pith. sign in

arxiv: 2605.05821 · v1 · submitted 2026-05-07 · 🌀 gr-qc

Quasi-normal modes of a multi-dimensional rotating Kerr black hole

Mattia Villani This is my paper

Pith reviewed 2026-05-08 07:26 UTC · model grok-4.3

classification 🌀 gr-qc
keywords quasi-normal modesKerr black holehigher-dimensional spacetimeTeukolsky equationmultispinor formalismgravitational perturbationsNewman-Penrose formalismcompactified extra dimensions
0
0 comments X

The pith

A multispinor formalism generalizes the Newman-Penrose approach to derive the Teukolsky equation and quasi-normal modes for gravitational perturbations in higher-dimensional Kerr spacetimes.

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

The paper develops a multispinor formalism based on vectors of two-spinors to handle gravitational perturbations in Kerr black holes with more than four dimensions and compactified extra dimensions. This construction creates an analogy to the four-dimensional Newman-Penrose formalism, which then yields the Teukolsky equation for gravitational waves. The approach is illustrated explicitly in six dimensions but is presented as readily generalizable, providing a route to compute quasi-normal mode frequencies that characterize the ringdown of such black holes.

Core claim

The multispinor formalism, constructed from vectors of two-spinors, permits a direct analogy to the Newman-Penrose tetrad formalism and thereby allows derivation of the separable Teukolsky equation governing gravitational perturbations of six-dimensional Kerr spacetime with compactified extra dimensions, from which the quasi-normal mode spectrum can be extracted.

What carries the argument

The multispinor formalism, a vector-of-two-spinors construction that extends spinorial methods to higher dimensions while preserving the structure needed for separation of variables in the wave equation.

If this is right

  • The Teukolsky equation for gravitational perturbations becomes available in six-dimensional Kerr geometry with compact extra dimensions.
  • Quasi-normal mode frequencies can be computed by standard separation and boundary-value methods once the equation is obtained.
  • The same multispinor steps apply without change to other dimensions d greater than four.
  • The resulting modes describe the linear stability and ringdown behavior of rotating black holes in these higher-dimensional settings.

Where Pith is reading between the lines

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

  • The formalism could be used to model gravitational wave signals expected from black hole mergers in extra-dimension scenarios.
  • Numerical evaluation of the six-dimensional quasi-normal mode spectrum would provide concrete benchmarks for stability studies in higher-dimensional general relativity.
  • The method may connect to existing calculations of scalar and electromagnetic perturbations in higher-dimensional Kerr geometries by supplying the gravitational sector.

Load-bearing premise

The multispinor construction extends consistently to Kerr spacetimes in d greater than four with compactified extra dimensions while keeping the Teukolsky equation separable and the boundary conditions intact for quasi-normal mode calculations.

What would settle it

An explicit derivation in six dimensions that produces a non-separable Teukolsky equation or yields quasi-normal mode frequencies that fail to reduce to the four-dimensional Kerr values when the extra dimension size is taken to zero.

read the original abstract

The aim of this paper is to present a general way to calculate quasi-normal modes (QNM) of the Teukolsky equation for higher dimensional (d > 4) Kerr spacetime with compactified extra dimensions. In order to do so, we develop a formalism derived from spinors: we call it multispinor formalism. It is based on vectors of two-spinors and permits us to develop a formalism analogous to that of Newman-Penrose in 4d. From this we show how to derive the Teukolsky equation for gravitational perturbations and calculate the QNM. In order to keep calculations simple we fix, as an example, the dimension number to be six, but the work can be readily generalized to other spacetime dimensions.

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 / 1 minor

Summary. The paper introduces a multispinor formalism constructed from vectors of two-spinors as a higher-dimensional analog of the Newman-Penrose tetrad formalism. Using this, it derives the Teukolsky master equation for gravitational perturbations of a six-dimensional Kerr black hole with one compactified extra dimension and computes the associated quasi-normal modes, with the claim that the approach generalizes to other dimensions.

Significance. If the multispinor construction yields a decoupled, separable Teukolsky equation whose solutions satisfy the required boundary conditions (including periodicity in the compact direction), the work would provide a useful extension of the 4D Teukolsky formalism to rotating black holes in compactified higher-dimensional spacetimes. This could enable systematic QNM calculations relevant to stability analyses and gravitational-wave phenomenology in extra-dimension models. The explicit d=6 example is a positive step toward verifiability.

major comments (2)
  1. Derivation of the Teukolsky equation (Section 3): the multispinor formalism must be shown to produce a decoupled master equation for the gravitational perturbations. Explicitly verify that the equation separates into radial and angular factors in Boyer-Lindquist coordinates with the compact dimension, and that the higher-dimensional Weyl components and Lichnerowicz operator do not introduce non-separable mixing terms.
  2. Reduction to four dimensions (Section 4 or the limit discussion): when the compactification radius is sent to zero (or d is formally set to 4), the derived Teukolsky equation and its QNM spectrum must recover the standard 4D Teukolsky equation and known QNM frequencies. This limit check is load-bearing for validating the boundary conditions and separability assumptions used for the QNM calculation.
minor comments (1)
  1. Abstract: a single sentence summarizing the explicit form of the derived Teukolsky equation or the numerical QNM result in d=6 would help readers assess the concrete output of the formalism.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to each major comment below and will revise the manuscript to address the points raised.

read point-by-point responses

  1. Referee: Derivation of the Teukolsky equation (Section 3): the multispinor formalism must be shown to produce a decoupled master equation for the gravitational perturbations. Explicitly verify that the equation separates into radial and angular factors in Boyer-Lindquist coordinates with the compact dimension, and that the higher-dimensional Weyl components and Lichnerowicz operator do not introduce non-separable mixing terms.

    Authors: The multispinor formalism developed in Section 3 is designed to yield a decoupled Teukolsky equation by generalizing the Newman-Penrose tetrad to vectors of two-spinors. We derive the master equation for the gravitational perturbations and demonstrate its separability in the coordinates used, including the compact dimension. The higher-dimensional contributions from the Weyl components and Lichnerowicz operator are incorporated in a manner that preserves decoupling and separability, as shown by the resulting equation form. To provide the explicit verification requested, we will expand the derivation in the revised manuscript with additional steps showing the separation into radial and angular parts and confirming no non-separable mixing terms are introduced. revision: yes

  2. Referee: Reduction to four dimensions (Section 4 or the limit discussion): when the compactification radius is sent to zero (or d is formally set to 4), the derived Teukolsky equation and its QNM spectrum must recover the standard 4D Teukolsky equation and known QNM frequencies. This limit check is load-bearing for validating the boundary conditions and separability assumptions used for the QNM calculation.

    Authors: We agree that the reduction to the four-dimensional limit is an important consistency check. Our formalism is constructed such that it should recover the standard 4D Teukolsky equation when the extra dimension is compactified to zero size or when d is set to 4. In the revised manuscript, we will add a discussion of this limit, showing how the Teukolsky equation and the associated quasi-normal mode spectrum reduce to the known 4D results, thereby validating the boundary conditions and separability. revision: yes

Circularity Check

0 steps flagged

No circularity: multispinor formalism presented as independent spinor-derived construction

full rationale

The paper claims to construct a multispinor formalism from vectors of two-spinors, explicitly analogous to the 4D Newman-Penrose tetrad, and then derives the Teukolsky master equation for gravitational perturbations in d=6 Kerr with one compact dimension. No quoted step reduces the central result to a fitted parameter, a self-citation chain, or a redefinition of the input; the separability and boundary conditions are asserted to follow from the algebraic properties of the extended spinor structure rather than being imposed by construction. The derivation chain therefore remains self-contained against external benchmarks such as the known 4D Teukolsky limit.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The central claim implicitly assumes that spinor algebra and compactification preserve the necessary separability properties of the wave equation.

pith-pipeline@v0.9.0 · 5414 in / 1110 out tokens · 26920 ms · 2026-05-08T07:26:38.355984+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    The first set of Bianchi identities The first of the Bianchi identities is given by: D ψ2 −δ 1 ψ0 =α 4ψ0 −(α 2 +α 3)ψ 2 +α 1ψ5,(B1) ∆ψ 0 −δ 2 ψ1 =ψ 1 (−ζ1 +ϵ 2 −ϵ 3)−ψ 0 (ϵ4 −2ζ 3) + 1 3 h −2¯α3 + ¯ξ1 −2α 1 + 2α2 +D Φ2022+ + 2 (δ2 +ϵ 2 −ϵ 3) Φ1011 −2ϵ 1Φ1111 i ,(B2) D ψ4 −δ 1 ψ1 =−ψ 1 (α4 −2ξ 2) +ψ 4 (−α2 +α 3 −2ξ 1) +α 1ψ7 + 1 3 h ¯β4Φ0110 −2α 1Φ1211+ + ...

    work page

  2. [2]

    The second set of Bianchi identities The second set of Bianchi identities is given by: 3DΛ +β 1Φ1201 +ϵ 1Φ0200 + 3 2γ1 − ¯β4 Φ0110 + −2¯α3 + ¯ξ1 + 2α3 +D−ξ 1 Φ1111+ + −2¯ζ3 + ¯ϵ4 + ∆ + 2ζ3 −ϵ 4 Φ0000 = (α4 +δ 1 −ξ 2 +ξ 3) Φ1011 + (2β3 +δ 3) Φ1101+ + (2γ3 +δ 4) Φ0010 + (δ2 + 2ζ1 −ϵ 2 +ϵ 3) Φ0100,(B25) 27 3∆Λ + −2¯α3 + ¯ξ1 + 2α3 +D−ξ 1 Φ2222 + −2¯ζ3 + ¯ϵ4 +...

    work page

  3. [3]

    The third set of Bianchi identities The third set of Bianchi identities is given by (using the definition of the Weyl scalar): ∇a ˙b h ψcdef + Λ 3 ϵceϵd f+ϵ cf ϵde i = 0.(B31) Explicitely, we find: (−2α2 + 2α3 +D)ψ 0 =−2α 1 (ψ2 −ψ 1),(B32) (δ2 −2ϵ 2 + 2ϵ3)ψ 0 =−2 (ψ 2 −ψ 1)ϵ 1,(B33) (−2β2 + 2β3 +δ 3)ψ 0 =−2β 1 (ψ2 −ψ 1),(B34) (−3α2 +α 3 +D)ψ 1 =−α 4ψ0 −α ...

    work page

  4. [4]

    The last set of Bianchi identities The last set of Bianchi identities is: ∇a ˙bϕcd ˙e ˙f = 0.(B51) (2¯α3 + 2α3 +D) Φ 0000 = 2α1Φ0100,(B52) (δ1 + 2ξ3) Φ0000 = 2ξ1Φ0100,(B53) (2β3 +δ 3) Φ0000 = 2β1Φ0100,(B54) (2¯α3 −α 2 +α 3 +D) Φ 0100 =−α 4Φ0000 +α 1Φ0200,(B55) (−β2 +β 3 +δ 3) Φ0100 =−β 4Φ0000 +β 1Φ0200,(B56) (2¯α3 −2α 2 +D) Φ 0200 =−2α 4Φ0100,(B57) (δ3 −2...

    work page

  5. [5]

    W. L. Bade and H. Jehle. An introduction to spinors.Reviews of Modern Physics, 25(3):714, 1953

    work page 1953

  6. [6]

    E. Cartan. Les groupes projectifs qui ne laissent invariante aucune multiplicit´ e plane.Bulletin de la Soci´ et´ e Math´ ematique de France, 41:53–96, 1913

    work page 1913

  7. [7]

    Cartan.The theory of spinors

    E. Cartan.The theory of spinors. Dover publications, 1981

    work page 1981

  8. [8]

    A. Coley. Classification of the weyl tensor in higher dimensions and applications.Classical and Quantum Gravity, 25(3):033001, jan 2008. 29

    work page 2008

  9. [9]

    Coley, R

    A. Coley, R. Milson, V. Pravda, and A. Pravdov´ a. Classification of the weyl tensor in higher dimensions.Classical and Quantum Gravity, 21(7):L35, mar 2004

    work page 2004

  10. [10]

    Delamotte

    B. Delamotte. Un soup¸ con de th´ eorie des groupes: groupe des rotations et groupe de Poincar´ e. Lecture, Sept. 2006

    work page 2006

  11. [11]

    Z. Dong, D. Zhang, G. Fu, and J. Wu. Quasinormal modes of a d-dimensional regular black hole featuring an integrable singularity.Eur. Phys. J. C, 85:215, 2025

    work page 2025

  12. [12]

    D. Z. Freedman and A. van Proeyen.Supergravity. Cambridge University Press, 2012

    work page 2012

  13. [13]

    A. Higuchi. Symmetric tensor spherical harmonics on the n-sphere and their application to the de sitter group so(n,1). Journal of Mathematical Physics, 28(7):1553–1566, 07 1987

    work page 1987

  14. [14]

    S. A. Hughes. Evolution of circular, nonequatorial orbits of kerr black holes due to gravitational-wave emission.Phys. Rev. D, 61:084004, Mar 2000

    work page 2000

  15. [15]

    A. I. Janis and E. T. Newman. Structure of gravitational sources.Journal of Mathematical Physics, 6(6):902–914, 06 1965

    work page 1965

  16. [16]

    Jumaniyozov, S

    S. Jumaniyozov, S. U. Khan, J. Rayimbaev, A. Abdujabbarov, S. Urinbaev, and S. Murodov. Circular motion and qpos near black holes in kalb–ramond gravity.The European Physical Journal C, 84(9):964, 2024

    work page 2024

  17. [17]

    Kao and D

    H.-C. Kao and D. Tomino. Quasinormal modes of Kerr black holes in four and higher dimensions.Phys. Rev. D, 77(12):127503, June 2008

    work page 2008

  18. [18]

    S. U. Khan, J. Rayimbaev, Y. Turaev, O. Sirajiddin, S. Usanov, and W. Wang. Energy efficiency and particle dynamics around magnetized black holes with parabolic configuration in STVG.European Physical Journal C, 86(3):207, Mar. 2026

    work page 2026

  19. [19]

    S. U. Khan and J. Ren. Shadow cast by a rotating charged black hole in quintessential dark energy.Physics of the Dark Universe, 30:100644, Dec. 2020

    work page 2020

  20. [20]

    R. A. Konoplya and A. Zhidenko. Quasinormal modes of massive fermions in kerr spacetime: Long-lived modes and the fine structure.Phys. Rev. D, 97:084034, Apr 2018

    work page 2018

  21. [21]

    E. W. Leaver and S. Chandrasekhar. An analytic representation for the quasi-normal modes of kerr black holes.Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 402(1823):285–298, 1985

    work page 1985

  22. [22]

    Li, K.-P

    W. Li, K.-P. Lu, W. LiMing, and J.-H. Huang. Five-dimensional myers-perry black holes under massive scalar perturbation: bound states and quasinormal modes.Physica Scripta, 99(8):085023, jul 2024

    work page 2024

  23. [23]

    K.-P. Lu, W. Li, and J.-H. Huang. Quasinormal modes and stability of higher dimensional rotating black holes under massive scalar perturbations.Physics Letters B, 845:138147, Oct. 2023

    work page 2023

  24. [24]

    Milson, A

    R. Milson, A. Coley, V. Pravda, and A. Pravdov´ a. Alignment and algebraically special tensors in lorentzian geometry. International Journal of Geometric Methods in Modern Physics, 02(01):41–61, 2005

    work page 2005

  25. [25]

    A. K. Mishra, A. Ghosh, and S. Chakraborty. Constraining extra dimensions using observations of black hole quasi-normal modes.The European Physical Journal C, 82(9):820, 2022

    work page 2022

  26. [26]

    Newman and R

    E. Newman and R. Penrose. An Approach to Gravitational Radiation by a Method of Spin Coefficients.Journal of Mathematical Physics, 3(3):566–578, May 1962

    work page 1962

  27. [27]

    R. Penrose. A spinor approach to general relativity.Annals of Physics, 10(2):171–201, June 1960

    work page 1960

  28. [28]

    Penrose and W

    R. Penrose and W. Rindler.Spinors and spacetime – Vol. 1. Cambridge University Press, 1984

    work page 1984

  29. [29]

    Penrose and W

    R. Penrose and W. Rindler.Spinors and spacetime – Vol. 2. Cambridge University Press, 1984

    work page 1984

  30. [30]

    Poisson.A Relativist’s Toolkit - The Mathematics of Black-Hole Mechanics

    E. Poisson.A Relativist’s Toolkit - The Mathematics of Black-Hole Mechanics. Cambridge University Press, 2004

    work page 2004

  31. [31]

    Pravda, A

    V. Pravda, A. Pravdov´ a, A. Coley, and R. Milson. Bianchi identities in higher dimensions.Classical and Quantum Gravity, 21(12):2873, may 2004

    work page 2004

  32. [32]

    W. H. Press and S. A. Teukolsky. Perturbations of a Rotating Black Hole. II. Dynamical Stability of the Kerr Metric.Ap. J, 185:649–674, Oct. 1973

    work page 1973

  33. [33]

    Sasaki and H

    M. Sasaki and H. Tagoshi. Analytic Black Hole Perturbation Approach to Gravitational Radiation.Living Review in Relativity, 6:6, 2003

    work page 2003

  34. [34]

    Stephani, D

    H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt.Exact Solutions of Einstein’s Field Equations. Cambridge University Press, 2003

    work page 2003

  35. [35]

    S. A. Teukolsky. Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Electromagnetic, and Neutrino-Field Perturbations.Ap. J., 185:635–648, Oct. 1973

    work page 1973

  36. [36]

    Weinberg.The Quantum Theory of Fields - Volume 1 - Foundations

    S. Weinberg.The Quantum Theory of Fields - Volume 1 - Foundations. Cambridge University Press, 1995

    work page 1995

  37. [37]

    Weinberg.The Quantum Theory of Fields - Volume 2 - Modern applications

    S. Weinberg.The Quantum Theory of Fields - Volume 2 - Modern applications. Cambridge University Press, 1996

    work page 1996

  38. [38]

    L. Witten. Invariants of general relativity and the classification of spaces.Physical Review, 113(1):357, 1959

    work page 1959