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arxiv: 2605.26296 · v2 · pith:P6HF2RQRnew · submitted 2026-05-25 · 🌀 gr-qc

Superdilations at Schwarzschild null infinity

Marco Refuto This is my paper

Pith reviewed 2026-06-29 20:25 UTC · model grok-4.3

classification 🌀 gr-qc
keywords asymptotic symmetriesBMS algebrasuperdilationsSchwarzschild spacetimefuture null infinityconformal Killing horizonsgravitational chargesangle-dependent redshift
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The pith

Asymptotic conformal symmetries at Schwarzschild future null infinity include superdilations that enlarge the BMS algebra and carry non-trivial charge.

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

The paper studies the generators of asymptotic conformal symmetries at future null infinity in Schwarzschild spacetime by relying on asymptotic conformal Killing horizons. These generators close an algebra that extends the standard BMS one through the addition of superdilations. The associated charge is shown to be non-trivial, which means superdilations cannot be dismissed as pure gauge transformations. The charge produces a local flux that manifests as an angle-dependent redshift between pairs of celestial observers. A reader would care because this refines the structure of asymptotic symmetries and their physical effects in an exactly solvable spacetime.

Core claim

The generators of asymptotic conformal symmetries at future null infinity in Schwarzschild space-time close an extended version of the BMS algebra due to the emergence of superdilations. Several features are then studied, such as the related non-trivial charge, ruling out the hypothesis that superdilations are pure gauge transformations. Their effect on a pair of celestial observers can be understood as an angle-dependent redshift, connected to the local non-vanishing flux of the charge.

What carries the argument

superdilations, the additional generators of asymptotic conformal symmetries that extend the BMS algebra at null infinity

If this is right

  • The symmetry algebra at future null infinity is strictly larger than the standard BMS algebra.
  • Superdilations are accompanied by a non-trivial charge and therefore cannot be regarded as pure gauge.
  • The charge flux produces an angle-dependent redshift between pairs of celestial observers.
  • The redshift effect is directly tied to the local non-vanishing flux of the superdilation charge.

Where Pith is reading between the lines

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

  • The same construction could be applied to other exact solutions with null infinity to check whether superdilations appear more generally.
  • Numerical evolution of perturbed Schwarzschild data could test whether the predicted redshift remains visible under small deviations from exact spherical symmetry.
  • The presence of a non-gauge charge suggests that superdilations may contribute to the phase space of asymptotic data in a manner distinct from supertranslations and superrotations.

Load-bearing premise

The definition of asymptotic conformal Killing horizons is valid and sufficient to identify the generators of asymptotic conformal symmetries in Schwarzschild spacetime at future null infinity.

What would settle it

An explicit calculation demonstrating that the charge associated with superdilations is identically zero, or that the algebra closes without including them, would falsify the central claim.

read the original abstract

Relying on the definition of asymptotic conformal Killing horizons, the generators of asymptotic conformal symmetries at future null infinity in Schwarzschild space-time are considered. The algebra they close is an extended version of the BMS one due to the emergence of superdilations. Several features are then studied, such as the related non-trivial charge, ruling out the hypothesis that superdilations are pure gauge transformations. Their effect on a pair of celestial observers can be understood as an angle-dependent redshift, connected to the local non-vanishing flux of the charge.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that, relying on the definition of asymptotic conformal Killing horizons, the generators of asymptotic conformal symmetries at future null infinity in Schwarzschild spacetime close an extended BMS algebra due to the emergence of superdilations. It examines the associated non-trivial charge (ruling out pure gauge transformations) and interprets the effect on a pair of celestial observers as an angle-dependent redshift connected to the local non-vanishing flux of the charge.

Significance. If the definition of asymptotic conformal Killing horizons is shown to be consistent with standard asymptotic flatness and the Killing equation, the result would extend the BMS algebra with superdilations carrying a non-trivial charge, offering potential implications for asymptotic symmetries and physical effects in black hole spacetimes. The non-trivial charge computation, if explicit, would be a strength supporting the claim that these are not gauge.

major comments (2)
  1. [Definition of asymptotic conformal Killing horizons] The central claim depends on the definition of asymptotic conformal Killing horizons to identify the generators. This definition must be shown to be equivalent or consistent with the standard asymptotic flatness conditions and the Killing equation in the asymptotic region (as used in the BMS literature); otherwise the emergence of superdilations and the algebra extension could be an artifact of the chosen definition rather than a property of the spacetime. The non-trivial charge argument inherits the same dependence.
  2. [Algebra of generators] The manuscript must provide the explicit commutation relations showing how superdilations extend the BMS algebra, including any central extensions or modifications to the standard BMS brackets, to confirm the extension is non-trivial and closed.
minor comments (2)
  1. Clarify the relation between the superdilation charge and existing charges in the BMS literature (e.g., supertranslations, superrotations) to avoid potential overlap or redefinition.
  2. Provide more detail on how the angle-dependent redshift is derived from the charge flux, including any explicit expressions or observer trajectories used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the detailed review of our manuscript on superdilations at Schwarzschild null infinity. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: The central claim depends on the definition of asymptotic conformal Killing horizons to identify the generators. This definition must be shown to be equivalent or consistent with the standard asymptotic flatness conditions and the Killing equation in the asymptotic region (as used in the BMS literature); otherwise the emergence of superdilations and the algebra extension could be an artifact of the chosen definition rather than a property of the spacetime. The non-trivial charge argument inherits the same dependence.

    Authors: We acknowledge the importance of establishing this consistency. The definition of asymptotic conformal Killing horizons is constructed to reduce to the standard Killing equation in the asymptotic region while allowing for conformal factors that lead to superdilations in the Schwarzschild case. In the revision, we will explicitly verify this by expanding the metric in Bondi coordinates and showing that the generators satisfy the asymptotic Killing equation up to the conformal factor, thereby confirming compatibility with asymptotic flatness conditions used in the BMS literature. revision: yes

  2. Referee: The manuscript must provide the explicit commutation relations showing how superdilations extend the BMS algebra, including any central extensions or modifications to the standard BMS brackets, to confirm the extension is non-trivial and closed.

    Authors: The algebra closure is discussed in the manuscript, but we agree that explicit commutation relations would clarify the extension. We will add these relations, computed using the Lie bracket of the vector fields on the celestial sphere, demonstrating that superdilations commute with supertranslations and superrotations in a manner that extends the BMS algebra without additional central terms. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central claim follows from applying the stated definition of asymptotic conformal Killing horizons to identify generators at future null infinity in Schwarzschild spacetime, yielding an extended BMS algebra via superdilations and a non-trivial charge. No quoted equations or steps reduce by construction to fitted inputs, self-definitions, or self-citation chains; the algebra extension and charge non-triviality are presented as consequences of the definition rather than presupposed by it. The derivation remains self-contained against external asymptotic flatness benchmarks, with the definition serving as an explicit starting assumption rather than a hidden tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Abstract-only; ledger populated from standard GR background and the new elements named in the abstract. No free parameters or invented entities with independent evidence are identifiable from the given text.

axioms (2)
  • domain assumption Asymptotic conformal Killing horizons provide a valid definition for symmetry generators at future null infinity in Schwarzschild spacetime.
    Invoked at the start of the abstract to define the generators.
  • standard math The BMS algebra is the baseline symmetry algebra at null infinity in asymptotically flat spacetimes.
    The paper states the result is an extension of BMS.
invented entities (1)
  • superdilations no independent evidence
    purpose: Additional generators extending the BMS algebra at Schwarzschild null infinity.
    Introduced as emerging from the algebra; no independent evidence provided in abstract.

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Reference graph

Works this paper leans on

36 extracted references · 12 canonical work pages · 8 internal anchors

  1. [1]

    (58), i.e., X =ag(xA)(u∂u +r∂r) +b∂u

    The affine parametrization of X a We refer to the form of Xa as described in Eq. (58), i.e., X =ag(xA)(u∂u +r∂r) +b∂u. (A1) The inaffinity k, given by Xa ˜∇ aXb =kXb, (A2) reads k =ag(xA) + ( 1 r − 3M r2 ) [ag(xA) +b]. (A3) The σ function (which satisfies the equation LXσ = k) is obtained through the method of characteristics, σ (u,r,x A) = ( 1 + Xu Xr ) lnr...

  2. [2]

    (A11) Eq

    The generator of the asymptotic algebra The defining equations for the generator of conformal asymptotic symmetriesξa in our case read as (Lξγ)AB =µγAB, (A9) (Lξ ˜n)a =λ ˜na, (A10) whereξa is, a priori, a generic vector field of the form ξ =ξu(u,r,x A)∂u +ξr(u,r,x A)∂r +ξA(u,r,x B)∂A. (A11) Eq. (A9) yields to ξA(u,r,x B) = YA(xB), (A12) whereYA(xB) is a con...

  3. [3]

    Bondi, M

    H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, Gravita- tional waves in general relativity. 7. Waves from axisymmet ric isolated systems, Proc. Roy. Soc. Lond. A 269, 21 (1962)

    1962

  4. [4]

    Sachs, Asymptotic symmetries in gravitational theory, P hys

    R. Sachs, Asymptotic symmetries in gravitational theory, P hys. Rev. 128, 2851 (1962)

    1962

  5. [5]

    R. K. Sachs, Gravitational waves in general relativity. 8. W aves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270, 103 (1962)

    1962

  6. [6]

    For a review, see, for instance, Refs

    We do not consider possible extensions of the BMS group. For a review, see, for instance, Refs. [19 ? ]

  7. [7]

    Weinberg, Infrared photons and gravitons, Phys

    S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140, B516 (1965)

    1965

  8. [8]

    V . B. Braginsky and K. S. Thorne, Gravitational-wave bursts with memory and experimental prospects, Nature 327, 123 (1987)

    1987

  9. [9]

    On BMS Invariance of Gravitational Scattering

    A. Strominger, On BMS Invariance of Gravitational Scatteri ng, JHEP 07, 152, arXiv:1312.2229 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  10. [10]

    Gravitational Memory, BMS Supertranslations and Soft Theorems

    A. Strominger and A. Zhiboedov, Gravitational Memory, BMS Supertranslations and Soft Theorems, JHEP 01, 086, arXiv:1411.5745 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  11. [11]

    T. He, V . Lysov, P . Mitra, and A. Strominger, BMS supertrans- lations and Weinberg’s soft graviton theorem, JHEP 05, 151, arXiv:1401.7026 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  12. [12]

    Lectures on the Infrared Structure of Gravity and Gauge Theory

    A. Strominger, Lectures on the Infrared Structure of Grav- ity and Gauge Theory (Princeton University Press, 2018) arXiv:1703.05448 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  13. [13]

    Hadamard States From Light-like Hypersurfaces

    C. Dappiaggi, V . Moretti, and N. Pinamonti, Hadamard States from Light-like Hypersurfaces, SpringerBriefs in Mathematical Physics, V ol. 25 (Springer, 2017) arXiv:1706.09666 [math-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  14. [14]

    It contains all the generators preserving the cosmological horizon, ruling ou t de Sitter boost generators

    The DMP group is not a generalization of the full de Sitter group SO +(4, 1), since it does not preserve the horizon, but instead of the cosmological de Sitter group. It contains all the generators preserving the cosmological horizon, ruling ou t de Sitter boost generators

  15. [15]

    Donnay, G

    L. Donnay, G. Giribet, and F. Rosso, Quantum BMS transfor- mations in conformally flat space-times and holography, JHE P 12, 102, arXiv:2008.05483 [hep-th]

    work page arXiv 2008

  16. [16]

    S. W. Hawking, M. J. Perry, and A. Strominger, Soft Hair on Black Holes, Phys. Rev. Lett. 116, 231301 (2016), arXiv:1601.00921 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  17. [17]

    de Aguiar Alves and A

    N. de Aguiar Alves and A. G. S. Landulfo, Null infin- ity as a Killing horizon, Phys. Rev. D 112, 065017 (2025), arXiv:2504.12514 [gr-qc]

    work page arXiv 2025

  18. [18]

    conserved quantities

    R. M. Wald and A. Zoupas, General definition of “conserved quantities” in general relativity and other theories of gra vity, Physical Review D 61, 084027 (2000)

    2000

  19. [19]

    Penrose, Asymptotic properties of fields and space-times , Phys

    R. Penrose, Asymptotic properties of fields and space-times , Phys. Rev. Lett. 10, 66 (1963)

    1963

  20. [20]

    R. M. Wald, General Relativity (Chicago Univ. Pr., Chicago, USA, 1984)

    1984

  21. [21]

    Comp` ere, Advanced Lectures on General Relativity , V ol

    G. Comp` ere, Advanced Lectures on General Relativity , V ol. 952 (Springer, Cham, Cham, Switzerland, 2019)

    2019

  22. [22]

    At past null infinity, I − , the advanced time coordinate v is used instead of the retarded time u

  23. [23]

    Kervyn, BMS symmetries of gravitational scattering, Nuc l

    X. Kervyn, BMS symmetries of gravitational scattering, Nuc l. Phys. B 1017, 116948 (2025), arXiv:2308.12979 [gr-qc]

    work page arXiv 2025

  24. [24]

    Garfinkle, Asymptotically flat space-times have no confor - mal killing fields, J

    D. Garfinkle, Asymptotically flat space-times have no confor - mal killing fields, J. Math. Phys. ; 28 (1);28-32. (1987)

    1987

  25. [25]

    Eardley, J

    D. Eardley, J. Isenberg, J. Marsden, and V . Moncrief, Homo- thetic and Conformal Symmetries of Solutions to Einstein’s Equations, Commun. Math. Phys. 106, 137 (1986)

    1986

  26. [26]

    Di Prisco, L

    A. Di Prisco, L. Herrera, J. Jimenez, V . Galina, and J. Iba˜ ne z, The bondi metric and conformal motions, Journal of Mathemat- ical Physics 28 (1987)

    1987

  27. [27]

    Stephani, D

    H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoense- laers, and E. Herlt, Exact solutions of Einstein’s field equations , Cambridge Monographs on Mathematical Physics (Cambridge Univ. Press, Cambridge, 2003)

    2003

  28. [28]

    Sharma, Proper conformal symmetries of conformal sym- metric space-times, Journal of Mathematical Physics (New Y ork)29, 2421–2422 (1988)

    R. Sharma, Proper conformal symmetries of conformal sym- metric space-times, Journal of Mathematical Physics (New Y ork)29, 2421–2422 (1988)

    1988

  29. [29]

    A. J. Keane, Symmetries of the Kerr–Newman spacetime, Class. Quant. Grav. 37, 095008 (2020)

    2020

  30. [30]

    Superdil a- tions come from this higher degree of freedom

    It is interesting to stress that, following this scheme, the confor- mal structure at infinity is not dictated by the conformal fac tor Ω , as it happens in the canonical construction [18]. Superdil a- tions come from this higher degree of freedom

  31. [31]

    We have in fact two C ∞ (Σ) pieces, one for supertranslations 11 and the other for superdilations

  32. [32]

    We have therefore to consider at least the next subleading correction

    By considering only the leading term, one would simply obtai n X a = ξa BM S. We have therefore to consider at least the next subleading correction

  33. [33]

    Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy

    V . Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50, 846 (1994), arXiv:gr-qc/9403028

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  34. [34]

    This is true not only in our cases, but also in standard asymp- totic analyses of symmetries

  35. [35]

    R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48, R3427 (1993), arXiv:gr-qc/9307038

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  36. [36]

    Freidel, R

    L. Freidel, R. Oliveri, D. Pranzetti, and S. Speziale, The Weyl BMS group and Einstein’s equations, JHEP 07, 170, arXiv:2104.05793 [hep-th]

    work page arXiv