Magnetized Reissner-Nordstrom-Taub-NUT spacetime and microscopic entropy
aa r X i v : . [ g r- q c ] F e b Magnetized Reissner-Nordstrom-Taub-NUT spacetimeand microscopic entropy
Haryanto M. Siahaan ∗ Center for Theoretical Physics,Department of Physics, Parahyangan Catholic University,Jalan Ciumbuleuit 94, Bandung 40141, Indonesia
Abstract
We present a novel solution describing magnetized spacetime outside a massiveobject with electric charge equipped with NUT parameter. To get the solution, weemploy the Ernst magnetization to the Reissner-Nordstrom-Taub-NUT spacetime asthe seed. Using the Kerr/CFT correspondence, we show that the extremal entropy ofa magnetized Reissner-Nordstrom-Taub-NUT black hole can be reproduced using theCardy formula. ∗ [email protected] Introduction
Reissner-Nordstrom black hole is an exact solution of the Einstein-Maxwell theory describingthe spacetime outside a collapsing matter with electric charge. Despite it is very unlikelyfor a collapsing matter to maintain a significant amount of electric charge in real world, stillthe Reissner-Nordstrom solution has been one of the most discussed topics in gravitationalresearches [1] especially black hole related [2, 3, 4, 5]. Moreover, inspired by the Kerr/CFTcorrespondence, the charged black hole/CFT holography has been investigated [6, 7, 8]where some properties of extremal or near extremal Reissner-Nordstrom black holes can bereproduced by using some two dimensional conformal field theory approaches.In addition to the black hole with electric charge, there also exist exact solutions de-scribing black hole immersed in external magnetic field. Solution where the magnetic fieldis considered as some perturbations in the spacetime was introduced by Wald [9], and forthe case of strong magnetic field was proposed by Ernst [10]. In the proposal by Ernst, themagnetized spacetime is obtained by using a Harrison type of transformation applied to aseed solution in Einstein-Maxwell theory. For example, one can use the Kerr-Newman solu-tion as the seed to obtain the magnetized Kerr-Newman spacetime [10]. Setting the mass,electric charge, and rotation parameters in the magnetized Kerr-Newman solution yields theMelvin magnetic universe [11]. Interestingly, the presence of external magnetic field doesnot change the area of embedded black hole, but rather to deform the surface as studiedin [12]. Various studies on black holes in external magnetic field can be found in literature[13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29].A black hole in Einstein-Maxwell theory can have the NUT parameter in addition tothe well known hairs of black hole, namely the mass m , rotation a , and electric charge q [1]. Sometime adding this NUT parameter to a known black hole solution in theory can beconsidered as a sort of NUT extension to the solution. For example, adding NUT parameterto the Schwarzschild spacetime yields the Taub-NUT metric of the vacuum Einstein system.One can also add NUT parameter to the Kerr-Newman solution to yield the Kerr-Newman-Taub-NUT spacetime. The Taub-NUT solution mentioned previously is just a neutral andstatic limits of Kerr-Newman-Taub-NUT family. Despite the obscure realization of NUTparameter in our real world, solutions containing this quantity have contributed significantlyin the gravitational studies. For example the motion of charged test particle was studied in[30], and the gravitomagnetism in this spacetime was investigated in [31]. Spacetime withNUT parameter can also exist in gravitational theories beyond Einstein-Maxwell, for examplein scalar-tensor [32], Randall-Sundrum braneworld [33], and low energy heterotic string [34]theories. In a recent work [35], the authors show that the Misner string contribution to theTaub-NUT-AdS entropy can be renormalized by introducing the Gauss-Bonnet term.Particularly, related to the seed solution to be magnetized in this paper, we considerthe Reissner-Nordstrom -Taub-NUT (RNTN) solution of Einstein-Maxwell theory describ-ing spacetime outside static charged collapsing object with NUT parameter. One can alsoconsider this as a Taub-NUT extension of the Reissner-Nordstrom solution. The magnetizedReissner-Nordstrom spacetime had been reported decades ago [12], where the corresponding1err/CFT correspondence study was carried out in [36].Here we would like to obtain the NUT extension of a magnetized Reissner-Nordstrom blackhole. To proceed, we employ the Ernst magnetization transformation to the RNTN space-time as the seed. A sort of similar problem was investigated in [40], where the authorsdiscussed a spacetime with NUT parameter surrounded by some weak magnetic fields. As-pects of the obtained spacetime solution in this paper are discussed, such as the black holearea and Hawking temperature. We then test the conjectured extremal Kerr/CFT corre-spondence in the new spacetime solution, where we are able to recover the extremal entropyby using the asymptotic symmetry group method [37, 38, 39].The organization of this paper is as follows. In the next section, we construct the mag-netized RNTN solution by employing the Ernst magnetization to the RNTN spacetime asthe seed. The area, entropy, and Hawking temperature of the black hole in the spacetimeare discussed in section 3. In section 4, we compute the entropy of extremal MRNTN blackhole using microscopic formula. Finally we give conclusions and discussions. We considerthe natural units c = ~ = k B = G = 1. Here we review the Ernst magnetization prescription, which basically is a type of Harrisontransformation. We consider the following stationary and axial symmetric line element,d s = f ( ω d t − d φ ) + f − (cid:0) e γ d χ d ¯ χ − ρ d t (cid:1) , (2.1)which is known as the Lewis-Papapetrou-Weyl (LPW) type of metric. In the metric above,the functions f , γ , and ω depend on χ , and we have used the − + ++ signs convention forthe spacetime. Now let us also consider two kind of complex potentials, namelyΦ = A φ + i ˜ A φ , (2.2)and E = f + Φ ¯Φ − i Ψ . (2.3)These complex functions are known as the electromagnetic and gravitational Ernst poten-tials, respectively. Note that the real part of Φ above is A φ is a component in the vectorpotential A µ , and the imaginary part comes from the vector field ˜ A µ which builds the dualfield strength tensor ˜ F µν = ∂ µ ˜ A ν − ∂ ν ˜ A µ where ˜ F µν = ε µναβ F αβ . To find the solution for A t , we can use the following equation − i ρf ∇ ˜ A φ = ∇ A t + ω ∇ A φ . (2.4)2he twist potential Ψ in gravitational potential (2.3) is related to the metric functions andvector potential by the equation − i ∇ Ψ = f ρ ∇ ω + 2 ¯Φ ∇ Φ . (2.5)Using these two potentials above, the Einstein-Maxwell equations R µν = 2 F µα F αν − g µν F αβ F αβ , (2.6)can be rewritten in the form of wave equations (cid:0) E + ¯ E + Φ ¯Φ (cid:1) ∇ E = 2 (cid:0) ∇E + 2 ¯Φ ∇ Φ (cid:1) · ∇E , (2.7) (cid:0) E + ¯ E + Φ ¯Φ (cid:1) ∇ Φ = 2 (cid:0) ∇E + 2 ¯Φ ∇ Φ (cid:1) · ∇ Φ , (2.8)known as the Ernst equations [1].There exist some transformations that leave the two equations (2.7) and (2.8) above tobe invariant. One example is the Harrison transformation [41]. Ernst showed that one canperform a magnetization of a known spacetime using a type of Harrison transformation,namely E → E ′ = Λ − E and Φ → Φ ′ = Λ − (Φ − b E ) , (2.9)where Λ = 1 − b Φ + b E . (2.10)Above, the constant b represents the strength of external magnetic field in the spacetime .The transformation (2.9) leaves equations (2.7) and (2.8) unchanged for the new potentials E ′ and Φ ′ from which a new set of solutions (cid:8) A ′ µ , g ′ µν (cid:9) to the Einstein-Maxwell equationscan be extracted.It can be shown that the magnetization transformation (2.9) acting on the potentials E and Φ built from a seed solution lead to the metric functions fulfilling f ′ = Re {E ′ } − Φ ′ ¯Φ ′ = f | Λ | , (2.11)and ∇ ω ′ = | Λ | ∇ ω + ρf (cid:0) Λ ∇ ¯Λ − ¯Λ ∇ Λ (cid:1) , (2.12)while γ remains unchanged. Since all the incorporating function in the metric (2.1) dependon ρ and z only, then the operator ∇ can be defined in the flat Euclidean spaced χ d ¯ χ = d ρ + d z , (2.13) Here we use a quite different label for the strength of magnetic field, where in some other works [42, 36, 43]is used B = 2 b instead. dχ = dρ + idz and operator ∇ = ∂ ρ + i∂ z accordingly.However, a typical spacetime solution in Einstein-Maxwell theory containing black holeis expressed by using the Boyer-Lindquist type coordinate { t, r, x = cos θ, φ } . Surely we canbring the LPW line element (2.1) into a Boyer-Lindquist type, where now we haved χ d ¯ χ = d r ∆ r + d x ∆ x , (2.14)with ∆ r = ∆ r ( r ) and ∆ x = ∆ x ( x ). Accordingly, the corresponding operator ∇ will read ∇ = √ ∆ r ∂ r + i √ ∆ x ∂ x , and since ρ = ∆ r ∆ x we then have ∂ r A t = ∆ x f ∂ x ˜ A φ − ω∂ r A φ , (2.15)and − ∂ x A t = ∆ r f ∂ r ˜ A φ + ω∂ x A φ , (2.16)from eq. (2.4). The last two equations are useful later in obtaining the A t component associ-ated to the magnetized spacetime according to (2.9). To end some details on magnetizationprocedure, another equations that we may require to complete the magnetized solution are ∂ r ω ′ = | Λ | ∂ r ω − ∆ x f Im (cid:8) Λ ∂ x ¯Λ − ¯Λ ∂ x Λ (cid:9) , (2.17)and ∂ x ω ′ = | Λ | ∂ x ω + ∆ r f Im (cid:8) Λ ∂ r ¯Λ − ¯Λ ∂ r Λ (cid:9) . (2.18)In the following section, the Ernst magnetization procedure reviewed here will be applied tothe RNTN solution, to obtain the magnetized version of the solution. We start with the metricd s = − ∆ r (d t + 2 lx d φ ) r + l + (cid:0) r + l (cid:1) ∆ x d φ + (cid:0) r + l (cid:1) (cid:18) d r ∆ r + d x ∆ x (cid:19) , (2.19)where ∆ r = r − mr + q − l and ∆ x = 1 − x . The metric above together with the vectorpotential A µ d x µ = qr (d t + 2 lx d φ ) r + l , (2.20)solve the Einstein-Maxwell equation (2.6). The line element (2.19) and vector (2.20) areknown as the RNTN solution. 4o get the magnetized version of the spacetime solution above, we need to bring theseed metric (2.19) into the LPW form (2.1). In doing this, we have the associated metricfunctions f = ( r + l ) + x (3 l − r + l { mr − r − q } )( r + l ) , (2.21) ω = 2 lx ∆ r ( r + l ) + x (3 l − r + l { mr − r − q } ) , (2.22) ρ = ∆ x ∆ r , (2.23)and e γ f − = r + l . (2.24)With this functions and vector solution (2.20), the corresponding gravitational potential is E = l (1 + 3 x ) + il (3 r ∆ x + 2 mx ) + l (6 mrx − x { r + q } − r ) + ir ( q x + r ∆ x ) l + ir , (2.25)while the electromagnetic one reads Φ = − iqx ( ir − l ) ir + l . (2.26)Furthermore, from (2.10), these two potentials give usΛ = 1 + i bqx ( ir − l ) ir + l + b ir + l (cid:8) l (cid:0) x (cid:1) + il (cid:0) r ∆ x + 2 mx (cid:1) + l (cid:0) mrx − x (cid:8) r + q (cid:9) − r (cid:1) + ir (cid:0) q x + r ∆ x (cid:1)(cid:9) . (2.27)Finally, we can get the magnetized Ernst potentials E ′ and Φ ′ after inserting (2.25), (2.26),and (2.27) into (2.9). The resulting line element can be written 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 ) , (2.28)where the functions appearing in metric above read f ′ = n ( r + l ) + x (3 l − r + l { mr − r − q } ) o P k =0 ¯ f k l k , (2.29)where ¯ f = r (cid:2) b r ∆ x + 2 b r ∆ x (cid:0) b q x (cid:1) + 1 + b q x (cid:0) b q x (cid:1)(cid:3) , ¯ f = − bqxr (cid:0) − b x q − b r x + 3 b mrx (cid:1) , f = 1 + b x q (cid:0) b x q − (cid:1) + r b (cid:0) x + 7 − x (cid:1) − r b mx (cid:0) x + 1 (cid:1) +4 r b (cid:0) b x q + 1 − x + 3 b x q + 9 b m x (cid:1) − rb mx (cid:0) b x q − (cid:1) , ¯ f = − b qx (cid:0) mx + 2 r (cid:1) , ¯ f = b (cid:2) b m x + 24 b rx ∆ x m − b x (cid:0) r + 2 q (cid:1) − x (cid:0) q b + 5 b r − (cid:1) + 7 b r + 2 (cid:3) , ¯ f = 0 , ¯ f = b (cid:0) x (cid:1) , and ω ′ = P k =0 ¯ ω k x k ( r + l ) + x (3 l − r + l { mr − r − q } ) , (2.30)where ¯ ω = b l (cid:0) l m + 4 r m − l mr − r + 6 r l − q r − q l + q + 3 l (cid:1) ∆ r , ¯ ω = − b q (cid:0) l m + r − rl (cid:1) ∆ r , ¯ ω = − b l (cid:0) b l − b l q + 6 b l r − b l mr − b q r − q + b r + 2 b r m (cid:1) ∆ r , ¯ ω = 2 bqr (cid:0) b q r − b l q − b r m − b l r + r + 6 b l mr + 4 b l + l (cid:1) , ¯ ω = − l (cid:0) b l − l b r − − b r (cid:1) ∆ r , ¯ ω = − rqb (cid:0) r + l (cid:1) (cid:0) b l + 1 − b r (cid:1) . Furthermore, the accompanying Maxwell vector field to this magnetized metric can be writ-ten in a compact form as A ′ µ d x µ = d t P k =0 ¯ c k x k + d φ P k =0 ¯ d k x k P k =0 ¯ f k x k . (2.31)The functions ¯ c k and ¯ d k appearing in the numerator of r.h.s. in eq. (2.31) are¯ c = b q (cid:8) q r − (cid:0) r − l m + 10 rl (cid:1) q − rl m + 16 r l m − r l − l m + r + 9 rl (cid:9) ∆ r , ¯ c = − b l (cid:8) q + (cid:0) r − mr − l (cid:1) q + 4 l m + 4 mr + 6 r l − l rm − r + 3 l (cid:9) ∆ r , ¯ c = qb (cid:8)(cid:0) b r + 8 b m (cid:1) l + (cid:0) r + 5 b r + 4 b r m − q b m + 12 b rm − q b r − m (cid:1) l + (cid:0)(cid:0) − r + 4 r m − q r + 16 mr − q r m + 9 q r (cid:1) b + 44 mr − r − rm + 6 q r + 8 q m (cid:1) l − r + 3 q b r + 12 q r m − mr − q b r − q r + 4 b mr + 2 q r − b r − q b mr (cid:9) , ¯ c = − b l (cid:8)(cid:0) − b l + 5 b r (cid:1) q + (cid:0) rb l m − mr + 4 b r − r b m + 6 r − b l r − l (cid:1) q (cid:0) r m − r − l r m + 16 l r m − l r + l + 5 l r (cid:1) b + r − r l + l + 4 r m + 4 l rm − mr (cid:9) , ¯ c = b qr (cid:8)(cid:0) l r − l r m − l r + 3 q r + 14 l r q − l q + 6 l rm + 9 l − r − mr (cid:1) b + (cid:0) r l − q r + 6 l q − l + 6 r + 4 mr − l rm (cid:1) b − q − r + 6 mr + 5 l (cid:9) , ¯ c = − bl (cid:8)(cid:0) l r q + 3 q r − l q + 3 l r − l r + 3 r − mr + l + 2 l rm − l r m (cid:1) b + (cid:0) l rm − mr − r l − q r − l q + 2 r + 2 l (cid:1) b + 2 mr + l − q − r (cid:9) , ¯ c = qr (cid:0) − b r l − b r + 5 b r l − b r l − l b + b r + b l − b r − l b (cid:1) , and ¯ d = − b n(cid:0) q r − mrq − mr + 36 r m + 15 r + 9 q (cid:1) l + r (cid:0) q − r (cid:1) l + (cid:0) m − q + 24 mr − r (cid:1) l (cid:9) , ¯ d = 6 b lq (cid:8) mr − q r + l m − r (cid:9) , ¯ d = b (cid:8)(cid:0) − l r q + 6 l r + 2 r + 8 l r m − q r + 6 l q − l − l rm + 30 l r (cid:1) b + r + l q − l rm + 6 r l − q r − l (cid:9) , ¯ d = 2 qrl (cid:0) b l (cid:1) , ¯ d = − b (cid:0) l + r (cid:1) (cid:0) b l + l + 6 b l r + b r + r (cid:1) , while the function ¯ f k in the denominator are¯ f = b (cid:8) l m (cid:0) r + l (cid:1) + 8 rl m (cid:0) l − r − q (cid:1) + r + (cid:0) l − q (cid:1) r + (cid:0) q + 12 l q − l (cid:1) r − l q + 9 l + 9 l q (cid:9) , ¯ f = 8 b lq (cid:8) q r − mr − l m + 2 r (cid:9) , ¯ f = 2 b (cid:8)(cid:0) l rm + 3 l − l r m + q r + 6 l r q − l r − l r − l q − r (cid:1) b − r + 3 q r + 8 l rm − r l + 3 l − l q (cid:9) , ¯ f = − bqrl (cid:8) b l (cid:9) , ¯ f = (cid:0) l + r (cid:1) (cid:8) l b + 6 b r l + 2 b l + 2 b r + b r + 1 (cid:9) . The solutions (2.28) and (2.31), which would be referred as the magnetized Reissner-Nordstrom -Taub-NUT (MRNTN) solution, can be considered as the Taub-NUT extensionof magnetized Reissner-Nordstrom black holes proposed in [10], and studied further in [36].As we expect for a black hole immersed in magnetic field, the area of horizon is just thesame to that of the non-magnetized one [42, 36, 43]. The similar situation is repeated here,where the radius of MRNTN black hole is just the same to that of RNTN spacetime, namely r ± = m ± p m − q + l . Accordingly, the extremal conditions between the magnetized andnon-magnetized RNTN black hole is also the same, i.e. m + l = q .7 Some aspects of the black hole
Here we discuss some aspects related to the MRNTN spacetime presented in the previoussection. These aspects are those which consider to have relations to what we will discuss inthe next section, namely the extremal charge black hole/CFT correspondence inspired bythe Kerr/CFT conjecture [37]. We start with the area of black hole by using the standardtextbook formula, A = π Z dφ Z − dx √ g φφ g xx = 4 π (cid:0) r + l (cid:1) . (3.1)This is just the area of a generic RNTN black hole [44], and consequently the correspondingentropy reads [44], S = A π (cid:0) r + l (cid:1) . (3.2)Now let us turn to the Hawking temperature, which can be computed using several ways.However, obtaining the Hawking temperature T H = κ π using the surface gravity κ = r −
12 ( ∇ µ ξ ν ) ( ∇ µ ξ ν ) , (3.3)is found to be troublesome, due to the complexity of spacetime metric. However the cal-culation turns out to be much simpler if we consider the complex path method in (1 + 1)dimension [45], where the case of Taub-NUT black hole is worked out in [46] and for a Vaidyablack hole in [47]. Note that the calculation in [46] applies to a general stationary and axialsymmetric black hole spacetime metric ds = − ˜ f ( r, x ) dt + dr ˜ g ( r, x ) + ˜ C ( r, x ) h ij ( r, x ) d ˜ x i d ˜ x j , (3.4)where ˜ x i = h x, ˜ φ i , and ˜ φ = φ − ω ′ t , and the line element (2.28) has this form. Now we considerthe geodesic with a fixed x = 0 and d ˜ φ = 0, which then yields only the ( t − r ) sector in themetric that matters. In such consideration, the reading of massless Klein-Gordon equation ∇ µ ∇ µ Φ can be written as (cid:18) ∂S∂r (cid:19) = 1˜ f ( r ) ˜ g ( r ) (cid:18) ∂S∂t (cid:19) , (3.5)after employing the Hamilton-Jacobi ansatz for the scalar field Φ = exp [ − iS ( t, r )]. Sincethe spacetime under discussion is stationary, it is allowed to consider S ( t, r ) = Et + ˜ S ( r ) , (3.6)8o solve eq. (3.5), which leads to the solution S ( t, r ) = E t ± r Z dr q ˜ f ( r ) ˜ g ( r ) . (3.7)Based on this solution, we have the ingoing and outgoing fieldsΦ in = exp − iE t + r Z dr q ˜ f ( r ) ˜ g ( r ) , (3.8)and Φ out = exp − iE t − r Z dr q ˜ f ( r ) ˜ g ( r ) , (3.9)respectively. By imposing that the probability of ingoing particle is unity, i.e. P in = | Φ in | =1, and by using the detailed balance principle P out = exp ( − E/T H ) P in , (3.10)we finally can get the Hawking temperature T H = 14 Im r Z dr q ˜ f ( r ) ˜ g ( r ) − . (3.11)Plugging the metric function (2.28) into the last equation, we have T H = 14 Im r Z ( r + l ) dr ( r − r + ) ( r − r − ) − , (3.12)which then gives us the Hawking temperature for a MRNTN black hole T H = r + − m π ( r + l ) . (3.13)Note that the temperature (3.13) is just the one for the generic RNTN black hole [44]. Thisis as what we expected, that the Hawking temperature for a magnetized black hole is exactlythe same to that of the non-magnetized one [42, 36, 43].9 Microscopic entropy for the extermal MRNTN blackhole
In this section, we extend the magnetized Reissner-Nordstrom /CFT correspondence re-ported in [36] to the case with the presence of NUT parameter. The first step is to obtainthe near horizon geometry for a MRNTN black hole, which can be achieved by performingthe transformation t → r tλ , r → r e + λr r , φ → φ + Ω extJ r λ t . (4.1)In equation above Ω J = ω ′ ( r + ), and Ω extJ is the corresponding quantity evaluated at ex-tremality.Note that from eq. (2.5) there exist a gauge freedom for the twist potential, namelyΨ ′ → Ψ ′ + Ψ ′ for a constant Ψ ′ , which leaves the Ernst equations to be invariant. Recallthat Ψ ′ is the magnetized twist potential obeying − i ∇ Ψ ′ = f ′ ρ ∇ ω ′ + 2 ¯Φ ′ ∇ Φ ′ (4.2)where f ′ , ω ′ are the magnetized metric functions, and Φ ′ is the magnetized electromagneticErnst potential. We then apply the near horizon transformation (4.1) above to the gaugedMRNTN metric, with Ψ ′ = 2 ml (1 + 2 b l ) b ( l − m ) . (4.3)The resulting near horizon metric of an extremal MRNTN black hole is found to bed 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 ) , (4.4)where r = √ m + l , α ( x ) = 1,Γ ( x ) = q (cid:16) b ( l − m ) x + q (1 + q b ) (cid:17) ( l − m ) , (4.5) γ ( x ) = q ∆ x Γ ( x ) , (4.6)and k = − q b (1 + q b ) m − l . (4.7)Moreover, the associated vector field is A µ d x µ = L ( x ) (d φ + kr d t ) , (4.8)10here L ( x ) = q x q q (1 + q b ) − b ( l − m ) Γ ( x ) . (4.9)One can find that the near horizon geometry (4.4) above possesses the SL (2 , R ) × U (1)isometry. The SL (2 , R ) symmetry is generated by the Killing 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)obeying [ K , K ± ] = ± K ± and [ K − , K + ] = K , and the U (1) symmetry is generated by ∂ φ .Interestingly, we find that the presence of NUT parameter l does not break the SL (2 , R ) × U (1) isometry of the near horizon geometry of a magnetized Reissner-Nordstrom black hole[36]. Furthermore, this isometry is the hint that the Kerr/CFT prescription to recover theentropy of extremal black hole should apply also to the MRNTN case.On the other hand, the authors of [38] managed to obtain the central charge for a classof near horizon spacetime and vector field solutions in Einstein-Maxwell theory by using theasymptotic symmetry group (ASG) method. This work was reviewed and then generalizedto a class of gravitational theories in [39]. Interestingly, the near horizon metric (4.4) andvector field (4.8) fall into the category discussed in [38]. Therefore, the ASG method toobtain the corresponding central charge derived in [38] should apply to the case of MRNTNblack hole studied in this paper.To ensure that the ASG method works for the extremal MRNTN black hole, first weneed to consider the following boundary condition to the near horizon 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 the condition below for the accompanying 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)Accordingly, the most general diffeomorphisms preserving the boundary condition for themetric is ζ µ ∂ µ = ε ( φ ) ∂ φ − r dε ( φ ) dφ ∂ r + subleading term (4.15)which may lead us the associated central charge [38, 36, 42] c = c grav + c gauge , (4.16)11here 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 (4.4) into eq. (4.16) gives c = − q b (1 + q b ) q − l , (4.19)and it agrees to the central charge associated the near horizon geometry of an extremalmagnetized Reissner-Nordstrom black hole [36] after taking b = B/ l = 0.Before we can employ the Cardy formula in recovering the extremal black hole entropy, weneed to get the associated near horizon temperature. Clearly the Hawking temperature (3.13)vanishes in extremal limit, which is typical for any other extremal black holes. However,since the Hawking temperature is measured by an observer at infinity, the near horizontemperature is not necessary to vanish even in extremal state. This temperature can becomputed in the following way 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)which gives us the Frolov-Thorne temperature near an extremal black hole under consider-ation. For an extremal MRNTN black hole, this temperature is T φ = 2 l − q πq b (1 + b q ) = 12 πk , (4.21)where the constant k is given in (4.7). This result is exactly what we look for so the Kerr/CFTcorrespondence can provide us the “holographic” entropy calculation of an extremal MRNTNblack hole by using the Cardy formula, S Cardy = π cT φ . (4.22)Plugging the central charge (4.19) and Frolov-Thorne temperature (4.21) into the last equa-tion gives us the entropy of an extremal MRNTN black hole, S ext . = A ext . πq . (4.23)This entropy is equal to the extremal case of (3.2), so we have recovered the entropy for theblack hole by using the microscopic Cardy formula.12 Conclusion
In this paper, we have presented a new solution in Einstein-Maxwell theory describing themagnetized spacetime outside a charged mass equipped with NUT parameter. The magneti-zation procedure is performed by using the Ernst prescription, applied to the RNTN solutionas the seed. As expected, some properties of the MRNTN black hole are just those of thenon magnetized one, for example the area of horizon and the Hawking temperature.Inspired by the Kerr/CFT proposal for some magnetized black holes [42, 36, 43], weextend the conjecture to the case of MRNTN case. To proceed, first we need to obtainthe corresponding near horizon and accompanying vector solution in an extremal MRNTNgeometry. It turns out that resulting near horizon metric and the vector field solution arecompatible with the general form that is used in the asymptotic symmetry group method[38]. Therefore, the general formula for the central charge and Frolov-Thorne temperatureachieved in [38] can apply. In section 4, we recover the extremal entropy for a MRNTN blackhole by using Cardy formula as prescribed by Kerr/CFT correspondence [37, 38, 39].The spacetime solution presented in this paper is a generalization to the novel solutionreported in our previous work, namely the magnetized Taub-NUT spacetime [48]. Obviously,the similar solution generating method should apply if one considers the Kerr-Taub-NUTsolution as a seed. Discussing the extremal Kerr/CFT correspondence to the obtained mag-netized Kerr-Taub-NUT black hole also worth our consideration, i.e. extending the works in[42, 43] to the case with NUT parameter. Investigating the surface deformation of a MRNTNblack hole can be an interesting project as well.
Acknowledgement
I thank Merry K. Nainggolan for her support and encouragement.
A Near horizon geometry and the twist potential
Note that one can “gauge” the twist potential Ψ in (2.3) by adding some constant Ψ to it. Itresults some changes in the magnetized solutions, and in particular for the metric functions f ′ → f ′ + ∆ f ′ and ω ′ → ω ′ + ∆ ω ′ . These changes are∆ ω ′ = − b Ψ ∆ x r + 2 l (1 − x ) r + l ( l + 3 l x − q x ) (cid:8) b ∆ r (cid:0) l m − rl − r (cid:1) x +2 lq (cid:0) q + 3 r − mr − l (cid:1) x + b (cid:0) rl − l Ψ + 2 r (cid:1) x − ql (cid:0) r + l (cid:1)(cid:9) , (A.1)and ∆ f ′ = − b Ψ Ξ Ξ Ξ Ξ , (A.2)where Ξ = ∆ x r + (cid:0) l − l x (cid:1) r + 8 mrl x − q l x + 3 l x + l , = 4 bl (cid:0) rl − l m − r − q r + 3 mr (cid:1) x + 4 q (cid:0) l − r (cid:1) x + b (cid:0) r + l (cid:1) (Ψ − rl ) , Ξ = ∆ x b r + b (cid:0) b x q + 7 l b − b l x + 15 b l x − b q x − x + 2 (cid:1) r − lb (cid:0) mlx b + 10 blx m + Ψ x b + Ψ b − x q (cid:1) r + (cid:0) l b x q + 4 l b +12 l b x q − lb mx q + b x q + 12 lb mx Ψ − b l x + 7 b l +Ψ b + 6 b x q − l b x − b qxC + 1 − l b x + 36 l b m x (cid:1) r +4 bl (cid:0) lb m (cid:8) l − q (cid:9) x + 2 b x q + (cid:8) lmb + 3 b Ψ l + 6 l b m − b q Ψ (cid:9) x − (cid:8) q + 4 l qb (cid:9) x − b Ψ l (cid:1) r + l b Ψ + (cid:0) l b qx − l b mx (cid:1) Ψ + l − l b x q − b l x q + 6 l b x + 2 b l + 6 b l x + 9 l b x q +4 l b m x − l b qx m + l b + 9 l b x − l b x q , Ξ = Ξ − b Ψ (cid:8) bl (cid:0) rl − l m − r − q r + 3 mr (cid:1) x + 4 q (cid:0) r − l (cid:1) x + b (cid:0) r + l (cid:1) (Ψ − rl ) (cid:9) . References [1] J. B. Griffiths and J. Podolsky, Cambridge University Press, 2009.[2] W. B. Feng, S. J. Yang, Q. Tan, J. Yang and Y. X. Liu, [arXiv:2009.12846 [gr-qc]].[3] X. Y. Wang and J. Jiang, JHEP , 161 (2020).[4] Y. W. Han, K. J. He and Y. Hong, Int. J. Theor. Phys. , no.5, 1537-1546 (2020).[5] Y. Mo, Y. Tian, B. Wang, H. Zhang and Z. Zhong, Phys. Rev. D , no.12, 124025(2018).[6] M. R. Garousi and A. Ghodsi, Phys. Lett. B , 79-83 (2010).[7] C. M. Chen, Y. M. Huang and S. J. Zou, JHEP , 123 (2010).[8] C. M. Chen, Y. M. Huang, J. R. Sun, M. F. Wu and S. J. Zou, Phys. Rev. D , 066003(2010).[9] R. M. Wald, Phys. Rev. D , 1680 (1974).[10] F. J. Ernst, J. Math. Phys. 17, 54 (1976).[11] M. A. Melvin, Phys. Lett. , 65 (1964).[12] W. J. Wild and R. M. Kerns, Phys. Rev. D (1980), 332-335.[13] F. J. Ernst and W. J. Wild, J. Math. Phys., 17, 182, 1976.1414] A. N. Aliev and D. V. Galtsov, Sov. Phys. Usp. , 75 (1989).[15] R. Brito, V. Cardoso and P. Pani, Phys. Rev. D , no. 10, 104045 (2014).[16] J. Bicak and F. Hejda, Phys. Rev. D , no. 10, 104006 (2015).[17] M. Koloˇs, Z. Stuchlik and A. Tursunov, Class. Quant. Grav. , no. 16, 165009 (2015).[18] A. Tursunov, M. Kolos, Z. Stuchlik and B. Ahmedov, Phys. Rev. D , no. 8, 085009(2014).[19] M. Astorino, G. Comp`ere, R. Oliveri and N. Vandevoorde, Phys. Rev. D , no. 2,024019 (2016).[20] I. Booth, M. Hunt, A. Palomo-Lozano and H. K. Kunduri, Class. Quant. Grav. , no.23, 235025 (2015).[21] G. W. Gibbons, A. H. Mujtaba and C. N. Pope, Class. Quant. Grav. , no. 12, 125008(2013).[22] G. W. Gibbons, Y. Pang and C. N. Pope, Phys. Rev. D , no. 4, 044029 (2014).[23] A. N. Aliev and D. V. Galtsov, Astrophys. Space Sci. , 181 (1989).[24] A. N. Aliev, D. V. Galtsov and A. A. Sokolov, Sov. Phys. J. , 179 (1980).[25] A. N. Aliev, D. V. Galtsov and V. I. Petrukhov, Astrophys. Space Sci. , 137 (1986).[26] A. N. Aliev and D. V. Galtsov, Sov. Phys. JETP , 1525 (1988).[27] A. N. Aliev and D. V. Galtsov, Sov. Phys. J. , 790 (1989).[28] D. V. Galtsov and V. I. Petukhov, Zh. Eksp. Teor. Fiz. , 801 (1978).[29] W. A. Hiscock, J. Math. Phys. , 1828 (1981).[30] H. Cebeci, N. ¨Ozdemir and S. S¸entorun, Phys. Rev. D (2016) no.10, 104031[31] D. Bini, C. Cherubini, R. T. Jantzen and B. Mashhoon, Class. Quant. Grav. (2003),457-468[32] A. Cisterna, A. Neira-Gallegos, J. Oliva and S. C. Rebolledo-Caceres, [arXiv:2101.03628[gr-qc]].[33] H. M. Siahaan, Phys. Rev. D (2020) no.6, 064022[34] H. M. Siahaan, Eur. Phys. J. C (2020) no.10, 10001535] L. Ciambelli, C. Corral, J. Figueroa, G. Giribet and R. Olea, Phys. Rev. D (2021)no.2, 024052[36] M. Astorino, JHEP , 016 (2015).[37] M. Guica, T. Hartman, W. Song and A. Strominger, Phys. Rev. D (2009), 124008[38] T. Hartman, K. Murata, T. Nishioka and A. Strominger, JHEP (2009), 019[39] G. Comp`ere, Living Rev. Rel. (2017) no.1, 1[40] V. P. Frolov, P. Krtous and D. Kubiznak, Phys. Lett. B (2017), 254-256[41] B. K. Harrison, J.Math.Phys.,9,1744, (1968).[42] H. M. Siahaan, Class. Quant. Grav. , no. 15, 155013 (2016).[43] M. Astorino, Phys. Lett. B , 96 (2015).[44] P. Pradhan, Mod. Phys. Lett. A (2015) no.35, 1550170[45] K. Srinivasan and T. Padmanabhan, Phys. Rev. D (1999), 024007[46] R. Kerner and R. B. Mann, Phys. Rev. D (2006), 104010[47] H. M. Siahaan and Triyanta, Int. J. Mod. Phys. A25