Magnetized Kerr-Newman-Taub-NUT spacetimes.

We find a new class of exact solutions in the Einstein-Maxwell theory by employing the Ernst magnetization process to the Kerr-Newman-Taub-NUT spacetimes. We study the solutions and find that they are regular everywhere. We also find the quasilocal conserved quantities for the spacetimes, the corresponding Smarr formula and the first law of thermodynamics.

Introduction

Finding the exact solutions to the Einstein–Maxwell theory is always fascinating, as it opens a door to explore the new aspects of the gravitational physics. The exact solutions to the aforementioned theory contain the black hole solutions, such as the Kerr–Newman family, to a more general spacetime solutions of Plebanski–Demianski [1]. Different aspects of those solutions have been studied and reported, in which, some can be related to the real astrophysical phenomena, and others are still in vague. Among the latter, is the spacetime solutions with the NUT parameter, which is considered as the extension of the mass parameter. We note that the conserved quantities, such as the mass and angular momentum in a spacetime with a particular boundary, can be computed, which are related to the symmetry of the spacetime. However, the NUT parameter is not associated to any symmetry of the spacetime, and yet it also leads to some peculiar properties in the spacetime, such as conical singularity and the regular invariants such as squared Riemann tensor, at the origin of the coordinate system. Nevertheless, spacetimes with the NUT parameter has helped to shape our understanding of some gravitational and thermodynamical aspects of gravity theories [2-14]

The Kerr–Newman spacetime is a well known black hole solution in the Einstein–Maxwell theory. The solutions can be extended to contain the NUT parameter, and usually referred to, as the Kerr–Newman–Taub-NUT spacetimes. Despite the conical singularities in the spacetimes, there are many research works to explore the different aspects of the Kerr–Newman–Taub-NUT black holes. We note the presence of the NUT parameter in the spacetime, leads to the loss of asymptotic flatness, if the corresponding null NUT counterpart has this asymptotic [1]. It can be shown that the Kerr–Newman–Taub-NUT spacetime is a special case of the Plebanski–Demianski spacetime which is considered as one of the most general solution in Einstein–Maxwell theory that can contain black holes [15].

In Einstein–Maxwell theory, it also exists an exact solution describing a universe filled by a homogeneous magnetic field known as the Melvin universe [16]. A black hole solution in this Melvin universe can be obtained by using the Ernst magnetization [17] applied to a known black hole spacetime in Einstein–Maxwell theory as a seed. In fact, performing Ernst magnetization to the Minkowski spacetime can give us the Melvin universe. In general, the magnetization can be done in two ways, namely at the level of perturbation as Wald introduced in [18], and as a strong field as Ernst proposed in [17]. In the Wald prescription, Maxwell field is introduced perturbatively by using the Killing vectors associated to the spacetime, while the presence of homogeneous magnetic field does not change the spacetime solution. We can infer that the magnetization by Wald does not change the asymptotic structure of the magnetized spacetime. The superradiant instability in this weakly magnetized black hole had been investigated in [19], and this type of magnetization for Kerr-NUT-AdS spacetime had been performed in [20].

This Ernst magnetization itself can be viewed as a type of Harrison transformation [21] which maps an old solution to a new one in the theory. A number of aspects of the known magnetized black hole solutions had been reported in literature [22-38], and this shows the importance of such solution in shaping our knowledge on gravity. The most recent ones are the magnetization to Reissner–Nordstrom -Taub-NUT [39] and Kerr–Taub-NUT [40]. The work presented in this paper extends the previous works to magnetizing the Kerr–Newman–Taub-NUT (KNTN) spacetimes, which we refer to as the Melvin–Kerr–Newman–Taub–Nut (MKNTN) spacetimes. Though the idea is straightforward and the mechanism is well understood, but incorporating the functions in the solution are quite challenging.

In this paper, we perform the magnetization procedure to the Kerr–Newman–Taub-NUT spacetimes. We expect to get the magnetized KNTN spacetime solution, whose massless, null NUT, static, and neutral limit, is the Melvin magnetic universe [16]. Some aspects of the spacetime are discussed, such as the deformation of the horizon and the quasilocal conserved quantities associated to the solution.

The organization of this paper is as follows. In Sect. 2, after reviewing the Ernst magnetization process, we construct the MKNTN solutions by employing the Ernst magnetization to the KNTN metric as the seed solution. In Sect. 3, we study some properties of the MKNTN spacetimes. In Sect. 4, we obtain the quasilocal thermodynamical quantities for the MKNTN black holes, as well as the Smarr equation for the MKNTNblack holes and verify the first law of thermodynamics. We consider the natural units .

Construction of the magnetized spacetimes

Ernst magnetization

Ernst magnetization is a transformation acting on a set of Ernst potentials which can be defined by using some functions appearing in the seed spacetime solution and the accompanying vector field in Einstein–Maxwell theory. The seed solution is typically expressed in the Lewis–Papapetrou–Weyl (LPW) formwhere f, , and are function of . Here we have used the signs convention for the spacetime, and notation representing the complex conjugation. Using the f function in the LPW line element above, accompanied by the vector , the gravitational Ernst potential,and the electromagnetic onecan be constructed. The component can be obtained after solvingNote that the imaginary part of is the vector field which constructs the dual field strength tensorwhere .

In equation above, the twist potential is given by the relationUsing the Ernst potentials, the following equations can be extracted from the equations of motion in Einstein–Maxwell theory,The last equation is known as the Ernst equations, and is invariant under some transformation [41]. We note that all the incorporating functions in the metric (2.1) depend on and z only, then the operator , in Eqs. (2.4), (2.6), (2.7) and (2.8) can be defined in the flat Euclidean spaceas , where we have set the complex coordinate . Moreover, as we explain explicitly in Appendix A, we find the following differential equations for the function ,andAccording to Ernst, one can magnetized the seed solution described by the line element (2.1 ) and vector solution above by transforming the corresponding Ernst potentialswhereHere, the constant b is interpreted as the external magnetic field strength in the spacetime.1 The transformation (2.12) leaves Eqs. (2.7) and (2.8) unchanged for the new potentials and . In other words, the new metric consisting the functions and , together with the new vector potentials and are also solutions to the Einstein–Maxwell field equations.

In particular, the transformed line element (2.1) resulting from the magnetization (2.12) has the componentsandwhile the function remains unchanged. In Appendix A, we present an example, which shows the differential equations for the function , and the invariance of the function under the Ernst magnetization process.

We note that typical black hole solutions in the Einstein–Maxwell theory, are more compact where they are expressed in the Boyer–Lindquist type coordinates . Consequently, the LPW type metric (2.1) with stationary and axial Killing symmetries will have the metric function that depend on r and x, and the corresponding flat metric line element readswhere and . Therefore, the corresponding operator will read . Furthermore we can have , then Eq. (2.4) gives usandThe last two equations are useful later in obtaining the component associated to the magnetized spacetime according to (2.12). To end some details on magnetization procedure, another equations which will be required to complete the metric areandIn the following section, we employ this magnetization scheme to the Taub-NUT spacetime.

The Melvin–Kerr–Newman–Taub-NUT spacetimes

To obtain the desired magnetized solution, we use the Ernst potentials that belong to Kerr–Newman–Taub-NUT system,andwhere , , and . In the form of LPW line element (2.1), the above spacetime metric, associates to the functionsand .

From this seed solution, one can construct the corresponding Ernst potentials as followsandwhereThe magnetized Ernst potentials can be obtained from the seed ones above, which yields to the magnetized metric with the new functions and , while is unchanged.

In fact, the differential equations for the function , are given byandSimilar equations hold for with , and . We explicitly check that Eqs. (2.30) and (2.31) and their counterparts for imply the metric function is the same as , which is given by (2.25).

To summarize the results, the Melvin–Kerr–Newman-NUT black hole is given bytogether with the Maxwell’s fieldwhere the metric functions and are given byandMoreover, the components of the Maxwell’s field (2.33) are given byand The expressions for , as functions of r, are given in Appendix B.

Some properties of the Melvin–Kerr–Newman–Taub-NUT spacetimes

The largest root of the metric function describes the outer event horizon of the black hole (2.32), which is given by which implies , to have a real value for the outer event horizon. The inner event horizon is located at . We note that the inner event horizon exists at a real positive (or zero) value , if . Combining the former and the latter inequalities, we find the following range for the summation of squares of the rotational parameter and the electric charge of the spacetime (2.32). The trace of the energy-momentum tensor for the Maxwell’s field (2.33) is identically zero. We have verified exactly that the metric (2.32) with the Maxwell’s field (2.33) satisfy exactly all the Einstein–Maxwell field equations. The Ricci scalar of the spacetime is identically zero and the Ricci square invariant is regular everywhere including . The expression for the is very long and so we don’t present it here. We also find the Kretschmann invariant where and are two functions with coefficients of the black hole parameters. Though the expression for is very complicated, howeverThe location of curvature/coordinate singularities for the black hole (2.32) can be determined by the equation . In fact, beside =0, we find the following equation for the location of singularities, which is expectedly independent of the magnetic fieldThe event horizon and are the roots of , where is regular and finite. Moreover, it seems or Eq. (3.4) leads to other singular points, however, we verify that at those points, the Kretschmann invariant remains completely finite.

We notice from Eq. (3.4) and Table 1, that the only magnetized spacetimes with the point singularity, at , are Melvin–Schwarzschild and Melvin–Reissner–Nordstrom space-times. All other magnetized spacetimes, i.e. Melvin, Melvin–Kerr, Melvin-NUT, Melvin–Kerr–Newmann, Melvin–Kerr-NUT, Melvin–Reissner–Norstrom-NUT and MKNTN are completely regular at .

Table 1

Singularity at r = 0 = x

Spacetime	Is r = 0 = x singularity?	M-Spacetime	Is r = 0 = x singularity?
Minkowski	No	Melvin	No
Schwarzschild	Yes	M-Schwarzschild	Yes
Kerr	Yes	M-Kerr	No
NUT	No	M-NUT	No
Reissner–Nordstrom	Yes	M-Reissner–Nordstrom	Yes
Kerr–Newman	Yes	M-Kerr–Newman	No
Kerr-NUT	No	M-Kerr-NUT	No
Reissner–Nordstorm-NUT	No	M-Reissner–Nordstrom-NUT	No
Kerr–Newman-NUT	No	M-Kerr–Newman-NUT	No

In the special case, where all the black hole parameters approach zero, we find the metric (2.32) reduces towhere The metric (3.5) describes the spacetime of axisymmetric universe filled by parallel magnetic forcelines known as the Melvin magnetic universe [16]. Interestingly, these forcelines do not contract and collapse under their own gravity, and the corresponding stability against radial perturbations had been investigated in [42]. In fact, the Maxwell’s field (2.33) becomeswhich generates the magnetic field

The Ricci scalar of the Melvin space-time (3.5) is identically zero, while the Kretschmann invariant is given byMoreover, we find the asymptotic behaviour of the black hole (2.32), where , by analyzing all the metric functions at large values of the radial coordinate. We find the asymptotic expressions for the metric functions, aswhere .

Using expressions (3.10)–(3.13), we find the asymptotic metric of the black hole (2.32), aswhere is the asymptotic of the Melvin universe (3.5). We note that the presence of the NUT charge makes an off-diagonal term to the asymptotic metric of the black hole (2.32).

We consider now the set of 3-dimensional surfaces at a fixed value for the radial coordinate . The induced metric on the surface , is given byThe determinant of the metric (3.15) isFrom (3.16), we notice that the surface describes a -dimensional space-time, if or . On the other hand for , the surface describes a 3-dimensional space. Of course for or , the surface becomes a null surface.

We note that due to the presence of the NUT charge as well as the magnetic field, the horizon geometry is a distorted sphere. In fact the two-dimensional horizon is given by the line elementThe two grand circles on the horizon, one at and the other passing through , have two different circumferences. The former circumference is given byIn Fig. 1, we plot the typical behaviour of the equatorial circumference versus the NUT charge l and the magnetic field b, where we set and .

Fig. 1

The equatorial circumference versus the NUT charge l and the magnetic field b, where we set the other black hole parameters and . The horizontal plane shows the equatorial circumference (which is equal to 87.4 in arbitrary unites), where the NUT charge and the magnetic field are zero

The latter circumference involves an integral, which we can’t find it explicitly as an exact form. In Fig. 2, we plot the result of numerical integration for the circumference, as a function of the magnetic field, where we set the other black hole parameters as and .

Fig. 2

The circumference of the great circle, passing through the north and south pole, versus the magnetic field b, where we set the other black hole parameters and . The circumference with no magnetic field is equal to 71.9 in arbitrary units

The other interesting surfaces for the black hole (2.32) are the stationary limit surfaces. The stationary limit surfaces are the roots of equationFor a generic black hole (2.32), the Eq. (3.19) turns out to bewhich of course, is not solvable by radicals. Hence, we consider the black hole with the same parameters and , which we considered before in this section. The inner and outer event horizons are at and , respectively.

In Fig. 3, we plot the function S(r, x) versus and . The outer stationary limit surfaces are the intersection of the curve with horizontal plane at 0, where . The outer ergoregion for the black hole (2.32), is the region between and , where .

Fig. 3

The scaled function S(r, x) versus and , where we set the black hole parameters and . The stationary limit surfaces are the intersection of the curve S(r, x) with the horizontal plane at zero

In Fig. 4, we plot the function S(r, x) versus and . The inner stationary limit surfaces are the intersection of the curve S(r, x) with the horizontal plane at 0, where . The inner ergoregion for the black hole (2.32), is the region between and , where . We also note that for , the function S(r, x) is positive everywhere, as shown in Fig. 5.

Fig. 4

The scaled function S(r, x) versus and , where we set the black hole parameters and . The stationary limit surface is the intersection of the function S(r, x) with the horizontal plane at zero

Fig. 5

The scaled function S(r, x) versus and , where we set the black hole parameters and

Now, we consider the rich structure of the electromagnetic fields, on and outside the event horizon. The electromagnetic field components are given bywhere , the Hodge dual of two-form F, is given in (2.5), and is the 4-velocity of the observer. We find the exact forms for the electromagnetic fields, though their expressions are quite long, and so we do not present them here. In Fig. 6, we plot the typical behaviour of the components of the magnetic field, outside the event horizon, for a black hole with parameters and . Moreover in Fig. 7, we plot the typical behaviour of the components of the electric field, outside the event horizon, for the same black hole parameters.

Fig. 6

The r (left) and x (right) components of the magnetic field, versus r and x, for a black hole which the event horizon is located at . We set the black hole parameters as and

Fig. 7

The r (left) and x (right) components of the electric field, versus r and x, for a black hole which the event horizon is located at . We set the black hole parameters as and

In Figs. 8 and 9, we plot the behaviour of the electromagnetic fields on the event horizon. We notice the minimum and maximum of and appear quite away from the equatorial plane, however the maximum of and the minimum of occurs almost on the equatorial plane.

Fig. 8

The r (left) and x (right) components of the magnetic field, on the event horizon, versus x, for a black hole which the event horizon is located at . We set the black hole parameters as and

Fig. 9

The r (left) and x (right) components of the electric field, on the event horizon, versus x, for a black hole which the event horizon is located at . We set the black hole parameters as and

We also plot the polar electromagnetic fields outside the even horizon in Figs. 10 and 11.

Fig. 10

The r (left) and x (right) components of the polar magnetic field, outside the event horizon, versus r, for a black hole which the event horizon is located at . We set the black hole parameters as and . In the left figure, the down curve is for and the up curve is for . In the right figure, the down curve is for and the up curve is for

Fig. 11

The r (left) and x (right) components of he polar electric field, outside the event horizon, versus r, for a black hole which the event horizon is located at . We set the black hole parameters as and . In both figures, the down curve is for and the up curve is for

We should notice that increasing behaviour in the polar electromagnetic fields doesn’t extend to large values of the radial coordinate. In fact, we find the asymptotic behaviour of the electromagnetic fields for , asfor an observer with the 4-velocity .

Thermodynamics of the Melvin–Kerr–Newman–Taub-NUT spacetimes

In this section, we discuss the thermodynamical quantities for the black hole (2.32) and then construct the mass of the black hole according to the Smarr relation.

We should emphasis that using the term “black hole” for the metric (2.32), with the NUT charge parameter l must be taken with some cautions, to avoid contradictions with the black hole uniqueness theorems. The event horizon is a global concept and, therefore, requires asymptotic flatness to be well defined. On the other hand, the NUT charge parameter l makes the spacetime (2.32) asymptotically locally flat, and so violates the global flatness condition. Hence, in general, a global event horizon does not exist. However, since the spacetime (2.32) possesses many physical quantities, similar to a black hole (such as event horizons,), we often refer to the spacetime (2.32), as a black hole.2

The surface gravity for the black hole (2.32) is given bywhere is the Killing vector and is the angular velocity of the horizon, which is given bywhereandWe note that for , we recover the angular velocity for the Kerr–Newman-NUT black hole [31]. In Figs. 12 and 13, we plot the angular velocity of the horizon versus different black hole parameters.

Fig. 12

The angular velocity of the horizon as function of (left) and (right), where we set the other black hole parameters to a set of fixed numbers

Fig. 13

The angular velocity of the horizon as function of (left) and (right), where we set the other black hole parameters to a set of fixed numbers

We find the surface gravity of the black holeis the same as the Kerr–Newman-NUT black hole, and so the Hawking temperature is

The Coulomb potential on the horizon is given bywhere we add a constant term to the potential to make it regular at . The expression for the is long and so we do not present it explicitly here. The area of the horizon is given bywhere is the horizon area for the Kerr–Newman-NUT black hole. To find the electric charge of the black hole (2.32), we use the well-known equation [31]where is the two-dimensional hypersurface, parameterized with in (2.32) and is the normal component of the electric field on . We findwhereandMoreover, the functions are given byIn the limit of , we recover exactly the results of [31] for the electric charge. In Figs. 14 and 15 we plot the electric charge (4.10) of the black hole, versus different black hole parameters.

Fig. 14

The electric charge of the black hole as function of (left) and (right), where we set the other black hole parameters to a set of fixed numbers

Fig. 15

The electric charge of the black hole as function of (left), and (right), where we set the other black hole parameters to a set of fixed numbers

We then find the angular momentum J of the black hole (2.32), according to [31]where is the Killing vector in -direction. We find the integrand in (4.14) is given byA straightforward and lengthy calculation shows that we getwhere and depend on black hole parameters a, l, m and q. The expressions for and are very long to present, so we don’t present them here. We verify that the expression (4.16) recovers exactly the results of [31] for Melvin–Kerr–Newman black holes, where the NUT charge l goes to zero. In Figs. 16 and 17 we plot the angular momentum (4.16) of the black hole, versus different black hole parameters.

Fig. 16

The angular momentum of the black hole as function of (left) and (right), where we set the other black hole parameters to a set of fixed numbers

Fig. 17

The angular momentum of the black hole as function of (left) and (right), where we set the other black hole parameters to a set of fixed numbers

Inspired by the dimensions of physical quantities for the black hole, as well as the presence of the NUT charge [43], we consider the total mass of the black hole (2.32) aswhere and ( and ) are the thermodynamic charge and the potential, due to the presence of the NUT charge on north (south) pole, and and are given by Eqs. (4.7), (4.10), (4.2), (4.5) and (4.8), respectively. The total angular momentum in Eq. (4.17) iswhere J is given by (4.16), and and are contributions to the angular momentum from NUT charges on the north and south poles. The total mass for the black hole (2.32) is given bywhere is the time-like Killing vector. Evaluating the integral in Eq. (4.19), where , we findMoreover, the thermodynamic potentials and are given by [43]where is the Killing vectorwhich generates the north pole Killing horizon, due to the presence of the NUT charge. The other Killing vector iswhich generates the south pole Killing horizon, due to the presence of the NUT charge. The functions and are given byUsing the form of the Killing vectors on the north and south pole horizons, we findandwhere is given byAfter substituting all the known functions , , , and in (4.28) and taking the limits in Eqs. (4.26) and (4.27), we findandAs expected, we notice that neither the magnetic field nor the other parameters of the black hole (except the NUT charge) contribute to the thermodynamic potentials and , where . The independence of and from other parameters of the black hole and the magnetic field, is consistent with the notion of considering them as the “surface gravity” over the NUT tubes [43], similar to as the surface gravity (4.1) over the horizon . The thermodynamic charges and are given bywhere the integrals are over the very narrow NUT tubes and , along the positive and negative z-axis, where and , respectively. We find the integrand in Eqs. (4.31) and (4.32), is given byEvaluating the integrals in (4.31) and (4.32), we find very long expressions for the and in terms of the black hole parameters a, l, m and q and the magnetic field parameter b which are unfeasible to present here. We verify that in the special cases, the thermodynamic NUT potentials reduce exactly to the well-known results in [43].

As we notice from Eq. (3.14), the NUT charge contributes to an off-diagonal term in the asymptotic of MKNTN black holes (2.32). As a result, we find two contributions to the total angular momentum (4.18) from the north and south NUT tubes, which are given byrespectively, where is the space-like Killing vector. We find the integrand in Eqs. (4.34) and (4.35), is given byEvaluating the integrals in (4.34) and (4.35), we find very long expressions for the and in terms of the black hole parameters a, l, m and q and the magnetic field parameter b which are unfeasible to present here. We verify that in the special cases, the thermodynamic NUT potentials reduce exactly to the well-known results in [43].

To complete the calculation, we also mention that the area and of the north and south NUT tubes are given byrespectively, where and are the Killing vectors (4.23) and (4.24), which generate the north and south pole tubes. The integrand in Eqs. (4.37) and (4.38), is given bywhere Z stands for N and S, respectively. After calculating the integrals in (4.37), (4.38), we find that and , are independent of the magnetic field parameter b, and are given bywhere is the length of the north (south) NUT tube along the positive (negative) z-axis.

Furnished by the results (4.17), (4.18), (4.29), (4.30), (4.31), (4.32), (4.34) and (4.35), we can verify that Eq. (4.17) is indeed the Smarr relation for the MKNTN black hole (2.32). Moreover, by construction of different thermodynamical quantities for the black hole (2.32), and the Smarr equation (4.17), we find the first law of thermodynamics, as given byIn fact, a tedious but straightforward calculation shows that Eq. (4.17) implies that for , we have the first law of thermodynamics, as it is given by (4.41). We also verify that Eq. (4.41) reduces exactly to the well-know first law of thermodynamics for the Kerr-NUT and Melvin–Kerr–Newman black holes in [43] and [31].

Concluding remarks

In this work, we have constructed a new class of the exact solutions to the Einstein–Maxwell theory in four dimensions which describes the immersion of the Kerr–Newman–Taub-NUT spacetimes in an external magnetic field. The solutions are obtained by applying the Ernst magnetization procedure to the four dimensional Kerr–Newman–Taub-NUT spacetime as the seed. We discuss the properties of the MKNTN spacetimes and show that they are completely regular at . Particularly, in addition to the extensive investigation on the space-time structure and the ergoregions for the MKNTN black hole, we also study the behaviour of the electromagnetic fields in the magnetized spacetime solutions. We find that the horizon has a non-trivial topology which leads to the eccentric horizon. The thermodynamical quasi-local conserved quantities of the spacetime are obtained, though they are generally quite complicated functions of the five independent parameters of MKNTN black holes, namely m, a, l, q and b. In this paper, we also establish the Smarr formula for the MKNTN black holes. Finally, we study the thermodynamics of the MKNTN spacetimes, and show the corresponding first law of thermodynamics.