Pith. sign in

REVIEW 4 major objections 4 minor 23 references

Rotating black holes in de Rham-Gabadadze-Tolley massive gravity: Newman-Janis Algorithm

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper reports the first analytic rotating hairy black hole in dRGT massive gravity, generated by the Newman-Janis algorithm, and claims the algorithm works in this theory.

desk verdict A clearly written letter offering a rotating KNdS-type metric as a dRGT solution, but the central verification is explicitly postponed to another paper; as submitted it is an ansatz, not a result. read the letter →

arxiv 2501.12586 v1 pith:GPP5MCDL submitted 2025-01-22 gr-qc

Ping Li , Jiang-he Yang This is my paper · ORCID
classification gr-qc MSC 83C5783C1583D05 PACS 04.70.-s04.50.Kd
keywords massivegravitydRGTtheoryrotatingblackholeshairyNewman-JanisalgorithmStückelbergfieldsreferencemetricno-hairconjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to settle three related questions in de Rham-Gabadadze-Tolley massive gravity: whether a rotating black hole exists that reproduces the known static spherically symmetric case, whether a hairy rotating solution can be written analytically, and whether the Newman-Janis algorithm works in a massive theory of gravity. Its answer to all three is yes. The route is a gauge lemma that identifies every nonunitary St\"uckelberg configuration with the Minkowski reference metric with a unitary configuration using a curved reference metric, followed by a cosmological-constant-modified Newman-Janis algorithm applied to the static solution. The paper's rotating metric has the same static limit when the spin parameter vanishes, and it carries a new hair parameter $S$ that enters alongside the electric charge $Q$.

What carries the argument

The machinery is the gauge lemma together with the modified Newman-Janis algorithm. The lemma shows that any nonunitary gauge with the Minkowski reference metric is equivalent to a unitary gauge with some curved reference metric, allowing the authors to work with $\varphi^a=x^a\delta^a_\alpha$ and a general reference metric $f_{ab}$. The Newman-Janis algorithm then takes the static spherically symmetric solution (10), rewrites it in Eddington-Finkelstein coordinates, complexifies $u$, $r$, and $\phi$ with spin parameter $a$, and rebuilds the metric from a null tetrad; the cosmological-constant version fixes $\Delta_\theta$ and the $\phi$-transformation. The output is metric (45), whose $\Delta_r$ carries mass $M$, charge $Q$, graviton mass $m$, hair constant $S$, effective cosmological parameter $\Lambda$, and spin $a$.

What would settle it

Take metric (45) with the reference-metric components determined by the five equations in Appendix A and substitute both into the field equation (7). If any component, particularly an off-diagonal equation, is nonzero, the rotating metric is not a solution; the paper does not perform this substitution, so this check is the minimal calculation that would settle the claim.

Watch

Extended reading notes

Core claim

The central object is metric (45), with $\Delta_r=(r^2+a^2)(1-\Lambda m^2 r^2/3)-2Mr+Q^2-m^2S^2$ and $\Delta_\theta=1+\Lambda m^2 a^2\cos^2\theta/3$. The paper presents this as a rotating, electrically charged hairy black hole in dRGT massive gravity, where the hair is the graviton-mass contribution encoded by the St\"uckelberg field through the constant $S$. It is constructed so that $a\to0$ returns the static solution $f(r)=1-2M/r+(Q^2-m^2S^2)/r^2-m^2\Lambda r^2/3$, and the graviton mass appears both in an effective cosmological term and as a modification of the $1/r^2$ charge term. The paper further states that the Newman-Janis algorithm is applicable in massive gravity, and that this is the first analytic hairy rotating black hole in this theory that can reduce to the non-rotating case.

Load-bearing premise

The rotating metric (45) is not demonstrated in this paper to satisfy the dRGT field equations; the authors defer that check to a separate paper, so the claim collapses if the substitution fails.

Editorial extensions

If this is right

  • If metric (45) is a genuine solution, dRGT massive gravity gains its first explicit analytic rotating hairy black hole that reduces to the static case as $a\to0$.
  • The Newman-Janis algorithm, with the cosmological-constant modification of Ref. [17], is thereby shown to work in a nonlinear massive theory, not just in Einstein or f(R) gravity.
  • The graviton mass enters the $1/r^2$ term as $-m^2S^2$, so the new solution is not simply Kerr-Newman with a cosmological constant; horizon radii and geodesics shift in a way controlled by the hair parameter $S$.
  • Taking $a\to0$ recovers metric (20), which gives a built-in consistency check that the rotating solution's static limit is exactly the previously known static hairy solution.

Reading between the lines

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

  • Editorial extension: the gauge lemma may be the paper's most reusable piece, converting the search for dRGT black holes into a search over curved reference metrics in unitary gauge and potentially yielding other rotating families beyond the one exhibited.
  • Editorial extension: because the charge term is $(Q^2-m^2S^2)/r^2$, the spacetime geometry alone cannot separate the electric charge from the graviton-mass hair; an astrophysical test would need an independent measurement of one of them.
  • Editorial extension: if the companion rigorous derivation passes, a natural next check is the shadow of metric (45), which would differ from Kerr by an amount controlled by $m^2S^2$ and by the effective cosmological term.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript claims three results in dRGT massive gravity: a lemma equating nonunitary gauges with a Minkowski reference metric to unitary gauges with a curved reference metric; a static, spherically symmetric hairy black hole solution (20) based on the assumed massive stress-tensor form (17)-(18); and a rotating generalization (45) obtained by a modified Newman-Janis algorithm, which the authors state is the first analytic rotating hairy black hole in dRGT massive gravity. The paper also claims that the Newman-Janis algorithm is applicable in massive gravity. The rotating metric is presented as the final result, but the manuscript does not verify that this metric satisfies the dRGT field equations (7); the Conclusion explicitly defers that verification to another paper.

Significance. If the central claim were established, the paper would be significant: an analytic rotating hairy black hole in dRGT massive gravity, with the graviton mass effectively shifting the charge term through Q^2 - m^2 S^2, would be a concrete extension of the static solutions in Ref. [6] and would give new evidence on the rotating applicability of the Newman-Janis algorithm in modified gravity. The paper is also honest in stating that NJA-generated metrics must be checked against the field equations. However, the significance is entirely conditional: metric (45) is a candidate ansatz, not a demonstrated solution, and the static parent solution itself is not derived in full detail. The paper contains no machine-checked proof or explicit field-equation verification, and its central claims therefore remain unsupported.

major comments (4)
  1. [Conclusion and Eq. (45)] The central claim of the paper is not demonstrated. The manuscript states in the Conclusion that 'To ensure that the rotation metric (45) is indeed the solution, we perform a rigorous mathematical derivation in another paper,' and no such derivation is included here. The transformations (31)-(33) and (43)-(44) act on coordinates, the metric, and the null tetrad only; they are never applied to the Stückelberg fields φ^a or to the reference metric f_ab, and no component check of T^(K)_μν in Eq. (7) is provided for the rotating metric. Metric (45) is therefore an ansatz, not a verified solution, and the statement that the Newman-Janis algorithm is 'applicable' in dRGT massive gravity is unsupported.
  2. [Eqs. (17)-(18) and Appendix A] The static solution on which the rotating metric is based is itself not fully established here. Equation (20) follows from (19) only after assuming the massive stress tensor has the special forms (17)-(18). Appendix A asserts that 'Five equations will uniquely determine a set of five unknowns' and that solving yields f00, f01, f11, f22, f33, but no explicit reference metric is displayed and no argument is given that the solution is real, regular, and compatible with the ansatz (15). No relation is given between Λ or S and the theory parameters m, α3, α4, so Λ and S are effectively free parameters inserted into T^(K). Since the claimed static limit of (45) depends on this solution, this gap affects the central claim as well.
  3. [Paragraph after Eq. (14)] The Lemma that any nonunitary gauge with the Minkowski reference metric is equivalent to a unitary gauge with some curved reference metric is a reparameterization rather than a substantive physical statement: given φ^a one may define f̄_ab = ∂_aφ^c ∂_bφ^d η_cd, and the unitary-gauge description follows by definition. The Lemma therefore cannot, by itself, justify the existence of black hole solutions with the assumed stress tensor (17)-(18) or the extension of such solutions to rotating configurations; those claims require an explicit Stückelberg/reference-metric configuration satisfying (7).
  4. [Introduction and Conclusion (NJA applicability)] The authors correctly note in the Introduction that the NJA operates on the metric and that any resulting metric must be checked against the field equations. However, the paper does not perform that check, and a single metric ansatz would not suffice to 'confirm that the Newman-Janis algorithm is applicable in the context of massive gravity,' particularly in view of the known failures in other modified gravity theories cited in Refs. [18-20]. A general statement of applicability needs either a theorem or a full field-equation verification; the present manuscript provides neither.
minor comments (4)
  1. [Introduction] There are several typographical and grammatical errors, including 'Dose' in the third numbered question, 'ans¨atz' in the section heading, and 'the Minkowski reference metric are equal' in the Lemma paragraph.
  2. [Appendix A] The trace notation used throughout, such as [K], [γ], and [√Ξ], is not explicitly defined, and Eq. (A4) appears to contain a typo: the second displayed trace should likely be [√Ξ_2] rather than [√Ξ_1].
  3. [Eqs. (33), (35), (38)] In the complexified dϕ transformation, a term -1 + (1+Λm^2a^2/3)/Δθ appears in (33) and (35), but the final tetrad component (38) appears to retain only the fractional term; the authors should clarify whether terms cancel or whether a term is missing.
  4. [Eq. (41) and Conclusion] The statement that the graviton mass term 'modifies the black hole's charge term' is premature and purely parametrical: it is the free constant S that enters as Q^2 - m^2S^2, and no physical derivation of S or its relation to Stückelberg fields is given.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the static derivation is self-contained, and the rotating metric is an unverified ansatz, which is a correctness gap rather than a circular reduction.

full rationale

The derivation chain is not circular in the sense prohibited by the review rules. The static solution (20) follows from the field equations (7) and the assumed Stückelberg stress-tensor form (17)-(18), with S and Lambda as free parameters solved for in Appendix A rather than fitted to the rotating metric. The Newman-Janis step is an explicit tetrad transformation (26)-(38) of the static metric; the requirement that h and Psi reduce to f(r) and r^2 in the limit a to 0 is imposed by equations (29)-(30), so the property "reduces to the non-rotating case" is a consistency condition built into the algorithm, not a prediction extracted from the rotating metric. The central physical claim that metric (45) is a genuine rotating solution of the dRGT field equations is not established in this manuscript: no rotating Stückelberg or reference-metric configuration is provided, and the Conclusion explicitly states, "To ensure that the rotation metric (45) is indeed the solution, we perform a rigorous mathematical derivation in another paper." This is a lack of verification and an unsupported claim, appropriately flagged as a correctness risk, but it is not an instance of a result being equivalent to its inputs by construction. The only self-citation, Ref. [6], is used for context and comparison ("This solution is very similar to some solution given by Ref. [6]"), and the static solution is rederived independently, so the self-citation is not load-bearing. No circular step was found; score 0.

Assumptions & free parameters 2 free parameters · 3 assumptions · 1 invented entities

The central construction relies on a reparameterization lemma (nonunitary to unitary gauge), an assumed stress-tensor form for the static solution, and an unverified transfer of the same structure to the rotating metric. The parameters S and Λ are introduced by hand and not derived from α3, α4.

free parameters (2)
  • S (Stückelberg hair parameter) = unspecified
    Introduced in Eq. (17) so that the massive-gravity stress tensor takes the simple form S^2/r^4 - Λ. It is not fixed by the model parameters α3, α4 in this paper and it is carried into the rotating solution without independent derivation.
  • Λ (effective cosmological-constant-like parameter) = unspecified
    Appears as the constant part of T^(K) in Eqs. (17) and (18). The text says it is related to α3 and α4, but no explicit relation is provided, so it functions as an adjustable parameter in the solution.
assumptions (3)
  • domain assumption Any nonunitary gauge with a Minkowski reference metric is equivalent to a unitary gauge with some curved reference metric.
    Stated as a lemma without proof in the SSS ansatz section. It justifies replacing the Stückelberg fields by a fixed unitary gauge with a general reference metric (15).
  • ad hoc to paper The massive-gravity stress tensor for the SSS background takes the assumed diagonal form T^(K)0_0 = T^(K)1_1 = S^2/r^4 - Λ and T^(K)2_2 = T^(K)3_3 = -S^2/r^4 - Λ.
    This form is imposed in Eqs. (17)-(18) to reduce the field equation to a single ODE. The existence of reference-metric components realizing it is asserted but not exhibited.
  • ad hoc to paper The rotating metric (45) generated by the modified Newman-Janis algorithm satisfies the dRGT field equations.
    The paper does not check the field equations for the rotating metric. The Conclusion states the rigorous derivation is performed in another paper, so this is an unverified assumption in the present manuscript.
invented entities (1)
  • Stückelberg hair parameter S
    purpose: Adds a 1/r^2 contribution to the metric that is multiplied by the graviton mass squared, effectively modifying the electromagnetic charge term.
    S is introduced as a new constant in the stress tensor with no independent observational or theoretical derivation provided in this paper. It is a new conserved hair parameter without external falsifiable handle.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Rotating black holes in de Rham-Gabadadze-Tolley massive gravity: Newman-Janis Algorithm." pith.science (2026). https://pith.science/paper/GPP5MCDL

@misc{pith2026250112586,
  author       = {Pith},
  title        = {Pith review of: Rotating black holes in de Rham-Gabadadze-Tolley massive gravity: Newman-Janis Algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GPP5MCDL}},
  note         = {Machine review of arXiv:2501.12586}
}
read the original abstract

We report the discovery of rotating black hole solutions within the framework of de Rham-Gabadadze-Tolley (dRGT) massive gravity. We demonstrate that any nonunitary gauge with the Minkowski reference metric are equal to a unitary gauge with some curved reference metric. Based on this Lemma, we revisit the process of deriving black hole solutions in dRGT theory. We explain how to obtain a static, spherically symmetric solution and then transform it into the corresponding rotating black hole using the Newman-Janis algorithm. For the first time, we provide an analytic expression for a hairy black hole that can reduce to the non-rotating case. Additionally, we confirm that the Newman-Janis algorithm is applicable in the context of massive gravity.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [6]

    T. M. Nieuwenhuizen, Exact Schwarzschild-de Sitter bla ck holes in a family of massive gravity models, Phys. Rev. D 84, 024038 (2011)

    work page 2011

  2. [1]

    For the massive extension, the dRGT theory is the most competitive candidate [ 1]

    What is the rotating black hole solution that can reproduce the st atic spherical sym- metry (SSS) case in massive gravity? Traditionally, Einstein’s gravity has been recognized as the only massless spin-2 theory. For the massive extension, the dRGT theory is the most competitive candidate [ 1]. In this theory, the primary contribution to the decoupling l...

    work page

  3. [2]

    For many years, resea rchers have sought to prove or disprove this conjecture

    Is the no-hair conjecture true for massive gravity theory? In Einstein’s theory of gravity, the no-hair conjecture, proposed by John Wheeler [ 9], suggests that all black holes in nature belong to the Kerr-Newman family. For many years, resea rchers have sought to prove or disprove this conjecture. There are two fundamenta lly different approaches to testi...

    work page 2014

  4. [3]

    By employing a set of complex coordinate transformations, the NJA serves as a f ast and efficient shortcut for deriving rotating solutions

    Dose the Newman-Janis algorithm (NJA) is also valid for the hairy bla ck holes in massive gravity? The NJA [ 14] is a powerful technique used to generate Kerr or Kerr- like metrics from the corresponding spherically symmetric solutions ( SSS). By employing a set of complex coordinate transformations, the NJA serves as a f ast and efficient shortcut for deri...

    work page

  5. [4]

    de Rham, G

    C. de Rham, G. Gabadadze, and A. J. Tolley, Resummation of Massive Gravity, Phys. Rev. Lett 106, 231101 (2011)

    work page 2011

  6. [5]

    de Rham, Massive Gravity, Living Rev

    C. de Rham, Massive Gravity, Living Rev. Relativity 17, 7 (2014)

    work page 2014

  7. [7]

    Y. F. Cai, D. A. Easson, C. Gao, E. N. Saridakis, Charged bl ack holes in nonlinear massive gravity, Phys. Rev. D 87, 064001 (2013)

    work page 2013

  8. [8]

    Berezhiani, G

    L. Berezhiani, G. Chkareuli, C. de Rham, G. Gabadadze, an d A. J. Tolley, On Black Holes in Massive Gravity, Phys. Rev. D 85, 044024 (2012)

    work page 2012

Show all 23 references
  1. [9]

    Ping Li, Xin-zhou Li, and Ping Xi, Black hole solutions in de Rham-Gabadadze-Tolley massive gravity, Phys. Rev. D 93, 064040 (2016)

    2016

  2. [10]

    Rong-Gen Cai, Ya-Peng Hu, Qi-Yuan Pan, and Yun-Long Zhan g, Thermodynamics of Black Holes in Massive Gravity, Phys. Rev. D 91, 024032 (2015)

    2015

  3. [11]

    Ghadir Jafari, M. R. Setare, and Hamid R. Bakhtiarizadeh , Static spherically symmetric black holes of de Rham-Gabadadze-Tolley massive gravity in arbit rary dimensions, Phys. Lett. B 773 (2017) 395-400

    2017

  4. [12]

    Ruffini and J

    R. Ruffini and J. A. Wheeler, Introducing the black hole, Phys. Today 24, 30 (1971)

    1971

  5. [13]

    M. S. Volkov and D. V Galtsov, Non-Abelian Einstein Yang -Mills black holes, JETP Lett. 50, 346 (1989)

    1989

  6. [14]

    M. S. Volkov and D. V Galtsov, Black holes in Einstein Yan g-Mills theory, Sov. J. Nucl. Phys. 51, 747 (1990)

    1990

  7. [15]

    C. A.R. Herdeiro and E. Radu, Kerr Black Holes with Scala r Hair, Phys. Rev. Lett 112, 221101 (2014)

    2014

  8. [16]

    C. A.R. Herdeiro and E. Radu, Asymptotically flat black h oles with scalar hair: A review, Int. J. Mod. Phys. D 24, 1542014 (2015)

    2015

  9. [17]

    E. T. Newman and A. I. Janis, J. Math. Phys. 6, 915 (1965). 10

    1965

  10. [18]

    Azreg-A ¨ ınou, Generating rotating regular black ho le solutions without complexification, Phys

    M. Azreg-A ¨ ınou, Generating rotating regular black ho le solutions without complexification, Phys. Rev. D 90, 064041 (2014)

    2014

  11. [19]

    Azreg-A ¨ ınou, From static to rotating to conformal static solutions: rotating imperfect fluid wormholes with(out) eletric or magnetic field, Eur

    M. Azreg-A ¨ ınou, From static to rotating to conformal static solutions: rotating imperfect fluid wormholes with(out) eletric or magnetic field, Eur. Phys. J. C (2014) 74:2865

    2014

  12. [20]

    Chaturvedi, U

    P. Chaturvedi, U. Kumar, U. Thattarampilly and V. Kakka t, Exact rotating black hole solu- tions for f (R) gravity by modified Newman Janis algorithm, Eur. Phys. J. C (2023) 83:1124

    2023

  13. [21]

    Cadoni and P

    M. Cadoni and P. Pani, Holography of charged dilatonic b lack branes at finite temperature. JHEP 04 (2011) 049

    2011

  14. [22]

    Hansen and N

    D. Hansen and N. Yunes, Applicability of the Newman-Jan is algorithm to black hole solutions of modified gravity theories. Phys. Rev. D 88 (2013) 104020

    2013

  15. [23]

    Ayzenberg and N

    D. Ayzenberg and N. Yunes, Slowly-rotating black holes in Einstein-Dilaton-Gauss-Bonnet Gravity: quadratic order in spin solutions. Phys. Rev. D 91 (2015) 069905. 11

    2015

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.