Center of Mass Energy of the Collision for Two General Geodesic Particles Around a Kerr-Newman-Taub-NUT Black Hole

In this paper, we investigate the center of mass energy of the collision for two neutral particles with different rest masses falling freely from rest at infinity in the background of a Kerr-Newman-Taub-NUT black hole. Further, we discuss the center of mass energy near the horizon(s) of an extremal and non-extremal Kerr-Newman-Taub-NUT black hole and show that an arbitrarily high center of mass energy is achievable under some restrictions. We will study the special case of the center of mass energy when the specific energy, specific angular momentum and Carter constant of both the particles are same.


I. INTRODUCTION
Banados, Silk and West (BSW) [1] studied the collision for two particles around a Kerr black hole and determined the center of mass energy (CME) in the equatorial plane. Subsequently, in Refs. [2,3], the authors further elucidated the BSW mechanism. They pointed out that the arbitrarily high CME might not be achievable in nature due to the astrophysical limitations i.e., the maximal spin and gravitational radiation. Lake [4,5] showed that the CME for two colliding particles is divergent at the inner horizon of a non-extremal Kerr black hole. Grib and Pavlov [6]- [8] showed that the scattering energy of particles in the centre of mass frame can be obtained very large values for an extremal and non-extremal Kerr black hole. The collision in the innermost stable circular orbit for a Kerr black hole was discussed in Ref. [9]. In Ref. [10], the author considered the collision for two neutral particles within the context of the near-horizon extremal Kerr black hole and demonstrated that the CME is finite for any admissible value of the particle parameters. In Ref. [11], the authors showed that the particle acceleration to arbitrary high energy is one of the universal basic properties of an extremal Kerr black hole not only in astrophysics but also in more general context. An explicit expression of the CME for two colliding general geodesic massive and massless particles at any spacetime point around a Kerr black hole was obtained in Ref [12]. They found that, in the direct collision scenario, the collision with an arbitrarily high CME can occur near the horizon of an extremal Kerr black hole not only at the equator but also on a belt centered at the equator. This belt lies between latitudes ±a cos( √ 3 − 1) ≃ ±42.94 • . In Ref. [13], the author argued the possibility of having infinite CME in the centre of mass frame of colliding particles is a generic feature of a Kerr black hole.
In Ref. [14], the authors investigated the CME in the background of a Kerr-Newman black hole. They pointed out that the unlimited CME requires three conditions: (1) the collision takes place at the horizon of an extremal black hole, (2) one of the colliding particles has critical angular momentum, and (3) the spin parameter a of an extremal black hole needs to satisfy 1 √ 3 ≤ a ≤ 1. In Ref. [15], the author studied the collision of two general geodesic particles around a Kerr-Newman black hole and get the CME of the non-marginally and marginally bound critical particles. The collision for a freely falling neutral particle with a charged particle revolving at the circular orbit near a Schwarzschild black hole was considered in Ref. [16]. In Ref. [17], the authors studied the collision for two particles with different rest masses moving in the equatorial plane of a Kerr-Taub-NUT black hole. They demonstrated that the CME depends on the spin parameter a and NUT (Newman-Unti-Tamburino) charge n.
In Ref. [18], the authors discussed the collision for two particles in the background of a stringy black hole. They found that the CME is arbitrarily high under two conditions: (1) the spin parameter a = 0, and (2) one of the colliding particles should have critical angular momentum.
The collision for two particles in the background of a charged black string was discussed in Ref. [19]. The author showed that the CME of the collision is arbitrarily high at the outer horizon if one of the colliding particles has critical charge. The particle acceleration mechanism in S 2 × R 1 topology, namely, in the spacetime of the five-dimensional compact black string, has been studied in Ref. [20]. They obtained that the scattering energy of particles in the center of mass frame can take arbitrarily large values not only for an extremal black string but also for a non-extremal black string. The CME for two colliding particles in the absence and presence of a magnetic field around a Schwarzschild-like black hole was investigated in Ref. [21]. In Ref. [22], the authors discussed the CME for two colliding neutral particles at the horizon of a slowly rotating black hole in the Horava-Lifshitz theory of gravity and 3+1 dimensional topological Lifshitz black hole remains finite. The collision for test charged particles in the vicinity of the event horizon of a weakly magnetized non-rotating black hole with gravitomagnetic charge has been studied in Ref. [23]. In Ref. [24], the author argued that the BSW effect exists for a non-rotating but charged black hole even for the simplest case of radial motion of particles in a Reissner-Nordström black hole.
In Ref. [25], the author gave simple and general explanation to the effect of unbound acceleration of particles for Reissner-Nordström and Kerr black holes. The CME of the collision for charged particles in a Bardeen black hole was studied in Ref. [26]. In Ref. [27], the authors investigated the CME near the horizon of a non-extremal Plebanski and Demianski black hole with zero NUT parameter.
A non-vacuum solution of the Einstein field equations is a Kerr-Newman- Taub-NUT (KNTN) black hole, which besides the spin parameter a and electric charge Q carries the NUT charge n, which plays the role of a magnetic mass. In this paper, we will study the four equations of motion for a neutral particle in the background of a KNTN black hole. We will discuss the detailed behavior of the CME for two neutral particles with different rest masses m 1 and m 2 falling freely from rest at infinity in the background of a KNTN black hole. We will find the CME when the collision occurs at some radial coordinate r and angle θ. We will show that the CME of the collision near the horizon(s) of an extremal and non-extremal KNTN black hole is arbitrarily high when the specific angular momentum of the any particle is equal to the critical angular momentum and the spin parameter a does not vanish. We will also show that the CME is fixed when the specific energy, specific angular momentum and Carter constant of both the particles are same.
The paper is organized as follows. In Sec. II, we will discuss the equations of motion for a neutral particle in the background of a KNTN black hole. In Sec. III, we will obtain the CME of the collision for two neutral particles and discuss the properties. In Sec. IV, we will give a brief conclusion. We use the system of units c = G = 1 throughtout this paper.

II. EQUATIONS OF MOTION IN THE BACKGROUND OF A KERR-NEWMAN-TAUB-NUT BLACK HOLE
In this section, we will study the equations of motion for a neutral particle in the background of a KNTN black hole. Let us first give a brief review of a KNTN black hole. The KNTN black hole is a geometrically stationary and axisymmetric non-vacuum object, which is an important solution of the Einstein field equations. The KNTN black hole is determined by the following parameters i.e., the mass M , spin parameter a, NUT parameter n and electric charge Q. The KNTN black hole can be described by the metric in the Boyer-Lindquist coordinates (t, r, θ, φ) as in [28] 1 , [29] and [30] where Σ, ∆ and χ are respectively defined by Σ = r 2 + (n + a cos θ) 2 , χ = a sin 2 θ − 2n cos θ.
The KNTN metric contains the following metrics as special cases: Kerr- Taub Now, let us discuss the equations of motion for a neutral particle of mass m in the background of a KNTN black hole. The motion of the particle can be determined by the Lagrangian where a overdot denotes an ordinary differentiation with respect to an affine parameter λ related to the proper time τ by τ = mλ. The normalization condition is 1 m 2 g µνẋ µẋν = κ, where κ = −1 for timelike geodesics, κ = 0 for null geodesics and κ = 1 for spacelike geodesics. For the massive particle, we have κ = −1. The 4-momentum of the particle is which is related to the 4-velocity by where u ν = dx ν dτ , τ is the proper time for timelike geodesics. By Eq. (4), we can expressẋ µ in terms of the 4-momentum asẋ µ = g µν P ν . The Hamiltonian is given by which satisfies the two Hamilton equationṡ By the definition of the Hamilton-Jacobi methods, the Hamilton-Jacobi equation is given by where S is the Jacobi action as a function of the affine parameter λ and coordinates x µ and The Hamilton-Jacobi equation allows separation of variables in the form where E and L are respectively the energy and angular momentum of the particle, S r and S θ are arbitrary functions of r and θ, respectively. Here, Solving Eqs. (11) and (12), we obtain By Eqs. (8) and (10), we obtain The left-hand side of Eq. (15) does not depend on r while the right-hand side does not depend on θ, hence each side must be a constant. This constant is termed as the Carter constant and it is a conserved quantity denoted by K. Therefore Using the relations u r = 1 m ∂S r ∂r and u θ = 1 m ∂S θ ∂θ , the remaining 4-velocity components are with The ± signs are independent from each other, but one must be consistent in that choice. The +(−) sign corresponds to the outgoing(ingoing) geodesics. Clearly, the Carter constant K vanishes for the equations of motion in the equatorial plane θ = π 2 . The radial equation of motion for the neutral particle moving along timelike geodesics is given by with the effective potential From Eqs. (19) and (22), we conclude that 2Σ 2 . Note that from Eqs. (18) and (19), for the allowed motion Θ ≥ 0 and R ≥ 0 must be satisfied. Hence, the allowed and prohibited regions for the effective potential are given by In the equatorial plane, the effective potential is given by The function R(r) can also be written in the form The leading term of the highest power of r on the right-hand side is positive if E > 1. Only in this case, the motion can be unbounded (infinite). For E < 1, the motion is bounded (finite) i.e., the particle cannot reach the horizon(s) of the black hole. For E = 1, the motion is marginally bounded i.e., the motion is either finite or infinite. In this case, the particle's motion depends on the black hole parameters and specific angular momentum for the allowed and prohibited regions of R(r) and Θ(θ) but the motion can be fully analysed by R(r) or V eff r, π 2 in the equatorial plane. The particle whose motion is bounded, unbounded and marginally bounded are called bound, unbound and marginally bound particle, respectively. For bound and marginally bound particles, we have V eff (r, θ) < 0 and V eff (r, θ) ≤ 0, respectively.
We need to impose the condition u t > 0 along the geodesic. This is called the"forward-in-time" condition which shows that the time coordinate t increases along the trajectory of the particle's motion. By Eq. (13), this condition reduces to For r → r + , Eq. (26) implies Here, we get the upper bound of the specific angular momentum at the outer horizon of the nonextremal KNTN black hole which is called the critical angular momentum and is denoted byL + i.e.,L Similarly, the critical angular momentum at the inner horizon of the non-extremal KNTN black hole is given byL For the extremal KNTN black hole, we use r + = M in Eq. (28), which gives the critical angular momentum at the horizon of the extremal KNTN black holê For a = 0, Eqs. (28), (29) and (30) become ill-defined, so we will assume a = 0 throughout our work.

III. CENTER OF MASS ENERGY FOR TWO NEUTRAL PARTICLES
In this section, we will study the CME of the collision for two neutral particles with different rest masses falling freely from rest at infinity towards a KNTN black hole. Let us consider that two neutral particles collide at some radial coordinate r which are not restricted in the equatorial plane. The 4-momentum of the ith particle is given by where i = 1, 2 and P µ i , u µ i and m i are respectively the 4-momentum, 4-velocity and rest mass (mass at rest at infinity) of the ith particle. The total 4-momentum of the two particles is Since the 4-momentum has zero spatial components in the center of mass frame, therefore the CME for the two particles is Simplifying and using u µ (i) u (i)µ = −1 in Eq. (33), we obtain For the KNTN metric (1), using Eqs. (13), (14), (18) and (19) into Eq. (34), we get the CME of the collision where F (r, θ), G(r, θ), H(r, θ) and I(r, θ) are given by F (r, θ) = ∆Σ sin 2 θ − (∆ − a 2 sin 2 θ)L 1 L 2 + (Σ + aχ) 2 sin 2 θ − χ 2 ∆ E 1 E 2 + χ∆ − a(Σ + aχ) sin 2 θ L 1 E 2 + L 2 E 1 , G(r, θ) = sin 2 θ R 1 (r)R 2 (r), Here, E i , L i and K i are respectively the specific energy, specific angular momentum and Carter constant of the ith particle. Clearly, the CME (35) is invariant under the interchange of the quantities L 1 ↔ L 2 , E 1 ↔ E 2 and m 1 ↔ m 2 . As a special case, the CME for the two particles having the same value of the specific energy E, specific angular momentum L and Carter constant K is found to be This result shows that the radial coordinate r of the black hole has no influence on the CME. Thus the CME is finite at the horizon(s) of the extremal and non-extremal KNTN black hole. Since the result is also independent of θ, therefore the CME is also finite on the curvature singularity of the KNTN black hole.
A. Near-horizon collision of particles around the non-extremal KNTN black hole Let us discuss the properties of the CME (35) as the particles approach the horizons r + and r − of the non-extremal KNTN black hole.
The conditions for the allowed region, R(r) ≥ 0 and Θ(θ) ≥ 0 give the upper and lower bounds for the Carter constant K given below where

Equatorial plane and collision
If one chooses θ = π 2 , the CME (40) at the outer horizon of the non-extremal KNTN black hole reduces to which is indeed finite for all values of L 1 and L 2 except when L 1 or L 2 is approximately equal to the critical angular momentumL +i , for which the neutral particles collide with an arbitrarily high CME. In the case of the same specific energies, the form of the CME (46) at r + reads In Figure 1, we plot the effective potential V eff r, π 2 of marginally bound particles for M = 1, a = 0.8, n = 0.4, Q = 0.7211 with different specific angular momenta L = −2, − 1, 0, 1, 2.25964 where 2.25964 is the critical angular momentumL + . Clearly, the effective potential V eff r, π 2 is negative corresponds to different L when r ≥ r + , therefore the particles can reach the outer horizon. Vertical lines in the subplot represent the locations of the outer and inner horizons. We also plot the CME of the collision for L 1 = −2, − 1, 0, 1 and L 2 =L + . Clearly, the CME blows up at the outer horizon r + = 1.00385.

Collision at the inner horizon
Similarly the terms F (r, θ) − G(r, θ) − H(r, θ) and I(r, θ) of right-hand side of Eq. (35) also vanish at r − . Using L'Hospital's rule and by simplifying the calculation, we get the CME for the two neutral particles at the inner horizon This is the CME formula for the two neutral particles, whereL −i is the critical angular momentum at the inner horizon, which can be written asL . An arbitrary high CME can be obtained by using the condition L i ≈L −i for either of the two particles. The critical angular momentum is same when both particles have the same specific energy and is given bŷ , while the CME (48) reduces to − 1 2 r 2 − + (n + a cos θ) 2 sin 2 θ cos 2 θL 1 L 2 + 2n cos θE(L 1 + L 2 ) + (a sin 2 θ − 2n cos θ) 2 − a 2 sin 2 θ E 2 + sin 2 θ Θ 1 (θ)Θ 2 (θ) Let us consider a marginally bound particle (E = 1) with the critical angular momentumL − .
The conditions for the allowed region, R(r) ≥ 0 and Θ(θ) ≥ 0 give where K (2) min and K (2) max are given by

Equatorial plane and collision
In the equatorial plane, Eq. (48) at the inner horizon takes the following form Clearly, the CME is finite for all values of L 1 and L 2 except when L 1 or L 2 is approximately equal to the critical angular momentum. For E 1 = E 2 = E, Eq. (54) gives We plot the effective potential V eff r, π 2 of marginally bound particles in Figure 2 for M = 2, a = 0.6, n = 0.1, Q = 0.6 with different specific angular momenta L = −2, − 1, 0, 1,L − wherê L − = 2.83264. Clearly, the effective potential V eff r, π 2 is negative corresponds to different L for all r ≥ r − , so the particles can reach the inner horizon after crossing outer horizon. The subplot shows the behaviour of V eff r, π 2 near the horizons and identify the location of the outer and inner horizons. We also plot the CME of the collision for L 1 = −2, − 1, 0, 1 and L 2 =L − . The CME is finite at the outer horizon and blows up at the inner horizon r − = 1.35969.

B. Near-horizon collision of particles around the extremal KNTN black hole
Let us study the properties of the CME (35) as the particles approach the horizon of the extremal KNTN black hole. In the case of the extremal KNTN black hole, the NUT charge n, This is the CME for the two neutral particles at the horizon of the extremal KNTN black hole.
The critical angular momentum at the horizon is given byL i = E i (2a 2 +Q 2 ) a , for the ith particle.

Equatorial plane and collision
Further, if the collision occurs in the equatorial plane, the CME (56) at the horizon of the extremal KNTN black hole reduces to which is indeed finite for all values of L 1 and L 2 except when L 1 or L 2 approaches the critical angular momentum, for which the CME is arbitrarily high. When the electric charge Q vanishes, Eq. (62) gives the result for the extremal Kerr Taub NUT black hole as obtained in Ref. [17].
When the specific energy of both the particles are exactly alike, then the CME (62) becomes There must exist intervals for the spin parameter a, NUT charge n and electric charge Q to ensure that the marginally bound particles with the critical angular momentumL reach the horizon of the extremal KNTN black hole and the collision exists at the horizon. Since the motion of the particle in the equatorial plane can be fully analysed by the effective potential V eff r, π 2 , so with the help of the effective potential given in Eq. (24), we can determine intervals of a and n corresponding to different values of Q. The effective potential for the marginally bound particle with the critical angular momentumL is given by Here, the condition for the particle falling freely from rest at infinity to reach the horizon can be expressed as which is equivalent to Combining with the condition 0 ≤ n 2 = a 2 + Q 2 − M 2 and set M = 1, we get intervals for a and n for different values of Q as shown in Table I. Note that, there are two different intervals for the spin parameter a corresponding to the co-rotating and counter-rotating orbits. For Q = 0, we get intervals for a and n as discussed earlier in Ref. [17]. With the increase of Q, the intervals for a and n become narrow.
The maximum and minimum value of L can be obtained by the conditions Then the interval L ∈ [L min , L max ] can be determined from it. The interval for the specific angular momentum for different values of a and Q are shown in Table II. Note that, with the increase of a and Q, the interval L ∈ [L min , L max ] becomes wider.  We plot the effective potential V eff r, π 2 in Figure 3 for L = −1 and L =L in the top and bottom, respectively. Clearly, for case (III) Q = 0.3, a = 1.6, the effective potential V eff r, π 2 is nonpositive for L = −1 but positive near the horizon r + = r − = 1 for L =L, so the particle cannot reach the horizon in this case for L =L. For cases (I) Q = 0.1, a = 1.2, (II) Q = 0.2, a = 1.3, (IV) Q = 0.4, a = 1.06 and (V) Q = 0, a = √ 2, V eff r, π 2 ≤ 0 when r ≥ M = 1 for the both specific angular momenta. So, the particle can reach the horizon in all the four cases for L = −1 and L =L. We also plot the CME of the collision in Figure 4 for L 1 = −1 and L 2 =L. For the case (III) Q = 0.3, a = 1.6, a does not belong to 0.95394, 1.36485 , the CME only exists for r ≥ 1.69668. This is because the collision for the two marginally bound particle with L 1 = −1 and L 2 =L cannot take place at r < 1.69668. For the case (I), (II), (IV) and (V), the CME is divergent at the horizon r + = r − = 1.

IV. CONCLUSION
In this paper, we have studied the CME of the collision for two neutral particles with different rest masses falling freely from rest at infinity in the background of a KNTN background. Further, we have discussed the CME when the collision takes place near the horizon(s) of an extremal and non-extremal KNTN black hole. We have found that an arbitrarily high CME is achievable with following conditions: (1) the collision occurs at the horizon(s) of an extremal and non-extremal KNTN black hole, (2) one of the colliding particles has critical angular momentum, and (3) the spin parameter a = 0. We have also found a new case when the CME is independent of radial coordinate r and angle θ, it happens when the specific energy, specific angular momentum and Carter constant of both the particles are same i.e, E cm = m 1 + m 2 for E 1 = E 2 , L 1 = L 2 and K 1 = K 2 . We discovered the upper and lower bounds of the Carter constant K for a marginally bound particle with the critical angular momentum in an extremal and non-extremal KNTN black hole. In the equatorial plane, we discovered that there exists intervals for the spin parameter a, NUT charge n and specific angular momentum L correspond to the electric charge Q for which not only two marginally bound particles reach the horizon of the extremal KNTN black hole but also the collision of these particles happens at the horizon.