Blandford-Znajek mechanism in the general axially-symmetric black-hole spacetime
aa r X i v : . [ g r- q c ] F e b Blandford–Znajek mechanism in the general axially-symmetric black-hole spacetime
R. A. Konoplya,
1, 2, ∗ J. Kunz, † and A. Zhidenko
3, 4, ‡ Research Centre for Theoretical Physics and Astrophysics,Institute of Physics, Silesian University in Opava,Bezručovo nám. 13, CZ-74601 Opava, Czech Republic Peoples Friendship University of Russia (RUDN University),6 Miklukho-Maklaya Street, Moscow 117198, Russian Federation Institute of Physics, University of Oldenburg, D-26111 Oldenburg, Germany Centro de Matemática, Computação e Cognição (CMCC), Universidade Federal do ABC (UFABC),Rua Abolição, CEP: 09210-180, Santo André, SP, Brazil
We consider the Blandford-Znajek process of electromagnetic extraction of energy from a generalaxially symmetric asymptotically flat slowly rotating black hole. Using the general parametrizationof the black-hole spacetime we construct formulas for the flux of the magnetic field and the rate ofenergy extraction, which are valid not only for the Kerr spacetime, but also for its arbitrary axiallysymmetric deformations. We show that in the dominant order these quantities depend only on asingle deformation parameter, which relates the spin frequency of a black hole with its rotationparameter.
PACS numbers: 04.50.Kd,04.70.-s
I. INTRODUCTION
Among a number of mechanisms of extracting rota-tional energy from rotating black holes, such as the Pen-rose process [1], superradiance [2], and others, it is theBlandford-Znajek process [3], which is characterized byhigh efficiency and could be realized in astrophysical en-vironment. If a rotating black hole is immersed in an ex-ternal magnetic field, a Lorentz transformation says thatthe electric field appears in a co-rotating frame whichinduces the separation of charges, that is, the electriccurrent in the inertial frame. This way the rotationalenergy of a black hole is transferred into the energy ofthe currents outside the black hole. This is the essence ofthe Blandford-Znajek process [3], which could explain thegamma-ray bursts [4]. In a stricter sense, if the force-freemagnetosphere is established as a result of the equationof state for the magnetized equatorial accretion disks,then the interaction between the magnetosphere and theblack hole’s ergosphere leads to extraction of the rota-tional energy.This process has been extensively studied during thepast four decades in the Kerr background, that is, for anaxially symmetric and asymptotically flat black hole inthe Einstein theory of gravity. At the same time, thereare a number of reasons to consider various, generallyspeaking not small [5], deviations from Kerr geometry,either because of the modification of the Einstein the-ory of gravity or due to taking into consideration sometidal forces in the vicinity of the black hole. Therefore, itwould be interesting to see whether the Blandford-Znajekprocess can be described in terms of the most general ∗ [email protected] † [email protected] ‡ [email protected] deformations of Kerr spacetime and whether there aresome general characteristics of this process, which areindependent on the background geometry. An attemptin this direction was made in [6]. However, there wereconsidered only particular deformations given by the socalled Johannsen metric [7]. This metric introduces somead hoc deformations, however, it does not represent thegeneral form of the axially symmetric asymptotically flatblack hole in a parametrized form and, therefore, one can-not judge about general properties of black holes withinthis approach. On the contrary, the general parametriza-tion for axially symmetric asymptotically flat black holes,which is suitable for any metric theory of gravity, wassuggested in [8]. This parametrization has superior con-vergence and strict hierarchy of parameters and, there-fore, frequently allows one to describe essential astro-physically relevant properties of black holes in terms ofonly a few parameters of deformation [9].Here we consider the Blandford-Znajek process in thebackground of this arbitrary parametrized axially sym-metric and asymptotically flat black hole [9]. Althoughpart of our calculations is performed for arbitrary rota-tion, the full solution of the equations of electrodynamicscan be found in analytic form only perturbatively, in theregime of slow rotation. In this regime we show that therate of energy extraction for the split monopole magneticfield remarkably depends only on its flux and the blackhole’s spin frequency, i. e. can be described by a singleparameter of deviation from the Kerr spacetime. Noticethat in this approach, only the rotation, but not the de-formation parameter, must be small. In other words, wepropose a universal description of the Blandford-Znajekprocess for arbitrary axially symmetric black holes in anymetric theory of gravity in the regime of slow rotation.The paper is organized as follows. In Sec. II we brieflyoutline the general parametrization of an axially symmet-ric black hole spacetime developed in [8]. In Sec. III wewrite out the values of the coefficients of the parametriza-tion in the equatorial plane. Sec. IV is devoted to descrip-tion of the Blandford-Znajek process in the low rotationregime for a general axisymmetric black holes and alsowith a particular example of Kerr solution. Finally, inthe Conclusions, we summarize our results and their im-plications for alternative theories of gravity. II. GENERAL ROTATING BLACK HOLE
A general axisymmetric (rotating) black hole in an ar-bitrary metric theory of gravity can be represented bythe following line element [8] ds = − N ( r, θ ) − W ( r, θ ) sin θK ( r, θ ) dt (1) − W ( r, θ ) r sin θdtdϕ + K ( r, θ ) r sin θdϕ +Σ( r, θ ) (cid:18) B ( r, θ ) N ( r, θ ) dr + r dθ (cid:19) , with Σ( r, θ ) ≡ a cos θr , (2)where a is the rotation parameter defined through theasymptotic mass M and angular momentum J as follows, a ≡ JM .
Within this ansatz the coordinate system is fully fixedin such a way that the coordinates r and θ are mutuallyorthogonal and orthogonal to the coordinates t and φ ,associated with the Killing vectors. Thus, the black holemetric is fully determined by the four metric functions N , B , W and K of r and θ .The event horizon is defined by the equation N ( r, θ ) = 0 , (3)and the ergoregion corresponds to < N ( r, θ ) < W ( r, θ ) . (4)It is convenient to define x ≡ − r r , y ≡ cos θ, (5)where r is the event horizon radius in the equatorialplane ( θ = π/ , y = 0 ).The parametrization consists in a double expansionto obtain a generic axisymmetric metric expression: acontinued-fraction expansion in terms of a compact ra-dial coordinate and a Taylor expansion in terms of thecosine of the polar angle for the polar dependence. Thesechoices lead to a superior convergence in the radial direc-tion and to the exact limit in the equatorial plane. Thus, the metric functions are represented as series expansionin terms of the polar coordinate y , N = xA ( x ) + ∞ P i =1 A i ( x ) y i ,B = 1 + ∞ P i =0 B i ( x ) y i ,W = ∞ P i =0 W i ( x ) y i Σ ,K = 1 + aWr + ∞ P i =0 K i ( x ) y i Σ , (6)where we introduced the following functions B i ( x ) = b i (1 − x ) + ˜ B i ( x )(1 − x ) , (7a) W i ( x ) = w i (1 − x ) + ˜ W i ( x )(1 − x ) , (7b) K i ( x ) = k i (1 − x ) + ˜ K i ( x )(1 − x ) , (7c) A ( x ) = 1 − ǫ (1 − x ) + ( a − ǫ + k )(1 − x ) + ˜ A ( x )(1 − x ) , (7d) A i> ( x ) = K i ( x ) + ǫ i (1 − x ) + a i (1 − x ) ++ ˜ A i ( x )(1 − x ) , (7e)The functions ˜ A i ( x ) , ˜ B i ( x ) , ˜ W i ( x ) , and ˜ K i ( x ) are de-fined via continued fractions in terms of the compact ra-dial coordinate x , ˜ A i ( x ) = a i a i x a i x . . . , (8a) ˜ B i ( x ) = b i b i x b i x . . . , (8b) ˜ W i ( x ) = w i w i x w i x . . . , (8c) ˜ K i ( x ) = k i k i x k i x . . . , (8d)where the coefficients are determined by comparing theseries expansions near the event horizon ( x = 0 ). Onthe other hand, the coefficients ǫ i , a i , b i , w i , k i for i = 0 , , , . . . are fixed in order to match the asymptoticbehavior near spatial infinity ( x = 1 ).This way the general parametrization of the black holespacetime is similar in the spirit to the parametrizedpost-Newtonian (PPN) formalism, satisfies the PPNasymptotics in the far region, but is valid in thewhole space outside the black hole. The above generalparametrization and its spherically symmetric limit havebeen recently applied to construction of analytical ap-proximations for various numerical black hole solutions[13–19]. III. EQUATORIAL-PLANE COEFFICIENTS
There are a few reasons to consider the values of the co-efficients of the parametrization in the equatorial plane.First of all, relations between some characteristics suchas, the surface gravity, spin frequency, asymptotic massand rotation parameter do not depend on the point inwhich they are found, so that the easiest way is to relatethese characteristics on the equatorial plane and furtheruse them for the whole space. The second observation isrelated to the slow rotation regime. It turns out that inthis case the dominant contributions are coming mainlyfrom the values of the coefficients on the equatorial plane.This should not come as a surprise, because the expan-sion in the polar direction is made around the equatorialplane and the slow rotation is associated with it, whilefor the faster rotation more terms of expansion in powersof y ≡ cos θ around the equatorial plane should be takeninto account.The asymptotic coefficients in the equatorial plane are• ǫ , which relates the asymptotic mass and the hori-zon radius, ǫ = 2 M − r r , (9)• a and b , which depend on the post-Newtonianparameters β and γ , a = ( β − γ ) 2 M r = ( β − γ )(1 + ǫ ) , (10) b = ( γ − Mr = ( γ − ǫ )2 , (11)• and k is fixed as follows, k = a r . (12)In addition we shall consider the Lense-Thirring-liketerm, which is defined by the asymptotic parameter w , w = 2 Jr = 2 M ar = JM (1 + ǫ ) . (13) We can also express some near-horizon characteristicsin terms of the equatorial-plane coefficients. The black-hole spin frequency Ω H is a constant at the event horizon Ω H = g tφ g tt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g rr =0 = − g tφ g φφ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g rr =0 = W ( r, θ ) rK ( r, θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N ( r,θ )=0 . (14)In particular, one can calculate Ω H by taking its value inthe equatorial plane, Ω H = W (0) r + aW (0) + r K (0) (15) = 2 M a + w r r ( r + a ) + 2 M a + aw . Therefore, one can interpret w as a parameter of de-viation of the black hole’s spin frequency from its Kerrvalue (see Appendix A).The event horizon is a Killing horizon, correspondingto the Killing vector ξ µ = (1 , , , Ω H ) , ξ µ ξ µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N ( r,θ )=0 = 0 . (16)The surface gravity at the event horizon, κ g = r
12 ( ∇ ν ξ µ )( ∇ µ ξ ν ) , (17)is finite iff ∂N ∂θ = 0 . (18)Therefore, we conclude that, for the physically relevantblack-hole spacetimes, the event horizon is located at theconstant radial coordinate r = r , i. e. r = r is a solutionof (3) in our coordinates for any θ , N ( r , θ ) = 0 . After some algebra we obtain for the surface gravity κ g = 12 B ( r , θ ) p Σ( r , θ ) K ( r , θ ) ∂N ( r , θ ) ∂r , (19)which is also a constant (does not depend on θ ).Again, considering θ = π/ , we can express the surfacegravity in terms of the equatorial-plane coefficients, κ g = A (0)(2 + 2 B (0)) p r + ar W (0) + r K (0) (20) = 1 + ( a /r ) − ǫ + a + a b + b ) p r + ar w + 2 a + a ǫ , where we have used that [8] k = 0 . (21)With the above relations at hand we are ready to con-sider the parametrized metric functions in the slow rota-tion regime.First we notice that, due to the symmetry of the lineelement with respect to substitution φ → − φ, J → − J, the functions A i ( x ) , B i ( x ) , K i ( x ) , and W i ( x ) /a dependon a . Moreover, since in the slow-rotation regime, thefunctions N , B , K , and W do not depend on the polarvariable, we conclude that A i ( x ) = O ( a ) , B i ( x ) = O ( a ) , K i ( x ) = O ( a ) , and W i> = O ( a ) , W ( x ) = O ( a ) . Therefore, by definition (15), Ω H = O ( a ) . (22)Hence it follows that Σ = 1 + O ( a ) = 1 + O (Ω H ) ,N = xA ( x ) + O ( a ) = xA ( x ) + O (Ω H ) ,B = 1 + B ( x ) + O ( a ) = 1 + B ( x ) + O (Ω H ) ,K = 1 + O ( a ) = 1 + O (Ω H ) , (23) W = W ( x ) + O ( a ) = W ( x ) + O (Ω H ) . Note that the first relation follows from the definition (2).
IV. BLANDFORD–ZNAJEK MECHANISM
In the original paper of Blandford and Znajek [3] theenergy- and momentum- extraction from the Kerr blackhole surrounded by a stationary, axisymmetric, force-free, magnetized plasma was studied using the Boyer-Lindquist coordinates. However, in these coordinates theelectromagnetic field requires appropriate boundary con-dition at the event horizon because the metric coefficient g rr in the Boyer-Lindquist diverges there, leading to asingular point in the Maxwell equations. Here we shallfollow the approach of McKinney and Gammie [10], whoused the Kerr-Schild coordinates. Since the correspond-ing line element is regular at the horizon, there is noneed to specify boundary conditions at this point. Forthe generic axisymmetric black-hole metric we introducethe ingoing Eddington–Finkelstein-like variables, dτ = dt + C ( r, θ ) dr,dφ = dϕ + C ( r, θ ) W ( r, θ ) rK ( r, θ ) dr, (24)where C ( r, θ ) = p Σ( r, θ ) K ( r, θ ) B ( r, θ ) N ( r, θ ) . In terms of the 1-forms (24) the line element (1) takesthe following form ds = − N ( r, θ ) − W ( r, θ ) sin θK ( r, θ ) dτ (25) − W ( r, θ ) r sin θdτ dφ + K ( r, θ ) r sin θdφ +Σ( r, θ ) r dθ + 2 B ( r, θ ) s Σ( r, θ ) K ( r, θ ) drdτ . Notice that τ ( t, r, θ ) and φ ( ϕ, r, θ ) are not smooth func-tions of the radial and polar coordinates. In particular,from (24) it follows that the mixed derivatives ∂ τ∂r∂θ and ∂ φ∂r∂θ do not exist. It is possible to prove that it is al-ways possible to find smooth coordinate transformations τ ( t, r, θ ) and φ ( ϕ, r, θ ) , such that the resulting metric ten-sor is regular at the event horizon. However, since nofunction depends on τ and φ , in this section we shall usethe metric (25) without loss of generality.The electromagnetic field tensor in a force-free magne-tosphere satisfies [11] F µν J ν = 0 , (26)where J ν is the current and F µν = ∂A µ ∂x ν − ∂A ν ∂x µ is the Faraday tensor. Assuming that the electromagneticfield depends on θ and r , this tensor takes the followingform [10] F µν = √− g − ωB θ ωB r ωB θ B φ − B θ − ωB r − B φ B r B θ − B r , (27)where g is the determinant of the metric tensor (25) g = − B ( r, θ )Σ ( r, θ ) r sin θ. Now we are in a position to calculate energy and an-gular momentum transport. Let T µν be the total energy-momentum tensor, T νµ = F µλ F νλ − δ νµ F κλ F κλ , ∇ ν T νµ = 0 . (28)Then the conserved electromagnetic energy and angularmomentum flux are [3] E ν ≡ − T ντ , L ν ≡ T νφ . (29)Substituting (27) and (25) into (28), for the radialfluxes we have E r = (cid:0) F τθ F θφ g rφ − F rθ F τθ g rr − F τθ g τr (cid:1) g θθ , (30) L r = (cid:0) F θφ g rφ − F rθ F θφ g rr − F tθ F θφg τr (cid:1) g θθ . (31)Finally we obtain E r = ω L r = N ( r, θ ) r sin θ − B r ( r, θ ) B φ ( r, θ ) ω ( r, θ )+( B r ( r, θ )) · C ( r, θ ) (cid:18) W ( r, θ ) rK ( r, θ ) − ω ( r, θ ) (cid:19)! . (32)The energy flux from the black hole is given by (32)taken on the event horizon F E ( θ ) ≡ E r ( r , θ ) , (33)and the total rate of energy extraction from the black holeis given by the integral over the event horizon surface P BZ = Z Z √− gF E ( θ ) dθdφ = 4 π π/ Z √− gF E ( θ ) dθ. (34)Taking into account (14) and (3), we obtain F E ( θ ) = ω (Ω H − ω )( B r ) · B √ Σ K · r sin θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r . (35)This a generalization of the Blandford-Znajek formula forthe rate of energy extraction from an arbitrary axisym-metric black hole.It is interesting to note that, although the expressionfor the energy flux (35) depends explicitly on the metricfunction B ( r , θ ) , the total energy extraction in terms ofthe magnetic field strength and spin frequency dependsonly on K ( r , θ ) at the horizon √− gF E ( θ ) = ω (Ω H − ω )( F θφ ) · s K ( r , θ )Σ( r , θ ) · sin θ, (36)while the other metric coefficients affect (36) through themagnetic field F θφ , which is a solution of the Maxwellequations in the vicinity of the black-hole.This leads to the universal behaviour of the energyextraction rate in the slow-rotating regime. It was shownin [12] that, for the split monopole magnetic field, B r ( r , θ ) = B r + O (Ω H ) . (37)Then, taking into account (23), the total magnetic fieldflux is given by the following expression Φ =
Z Z √− gB r dθdφ = 4 π π/ Z B r · B · Σ · r sin θdθ = 4 πB r r (1 + B (0)) + O (Ω H ) . (38) Notice that by definition Φ differs from [12] by the factor 2. The maximum rate of energy extraction is achieved for ω = Ω H / , P BZ = 4 π π/ Z Ω H B r ) (1 + B (0)) r sin θdθ + O (Ω H ) = Ω H π · (cid:18) Φ2 (cid:19) + O (Ω H ) . (39)Thus, we conclude that, in the slow-rotating regime,the energy extraction rate depends on the black-hole ge-ometry only through the value of Ω H (15). Thus, forslow rotation, the difference in the Blandford-Znajek ef-fect for various black hole spacetimes will be stipulatedby the single deformation coefficient w , which is thedeviation of the spin frequency from its Kerr value. Thisdependence has been discussed for two different deforma-tions given by the Johannsen metric [6], which does notrepresent the most general rotating black-hole geometry.We have shown that for an arbitrary axisymmetric blackhole in a metric theory of gravity the maximum energyextraction rate in the slow rotating regime is the same(39).For faster rotation the near-horizon geometry alsomanifests through the magnetic-field configuration,which is a solution of the Maxwell equation in the par-ticular curved spacetime, which can be obtained numer-ically or perturbatively in terms of Ω H [12]. V. CONCLUSIONS
The Blandford-Znajek process [3] is an astrophysicallyrelevant mechanism of transformation of the rotationalenergy of a black hole into the electromagnetic one, whichis important for gamma-ray bursts [4]. This processhas been extensively studied for the Kerr spacetime, butnot for alternative theories of gravity [6]. Here we havestudied the Blandford-Znajek process using the generalparametrization of the axially-symmetric and asymptot-ically flat black holes in the regime of slow rotation asthe background. It turns out that for the slow rota-tion, the qualitative characteristics of the process, suchas the rate of extraction of energy and momentum, de-pend only on the three parameters: mass, angular mo-mentum and a single deformation parameter - the de-viation of the black hole’s spin frequency from its Kerrvalue w . This way we found universal relations for var-ious quantities describing the Blandford-Znajek processof axially-symmetric black holes in arbitrary metric the-ories of gravity in the regime of slow rotation. Our workcould, in principle, be continued to the higher orders ofexpansion in the polar direction, allowing to describe thefast rotation regime. ACKNOWLEDGMENTS
R. A. K. acknowledges the support of the grant 19-03950S of Czech Science Foundation (GAČR). A. Z. wassupported by the Alexander von Humboldt Foundation,Germany. J. K. acknowledges support by the DFG Re-search Training Group 1620 “Models of Gravity” and theCOST Actions CA15117 and CA16104.
Appendix A: Kerr black hole
For the Kerr black hole the only nonzero asymptoticcoefficients are [8] ǫ = a r , (A1) w = ar + a r , (A2) k = a r , (A3) a = a r + a r . (A4)Therefore, from (9) it follows that the mass is given by M = r + a r ; (10) and (11) imply that the post-Newtonian parameters β = γ = 1 . The nonzero near-horizon coefficients are a = − a r , (A5) k = − a r , (A6) k = − a r , (A7) k = a r . (A8)Since w = 0 , (15) reads Ω ( Kerr ) H = ar + a = a M r . (A9)Taking into account that a = b = 0 , from (20) wefind the surface gravity, κ ( Kerr ) g = r − a r ( r + a ) = r − M M r . (A10)Finally, substituting the coefficients into (6), we obtainthe explicit form for the metric functions, N ( r, θ ) = ( r − r rr − a ) r r ,B ( r, θ ) = 1 ,W ( r, θ ) = ar · r + a r + a cos θ ,K ( r, θ ) = 1 + a r + a ( a + r ) sin θrr ( r + a cos θ ) . (A11)Using the explicit expressions (A11), we find that forthe Kerr black hole (36) takes the form (cf. (4.5) of [3]) F E ( θ ) = ω (Ω H − ω ) (cid:18) F θφ ( r , θ ) r + a cos θ (cid:19) · ( r + a ) . (A12) [1] R. Penrose and R. M. Floyd, "Extraction of RotationalEnergy from a Black Hole", Nature Physical Science 229,177 (1971).[2] A. A. Starobinsky, Sov. Phys. JETP , no. 1, 28 (1973)[Zh. Eksp. Teor. Fiz. , 48 (1973)].[3] R. D. Blandford and R. L. Znajek, Mon.Not. Roy. Astron. Soc. , 433-456 (1977)doi:10.1093/mnras/179.3.433.[4] H. K. Lee, R. A. M. J. Wijers and G. E. Brown,Phys. Rept. (2000), 83-114 doi:10.1016/S0370-1573(99)00084-8 [arXiv:astro-ph/9906213 [astro-ph]].[5] R. Konoplya and A. Zhidenko, Phys. Lett. B , 350 (2016) doi:10.1016/j.physletb.2016.03.044[arXiv:1602.04738 [gr-qc]]. [6] G. Pei, S. Nampalliwar, C. Bambi and M. J. Mid-dleton, Eur. Phys. J. C , no.10, 534 (2016)doi:10.1140/epjc/s10052-016-4387-z [arXiv:1606.04643[gr-qc]].[7] T. Johannsen, Phys. Rev. D , no.4, 044002 (2013)doi:10.1103/PhysRevD.88.044002 [arXiv:1501.02809 [gr-qc]].[8] R. Konoplya, L. Rezzolla and A. Zhidenko,Phys. Rev. D , no.6, 064015 (2016)doi:10.1103/PhysRevD.93.064015 [arXiv:1602.02378[gr-qc]].[9] R. A. Konoplya and A. Zhidenko, Phys. Rev. D ,no. 12, 124004 (2020) doi:10.1103/PhysRevD.101.124004[arXiv:2001.06100 [gr-qc]]. [10] J. C. McKinney and C. F. Gammie, Astro-phys. J. , 977-995 (2004) doi:10.1086/422244[arXiv:astro-ph/0404512 [astro-ph]].[11] I. Okamoto, Mon. Not. Roy. Astron. Soc. , 457-474(1974) doi:10.1093/mnras/167.3.457.[12] A. Tchekhovskoy, R. Narayan and J. C. McKin-ney, Astrophys. J. , 50-63 (2010) doi:10.1088/0004-637X/711/1/50 [arXiv:0911.2228 [astro-ph.HE]].[13] R. A. Konoplya, A. F. Zinhailo and Z. Stuch-lik, Phys. Rev. D , no. 4, 044023 (2020)doi:10.1103/PhysRevD.102.044023 [arXiv:2006.10462[gr-qc]].[14] R. A. Konoplya, T. Pappas and A. Zhi-denko, Phys. Rev. D , no. 4, 044054 (2020)doi:10.1103/PhysRevD.101.044054 [arXiv:1907.10112[gr-qc]].[15] R. A. Konoplya, Z. Stuchlík and A. Zhi- denko, Phys. Rev. D , no. 8, 084044 (2018)doi:10.1103/PhysRevD.97.084044 [arXiv:1801.07195[gr-qc]].[16] K. D. Kokkotas, R. A. Konoplya and A. Zhi-denko, Phys. Rev. D , no. 6, 064004 (2017)doi:10.1103/PhysRevD.96.064004 [arXiv:1706.07460 [gr-qc]].[17] K. Kokkotas, R. A. Konoplya and A. Zhi-denko, Phys. Rev. D , no. 6, 064007 (2017)doi:10.1103/PhysRevD.96.064007 [arXiv:1705.09875[gr-qc]].[18] R. A. Hennigar and R. B. Mann, Phys. Rev. D ,no. 6, 064055 (2017) doi:10.1103/PhysRevD.95.064055[arXiv:1610.06675 [hep-th]].[19] R. A. Konoplya and A. Zhidenko, Phys. Rev. D100