aa r X i v : . [ g r- q c ] M a r Magnetized Taub-NUT spacetime
Haryanto M. Siahaan ∗ Center for Theoretical Physics,Department of Physics, Parahyangan Catholic University,Jalan Ciumbuleuit 94, Bandung 40141, Indonesia
Abstract
We find an exact solution describing the spacetime outside a massive object with NUTparameter embedded in an external magnetic field. To get the solution, we employ the Ernstmagnetization transformation to the Taub-NUT metric as the seed solution. The massless limitof this new solution is the Taub-NUT Melvin spacetime. Some aspects in the magnetized Taub-NUT spacetime are investigated, such as the surface geometry and the existence of closed timelike curve. ∗ [email protected] Introduction
Exact solutions in the Einstein-Maxwell theory are always fascinating objects to be studied [1,2, 3, 4] starting from its mathematical aspects to the possible astrophysical-related phenomenon.In the Einstein-Maxwell theory the spacetime can contain electromagnetic fields, and sometime isknown as the electrovacuum system. The most general asymptotically flat spacetime solution inEinstein-Maxwell theory that contains a black hole is known as the Kerr-Newman solution. Thissolution describes a black hole with rotation and electric charge as well. Despite being very unlikelyfor a collapsing object to maintain a significant amount of electric charge, this type of solution hasbeen discussed vastly in literature [1, 3, 5].Another interesting solution of black hole spacetime containing electromagnetic fields is themagnetized solution proposed by Wald [6]. The solution by Wald describes a black hole immersedin a uniform magnetic field, where the vector field solution is generated by the associated Killingsymmetries of spacetime. However, in Wald’s construction, the Maxwell field is considered just assome perturbations in spacetime. The nonperturbative consideration of a black hole immersed ina homogeneous magnetic field was introduced by Ernst [7]. Ernst method uses a Harison type oftransformation [8] applied to a known seed solution of Einstein-Maxwell theory. Recall that theHarison transformation leaves the Ernst equation for the Einstein-Maxwell system to be invariant.Nevertheless, the resulting magnetized spacetimes are no longer flat at asymptotic despite comingfrom an asymptotically flat seed metric. This can be understood since the transformed spacetimeis now filled by some homogeneous magnetic field up to infinity. Various aspects of black holesimmersed in external magnetic field had been studied extensively in literature [9, 10, 11, 12, 13,14, 15, 16, 17, 18, 19, 20, 21, 22].Despite the loss of asymptotic flatness of a magnetized spacetime containing a black hole, thissolution has been considered to have some astrophysical uses especially in modeling the spacetimenear rotating supermassive black hole surrounded by hot moving plasma [9]. Indeed, a full compre-hension of the interaction of a black hole with the surrounding magnetic field from the accretiondisc requires a sophisticated general relativistic treatment at least at the level of a costly numericalapproach. If the full general relativistic or the comprehensive numerical treatment is not necessary,in the sense that we just demand some approximate qualitative explanations, the perturbative pic-ture of the magnetic field around black hole introduced by Wald or even the non-perturbative modelby Ernst can be the alternatives. For example, these models can explain the charge induction by ablack hole immersed in the magnetic field, or the Meissner-like effect that may occur for this typeof black hole [23]. In particular, the superradiant instability of a magnetized rotating black holeis studied in [24]. The Kerr/CFT correspondence for some magnetized black holes are studied in[25, 26, 27], and the corresponding conserved quantities are proposed in [28].Another type of solution in Einstein-Maxwell theory is the Taub-NUT extension of a memberin the Kerr-Newman spacetime family. In a vacuum system, the Taub-NUT solution generalizesSchwarzschild solution to include the so-called NUT (Newman-Unti-Tamburino) parameter l . Thisparameter is interpreted as a “gravitomagnetic” mass in an analogy to the magnetic monopolein electromagnetic theory. However, the presence of this NUT parameter yields some interestingfeatures in spacetime [5]. First, the spacetime is not asymptotically flat which requires specialtreatment to define the conserved quantities in the spacetime. Second, the non-existence of physicalsingularity at the origin. This is interesting since it leads to the question of defining black hole insuch spacetime, which normally we understand as a true singularity covered by a horizon. Instead1f at the origin, the singularity in a spacetime with NUT parameter exists on its axis of symmetry.This is known as the conical singularity which then gives rise to a problem in describing theblack hole horizon. Despite these issues regarding black hole picture in a spacetime with NUTparameter, discussions on Kerr-Newman-Taub-NUT black hole family are still among the activeareas in gravitational studies[29, 30, 31, 32, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. In particular,a discussion related to the circular motion in a spacetime with NUT parameter [43]. The authorsfound that the existence of NUT parameter leads to a new non trivial constraint to the equatorialcircular motion for a test body. This problem also occurs in gravitational theories beyond Einstein-Maxwell, for example in low energy string theory [44], braneworld scenario [45]. In a recent work[46], the authors show that the Misner string contribution to the entropy of Taub-NUT-AdS can berenormalized by introducing the Gauss-Bonnet term, and in [47] the authors show how to embedTaub-NUT solutions in general scalar-tensor theories.The magnetization of some well-known solution in Einstein-Maxwell theory has appeared in theliterature [5, 9] and aspects of this type of solution have been studied extensively. In this work, weintroduce a new solution namely the magnetized extension of Taub-NUT spacetime. The idea isstraightforward, i.e. performing the magnetization transformation to the Taub-NUT metric as theseed solution. One key aspect here is the compatibility of the Taub-NUT metric to be expressedin the Lewis-Papapetrou-Weyl (LPW) line element. Following Ernst [7], the Ernst potential isdefined incongruence to ∂ φ before the Ernst magnetization is applied. The obtained solution is aspacetime describing an object with mass and NUT parameter embedded in an external magneticfield. A quite similar idea where the weak external magnetic field exists outside an object withNUT parameter has been performed in [48].The properties of the event horizon under the influence of some external magnetic fields are alsoan interesting aspect to be investigated [49]. It is known for the magnetized Schwarzschild solution,the scalar curvature of the horizon varies depending on the strength of the external magnetic field.It can take positive, zero, or negative values, which indeed each of them associates to some differentphysics. Recall that a “normal” horizon such as the one of Schwarzschild black hole has a positivecurvature, which is understood due to its spherical form. However, despite the shape of the horizonchanging due to the presence of an external magnetic field, the total area of the horizon is invariant.The organization of this paper is as follows. In the next section, we provide some reviews ofthe Ernst magnetization procedure by using a complex differential operator. In section 3, afteremploying the magnetization procedure to the Taub-NUT spacetime, we obtain the magnetizedTaub-NUT solution. The surface geometry and closed timelike curve in this new spacetime arediscussed in section 4. Finally, we give some conclusions and discussions. We consider the naturalunits c = ~ = k B = G = 1. To get a magnetized spacetime solution, one can employ the Ernst magnetization procedure to aknown seed solution, which basically is a Harison type of transformation. Explicitly, it is the Ernstpotentials which transform into a new set of magnetized ones, where the potential is defined usingthe metric component of the Lewis-Papapetrou-Weyl (LPW) type ds = − f − (cid:0) ρ dt − e γ dζdζ ∗ (cid:1) + f ( dφ − ωdt ) . (2.1)2bove, f , γ , and ω are functions of a complex coordinate ζ . Here we are using the − + ++ signsconvention for the spacetime, and ∗ notation represents the complex conjugation. In Einstein-Maxwell theory, the metric (2.1) together with a vector solution A = A t dt + A φ dφ obey the fieldequations R µν = 2 F µα F αν − g µν F αβ F αβ , (2.2)where R µν is Ricci tensor, and F µν = ∂ µ A ν − ∂ ν A µ is the Maxwell field-strength tensor.Interestingly, Ernst [50, 51] showed that the last equations can be re-expressed in terms of Ernstgravitational and electromagnetic potentials in the form of wave like equations. Using the metricfunctions in (2.1), we can construct the gravitational Ernst potential E = f + | Φ | − i Ψ , (2.3)where the electromagnetic Ernst potential Φ consists ofΦ = A φ + i ˜ A φ . (2.4)Note that the real part of Φ above is A φ instead of A t as introduced in [51] since the gravitationalErnst potential E is defined with respect to the Killing vector ∂ φ in (2.1) line element. The imaginarypart of Φ is the vector field which constructs the dual field strength tensor 2 ˜ F µν = ε µναβ F αβ , i.e.˜ F µν = ∂ µ ˜ A ν − ∂ ν ˜ A µ . The relation between these vectors is given by ∇ A t = − ω ∇ A φ − i ρf ∇ ˜ A φ , (2.5)where the twist potential Ψ satisfies ∇ Ψ = if ρ ∇ ω + 2 i Φ ∗ ∇ Φ . (2.6)Normally, the last equation is useful to obtain the potential A t based on the given Ernst potentials E and Φ, eqs. (2.3) and (2.4) respectively. Interestingly, as it was first shown by Ernst [50, 51], theEinstein-Maxwell eq. (2.2) dictates the Ernst potentials to obey the nonlinear complex equations (cid:16) Re {E} + | Φ | (cid:17) ∇E = ( ∇E + 2Φ ∗ ∇ Φ) · ∇E , (2.7)and (cid:16) Re {E} + | Φ | (cid:17) ∇ Φ = ( ∇E + 2Φ ∗ ∇ Φ) · ∇ Φ . (2.8)The magnetization procedure according to Ernst can be written as the following, 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 is a magnetic field parameter, representing its strength in the spacetime .The transformation (2.9) leaves the two equations (2.7) and (2.8) remain unchanged for the new For economical reason, we prefer to express the magnetic parameter as b instead of B/ B = 2 b . E ′ and Φ ′ . In other words, the new metric consisting the new functions f ′ and ω ′ , togetherwith the new vector potentials A ′ t and A ′ φ , are also solutions to the field equations (2.2).In particular, the transformed line element (2.1) resulting from the magnetization (2.9) has thecomponents f ′ = Re (cid:8) E ′ (cid:9) − (cid:12)(cid:12) Φ ′ (cid:12)(cid:12) = | Λ | − f , (2.11)and ω ′ obeying ∇ ω ′ = | Λ | ∇ ω − ρf (Λ ∗ ∇ Λ − Λ ∇ Λ ∗ ) , (2.12)while the function γ in (2.1) remains unchanged. Since all the incorporated functions in the metric(2.1) depend only on ρ and z coordinates, then the operator ∇ can be defined in the flat Euclideanspace dζdζ ∗ = dρ + dz , (2.13)where we have set the complex coordinate dζ = dρ + idz . Here we have ∇ = ∂ ρ + i∂ z , accordingly.Indeed, the spacetime solutions in Einstein-Maxwell theory that contain a black hole are morecompact to be expressed in the Boyer-Lindquist type coordinate { t, r, x = cos θ, φ } . Consequently,the LPW type metric (2.1) with stationary and axial Killing symmetries will have the metricfunction that depend only on r and x coordinates, and the corresponding flat metric line elementreads dζdζ ∗ = dr ∆ r + dx ∆ x , (2.14)where ∆ r = ∆ r ( r ) and ∆ x = ∆ x ( x ). Therefore, the corresponding operator ∇ will read ∇ = √ ∆ r ∂ r + i √ ∆ x ∂ x . Furthermore, here we have ρ = ∆ r ∆ x , which then allows us to write thecomponents of eq. (2.5) as ∂ r A t = − ω∂ r A φ + ∆ x f ∂ x ˜ A φ , (2.15)and ∂ x A t = − ω∂ x A φ − ∆ r f ∂ r ˜ A φ . (2.16)The last two equations are useful later in obtaining the A t component associated to the magnetizedspacetime according to (2.9). To end some details on magnetization procedure, another equationsto complete the magnetized metric are ∂ r ω ′ = | Λ | ∂ r ω + ∆ x f Im { Λ ∗ ∂ x Λ − Λ ∂ x Λ ∗ } , (2.17)and ∂ x ω ′ = | Λ | ∂ x ω − ∆ r f Im { Λ ∗ ∂ r Λ − Λ ∂ r Λ ∗ } . (2.18)In the next section, we will employ this magnetization prescription to the Taub-NUT spacetime. Taub-NUT spacetime is a non-trivial extension of Schwarzschild solution, where in addition to massparameter M , the solution contains an extra parameter l known as the NUT parameter. However,4nlike the mass M which can be considered as a conserved quantity due to the timelike Killingsymmetry ∂ t [3], the NUT parameter cannot be viewed in analogous way as a sort of conservedcharge associated to a symmetry in the spacetime. The line element of Taub-NUT spacetime canbe expressed as [5] ds = − ∆ r r + l ( dt − lxdφ ) + (cid:0) r + l (cid:1) (cid:18) dr ∆ r + dx ∆ x (cid:19) + (cid:0) r + l (cid:1) ∆ x dφ , (3.19)where ∆ r = r − M r − l and ∆ x = 1 − x . Matching this line element to the LPW form (2.1)gives us f = ∆ x (cid:0) r + l (cid:1) − r l x r + l , (3.20) ω = − r lx r l x − ∆ x ( r + l ) , (3.21)and e γ = ∆ x (cid:0) r + l (cid:1) − r l x . (3.22)From eqs. (2.13) and (2.14), one can find ρ = ∆ r ∆ x and z = rx . Using eq. (2.6), the associatedtwist potential for Taub-NUT metric can be found asΨ = − l (cid:0) r + rl + r x − rl x + M l x − M r x (cid:1) r + l . (3.23)Accordingly, the gravitational Ernst potential (2.7) defined with respect to ∂ φ for Taub-NUT space-time (3.19) is E = 6 M rlx − r l + l − lr x + 3 l x + i (cid:0) ∆ x (cid:8) r + 3 rl (cid:9) + 2 M l x (cid:1) l + ir , (3.24)and the electromagnetic Ernst potential Φ vanishes. Furthermore, from (2.10) we haveΛ = δ x b l + (cid:0) − b r δ x + 6 b M rx (cid:1) l + i (cid:8)(cid:0) b r ∆ x + 2 b M x (cid:1) l + r + b r ∆ x (cid:9) l + ir (3.25)where δ x = 1 + 3 x .Now, let us obtain the magnetized spacetime by using this Taub-NUT metric as the seedsolution. Following (2.9), the corresponding magnetized Ernst gravitational potential from (3.24)and (3.25) can be written as E ′ = 6 M rlx − r l + l − lr x + 3 l x + i (cid:0) ∆ x (cid:8) r + 3 rl (cid:9) + 2 M l x (cid:1) δ x b l + (1 − b r δ x + 6 b M rx ) l + i { (3 b r ∆ x + 2 b M x ) l + r + b r ∆ x } . (3.26)On the other hand, the resulting electromagnetic Ernst potential simply readsΦ ′ = − b E ′ . (3.27)This is obvious from (2.9) since the seed metric (3.19) has no associated electromagnetic Ernstpotential, i.e. Φ = 0. Consequently, the magnetized metric function f ′ = Re (cid:8) E ′ (cid:9) − (cid:12)(cid:12) Φ ′ (cid:12)(cid:12) (3.28)5hich is related to the seed function f as f ′ = | Λ | − f can be expressed as f ′ = ∆ x (cid:0) r + l (cid:1) − r l x Ξ . (3.29)In the last equation, we have used Ξ = d l + d l + d l + d where d = r (cid:0) r b ∆ x (cid:1) , d = b δ x ,d = b (cid:0) r b + 24 b x M r + 24 b M rx + 6 x + 2 − r b x + 4 b M x − b r x (cid:1) , and d = 1+ (cid:0) M r x − r x M − r x + 15 x r + 7 r − r M x (cid:1) b + (cid:0) M rx r − r x (cid:1) b . Note that the new twist potential Ψ ′ associated to the transformed Ernst potential E ′ readsΨ ′ = − l (cid:0) r + rl + r x − rl x + M l x − M r x (cid:1) Ξ . (3.30)Furthermore, integrating out (2.17) and (2.18) gives us ω ′ = 2 lx ∆ r (cid:8) c l + c l + c (cid:9) ∆ x ( r + l ) − r l x , (3.31)where c = − b δ x ∆ x , (3.32) c = 2 b (cid:0) r x − x M r + 2 M x + 3 r − r x + 2 x M r (cid:1) , (3.33)and c = 1 + b r ∆ x (cid:0) rx + 3 r − M x (cid:1) . (3.34)Obviously, ω ′ reduces to (3.21) as one considers the limit b →
0. Using the obtained ω ′ and f ′ functions above, the metric after magnetization now becomes ds = 1 f ′ (cid:26) − ∆ r ∆ x dt + (cid:0) r + l (cid:1) (cid:18) dr ∆ r + dx ∆ x (cid:19)(cid:27) + f ′ (cid:0) dφ − ω ′ dt (cid:1) . (3.35)On the other hand, the accompanying vector field in solving the Einstein-Maxwell equations(2.2) can be obtained from the electromagnetic Ernst potential Φ ′ = A φ + i ˜ A φ , where the vectorcomponent A t can be found after integrating (2.15) and (2.16). Explicitly, these vector componentsread A φ = − b Ξ (cid:8) b ∆ x r + (cid:0) b l x − x − b l x + 7 b l (cid:1) r − b M l x (cid:0) x + 1 (cid:1) r + l (cid:0) − x − b l x + 36 b M x + 7 b l − b l x (cid:1) r + 8 M l x (cid:0) b l (cid:8) x (cid:9)(cid:1) r + l (cid:0) b M x + 6 b l x + 3 x + b l + 1 + 9 b l x (cid:1)(cid:9) (3.36)and A t = − lbx ∆ r Ξ (cid:8) b ∆ x (cid:0) x (cid:1) r − b x M ∆ x r + 2 b (cid:0) x + 3 b ∆ x (cid:1) r K ∗ s = M − R µναβ R µναβ . − b x M (cid:0) − b l ∆ x (cid:1) r + 4 l b M x − (cid:0) b l ∆ x (cid:1) (cid:0) b l x + 1 + b l (cid:1)(cid:9) . (3.37)It can be verified that this vector solution obeys the source-free condition, ∇ µ F µν = 0. Moreover,one can consider the massless limit of (3.35) together with the vector A µ with the components givenin (3.36) and (3.37). The result is can be regarded as the Melvin-Taub-NUT universe , namely theTaub-NUT extension of the Melvin magnetic universe discovered in [52, 5]. In this section, let us study some aspects of the magnetized spacetime solution introduced in theprevious section, namely the horizon surface deformation and the closed timelike curve in thespacetime. Before we discuss these features, let us examine the Kretschmann scalar in spacetime,to justify whether the true singularity can exist in spacetime with NUT parameter in the presenceof magnetic fields or not. However, the complexity of the spacetime solution (3.35) hinders us toexpress the Kretschmann scalar explicitly. Therefore, we will perform some numerical evaluationsand see whether the Kretschmann scalar can be singular at the origin. As we pointed out in theintroduction, spacetime with a NUT parameter has the conical singularity instead of a true oneat the origin. This is known from the fact the typical spacetime with NUT parameter has a non-singular Kretschmann scalar at r = 0. In the absence of NUT parameter, Kretschmann scalar for aspacetime containing a black hole blows up at origin even in the presence of an external magneticfield as depicted in fig. 1. However, the typical plots of Kretschmann scalar for a spacetime withNUT parameter appear in figs. 2 and 3, which allow us to infer that the magnetized Taub-NUTspacetime does not possess a true singularity at the origin. The solution is given in appendix A. l = M/ K ∗ s = M − R µναβ R µναβ .Figure 3: Some numerical evaluations for the Kretschmann scalar in the magnetized Taub-NUTspacetime with l = M at equator, where K ∗ s = M − R µναβ R µναβ .8s one would expect for a magnetized spacetime obtained by using Ersnt method, the existenceof an external magnetic field does not affect the radius of the event horizon. It is the zero of∆ r which happens to be the same as in the non-magnetized one, namely r + = M + √ M + l .Furthermore, the total area of horizon reads A = π Z Z − √ g xx g φφ dxdφ = 4 π (cid:0) r + l (cid:1) , (4.38)which is equal to the area of the generic Taub-NUT black hole. Consequently, the entropy ofa magnetized Taub-NUT black hole will be the same as that of a non-magnetized case, namely S = A/
4. However, the external magnetic field can distort the horizon of black hole, as reportedin [49]. In getting to this conclusion, one can study the Gaussian curvature K = R of the twodimensional surface of the horizon, where R is the scalar curvature. For the magnetized Taub-NUTblack hole, the corresponding two dimensional surface of horizon reads ds = (cid:0) r + l (cid:1) dx f + ′ ∆ x + f + ′ dφ , (4.39)where f ′ + is f ′ evaluated at r + . The Gaussian curvature on equator can be found as K x =0 = (cid:8) b l + 4 r + b (4 M − r + ) l − (cid:0) r b + 32 r M b + 3 (cid:1) l (cid:0) r + l (cid:1) (cid:16) b l + 2 l b (cid:8) b r (cid:9) + (cid:8) b r (cid:9) (cid:17) − r + (cid:0) b M r − r b − r + + 4 M (cid:1) l + r (cid:0) − b r (cid:1)(cid:9) . (4.40)Taking the limit l → x = 0 forSchwarzschild black hole in magnetic field [49] K l =0 ,x =0 = 1 − b M M (1 + b M ) . (4.41)Note that in the absence of external magnetic field, this curvature takes the form K Taub − NUT ,x =0 = 12 r + √ M + l , (4.42)which is always positive just like in the Schwarzschild case.Moreover, in the case of magnetized Schwarzschild black holes, the scalar curvature can benegative or zero, depending on the magnitude of external magnetic fields. In particular for themagnetized Schwarzschild spacetime, the scalar curvature at horizon vanishes for 2 bM = 1, becomesnegative for 2 bM >
1, and positive for 2 bM <
1. In the magnetized Taub-NUT case, the zeroscalar curvature at horizon and on equator occurs for b = b = r (cid:16) M { M + 6 l M + 3 l } √ M + l + 8 M + 16 l M + 11 l M + 2 l (cid:17) . (4.43)9igure 4: Gaussian curvature (4.40) evaluated for some magnetic field strength b .Obviously, if we set l → b < b , and the negative one for b > b . Nevertheless, the expression of (4.43) is quite complicated,which yields the job to evaluate eq. (4.40) is troublesome. Again, we can study some numericalexamples related to (4.40), to see how the curvature varies as the magnetic field parameter b orNUT parameter l change. This is presented in figs 4 and 5, where we can learn that for the nullNUT parameter the curvature can take the positive, negative, or zero values depending on theexternal magnetic field strength. This is in agreement to the results reported in [49]. Furthermore,the curvature (4.40) vanishes as we consider l → ∞ which agrees to the plots presented in fig. 4.On the other hand, plots in fig. 5 confirm that the curvature (4.40) can vary as the magnetic fieldincreases.Another way to see how the external magnetic field deforms the horizon can be done by studyingthe shape of horizon as performed in [49] and [53]. Surely, the generic Schwarzschild black holehorizon is a sphere. However, the horizon of a magnetized Schwarzschild black hole can form anoval shape, or even an hourglass appearance for a sufficiently large magnetic field [53]. We showhere that the horizon in the spacetime with NUT parameter also exhibits this effect, which can beunderstood by the previous finding that the Gaussian curvature at equator can become negative for b > b . To illustrate this prolateness effect, let us compute the equatorial circumference of horizon C e and also the polar one C p . Since we will compute some integration in a full cycle, let us returnto the standard Boyer-Lindquist type coordinate ( t, r, θ, φ ). The standard textbook definition for10igure 5: Gaussian curvature (4.40) evaluated for some values of NUT parameter l .these circumferences are C p = π Z √ g θθ dθ , (4.44)and C e = π Z √ g φφ dφ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ = π/ . (4.45)Following [49], we can define the quantity δ which denotes the prolateness or non-spherical degreeof horizon as a function of magnetic field b , δ = C p − C e C e . (4.46)Note that, for the seed solution (3.19), this quantity vanishes, namely it is spherical. In terms of γ and f ′ functions expressed in θ instead of x , the corresponding metric functions incorporated in C p and C e are g θθ = e γ ( r + ,θ ) f ′ ( r + , θ ) and g φφ = f ′ (cid:0) r + , π (cid:1) . (4.47)To illustrate the shape of horizon, we provide numerical plots for some cases of NUT parametersin figs 6 and 7. 11igure 6: Plots of deviation from the spherical form of the surface as dictated by (4.46). We canobserve that the deviation increases for the larger NUT parameter.Figure 7: Solid lines are for subscript i = e and dashed ones are for i = p . Here r ∗ = r/M andthe red, black, and blue colors denote the cases of l = 0, l = M/
2, and l = M , respectively. Herewe learn that the gap between C p and C e gets bigger as the value of NUT parameter l increases,confirming the results presented in fig. 6. 12igure 8: Evaluation of g φφ in the absence of external magnetic field and for some values of l ’s.Now let us turn to the discussion on another aspect in the magnetized Taub-NUT spacetime,namely the existence of a closed timelike curve (CTC). It is well known that the Taub-NUT space-time possesses the CTC [5], which can be understood from the fact of g φφ in the seed metric (3.19)being negative for x < l ∆ r − (cid:0) r + l (cid:1) l ∆ r + ( r + l ) . (4.48)Then it is natural to ask whether the CTC can also occur in the magnetized version of (3.19). Ifit exists, then how would the external magnetic field influence the existing CTC? Related to themagnetized line element, the ( φ, φ ) component of the metric changes in the form g φφ → g ′ φφ = | Λ | − g φφ . (4.49)However, since | Λ | − ≥ b ≥
0, then we can insert that the condition for CTC existencein magnetized Taub-NUT spacetime is just the same as the non-magnetized one, i.e. eq. (4.48).Clearly it is troublesome to express a condition for CTC occurrence for the magnetized Taub-NUTthat is analogous to eq. (4.48) of the non-magnetized one. Therefore, we provide figs. 8 and 9 assome numerical evaluations for the magnitude of g φφ over some angles x , for some particular valuesof l and b , evaluated at the position r = 2 r + . In fig. 8, we confirm eq. (4.48) that the magnitudeof g φφ can be negative for a range of angles x . Note that the plots in fig. 9 resemble this behavior,just the slope becomes smaller as the magnetic field strength increases. This fact is understoodsince g φφ → b → ∞ . So we can conclude that the CTC can also occur in the magnetizedTaub-NUT spacetime, just like in another spacetime with NUT parameter.13igure 9: Evaluation of g φφ for l = M and some values of b , over the same angles as in fig. 8. In this paper, we have presented a novel solution in Einstein-Maxwell theory namely the Taub-NUT extension of magnetized black hole reported in [7]. To get the solution, we have employed theErnst magnetization to the Taub-NUT spacetime. Typical for a spacetime with NUT parameter,the equatorial Kretschmann scalar of the magnetized Taub-NUT spacetime does not blow up at theorigin. Moreover, we find that the black hole surface in magnetized Taub-NUT spacetime deformsdue the presence of an external magnetic field, similar to the magnetized Schwarzschild case asreported in [49]. Furthermore, the existence of NUT parameter leads to the occurrence of a closedtimelike curve in the spacetime as shown in fig 9.Related to this new spacetime solution, there are several interesting future problems that wecan investigate. Extending the solution to a rotating and charged case would be a challenging job,considering the complexity of involved functions in the metric and vector solutions. In particular,it is associated with the Taub-NUT extension of the Melvin magnetic universe [52] as given in theappendix. The distribution of energy in Melvin spacetime had been studied in [54, 55], and itsstability against some perturbations was investigated in [56]. These are interesting open problemsrelated to the magnetized spacetime, and similar studies for the Taub-NUT magnetized spacetimeworth considerations.
Acknowledgement
I thank Merry K. Nainggolan for her support and encouragement.14
Melvin Taub NUT spacetime
In this appendix we provide the solution describing the Taub-NUT extension of the Melvin magneticuniverse. The metric components are g rr = Υ r − l = ∆ x g xx r − l , (A.50) g tt = Υ − n(cid:0) b r ∆ x (cid:1) + b ∆ x δ x (cid:0) x + 10 x + 1 (cid:1) l − b ∆ x (cid:0) b r x (cid:8) x (cid:9) − x +21 x b r − x − b r − (cid:1) l + 2 b ∆ x (cid:0) b x r + 105 b r x − r b x − b r x − x b r + 5 x + 14 b r + 3 + 19 b r (cid:1) l − b ∆ x (cid:0) r b x − b r x − r b x + 13 r b x + 9 x b r + 6 b r x − b r − r b − b r − (cid:1) l (cid:9) , (A.51) g tφ = 2 lx Υ (cid:0) l − r (cid:1) (cid:0) − b r x (cid:8) x (cid:9) + 3 b r + 6 l b r ∆ x + 3 l b x − l b (cid:8) x (cid:9)(cid:1) , (A.52) g φφ = Υ − (cid:0) r − r x + 2 r l + l − r l x + 3 l x (cid:1) , (A.53)where Υ = b δ x l + (cid:8) b r (cid:0) b r + 15 r b x − x b r + 4 − x (cid:1)(cid:9) l − b (cid:0) r b x + 30 x b r − b r − − x (cid:1) l + r (cid:0) b r ∆ x (cid:1) , (A.54)with δ x as introduced in (3.25). The associated vector components are A t = 2 lbx (cid:0) r − l (cid:1) Υ − (cid:8) b r x + 2 b r x − b r − l r b x + 12 l b r x − x b r − l b r − b r + 2 l b x + 2 b l + l b + 2 b l x − l b x (cid:9) , (A.55)and A φ = − b Υ − (cid:8) r x b − r b l x − r b l x + l b + 7 r b l + 7 r b l − r b x + 6 l b x +9 b l x − r l x b + 15 b l r x − r l x + 2 r l − r x + 3 l x + r + l + r b (cid:9) . (A.56)This solution reduces to that of the Melvin universe [52] as the limit l → References [1] H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers and E. Herlt, “Exact solutionsof Einstein’s field equations,” Cambridge University Press, 2004.[2] Islam, J. N., “Rotating fields in general relativity,” Cambridge University Press (1985).[3] R. M. Wald, ‘General Relativity,” Chicago, Usa: Univ. Pr. ( 1984) 491p.[4] C. W. Misner, K. S. Thorne and J. A. Wheeler, “Gravitation,” San Francisco 1973, 1279p.[5] J. B. Griffiths and J. Podolsky, Cambridge University Press, 2009.156] R. M. Wald, Phys. Rev. D , 1680 (1974).[7] F. J. Ernst, J. Math. Phys. 17, 54 (1976).[8] B. K. Harrison, J.Math.Phys.,9,1744, (1968).[9] A. N. Aliev and D. V. Galtsov, Sov. Phys. Usp. , 75 (1989).[10] M. Koloˇs, Z. Stuchlik and A. Tursunov, Class. Quant. Grav. , no. 16, 165009 (2015).[11] A. Tursunov, M. Kolos, Z. Stuchlik and B. Ahmedov, Phys. Rev. D , no. 8, 085009 (2014).[12] A. N. Aliev and D. V. Galtsov, Astrophys. Space Sci. , 181 (1989).[13] A. N. Aliev, D. V. Galtsov and A. A. Sokolov, Sov. Phys. J. , 179 (1980).[14] A. N. Aliev, D. V. Galtsov and V. I. Petrukhov, Astrophys. Space Sci. , 137 (1986).[15] A. N. Aliev and D. V. Galtsov, Sov. Phys. JETP , 1525 (1988).[16] A. N. Aliev and D. V. Galtsov, Sov. Phys. J. , 790 (1989).[17] D. V. Galtsov and V. I. Petukhov, Zh. Eksp. Teor. Fiz. , 801 (1978).[18] W. A. Hiscock, J. Math. Phys. , 1828 (1981).[19] K. Orekhov, J. Geom. Phys. , 242 (2016).[20] G. W. Gibbons, A. H. Mujtaba and C. N. Pope, Class. Quant. Grav. , no. 12, 125008 (2013).[21] G. W. Gibbons, Y. Pang and C. N. Pope, Phys. Rev. D , no. 4, 044029 (2014).[22] M. Rogatko, Phys. Rev. D , no. 4, 044008 (2016).[23] J. Bicak and F. Hejda, Phys. Rev. D , no. 10, 104006 (2015).[24] R. Brito, V. Cardoso and P. Pani, Phys. Rev. D , no. 10, 104045 (2014).[25] H. M. Siahaan, Class. Quant. Grav. , no. 15, 155013 (2016).[26] M. Astorino, JHEP , 016 (2015).[27] M. Astorino, Phys. Lett. B , 96 (2015).[28] M. Astorino, G. Comp`ere, R. Oliveri and N. Vandevoorde, Phys. Rev. D , no. 2, 024019(2016).[29] C. Chakraborty and S. Bhattacharyya, Phys. Rev. D , no. 4, 043021 (2018).[30] P. Pradhan, Class. Quant. Grav. , no. 16, 165001 (2015).[31] M. F. A. R. Sakti, A. Suroso and F. P. Zen, Int. J. Mod. Phys. D (2018) no.12, 1850109.[32] M. F. A. R. Sakti, A. Suroso and F. P. Zen, arXiv:1901.09163 [gr-qc].1633] A. N. Aliev, H. Cebeci and T. Dereli, Phys. Rev. D , 124022 (2008).[34] K. D¨uzta¸s, Class. Quant. Grav. , no. 4, 045008 (2018).[35] H. Cebeci, N. ´’Ozdemir and S. S¸entorun, Phys. Rev. D , no. 10, 104031 (2016).[36] A. Zakria and M. Jamil, JHEP , 147 (2015).[37] S. Mukherjee, S. Chakraborty and N. Dadhich, Eur. Phys. J. C (2019) no.2, 161.[38] A. A. Abdujabbarov, B. J. Ahmedov, S. R. Shaymatov and A. S. Rakhmatov, Astrophys.Space Sci. (2011) 237.[39] A. A. Abdujabbarov, A. A. Tursunov, B. J. Ahmedov and A. Kuvatov, Astrophys. Space Sci. (2013) 173.[40] B. J. Ahmedov, A. V. Khugaev and A. A. Abdujabbarov, Astrophys. Space Sci. (2012)19.[41] A. A. Abdujabbarov, B. J. Ahmedov and V. G. Kagramanova, Gen. Rel. Grav. (2008)2515.[42] A. Abdujabbarov, F. Atamurotov, Y. Kucukakca, B. Ahmedov and U. Camci, Astrophys.Space Sci. (2013) 429.[43] P. Jefremov and V. Perlick, Class. Quant. Grav. (2016) no.24, 245014.[44] H. M. Siahaan, Eur. Phys. J. C (2020) no.10, 1000.[45] H. M. Siahaan, Phys. Rev. D (2020) no.6, 064022.[46] L. Ciambelli, C. Corral, J. Figueroa, G. Giribet and R. Olea, Phys. Rev. D (2021) no.2,024052[47] A. Cisterna, A. Neira-Gallegos, J. Oliva and S. C. Rebolledo-Caceres, [arXiv:2101.03628 [gr-qc]].[48] V. P. Frolov, P. Krtous and D. Kubiznak, Phys. Lett. B (2017), 254-256.[49] W. J. Wild and R. M. Kerns, Phys. Rev. D (1980), 332-335.[50] F. J. Ernst, Phys. Rev. , 1175 (1968).[51] F. J. Ernst, Phys. Rev. , 1415 (1968).[52] M. A. Melvin, Phys. Lett. , 65 (1964).[53] I. Booth, M. Hunt, A. Palomo-Lozano and H. K. Kunduri, Class. Quant. Grav. , no. 23,235025 (2015).[54] S. S. Xulu, Int. J. Mod. Phys. A (2000), 4849-4856.[55] S. S. Xulu, Int. J. Mod. Phys. A (2000), 2979-2986.[56] K. Thorne, Phys. Rev.139