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arxiv: 2511.07111 · v4 · submitted 2025-11-10 · ✦ hep-th · gr-qc

From BTZ Perturbations to Schwarzian Modes: A Geometrical and Perturbative Analysis

Lucas Acito , Mat\'ias N. Semp\'e This is my paper

Pith reviewed 2026-05-17 23:39 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords BTZ black holeSchwarzian modesmetric perturbationsKerr-Schild constructionfinite temperatureAdS3 geometryblack hole perturbations
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The pith

Schwarzian modes emerge from the general perturbative solution of the BTZ black hole at finite temperature under conditions that exclude rotational modes, and they are recovered equivalently via a Kerr-Schild geometric construction.

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

The paper derives the Schwarzian modes explicitly from the full geometry of the BTZ black hole at finite temperature. It identifies the precise conditions under which these modes appear in the general solution and demonstrates that rotational modes are absent. A sympathetic reader would care because this clarifies how certain low-energy modes arise in black hole perturbations and shows an equivalence between perturbative and purely geometric methods. The work also recovers the same modes through a Kerr-Schild construction, highlighting a link between the two approaches.

Core claim

In the full geometry of the BTZ black hole at finite temperature, the Schwarzian modes arise from the general perturbative solution under specific conditions that confirm the absence of rotational modes. The identical modes can be obtained through a purely geometric Kerr-Schild construction, which supplies an equivalent description of the Schwarzian sector and indicates a correspondence with double copy ideas.

What carries the argument

The perturbative ansatz for metric perturbations together with the Kerr-Schild null-vector addition, which together isolate the Schwarzian modes from the general solution in the BTZ geometry.

If this is right

  • The Schwarzian modes appear in the complete BTZ geometry without rotational contributions under the derived conditions.
  • Perturbative and geometric Kerr-Schild methods produce equivalent descriptions of the Schwarzian sector.
  • The correspondence between the two approaches extends the understanding of how Schwarzian modes arise at finite temperature.
  • The construction points to a possible connection with double copy relations for these modes.

Where Pith is reading between the lines

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

  • The equivalence of methods may allow simpler computations in related asymptotically AdS geometries by selecting the more convenient approach.
  • Similar geometric constructions could be tested in other black hole solutions to check whether Schwarzian modes emerge universally under finite temperature.
  • The absence of rotational modes might constrain the spectrum in holographic models that rely on BTZ backgrounds.

Load-bearing premise

The chosen perturbative ansatz and the Kerr-Schild null-vector addition both faithfully capture the same physical sector of the BTZ solution without additional hidden constraints or gauge choices that would alter the mode spectrum.

What would settle it

A calculation of the full mode spectrum from the general BTZ perturbation solution that either produces rotational modes or fails to recover the Schwarzian modes when the stated conditions are applied would falsify the central claim.

read the original abstract

We provide a detailed derivation of the Schwarzian modes in the full geometry of the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole at finite temperature, establishing the precise conditions under which they emerge from the general solution, thereby clarifying the absence of rotational modes in the full geometry. In addition, we demonstrate that the same modes can be recovered through a purely geometric Kerr-Schild construction. This equivalent approach offers a new geometric understanding of the Schwarzian sector and highlights the correspondence between perturbative and pure geometric approaches, additionally it provides a connection with double copy.

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 derives Schwarzian modes from linearized metric perturbations on the finite-temperature BTZ black hole, identifying the precise conditions under which these modes appear in the general solution while proving the absence of rotational modes. It further recovers the identical spectrum via a Kerr-Schild null-vector deformation of the BTZ geometry, establishing an equivalence between the perturbative and purely geometric approaches and noting a connection to the double-copy construction.

Significance. If the central derivations hold and the perturbative ansatz is shown to be unrestricted, the result would clarify the geometric origin of Schwarzian dynamics in AdS3 gravity at finite temperature, strengthening links between near-extremal black-hole thermodynamics, holographic models, and double-copy relations. The dual geometric construction offers an independent route that could aid generalization beyond three dimensions.

major comments (2)
  1. [Perturbative analysis section (around the linearized equations)] The perturbative analysis must demonstrate that the chosen ansatz for metric perturbations is the most general solution compatible with the BTZ isometries, asymptotic AdS3 fall-off, and linearized Einstein equations; if residual gauge fixing or boundary conditions implicitly set rotational (off-diagonal) components to zero before solving, the reported absence of rotational modes and the emergence conditions for Schwarzian modes would be artifacts rather than intrinsic results. This is load-bearing for the central claim that the modes emerge from the general solution.
  2. [Kerr-Schild construction section] In the Kerr-Schild construction, the specific choice of null vector must be shown not to impose hidden constraints that restrict the mode spectrum to match the perturbative result by construction; an explicit comparison of the resulting metric components and their fall-off behavior with the unrestricted perturbative solution is required to confirm equivalence.
minor comments (2)
  1. [Abstract and conclusions] The abstract asserts 'precise conditions' and 'detailed derivation' but the main text should include explicit error estimates or numerical checks confirming that higher-order terms do not reintroduce rotational modes.
  2. [Throughout] Notation for the Schwarzian parameter and the temperature should be unified across the perturbative and geometric sections to avoid confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We have carefully considered the major comments and provide point-by-point responses below. We believe these clarifications strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Perturbative analysis section (around the linearized equations)] The perturbative analysis must demonstrate that the chosen ansatz for metric perturbations is the most general solution compatible with the BTZ isometries, asymptotic AdS3 fall-off, and linearized Einstein equations; if residual gauge fixing or boundary conditions implicitly set rotational (off-diagonal) components to zero before solving, the reported absence of rotational modes and the emergence conditions for Schwarzian modes would be artifacts rather than intrinsic results. This is load-bearing for the central claim that the modes emerge from the general solution.

    Authors: We agree that establishing the generality of the ansatz is crucial for the validity of our claims. In the original manuscript, we derived the perturbations starting from the linearized Einstein equations with the BTZ background, imposing only the isometries and AdS3 fall-offs. However, to make this explicit and address the referee's concern, we have revised the perturbative analysis section to include a more detailed parameterization of the general metric perturbation, including potential off-diagonal rotational components. We solve the equations without prior gauge fixing that eliminates them and show that the equations and boundary conditions naturally lead to their absence at finite temperature. This confirms that the Schwarzian modes emerge from the general solution as stated. We have added this demonstration in the revised version. revision: yes

  2. Referee: [Kerr-Schild construction section] In the Kerr-Schild construction, the specific choice of null vector must be shown not to impose hidden constraints that restrict the mode spectrum to match the perturbative result by construction; an explicit comparison of the resulting metric components and their fall-off behavior with the unrestricted perturbative solution is required to confirm equivalence.

    Authors: We thank the referee for this suggestion, which helps to solidify the equivalence between the two approaches. The choice of null vector in the Kerr-Schild construction is guided by the requirement to preserve the asymptotic AdS3 structure and is consistent with the double-copy framework. In the revised manuscript, we have included an explicit side-by-side comparison of the metric components and their asymptotic fall-off behaviors between the Kerr-Schild deformed metric and the general perturbative solution. This comparison shows that the spectra match without the null vector imposing additional restrictions beyond the geometric requirements. We have expanded the relevant section accordingly. revision: yes

Circularity Check

0 steps flagged

No circularity: independent perturbative derivation and geometric construction

full rationale

The paper derives Schwarzian modes by solving the linearized equations on the finite-temperature BTZ background starting from a general solution and separately recovers the same modes via a Kerr-Schild null-vector deformation. No equations or claims in the provided abstract or description reduce a prediction to a fitted input, rename a known result, or rely on a self-citation chain for the central uniqueness or emergence conditions. The two routes are presented as cross-verifying independent approaches without evident self-definitional or ansatz-smuggling steps, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions of three-dimensional Einstein gravity with negative cosmological constant and on the validity of linear perturbation theory around the BTZ background; no new free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption BTZ metric is an exact solution of 3D Einstein gravity with negative cosmological constant
    Invoked as the background geometry for all perturbations and constructions.
  • domain assumption Linearized perturbations around BTZ capture the relevant low-energy modes
    Required for the perturbative derivation to isolate Schwarzian modes.

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Forward citations

Cited by 1 Pith paper

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

  1. Revisiting near-extremal and near-BPS black holes in AdS3 supergravity

    hep-th 2026-04 unverdicted novelty 6.0

    In AdS3 supergravity, the gravitational path integral at low temperatures in the near-horizon region is inequivalent to that of the BTZ background, with distinct contributions from bosonic fluctuations, Chern-Simons f...

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages · cited by 1 Pith paper · 12 internal anchors

  1. [1]

    G. W. Gibbons and S. W. Hawking,Action integrals and partition functions in quantum gravity,Phys. Rev. D15(1977) 2752–2756

    work page 1977

  2. [2]

    Supergravity spectrum on AdS_2 x S^2

    J. Michelson and M. Spradlin,Supergravity spectrum onAdS(2)×S 2,JHEP09(1999) 029, [hep-th/9906056]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  3. [3]

    D. N. Page,Thermodynamics of near extreme black holes, 2000

    work page 2000

  4. [4]

    Banerjee, R

    S. Banerjee, R. Gupta and A. Sen,Logarithmic corrections to extremal black hole entropy from quantum entropy function,JHEP03(2011) 147

    work page 2011

  5. [5]

    Banerjee, R

    S. Banerjee, R. Gupta, I. Mandal and A. Sen,Logarithmic corrections toN= 4andN= 8 black hole entropy: a one loop test of quantum gravity,JHEP11(2011) 143

    work page 2011

  6. [6]

    Logarithmic Corrections to N=2 Black Hole Entropy: An Infrared Window into the Microstates

    A. Sen,Logarithmic Corrections to N=2 Black Hole Entropy: An Infrared Window into the Microstates,Gen. Rel. Grav.44(2012) 1207–1266, [1108.3842]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  7. [7]

    Logarithmic Corrections to Rotating Extremal Black Hole Entropy in Four and Five Dimensions

    A. Sen,Logarithmic Corrections to Rotating Extremal Black Hole Entropy in Four and Five Dimensions,Gen. Rel. Grav.44(2012) 1947–1991, [1109.3706]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  8. [8]

    Logarithmic Corrections to Schwarzschild and Other Non-extremal Black Hole Entropy in Different Dimensions

    A. Sen,Logarithmic Corrections to Schwarzschild and Other Non-extremal Black Hole Entropy in Different Dimensions,JHEP04(2013) 156, [1205.0971]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  9. [9]

    Ghosh, H

    A. Ghosh, H. Maxfield and G. J. Turiaci,A universal Schwarzian sector in two-dimensional conformal field theories,JHEP05(2020) 104, [1912.07654]

    work page arXiv 2020

  10. [10]

    L. V. Iliesiu and G. J. Turiaci,The statistical mechanics of near-extremal black holes,JHEP 05(2021) 145, [2003.02860]

    work page arXiv 2021

  11. [11]

    L. V. Iliesiu, S. Murthy and G. J. Turiaci,Revisiting the logarithmic corrections to the black hole entropy,JHEP07(2025) 058, [2209.13608]

    work page arXiv 2025

  12. [12]

    H. K. Kunduri, J. Lucietti and H. S. Reall,Near-horizon symmetries of extremal black holes, Class. Quant. Grav.24(2007) 4169–4190, [0705.4214]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  13. [13]

    Kapec, A

    D. Kapec, A. Sheta, A. Strominger and C. Toldo,Logarithmic Corrections to Kerr Thermodynamics,Phys. Rev. Lett.133(2024) 021601, [2310.00848]

    work page arXiv 2024

  14. [14]

    Cassani and S

    D. Cassani and S. Murthy,Quantum black holes: supersymmetry and exact results, 2025

    work page 2025

  15. [15]

    Maulik, L

    S. Maulik, L. A. Pando Zayas, A. Ray and J. Zhang,Universality in logarithmic temperature corrections to near-extremal rotating black hole thermodynamics in various dimensions, JHEP06(2024) 034, [2401.16507]

    work page arXiv 2024

  16. [16]

    Rakic, M

    I. Rakic, M. Rangamani and G. J. Turiaci,Thermodynamics of the near-extremal Kerr spacetime,JHEP06(2024) 011, [2310.04532]

    work page arXiv 2024

  17. [17]

    One loop corrections to the thermodynamics of near-extremal Kerr-(A)dS black holes from Heun equation,

    P. Arnaudo, G. Bonelli and A. Tanzini,One loop corrections to the thermodynamics of near-extremal Kerr-(A)dS black holes from Heun equation,2506.08959

    work page arXiv

  18. [18]

    Looking at extremal black holes from very far away,

    M. Kolanowski, D. Marolf, I. Rakic, M. Rangamani and G. J. Turiaci,Looking at extremal black holes from very far away,JHEP04(2025) 020, [2409.16248]

    work page arXiv 2025

  19. [19]

    Castro, R

    A. Castro, R. Mancilla and I. Papadimitriou,Near-extremal dynamics away from the horizon,2507.01126

    work page arXiv

  20. [20]

    The Black Hole in Three Dimensional Space Time

    M. Banados, C. Teitelboim and J. Zanelli,The Black hole in three-dimensional space-time, Phys. Rev. Lett.69(1992) 1849–1851, [hep-th/9204099]. – 28 –

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  21. [21]

    Tweaking one-loop determinants in AdS$_3$

    A. Castro, C. Keeler and P. Szepietowski,Tweaking one-loop determinants in AdS 3,JHEP 10(2017) 070, [1707.06245]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  22. [22]

    Higher spin quasinormal modes and one-loop determinants in the BTZ black hole

    S. Datta and J. R. David,Higher Spin Quasinormal Modes and One-Loop Determinants in the BTZ black Hole,JHEP03(2012) 079, [1112.4619]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  23. [23]

    Acosta, A

    J. Acosta, A. Garbarz, A. Goya and M. Leston,One-loop partition function, gauge accessibility and spectra in AdS 3 gravity,JHEP12(2021) 097, [2109.06938]

    work page arXiv 2021

  24. [24]

    Most general AdS_3 boundary conditions

    D. Grumiller and M. Riegler,Most general AdS 3 boundary conditions,JHEP10(2016) 023, [1608.01308]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  25. [25]

    Del Duca, L

    V. Del Duca, L. Dixon and F. Maltoni,New color decompositions for gauge amplitudes at tree and loop level,Nuclear Physics B571(2000)

    work page 2000

  26. [26]

    Monteiro, D

    R. Monteiro, D. O’Connell and C. D. White,”black holes and the double copy”,Journal of High Energy Physics56(2014)

    work page 2014

  27. [27]

    Carrillo-Gonzalez, R

    M. Carrillo-Gonzalez, R. Penco and M. Trodden,”the classical double copy in maximally symmetric spacetimes”, 2018

    work page 2018

  28. [28]

    The classical double copy in three spacetime dimensions

    M. Carrillo Gonz´ alez, B. Melcher, K. Ratliff, S. Watson and C. D. White,The classical double copy in three spacetime dimensions,JHEP07(2019) 167, [1904.11001]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  29. [29]

    Balasubramanian and P

    V. Balasubramanian and P. Kraus,A stress tensor for anti-de Sitter gravity, Communications in Mathematical Physics2(1999) 208, [9902121]

    work page 1999

  30. [30]

    C. T. M. Ba˜ nados and J. Zanelli,Black hole in three-dimensional spacetime,Phys. Rev. Lett. 69(1992)

    work page 1992

  31. [31]

    Quasinormal Corrections to Near-Extremal Black Hole Thermodynamics,

    D. Kapec, Y. T. A. Law and C. Toldo,Quasinormal corrections to near-extremal black hole thermodynamics,JHEP06(2025) 069, [2409.14928]

    work page arXiv 2025

  32. [32]

    S. M. Christensen and M. J. Duff,Quantizing Gravity with a Cosmological Constant,Nucl. Phys. B170(1980) 480–506

    work page 1980

  33. [33]

    S. M. Christensen and M. J. Duff,New Gravitational Index Theorems and Supertheorems, Nucl. Phys. B154(1979) 301–342

    work page 1979

  34. [34]

    Lagrangian formulation of irreducible massive fields of arbitrary spin in 2+1 dimensions

    I. Tyutin and M. Vasiliev,Lagrangian formulation of irreducible massive fields of arbitrary spin in 2+1 dimensions,hep-th/9704132

    work page internal anchor Pith review Pith/arXiv arXiv

  35. [35]

    Notes on gauge fields and discrete series representations in de Sitter spacetimes,

    A. Rios Fukelman, M. Semp´ e and G. A. Silva,Notes on gauge fields and discrete series representations in de Sitter spacetimes,JHEP01(2024) 011, [2310.14955]

    work page arXiv 2024

  36. [36]

    Z. Bern, J. J. M. Carrasco and H. Johansson,Perturbative quantum gravity as a double copy of gauge theory,Physical Review Letters105(2010)

    work page 2010

  37. [37]

    J. D. Brown and M. Henneaux,Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,Commun. Math. Phys.104 (1986) 207–226

    work page 1986

  38. [38]

    Comp` ere, W

    G. Comp` ere, W. Song and A. Strominger,New boundary conditions for ads3,Journal of High Energy Physics2013(2013)

    work page 2013

  39. [39]

    Ghosh, H

    A. Ghosh, H. Maxfield and G. J. Turiaci,A universal schwarzian sector in two-dimensional conformal field theories,Journal of High Energy Physics2020(2020)

    work page 2020

  40. [40]

    Castro, F

    A. Castro, F. Larsen and I. Papadimitriou,5d rotating black holes and the nads2/ncft1 correspondence,Journal of High Energy Physics2018(2018) . – 29 –

    work page 2018

  41. [41]

    Moitra, S

    U. Moitra, S. K. Sake, S. P. Trivedi and V. Vishal,Jackiw-teitelboim gravity and rotating black holes,Journal of High Energy Physics2019(2019)

    work page 2019

  42. [42]

    Kapec, Y

    D. Kapec, Y. T. A. Law and C. Toldo,Quasinormal corrections to near-extremal black hole thermodynamics,Journal of High Energy Physics2025(2025)

    work page 2025

  43. [43]

    Taub,Generalised kerr-schild space-times,Annals of Physics134(1981) 326–372

    A. Taub,Generalised kerr-schild space-times,Annals of Physics134(1981) 326–372

    work page 1981

  44. [44]

    One-loop Partition Functions of 3D Gravity

    S. Giombi, A. Maloney and X. Yin,One-loop Partition Functions of 3D Gravity,JHEP08 (2008) 007, [0804.1773]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  45. [45]

    Exact solu- tions of Einstein’s field equations,

    H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers and E. Herlt,Exact solutions of Einstein’s field equations. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, 2003, 10.1017/CBO9780511535185

    work page doi:10.1017/cbo9780511535185 2003

  46. [46]

    Farina, J

    C. Farina, J. Gamboa and A. J. Segui-Santonja,Motion and trajectories of particles around three-dimensional black holes,Classical and Quantum Gravity10(1993) L193

    work page 1993

  47. [47]

    N. Cruz, C. Mart´ ınez and L. Pe˜ na,Geodesic structure of the (2 + 1)-dimensional btz black hole,Classical and Quantum Gravity11(1994) 2731

    work page 1994

  48. [48]

    Blau,Lecture Notes on General Relativity

    M. Blau,Lecture Notes on General Relativity. Bern University, 2025

    work page 2025

  49. [49]

    S. W. Hawking and G. F. R. Ellis,The Large Scale Structure of Space-Time. Cambridge University Press, 1973

    work page 1973

  50. [50]

    R. M. Wald,General Relativity. Chicago University Press, 1984

    work page 1984

  51. [51]

    Deser, R

    S. Deser, R. Jackiw and G. ’t Hooft,Three-Dimensional Einstein Gravity: Dynamics of Flat Space,Annals Phys.152(1984) 220

    work page 1984

  52. [52]

    B. C. Xanthopoulos,Exact vacuum solutions of Einstein’s equation from linearised solutions., Journal of Mathematical Physics19(Jan., 1978) 1607–1609

    work page 1978

  53. [53]

    Witten,(2+1)-Dimensional Gravity as an Exactly Soluble System,Nucl

    E. Witten,(2+1)-Dimensional Gravity as an Exactly Soluble System,Nucl. Phys. B311 (1988) 46

    work page 1988

  54. [54]

    A. C. A. Bac and D. Jain,in preparation,

    work page

  55. [55]

    Beetar, M

    C. Beetar, M. Carrillo Gonz´ alez, S. Jaitly and T. Keseman,Double copy in AdS 3 from minitwistor space,JHEP03(2025) 125, [2410.23342]

    work page arXiv 2025

  56. [56]

    Lescano and J

    E. Lescano and J. A. Rodr´ ıguez,Constructing conformal double field theory through a double copy map,Phys. Rev. D112(2025) 026004, [2408.11892]. – 30 –

    work page arXiv 2025