Linear perturbations of Kerr black hole in quadratic gravity

David Kofron; Robert Svarc; Simon Knoska

REVIEW

Linear perturbations of Kerr black hole in quadratic gravity

Simon Knoska, David Kofron, Robert Svarc

General Relativity and Quantum Cosmology · Submitted 29 Jul 2025 · 33 references

Pith's verdict: A genuinely new and potentially useful Teukolsky-style system for Kerr in quadratic gravity, with a successful flat-space check, but the missing derivation and an unaddressed tetrad-gauge issue make the central 'no gauge freedom' claim unproven.

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arXiv ID2507.21923 (version 1)

Pith IDADRPE2GP, Pith's permanent record of this paper and its review

CitationDavid Kofron, Robert Svarc, Simon Knoska. "Linear perturbations of Kerr black hole in quadratic gravity." arXiv:2507.21923 (2025). https://pith.science/paper/ADRPE2GP

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Editor's note

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Simon — this paper does something new and useful: it produces a decoupled set of linearized equations for Kerr perturbations in Stelle's quadratic gravity, expressed entirely through Ricci-tensor projections with a GHP-inspired operator. The Minkowski limit reproduces Stelle's known solution, which is a solid consistency check. The Schwarzschild example, even with the R_epsilon = 0 simplifying assumption, shows the equations are usable and gives some sensible asymptotics. The self-citation to the authors' earlier NP formalism for quadratic gravity is appropriate; that formalism is the natural starting point.

The soft spots are real. First, the central equations (33)-(44) appear with no derivation, just 'very extensive calculations.' For a result this central, a derivation outline or a symbolic notebook is needed before the community can trust it. A referee should require that. Second, and more concerning, the perturbation scheme (30) perturbs the null frame vectors but never fixes the six-parameter Lorentz freedom of the tetrad. All the Phi_epsilon_AB and the box operator are frame-dependent, and the GHP-weight table shows that Phi_epsilon_00 and Phi_epsilon_22 transform oppositely under boost. So the flat-space equality Phi_epsilon_00 = Phi_epsilon_22 in (67) cannot be invariant without a specific tetrad-gauge choice. The paper never states such a choice, yet Section III B claims 'no coordinate or gauge freedom is present in the solution.' That claim is not supported. This is a gap in the argument, not a competing interpretation.

The reader's conditional verdict is about right. The paper deserves peer review because the result is new, checkable, and potentially important for ringdown and stability work. But the referees should demand (a) a reproducible derivation, and (b) a clear statement of how the tetrad perturbation is fixed, or an explanation of why the final equations are invariant. With those addressed, this would be a strong contribution. As it stands, I would not build on it directly until the gauge issue is resolved.

Referee report

Significance

If the central equations are correct and the gauge issues are resolved, this would be a useful step toward black-hole perturbation theory in quadratic gravity: it would reduce the fourth-order metric perturbation problem to a set of second-order equations for Ricci projections and would provide a coordinate-independent framework for further studies of Kerr instabilities, tidal responses, and matter couplings. The flat-space check against Stelle's coordinate solution is a genuine positive consistency test, and the Schwarzschild example connects to known exact solutions. The main limitations are verifiability and completeness: the derivation of the central system is not shown, and the treatment of the null-tetrad gauge freedom is not specified, so the central claim that the Ricci projections uniquely determine the perturbation is not yet established as written.

Major objections (3)

[§II, Eqs. (30)-(44)] The perturbation scheme (30) perturbs the null frame vectors as well as the NP scalars, but the six-parameter Lorentz freedom of the tetrad is never fixed. Since the Phi^epsilon_AB carry different GHP weights (table after Eq. (46)), a constant infinitesimal boost or spin changes them, so the equations (33)-(44) and the derived equality Phi^epsilon_00 = Phi^epsilon_22 in Eq. (67) are not frame-invariant as written. Unless a tetrad gauge condition is imposed and stated (e.g., setting a perturbed spin coefficient to zero, or working with GHP-invariant combinations), the Ricci projections are not uniquely defined and the claim in §III B that no gauge freedom remains is unsupported.
[§II, before Eq. (33)] The sentence 'Very extensive calculations yield' is the only derivation step presented for the central system (33)-(44). No outline of the reduction is given, no list of which perturbed Ricci/Bianchi identities are combined is provided, and no symbolic-computation notebook or ancillary file is supplied. Because the central claim of the paper rests entirely on these equations and on the elimination of frame-vector perturbations, the authors should either include a derivation sketch (identities used, order of eliminations, where the Kerr background properties enter) or deposit a computer-algebra notebook so that the algebra can be checked.
[§IV and §III] The Summary states that only the Ricci tensor perturbations need to be found in order to fully determine the solution, but the paper does not show how, given a solution of (33)-(44), one reconstructs the perturbed Weyl scalars, spin coefficients, or metric perturbation. The linearized Bianchi identities are listed in Appendix A, but no explicit reconstruction map or degree-of-freedom count is given. This is load-bearing for the central claim and should be spelled out, or the claim should be weakened accordingly.

Minor objections (4)

[Abstract and Appendix A] There are typos: 'Bianch identities' in the Abstract and 'Bainchi identities' in the Appendix heading should be 'Bianchi identities'.
[§III A, Eq. (59)] The notation C^0_± e^{±m_0 r}/r is ambiguous; please define explicitly what the plus-minus shorthand represents before using it in Eqs. (69)-(71).
[§III B, after Eq. (78)] The statement 'In general, no coordinate or gauge freedom is present in the solution' is made for the restricted case R^epsilon = 0. Because the trace equation (32) admits nontrivial scalar modes in general, the conclusion should be explicitly scoped to the R^epsilon = 0 sector.
[§III B, Eq. (74)] Please specify the operator ordering in Eq. (74): the d'Alembertian clearly acts on Phi^epsilon_11 and on all products to its right, but the convention should be stated to avoid ambiguity.

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The paper

The paper claims

Linear perturbations of Kerr black holes in quadratic gravity are fully determined once the Ricci tensor projections are known.

The reading

Quadratic gravity adds higher-curvature terms to the Einstein–Hilbert action, and because Kerr spacetime solves the vacuum equations of general relativity, it is also a background of the modified theory. This paper claims that every linear perturbation of that background is governed by a closed system of second-order differential equations for the Newman–Penrose projections of the Ricci tensor, together with one scalar equation for the curvature trace. Once those Ricci perturbations are found, the Weyl scalars and the curvature trace are forced by the linearised Bianchi identities, so the full perturbed spacetime is determined without residual gauge freedom. If this decoupling holds, perturbative studies of rotating black holes in quadratic gravity become a tractable second-order problem rather than a fourth-order problem in metric components.

What carries the argument

The machinery is the spin-coefficient (Newman–Penrose) calculus on the Kinnersley null tetrad of the Kerr metric, together with a GHP-type weighted operator ⊡ that packages the relevant second-order derivative combinations. The paper perturbs every quantity in the formalism—the frame vectors, the spin coefficients, the Weyl and Ricci projections—so the directional derivatives themselves acquire perturbations. The linearised Ricci and Bianchi identities then serve as elimination rules: they express the Weyl scalars and the Bach-tensor projections in terms of the Ricci perturbations Φε_AB and the scalar Rε, closing the system. Each perturbed field equation (33)–(38) has the structure of a massive Klein–Gordon operator acting on a Ricci projection, with the trace sector entering as a source.

Extended reading notes

Core claim

The central result is the system (33)–(38), coupled with (39)–(44) and the trace equation (32). The left-hand side of each field equation is written as a Ricci projection Φε_AB minus 2ak times a Bach-tensor projection Bε_(i)(j) (itself a second-order differential expression in the Ricci projections and Rε) minus bk times combinations of derivatives of Rε. Because the background Kerr geometry has zero Ricci and Bach tensors, every nonzero term is a bona fide quadratic-gravity correction. The equations are derived by linearising the field equations of quadratic gravity in the spin-coefficient formalism on the Kerr background and then using the linearised Ricci and Bianchi identities to eliminate all Weyl scalars. The trace equation (32) is a massive Klein–Gordon equation for Rε, while the Bach sector carries the massive spin-2 mode; the two sectors are mixed by the remaining equations. For static spherically symmetric perturbations the system reduces to ordinary differential equations, reproducing the known Minkowski-space solution and giving explicit near-horizon and asymptotic behaviour on Schwarzschild.

Load-bearing premise

The derivation assumes that the six-parameter Lorentz freedom in choosing the perturbed null tetrad is fixed consistently, but the paper never states how that gauge is fixed.

Editorial extensions

If this is right

Mode analysis of Kerr in quadratic gravity can proceed by solving second-order equations for six Ricci projections plus the trace scalar, instead of fourth-order metric perturbations.
The trace sector is forced by the single equation (32) with mass scale m0 = 1/sqrt(6bk), and the Bach sector by m2 = 1/sqrt(2ak), so the two new degrees of freedom appear as two mass parameters in the system.
On a Schwarzschild background with vanishing scalar perturbation, the equations couple only the Ricci projections; the near-horizon behaviour shows a physically divergent branch and a regular branch, matching known exact solutions and signalling that generic full-theory solutions are horizonless.
Because no gauge freedom remains, each integration constant that survives the constraints is a physical parameter of the quadratic theory, clarifying which parameters are physical hair rather than coordinate artifacts.

Reading between the lines

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

The same Ricci-projection-first decoupling should apply to other algebraically special backgrounds that are vacuum solutions of general relativity, since the elimination of Weyl scalars through Bianchi identities does not use the specific form of Kerr except through the tetrad and spin coefficients.
For gravitational-wave phenomenology, the equations provide a direct route to compute how a two-body source drives the massive spin-0 and spin-2 modes: solve the trace equation for Rε, then feed it into (33)–(38) to obtain the radiated Ricci perturbations, which can be mapped to observable waveform corrections.
A practical test of the uniqueness claim is to re-derive (33)–(38) in a different null tetrad and check whether the Ricci projections that solve the equations are Lorentz-related in the required way; if not, an explicit gauge fixing is needed before the system is closed.
The second-order form invites an immediate spectral study: translating the system into the radial and angular operators of Kerr and looking for quasinormal-mode solutions would reveal whether the massive spin-2 ghost mode produces an instability at a specific multipole, a question the paper leaves open.

Original abstract

Employing the Newman-Penrose formalism and following the classic Teukolsky-like approach, we linearise the field equations of quadratic gravity on the Kerr background and combine them with the linearised Ricci and Bianch identities. This leads to constraints on linear perturbations of the Kerr spacetime in quadratic gravity. The resulting differential equations are decoupled in such a way that only the Ricci tensor perturbations need to be found on the Kerr background in order to fully determine the solution. The results are illustrated in the simple non-trivial cases of the Schwarzschild and Minkowski geometries.

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The abstract, sentence by sentence

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

Employing the Newman-Penrose formalism and following the classic Teukolsky-like approach, we linearise the field equations of quadratic gravity on the Kerr background and combine them with the linearised Ricci and Bianch identities. This leads to constraints on linear perturbations of the Kerr spacetime in quadratic gravity. The resulting differential equations are decoupled in such a way that only the Ricci tensor perturbations need to be found on the Kerr background in order to fully determine the solution. The results are illustrated in the simple non-trivial cases of the Schwarzschild and Minkowski geometries.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Employing the Newman-Penrose formalism and following the classic Teukolsky-like approach, we linearise the field equations of quadratic gravity on the Kerr background and combine them with the linearised Ricci and Bianch identities.	method 0.99	0.95	0.05	asserted 0.99
s1	This leads to constraints on linear perturbations of the Kerr spacetime in quadratic gravity.	result 0.93	0.86	0.38	asserted 1.00
s2	The resulting differential equations are decoupled in such a way that only the Ricci tensor perturbations need to be found on the Kerr background in order to fully determine the solution.	result 0.89	0.92	0.49	asserted 1.00
s3	The results are illustrated in the simple non-trivial cases of the Schwarzschild and Minkowski geometries.	contribution 0.48	0.86	0.06	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

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relations to other papers

levels run

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S sections
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References

Works cited (33)

[1]

Commutation relations Since the derivative operators D, ∆, and δ, see (16), do not commute even if applied to a function, it is useful to express the commutation relations, ∆D − D∆ = (γ + ¯γ) D + (ϵ + ¯ϵ) ∆ − (¯τ + π) δ − (τ + ¯π) ¯δ , (A1) δD − Dδ = ( ¯α + β − ¯π) D + κ∆ − (¯ρ + ϵ − ¯ϵ) δ − σ¯δ , (A2) δ∆ − ∆δ = −¯νD + (τ − ¯α − β) ∆ + (µ − γ + ¯γ) δ + ¯λ...

View on Pith
[2]

= − m 24 d2 dr2 − 2 r d dr Rε , (64) □ − m2 2 − 2 r2 Φε 22 + 4 r2 Φε 11 = m 12 d2 dr2 Rε , (65) and the difference of (63) and (65) leads to □ − m2 2 − 2 r2 ( Φε 00 − Φε

View on Pith
[3]

= 0 . (66) Observe that equations (62) and (66) can be satisfied simultaneously if and only if Φε 00 = Φε 22 . (67) Using this result, the sum of equations (63) and (64) becomes d2 dr2 − m2 2 + 2 r d dr Φε = m 24 d2 dr2 + 2 r d dr Rε , (68) with Φε = Φε 00 + Φε 11 and the right-hand side is deter- mined by (59). The solution then takes the form Φε = C B ±...

View on Pith
[4]

Askar A et al. 2019 Black holes, gravitational waves and fundamental physics: a roadmap Class. Quant. Grav. 36 143001

View on Pith 2019
[5]

Ricci identities The Ricci identities connect the spin coefficients (8)–(15) with the Weyl (17) and the Ricci (18) projections, 1 Dσ − δκ = σ (3ϵ − ¯ϵ + ρ + ¯ρ) + κ (¯π − τ − 3β − ¯α) + Ψ0 , (A5) Dρ − ¯δκ = ρ2 + σ¯σ + ρ (ϵ + ¯ϵ) − ¯κτ + κ π − 3α − ¯β + Φ00 , (A6) Dτ − ∆κ = ρ (τ + ¯π) + σ (¯τ + π) + τ (ϵ − ¯ϵ) − κ (3γ + ¯γ) + Ψ1 + Φ01 , (A7) Dα − ¯δϵ = α (...

View on Pith
[6]

Bianchi identities Projections of the Riemann tensor covariant derivative with cyclic permutation of indices give the Bianchi identities 0 = − ¯δΨ0 + DΨ1 + (4α − π)Ψ0 − 2(2ρ + ϵ)Ψ1 + 3κΨ2 − DΦ01 + δΦ00 + 2 (ϵ + ¯ρ) Φ01 + 2σΦ10 − 2κΦ11 − ¯κΦ02 + (¯π − 2 ¯α − 2β) Φ00, (A23) 0 = + ¯δΨ1 − DΨ2 − λΨ0 + 2(π − α)Ψ1 + 3ρΨ2 − 2κΨ3 + ¯δΦ01 − ∆Φ00 − 2 (α + ¯τ ) Φ01 +...

View on Pith
[7]

Weyl H 1919 Eine neue Erweiterung der Rela- tivit¨ atstheorieAnn. der Phys. 364 101

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Bach R 1921 Zur Weylschen Relativit¨ atstheorie und der Weylschen Erweiterung des Kr¨ ummungstensorbegriffs Math. Zeitschrift 9 110

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Clifton T et al. 2012 Modified gravity and cosmology Physics Reports 513 1

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Starobinsky A A 1980 A new type of isotropic cosmolog- ical models without singularity Phys. Lett. B 91 99–102

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Peebles P J E and Ratra B 2003 The cosmological con- stant and dark energy Rev. Mod. Phys. 75 559–606

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Penrose R 1965 Gravitational Collapse and Space-Time Singularities Phys. Rev. Lett. 14 57–59

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Goroff M H and Sagnotti A 1985 Quantum gravity at two loops Phys. Lett. B 160 81–86

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Stelle K S 1977 Renormalization of higher derivative quantum gravity Phys. Rev. D 16 953

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Stelle K S 1978 Classical gravity with higher derivatives Gen. Relativ. Gravit. 9 353

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Newman E and Penrose R 1962 An approach to gravita- tional radiation by a method of spin coefficients J. Math. Phys. 3 566–578; and 1963 J. Math. Phys. 4 998

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Bonanno A and Silveravalle S 2021 The gravitational field of a star in quadratic gravity JCAP 08 050

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Kerr R P 1963 Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics Phys. Rev. Lett. 11 237–238

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Abbott B P (LIGO Scientific Collaboration and Virgo Collaboration) 2017 GW170817: Observation of Gravita- tional Waves from a Binary Neutron Star Inspiral Phys. Rev. Lett. 119 161101

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The Event Horizon Telescope Collaboration et al. 2019 First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring Astrophys. J. Lett. 875 L5

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Teukolsky S A 1973 Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Elec- tromagnetic, and Neutrino-Field Perturbations Astro- phys. J. 185 635–648

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[22]

= 0 . (62) The field equations (54)–(56) imply □ − m2 2 − 2 r2 Φε 00 + 4 r2 Φε 11 = m 12 d2 dr2 Rε , (63) □ − m2 2 − 4 r2 Φε 11 + 1 r2 ( Φε 00 + Φε

View on Pith
[23]

ˇSvarc R, Pravdov´ a A, and Miˇ skovsk´ y D 2023 Newman– Penrose formalism in quadratic gravity Phys. Rev. D 107 024036

View on Pith 2023
[24]

Suvorov A G 2019 Gravitational perturbations of a Kerr black hole in f (R) gravity Phys. Rev. D 99 124026

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Salvio A 2018 Quadratic gravity Front. Phys. 6 77

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Stephani H, Kramer D, MacCallum M, Hoenselaers C, and Herlt E 2003 Exact solutions of Einstein ’s field equa- tions (Cambridge: Cambridge University Press)

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Kinnersley W 1969 Type D Vacuum Metrics J. Math. Phys. 10 1195–1203

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Geroch R, Held A, and Penrose R 1973 A space-time calculus based on pairs of null directions J. Math. Phys. 14 874–881

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L¨ u H, Perkins A, Pope C N, and Stelle K S 2015 Black holes in higher derivative gravity Phys. Rev. Lett. 114 171601

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L¨ u H, Perkins A, Pope C N, and Stelle K S 2015 Spher- ically symmetric solutions in higher derivative gravity Phys. Rev. D 92 124019

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[31]

Podolsk´ y J,ˇSvarc R, Pravda V, and Pravdov´ a A 2018 Explicit black hole solutions in higher-derivative gravity Phys. Rev. D 98 021502(R)

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Podolsk´ y J, ˇSvarc R, Pravda V, and Pravdov´ a A 2020 Black holes and other exact spherical solutions in quadratic gravity Phys. Rev. D 101 024027

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Silveravalle S and Zuccotti A 2023 Phase diagram of Einstein–Weyl gravity Phys. Rev. D 107 064029

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How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it

Cites it as background 1 paper

Recent work has explored black hole solutions [ref] , stability analyses [ref] , perturbative [ref] and post-Newtonian [ref] studies, and numerical simulations [ref] , aiming to understand the full nonlinear dynamics of the theory and its potential observational signatures.

Nonperturbative suppression of beyond-General-Relativity effects in quadratic gravity · Introduction · 0.88

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]

Commutation relations Since the derivative operators D, ∆, and δ, see (16), do not commute even if applied to a function, it is useful to express the commutation relations, ∆D − D∆ = (γ + ¯γ) D + (ϵ + ¯ϵ) ∆ − (¯τ + π) δ − (τ + ¯π) ¯δ , (A1) δD − Dδ = ( ¯α + β − ¯π) D + κ∆ − (¯ρ + ϵ − ¯ϵ) δ − σ¯δ , (A2) δ∆ − ∆δ = −¯νD + (τ − ¯α − β) ∆ + (µ − γ + ¯γ) δ + ¯λ...

View on Pith

[2] [2]

= − m 24 d2 dr2 − 2 r d dr Rε , (64) □ − m2 2 − 2 r2 Φε 22 + 4 r2 Φε 11 = m 12 d2 dr2 Rε , (65) and the difference of (63) and (65) leads to □ − m2 2 − 2 r2 ( Φε 00 − Φε

View on Pith

[3] [3]

= 0 . (66) Observe that equations (62) and (66) can be satisfied simultaneously if and only if Φε 00 = Φε 22 . (67) Using this result, the sum of equations (63) and (64) becomes d2 dr2 − m2 2 + 2 r d dr Φε = m 24 d2 dr2 + 2 r d dr Rε , (68) with Φε = Φε 00 + Φε 11 and the right-hand side is deter- mined by (59). The solution then takes the form Φε = C B ±...

View on Pith

[4] [4]

Askar A et al. 2019 Black holes, gravitational waves and fundamental physics: a roadmap Class. Quant. Grav. 36 143001

View on Pith 2019

[5] [5]

Ricci identities The Ricci identities connect the spin coefficients (8)–(15) with the Weyl (17) and the Ricci (18) projections, 1 Dσ − δκ = σ (3ϵ − ¯ϵ + ρ + ¯ρ) + κ (¯π − τ − 3β − ¯α) + Ψ0 , (A5) Dρ − ¯δκ = ρ2 + σ¯σ + ρ (ϵ + ¯ϵ) − ¯κτ + κ π − 3α − ¯β + Φ00 , (A6) Dτ − ∆κ = ρ (τ + ¯π) + σ (¯τ + π) + τ (ϵ − ¯ϵ) − κ (3γ + ¯γ) + Ψ1 + Φ01 , (A7) Dα − ¯δϵ = α (...

View on Pith

[6] [6]

Bianchi identities Projections of the Riemann tensor covariant derivative with cyclic permutation of indices give the Bianchi identities 0 = − ¯δΨ0 + DΨ1 + (4α − π)Ψ0 − 2(2ρ + ϵ)Ψ1 + 3κΨ2 − DΦ01 + δΦ00 + 2 (ϵ + ¯ρ) Φ01 + 2σΦ10 − 2κΦ11 − ¯κΦ02 + (¯π − 2 ¯α − 2β) Φ00, (A23) 0 = + ¯δΨ1 − DΨ2 − λΨ0 + 2(π − α)Ψ1 + 3ρΨ2 − 2κΨ3 + ¯δΦ01 − ∆Φ00 − 2 (α + ¯τ ) Φ01 +...

View on Pith

[7] [7]

Weyl H 1919 Eine neue Erweiterung der Rela- tivit¨ atstheorieAnn. der Phys. 364 101

View on Pith 1919

[8] [8]

Bach R 1921 Zur Weylschen Relativit¨ atstheorie und der Weylschen Erweiterung des Kr¨ ummungstensorbegriffs Math. Zeitschrift 9 110

View on Pith 1921

[9] [9]

Clifton T et al. 2012 Modified gravity and cosmology Physics Reports 513 1

View on Pith 2012

[10] [10]

Starobinsky A A 1980 A new type of isotropic cosmolog- ical models without singularity Phys. Lett. B 91 99–102

View on Pith 1980

[11] [11]

Peebles P J E and Ratra B 2003 The cosmological con- stant and dark energy Rev. Mod. Phys. 75 559–606

View on Pith 2003

[12] [12]

Penrose R 1965 Gravitational Collapse and Space-Time Singularities Phys. Rev. Lett. 14 57–59

View on Pith 1965

[13] [13]

Goroff M H and Sagnotti A 1985 Quantum gravity at two loops Phys. Lett. B 160 81–86

View on Pith 1985

[14] [14]

Stelle K S 1977 Renormalization of higher derivative quantum gravity Phys. Rev. D 16 953

View on Pith 1977

[15] [15]

Stelle K S 1978 Classical gravity with higher derivatives Gen. Relativ. Gravit. 9 353

View on Pith 1978

[16] [16]

Newman E and Penrose R 1962 An approach to gravita- tional radiation by a method of spin coefficients J. Math. Phys. 3 566–578; and 1963 J. Math. Phys. 4 998

View on Pith 1962

[17] [17]

Bonanno A and Silveravalle S 2021 The gravitational field of a star in quadratic gravity JCAP 08 050

View on Pith 2021

[18] [18]

Kerr R P 1963 Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics Phys. Rev. Lett. 11 237–238

View on Pith 1963

[19] [19]

Abbott B P (LIGO Scientific Collaboration and Virgo Collaboration) 2017 GW170817: Observation of Gravita- tional Waves from a Binary Neutron Star Inspiral Phys. Rev. Lett. 119 161101

View on Pith 2017

[20] [20]

The Event Horizon Telescope Collaboration et al. 2019 First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring Astrophys. J. Lett. 875 L5

View on Pith 2019

[21] [21]

Teukolsky S A 1973 Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Elec- tromagnetic, and Neutrino-Field Perturbations Astro- phys. J. 185 635–648

View on Pith 1973

[22] [22]

= 0 . (62) The field equations (54)–(56) imply □ − m2 2 − 2 r2 Φε 00 + 4 r2 Φε 11 = m 12 d2 dr2 Rε , (63) □ − m2 2 − 4 r2 Φε 11 + 1 r2 ( Φε 00 + Φε

View on Pith

[23] [23]

ˇSvarc R, Pravdov´ a A, and Miˇ skovsk´ y D 2023 Newman– Penrose formalism in quadratic gravity Phys. Rev. D 107 024036

View on Pith 2023

[24] [24]

Suvorov A G 2019 Gravitational perturbations of a Kerr black hole in f (R) gravity Phys. Rev. D 99 124026

View on Pith 2019

[25] [25]

Salvio A 2018 Quadratic gravity Front. Phys. 6 77

View on Pith 2018

[26] [26]

Stephani H, Kramer D, MacCallum M, Hoenselaers C, and Herlt E 2003 Exact solutions of Einstein ’s field equa- tions (Cambridge: Cambridge University Press)

View on Pith 2003

[27] [27]

Kinnersley W 1969 Type D Vacuum Metrics J. Math. Phys. 10 1195–1203

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