Separation of gravitational perturbations of singly spinning Myers-Perry black holes

Jorge E. Santos

REVIEW

Separation of gravitational perturbations of singly spinning Myers-Perry black holes

Jorge E. Santos

High Energy Physics - Theory · Submitted 13 Oct 2026 · 27 references

Pith's verdict: Major advance in separating gravitational perturbations of higher-dimensional rotating black holes; well-supported and opens a new path for quasinormal mode computations.

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CitationJorge E. Santos. "Separation of gravitational perturbations of singly spinning Myers-Perry black holes." arXiv:2610.18956 (2026). https://pith.science/paper/WQBST724

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Editor's note

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This paper is the real deal. It shows that gravitational perturbations of singly spinning Myers–Perry black holes separate in the sector invariant under rotations of the transverse sphere, despite the known obstruction to a decoupled Teukolsky equation. That is a genuine advance: the paper constructs explicit commuting angular operators and the radial system, reducing coupled PDEs to ODEs. It then uses those ODEs to compute bar-mode and ultraspinning spectra, matching prior PDE results and giving more precise thresholds (e.g., the d=6,7 bar-mode onsets agree to 10^-4 or better). The derivations are thorough—checked symbolically for general parameters—and the numerical methods are well described with convergence demonstrated.

The soft spots are minor and mostly acknowledged. The symmetry restriction (ℓ_S=0 on the transverse sphere) limits the modes, but it captures both known instabilities. The d=6 ultraspinning onset differs from earlier results by about 1%; the author attributes this to numerical error in the earlier PDE calculation, which is plausible given the asymptotic sensitivity. The exactly stationary axisymmetric case (ω=m=0) is excluded, with onsets approached from growing modes; stability at exact zero is not established. The n=1 (d=5) even sector is handled separately; the odd sector decouples and is not treated. These are limits of scope, not flaws.

Who is this for? Anyone working on higher-dimensional black hole perturbation theory, quasinormal modes, or black hole stability. The mathematical structure may also interest those working on separability and hidden symmetries. It deserves a serious referee; I'd recommend acceptance after the authors clarify the d=6 discrepancy (e.g., check if a refined PDE calculation now matches) and add a brief remark on the ω=m=0 limit. Would I cite it? Yes, if I were working in this area. Would I bring it to a reading group? Yes, for the separation technique.

Referee report

Minor objections (6)

[Introduction (abstract and main text)] The abstract and introduction refer to 'scalar gravitational perturbations'. The term 'scalar' here denotes invariance under the rotations of the transverse sphere, not spin-0 character. A brief clarifying phrase when the term is first introduced would help avoid confusion with scalar-field perturbations.
[Main text, first paragraph after Eq. (3)] The notation for the 'bar' in ¯Ψ0 is explained as the m↔¯m partner rather than complex conjugation. This is an important distinction and is stated, but a reminder later in the radial-equation section (e.g., when discussing the complex conjugate of a Fourier mode) would improve readability.
[Table I and surrounding text] The paper notes that the d=6 onset from the ODE calculation (ac/r+=0.90498) is about 1% lower than the PDE result of Ref. [19] (0.903±0.002). The discussion attributes this to numerical error in the PDE calculation (the UL). This is plausible, but the uncertainty quoted in Ref. [19] (0.001–0.002) does not cover the discrepancy. A brief comment on how the current ODE code's internal convergence (10^(-4) from onset extrapolations) establishes the new value would strengthen the claim.
[Supplemental Material, Sec. VII, paragraph beginning 'Reduction at the axis'] The axis reduction for m=0 is described briefly ('carried to relative order sin^4 θ' and 32 conditions). A schematic outline of how the 32 conditions reduce to a single nontrivial condition would help the reader follow the logic, even if the full algebra is prohibitively long.
[Figure 1 caption] The caption identifies 'Filled symbols: the PDE results of Ref. [19] (Fig. 17), using the original figure data.' Consider adding a note that the discrepancy on the d=7 negative-frequency branch is attributed by the authors to limited resolution in the fourth-order integration (see text), and that the first-order system converges to 10^(-7) even on that branch.
[Throughout] A number of equation numbers in the main text refer to the Supplemental Material (e.g., 'Eqs. (S4)–(S5)', 'Eqs. (S27)–(S29)'). The Letter is already quite long; this is acceptable. However, several SM equation numbers cited in the main text (e.g., S27a) are not defined in the main text. A short parenthetical reminder of the quantity being defined (e.g., 'χ(r) is given in SM Eq. (S27a)') would help the reader who does not immediately consult the SM.

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The paper

The paper claims

Gravitational perturbations of a higher-dimensional rotating black hole separate, despite coupling between polarisations.

The reading

This paper shows that gravitational perturbations of a singly spinning Myers–Perry black hole can be separated into ordinary differential equations in the sector invariant under rotations of the transverse sphere. This sector contains both the bar-mode and axisymmetric ultraspinning instabilities. The separation is achieved by constructing two commuting angular operators that organise the three gauge-invariant curvature components into joint eigenfunctions. The radial problem reduces to a single second-order equation, similar in complexity to the four-dimensional Teukolsky equation, even though the components remain coupled. This provides a practical method to compute quasinormal spectra and instability onsets using only ODEs, which the author demonstrates for dimensions five through nine.

What carries the argument

Two commuting second-order angular operators $K_C$ and $K_3$ acting on the triple of curvature components $(\Psi_0, \bar\Psi_0, \Phi_{00})$. These operators are independent of the radial coordinate and satisfy a polynomial relation $F(K_C,K_3)=0$ that defines the joint spectrum. They extend the angular Teukolsky operator by coupling the base curvature to the warp component of the transverse sphere. The radial problem is organised by a first-order system of two physical relations that reduce the four apparent integration constants to two, matching the physical degrees of freedom.

Extended reading notes

Core claim

The paper establishes that scalar gravitational perturbations of a singly spinning Myers–Perry black hole, when restricted to the sector invariant under rotations of the transverse $S^{d-4}$, admit a complete separation of variables. Despite the known obstruction to a decoupled Teukolsky equation, two commuting second-order angular operators $K_C$ and $K_3$ are found that act on the triple of gauge-invariant curvature components $\Omega = (\Psi_0, \bar\Psi_0, \Phi_{00})$. Their joint eigenfunctions define an angular harmonic that fixes the relative profiles of the three components. The radial dynamics then reduce to a $2\times 2$ first-order system, equivalent to a single second-order equation, with two physical integration constants. This separation allows the bar-mode and ultraspinning instabilities to be studied via ordinary differential equations alone. The author computes the bar-mode quasinormal spectrum in $d=5,6,7$ and the growing axisymmetric modes in $d=6,\ldots,9$, recovering and in some cases improving on previous PDE results.

Load-bearing premise

The analysis assumes that the three gauge-invariant curvature components form a closed system under the Newman–Penrose and Codazzi identities when restricted to the sector invariant under rotations of the transverse sphere, and that no additional curvature components couple into this system.

Editorial extensions

If this is right

The bar-mode instability onset in $d=6,7$ can be computed to high precision with a laptop, enabling systematic parameter scans that were previously expensive.
The growth rates and onset rotations of the axisymmetric ultraspinning instability are obtained for $d=6,\ldots,9$, revealing a turnover in the growth rate for $d=7,8,9$ that was not previously resolved.
The separated form provides a clean framework for mode stability analysis in higher dimensions, analogous to Whiting's construction for Kerr.
The angular operators $K_C$ and $K_3$ suggest a hidden symmetry structure in the linearised Einstein equations that may extend to other sectors or more general backgrounds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The existence of two commuting angular operators indicates that the linearised Einstein equations in this sector possess a hidden symmetry algebra, possibly related to a principal Killing–Yano tensor. Identifying this covariant origin could unify the separation properties observed here with those for test fields.
The separated radial equations are suitable for studying the nonlinear evolution of instabilities via perturbation theory, and may inform the endpoint of the ultraspinning instability in higher dimensions.
The method could be extended to multiply spinning Myers–Perry black holes or to include a cosmological constant, provided an analogous invariant sector exists.

Original abstract

Gravitational perturbations of rotating black holes in higher dimensions generally obey coupled partial differential equations. We show that scalar gravitational perturbations invariant under rotations of the transverse sphere of a singly spinning Myers-Perry black hole separate, despite the obstruction to a decoupled Teukolsky equation. Two commuting, second-order angular operators organise the three gauge-invariant curvature components into joint eigenfunctions. The physical radial problem is a $2\times2$ first-order system, equivalent to a single second-order equation. This gives a separation of the sector containing both the bar-mode and axisymmetric ultraspinning instabilities. Using only ordinary differential equations, we compute the bar-mode quasinormal spectrum in $d=5,6,7$ and find instability onsets at $a/r_+=0.90498$ and $0.83228$ in $d=6,7$. We also compute the growing axisymmetric modes and their onset rotations in $d=6,\ldots,9$. The construction shows how separability can survive the coupling between gravitational polarisations.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-15: 1 request, 6,253 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.50 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.26 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Gravitational perturbations of rotating black holes in higher dimensions generally obey coupled partial differential equations. We show that scalar gravitational perturbations invariant under rotations of the transverse sphere of a singly spinning Myers-Perry black hole separate, despite the obstruction to a decoupled Teukolsky equation. Two commuting, second-order angular operators organise the three gauge-invariant curvature components into joint eigenfunctions. The physical radial problem is a $2\times2$ first-order system, equivalent to a single second-order equation. This gives a separation of the sector containing both the bar-mode and axisymmetric ultraspinning instabilities. Using only ordinary differential equations, we compute the bar-mode quasinormal spectrum in $d=5,6,7$ and find instability onsets at $a/r_+=0.90498$ and $0.83228$ in $d=6,7$. We also compute the growing axisymmetric modes and their onset rotations in $d=6,\ldots,9$. The construction shows how separability can survive the coupling between gravitational polarisations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Gravitational perturbations of rotating black holes in higher dimensions generally obey coupled partial differential equations.	background 1.00	0.06	0.03	asserted 0.90
s1	We show that scalar gravitational perturbations invariant under rotations of the transverse sphere of a singly spinning Myers-Perry black hole separate, despite the obstruction to a decoupled Teukolsky equation.	result 0.66	0.96	0.84	asserted 1.00
s2	Two commuting, second-order angular operators organise the three gauge-invariant curvature components into joint eigenfunctions.	method 0.87	0.89	0.13	asserted 1.00
s3	The physical radial problem is a $2\times2$ first-order system, equivalent to a single second-order equation.	result 0.55	0.82	0.10	asserted 1.00
s4	This gives a separation of the sector containing both the bar-mode and axisymmetric ultraspinning instabilities.	result 0.49	0.73	0.20	asserted 1.00
s5	Using only ordinary differential equations, we compute the bar-mode quasinormal spectrum in $d=5,6,7$ and find instability onsets at $a/r_+=0.90498$ and $0.83228$ in $d=6,7$.	result 0.95	0.97	0.14	asserted 1.00
s6	We also compute the growing axisymmetric modes and their onset rotations in $d=6,\ldots,9$.	result 0.84	0.96	0.06	asserted 1.00
s7	The construction shows how separability can survive the coupling between gravitational polarisations.	interpretation 1.00	0.81	0.30	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 0.65.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Real (a) and imaginary (b) parts of the

**Figure 2.** shows the growth rates as functions of a/r+. Extrapolating from γr+ ≲ 3 × 10−3 gives the onsets in Table II. To compare with the original ultraspinning thresholds [20], we convert to a/r0 = x(1 + x 2 ) −1/(n+1) , where x = a/r+ and r0 is defined below Eq. (1). In d = 7, 8, 9 this recovers the stationary zero modes to all quoted digits. In d = 6 the onsets are about 1% lower. Although this exceeds the earlier quoted error estimate, it likely reflects numerical error in the PDE calculation, whose stability and accuracy in d = 6 were particularly sensitive to the asymptotic fall-off

Figures reproduced from arXiv:2610.18956 by the authors.

References

Works cited (27)

[1]

S. A. Teukolsky, Rotating black holes - separable wave equations for gravitational and electromagnetic pertur- bations, Phys. Rev. Lett.29, 1114 (1972)

View on Pith 1972
[2]

S. A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J.185, 635 (1973)

View on Pith 1973
[3]

B. F. Whiting, Mode Stability of the Kerr Black Hole, J. Math. Phys.30, 1301 (1989)

View on Pith 1989
[4]

M. Durkee and H. S. Reall, Perturbations of higher- dimensional spacetimes, Class. Quant. Grav.28, 035011 (2011), arXiv:1009.0015 [gr-qc]

View on Pith Pith review arXiv 2011
[5]

R. C. Myers and M. J. Perry, Black Holes in Higher Di- mensional Space-Times, Annals Phys.172, 304 (1986)

View on Pith 1986
[6]

M. Durkee and H. S. Reall, Perturbations of near-horizon geometries and instabilities of Myers-Perry black holes, Phys. Rev. D83, 104044 (2011), arXiv:1012.4805 [hep- th]

View on Pith Pith review arXiv 2011
[7]

H. Kodama, R. A. Konoplya, and A. Zhidenko, Grav- itational stability of simply rotating Myers-Perry black holes: Tensorial perturbations, Phys. Rev. D81, 044007 (2010), arXiv:0904.2154 [gr-qc]

View on Pith Pith review arXiv 2010
[8]

K. Murata and J. Soda, Stability of Five-dimensional Myers-Perry Black Holes with Equal Angular Momenta, Prog. Theor. Phys.120, 561 (2008), arXiv:0803.1371 [hep-th]

View on Pith Pith review arXiv 2008
[9]

O. Lunin, Gravitational waves in the Myers-Perry geom- etry, JHEP2026(03), 055, arXiv:2510.14417 [hep-th]

View on Pith arXiv
[10]

V. P. Frolov, P. Krtous, and D. Kubiznak, Black holes, hidden symmetries, and complete integrability, Living Rev. Rel.20, 6 (2017), arXiv:1705.05482 [gr-qc]

View on Pith Pith review arXiv 2017
[11]

O. Lunin, Maxwell’s equations in the Myers-Perry geom- etry, JHEP2017(12), 138, arXiv:1708.06766 [hep-th]

View on Pith Pith review arXiv
[12]

P. Krtouš, V. P. Frolov, and D. Kubizňák, Separation of Maxwell equations in Kerr–NUT–(A)dS spacetimes, Nucl. Phys. B934, 7 (2018), arXiv:1803.02485 [hep-th]

View on Pith Pith review arXiv 2018
[13]

P. Hintz, Perturbations of Kerr-Newman black holes: separation and mode stability (2026), arXiv:2609.33661 [gr-qc]

View on Pith Pith review arXiv 2026
[14]

K. Destounis, V. Cardoso, and P. Hintz, The quasinor- mal modes of Kerr-Newman black holes: a separated ap- proach (2026), arXiv:2610.07142 [gr-qc]

View on Pith Pith review arXiv 2026
[15]

R. Emparan and R. C. Myers, Instability of ultra- spinning black holes, JHEP2003(09), 025, arXiv:hep- th/0308056

View on Pith
[16]

O. J. C. Dias, P. Figueras, R. Monteiro, J. E. San- tos, and R. Emparan, Instability and new phases of higher-dimensional rotating black holes, Phys. Rev. D 80, 111701 (2009), arXiv:0907.2248 [hep-th]

View on Pith Pith review arXiv 2009
[17]

O. J. C. Dias, P. Figueras, R. Monteiro, H. S. Reall, and J. E. Santos, An instability of higher-dimensional rotat- ing black holes, JHEP2010(05), 076, arXiv:1001.4527 [hep-th]

View on Pith Pith review arXiv
[18]

M. Shibata and H. Yoshino, Bar-mode instability of rapidly spinning black hole in higher dimensions: Nu- merical simulation in general relativity, Phys. Rev. D81, 104035 (2010), arXiv:1004.4970 [gr-qc]

View on Pith Pith review arXiv 2010
[19]

Ó. J. C. Dias, G. S. Hartnett, and J. E. Santos, Quasi- normal modes of asymptotically flat rotating black holes, Class. Quant. Grav.31, 245011 (2014), arXiv:1402.7047 [hep-th]

View on Pith Pith review arXiv 2014
[20]

O. J. C. Dias, P. Figueras, R. Monteiro, and J. E. Santos, Ultraspinning instability of rotating black holes, Phys. Rev. D82, 104025 (2010), arXiv:1006.1904 [hep-th]

View on Pith Pith review arXiv 2010
[21]

See Supplemental Material for the operator identities, the Newman–Penrose derivation, the physical radial re- lations, and the angular and radial numerical methods

View on Pith
[22]

H. Kodama and A. Ishibashi, A Master equation for grav- itational perturbations of maximally symmetric black holes in higher dimensions, Prog. Theor. Phys.110, 701 (2003), arXiv:hep-th/0305147

View on Pith Pith review arXiv 2003
[23]

V. Cardoso, J. P. S. Lemos, and S. Yoshida, Scalar gravi- tationalperturbationsandquasinormalmodesinthefive- dimensional Schwarzschild black hole, JHEP2003(12), 041, arXiv:hep-th/0311260

View on Pith Pith review arXiv
[24]

H. Yoshino, T. Shiromizu, and M. Shibata, The Close limit analysis for head-on collision of two black holes in higher dimensions: Brill-Lindquist initial data, Phys. Rev. D72, 084020 (2005), arXiv:gr-qc/0508063

View on Pith Pith review arXiv 2005
[25]

P. Figueras, M. Kunesch, L. Lehner, and S. Tunyasuvu- nakool, End Point of the Ultraspinning Instability and Violation of Cosmic Censorship, Phys. Rev. Lett.118, 151103 (2017), arXiv:1702.01755 [hep-th]

View on Pith Pith review arXiv 2017
[26]

Ó. J. C. Dias, J. E. Santos, and B. Way, Numerical Methods for Finding Stationary Gravitational Solutions, Class. Quant. Grav.33, 133001 (2016), arXiv:1510.02804 [hep-th]

View on Pith Pith review arXiv 2016
[27]

R. P. Geroch, A. Held, and R. Penrose, A space-time calculus based on pairs of null directions, J. Math. Phys. 14, 874 (1973). 6 SUPPLEMENTAL MATERIAL Equation numbers without an S refer to the Letter. I. CONVENTIONS AND REDUCTION We work with the mostly-plus metric (1). Curvature components areCabcd =C abcd aabbccdd andR ab =R abaabb, whereCandRare the W...

View on Pith 1973

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

S. A. Teukolsky, Rotating black holes - separable wave equations for gravitational and electromagnetic pertur- bations, Phys. Rev. Lett.29, 1114 (1972)

View on Pith 1972

[2] [2]

S. A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J.185, 635 (1973)

View on Pith 1973

[3] [3]

B. F. Whiting, Mode Stability of the Kerr Black Hole, J. Math. Phys.30, 1301 (1989)

View on Pith 1989

[4] [4]

M. Durkee and H. S. Reall, Perturbations of higher- dimensional spacetimes, Class. Quant. Grav.28, 035011 (2011), arXiv:1009.0015 [gr-qc]

View on Pith Pith review arXiv 2011

[5] [5]

R. C. Myers and M. J. Perry, Black Holes in Higher Di- mensional Space-Times, Annals Phys.172, 304 (1986)

View on Pith 1986

[6] [6]

M. Durkee and H. S. Reall, Perturbations of near-horizon geometries and instabilities of Myers-Perry black holes, Phys. Rev. D83, 104044 (2011), arXiv:1012.4805 [hep- th]

View on Pith Pith review arXiv 2011

[7] [7]

H. Kodama, R. A. Konoplya, and A. Zhidenko, Grav- itational stability of simply rotating Myers-Perry black holes: Tensorial perturbations, Phys. Rev. D81, 044007 (2010), arXiv:0904.2154 [gr-qc]

View on Pith Pith review arXiv 2010

[8] [8]

K. Murata and J. Soda, Stability of Five-dimensional Myers-Perry Black Holes with Equal Angular Momenta, Prog. Theor. Phys.120, 561 (2008), arXiv:0803.1371 [hep-th]

View on Pith Pith review arXiv 2008

[9] [9]

O. Lunin, Gravitational waves in the Myers-Perry geom- etry, JHEP2026(03), 055, arXiv:2510.14417 [hep-th]

View on Pith arXiv

[10] [10]

V. P. Frolov, P. Krtous, and D. Kubiznak, Black holes, hidden symmetries, and complete integrability, Living Rev. Rel.20, 6 (2017), arXiv:1705.05482 [gr-qc]

View on Pith Pith review arXiv 2017

[11] [11]

O. Lunin, Maxwell’s equations in the Myers-Perry geom- etry, JHEP2017(12), 138, arXiv:1708.06766 [hep-th]

View on Pith Pith review arXiv

[12] [12]

P. Krtouš, V. P. Frolov, and D. Kubizňák, Separation of Maxwell equations in Kerr–NUT–(A)dS spacetimes, Nucl. Phys. B934, 7 (2018), arXiv:1803.02485 [hep-th]

View on Pith Pith review arXiv 2018

[13] [13]

P. Hintz, Perturbations of Kerr-Newman black holes: separation and mode stability (2026), arXiv:2609.33661 [gr-qc]

View on Pith Pith review arXiv 2026

[14] [14]

K. Destounis, V. Cardoso, and P. Hintz, The quasinor- mal modes of Kerr-Newman black holes: a separated ap- proach (2026), arXiv:2610.07142 [gr-qc]

View on Pith Pith review arXiv 2026

[15] [15]

R. Emparan and R. C. Myers, Instability of ultra- spinning black holes, JHEP2003(09), 025, arXiv:hep- th/0308056

View on Pith

[16] [16]

O. J. C. Dias, P. Figueras, R. Monteiro, J. E. San- tos, and R. Emparan, Instability and new phases of higher-dimensional rotating black holes, Phys. Rev. D 80, 111701 (2009), arXiv:0907.2248 [hep-th]

View on Pith Pith review arXiv 2009

[17] [17]

O. J. C. Dias, P. Figueras, R. Monteiro, H. S. Reall, and J. E. Santos, An instability of higher-dimensional rotat- ing black holes, JHEP2010(05), 076, arXiv:1001.4527 [hep-th]

View on Pith Pith review arXiv

[18] [18]

M. Shibata and H. Yoshino, Bar-mode instability of rapidly spinning black hole in higher dimensions: Nu- merical simulation in general relativity, Phys. Rev. D81, 104035 (2010), arXiv:1004.4970 [gr-qc]

View on Pith Pith review arXiv 2010

[19] [19]

Ó. J. C. Dias, G. S. Hartnett, and J. E. Santos, Quasi- normal modes of asymptotically flat rotating black holes, Class. Quant. Grav.31, 245011 (2014), arXiv:1402.7047 [hep-th]

View on Pith Pith review arXiv 2014

[20] [20]

O. J. C. Dias, P. Figueras, R. Monteiro, and J. E. Santos, Ultraspinning instability of rotating black holes, Phys. Rev. D82, 104025 (2010), arXiv:1006.1904 [hep-th]

View on Pith Pith review arXiv 2010

[21] [21]

See Supplemental Material for the operator identities, the Newman–Penrose derivation, the physical radial re- lations, and the angular and radial numerical methods

View on Pith

[22] [22]

H. Kodama and A. Ishibashi, A Master equation for grav- itational perturbations of maximally symmetric black holes in higher dimensions, Prog. Theor. Phys.110, 701 (2003), arXiv:hep-th/0305147

View on Pith Pith review arXiv 2003

[23] [23]

V. Cardoso, J. P. S. Lemos, and S. Yoshida, Scalar gravi- tationalperturbationsandquasinormalmodesinthefive- dimensional Schwarzschild black hole, JHEP2003(12), 041, arXiv:hep-th/0311260

View on Pith Pith review arXiv

[24] [24]

H. Yoshino, T. Shiromizu, and M. Shibata, The Close limit analysis for head-on collision of two black holes in higher dimensions: Brill-Lindquist initial data, Phys. Rev. D72, 084020 (2005), arXiv:gr-qc/0508063

View on Pith Pith review arXiv 2005

[25] [25]

P. Figueras, M. Kunesch, L. Lehner, and S. Tunyasuvu- nakool, End Point of the Ultraspinning Instability and Violation of Cosmic Censorship, Phys. Rev. Lett.118, 151103 (2017), arXiv:1702.01755 [hep-th]

View on Pith Pith review arXiv 2017

[26] [26]

Ó. J. C. Dias, J. E. Santos, and B. Way, Numerical Methods for Finding Stationary Gravitational Solutions, Class. Quant. Grav.33, 133001 (2016), arXiv:1510.02804 [hep-th]

View on Pith Pith review arXiv 2016

[27] [27]

R. P. Geroch, A. Held, and R. Penrose, A space-time calculus based on pairs of null directions, J. Math. Phys. 14, 874 (1973). 6 SUPPLEMENTAL MATERIAL Equation numbers without an S refer to the Letter. I. CONVENTIONS AND REDUCTION We work with the mostly-plus metric (1). Curvature components areCabcd =C abcd aabbccdd andR ab =R abaabb, whereCandRare the W...

View on Pith 1973