Magnetized Kerr-Taub-NUT spacetime and Kerr/CFT correspondence
aa r X i v : . [ g r- q c ] F e b Magnetized Kerr-Taub-NUT spacetime and Kerr/CFTcorrespondence
Haryanto M. Siahaan ∗ Center for Theoretical Physics,Department of Physics, Parahyangan Catholic University,Jalan Ciumbuleuit 94, Bandung 40141, Indonesia
Abstract
We present a new solution in Einstein-Maxwell theory which can be considered asthe magnetized version of Kerr-Taub-NUT solution. Some properties of the spacetimeare discussed. We also compute the entropy of extremal black hole in the spacetimeby using the Kerr/CFT correspondence approach. ∗ [email protected] Introduction
Kerr spacetime is an exact solution to the vacuum Einstein equations describing the space-time outside a rotating mass [1, 2]. Adding the NUT parameter to this solution yields theKerr-Taub-NUT (KTN) spacetime which also belongs to the vacuum Einstein system. In-terestingly, the presence of NUT parameter in the solution leads to a regular Kretschmannscalar at the origin, in contrast to the generic Kerr black hole solution that possesses a ringsingularity inside the horizon. However, despite the non existence of ring singularity hidinginside the horizon of KTN black hole, KTN spacetime suffers the so-called conical singular-ity [2] which is related to the non 2 π periodicity of angular coordinate φ . In addition tothis conic singularity, another peculiar feature of a spacetime equipped with NUT param-eter is the closed timelike curve (CTC) [2] where a spacelike coordinate with a periodicitycan become timelike. Nonetheless, in spite of these obscure properties of KTN spacetime,numerous studies have been performed to explore its aspects [3, 4, 5, 6, 7, 8, 9]. Even aspacetime solution with NUT parameter can also exist in theories beyond Einstein-Maxwell[10, 11, 12].Adding electromagnetic field into the spacetime of Einstein-Maxwell theory yield an elec-trovacuum system. One of the popular exact solutions to the Einstein-Maxwell equations isthe Kerr-Newman spacetime. This solution describes spacetime outside a rotating chargedand massive object. Moreover, this solution can contain a black hole provided that the ro-tating collapsing object can maintain some amount of electric charge during its gravitationalcontraction [1]. Another way to add electromagnetic fields in the spacetime is by magnetiz-ing a solution following the Ersnt formalism [14]. Basically, this Ersnt magnetization is atype of Harrison transformation acting on some known seed solutions in Einstein-Maxwelltheory. The review on Ersnt magnetization can be found in literature [15, 16, 17], and onecan view the Melvin magnetic universe [18] can be obtained by magnetizing the Minkowskispacetime.Particularly, applying the Ernst magnetization to the Kerr spacetime gives us the mag-netized Kerr solution [14] which is interpreted as the spacetime outside a Kerr black holeimmersed in a strong magnetic field. Unlike the generic Kerr solution which has the asymp-totic flatness, the magnetized version is no longer flat at asymptotic due to the presence ofhomogeneous external magnetic field filling in the spacetime. Indeed, this nature of mag-netized Kerr solution is found quite unrealistic to be associated to any known astrophysicalsystem, but one can still consider it as some approximate way to model a black hole im-mersed in a strong magnetic field coming from the accretion disc [16]. Therefore, it can beunderstood why this solution is still worth further investigations, for example in its relationto the Meisner effect [19] and the Kerr/CFT holography [20, 21]. Some other aspects ofmagnetized black holes can be found in literature [17, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]The Kerr/CFT correspondence has been used as a tool to compute the entropy of ex-tremal Kerr black hole using some two dimensional conformal field theory (CFT ) pointof views [32, 33, 34]. The SL (2 , R ) × U (1) isometry of the near horizon geometry of theextremal black hole is the window where the CFT method can apply. The obtained dif-1eomorphisms subject to some boundary conditions can be brought to be a set of Virasorogenerators of a CFT whose commutator incorporates a specific central charge. In turn,plugging the obtained central charge and Frolov-Thorne temperature in Cardy formula, theentropy for extremal black hole can be reproduced. This Kerr/CFT holography has beenapplied to many type of rotating or charged black holes in the Einstein-Maxwell theory [34]and beyond [35, 36].Despite the real world application of the NUT parameter or external magnetic field in thespacetime obeying the Einstien-Maxwell equations is still vague, the discussions of spacetimeexhibiting these properties have extended our understanding of some aspects in gravitationaltheories [37, 38, 39, 40, 41, 42]. In particular in how we define the conserved quantities inthe spacetime such as mass and angular momentum, and also the geodesics of test particlesin the spacetime [6]. Also in a recent work [43], the authors show that the Misner stringcontribution to the entropy of Taub-NUT-AdS can be renormalized by introducing the Gauss-Bonnet term. Therefore, one may wonder what the properties of spacetime with both NUTparameter and external magnetic field. This is exactly the motivation in [44] where theauthor introduced the magnetized version of the Taub-NUT solution, which was extendedto the charged case namely the Reissner–Nordstrom -Taub-NUT spacetime [45]. In thispaper, still in the same spirit, we would like apply the magnetization scheme to the case ofKTN spacetime to obtain the magnetized KTN (MKTN) solution which is to the best of ourknowledge does not exist in literature.The organization of this paper is as follows. In the next section, we construct the MKTNsolution by employing the Ernst magnetization to the KTN metric as the seed solution. Someaspects of the obtained spacetime such as the surface area, squared of Riemann tensor atequator, and the Hawking temperature are studied in section 3. In section 4, the near horizongeometry of the black hole is obtained, followed by the microscopic entropy calculationaccording to Kerr/CFT correspondence. Finally we give conclusions and discussions. Abrief review of Ernst magnetization is presented in the appendix A. We consider the naturalunits c = ~ = k B = G = 1. The MKTN spacetime presented in this section is a result of Ernst magnetization to theKTN line element ds = − ∆ r ( dt − ( a ∆ x − lx ) dφ ) Ξ + ρ (cid:18) dr ∆ r + dx ∆ x (cid:19) + ∆ x Ξ (cid:0) adt − (cid:0) r + a + l (cid:1) dφ (cid:1) (2.1)as the seed. Above we have Ξ = r + ( l + ax ) , ∆ r = r − mr + a − l , and ∆ x = 1 − x .Note that the KTN (2.1) solution above solves the vacuum Einstein equations, so there isno electromagnetic field associated to the system. To apply the Ersnt magnetization, firstwe need to rewrite the last metric into the Lewis-Papapetrou-Weyl form (A.1), where the2orresponding functions can be found as the followings, f = ∆ x ( r + l + a ) − ∆ r ( a ∆ x − lx ) Ξ , (2.2) ω = ∆ r ( a ∆ x − lx ) − a ∆ x ( r + l + a )∆ r ( a ∆ x − lx ) − ∆ x ( r + l + a ) , (2.3) e γ = ∆ x (cid:0) r + l + a (cid:1) − ∆ r ( a ∆ x − lx ) , (2.4)and ρ = ∆ x ∆ r . In the absence of electromagnetic field, the existing Ernst potential is only E , where the real and imaginary components are given byRe E = 1Ξ n ∆ x r + (cid:0) a + 4 lax − l x − a x + 2 l − ax l (cid:1) r + 2 m (cid:0) lx + ax − a (cid:1) r − ( l + ax ) (cid:0) a x − a x − ax l − l x + 3 a x l + 5 l ax − a l − l (cid:1)(cid:9) , (2.5)andIm E = 1 a Ξ (cid:8) a l (cid:0) x + 1 (cid:1) r − m (cid:0) a x − a x − a l − l + 3 a x l (cid:1) r + 2 a l (cid:0) a x − ax l + l + 6 lax − a − l x (cid:1) r + 2 m (cid:0) a + l (cid:1) ( l + ax ) (cid:0) a x + a + lax + l (cid:1)(cid:9) , (2.6)respectively. Following (A.9), the Λ function associated to this Ernst potential is given byΛ = ˜Λ R + i ˜Λ I a ( l + ax + ir ) , (2.7)where ˜Λ R = − a b (cid:0) l + 3 lx − ax ∆ x (cid:1) r + 2 b m (cid:0) a x − a x − l − a l + 3 a lx (cid:1) r − a (cid:8)(cid:0) ab x + 3 b x l (cid:1) (cid:0) a − l (cid:1) − ax (cid:0) b a + 1 − b l (cid:1) − l (cid:0) b a + 1 + b l (cid:1)(cid:9) , (2.8)and ˜Λ I = b a ∆ x r + (cid:8) b (cid:0) l + a (cid:1) x − ab lx − b a − b l − (cid:9) r b m (cid:0) l + a (cid:1) (cid:0) a x + a + lax + l (cid:1) . (2.9)Then the transformed Ernst potentials, which give us a set of magnetized solution forthe metric and vector potential, can be found using (A.8). The resulting magnetized metriccan be expressed asd s = 1 f ′ (cid:26) − ρ d t + e γ (cid:18) d r ∆ r + d x ∆ x (cid:19)(cid:27) + f ′ (d φ − ω ′ d t ) , (2.10)3here f ′ = | Λ | f , and ω ′ = P j =0 c j x j P k =0 d k x k , (2.11)with the corresponding c j ’s are given by c = 2 (cid:0) l a + 3 mra − a m + 3 l mr − r m − r l − l − l m (cid:1) a b ∆ r ,c = − (cid:0) a − l a + 8 a m − r a − mra − l mr + 3 l − r + 6 r l + 4 r m + 8 l m (cid:1) a b l ∆ r ,c = − (cid:0) l m − l a − mrl + 16 a l − mrl a − r l a + 14 m a l − mr a − mra + 4 m a (cid:1) a b ∆ r ,c = 4 (cid:0) l a + 6 mrl − l m − a l + 8 mrl a + 6 r l a − m a l + 2 mr l + a + 2 mra + a r + 6 mr a − r a − m a (cid:1) a b l ∆ r ,c = − a mb r − a b (cid:0) l − m (cid:1) r − a b m (cid:0) a + 4 l − l a (cid:1) r − a b (cid:0) a l − l m − m a − m a l (cid:1) r − m (cid:8) m b (cid:0) l + 3 a (cid:1) (cid:0) − l − l a + a (cid:1) a (cid:0) a b − a − a b l + 34 a b l − b l (cid:1)(cid:9) r ,c = 2 l ∆ r n − r a b − a b m (cid:0) l + 2 a (cid:1) r + 6 a b (cid:0) a − l (cid:1) r − a b m (cid:0) a + l (cid:1) r + a b + b a l + 18 a b l + 8 a b m − a b m l − a − b m l − a b m l (cid:9) ,c = 2 a (cid:8) − a mb r − a b m (cid:0) l + 7 l a + a (cid:1) r + a b (cid:0) a l − m a l − l m − m a (cid:1) r + m (cid:0) a b − a b m l − a − a b l − b a l − b l a − b m l − a b m l (cid:1) r − b m l + 4 a b (cid:0) a − m (cid:1) l + (cid:0) a b − a b m (cid:1) l + (cid:0) a b + 10 a b m − a (cid:1) l + 2 a b m (cid:9) , and the corresponding d k ’s are d = a ∆ r ,d = 4 la ∆ r ,d = − (cid:0) l mr + a − mra − r − l a + 3 l − r l (cid:1) a ,d = − la ∆ r ,d = − (cid:0) mra + r a + 3 l a + r + l + 2 r l (cid:1) a . Note that the magnetized metric (2.10) possesses the Killing vectors ζ µ ( t ) ∂ µ = ∂ t and ζ µ ( φ ) ∂ µ = ∂ φ just like the seed solution (2.1).Unlike the seed solution which belongs to the vacuum Einstein, the magnetized versionis now contain electromagnetic field where the corresponding Ernst potential is given byΦ ′ = F R + i F I G R + i G I , (2.12)4here F R = b (cid:8) a (cid:0) l − ax + ax + 3 lx (cid:1) r − m (cid:0) − l + a x − a x − a l + 3 a lx (cid:1) r + a (cid:0) a x − l − al x + 5 al x − l x − a x + 3 a lx − a l (cid:1)(cid:9) , (2.13) F I = − b (cid:8) a ∆ x r − a (cid:0) l x − a ∆ x − l − lax (cid:1) r + m (cid:0) a + l (cid:1) (cid:0) a x + a + lax + l (cid:1)(cid:9) , (2.14) G R = − a b (cid:0) l − ax ∆ x + 3 lx (cid:1) r + 2 b m (cid:0) a lx − l + a x − a x − a l (cid:1) r − a (cid:0) b a x − l b − b a x − b a l − x l b − l − b al x + 5 b al x + 3 b a x l − ax (cid:1) , (2.15) G I = b a ∆ x r − a (cid:0) x b l − ab lx − l b − b a − b a x (cid:1) r + 2 b m (cid:0) a + l (cid:1) (cid:0) a x + a + lax + l (cid:1) . (2.16)From this potential, the components of vector field A µ can be obtained using eq. (A.2),(A.13), and (A.14) accordingly. However, the full expression of A φ and ˜ A φ are tedious, andwe provide them in appendix B. Furthermore, getting the full expression for A t is troublesomesince we need to integrate A φ and ˜ A φ . Therefore, since in this present work we do not needthe full mathematical expression for A µ , we will omit to present the complete result for A t .In using some symbolic manipulation programs we can show the obtained solution solves theEinstein-Maxwell equations by using the field strength tensor F µν constructed from the A φ solution in (B.1) and ∂ r A t together with ∂ x A t obeying (A.13) and (A.14), respectively. The spacetime (2.10) introduced in this paper can contain a black hole with radius r + = m + √ m − a + l , exactly the same as the radius of KTN horizon. The total area of theblack hole then can be computed using the standard textbook formula A = π Z dφ Z − dx √ g φφ g xx = 4 π (cid:0) r + a + l (cid:1) , (3.1)which is equal to the area of KTN black hole [8]. Following the relation where the entropyof a black hole is a quarter of its area [8], the entropy of a MKTN black hole reads S = A π (cid:0) r + a + l (cid:1) . (3.2)To grasp more aspects of this new spacetime, let us here examine the squared of corre-sponding Riemann curvature tensor. Surely expressing this quantity in its full complicatedand lengthy expression, even on equator, is unnecessary. Then we would rather to presentsome of its numerical evaluations as presented in fig. 3.1. The typical curvature in thespacetime with NUT parameter is obvious in the red and black lines, where they intersect5igure 3.1: Plots of dimensionless squared Riemann tensor K ∗ = m − R αβµν R αβµν at x = 0for the spacetime (2.10) where we have considered the numerical value a = 0 . m .the vertical axis of r = 0 at some finite values of K ∗ = m − R αβµν R αβµν . We can compareto the blue line which represents the curvature in generic Kerr spacetime, where there existsingularity at the origin. This confirms our earlier statement that rotating spacetimes withNUT parameter, including MKTN, does not possess the true ring singularity as the nullNUT parameter counterpart exhibits.Now let us turn to a semiclassical aspect where we would like to compute the Hawkingtemperature associated to a black hole in MKTN spacetime. Here we adopt the complexpath method proposed in [46] which has been applied to a Taub-NUT black hole in [47]. Westart with a general form of metric ds = − ˜ f ( r, x ) dt + dr ˜ g ( r, x ) + ˜ C ( r, x ) h ij ( r, x ) d ˜ x i d ˜ x j , (3.3)where ˜ x i = h x, ˜ φ i , and ˜ φ = φ − ω ′ t . Obviously the metric (2.10) can be rewritten in theform (3.3) above, hence the complex path approach for Hawking radiation presented in [47]can apply. Now we consider the geodesic dx = d ˜ φ = 0, for a fixed x = 0, then the ( t − r )sector of metric above which matters. In such consideration and using the Hamilton-Jacobiansatz for the scalar field Φ = exp [ − iS ( t, r )], the reading of massless Klein-Gordon equation ∇ µ ∇ µ Φ can be written in the form (cid:18) ∂S∂r (cid:19) = 1˜ f ( r ) ˜ g ( r ) (cid:18) ∂S∂t (cid:19) . (3.4)6s the spacetime being stationary, one can look for a solution S ( t, r ) = Et + ˜ S ( r ) . (3.5)Then the solution to eq. (3.4) can be written as S ( t, r ) = E t ± r Z dr q ˜ f ( r ) ˜ g ( r ) . (3.6)Plugging the last result into our Hamilton-Jacobi ansatz yields the ingoing and outgoingmodes Φ in = exp − iE t + r Z dr q ˜ f ( r ) ˜ g ( r ) , (3.7)and Φ out = exp − iE t − r Z dr q ˜ f ( r ) ˜ g ( r ) , (3.8)respectively.Obviously, the probability of ingoing particle is unity, i.e. P in = | Φ in | = 1. Then byusing the detailed balance principle P out P in = exp (cid:20) − ET H (cid:21) , (3.9)one can write the Hawking temperature T H = 14 Im r Z dr q ˜ f ( r ) ˜ g ( r ) − . (3.10)Plugging the corresponding functions from the MKTN metric (2.10) into the last equationgives us T H = 14 Im r Z q ( r + a + l ) − a ∆ r ( r − r + ) ( r − r − ) − , (3.11)which yields the Hawking temperature for a MKTN black hole can be written as T H = r + − r − π ( r + a + l ) . (3.12)7s we have expected, this Hawking temperature is the same to that of KTN case [8]. As analternative, the Hawking temperature above can also be achieved by using the straightfor-ward calculation of surface gravity κ = −
12 ( ∇ µ ξ ν ) ( ∇ µ ξ ν ) . (3.13)The result is κ = r + − r − r + a + l ) , (3.14)and after using the relation T H = κ π , we can confirm the Hawking temperature (3.12) above. In this section, we investigate the Kerr/CFT correspondence for a MKTN black hole. Wewould like to extent the previous works [20, 21] where the authors show how to compute theentropy of an extremal magnetized Kerr black hole using the Kerr/CFT holography method.The initial step is to find the near horizon geometry of the extremal black hole which possessesthe SL (2 , R ) × U (1) isometry. This near horizon metric together with the correspondingnear horizon vector field solution solve the Eintein-Maxwell equations. Luckily, the generalformulation of asymptotic symmetry group (ASG) method in Kerr/CFT correspondence forthe Einstein-Maxwell theory has been discussed in [33], and had been extended to a classof gravitational theory in [34]. Hence, in this section we just need to obtain a near horizongeometry which falls into the category considered in [33], and employ the general formulafor central charge given there.The near horizon geometry of a MKTN black hole can be achieved by performing thefollowing coordinate transformation t → r tλ , r → r e + λr r , φ → φ + Ω extJ r λ t . (4.1)In the equations above we have Ω J = ω ′ , and Ω extJ is the corresponding quantity evaluatedat extremality, i.e. m + l = a . By applying this near horizon transformation to the metric(2.10), one can obtain the near horizon line elementd s = Γ ( x ) (cid:26) − r d t + d r r + α ( x ) d x ∆ x (cid:27) + γ ( x ) (d φ + kr d t ) , (4.2)after setting r = √ a . In equation ( 4.2) we have α ( x ) = 1,Γ ( x ) = a − (cid:8)(cid:2) (cid:0) a − l − l a − l a + 9 l a (cid:1) b − a (cid:0) a b − (cid:1)(cid:3) ax + (cid:2) (cid:0) l a − l + 8 a − l a − l a (cid:1) b + a (cid:3) lx + 4 a (cid:0) a − l − l a − l a + 9 l a (cid:1) b + a (cid:0) a b + 1 (cid:1)(cid:9) , (4.3)8 ( x ) = 4 a ∆ x Γ ( x ) , (4.4)and k = − √ a − l a (cid:8) a b (cid:0) a + 3 l (cid:1) − a (cid:0) − b l (cid:1) + 4 l b (cid:0) a + l (cid:1)(cid:9) . (4.5)To complete the metric solution above in solving the Einstein-Maxwell equations, theaccompanying vector field is given by A µ d x µ = L ( x ) (d φ + kr d t ) , (4.6)where L ( x ) = L x + L Γ ( x ) { b (9 a l − a l − a l + 4 a − l ) + a (1 − a b ) } , (4.7)where L = a (cid:0) a b l − a b l − a b − a l b + a − l b + 16 a b (cid:1) , (4.8)and L = l (cid:0) a b l + 32 a b − a l b + a − a b l − l b (cid:1) . (4.9)As we aimed, this near horizon geometry possesses the SL (2 , R ) × U (1) isometry which isvital for Kerr/CFT correspondence to apply. The SL (2 , R ) symmetry is generated by theKilling vectors K − = ∂ t , (4.10) K = t∂ t − r∂ r , (4.11) K + = (cid:18) r + t (cid:19) ∂ t − tr∂ r − kr ∂ φ , (4.12)which obey the commutation relations [ K , K ± ] = ± K ± and [ K − , K + ] = K , while the U (1)symmetry is generated by ∂ φ . Here we learn that the presence of NUT parameter does notbreak the SL (2 , R ) × U (1) symmetry of the near horizon extremal magnetized Kerr blackhole studied in [20, 21].Moreover, the form of the near horizon metric (4.2) and vector fields (4.6) match thegeneral forms of near horizon Einstein-Maxwell fields discussed in [33, 48]. Hence, we canapply the ASG method constructed in [33, 34] to our present case. Note that we also needto impose the following boundary condition for the spacetime metric h µν ∼ O ( r ) O ( r ) O ( r − ) O ( r − ) O (1) O ( r − ) O ( r − ) O ( r − ) O ( r − ) O ( r − ) , (4.13)and for the Maxwell vector field a µ d x µ ∼ O ( r ) d t + O (cid:0) r − (cid:1) d r + O (1) d x + O (cid:0) r − (cid:1) d φ , (4.14)9o the formalism examined in [33, 34] can apply. Accordingly, the most general diffeomor-phisms preserving the boundary condition for the spacetime fluctutation can be writtenas K ε = ε ( φ ) ∂ φ − r dε ( φ ) dφ ∂ r + subleadingterm (4.15)Now, from the near horizon geometry we have in (4.2), the associated central charge canbe computed using the general formula [33] c = c grav + c gauge , (4.16)where c grav = 3 k +1 Z − dx p Γ ( x ) α ( x ) γ ( x ) , (4.17)and c gauge = 0 . (4.18)Inserting the metric component (4.2) into eq. (4.17) gives us the result c = 12 m { a − b (5 l a + 7 l a + 3 l a + l + 4 a ) } a . (4.19)As we expected, setting b = B/ l = 0 to the last equation gives us the central chargeused in the magnetized Kerr/CFT correspondence investigated in [20, 21].Recall that in section 3 we have obtained the Hawking tempereture for a MKTN blackhole, which happens to be the same to the temperature of a KTN black hole. In extremalstate, this Hawking temperature vanishes, as occurred for the another extremal black holes.However, one can still find the non-vanishing temperature near the horizon, namely theFrolov-Thorne temperature obtained by using the formula [34] T φ = lim r + → m T H Ω ext J − Ω J = − ∂T H / ∂r + ∂ Ω J / ∂r + (cid:12)(cid:12)(cid:12)(cid:12) r + = m . (4.20)It turns out the Frolov-Thorne temperature for our MKTN case is T φ = a π √ a − l { a − b (5 l a + 7 l a + 3 l a + l + 4 a ) } = 12 πk , (4.21)where the constant k is given in (4.5).Now, having obtained the central charged associated to the symmetry of near horizonof extremal MKTN black hole and the corresponding Frolov-Thorne temperature, we canproceed to get the entropy using the Cardy formula which reads S Cardy = π cT . (4.22)10fter plugging the central charge c in (4.19) and temperature (4.21) into (4.22), we recoverthe entropy of extremal MKTN black hole S ext . = A ext . πa . (4.23)The last equation is just the extremal limit of the entropy given in (3.2), i.e. after setting r + = m and m + l = a . Therefore, here we have managed to reproduce the entropy of anextremal MKTN black hole by using microscopic formula (4.22) according to the Kerr/CFTcorrespondence [32]. In this paper we have presented the magnetized version of the KTN spacetime solution of theEinstein-Maxwell theory. The solution is obtained by employing the Ernst magnetizationintroduced in [14], which has been used to magnetized some known solutions of the Einstein-Maxwell equations. The novel solution presented in this paper can be considered as a Taub-NUT extension of the previous solution known in literature as the magnetized Kerr spacetime[14]. Some aspects of the spacetime are discussed, namely the area of black hole, squared ofRiemann tensor on equator, and the corresponding Hawking temperature.In addition to the studies on some aspects of the new solution, we also discussed theKerr/CFT holography associated to the spacetime. The agreement of near horizon metric(4.2) to the general form discussed in [33, 34] allows us to employ the method developed inthat works to obtain the corresponding central charge and Frolov-Thorne temperature. Inturn, these quantities give us the extremal black hole entropy after using the Cardy formula,which can be viewed as the microscopic calculation for the black hole entropy. This resultis a generalization of some previous works on the magnetized Kerr/CFT correspondence[20, 21], where now the spacetime is equipped with the NUT parameter.There are several future projects that can be pursued based on the results presentedin this paper. First is to extend the solution obtained here to the charged one, namelythe magnetized Kerr-Newman-Taub-NUT solution [50]. The method would be similar, butthe incorporating functions and the results would be more complicated compared to thoseappearing in this paper. Also discussing the associated Meissner effect as that investigatedin [19] for an extremal magnetized black holes with NUT parameter would be interesting.
A Ernst magnetization
In this appendix, let us review briefly the Ernst magnetization procedure employed in section2. The seed metric should be expressible in the form of Lewis-Papapetrou-Weyl (LPW) lineelement, namely ds = f ( dφ − ωdt ) − f − (cid:0) ρ dt − e γ dζ dζ ∗ (cid:1) . (A.1)11bove, f , γ , and ω are functions of ζ . Moreover, we have used − + ++ sign convention forthe spacetime, and ∗ notation represents the complex conjugation. Together with the vectorsolution A µ dx µ = A t dt + A φ dφ , the line element above construct the Ernst potentialsΦ = A φ + i ˜ A φ , (A.2)and E = f + | Φ | − i Ψ . (A.3)The first potential above is known as the electromagnetic one, and the latter one is gravita-tional potential. Particularly for the electromagnetic potential (A.2), the imaginary part isobtained by solving ∇ A t + ω ∇ A φ = − i ρf ∇ ˜ A φ . (A.4)On the other hand, the twist potential Ψ in (A.3) obeys the differential equations ∇ Ψ = if ρ ∇ ω + 2 i Φ ∗ ∇ Φ . (A.5)In terms of Ernst potentials (A.3) and (A.2), the Einstein-Maxwell field equations canbe written as (cid:0) Re {E } + | Φ | (cid:1) ∇E = ( ∇E + 2Φ ∗ ∇ Φ) · ∇E , (A.6)and (cid:0) Re {E } + | Φ | (cid:1) ∇ Φ = ( ∇E + 2Φ ∗ ∇ Φ) · ∇ Φ , (A.7)which are known as the Ernst equations. The last two equations are invariant under a typeof Harrison transformation E → E ′ = Λ − E and Φ → Φ ′ = Λ − (Φ − b E ) , (A.8)where Λ = 1 − b Φ + b E . (A.9)The constant b in the equation above represent the strength of external magnetic field wherethe black hole is embedded.Furthermore, the magnetized line element (A.1) resulting from the magnetization (A.8)will contain f ′ = Re {E ′ } − | Φ ′ | = | Λ | − f , (A.10)and ∇ ω ′ = | Λ | ∇ ω − ρf (Λ ∗ ∇ Λ − Λ ∇ Λ ∗ ) , (A.11)while the function γ remains unaffected. A little bit more detail, since we would like toexpress the metric in a Boyer-Lindquist type, the term dζ dζ ∗ in (A.1) can be written as dζ dζ ∗ = dr ∆ r + dx ∆ x , (A.12)12here ∆ r = ∆ r ( r ) and ∆ x = ∆ x ( x ). Note that the corresponding operator ∇ would takethe form ∇ = √ ∆ r ∂ r + i √ ∆ x ∂ x , and we would have ρ = ∆ r ∆ x . Consequently, (A.4) givesus ∂ r A t = − ω∂ r A φ + ∆ x f ∂ x ˜ A φ , (A.13)and ∂ x A t = − ω∂ x A φ − ∆ r f ∂ r ˜ A φ . (A.14)The last two equations are useful to obtain the A t component associated to the magnetizedspacetime according to (A.8). To end this appendix, another equations which are importantto complete the metric solution are ∂ r ω ′ = | Λ | ∂ r ω + ∆ x f Im { Λ ∗ ∂ x Λ − Λ ∂ x Λ ∗ } , (A.15)and ∂ x ω ′ = | Λ | ∂ x ω − ∆ r f Im { Λ ∗ ∂ r Λ − Λ ∂ r Λ ∗ } . (A.16) B Components of electromagnetic potential
Here we present the component of electromagnetic Ernst potential Φ ′ = A φ + i ˜ A φ in (2.12).The real part reads A φ = b P j =0 a j x j P k =0 b k x k (B.1)where the function a j ’s are a = − a b ∆ r ,a = 6 a b l ∆ r ,a = a (cid:8) − r b − b r l − b m (cid:0) a − l (cid:1) r + (cid:0) b a m − b l m + 9 b l − b a l + 3 b a + a (cid:1) r + − m (cid:0) b l + a − b a l + 6 b a (cid:1) r − b (cid:0) l + a (cid:1) m + 28 b a l − b l + 2 a b + a − b a l − l a o ,a = 4 a l (cid:8) b a r − b a mr (cid:0) b a − b a l + 22 b a m + a + 2 b l m (cid:1) r − m (cid:0) b a − b a l + a − b l (cid:1) r − b (cid:0) l + a (cid:1) m − l a + a − b a l + 7 b a l + 3 a b o ,a = a (cid:8) a b r + a (cid:0) l b + 1 + 3 b a (cid:1) r − b m (cid:0) l − a + 2 l a (cid:1) r + (cid:0) l a + 24 b m l +18 b a l + 48 a b l m − b a m + 30 b a l (cid:1) r +4 m (cid:0) a b + 6 b l + a − b a l − l a (cid:1) r b (cid:0) a + 3 l (cid:1) (cid:0) l + a (cid:1) m − a (cid:0) b a l − b a l + a b + 3 l + a − l a + 6 b l (cid:1)o ,a = − la (cid:8) a b r + 4 a r b m + 2 a (cid:0) b l m + 12 b a m + 3 b a + a + 9 b a l (cid:1) r − ma (cid:0) a − b a l − b l + 3 b a (cid:1) r + 4 b (cid:0) l + a (cid:1) m + a (cid:0) a − b l − l − b a l + 3 b a (cid:1)o ,a = − (cid:8) a b r + a (cid:0) l b + 2 b a (cid:1) r + 4 b a m (cid:0) a + 4 l a + 2 l (cid:1) r + (cid:0) a b m l + a b +16 a b m l + 7 a b l + a + 2 l a + 4 m l b (cid:1) r +2 ma (cid:0) a b + 4 b a l + 4 b l − b a l + a (cid:1) r b (cid:0) l + a (cid:1) m + l a (cid:0) a + l (cid:1) (cid:0) b a + l b + 1 (cid:1)o , and the corresponding b k ’s are b = b a ∆ r ,b = 6 la b ∆ r ,b = − b a (cid:8) a b + 2 a + 84 rb ml a + 40 b mr l − b mrl − m l r b − r b l +9 l b r − b m l − b a mr − mra − b a r l + 24 b a m r − a b l m − b a mr − b a m − b a l + 3 b a r + 2 r a − l a + 28 b a l − l b − b r (cid:9) ,b = 4 b la (cid:8) b a mr + 4 mra + 12 b a mr − rb ml a − b a m r − b mrl − m l r b − b a r + 2 b a r l + 4 a b l m + 2 b m l − b a r + 2 b a m − r a − a b − a + 10 b a l + 2 l a − b a l (cid:9) ,b = a (cid:8) b a l − b a r − b a l − a b l + a b − ra b m + 132 rl b a m + 16 rb ml a − b m l r − b r l a + 36 b a m r − b r a − r a b + 8 b r l a m + 2 a b − a b m l r − b a r l − b a mr − r b a m + a + 12 b m l + 32 b m l a + 28 b m a l +8 a b m − l b mr + 8 l b mr − b a r l − b a r l + 6 b l a + 37 b a l (cid:9) ,b = 2 la (cid:8) a + 4 b m l − ra b m − b a mr + 24 b a m r + 36 rl b a m + 4 r b a m + 24 b l a mr +12 a b m l r + 3 r a b + 18 b a r l + 12 b m a l + 12 b m l a + 6 r b a + 4 a b m +4 b a r + 3 a b + 4 a b − a b l − b a l − b a l (cid:9) ,b = 4 b m l + b (cid:0) mra + a + 4 m r + 16 a m (cid:1) l + b a (cid:0) r a b + 8 b mr + 2 a +8 b a mr + 6 b a + 16 b m r + 24 b a m (cid:1) l + a (cid:0) b mr + 9 b a + 6 b a + 4 b r + 7 b r +16 b m r − rb a m + 16 b m a + 1 (cid:1) l + a (cid:0) r + b a r + b r + 2 b a m (cid:1) . The imaginary one can be written as˜ A φ = − a b P j =0 ˜ a j x j P k =0 ˜ b k x k , (B.2)14here we have ˜ a j ’s as the followings˜ a = a (cid:0) ma − mr + ml − rl (cid:1) , ˜ a = a l (cid:0) r − mr + 2 ml − rl + 2 ma + 3 ra (cid:1) , ˜ a = a (cid:0) a mr + a m + 3 a ml + 2 ml + 6 a rl (cid:1) , ˜ a = l (cid:0) a rl + a r + a m + 2 a ml + ml + 2 a mr + l mr − a r (cid:1) , and the associated ˜ b k ’s read ˜ b = b a ∆ r , ˜ b = 6 la b ∆ r , ˜ b = − b a (cid:8) a b + 2 a + 84 rb ml a + 40 b mr l − b mrl − m l r b − r b l +9 l b r − b m l − b a mr − mra − b a r l + 24 b a m r − a b l m − b a mr − b a m − b a l + 3 b a r + 2 r a − l a + 28 b a l − l b − b r (cid:9) , ˜ b = − b la (cid:8) − b a mr − mra − b a mr + 24 rb ml a + 22 b a m r + 4 b mrl +2 m l r b + 3 b a r − b a r l − a b l m − b m l + 6 b a r − b a m +2 r a + 3 a b + 2 a − b a l − l a + 7 b a l (cid:9) , ˜ b = a (cid:8) b a l − b a r − b a l − a b l + a b − ra b m + 132 rl b a m + 16 rb ml a − b m l r − b r l a + 36 b a m r − b r a − r a b + 8 b r l a m + 2 a b − a b m l r − b a r l − b a mr − r b a m + a + 12 b m l + 32 b m l a + 28 b m a l + 8 a b m − l b mr + 8 l b mr − b a r l − b a r l + 6 b l a + 37 b a l (cid:9) , ˜ b = 2 la (cid:8) a + 4 b m l − ra b m − b a mr + 24 b a m r + 36 rl b a m + 4 r b a m +24 b l a mr + 12 a b m l r + 3 r a b + 18 b a r l + 12 b m a l + 12 b m l a + 6 r b a +4 a b m + 4 b a r + 3 a b + 4 a b − a b l − b a l − b a l (cid:9) , ˜ b = 4 b m l + b (cid:0) mra + a + 4 m r + 16 a m (cid:1) l + b a (cid:0) r a b + 8 b mr + 2 a +8 b a mr + 6 b a + 16 b m r + 24 b a m (cid:1) l + a (cid:0) b mr + 9 b a + 6 b a + 4 b r + 7 b r +16 b m r − rb a m + 16 b m a + 1 (cid:1) l + a (cid:0) r + b a r + b r + 2 b a m (cid:1) . eferences [1] R. 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