The construction extends to Kerr–Newman black holes (rotating and electrically charged), the Kerr absorption probabilities independently exhibit the universal CFT form, and near-horizon soft-hair algebras generalize to a broad class of horizons. Perry's 2020 review recounts this derivation and closes with a list of unresolved issues that together ask whether the horizon CFT is the complete story of black hole entropy and information; the present problem is that closing list.

The stakes are the black hole information paradox: Hawking's semiclassical calculation makes the evaporation radiation exactly thermal, so the final state appears mixed, in conflict with the unitary time evolution of a closed quantum system. If all information about the infalling matter were encoded in horizon CFT states and the radiation could be shown to carry it away, the paradox would be resolved without firewalls (hypothetical destructive structures at the horizon, proposed by Almheiri, Marolf, Polchinski and Sone (AMPS) as an alternative to accepting evaporation unitarity). Known obstacles exist: Mirbabayi and Porrati (2016) and Bousso and Porrati (2017) argued from infrared factorization of the scattering matrix that soft modes decouple from hard dynamics, so soft hair cannot carry the 'hard' information needed for unitarity — a critique the program has not conclusively answered. In parallel, the island/replica-wormhole computations (Penington 2020; Almheiri et al. 2020) derive unitary Page curves (the entanglement entropy shared between the emitted radiation and the remaining hole, evolving unitarily in time rather than rising monotonically as Hawking's calculation predicts) for evaporating black holes in anti-de Sitter spacetime (the constant-negative-curvature solution of Einstein's equations) and in two-dimensional settings without identifying any horizon CFT microstates; the relation between that route and the soft-hair route is itself unresolved.

Problem Statement

Does the soft-hair horizon conformal field theory give a complete account of black hole entropy and evaporation? The source (Perry 2020, closing section) poses this as the following linked unresolved questions, which jointly constitute the problem:

  1. Completeness of the encoding: does the horizon CFT contain a complete description of the black hole formation process — i.e., is all relevant information about the infalling matter encoded in the horizon CFT states?
  2. Unitarity mechanism: by what mechanism does the outgoing Hawking radiation encode this information so that time evolution is unitary?
  3. Spectrum independence: the horizon-CFT Cardy counting yields S=A/(4G)S=A/(4G) regardless of the elementary-particle spectrum, yet the number of distinct ways a black hole can form depends strongly on that spectrum; why is the entropy insensitive to the spectrum?
  4. No-cloning consistency: if a particle falling through the horizon leaves an imprint in the horizon CFT while also crossing into the interior, how is this consistent with the quantum no-cloning theorem (the result of quantum mechanics forbidding the copying of an arbitrary unknown state into a second, independently accessible copy) — precisely which locality assumption of the theorem fails in gravity?

The problem asks whether the horizon CFT gives a complete account: a resolution would either demonstrate that the horizon CFT states encode the formation information and yield unitary evaporation, together with explanations of spectrum-independence and no-cloning consistency, or establish that they cannot. Because the four sub-questions are jointly necessary for completeness, the affirmative branch requires all of (1)–(4) together, while establishing that the horizon CFT fails at least one of (1)–(4) suffices for the negative branch. The existing soft-decoupling arguments (Mirbabayi–Porrati, Bousso–Porrati) remain contested and do not by themselves settle the question.

The review's closing list also raises the resolution of the classical singularity and the final stages of black hole evaporation, and how global symmetries (e.g., baryon number) fit into a picture in which all symmetries are gauge symmetries. These items are recorded as context only: they constitute separable general quantum-gravity questions, while the coherent entropy-and-information core of the problem is (1)–(4), and a resolution of (1)–(4) is what would bear on the black hole information paradox.

References

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  2. Stephen W. Hawking, Malcolm J. Perry, Andrew Strominger, "Soft Hair on Black Holes", Phys. Rev. Lett. 116 (2016) 231301, arXiv:1601.00921, DOI:10.1103/PhysRevLett.116.231301, https://arxiv.org/abs/1601.00921
  3. Sasha Haco, Stephen W. Hawking, Malcolm J. Perry, Andrew Strominger, "Black Hole Entropy and Soft Hair", JHEP 12 (2018) 098, arXiv:1810.01847, DOI:10.1007/JHEP12(2018)098, https://arxiv.org/abs/1810.01847
  4. Sasha Haco, Malcolm J. Perry, Andrew Strominger, "Kerr-Newman Black Hole Entropy and Soft Hair", arXiv preprint arXiv:1902.02247 (no journal version as of this review), https://arxiv.org/abs/1902.02247
  5. Mehrdad Mirbabayi, Massimo Porrati, "Shaving off Black Hole Soft Hair" (published as "Dressed Hard States and Black Hole Soft Hair"), Phys. Rev. Lett. 117 (2016) 211301, arXiv:1607.03120, DOI:10.1103/PhysRevLett.117.211301, https://arxiv.org/abs/1607.03120
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  11. Francesco Di Filippo, Naoki Ogawa, Shinji Mukohyama, Takahiro Waki, "Soft hair, dressed coordinates, and information loss paradox", Phys. Rev. D 108 (2023) 044034, arXiv:2305.15800, DOI:10.1103/PhysRevD.108.044034, https://arxiv.org/abs/2305.15800
  12. Chethan Krishnan, Jude Pereira, "Soft Hair on Schwarzschild: A Wrinkle in Birkoff's Theorem", JHEP 08 (2025) 033, arXiv:2304.01607, DOI:10.1007/JHEP08(2025)033, https://arxiv.org/abs/2304.01607
  13. Geoffrey Penington, "Entanglement Wedge Reconstruction and the Information Paradox", JHEP 09 (2020) 002, arXiv:1905.08255, DOI:10.1007/JHEP09(2020)002, https://arxiv.org/abs/1905.08255
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