On some novel features of the Kerr–NewmanNUT spacetime
Abstract
In this work we have presented a special class of Kerr–NewmanNUT black hole, having its horizon located precisely at \(r=2M\), for \(Q^{2}=l^{2}a^{2}\), where M, l, a and Q are respectively mass, NUT, rotation and electric charge parameters of the black hole. Clearly this choice radically alters the causal structure as there exists no Cauchy horizon indicating spacelike nature of the singularity when it exists. On the other hand, there is no curvature singularity for \(l^2 > a^2\), however it may have conical singularities. Furthermore there is no upper bound on specific rotation parameter a / M, which could exceed unity without risking destruction of the horizon. To bring out various discerning features of this special member of the Kerr–NewmanNUT family, we study timelike and null geodesics in the equatorial as well as off the equatorial plane, energy extraction through superradiance and Penrose process, thermodynamical properties and also the quasiperiodic oscillations. It turns out that the black hole under study radiates less energy through the superradiant modes and Penrose process than the other black holes in this family.
1 Introduction
Duality of Maxwell’s equations in the presence of magnetic monopole has far reaching consequences. Therefore, it seems legitimate to explore whether there can exist any such duality for gravitational dynamics as well. Surprisingly, it turns out that there is indeed such a duality in the realm of gravitational field of a Kerr–Newman black hole. For asymptotically flat spacetimes, the unique solutions of the Einstein–Maxwell field equations are the black holes in Kerr–Newman family [1]. However if the condition of asymptotic flatness is dropped, then one can have an additional hair on the black hole, known as the NUT charge and the black holes are referred to as the KerrNUT solutions [2, 3]. We also refer our readers to Refs. [4, 5, 6, 7, 8] for a descriptive overview on NUT solutions and their various implications in Einstein as well as alternative theories of gravity. Besides admitting separable Hamilton–Jacobi and Klein–Gordon equations [9], the above family also shares a very intriguing duality property: the spacetime structure is invariant under the transformation \(\text {mass}\leftrightarrow \text {NUT charge}\) and \(\text {radius}\leftrightarrow \text {angular coordinate}\) [10]. Given this duality transformation one can associate a physical significance to the NUT parameter, namely a measure of gravitational magnetic charge. Even though it is possible to arrive at a NUT solution without having rotation, the above duality only works if the rotation parameter is nonzero [11]. Thus in order to have a concrete theoretical understanding of the present scenario rotation is necessary.
On the other hand, even though there is no observational evidence whatsoever for the existence of gravitomagnetic mass [12], investigation of the geodesics in Kerr–NewmanNUT spacetime has significance from both theoretical as well as conceptual points of view. The observational avenues to search for the gravitomagnetic monopole includes, understanding the spectra of supernovae, quasars and active galactic nuclei [12]. All of these scenarios require presence of thin accretion disk [13] and can be modelled if circular geodesics in the spacetime are known. Thus a proper understanding of the geodesic motion in the presence of NUT parameter is essential. Following such implications in mind, there have been attempts to study circular timelike geodesics in presence of NUT parameter [14, 15, 16]^{1} as well as motion of charged particles in this spacetime [17]. Various weak field tests, e.g., perihelion precession, Lense–Thirring effect has also been discussed [18, 19, 20] (for a taste of these weak field tests in theories beyond general relativity, see [21, 22, 23, 24, 25]). We would like to emphasize that most of these studies on the geodesic motion crucially hinges on the equatorial plane, however there can also be interesting phenomenon when offtheequatorial plane motion is considered.
Besides understanding the geodesic structure, it is of utmost importance to explore the possibilities of energy extraction from black holes as well. In particular, the phenomenon of Penrose process [26], superradiance [27, 28] and the Bañados–Silk–West effect are well studied in the context of Kerr spacetime [29]. Implications and modifications to these energy extraction processes in presence of NUT charge is another important issue to address. It will be interesting to see how the efficiency of energy extraction in the Penrose process depends on the gravitomagnetic charge inherited by the spacetime. Further, being asymptotically nonflat, whether some nontrivial corrections to the energy extraction process appear is something to wonder about. Moreover, it is expected that the phenomenon of superradiance and the counterintuitive Banados–Silk–West effect will inherit modifications over and above the Kerr spacetime due to presence of the NUT charge. In particular, for what values of angular momentum the centerofmass energy of a system of particles diverges is an interesting question in itself.
The uniqueness theorems dictates that a black hole can have only three hairs, mass, angular momentum and electric charge in an asymptotically flat spacetime. If the requirement of asymptotic flatness is relaxed, it can have a new hair, namely the NUT parameter. Each of these black hole hairs must be tested on the anvil of astrophysical observations. So the question is, could we work out an observationally testable effect that could put bounds on NUT paramater. For that we have studied quasi periodic oscillations for the black hole in question where fundamental frequency of oscillations depends upon it. This could be one of the possible observational tests to unveil the existence of NUT parameter.
Finally we should say a word about our choice of the metric for this investigation. In the Kerr–NewmanNUT metric, rotation, NUT charge and the electric charge parameters, \(a^2\), \(l^2\) and \(Q^2\), appear linearly in \(\Delta = r^22Mrl^2+a^2+Q^2\). Thus if we make the following unusual choice: \(Q^2+a^{2}=l^2\) [30],^{2} \(\Delta \) becomes simply \(r^22Mr\) and thereby black hole horizon is entirely determined by mass alone and coincides with that of the Schwarzschild’s. Despite this the black hole is having both electric and NUT charge and also rotating. This happens because the electric charge appears only in \(\Delta \) and nowhere else in the metric, while NUT and rotation parameters also define geometrical symmetry of the spacetime. This is why it could be simply added or subtracted, i.e., in \(\Delta \) of KerrNUT metric, simply add \(Q^2\) to obtain the Kerr–NewmanNUT solution of the Einstein–Maxwell equations. It is noteworthy that despite presence of rotation and electric charge, the singularity is not timelike but rather spacelike, as of the Schwarzschild black hole. That is, the choice of \(Q^2+a^2=l^2\) indicates that repulsive effect due to charge and rotation is fully balanced by attractive effect due to the NUT parameter. This is why the causal structure of spacetime has been radically altered [31]. Unlike any other black holes in this family, it would have spacelike singularity when it exists. For existence of singularity, one must have \(r^2 + (l + a \cos \theta )^2 = 0\), which will never be so for \(l>a\). Thus spacelike singularity will only exist for \(l \le a\), else for \(l>a\) it would be free of the ring singularity at \(r=0\). Another remarkable feature is that the specific rotation parameter a / M could have any values even exceeding unity without risking the singularity turning naked. All these are very novel and interesting features, and their exposition is the main aim of this paper. With the choice \(l>a\), there occurs no ring singularity as reflected in the fact that \(r^2 + (l + a \cos \theta )^2 \ne 0\) for any choices of r and \(\theta \). It is therefore a very interesting special case of the Kerr–NewmanNUT family of spacetimes, whose structure we wish to understand in this paper for studying its various interesting properties.
The paper is organized as follows: In Sect. 2 we have elaborated the spacetime structure we will be considering in this work. Subsequently in Sect. 3 the trajectory of a particle in both equatorial and nonequatorial plane has been presented. Various energy extraction processes in this spacetime have been illustrated in Sect. 4 and finally thermodynamics of black holes in presence of NUT charge has been jotted down in Sect. 6. We finish with a discussion on the results obtained in Sect. 7.
Notations and conventions Throughout this paper we have set the fundamental constants \(c=1=G\). All the Greek indices run over four dimensional spacetime coordinates, while the roman indices run over spatial three dimensional coordinates.
2 The spacetime structure
3 Trajectory of massive and massless particles
3.1 Orbits confined on a given plane
The addition of the NUT charge will introduce nontrivial difficulties while obtaining the orbital dynamics for a massive or even a massless particle. This can be understood by employing the angular equations given in Eqs. (10) and (12) respectively. For any trajectory to be confined on a particular plane \(\theta =\theta _{0}\), we bound to have \({\dot{\theta }}=\ddot{\theta }=0\). While the condition \({\dot{\theta }}=0\) indicates a trajectory moving on a constant plane, the additional second derivative ensures that the first derivative remains null throughout its motion. Therefore both of these conditions are essentials to determine any planner orbit in the presence of a NUT charge and unlike the Kerr black hole, this constraint would introduce stringent bound on the particle trajectories in Kerr–NewmanNUT spacetime. In the upcoming discussions, we shall explore any possible scenarios in which the conditions for a planner orbit can be satisfied in a certain ranges of parameters. Regarding that, we first introduce the massless particles and following that, the phenomenon involving massive particles will be addressed.
3.1.1 The massless particles
3.1.2 The timelike geodesics
Having described this particular case in which the impact parameter is directly related to black hole spin, let us now concentrate on the motion of massive particles moving in circular orbits.
Circular orbits for timelike particles In this section, we shall discuss the possible existence of timelike circular orbits and hence compute the exact expressions for conserved quantities associated with it. In this case as well, the conditions for circular orbit are given by \(V_{\mathrm{eff}}=0\) and \(V_{\mathrm{eff}}'=0\), see Eq. (16). To see the existence of circular orbits in this spacetime in a direct manner, we have plotted the effective potential \(V_{\mathrm{eff}}\), rescaled by \((r^{2}+l^{2})^{2}\), for specific choices of various constants of motion and parameters appearing in the problem in Fig. 2. In particular, we have considered only marginally bound orbits, i.e., orbits with \({\tilde{E}}=1\) and the situation when the rotation parameter and NUT charge of the black hole coincides, implying \(l=a\) and \({\tilde{\lambda }}=M^2\). As evident from Fig. 2 the effective potential indeed exhibits a minima for various choices of the (a / M) as well as \(({\tilde{L}}/M)\) ratio and hence allows existence of stable circular orbits in the spacetime. Thus we have explicitly demonstrated the existence of stable circular orbits in this spacetime, which we will now explore analytically.
This finishes our discussion on trajectory of a massive as well as massless particle confined on a given plane. We will next take up the computation on the motion of a particle on an arbitrary plane.
3.2 Nonequatorial plane
In the previous section we have described the motion of both massive and massless particles in the equatorial plane of a Kerr–NewmanNUT black hole. However to understand some other subtle features associated with this spacetime it is important that we consider motion in nonequatorial plane as well. In this case, both for massive and massless particles the Carter constant will be nonzero and will play a significant role in determining various properties of the trajectory.
(A) For \(\eta =0\):

\(P^{\theta }\) vanishes for \(\theta =\theta _0\), \(\theta _1\) and \(\theta _2\) which are referred to as the turning points in the \(\theta \) direction. In addition, the potential attains its local maximum value on the equatorial plane. Hence, a particle would remain on the equatorial plane unless acted upon by an external perturbation (see Fig. 5a), even though it is not a stable equilibrium point.

Along with the confined motion on the equatorial plane, a massless particle can have off equatorial trajectories whenever \(V(\theta )\) is negative. This is only possible if it follows, \(\theta _2<\theta <\theta _0\) and \(\theta _0<\theta <\theta _1\). The particle never reach the equator as it asymptotically approaching the equatorial plane. This is shown in Fig. 5a.

For \(\xi =(L/E)=M\), one has \(L/aE>1\) and hence in this case the potential has a minima and it vanishes only on the equatorial plane. Since it never becomes negative, motion is only allowed on the equatorial plane with vanishing momentum along the angular direction (see Fig. 5b), i.e, \(P^{\theta }=0\).

For \((L/aE)<1\), the potential \(V_{\mathrm{ang}}\) is negative within the region \(\theta _2<\theta <\theta _1\). Since \(\eta >0\) it follows that the momentum \(P^{\theta }\) has two turning points located at \(\theta ^{\pm }\), satisfying: \(\theta _{}<\theta _{2}\) and \(\theta _{+}>\theta _{1}\) respectively. As evident from Fig. 6a the particle oscillates about the equatorial plane.

For \((L/aE)>1\), unlike the previous case, here the particle can travel beyond the equatorial plane upto the point where \(\{P^{\theta }\}^{2}\) vanishes. Thus in this case the particle can travel away from the equatorial plane. This is depicted in Fig. 6b.
(C) For \(\eta <0\):

In this case with \((L/aE)<1\), not only \(V_{\mathrm{ang}}\) has to be negative, one has to ensure that \(V_{\mathrm{ang}}>\lambda \). Thus on the equatorial plane \(V_{\mathrm{ang}}\) vanishes and hence there can be no physical motion on the equatorial plane. Thus in this case the particle has to travel off the equatorial plane. In particular, the particle has to obey either of these two conditions – (a) \(\theta _2<\theta ^{}<\theta<\theta _{\pi /2}^{}<(\pi /2)\) or, (b) \((\pi /2)<\theta _{\pi /2}^{+}<\theta<\theta ^{+}<\theta _1\), where \(\theta _{\mathrm{\pi /2}}^{\pm }\) and \(\theta ^{\pm }\) are the turning points of the momentum \(P^{\theta }\). A qualitative description can be found in Fig. 7a.

On the other hand, for \((L/aE)>1\) it follows that \(V_{\mathrm{ang}}\) is always positive. Thus in this case there is absolutely no phase space available for the particle. Hence this corresponds to a unphysical situation, as presented in Fig. 7b.
3.2.1 Massless particles in Kerr–NewmanNUT black holes
(A) Angular motion:
 If the rotation parameter a, NUT charge l and the specific angular momentum \(\xi \) are such that the following condition is satisfiedthe solutions to Eq. (49) can be written as,$$\begin{aligned}&{\mathcal {G}}\equiv \{a^444 a^2 l^2+2 a^3 \xi (4 l^2+\xi ^2)^2\nonumber \\&\quad +a^2 (56 l^2 \xi 2 \xi ^3) \}>0 \end{aligned}$$(51)$$\begin{aligned} \mu _{1}&=\dfrac{4 l}{3 a}+\dfrac{(3 a^2+4 l^23\xi ^2)^{1/2}}{3 a} \left\{ 2 \cos \alpha \right\} \end{aligned}$$(52)Here \(\alpha \) is an angle within the range \((0,2\pi )\) and can be explicitly written as$$\begin{aligned} \mu _{2,3}&=\dfrac{4l}{3a}\dfrac{(3a^2+4 l^23 \xi ^2)^{1/2}}{3a}\left( \cos \alpha \mp \sqrt{3}\sin \alpha \right) . \end{aligned}$$(53)with A and B defined as, \(A=36 a^5 l54 a^4 l \xi +2 a^3 (4 l^3+9 l \xi ^2)\) and \(B=a^3(a\xi )\sqrt{{\mathcal {G}}}\). For \(l>0\), the order of the solutions for the angular variables are, \(\theta _3>\pi>\theta _2>\theta _0(=\pi /2) >\theta _1\). Here, \(\theta _{i}=\cos ^{1}(\mu _i)\) with i running from 0 to 3. Thus depending on the value of the Carter constant, one can have different motion in the angular direction following Eq. (48). Note that for \(l=0\), one arrives at \(\alpha =\pi /6\) and hence one gets three solutions such as, \(\mu =0,\pm \sqrt{1(\xi /a)^2}\). This is consistent with the corresponding result for Kerr black hole. (Note that the angular motion is independent of the choice \(Q^{2}=l^{2}a^{2}\) and hence the results derived above will be applicable even in the general situation. This is why we have discussed the \(l=0\) limit in the context of angular motion.)$$\begin{aligned} \alpha =\dfrac{1}{3}\arctan \left( \dfrac{3 \sqrt{3}B}{A}\right) , \end{aligned}$$(54)
 On the other hand if we have, \({\mathcal {G}}<0\) there will be two solutions. One of them corresponds to the usual equatorial plane while the other one is at \(\mu =\mu ^{\prime }\) and given bywhere, A is already mentioned earlier and \(B^{\prime }\) is given as, \(B^{\prime }=a^3(a\xi )\sqrt{{\mathcal {G}}}\). Interestingly, for \(l=0\), \(\mu ^{\prime }\) become zero with the constraint \(\xi >a\). This matches with the results discussed earlier in the context of Kerr–Newman black hole.$$\begin{aligned} \mu ^{\prime }= & {} \dfrac{4l}{3a}+\dfrac{1}{3 a^2} \biggl \{(A+3\sqrt{3}B^{\prime })^{1/3}\nonumber \\&+\dfrac{a^2 (3 a^2+4 l^23 \xi ^2)}{(A+3\sqrt{3}B^{\prime })^{1/3}}\biggr \}, \end{aligned}$$(55)
 1.For \(\eta =0\): In the case of vanishing Carter constant, as evident from Eq. (48), the potential \(V_{\mathrm{ang,gen}}\) has to be negative. This results into the following behaviours:

For \({\mathcal {G}}>0\), the angular potential can be negative only if \(\theta _{1}<\theta <\theta _{0}\) or, \(\theta _{2}<\theta <\pi \) (see Fig. 8a). On the other hand with \({\mathcal {G}}<0\), one has \(\theta ^{\prime }<\theta <\theta _0(\pi /2)\), where \(\theta ^{\prime }=\cos ^{1}(\mu ^{\prime })\) (see Fig. 8b).

Unlike the angular motion in the context of Kerr–Newman black hole, \(V_{\mathrm{ang,gen}}\) has neither a maxima nor a minima on the equatorial plane located at \(\theta =\theta _{0}=\pi /2\). The motion is depicted in Fig. 8 for a particular set of parameters.

 2.For \(\eta >0\): In the case of positive Carter constant, the potential can take both positive and negative values. In the case of positive potential, motion along angular direction is possible only if numerical value of Carter constant is larger than the potential.

For positive values of \(V(\theta )\) with \({\mathcal {G}}>0\), the particle has to be within the angular range: \(\theta ^{}<\theta <\theta _{1}\) or, \(\theta _{0}<\theta <\theta _{2}\) (see Fig. 9a). While for \({\mathcal {G}}<0\), orbits with exist, provided the angular coordinate satisfy: \(\theta ^{}<\theta <\theta ^{\prime }\) and \(\theta _{0}<\theta <\theta ^{+}\) (see Fig. 9b). Here \(\theta ^{\pm }\) are two turning points in the presence of positive Carter constant \(\lambda \).

\(V(\theta )\) can also take negative values. In this case for \({\mathcal {G}}>0\), the only possibilities are: \(\theta _{1}<\theta <\theta _{0}\) and \(\theta _2<\theta <\pi \) (see Fig. 9a). Otherwise, with \({\mathcal {G}}<0\), one must have \(\theta ^{\prime }<\theta <\theta _{0}\) (see Fig. 9b).

 3.For \(\eta <0\): With negative Carter constant the potential can only be negative and also should have a magnitude larger than the Carter constant. This results into the following situation:

In this case, for \({\mathcal {G}}>0\), one has to ensure either \(\theta _{1}<\theta ^{}<\theta<\theta _{\mathrm{\pi /2}}^{}<\theta _0\) or \(\theta _2<\theta _{\mathrm{\pi /2}}^{+}<\theta <\pi \). Here \(\theta ^{\pm }\) and \(\theta _{\mathrm{\pi /2}}^{\pm }\) are turning points of the momentum along the angular direction (see Fig. 10a).

For \({\mathcal {G}}<0\), we need to have \(\theta ^{\prime }<\theta ^{}<\theta<\theta ^{+} <\theta _0(\pi /2)\). The corresponding situation is depicted in Fig. 10b.

3.2.2 Massive particles in Kerr–NewmanNUT black holes
 In the first case, where we have all the solutions for \(\theta \) originating from setting \(\Theta _{\mathrm{m}}=0\) in Eq. (59), it is necessary that the parameters associated with the black hole satisfies the following identity,Unlike the case for massless particles, in the present context the angular potential does not vanish at \(\theta =\pi /2\), as evident from Eq. (59). However the angular potential being a cubic expression in terms of \(\cos \theta \equiv \mu \), there will be at most three solutions for which the potential vanishes. The first solution takes the following form,$$\begin{aligned} {\mathcal {F}}&=\{16 a^4 l^296 a^3 l^2 \xi 3 (l^2\xi ^2)(3 l^2+\xi ^2)^2 \nonumber \\&+a^2 (72 l^4+180 l^2 \xi ^2+\xi ^4)\nonumber \\&+4 a (18 l^4 \xi 29 l^2 \xi ^3\xi ^5)\}>0 \end{aligned}$$(60)where \(\chi ^2\equiv 12 a^2 l^224 a l^2 \xi +(3 l^2+\xi ^2)^2\) and \(\beta \) is an angle taking values within the range \((0,2\pi )\) whose explicit expression can be given by,$$\begin{aligned} \mu _{1}=\dfrac{3 l^2+\xi ^2}{6 a l}+\dfrac{\chi }{3 a l}\cos \beta . \end{aligned}$$(61)Here \({\mathcal {D}}\) and \({\mathcal {C}}\) are mathematical quantities having the following expressions,$$\begin{aligned} \beta =\dfrac{1}{3}\arctan \Bigg (\dfrac{6\sqrt{3}{\mathcal {D}}}{{\mathcal {C}}}\Bigg ), \end{aligned}$$(62)The equation \(\Theta _{\mathrm{m}}=0\) is a cubic equation for \(\mu =\cos \theta \) and there should be three independent solutions at most. One of the solution is given above by \(\mu =\mu _{1}\), while the other two solutions take the following form,$$\begin{aligned} {\mathcal {C}}= & {} 27 l^627 l^4 \xi ^29 l^2 \xi ^4\xi ^6+36 a l^2 \xi (3 l^2+\xi ^2)\nonumber \\&18 a^2 l^2 (6 l^2+\xi ^2),~\text {and}~{\mathcal {D}}=a l^2 \sqrt{{\mathcal {F}}}. \end{aligned}$$(63)The parameters introduced above have their usual meanings.$$\begin{aligned} \mu _{2,3}=\dfrac{3 l^2+\xi ^2}{6 a l}\dfrac{\chi }{6 a l}\left( \cos \beta \pm \sqrt{3}\sin \beta \right) . \end{aligned}$$(64)
 The other possibility corresponds to \({\mathcal {F}}<0\). In this case two of the three solutions does not exist as they become complex. The only angular coordinate \(\theta '\), where the potential associated with angular motion of a geodesic vanishes corresponds to,The quantity \({\mathcal {C}}\) appearing in the above equation has already been defined while, \({\mathcal {D}}^{\prime }=a l^2 ({\mathcal {F}})^{1/2}\). Thus in this case there is only a single angular coordinate where the potential vanishes. This is certainly different from the corresponding situation in Kerr–Newman spacetime.$$\begin{aligned} \cos \theta '\equiv \mu ^{\prime }= & {} \dfrac{3 l^2+\xi ^2}{6 a l}\dfrac{1}{6 a l}\Big \{\Big ({\mathcal {C}}+6 \sqrt{3} {\mathcal {D}}^{\prime }\Big )^{1/3}\nonumber \\&+\dfrac{12 a^2 l^224 a l^2 \xi +(3 l^2+\xi ^2)^2}{({\mathcal {C}}+6 \sqrt{3}{\mathcal {D}}^{\prime })^{1/3}}\Big \} \end{aligned}$$(65)
4 Energy extraction from Kerr–NewmanNUT black hole
The origin of high energy particles in the universe is a long standing problem. Even though, there have been several explorations to model such phenomena [39, 40], it can be theoretically intriguing if it has its roots back to some exotic objects, such as black hole or neutron star. Historically, many high energetic events in the universe has their connections one way or another into black hole or stars, such as formation of jets from rotating objects as a result of gamma ray burst [41] or active galactic nuclei [42]. More recently, it has been proposed that black holes could also be used as a system to accelerate particles, giving rise to arbitrary large energy Debris [43, 44, 45], which in principle dictates a modified version of the Penrose process. This idea was originally suggested by Penrose and Floyed in the late seventies [26], concerning energy extraction from a black hole in the presence of a ergoregion. Since then, many investigations have been carried out in many aspects to examine the implications of Penrose process in various astrophysical domains [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56].
In the present context, we would reconsider the possibilities of energy extraction from a Kerr–NewmanNUT black hole constrained with \(\Delta =r^22Mr\) and \(Q^2=l^2a^2\). We start with the original Penrose process and study the bounds from Wald inequality. Afterward, we investigate the implication of collisional Penrose process followed by a survey of recent Bañados–Silk–West effect regarding the divergence of collisional energy in the center of mass frame. Finally, we will address the superradiance phenomenon in the Kerr–NewmanNUT black hole and discuss the advantages over other spacetimes such as Kerr or Kerr–Newman.
4.1 The original Penrose process
The numerical value of the minimum velocity \(\varvec{{\mathcal {V}}}_{\mathrm{min}}\) introduced above is of importance to the energy extraction process. In this table we have provided numerical estimates for the minimum velocity \(\varvec{{\mathcal {V}}}_{\mathrm{min}}\) with \(l=a\) and various choices of the ratio a / M. As evident for larger values of a / M, the minimum velocity decreases from being unity
Angular momentum (a / M)  \(\varvec{{\mathcal {V}}}_{\mathrm{min}}\) 

1.0  0.912871 
3.0  0.768706 
5.0  0.732828 
7.0  0.720838 
9.0  0.715575 
4.2 Bañados–Silk–West process
4.3 Superradiance in Kerr–NewmanNUT spacetime
Superradiance is another way of extracting energy from a rotating object originally proposed by Zel’dovich in the early seventies. He suggested that in a particular limit, the amplitude of the reflected wave scattered by a rotating object can be larger than the amplitude of the incident wave. However, this rotating object has to have a well defined boundary and Zel’dovich had conducted his experiment with a rotating cylinder [27]. Afterwards, the idea to include the model of a black hole spacetime to explain the superradiance was investigated in Refs. [61, 62, 63] and investigated by many others [64, 65, 66, 67]. In the present context, we use the Kerr–NewmanNUT spacetime with \(\Delta =r^22 M r\) and explore the possibilities of energy extraction via superradiance.
5 Astrophysical signatures of Kerr–NewmanNUT black hole: quasiperiodic oscillations
Having addressed most of the energy extraction processes associated with the Kerr–NewmanNUT black hole, for completeness let us also mention one astrophysical implication of the same. There exist several possibilities to be explored, including luminosity from a black hole in the Kerr–NewmanNUT family and ironline spectroscopy of the radiation originating from accretion disc around a Kerr–NewmanNUT black hole. However, in this work we will concentrate on the quasiperiodic oscillation from a accreting black hole which is described by the Kerr–NewmanNUT spacetime. Quasiperiodic oscillations (henceforth QPO) are related to very fast flux variability associated with matter accreting onto a black hole located very close to the innermost stable circular orbit. These QPOs are essentially believed to probe the geodesic motion of a particle in the strong field regime [68, 69, 70]. Till now, there are several black hole candidates, varying from stellar mass black hole to a supermassive one [71, 72] for which such QPOs were observed. The frequencies of these QPOs are related to the fundamental frequencies associated with the motion of accreting matter in the strong gravity regime [68, 73, 74] and may probe the presence of a NUT charge if the background is given by Kerr–NewmanNUT spacetime. Further using the relativistic precession model, one can indeed predict correct values of black hole hairs (namely mass and angular momentum in the context of Kerr black hole) starting from the observations of QPOs [72, 75]. Thus it is important to ask, whether it is possible to test the NUT hair as well using QPOs. In this section we will describe how the frequencies of QPOs depend on the NUT charge, while a numerical estimation by invoking real measurements will be done elsewhere [76].
6 Thermodynamics of Kerr–NewmanNUT black hole
7 Concluding remarks
The duality between gravitational mass and NUT charge makes the Kerr–NewmanNUT spacetime an interesting testbed to understand gravitational physics. It has been shown in [9, 10] that KerrNUT metric is invariant under the duality transformation: \(M \leftrightarrow il\), \(r \leftrightarrow i(l + a \cos \theta )\), exhibiting duality between gravoelectric (M) and gravomagnetic (l) charge [12], and correspondingly between radial and angular coordinates. In the Kerr–NewmanNUT metric, (M, Q) are gravoelectric charges (which are purely coloumbic in nature in the sense that they appear only in \(\Delta \)) while (l, a) are gravomagnetic, which in addition to \(\Delta \) (since energy in any form must gravitate) also appear in the metric defining the geometric symmetry of the spacetime.
In generic situations the Kerr–NewmanNUT spacetime inherits two horizons and a timelike singularity. However, for a particular choice of the charge parameter, namely, \(Q^{2}+a^{2}=l^{2}\), where l and a are the NUT charge and the black hole rotation parameter respectively, the horizon sits at \(r=2M\). This particular relation between the black hole hairs, namely the electric charge Q, NUT charge l and rotation parameter a ensures that the amount of repulsion offered by a and Q is being exactly balanced by the attraction due to the NUT charge and hence as a consequence the horizon is located at a position as if none of these hairs are present. Note that the horizon can appear at \(r=2M\) even for \(l=a, Q=0\), a particular case of the KerrNUT spacetime, where the two magnetic hairs l and a are equal. Interestingly, for \(l=a\) the curvature singularity is spacelike in stark contrast to the generic Kerr–NewmanNUT spacetimes. The radical alteration in casual structure due to the above choice is intriguing and has remained unnoticed in the literature. On the other hand, for \(l^2 > a^2\) it follows that \( r^2 + (l + a \cos \theta )^2 \ne 0\) for any real value of r and hence remarkably the solution presents a black hole solution free of any curvature singularity. The above prescription squarely balances coloumbic gravitational effects of charge, rotation and NUT parameter leaving mass alone to determine the horizon and the nature of curvature singularity. However l and a have not been fully eliminated as they also occur in the metric in their magnetic role as in \(\rho ^2\). It is their magnetic role which is very interesting and would require a separate detailed and deeper investigation. However the key fact derived here corresponds to the result that when NUT parameter is dominant over rotation, singularity is avoided. That means magnetic contribution of l and a dominates over coloumbic contribution due to mass. On the other hand, when \(l \le a\), mass dominates and a spacelike singularity arises. Since location of horizon is free of NUT, rotation and charge parameters, they could have any value, even greater than unity. For the generic Kerr–NewmanNUT family, one of the interesting extremal configuration could be equality of gravoelectric and gravomagnetic charges, i.e., \(M=Q, l=a\), separately. It should be interesting to study this kind of extremal Kerr–NewmanNUT black hole in an extensive scale.
Given all these distinguishing features associated with the above black hole spacetime, we have studied the trajectories of massive as well as massless particles in this spacetime. As we have explicitly demonstrated, in general circular geodesics cannot be confined to the equatorial plane. Following which we have determined the conditions on the angle \(\theta \) for which circular geodesics are possible and it turns out that nonzero Carter constant plays a very important role in this respect. Besides we have also discussed geodesic motion on nonequatorial planes to understand its physical properties in a more comprehensive manner. In particular, we have studied the photon circular orbits as well as circular orbits for massive particles on a fixed \(\theta =\text {constant}\) plane and have determined the possible conditions on the energy for their existence. The results obtained thereafter explicitly demonstrate the departure of the present context from the usual Kerr–NewmanNUT scenario, e.g., the innermost stable circular orbits are located at a completely different angular plane and also at different location compared to the corresponding radius in Schwarzschild spacetime. This may have interesting astrophysical implications, e.g., this will affect the structure of accretion disk around the black hole, which in turn will affect the observed luminosity from the accretion disk. Besides the above, we have also studied various energy extraction processes in this spacetime. It turns out that in both the Penrose process and superradiance the amount of energy extracted is less in comparison to the corresponding situation with Kerr black hole. On the other hand, in the black hole spacetime under consideration, the center of mass energy of a pair of colliding particles can be very large (the Banados–Silk–West effect) for a much wider class of angular momentum of the incoming particles. This is also in sharp contrast with the corresponding scenario for Kerr spacetime. In addition, the fundamental QPO frequencies for a geodesic trajectory orbiting in nearly circular orbits on the equatorial plane are also carried out in detail pointing out its astrophysical significance. It is also stressed that the domain of the black hole parameters l and a leads to a much large parameter space, which could be spanned to look for any signature of the NUT charge in astrophysical scenarios. This may provide a better scope for estimating the parameters in order to match with the observational data. Further we have also commented on the thermodynamical aspects, in which case unlike the general Kerr–NewmanNUT spacetime, the black hole temperature does not vanish for any parameter space of the NUT charge and the rotation parameter. We would like to emphasize that the results derived in this work are qualitatively different from the results presented in the earlier literature, see [32, 33, 34]. Since the case \(\ell =a\) with \(Q=0\) has not been studied extensively earlier in the literature, the results presented in this work has possibly shed some light into this parameter space of the Kerr–NewmanNUT solution and has filled a gap in the literature. However in certain arenas, e.g., study of photon circular orbits, one can use the results derived in [33] to immediately observe that our claim regarding nonequatorial motion are in direct consonance with earlier literatures. This provides yet another verification of our result presented in this work.
Finally, we would like to point out that for this particular relation among black hole hairs, namely, \(Q^{2}+a^{2}=l^{2}\), the ratio (a / M) is completely free and can even take values larger than unity, while at the same time if \(l>a\), the black hole will be free of any curvature singularity. Thus any observational evidence that indicates a possibility of having a superrotating black hole with \((a/M)>1\), need not necessarily be a signature of naked singularity but instead it could as well be a case of black hole having a NUT charge. This, as well as all other features mentioned above would indeed make a good case for studying the role of NUT parameter in high energy astrophysical setting and phenomena.
Footnotes
Notes
Acknowledgements
Research of S.C. is supported by the INSPIRE Faculty Fellowship (Reg. no. DST/INSPIRE/04/2018/000893) from Department of Science and Technology, Government of India. S.C. and N.D. acknowledge the warm hospitality provided by the Albert Einstein Institute, Golm, Germany. Finally, S.C. and S.M. would like to thank the InterUniversity Centre for Astronomy and Astrophysics (IUCAA), Pune, India where parts of this work were carried out during short visits.
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