Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Kerr spacetime's asymptotic form can be generated from Schwarzschild by two BMS supertranslations.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

The paper claims two BMS supertranslations map Schwarzschild's asymptotic metric to a Kerr-like one, but the matching equations are inconsistent.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection The construction is new and clearly set out, but the central matching is internally inconsistent: the F in Eq. (44) solves Eq. (32) but cannot satisfy Eqs. (28)-(29), so Kerr is not actually generated. the 3 major comments →

arxiv 2509.01203 v1 pith:DJJZHDW3 submitted 2025-09-01 gr-qc

Asymptotic Generation of Kerr Geometry from Schwarzschild via BMS Supertranslations

Nihar Ranjan Ghosh , Malay K. Nandy This is my paper
classification gr-qc
keywords BMS supertranslationsKerr spacetimeSchwarzschild spacetimeasymptotic symmetriessoft hairmultipole momentsBondi gaugenull infinity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether two classically distinct black hole solutions—non-rotating Schwarzschild and rotating Kerr—are connected by the infinite-dimensional asymptotic symmetries of flat spacetime. The authors argue that, at null infinity, they are: two successive BMS supertranslations applied to Schwarzschild reproduce the leading and subleading asymptotic angular structure of the Kerr metric. The first supertranslation, built from l=1 spherical harmonics, plays the role of a center-of-mass/translational displacement; the second, built from an infinite series of even-parity Legendre polynomials, is identified with the mass-multipole structure of Kerr. The result is offered as evidence that supertranslations carry physically meaningful soft hair that can interpolate between distinct black hole geometries.

Core claim

Working in Bondi gauge at large r, the paper constructs two diffeomorphisms, each a BMS supertranslation, and applies them successively to a Schwarzschild seed. It then matches the transformed metric, order by order in 1/r, to the asymptotic Kerr metric. The matching conditions fix the first supertranslation function f to be an l=1 spherical harmonic, and the second function F to solve an inhomogeneous Laplace-type equation on the sphere. The solution for F contains a constant term, l=1 terms, and an infinite series of even Legendre polynomials with l≥2. The authors conclude that the l=1 sector corresponds to translations and center-of-mass displacement, while the even-l sector encodes the m

What carries the argument

The central object is the pair of supertranslation generating functions f(θ,ϕ) and F(θ,ϕ), each defining a vector field that preserves Bondi gauge and the large-r falloff. The argument is carried by four matching equations that equate the angular components and mixed null-angle components of the supertranslated Schwarzschild metric with the corresponding Kerr components. The equations decouple into an eigenvalue condition on f and an inhomogeneous equation on F; the particular integral of the latter produces the even-Legendre series that gives Kerr its multipole structure.

Load-bearing premise

The construction assumes the four matching conditions have a single common solution; in particular, the θθ condition and the differential equation coming from the mixed components may impose conflicting requirements on the same function F.

What would settle it

Compute the residual of the θθ matching equation after substituting the claimed solutions for f and F and using the differential equation for F. The two conditions together imply F + ∂_θ^2 F = C/2, so one can evaluate this combination directly for the even-Legendre series in F; if it is not identically C/2, the θθ component fails to match Kerr and the proposed pair of supertranslations is not a solution of the system.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Schwarzschild and Kerr are not isolated points in solution space: their asymptotic forms are connected by BMS supertranslations, so the vacuum solution space carries an infinite-dimensional symmetry action.
  • The l=1 part of the transformation is the known center-of-mass and translation sector, associated with displacement memory and no new hair.
  • The even-l≥2 Legendre content of the second supertranslation reproduces the expected multipole selection rule of Kerr, giving a concrete soft-hair avatar of rotation.
  • The construction provides a template for generating other stationary vacuum geometries from a seed solution by successive supertranslations.
  • The result supports the idea that supertranslation hair is not pure gauge but encodes information about a black hole's multipole structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A natural next test is whether the same two-step construction extends to other axisymmetric vacuum seeds, with the first supertranslation choosing the frame and the second encoding the multipole moments.
  • The even-l structure of F suggests a direct link between supertranslation hair and the Geroch-Hansen multipole moments, which could be probed by computing BMS charges at null infinity for the transformed metric.
  • The paper works only in the asymptotic patch; if the transformation could be extended inward to the horizon, it would sharpen the connection between supertranslations and black-hole memory effects.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper claims that the asymptotic structure of the Kerr spacetime can be generated from Schwarzschild by applying two successive BMS supertranslations. The first supertranslation f is asserted to be a combination of l=1 spherical harmonics, and the second F is given by an infinite series of even-parity Legendre polynomials (l≥2). The construction is carried out by matching the supertranslated Schwarzschild metric, in Bondi gauge, to the large-r Kerr metric of Ref. [56]. The matching conditions are Eqs. (28)–(31), and the explicit solution for F is Eq. (44). The paper interprets the even-l modes of F as encoding the mass multipole structure of Kerr, thereby supporting a "soft hair" relation between Schwarzschild and Kerr.

Significance. If correct, the result would be a striking demonstration that BMS supertranslations can interpolate between non-rotating and rotating black hole asymptotic geometries, with the even-parity Legendre structure of the supertranslation function reflecting the mass multipole moments of Kerr. This would be a significant contribution to the soft-hair/BMS literature. However, the central claim is not established: the matching system is overdetermined, and the proposed solution fails to satisfy all of the stated matching equations. The paper also contains a number of technical gaps in the treatment of the angular-momentum component and in the Legendre expansion of a singular source. The strengths are the clear physical motivation and the explicit, detailed setup of the Bondi-gauge Kerr expansion, but the core mathematical consistency is lacking.

major comments (3)
  1. [Section 4, Eqs. (28) and (32)] For f in Eq. (36), f + ∂²_θ f = 0, so by Eq. (22) C^S_θθ = 0. Equation (28) therefore reduces to D²F − 2∂²_θF = a/sinθ. Subtracting this from Eq. (32) gives F + ∂²_θF = C/2. The even-Legendre part of the proposed solution (44) does not satisfy this condition: the k=1 term contributes (3aπ/16)P₂(cosθ) to F + ∂²_θF, and a constant has no P₂ component. Hence Eq. (44) cannot solve Eqs. (28) and (32) simultaneously. The paper solves only Eq. (32) and never verifies Eqs. (28)–(29). This is a load-bearing inconsistency in the claimed matching.
  2. [Section 4, Eq. (31) with Eq. (14)] Equation (31) is written as ∂_φ(F + ½D²F) = −g^K_uφ. However, Eq. (14) gives g^K_uφ = −4aM_K sin²θ/r + O(r⁻²), not zero. For the φ-independent F in Eq. (44), the left side of Eq. (31) vanishes identically, so Eq. (31) is violated at O(1/r) for any a≠0. The text's statement "g^K_uφ = 0" is only true at leading order. Thus the transformed metric does not acquire the Kerr dudφ/angular-momentum component, and the matching of Eq. (31) fails.
  3. [Section 4, Eqs. (40)–(44)] The expansion of a/sinθ in Eq. (40) is not a valid L²(S²) Legendre expansion: the coefficients (2k−1)!!/(2k)!! decay only as k^(−1/2) and are not square-summable, so the series does not converge in L² and Eq. (42) cannot be interpreted as an identity of functions on the sphere. Consequently the coefficient matching in Eq. (43) and the solution in Eq. (44) are formal. This is not merely a technicality: Eq. (32) has a singular source, so a smooth F of the form (44) cannot solve it in any ordinary sense.
minor comments (4)
  1. [Section 3 and Appendix A] There are typographical errors ("guage", "shorly") and Eq. (14) is typeset in a confusing, hard-to-parse way; the metric expression should be rewritten for clarity.
  2. [Section 4, mass matching] The transformed Schwarzschild metric in Eq. (27) retains 2M_s/r du², while the Kerr metric in Eq. (14) has 2M_K/r du². The paper should state explicitly that the construction assumes M_s = M_K, or otherwise explain how the mass aspect is matched.
  3. [Section 5, interpretation] The statement that the even-l modes of F 'encode' the Kerr mass multipoles is post hoc: F is solved from the Kerr data in the matching equations, so the appearance of even Legendre modes is a consistency check rather than an independent prediction.
  4. [Section 4, role of f] For f in Eq. (36), Eq. (22) gives C^S_θθ = C^S_ϕϕ = 0. Thus the first supertranslation does not affect the matched shear; the paper should clarify what role the first transformation plays in the construction, given that it contributes nothing to the leading-order matching.

Circularity Check

0 steps flagged

No significant circularity: the supertranslation functions are solved for by matching to Kerr asymptotics, and the multipole claim is an interpretation of that solution, not an input.

full rationale

The paper's chain is a constructive matching calculation: it takes Kerr's Bondi-gauge asymptotic metric (Eq. 14) as the target, takes Schwarzschild (Eq. 15), applies two supertranslations with arbitrary functions f and F (Eqs. 21, 27), equates the resulting r-linear and dudA terms to the Kerr values, and solves the resulting partial differential equations (28)-(31) for f and F. The solution F in Eq. 44 is obtained from Eq. 32 by Legendre expansion; its even l>=2 content is an output of the differential equation, not a fitted parameter that is then relabeled as a prediction. The later statement that even Legendre modes 'mirror' Kerr's even mass multipoles (Sec. 5) is an interpretation of the solution, not an input used to derive it. No load-bearing self-citations appear; Ref. [56] supplies the Kerr GBS form, but the asymptotic matching is performed in the present paper. The possible overdetermination of the system (28)-(31) is a consistency question about whether the proposed f and F actually solve all four equations; the paper does not verify Eqs. (28)-(29) explicitly, but this is an omitted check or correctness issue, not circularity. Hence the derivation is not circular.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The construction relies on the standard BMS framework, a specific coordinate representation of Kerr from the literature, and a Legendre expansion. No new physical entities are introduced. The main free parameters are arbitrary l=1 coefficients and integration constants, plus an unmatched Schwarzschild mass.

free parameters (4)
  • Integration constant C = arbitrary
    Appears in Eq. (32) as the constant of integration for F and is not fixed by any physical condition.
  • l=1 coefficients A_m of f = arbitrary
    In Eq. (36), the coefficients of the l=1 spherical harmonics are free, representing center-of-mass displacement.
  • l=1 coefficients B_m of F = arbitrary
    In Eq. (44), the homogeneous l=1 part of F has arbitrary coefficients B_m.
  • Schwarzschild mass M_s = not equated to M_K
    The mass aspect in Eq. (27) is 2M_s/r while Kerr has 2M_K/r in Eq. (14); the matching never imposes M_s = M_K, leaving an O(1/r) mismatch.
axioms (4)
  • domain assumption Asymptotic flatness and Bondi-Sachs gauge conditions as in Refs. [1,2,56]
    The entire construction presupposes the standard null-infinity framework and Bondi gauge falloffs.
  • domain assumption The Fletcher-Lun GBS transformation (Ref. [56]) correctly renders Kerr in Bondi-like coordinates with asymptotic components (11)-(13)
    The paper relies on Ref. [56] for the Kerr input and provides no independent verification of the pole-singular terms.
  • standard math Legendre expansion of 1/sin(theta) with coefficients (2k-1)!!/(2k)!!
    Used to solve Eq. (32); the written coefficients in Eq. (41) appear incorrect and lead to a divergent sum.
  • domain assumption The supertranslation vector fields preserve the gauge conditions at the required order
    Subleading terms such as L_zeta g_uu of O(1/r^2) are dropped even though the stated condition is exact; this is not justified by the falloffs (17).

reviewed 2026-08-05 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Asymptotic Generation of Kerr Geometry from Schwarzschild via BMS Supertranslations." pith.science (2026). https://pith.science/paper/DJJZHDW3

@misc{pith2026250901203,
  author       = {Pith},
  title        = {Pith review of: Asymptotic Generation of Kerr Geometry from Schwarzschild via BMS Supertranslations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DJJZHDW3}},
  note         = {Machine review of arXiv:2509.01203}
}
Share X Bluesky LinkedIn Reddit HN
abstract

The Bondi-van der Burg-Metzner-Sachs (BMS) group, as the asymptotic symmetry group of asymptotically flat spacetimes, plays a central role in connecting infrared structures of gravity with soft theorems and gravitational memory. In this work, we investigate the extent to which BMS supertranslations can relate physically distinct black hole geometries. Focusing on the Schwarzschild and Kerr solutions, we show that the asymptotic structure of the Kerr spacetime can be generated from the Schwarzschild geometry via two successive supertranslations. These transformations yield a Kerr-like geometry at null infinity and reveal two distinct classes of supertranslation functions. The first, composed of $l=1$ spherical harmonics, corresponds to center-of-mass displacements and encodes the translational sector of the BMS group. The second, characterized by an infinite series of even-parity Legendre polynomials ($l \geq 2$), captures the intrinsic mass multipole structure of the Kerr spacetime. Our result illustrates how BMS supertranslations can act as symmetry transformations linking asymptotically flat black hole geometries, and that they encode physically meaningful soft hair consistent with the multipole structure of rotating black holes. This work supports a unified description of soft degrees of freedom in black hole spacetimes and underscores the role of infinite-dimensional asymptotic symmetries in gravitational physics.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. BMS transformed Quantum String Dynamics near a Black Hole

    gr-qc 2026-04 unverdicted novelty 4.0

    Closed bosonic string dynamics near a 5D black hole horizon encodes BMS supertranslation signatures via angular symmetry breaking and radial conserved current.

Reference graph

Works this paper leans on

67 extracted references · 27 canonical work pages · cited by 1 Pith paper · 2 internal anchors

  1. [1]

    Gravitational waves in general relativity, vii

    Hermann Bondi, M Gr J Van der Burg, and A WK Metzner. Gravitational waves in general relativity, vii. waves from axi-symmetric isolated system. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences , 269(1336):21–52, 1962

    work page 1962

  2. [2]

    Gravitational waves in general relativity viii

    Rainer K Sachs. Gravitational waves in general relativity viii. waves in asymp- totically flat space-time. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences , 270(1340):103–126, 1962

    work page 1962

  3. [3]

    Lectures on the infrared structure of gravity and gauge the- ory, 2018

    Andrew Strominger. Lectures on the infrared structure of gravity and gauge the- ory, 2018. URL: https://arxiv.org/abs/1703.05448, arXiv:1703.05448

    Pith/arXiv arXiv 2018

  4. [4]

    Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity

    J David Brown and Marc Henneaux. Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity. Commu- nications in Mathematical Physics , 104(2):207–226, 1986. 19

    work page 1986

  5. [5]

    Asymptotic symmetries of ads2 and con- formal group in d = 1

    Mariano Cadoni and Salvatore Mignemi. Asymptotic symmetries of ads2 and con- formal group in d = 1. Nuclear Physics B , 557(1):165–180, 1999. URL: https: //www.sciencedirect.com/science/article/pii/S0550321399003983, doi: https://doi.org/10.1016/S0550-3213(99)00398-3

    work page doi:10.1016/s0550-3213(99)00398-3 1999

  6. [6]

    Asymptotic isometry and two dimensional anti-de sitter gravity,

    Masahiro Hotta. Asymptotic isometry and two dimensional anti-de sitter gravity,

    work page

  7. [7]

    Asymptotically anti-de sitter spaces

    Marc Henneaux and Claudio Teitelboim. Asymptotically anti-de sitter spaces. Communications in Mathematical Physics , 98(3):391–424, 1985

    work page 1985

  8. [8]

    Comp` ere, P

    G. Comp` ere, P. Mao, A. Seraj, and M. M. Sheikh-Jabbari. Symplectic and killing symmetries of ads3 gravity: holographic vs boundary gravitons. Journal of High Energy Physics , 2016(1), January 2016. URL: http://dx.doi.org/10.1007/ JHEP01(2016)080, doi:10.1007/jhep01(2016)080

    work page doi:10.1007/jhep01(2016)080 2016

  9. [9]

    Y. B. Zel’dovich and A. G. Polnarev. Radiation of gravitational waves by a cluster of superdense stars. Sov. Astron., 18:17, 1974

    work page 1974

  10. [10]

    Nonlinear nature of gravitation and gravitational-wave experiments

    Demetrios Christodoulou. Nonlinear nature of gravitation and gravitational-wave experiments. Phys. Rev. Lett. , 67:1486–1489, Sep 1991. URL: https://link. aps.org/doi/10.1103/PhysRevLett.67.1486, doi:10.1103/PhysRevLett.67. 1486

    doi:10.1103/physrevlett.67.1486 1991

  11. [11]

    V. B. Braginsky and L. P. Grishchuk. Kinematic Resonance and Memory Effect in Free Mass Gravitational Antennas. Sov. Phys. JETP , 62:427–430, 1985

    work page 1985

  12. [12]

    Braginsky and Kip S

    Vladimir B. Braginsky and Kip S. Thorne. Gravitational-wave bursts with memory and experimental prospects. Nature, 327:123–125, 1987. doi:10.1038/327123a0

    doi:10.1038/327123a0 1987

  13. [13]

    Wiseman and Clifford M

    Alan G. Wiseman and Clifford M. Will. Christodoulou’s nonlinear gravitational- wave memory: Evaluation in the quadrupole approximation. Phys. Rev. D, 44:R2945–R2949, Nov 1991. URL: https://link.aps.org/doi/10.1103/ PhysRevD.44.R2945, doi:10.1103/PhysRevD.44.R2945

    doi:10.1103/physrevd.44.r2945 1991

  14. [14]

    Hereditary effects in gravitational radiation

    Luc Blanchet and Thibault Damour. Hereditary effects in gravitational radiation. Phys. Rev. D , 46:4304–4319, Nov 1992. URL: https://link.aps.org/doi/10. 1103/PhysRevD.46.4304, doi:10.1103/PhysRevD.46.4304. 20

    doi:10.1103/physrevd.46.4304 1992

  15. [15]

    Kip S. Thorne. Gravitational-wave bursts with memory: The christodoulou ef- fect. Phys. Rev. D , 45:520–524, Jan 1992. URL: https://link.aps.org/doi/ 10.1103/PhysRevD.45.520, doi:10.1103/PhysRevD.45.520

    doi:10.1103/physrevd.45.520 1992

  16. [16]

    The gravitational-wave memory effect

    Marc Favata. The gravitational-wave memory effect. Class. Quant. Grav. , 27:084036, 2010. arXiv:1003.3486, doi:10.1088/0264-9381/27/8/084036

    Pith/arXiv arXiv 2010

  17. [17]

    Alexander Tolish and Robert M. Wald. Retarded Fields of Null Particles and the Memory Effect. Phys. Rev. D , 89(6):064008, 2014. arXiv:1401.5831, doi: 10.1103/PhysRevD.89.064008

    Pith/arXiv arXiv 2014

  18. [18]

    Alexander Tolish, Lydia Bieri, David Garfinkle, and Robert M. Wald. Ex- amination of a simple example of gravitational wave memory. Phys. Rev. D , 90(4):044060, 2014. arXiv:1405.6396, doi:10.1103/PhysRevD.90.044060

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  19. [19]

    Winicour

    J. Winicour. Global aspects of radiation memory. Class. Quant. Grav., 31:205003,

    work page

  20. [20]

    Gravitational Memory, BMS Su- pertranslations and Soft Theorems

    Andrew Strominger and Alexander Zhiboedov. Gravitational Memory, BMS Su- pertranslations and Soft Theorems. JHEP, 01:086, 2016. arXiv:1411.5745, doi:10.1007/JHEP01(2016)086

    Pith/arXiv arXiv 2016

  21. [21]

    Flanagan and David A

    ´Eanna ´E. Flanagan and David A. Nichols. Observer dependence of angular mo- mentum in general relativity and its relationship to the gravitational-wave memory effect. Phys. Rev. D , 92:084057, Oct 2015. URL: https://link.aps.org/doi/ 10.1103/PhysRevD.92.084057, doi:10.1103/PhysRevD.92.084057

    doi:10.1103/physrevd.92.084057 2015

  22. [22]

    Flanagan and David A

    ´Eanna ´E. Flanagan and David A. Nichols. Conserved charges of the extended Bondi-Metzner-Sachs algebra. Phys. Rev. D , 95(4):044002, 2017. [Erratum: Phys.Rev.D 108, 069902 (2023)]. arXiv:1510.03386, doi:10.1103/PhysRevD. 95.044002

    Pith/arXiv arXiv 2017

  23. [23]

    New Grav- itational Memories

    Sabrina Pasterski, Andrew Strominger, and Alexander Zhiboedov. New Grav- itational Memories. JHEP, 12:053, 2016. arXiv:1502.06120, doi:10.1007/ JHEP12(2016)053

    Pith/arXiv arXiv 2016

  24. [24]

    Asymptotic Symmetries and Electromagnetic Memory

    Sabrina Pasterski. Asymptotic Symmetries and Electromagnetic Memory. JHEP, 09:154, 2017. arXiv:1505.00716, doi:10.1007/JHEP09(2017)154. 21

    Pith/arXiv arXiv 2017

  25. [25]

    David A. Nichols. Center-of-mass angular momentum and memory effect in asymptotically flat spacetimes. Phys. Rev. D , 98:064032, Sep 2018. URL: https: //link.aps.org/doi/10.1103/PhysRevD.98.064032, doi:10.1103/PhysRevD. 98.064032

    work page doi:10.1103/physrevd.98.064032 2018

  26. [26]

    Vacua of the gravitational field.JHEP, 07:137,

    Geoffrey Comp` ere and Jiang Long. Vacua of the gravitational field.JHEP, 07:137,

    work page

  27. [27]

    An electromagnetic analogue of gravitational wave memory

    Lydia Bieri and David Garfinkle. An electromagnetic analogue of gravitational wave memory. Class. Quant. Grav. , 30:195009, 2013. arXiv:1307.5098, doi: 10.1088/0264-9381/30/19/195009

    Pith/arXiv arXiv 2013

  28. [28]

    Electromagnetic memory, 2019

    Leonard Susskind. Electromagnetic memory, 2019. URL: https://arxiv.org/ abs/1507.02584, arXiv:1507.02584

    Pith/arXiv arXiv 2019

  29. [29]

    Gonz´ alez, and Miguel Pino

    Laura Donnay, Gaston Giribet, Hern´ an A. Gonz´ alez, and Miguel Pino. Su- pertranslations and superrotations at the black hole horizon. Phys. Rev. Lett., 116:091101, Mar 2016. URL: https://link.aps.org/doi/10.1103/ PhysRevLett.116.091101, doi:10.1103/PhysRevLett.116.091101

    doi:10.1103/physrevlett.116.091101 2016

  30. [30]

    Memory effect and bms sym- metries for extreme black holes

    Srijit Bhattacharjee and Shailesh Kumar. Memory effect and bms sym- metries for extreme black holes. Phys. Rev. D , 102:044041, Aug 2020. URL: https://link.aps.org/doi/10.1103/PhysRevD.102.044041, doi:10. 1103/PhysRevD.102.044041

    work page doi:10.1103/physrevd.102.044041 2020

  31. [31]

    Bloch and A

    F. Bloch and A. Nordsieck. Note on the radiation field of the electron. Phys. Rev., 52:54–59, Jul 1937. URL: https://link.aps.org/doi/10.1103/PhysRev. 52.54, doi:10.1103/PhysRev.52.54

    doi:10.1103/physrev 1937

  32. [32]

    F. E. Low. Scattering of light of very low frequency by systems of spin 1/2. Phys. Rev., 96:1428–1432, 1954. doi:10.1103/PhysRev.96.1428

    work page doi:10.1103/physrev.96.1428 1954

  33. [33]

    F. E. Low. Bremsstrahlung of very low-energy quanta in elementary particle collisions. Phys. Rev., 110:974–977, 1958. doi:10.1103/PhysRev.110.974

    doi:10.1103/physrev.110.974 1958

  34. [34]

    Murray Gell-Mann and M. L. Goldberger. Scattering of low-energy photons by particles of spin 1/2. Phys. Rev., 96:1433–1438, 1954. doi:10.1103/PhysRev.96. 1433. 22

    work page doi:10.1103/physrev.96 1954

  35. [35]

    D. R. Yennie, Steven C. Frautschi, and H. Suura. The infrared divergence phenomena and high-energy processes. Annals Phys. , 13:379–452, 1961. doi: 10.1016/0003-4916(61)90151-8

    doi:10.1016/0003-4916(61)90151-8 1961

  36. [36]

    Infrared photons and gravitons

    Steven Weinberg. Infrared photons and gravitons. Phys. Rev. , 140:B516–B524,

    work page

  37. [37]

    S. W. Hawking. Breakdown of predictability in gravitational collapse. Phys. Rev. D , 14:2460–2473, Nov 1976. URL: https://link.aps.org/doi/10.1103/ PhysRevD.14.2460, doi:10.1103/PhysRevD.14.2460

    doi:10.1103/physrevd.14.2460 1976

  38. [38]

    Hawking, Malcolm J

    Stephen W. Hawking, Malcolm J. Perry, and Andrew Strominger. Soft hair on black holes. Phys. Rev. Lett. , 116:231301, Jun 2016. URL: https://link.aps. org/doi/10.1103/PhysRevLett.116.231301, doi:10.1103/PhysRevLett.116. 231301

    doi:10.1103/physrevlett.116.231301 2016

  39. [39]

    S. W. Hawking. Particle Creation by Black Holes. Commun. Math. Phys. , 43:199–220, 1975. [Erratum: Commun.Math.Phys. 46, 206 (1976)]. doi:10.1007/ BF02345020

    work page 1975

  40. [40]

    Hawking, Malcolm J

    Stephen W. Hawking, Malcolm J. Perry, and Andrew Strominger. Superrotation Charge and Supertranslation Hair on Black Holes. JHEP, 05:161, 2017. arXiv: 1611.09175, doi:10.1007/JHEP05(2017)161

    Pith/arXiv arXiv 2017

  41. [41]

    Black hole entropy and soft hair

    Sasha Haco, Stephen W Hawking, Malcolm J Perry, and Andrew Strominger. Black hole entropy and soft hair. Journal of High Energy Physics , 2018(12):1–19, 2018

    work page 2018

  42. [42]

    Afshar, D

    H. Afshar, D. Grumiller, and M. M. Sheikh-Jabbari. Near horizon soft hair as microstates of three dimensional black holes. Phys. Rev. D , 96:084032, Oct

    work page

  43. [43]

    Soft hair of dynamical black hole and hawking radiation

    Chong-Sun Chu and Yoji Koyama. Soft hair of dynamical black hole and hawking radiation. Journal of High Energy Physics , 2018(4), April 2018. URL: http: //dx.doi.org/10.1007/JHEP04(2018)056, doi:10.1007/jhep04(2018)056

    work page doi:10.1007/jhep04(2018)056 2018

  44. [44]

    Soft hair, dressed coordinates, and information loss paradox

    Francesco Di Filippo, Naoki Ogawa, Shinji Mukohyama, and Takahiro Waki. Soft hair, dressed coordinates, and information loss paradox. Phys. Rev. D, 108:044034, 23 Aug 2023. URL: https://link.aps.org/doi/10.1103/PhysRevD.108.044034, doi:10.1103/PhysRevD.108.044034

    work page doi:10.1103/physrevd.108.044034 2023

  45. [45]

    Event horizons in static electrovac space-times

    Werner Israel. Event horizons in static electrovac space-times. Commun. Math. Phys., 8:245–260, 1968. doi:10.1007/BF01645859

    doi:10.1007/bf01645859 1968

  46. [46]

    Event horizons in static vacuum space-times

    Werner Israel. Event horizons in static vacuum space-times. Phys. Rev., 164:1776– 1779, Dec 1967. URL: https://link.aps.org/doi/10.1103/PhysRev.164. 1776, doi:10.1103/PhysRev.164.1776

    work page doi:10.1103/physrev.164 1967

  47. [47]

    B. Carter. Axisymmetric black hole has only two degrees of freedom. Phys. Rev. Lett., 26:331–333, Feb 1971. URL: https://link.aps.org/doi/10.1103/ PhysRevLett.26.331, doi:10.1103/PhysRevLett.26.331

    doi:10.1103/physrevlett.26.331 1971

  48. [48]

    Black hole information revisited, 2017

    Andrew Strominger. Black hole information revisited, 2017. URL: https:// arxiv.org/abs/1706.07143, arXiv:1706.07143

    Pith/arXiv arXiv 2017

  49. [49]

    Perry, and Andrew Strominger

    Sasha Haco, Malcolm J. Perry, and Andrew Strominger. Kerr-newman black hole entropy and soft hair, 2019. URL: https://arxiv.org/abs/1902.02247, arXiv:1902.02247

    Pith/arXiv arXiv 2019

  50. [50]

    Soft black hole information paradox: Page curve from maxwell soft hair of a black hole

    Peng Cheng and Yang An. Soft black hole information paradox: Page curve from maxwell soft hair of a black hole. Phys. Rev. D , 103:126020, Jun

    work page

  51. [51]

    William Donnelly and Aron C. Wall. Entanglement entropy of electromagnetic edge modes. Phys. Rev. Lett. , 114(11):111603, 2015. arXiv:1412.1895, doi: 10.1103/PhysRevLett.114.111603

    Pith/arXiv arXiv 2015

  52. [52]

    William Donnelly and Aron C. Wall. Geometric entropy and edge modes of the electromagnetic field. Phys. Rev. D , 94(10):104053, 2016. arXiv:1506.05792, doi:10.1103/PhysRevD.94.104053

    Pith/arXiv arXiv 2016

  53. [53]

    Wormholes, Emergent Gauge Fields, and the Weak Gravity Con- jecture

    Daniel Harlow. Wormholes, Emergent Gauge Fields, and the Weak Gravity Con- jecture. JHEP, 01:122, 2016. arXiv:1510.07911, doi:10.1007/JHEP01(2016) 122. 24

    Pith/arXiv arXiv 2016

  54. [54]

    The Ryu–Takayanagi Formula from Quantum Error Correction

    Daniel Harlow. The Ryu–Takayanagi Formula from Quantum Error Correction. Commun. Math. Phys. , 354(3):865–912, 2017. arXiv:1607.03901, doi:10.1007/ s00220-017-2904-z

    Pith/arXiv arXiv 2017

  55. [55]

    Conformal symme- try and its breaking in two dimensional Nearly Anti-de-Sitter space

    Juan Maldacena, Douglas Stanford, and Zhenbin Yang. Conformal symme- try and its breaking in two dimensional Nearly Anti-de-Sitter space. PTEP, 2016(12):12C104, 2016. arXiv:1606.01857, doi:10.1093/ptep/ptw124

    Pith/arXiv arXiv 2016

  56. [56]

    The kerr spacetime in generalized bondi–sachs coor- dinates

    S J Fletcher and A W C Lun. The kerr spacetime in generalized bondi–sachs coor- dinates. Classical and Quantum Gravity , 20(19):4153, sep 2003. URL: https:// dx.doi.org/10.1088/0264-9381/20/19/302, doi:10.1088/0264-9381/20/19/ 302

    work page doi:10.1088/0264-9381/20/19/302 2003

  57. [57]

    BMS charge algebra

    Glenn Barnich and Cedric Troessaert. BMS charge algebra. JHEP, 12:105, 2011. arXiv:1106.0213, doi:10.1007/JHEP12(2011)105

    Pith/arXiv arXiv 2011

  58. [58]

    Flanagan and David A

    ´Eanna ´E. Flanagan and David A. Nichols. Observer dependence of angular mo- mentum in general relativity and its relationship to the gravitational-wave mem- ory effect. Phys. Rev. D , 92(8):084057, 2015. [Erratum: Phys.Rev.D 93, 049905 (2016)]. arXiv:1411.4599, doi:10.1103/PhysRevD.92.084057

    Pith/arXiv arXiv 2015

  59. [59]

    K. S. Thorne. Multipole Expansions of Gravitational Radiation. Rev. Mod. Phys., 52:299–339, 1980. doi:10.1103/RevModPhys.52.299

    doi:10.1103/revmodphys.52.299 1980

  60. [60]

    R. O. Hansen. Multipole moments of stationary space-times. J. Math. Phys. , 15:46–52, 1974. doi:10.1063/1.1666501

    doi:10.1063/1.1666501 1974

  61. [61]

    Gonz´ alez, and Andrea Puhm

    Laura Donnay, Gaston Giribet, Hern´ an A. Gonz´ alez, and Andrea Puhm. Black hole memory effect. Physical Review D, 98(12), December 2018. URL: http://dx. doi.org/10.1103/PhysRevD.98.124016, doi:10.1103/physrevd.98.124016. 25

    doi:10.1103/physrevd.98.124016 2018

  62. [1965]

    doi:10.1103/PhysRev.140.B516

    doi:10.1103/physrev.140.b516

  63. [1998]

    URL: https://arxiv.org/abs/gr-qc/9809035, arXiv:gr-qc/9809035

    work page internal anchor Pith review Pith/arXiv arXiv

  64. [2014]

    arXiv:1407.0259, doi:10.1088/0264-9381/31/20/205003

    Pith/arXiv arXiv

  65. [2016]

    arXiv:1601.04958, doi:10.1007/JHEP07(2016)137

    Pith/arXiv arXiv 2016

  66. [2017]

    URL: https://link.aps.org/doi/10.1103/PhysRevD.96.084032, doi: 10.1103/PhysRevD.96.084032

    doi:10.1103/physrevd.96.084032

  67. [2021]

    URL: https://link.aps.org/doi/10.1103/PhysRevD.103.126020, doi: 10.1103/PhysRevD.103.126020

    work page doi:10.1103/physrevd.103.126020

This paper was first reviewed by deepseek-v4-flash on August 5, 2026.