aa r X i v : . [ g r- q c ] M a r Rotating black hole and quintessence
Sushant G. Ghosh a, b, ∗ a Astrophysics and Cosmology Research Unit, School of Mathematics,Statistics and Computer Science, University of KwaZulu-Natal,Private Bag 54001, Durban 4000, South Africa and b Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India (Dated: October 15, 2018)
Abstract
We discuss spherically symmetric exact solutions of the Einstein equations for quintessentialmatter surrounding a black hole, which has an additional parameter ( ω ) due to the quintessentialmatter, apart from the mass ( M ). In turn, we employ the Newman − Janis complex transformationto this spherical quintessence black hole solution and present a rotating counterpart that is identi-fied, for α = − e = 0 and ω = 1 /
3, exactly as the Kerr − Newman black hole, and as the Kerr blackhole when α = 0. Interestingly, for a given value of parameter ω , there exists a critical rotationparameter ( a = a E ), which corresponds to an extremal black hole with degenerate horizons, whilefor a < a E , it describes a non-extremal black hole with Cauchy and event horizons, and no blackhole for a > a E . We find that the extremal value a E is also influenced by the parameter ω and sois the ergoregion . PACS numbers: 04.50.Kd, 04.20.Jb, 04.40.Nr, 04.70.Bw ∗ Electronic address: [email protected], [email protected] − momentumobtained by Kislev [1] and was also rigorously analyzed by himself and others [1–8]. Let uscommence with the general spherically symmetric spacetime ds = g ab ⊗ dx a ⊗ dx b , ( a, b = 0 , , , , (1)with g ab = diag( − f ( r ) , f ( r ) − , r , r sin θ ), the energy − momentum tensor for thequintessence [1] is given by T tt = T rr = ρ q ,T θθ = T φφ = − ρ q (3 ω + 1) . (2)On using the Einstein equations G ab = T ab , one obtains f ( r ) = 1 − r g r + βr ω +1 , (3)where β and r g are the normalization factor. The density of quintessence matter ρ q is givenby ρ q = β ωr ω +1) . (4)The sign of the normalization constant should coincide with the sign of the matter stateparameter, i.e. βω ≥ β is negative for the quintessence and hence we choose α = − β . Thus a metric of exact spherically symmetric solutions for the Einstein equationsdescribing black holes surrounded by quintessential matter with the energy − momentumtensor (2) is given by ds = (cid:20) − Mr − αr ω +1 (cid:21) dt − dr (cid:2) − Mr − αr ω +1 (cid:3) − r d Ω . (5)Here r g is related to the black hole mass via r g = 2 M , and ω is the quintessential stateparameter. The Ricci scalars for the solution reads R = R abab = 9 α ω (9 ω + 6 ω + 3)2 r ω +1) , (6)2ndicating scalar polynomial singularity at r = 0 if ω = { , , − } . Thus we have a generalform of exact spherically symmetric solutions for the Einstein equations describing blackholes surrounded by quintessential matter. The parameter ω has to have the range, − <ω < − / − / < ω < α = 0, itreduces to the Schwarzschild solution. The case with the relativistic matter state parameter ω = 1 /
3, with α = − e , corresponds to Reissner − Nordstrom black hole with f ( r ) = 1 − r g r + e r . (7)The solution for the Reissner − Nordstrom black hole surrounded by the quintessence gives f ( r ) = 1 − r g r + e r − αr ω +1 . (8)The borderline case of ω = − − like metric. The Kerr metric [9] is beyond question the mostextraordinary exact solution in the Einstein theory of general relativity, which representsthe prototypic black hole that can arise from gravitational collapse, which contains an eventhorizon [10]. It is well known that Kerr black hole enjoy many interesting properties distinctfrom its non-spinning counterpart, i.e., from Schwarzschild black hole. However, there is asurprising connection between the two black holes of Einstein theory, and was analyzed byNewman and Janis [11–14] that the Kerr metric [9] could be obtained from the Schwarzschildmetric using a complex transformation within the framework of the Newman − Penrose for-malism [15]. A similar procedure was applied to the Reissner − Nordstrm metric to generatethe previously unknown Kerr − Newman metric [12]. It is an ingenious algorithm to con-struct a metric for rotating black hole from static spherically symmetric solutions in Ein-stein gravity. The Newman − Janis method has proved to be prosperous in generating newstationary solutions of the Einstein field equations and the method have also been studiedoutside the domain of general relativity [17–26], although it may lead to additional stresses[20, 21, 26, 27]. For possible physical interpretations of the algorithm, see [28, 29], and fordiscussions on more general complex transformations, see [16, 28, 29]. For a review on theNewman − Janis algorithm see, e.g., [30]. 3ext, we wish to derive a rotating analogue of the static spherically symmetricquintessence solution (5) by employing the Newman − Janis [11] complex transformation.To attempt the similar for static quintessence solution (5) to generate rotating counterpart,we take the quintessence solution (5) and perform the Eddington − Finkelstein coordinatetransformation, du = dt − (cid:20) − Mr − αr ω +1 (cid:21) dr, so that the metric takes the form ds = (cid:20) − Mr − αr ω +1 (cid:21) du − dudr − r d Ω . (9)Following the Newman − Janis prescription [11, 25], we can write the metric in terms of nulltetrad, Z a = ( l a , n a , m a , ¯ m a ), as g ab = l a n b + l b n a − m a ¯ m b − ¯ m a m b , (10)where null tetrad are l a = δ ar ,n a = δ au − (cid:20) − Mr − αr ω +1 (cid:21) δ ar ,m a = 1 √ r (cid:18) δ aθ + i sin θ δ aφ (cid:19) . The null tetrad are orthonormal and obey the metric conditions l a l a = n a n a = ( m ) a ( m ) a = ( ¯ m ) a ( ¯ m ) a = 0 ,l a ( m ) a = l a ( ¯ m ) a = n a ( m ) a = n a ( ¯ m ) a = 0 ,l a n a = 1 , ( m ) a ( ¯ m ) a = 1 , (11)Now we allow for some r factor in the null vectors to take on complex values. We rewritethe null vectors in the form [25, 26] l a = δ ar ,n a = " δ au − " − M (cid:18) r + 1¯ r (cid:19) − α ( r ¯ r ) ω +12 δ ar ,m a = 1 √ r (cid:18) δ aθ + i sin θ δ aφ (cid:19) . r being the complex conjugate of r . Following the Newman − Janis prescription [11],we now write, x ′ a = x a + ia ( δ ar − δ au ) cos θ → u ′ = u − ia cos θ,r ′ = r + ia cos θ,θ ′ = θ,φ ′ = φ. (12)where a is the rotation parameter. Simultaneously, let null tetrad vectors Z a undergo atransformation Z a = Z ′ a ∂x ′ a /∂x b in the usual way, we obtain l a = δ ar ,n a = (cid:20) δ au − (cid:20) − M r Σ − α Σ ω +12 (cid:21) δ ar (cid:21) ,m a = 1 √ r + ia cos θ ) × (cid:18) ia ( δ au − δ ar ) sin θ + δ aθ + i sin θ δ aφ (cid:19) , (13)where ρ = r + a cos θ and we have dropped the primes. Using tetrad (13), the non zerocomponent of the inverse of a new metric can be written as g uu = − a sin ( θ )Σ( r, θ ) , g uφ = − a Σ( r, θ ) ,g φφ = − r, θ ) sin θ , g θθ = − r, θ ) ,g rr = − a sin θ Σ( r, θ ) − ζ ( r, θ ) , g rφ = a Σ( r, θ ) ,g ur = a sin ( θ )Σ( r, θ ) + 1 , (14)where, ζ ( r, θ ) = 1 − M r Σ − α Σ ω +12 . (15)From the new null tetrad, a new metric is constructed using (10), which takes the form ds = ζ ( r, θ ) du + 2 dudr − a sin θdrdφ − Σ( r, θ ) dθ − (cid:2) a ( ζ ( r, θ ) − − Σ( r, θ ) (cid:3) × sin θdφ − a (1 − ζ ( r, θ )) sin θ dudφ, with Σ( r, θ ) = r + a cos θ . 5 .0 0.5 1.0 1.5 - - r D Θ=Π (cid:144) Ω=- a = a = a E a = a = - - r D Θ=Π (cid:144) Ω=- a = a = a E a = a = - - r D Θ=Π (cid:144) Ω=- a = a = a E a = a = - - r D Θ=Π (cid:144) Ω=- a = a = a E a = a = - - r D Θ=Π (cid:144) Ω=- a = a = a E a = a = - - r D Θ=Π (cid:144) Ω=- a = a = a E a = a = - - r D Θ=Π (cid:144) Ω=- a = a = a E a = a = - - r D Θ=Π (cid:144) Ω=- a = a = a E a = a = FIG. 1: Plot showing the behavior of ∆ vs. r for fixed values of α = − .
1, and M = 1 by varying a . The case a = a E corresponds to an extremal black hole .0 0.5 1.0 1.5 2.0 - r g tt Θ=Π (cid:144) Ω=- a = a = a = a = - r g tt Θ=Π (cid:144) Ω=- a = a = a = a = - r g tt Θ=Π (cid:144) Ω=- a = a = a = a = - r g tt Θ=Π (cid:144) Ω=- a = a = a = a = - r g tt Θ=Π (cid:144) Ω=- a = a = a = a = - r g tt Θ=Π (cid:144) Ω=- a = a = a = a = - r g tt Θ=Π (cid:144) Ω=- a = a = a = a = - r g tt Θ=Π (cid:144) Ω=- a = a = a = a = FIG. 2: Plot showing the behavior of g tt vs. r for fixed values of α = − M = 1 by varying a - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=- - - - - - - - r sin Θ cos Φ r c o s Θ a = Ω=-
FIG. 3: Plot showing the variation of the shape of ergoregion in xz -plane with parameter ω , fordifferent values of a , of the rotating black hole. The blue and the red lines correspond, respectively,to static limit surfaces and horizons. The outer blue line corresponds to the static limit surface,whereas the two red lines correspond to the two horizons ABLE I: The effect of quintessence parameter ω on the extremal rotation parameter ( a E ) andextremal horizon ( r E ) θ = π/ θ = π/ ω a = a E r E a E r E -0.50 0.9270821091720760 0.884066 0.9556124089886700 0.894288-0.66 0.9243197545134466 0.860520 0.9578436527014000 0.883037-0.77 0.9228228129512000 0.845995 0.9593257246846765 0.877225-0.88 0.9215809590176600 0.832531 0.9607794407624000 0.872548 Thus we have obtained rotating analogue of the static black hole metric (5) ds = ζ ( r, θ ) dt + Σ( r, θ )∆( r ) dr +2 (1 − ζ ( r, θ )) sin θdtdφ − Σ( r, θ ) dθ − sin θ (cid:2) a (2 − ζ ( r, θ )) sin θ + Σ( r, θ ) (cid:3) dφ . (16)In order to simplify the notation we introduce the following quantities∆( r ) = ζ ( r, θ )Σ( r, θ ) + a sin θ = r + a − M r − α Σ ω − , (17)inside the metric and we write down the line element explicitly in Boyer − Lindquist coordi-nates defined by the coordinate transformation du = dt − (cid:18) r + a ∆ (cid:19) dr, dφ = dφ ′ − a ∆ dr. In above and henceforth, we omit writing the dependent on θ and r in the function ∆ as wellas in Σ. Then, this metric could be cast in the more familiar Boyer − Lindquist coordinatesto read as ds = ∆ − a sin θ Σ dt − Σ∆ dr +2 a sin θ (cid:18) − ∆ − a sin θ Σ (cid:19) dt dφ − Σ dθ − sin θ (cid:20) Σ + a sin θ (cid:18) − ∆ − a sin θ Σ (cid:19)(cid:21) dφ , (18)9 ABLE II: The Cauchy and event horizons of the black hole for different values of ω and a ( θ = π/ ω = − . ω = − . ω = − . ω = − . a < a E r − r + r − r + r − r + r − r + with ∆ and Σ as defined above. This is a rotating black hole metric which for α = 0 reducesto Kerr black hole, while in the particular case a = 0, it reconstruct the Schwarzschildsolution surrounded by the quintessence, and for definiteness, we call the metric (18) asrotating quintessence black hole which is stationary and axisymmetric with Killing vectors.However, like the Kerr metric, the rotating quintessence black hole metric (18) is alsosingular at r = 0. The metric (18) generically must have two horizons, viz., the Cauchyhorizon and the event horizon. The surface of no return is known as the event horizon. Thezeros of ∆ = 0 gives the horizons of black hole, i.e., the roots of∆ = r + a − M r − α Σ ω − = 0 . (19)This depends on a, α , ω , and θ , and it is different from the Kerr black hole where it is θ independent. The numerical analysis of Eq. (19) suggests the possibility of two roots for aset of values of parameters which corresponds to the two horizons of a rotating quintessenceblack hole metric (18). The larger and smaller roots of the Eq. (19) correspond, respectively,to the event and Cauchy horizons. An extremal black hole occurs when ∆ = 0 has a doubleroot, i.e., when the two horizons coincide. When ∆ = 0 has no root, i.e., no horizon exists,which mean there is no black hole (cf. Fig. 1 ). We have explicitly shown that, for each ω ,one gets two horizons for a < a E , and when a = a E the two horizons coincide, i.e., we havean extremal black hole with degenerate horizons (Fig. 1 and Table I). Further, for a < a E ,Eq. (19) admits two positive roots which are ω dependent (Fig. 1 and Tables II and III).In the case α = 0, when the Kerr black hole solution is recovered, there is an eventhorizon with spherical topology, which is biggest root of the equation ∆ = 0, given by r ± = M ± √ M − a , (20)for a ≤ M . Beyond this critical value of the spin there is no event horizon and causality10 ABLE III: The Cauchy and event horizons of the black hole for different values of ω and a ( θ = π/ ω = − . ω = − . ω = − . ω = − . a < a E r − r + r − r + r − r + r − r + violations are present in the whole spacetime, with the appearance of a naked singularity.While the case α = − e = 0, and ω = 1 / r + a − M r + e and the roots r ± = M ± √ M − a − e , (21)correspond to outer and inner horizons of Kerr − Newman black hole.In general, as envisaged black hole horizon is spherical and it is given by ∆ = 0, whichhas two positive roots giving the usual outer and inner horizon and no negative roots. Thenumerical analysis of the algebraic equation ∆ = 0 reveals that it is possible to find non-vanishing values of parameters a, ω and α for which ∆ has a minimum, and that ∆ = 0admits two positive roots r ± (cf. Fig. 1).The static limit or ergo surface is given by g tt = 0, i.e.,( r + a cos θ ) − M r − α Σ ω − = 0 . (22)The behavior of static limit surface is shown in Fig. 2. The two surfaces, viz. event horizonand static limit surface, meet at the poles and the region between them give the ergoregion admitting negative energy orbits, i.e., the region between r EH + < r < r SLS + is called ergore-gion , where the asymptotic time translation Killing field ξ a = ( ∂∂t ) a becomes spacelike andan observer follow orbit of ξ a . It turns out that the shape of ergoregion , therefore, dependson the spin a , and parameter ω . Interestingly, the quintessence matter does influence theshape of ergoregion as described in the Fig. 3 when compared with the analogous situationof the Kerr black hole. Indeed, we have demonstrated that the ergoregion is vulnerable tothe parameter ω and ergoregion becomes more prolate, and ergoregion area increases as thevalue of the parameter ω increases. Further, we find that for a given value of ω , one canfind critical parameter a C such that the horizons are disconnected for a > a C (cf. Fig. 3).11enrose [31] surprised everyone when he suggested that energy can be extracted from arotating black hole. The Penrose process [31] relies on the presence of an ergoregion , whichfor the solution (18) grows with the increase of parameter ω as well with spin a (cf. Fig. 3).Thus the parameter ω is likely to have impact on energy extraction. It will be also usefulto further study the geometrical properties, causal structures and thermodynamics of thisblack hole solution. All these and related issues are being investigated.In this letter, we have used the complex transformations pointed out by Newman andJanis [11], for to obtain rotating solutions from the static counterparts for the quintessentialmatter surrounding a black hole. Interestingly, the limit as a → Acknowledgments
We would like to thank SERB-DST Research Project Grant NO SB/S2/HEP-008/2014,to M. Amir for help in plots. We also thanks IUCAA for hospitality while this work wasbeing done and ICTP for grant No. OEA-NET-76.
Note added in proof:
After this work was completed, we learned a similar work by Toshmatov et al [32], which appeared in arXiv a couple of days before. [1] V.V. Kiselev, Class. Quant. Grav. , 1187 (2003)[2] S.b. Chen and J.l. Jing, Class. Quant. Grav. , 4651 (2005)[3] Y. Zhang and Y.X. Gui, Class. Quant. Grav. , 6141 (2006)[4] S. Chen, B. Wang and R. Su, Phys. Rev. D , 124011 (2008)[5] Y.H. Wei and Z.H. Chu, Chin. Phys. Lett. , 100403 (2011).
6] B.B. Thomas, M. Saleh and T.C. Kofane, Gen. Rel. Grav. , 2181 (2012).[7] S. Fernando, Mod. Phys. Lett. A , 1350189 (2013)[8] R. Tharanath and V.C. Kuriakose, Mod. Phys. Lett. A , 1350003 (2013).[9] R.P. Kerr, Phys. Rev. Lett. D , 237 (1963).[10] B. Carter, in Black Holes (Gordon and Breach, New York, 1973).[11] E.T. Newman and A I Janis, J. Math. Phys. , 915 (1965)[12] E.T. Newman, E. Couch, K. Chinnapared, A. Exton, A. Prakash and R. Torrence, J. Math.Phys. , 918 (1965).[13] E.T. Newman, J. Math. Phys. , 774 (1973).[14] E.T. Newman, The remarkable efficacy of complex methods in general relativity Highlights inGravitation and Cosmology (Cambridge: Cambridge University Press) p 67 ( 1988).[15] E. Newman and R. Penrose, J. Math. Phys. , 566 (1962)[16] S.P. Drake and P. Szekeres, Gen. Relativ. Gravit.
445 (2000)[17] C. Bambi and L. Modesto, Phys. Lett. B , 329 (2013).[18] B. Toshmatov, B. Ahmedov, A. Abdujabbarov and Z. Stuchlik, Phys. Rev. D , no. 10,104017 (2014).[19] A. Larranaga, A. Cardenas-Avendano and D. A. Torres, Phys. Lett. B , 492 (2015).[20] S. G. Ghosh and S. D. Maharaj, Eur. Phys. J. C , no. 1, 7 (2015).[21] J. C. S. Neves and A. Saa, Phys. Lett. B , 44 (2014).[22] M. Azreg-Ainou, Phys. Lett. B , 95 (2014).[23] S. G. Ghosh, Eur. Phys. J. C , no. 11, 532 (2015).[24] S. G. Ghosh and U. Papnoi, Eur. Phys. J. C , no. 8, 3016 (2014)[25] S. Capozziello, 2M. De laurentis and A. Stabile, Class. Quant. Grav. , 165008 (2010)[26] D.J. Cirilo Lombardo, Class. Quant. Grav. , 1407 (2004).[27] M. Carmeli and M. Kaye, Annals Phys. , 97 (1977).[28] E.J. Flaherty, Hermitian and Khlerian Geometry in Relativity Lecture Notes in Physics(Berlin: Springer) (1976)[29] E.J. Flaherty, Complex variables in relativity vol 2, ed A Held General Relativity and Gravi-tation (New York: Plenum) p 207 (1980)[30] R. dInverno,Introducing Einsteins Relativity (Clarendon, Oxford, 1992)[31] R. Penrose, Riv. Nuovo Cimento , 252 (1969).
32] B. Toshmatov, Z. Stuchlk and B. Ahmedov, arXiv:1512.01498 [gr-qc].32] B. Toshmatov, Z. Stuchlk and B. Ahmedov, arXiv:1512.01498 [gr-qc].