Hidden Symmetries and Gyromagnetic Ratio of Kerr-Newman Black Holes in $f(R)$ Gravity

G\"oksel Daylan Esmer; Saliha T\"urkmen

arxiv: 2604.14066 · v2 · submitted 2026-04-15 · 🌀 gr-qc

Hidden Symmetries and Gyromagnetic Ratio of Kerr-Newman Black Holes in f(R) Gravity

G\"oksel Daylan Esmer , Saliha T\"urkmen This is my paper

Pith reviewed 2026-05-10 12:45 UTC · model grok-4.3

classification 🌀 gr-qc

keywords f(R) gravityKerr-Newman black holehidden symmetriesKilling-Yano tensorgyromagnetic ratiomodified gravityHamilton-Jacobi equation

0 comments

The pith

Kerr-Newman black holes in f(R) gravity retain the gyromagnetic ratio g=2 due to preserved hidden symmetries.

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

The paper derives the Killing and Killing-Yano tensors for charged rotating black holes in f(R) gravity to show that the spacetime structure supports the same hidden symmetries as in standard general relativity. This allows calculation of the gyromagnetic ratio, which remains exactly 2 for these four-dimensional black holes. A reader might care because this indicates that key electromagnetic and rotational properties of black holes are robust to changes in the underlying gravitational theory, which could affect how we model black hole mergers and gravitational waves.

Core claim

By deriving the Killing and Killing-Yano tensors, the role of hidden symmetries in the spacetime structure of Kerr-Newman black holes in f(R) gravity is established. The gyromagnetic ratio retains its universal value of g=2, consistent with all four-dimensional black holes. This highlights the connection between hidden symmetries and the separability of the Hamilton-Jacobi equation in this modified gravity setting.

What carries the argument

Killing and Killing-Yano tensors that encode the hidden symmetries and enable separation of the Hamilton-Jacobi equation while fixing the gyromagnetic ratio at 2.

If this is right

The Hamilton-Jacobi equation remains separable for these black holes in f(R) gravity.
The gyromagnetic ratio stays at the universal value g=2 without additional constraints from the f(R) modification.
These results apply to electrically charged rotating black holes and extend the consistency seen in general relativity.
Implications arise for theoretical studies of black holes in modified gravity and observational aspects like gravitational wave analyses.

Where Pith is reading between the lines

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

If the metric form holds for other f(R) functions, the same symmetries and ratio would apply more broadly.
This preservation might indicate a deeper geometric reason independent of the specific gravity theory.
Future work could test this by examining the motion of charged particles or the black hole's response in f(R) cosmologies.

Load-bearing premise

The standard Kerr-Newman metric form and the usual method for building Killing and Killing-Yano tensors continue to work without changes imposed by the particular f(R) function chosen.

What would settle it

Explicitly computing the gyromagnetic ratio using a specific f(R) model where the spacetime metric differs from the Kerr-Newman form and finding it deviates from 2 would falsify the claim.

read the original abstract

We explore hidden symmetries in electrically charged, four-dimensional rotating Kerr-Newman black hole within $f(R)$ gravity. By deriving the Killing and Killing-Yano tensors, we establish their role in the spacetime structure. The gyromagnetic ratio is calculated and shown to retain its universal value of $g = 2$, consistent with all four-dimensional black holes. These findings show that the gyromagnetic ratio remains consistent in this modified gravity setting. Moreover, they highlight the connection between hidden symmetries and the ability to separate the Hamilton-Jacobi equation in $f(R)$ gravity. This work advances the study of black holes in modified gravity, offering implications for both theoretical frameworks and observational cosmology, such as gravitational wave analyses.

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.

Desk Editor's Note private letter to a colleague

The paper extends the usual Killing-Yano and gyromagnetic-ratio calculations to Kerr-Newman in f(R) but does not verify that the metric solves the modified equations for general f.

read the letter

The core claim is that the Kerr-Newman solution keeps its hidden symmetries and g=2 gyromagnetic ratio when placed in f(R) gravity. They derive the Killing and Killing-Yano tensors explicitly and use them to separate the Hamilton-Jacobi equation, then extract the ratio from the asymptotic charges in the standard way. That part is a clean, direct application of the GR toolkit to the new setting and reproduces the expected universal value without extra fitting or redefinitions. The calculation itself looks reproducible from the abstract and the usual procedure. What is new is simply the confirmation that these tensors survive and give the same g in this modified-gravity background. The soft spot is the metric's status. Outside the horizon R vanishes identically, so the trace of the f(R) field equations reduces to the constraint 3□f' - 2f = 0. Arbitrary f(R) does not satisfy this, yet the paper treats the Kerr-Newman line element as an exact solution without stating which f(R) works or showing that the fourth-order equations are satisfied. That leaves the domain of the result unclear. The rest of the argument follows once the metric is granted, but the grant is not automatic. Readers working on black-hole symmetries or gravitational-wave templates in modified gravity would get a short, usable reference calculation from this. It is not a broad advance, but the symmetry part is solid enough that a referee could check the tensor derivations and the constraint issue in one pass. I would send it to review rather than desk-reject.

Referee Report

1 major / 1 minor

Summary. The manuscript explores hidden symmetries of the electrically charged rotating Kerr-Newman black hole in f(R) gravity. It derives the Killing and Killing-Yano tensors, uses them to establish the spacetime structure and separability of the Hamilton-Jacobi equation, and computes the gyromagnetic ratio, reporting that it retains the universal value g=2 found for all four-dimensional black holes.

Significance. If the derivations hold and the metric is shown to solve the f(R) equations, the result would establish that hidden symmetries and the gyromagnetic ratio g=2 are robust against this class of modifications to general relativity. This would strengthen the case for universality of these properties across four-dimensional black-hole solutions and could inform both analytic separability techniques and observational probes such as gravitational-wave ringdown.

major comments (1)

[Abstract and metric/tensor sections] The central claim that the Kerr-Newman metric and its Killing-Yano tensor can be used unchanged to obtain g=2 rests on the unverified assumption that this geometry solves the f(R) field equations for the chosen f. Outside the horizon R=0 identically, so the trace of the field equations reduces to the constraint 3□f'(R)−2f(R)=0. This is not satisfied for arbitrary f(R) and forces f to obey a specific differential relation (e.g., f''(R)=0). The manuscript applies the standard GR construction of the tensors and gyromagnetic ratio without demonstrating that the adopted f(R) satisfies the constraint or that the metric solves the full fourth-order equations. (Abstract; metric ansatz and tensor derivation sections.)

minor comments (1)

[Abstract] The abstract repeats the statement that the gyromagnetic ratio remains consistent; a single concise sentence would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. The major concern regarding the status of the Kerr-Newman metric as a solution of the f(R) field equations is addressed point by point below. We agree that clarification is needed and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract and metric/tensor sections] The central claim that the Kerr-Newman metric and its Killing-Yano tensor can be used unchanged to obtain g=2 rests on the unverified assumption that this geometry solves the f(R) field equations for the chosen f. Outside the horizon R=0 identically, so the trace of the field equations reduces to the constraint 3□f'(R)−2f(R)=0. This is not satisfied for arbitrary f(R) and forces f to obey a specific differential relation (e.g., f''(R)=0). The manuscript applies the standard GR construction of the tensors and gyromagnetic ratio without demonstrating that the adopted f(R) satisfies the constraint or that the metric solves the full fourth-order equations. (Abstract; metric ansatz and tensor derivation sections.)

Authors: We appreciate the referee highlighting this important point. The Kerr-Newman metric is considered in our work as an exact solution within the class of f(R) gravity models for which the Ricci scalar vanishes identically (R=0), as is the case for the Kerr-Newman spacetime both inside and outside the horizon. Under this condition the trace equation reduces to 3□f'(R)−2f(R)=0. Because R is a constant, f'(R) is constant and □f'(R)=0, so the equation is satisfied provided f(0)=0. This is a standard requirement for f(R) models that admit general-relativity solutions as a limit and does not impose a differential relation such as f''(R)=0. The hidden symmetries (Killing and Killing-Yano tensors) and the gyromagnetic ratio g=2 are geometric properties of the metric and the electromagnetic field; they therefore hold for any f(R) that admits the Kerr-Newman geometry. We will revise the abstract, introduction, and the metric/tensor sections to explicitly state the condition f(0)=0 and to confirm that the metric satisfies the f(R) field equations under this restriction. No change is required to the derivations themselves. revision: yes

Circularity Check

0 steps flagged

No significant circularity; g=2 follows from standard tensor construction on assumed metric

full rationale

The paper derives Killing and Killing-Yano tensors explicitly for the Kerr-Newman form and computes the gyromagnetic ratio from asymptotic charges, yielding g=2 by the same algebraic steps used in GR. No self-definitional loop, no fitted parameter renamed as prediction, and no load-bearing self-citation chain is visible in the abstract or described procedure. The result is an independent calculation once the metric is granted; any question of whether that metric solves the f(R) equations is an applicability issue, not a reduction of the derivation to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definitions of Killing and Killing-Yano tensors and on the assumption that the Kerr-Newman line element continues to solve the modified field equations for some f(R). No new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Killing and Killing-Yano tensors exist and can be constructed by the same algebraic procedure used in general relativity
Invoked when the paper states it derives these tensors for the f(R) spacetime.
domain assumption The spacetime metric retains the Kerr-Newman form in f(R) gravity
Required for the black-hole solution to be called Kerr-Newman and for the gyromagnetic-ratio calculation to proceed.

pith-pipeline@v0.9.0 · 5424 in / 1388 out tokens · 51073 ms · 2026-05-10T12:45:58.827080+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

70 extracted references · 70 canonical work pages

[1]

dt− asin 2 θ Ξ dϕ #2 + Σ ∆r dr2 + Σ ∆θ dθ2 + ∆θ sin2 θ Σ

also found the same value for four-dimensional rotation charged black hole solution in 3 heterotic string theory. [50] showed that the gyromagnetic ratio of four-dimensional black holes corresponding tog= 2 is a universal property. The gyromagnetic ratio for higher dimensional black holes is not universal as it is for four-dimensional black holes. While t...

work page
[2]

A. G. Riesset al., Astron. J.116, 1009 (1998)

work page 1998
[3]

Perlmutteret al., Astrophys

S. Perlmutteret al., Astrophys. J.517, 565 (1999)

work page 1999
[4]

J. L. Tonryet al., Astrophys. J.594, 1 (2003)

work page 2003
[5]

D. N. Spergelet al., Astrophys. J. Suppl. Ser.170, 377 (2007)

work page 2007
[6]

Komatsuet al., Astrophys

E. Komatsuet al., Astrophys. J. Suppl. Ser.180, 330 (2009)

work page 2009
[7]

Komatsuet al., Astrophys

E. Komatsuet al., Astrophys. J. Suppl. Ser.192, 18 (2011)

work page 2011
[8]

P. A. R. Adeet al., A&A571, A1 (2014)

work page 2014
[9]

P. A. R. Adeet al., A&A571, A16 (2014)

work page 2014
[10]

P. A. R. Adeet al., A&A594, A13 (2016)

work page 2016
[11]

A. H. Guth, Phys. Rev. D23, 347 (1981)

work page 1981
[12]

Nojiri and S

S. Nojiri and S. D. Odintsov, Phys. Rev. D68, 123512 (2003)

work page 2003
[13]

Nojiri and S

S. Nojiri and S. D. Odintsov, Gen. Relat. Gravit.36, 1765 (2004). 13

work page 2004
[14]

S. M. Carroll, V. Duvvuri, M. Trodden and M. S. Turner, Phys. Rev. D70, 043528 (2004)

work page 2004
[15]

Dobado and A

A. Dobado and A. L. Maroto, Phys. Rev. D52, 1895 (1995)

work page 1995
[16]

J. A. R. Cembranos, Phys. Rev. Lett.102, 141301 (2009)

work page 2009
[17]

Lovelock, J

D. Lovelock, J. Math. Phys.12, 498 (1971)

work page 1971
[18]

Nojiri and S

S. Nojiri and S. D. Odintsov, Phys. Rept.505, 59 (2011)

work page 2011
[19]

Nojiri, S

S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Phys. Rept.692, 1 (2017)

work page 2017
[20]

Capozziello and M

S. Capozziello and M. De Laurentis, Phys. Rept.509, 167 (2011)

work page 2011
[21]

T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys.82, 451 (2010)

work page 2010
[22]

De Felice and S

A. De Felice and S. Tsujikawa, Living Rev. Rel.13, 3 (2010)

work page 2010
[23]

de la Cruz-Dombriz and A

A. de la Cruz-Dombriz and A. Dobado, Phys. Rev. D74, 087501 (2006)

work page 2006
[24]

P. K. S. Dunsby, E. Elizalde, R. Goswami, S. Odintsov and D. Saez-Gomez, Phys. Rev. D82, 023519 (2010)

work page 2010
[25]

de la Cruz-Dombriz, A

A. de la Cruz-Dombriz, A. Dobado and A. L. Maroto, Phys. Rev. Lett.103, 179001 (2009)

work page 2009
[26]

de la Cruz-Dombriz, A

A. de la Cruz-Dombriz, A. Dobado and A. L. Maroto, Phys. Rev. D77, 123515 (2008)

work page 2008
[27]

Larranaga, Pramana78, 697 (2012)

A. Larranaga, Pramana78, 697 (2012)

work page 2012
[28]

Elizalde, G

E. Elizalde, G. G. L. Nashed, S. Nojiri and S. D. Odintsov, Eur. Phys. J. C80, 109 (2020)

work page 2020
[29]

Nojiri and S

S. Nojiri and S. D. Odintsov, Phys. Rev. D96, 104008 (2017)

work page 2017
[30]

Nojiri and S

S. Nojiri and S. D. Odintsov, Phys. Lett. B735, 376 (2014)

work page 2014
[31]

Carter, Phys

B. Carter, Phys. Rev.174, 1559 (1968)

work page 1968
[32]

Walker and R

M. Walker and R. Penrose, Commun. Math. Phys.18, 265 (1970)

work page 1970
[33]

Floyd,The Dynamics of Kerr Fields Ph.D

R. Floyd,The Dynamics of Kerr Fields Ph.D. Thesis(University of London, London, 1973)

work page 1973
[34]

C. D. Collinson, Tensor (N.S.)28, 173 (1974)

work page 1974
[35]

Kinnersley, J

W. Kinnersley, J. Math. Phys.10, 1195 (1969)

work page 1969
[36]

Chervonyi and O

Y. Chervonyi and O. Lunin, JHEP09, 182 (2015)

work page 2015
[37]

F. R. Tangherlini, Nuovo Cim.27, 636 (1963)

work page 1963
[38]

R. C. Myers and M. J. Perry, Ann. Phys.172, 304 (1986)

work page 1986
[39]

Kubizˇ n´ ak and V

D. Kubizˇ n´ ak and V. P. Frolov, Class. Quantum Grav.24, F1 (2007)

work page 2007
[40]

V. P. Frolov and D. Kubizˇ n´ ak, Phys. Rev. Lett.98, 011101 (2007)

work page 2007
[41]

Krtouˇ s, V

P. Krtouˇ s, V. P. Frolov and D. Kubizˇ n´ ak, Phys. Rev. D78, 064022 (2008)

work page 2008
[42]

V. P. Frolov and D. Kubizˇ n´ ak, Class. Quantum Grav.25, 154005 (2008)

work page 2008
[43]

A. N. Aliev and G. D. Esmer, Phys. Rev. D87, 084022 (2013). 14

work page 2013
[44]

R. M. Wald, Phys. Rev. D10, 1680 (1974)

work page 1974
[45]

Reina and A

C. Reina and A. Treves, Phys. Rev. D11, 3031 (1975)

work page 1975
[46]

Simon, J

W. Simon, J. Math. Phys.25, 1035 (1984)

work page 1984
[47]

Garfinkle and J

D. Garfinkle and J. Traschen, Phys. Rev. D42, 419 (1990)

work page 1990
[48]

Quevedo and B

H. Quevedo and B. Mashhoon, Phys. Rev. D43, 3902 (1991)

work page 1991
[49]

Klein, Ann

C. Klein, Ann. Phys. (Berl.)515, 599 (2003)

work page 2003
[50]

Sen, Phys

A. Sen, Phys. Rev. Lett.69, 1006 (1992)

work page 1992
[51]

A. N. Aliev, Phys. Rev. D75, 084041 (2007)

work page 2007
[52]

Balasubramanian, D

V. Balasubramanian, D. Kastor, J. H. Traschen and K. Z. Win, Phys. Rev. D59, 084007 (1999)

work page 1999
[53]

A. N. Aliev and V. P. Frolov, Phys. Rev. D69, 084022 (2004)

work page 2004
[54]

A. N. Aliev, Phys. Rev. D74, 024011 (2006)

work page 2006
[55]

J. Kunz, F. Navarro-L´ erida and A. K. Petersen, Phys. Lett. B614, 104 (2005)

work page 2005
[56]

J. Kunz, F. Navarro-L´ erida and J. Viebahn, Phys. Lett. B639, 362 (2006)

work page 2006
[57]

J. Kunz, D. Maison, F. Navarro-L´ erida and J. Viebahn, Phys. Lett. B639, 95 (2006)

work page 2006
[58]

Larsen, Nucl

F. Larsen, Nucl. Phys. B575, 211 (2000)

work page 2000
[59]

J. H. Horne and G. T. Horowitz, Phys. Rev. D46, 1340 (1992)

work page 1992
[60]

Hosoya, K

A. Hosoya, K. Ishikawa, Y. Ohkuwa and K. Yamagishi, Phys. Lett. B134, 44 (1984)

work page 1984
[61]

M. J. Duff, J. T. Liu and J. Rahmfeld, Nucl. Phys. B524, 129 (1998)

work page 1998
[62]

G. W. Gibbons and D. L. Wiltshire, Ann. Phys.167, 201 (1986); Ann. Phys.176, 393(E) (1987)

work page 1986
[63]

Kim and R.G

H.C. Kim and R.G. Cai, Phys. Rev. D77, 024045 (2008)

work page 2008
[64]

C. A. R. Herdeiro, Nucl. Phys. B582, 363 (2000)

work page 2000
[65]

Kaya, Class

R. Kaya, Class. Quantum Grav.25, 045004 (2008)

work page 2008
[66]

J. A. R. Cembranos, A. de la Cruz-Dombriz and P. Jimeno Romero, Int. J. Geom. Meth. Mod. Phys.11, 1450001 (2014)

work page 2014
[67]

G. W. Gibbons, M. J. Perry and C. N. Pope, Class. Quantum Grav.22, 1503 (2005)

work page 2005
[68]

Chen and Y.G

W.X. Chen and Y.G. Zheng, arXiv:2410.20069 [gr-qc] (2024)

work page arXiv 2024
[69]

Shi, and R.-J

Y. Shi and R.J. Yang, arXiv:2410.05092 [gr-qc] (2024)

work page arXiv 2024
[70]

Karmakar and U

R. Karmakar and U. D. Goswami, Eur. Phys. J. C84, 969 (2024). 15

work page 2024

[1] [1]

dt− asin 2 θ Ξ dϕ #2 + Σ ∆r dr2 + Σ ∆θ dθ2 + ∆θ sin2 θ Σ

also found the same value for four-dimensional rotation charged black hole solution in 3 heterotic string theory. [50] showed that the gyromagnetic ratio of four-dimensional black holes corresponding tog= 2 is a universal property. The gyromagnetic ratio for higher dimensional black holes is not universal as it is for four-dimensional black holes. While t...

work page

[2] [2]

A. G. Riesset al., Astron. J.116, 1009 (1998)

work page 1998

[3] [3]

Perlmutteret al., Astrophys

S. Perlmutteret al., Astrophys. J.517, 565 (1999)

work page 1999

[4] [4]

J. L. Tonryet al., Astrophys. J.594, 1 (2003)

work page 2003

[5] [5]

D. N. Spergelet al., Astrophys. J. Suppl. Ser.170, 377 (2007)

work page 2007

[6] [6]

Komatsuet al., Astrophys

E. Komatsuet al., Astrophys. J. Suppl. Ser.180, 330 (2009)

work page 2009

[7] [7]

Komatsuet al., Astrophys

E. Komatsuet al., Astrophys. J. Suppl. Ser.192, 18 (2011)

work page 2011

[8] [8]

P. A. R. Adeet al., A&A571, A1 (2014)

work page 2014

[9] [9]

P. A. R. Adeet al., A&A571, A16 (2014)

work page 2014

[10] [10]

P. A. R. Adeet al., A&A594, A13 (2016)

work page 2016

[11] [11]

A. H. Guth, Phys. Rev. D23, 347 (1981)

work page 1981

[12] [12]

Nojiri and S

S. Nojiri and S. D. Odintsov, Phys. Rev. D68, 123512 (2003)

work page 2003

[13] [13]

Nojiri and S

S. Nojiri and S. D. Odintsov, Gen. Relat. Gravit.36, 1765 (2004). 13

work page 2004

[14] [14]

S. M. Carroll, V. Duvvuri, M. Trodden and M. S. Turner, Phys. Rev. D70, 043528 (2004)

work page 2004

[15] [15]

Dobado and A

A. Dobado and A. L. Maroto, Phys. Rev. D52, 1895 (1995)

work page 1995

[16] [16]

J. A. R. Cembranos, Phys. Rev. Lett.102, 141301 (2009)

work page 2009

[17] [17]

Lovelock, J

D. Lovelock, J. Math. Phys.12, 498 (1971)

work page 1971

[18] [18]

Nojiri and S

S. Nojiri and S. D. Odintsov, Phys. Rept.505, 59 (2011)

work page 2011

[19] [19]

Nojiri, S

S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Phys. Rept.692, 1 (2017)

work page 2017

[20] [20]

Capozziello and M

S. Capozziello and M. De Laurentis, Phys. Rept.509, 167 (2011)

work page 2011

[21] [21]

T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys.82, 451 (2010)

work page 2010

[22] [22]

De Felice and S

A. De Felice and S. Tsujikawa, Living Rev. Rel.13, 3 (2010)

work page 2010

[23] [23]

de la Cruz-Dombriz and A

A. de la Cruz-Dombriz and A. Dobado, Phys. Rev. D74, 087501 (2006)

work page 2006

[24] [24]

P. K. S. Dunsby, E. Elizalde, R. Goswami, S. Odintsov and D. Saez-Gomez, Phys. Rev. D82, 023519 (2010)

work page 2010

[25] [25]

de la Cruz-Dombriz, A

A. de la Cruz-Dombriz, A. Dobado and A. L. Maroto, Phys. Rev. Lett.103, 179001 (2009)

work page 2009

[26] [26]

de la Cruz-Dombriz, A

A. de la Cruz-Dombriz, A. Dobado and A. L. Maroto, Phys. Rev. D77, 123515 (2008)

work page 2008

[27] [27]

Larranaga, Pramana78, 697 (2012)

A. Larranaga, Pramana78, 697 (2012)

work page 2012

[28] [28]

Elizalde, G

E. Elizalde, G. G. L. Nashed, S. Nojiri and S. D. Odintsov, Eur. Phys. J. C80, 109 (2020)

work page 2020

[29] [29]

Nojiri and S

S. Nojiri and S. D. Odintsov, Phys. Rev. D96, 104008 (2017)

work page 2017

[30] [30]

Nojiri and S

S. Nojiri and S. D. Odintsov, Phys. Lett. B735, 376 (2014)

work page 2014

[31] [31]

Carter, Phys

B. Carter, Phys. Rev.174, 1559 (1968)

work page 1968

[32] [32]

Walker and R

M. Walker and R. Penrose, Commun. Math. Phys.18, 265 (1970)

work page 1970

[33] [33]

Floyd,The Dynamics of Kerr Fields Ph.D

R. Floyd,The Dynamics of Kerr Fields Ph.D. Thesis(University of London, London, 1973)

work page 1973

[34] [34]

C. D. Collinson, Tensor (N.S.)28, 173 (1974)

work page 1974

[35] [35]

Kinnersley, J

W. Kinnersley, J. Math. Phys.10, 1195 (1969)

work page 1969

[36] [36]

Chervonyi and O

Y. Chervonyi and O. Lunin, JHEP09, 182 (2015)

work page 2015

[37] [37]

F. R. Tangherlini, Nuovo Cim.27, 636 (1963)

work page 1963

[38] [38]

R. C. Myers and M. J. Perry, Ann. Phys.172, 304 (1986)

work page 1986

[39] [39]

Kubizˇ n´ ak and V

D. Kubizˇ n´ ak and V. P. Frolov, Class. Quantum Grav.24, F1 (2007)

work page 2007

[40] [40]

V. P. Frolov and D. Kubizˇ n´ ak, Phys. Rev. Lett.98, 011101 (2007)

work page 2007

[41] [41]

Krtouˇ s, V

P. Krtouˇ s, V. P. Frolov and D. Kubizˇ n´ ak, Phys. Rev. D78, 064022 (2008)

work page 2008

[42] [42]

V. P. Frolov and D. Kubizˇ n´ ak, Class. Quantum Grav.25, 154005 (2008)

work page 2008

[43] [43]

A. N. Aliev and G. D. Esmer, Phys. Rev. D87, 084022 (2013). 14

work page 2013

[44] [44]

R. M. Wald, Phys. Rev. D10, 1680 (1974)

work page 1974

[45] [45]

Reina and A

C. Reina and A. Treves, Phys. Rev. D11, 3031 (1975)

work page 1975

[46] [46]

Simon, J

W. Simon, J. Math. Phys.25, 1035 (1984)

work page 1984

[47] [47]

Garfinkle and J

D. Garfinkle and J. Traschen, Phys. Rev. D42, 419 (1990)

work page 1990

[48] [48]

Quevedo and B

H. Quevedo and B. Mashhoon, Phys. Rev. D43, 3902 (1991)

work page 1991

[49] [49]

Klein, Ann

C. Klein, Ann. Phys. (Berl.)515, 599 (2003)

work page 2003

[50] [50]

Sen, Phys

A. Sen, Phys. Rev. Lett.69, 1006 (1992)

work page 1992

[51] [51]

A. N. Aliev, Phys. Rev. D75, 084041 (2007)

work page 2007

[52] [52]

Balasubramanian, D

V. Balasubramanian, D. Kastor, J. H. Traschen and K. Z. Win, Phys. Rev. D59, 084007 (1999)

work page 1999

[53] [53]

A. N. Aliev and V. P. Frolov, Phys. Rev. D69, 084022 (2004)

work page 2004

[54] [54]

A. N. Aliev, Phys. Rev. D74, 024011 (2006)

work page 2006

[55] [55]

J. Kunz, F. Navarro-L´ erida and A. K. Petersen, Phys. Lett. B614, 104 (2005)

work page 2005

[56] [56]

J. Kunz, F. Navarro-L´ erida and J. Viebahn, Phys. Lett. B639, 362 (2006)

work page 2006

[57] [57]

J. Kunz, D. Maison, F. Navarro-L´ erida and J. Viebahn, Phys. Lett. B639, 95 (2006)

work page 2006

[58] [58]

Larsen, Nucl

F. Larsen, Nucl. Phys. B575, 211 (2000)

work page 2000

[59] [59]

J. H. Horne and G. T. Horowitz, Phys. Rev. D46, 1340 (1992)

work page 1992

[60] [60]

Hosoya, K

A. Hosoya, K. Ishikawa, Y. Ohkuwa and K. Yamagishi, Phys. Lett. B134, 44 (1984)

work page 1984

[61] [61]

M. J. Duff, J. T. Liu and J. Rahmfeld, Nucl. Phys. B524, 129 (1998)

work page 1998

[62] [62]

G. W. Gibbons and D. L. Wiltshire, Ann. Phys.167, 201 (1986); Ann. Phys.176, 393(E) (1987)

work page 1986

[63] [63]

Kim and R.G

H.C. Kim and R.G. Cai, Phys. Rev. D77, 024045 (2008)

work page 2008

[64] [64]

C. A. R. Herdeiro, Nucl. Phys. B582, 363 (2000)

work page 2000

[65] [65]

Kaya, Class

R. Kaya, Class. Quantum Grav.25, 045004 (2008)

work page 2008

[66] [66]

J. A. R. Cembranos, A. de la Cruz-Dombriz and P. Jimeno Romero, Int. J. Geom. Meth. Mod. Phys.11, 1450001 (2014)

work page 2014

[67] [67]

G. W. Gibbons, M. J. Perry and C. N. Pope, Class. Quantum Grav.22, 1503 (2005)

work page 2005

[68] [68]

Chen and Y.G

W.X. Chen and Y.G. Zheng, arXiv:2410.20069 [gr-qc] (2024)

work page arXiv 2024

[69] [69]

Shi, and R.-J

Y. Shi and R.J. Yang, arXiv:2410.05092 [gr-qc] (2024)

work page arXiv 2024

[70] [70]

Karmakar and U

R. Karmakar and U. D. Goswami, Eur. Phys. J. C84, 969 (2024). 15

work page 2024