Particle Acceleration on the Background of the Kerr-Taub-NUT Spacetime

We study the collision of two particles with the different rest masses moving in the equatorial plane of a Kerr-Taub-NUT spacetime and get the center-of-mass (CM) energy for the particles. We find that the CM energy depends not only on the rotation parameter, $a$, but also on the NUT charge of the Kerr-Taub-NUT spacetime, $n$. Especially, for the extremal Kerr-Taub-NUT spacetime, an unlimited CM energy can be approached if the parameter $a$ is in the range $[1,\sqrt{2}]$, which is different from that of the Kerr and Kerr-Newman black holes.


I. INTRODUCTION
Recently, Banados, Silk and West (BSW) [1] studied the collision of two particles near a rotating black hole and found that CM energy E cm of the two particles moving along the equatorial plane can be arbitrarily high in the limiting case of maximal black hole spin.
This means that the extremal rotating black hole could be regarded as a Planck-energy-scale collider, which might bring us some visible signals from ultra high energy collisions, such as, dark matter particles. Thus, the BSW mechanism about the collision of two particles near a rotating black hole has been attracted much attention in the recent years. In Refs. [2,3] Berti et al pointed out that the arbitrarily high CM energies E cm for a Kerr black hole might not be achievable in nature due to the astrophysical limitation, such as the maximal spin [4], gravitational radiation. Subsequently, Lake [5] investigated the CM energy of the collision occurring at the inner horizon of the non-extremal Kerr black hole and found the CM energy is limited [5]. Grib and Pavlov [6][7][8][9] argued that the CM energy E cm for two particles collision can be unlimited even in the non-maximal rotation if one considers the multiple scattering, and they also evaluated extraction of energy after the collision. The collision in the innermost stable circular orbit was considered in Ref. [10]. The similar BSW mechanism had also been found in other kinds of black holes, e.g. Stringy Black Hole [11], Kerr-Newman black holes [12] and Kaluza-Klein Black Hole [13]. In Refs. [14][15][16], the author elucidated the universal property of acceleration of particles for rotating black holes and try to give a general explanation of this BSW mechanism for the general rotating black holes. The BSW mechanism stimulated some implications concerning the effects of gravity generated by colliding particles in Ref. [17] and the emergent flux from particle collision near the Kerr black holes [18].
Another interesting stationary axisymmetric object is the Kerr-Taub-NUT (KTN) spacetime [19,20], which is an important solution of Einstein-Maxwell equations for electrovacuum spacetime possessing with gravitomagnetic monopole and dipole moments. Besides the mass M and the rotation parameter a, the KTN spacetime carries with the NUT charge, n, which plays the role of a magnetic mass inducing a topology in the Euclidean section. The presence of the NUT charge brings this spacetime some special spacetime structure, such as, Misner singularity. The KNT spacetime serves as an attractive example of spacetimes with asymptotic non-flat structure for exploring various physical phenomena in general relativity e.g. gravitomagnetism [21]. The KNT solutions representing relativistic thin disks [22] are of great astrophysical importance since they can be used as models for certain galaxies, accretion disks, and the superposition of a black holes and a galaxy or an accretion disk as in the case of quasars. The main purpose of this paper is to study the collision of two particles with the different rest masses in the background of the KTN spacetime and to see what effects of the NUT charge on the CM energy for the particles in the near-horizon collision. This paper is organized as follows. In Sec. II we derive briefly particle orbits in the KTN spacetime. In Sec. III, we study the collision of two particles with the different rest masses moving in the equatorial plane of the Kerr-Taub-NUT spacetime and discuss the center-of-mass (CM) energy for the particles. Sec.IV is devoted to a brief summary. We use the units c = G = 1 throughout the paper.

II. PARTICLE ORBITS IN KERR-TAUB-NUT SPACETIME
The metric of the KTN spacetime in the Boyer-Lindquist coordinates can be expressed as [19,20] with Σ = r 2 + (n + a cos θ) 2 , ∆ = r 2 − 2Mr − n 2 + a 2 , Ξ = a sin 2 θ − 2n cos θ, we have the following conserved quantities along a geodesic on the equatorial plane where u b is the four velocity defined by u b = dx b dτ , τ is the proper time for timelike geodesics. Moreover, we can introduce a new conserved parameter κ defined as (2.6) whose values are given by κ = −1, 0, 1 corresponding to timelike geodesics, null geodesics, and spacelike geodesics, respectively.
With the help of Eqs. (2.4), (2.5) and (2.6), we can obtain where the quantities E and L are the specific energy and angular momentum of the particle, respectively. And then the radial equation for the timelike particle moving along geodesics in the equatorial plane is described by with the effective potential (2.11) The circular orbit of the particle is defined by Since u t > 0, the condition 13) must be satisfied. As r → r H for the timelike particle, this condition reduce to In this section, we will study the CM energy for the collision of two particles moving in the equatorial plane of the KTN spacetime. Let us now consider two colliding particles with rest masses m 1 and m 2 . We assume that two particles 1 and 2 are at the same spacetime point with the four momenta where p a i and u a i are the four momentum and the four velocity of particle i (i = 1, 2). The sum of the above two momenta is given by Then the CM energy E cm of the two particles is given by which can be rewritten as In the case of m 1 = m 2 , the above the equation reduce the result in Refs. [1], [8].
where E i and L i are the specific energy E and the angular momentum L for particle i.
Obviously, the result confirms that the NUT charge n indeed has influence on the CM energy.

A. Near-horizon collision in non-extremal KTN spacetime
We are now in the position to study the properties of the CM energy vanishes at r H . Using l'Hospital's rule and taking into account r 2 H − 2r H − n 2 + a 2 = 0, the value of E 2 cm at r H is given by By implementing the calculation and taking the limit, we reach This is the formula for the CM energy of two particles along the general geodesic orbits on the outer horizon. The critical angular momenta L Hi (i = 1, 2) can be written as In fact, as we will prove in section III B, Eq. (3.9) is valid even in the extremal KTN spacetime simply by taking the near-extremal limit a → √ 1 + n 2 . The necessary condition for obtaining an arbitrarily high E cm is therefore L ≈ L H or Ω H L ≈ E for either of the two particles. For E 1 = E 2 = E, we denote L H1 = L H2 = L H = 2(n 2 +r H ) a and Eq. (3.9) reduces to When NUT charge varnish (n = 0), the above equation reduces to the result in Ref. [9] for the kerr black hole. As mention in Ref. [12], to obtain an arbitrarily high CM energy E cm , one of the colliding particles should have critical angular momentum L H . We assume that (3.11) Now we will discuss how the rotating parameter a, and the NUT charge n affect the CM energy E cm . We denote the small parameter ξ = a max − a with a max = √ 1 + n 2 . For fixed  Table  I. From the table, we can see that, for the KTN spacetime with rotating parameter a less than a max there will be an upper bound for the CM energy. It is also suggested that the CM energy grows very slowly as the maximally spinning case (ξ → 0) is approached. For fixed parameter ξ, the value of CM energy decreases with the increase of the charge n. For the case n = 0, it reduces to result for the Kerr black hole and we recover the numerical result in Ref. [3].

B. Near-horizon collision in the extremal KTN spacetime
For the extremal KTN spacetime, the rotating parameter a and NUT charge n satisfy the relation n 2 = a 2 − 1, the numerator A(r) − B(r) and the denominator C(r) both must have a second-order zero at r = r H . Using l'Hospital's rule twice, we obtain where the second-order derivatives are given by C ′′ (r)| r=r H = 2(n 2 + r H (5r H + 4)) After tedious calculation, we obtain the CM energy at the outer horizon for the extremal KTN spacetime For the case E 1 = E 2 = 1, Eq. (3.13) reduces to where the critical angular momentumĽ H = 2a. For the special case m 1 = m 2 and n = 0, the above equation recovers the result obtained by BSW [1]. The expression (3.14) shows that the unlimited CM energy can be approached if one of the colliding particles has critical angular momentumĽ H which ensures the particle can reach the outer horizon. Since the particles move along the stable circular orbits in the equatorial plane, there must exist a restriction for the angular momentum, which is also shown in Table II. With the help the effective potential (2.11), we can determine the range of the rotating parameter a for the extremal KTN spacetime. The effective potential for a particle with critical angular momentumĽ H is As expected, the effective potential V eff(L→Ľ H ) approaches 0 at spatial infinity. Obviously, the condition for the particle falling freely from rest at infinity to the horizon can be expressed as V eff(L→Ľ H ) ≤ 0 for any r ≥ 1. (3.16) Combing with the condition 1 + n 2 ≥ a 2 , we can solve Eq. (3.15) and obtain the range for the parameters a and n 1 ≤ a ≤ √ 2, |n| ≤ 1, (3.17) which means that for the KTN extremal spacetime with a ∈ [1, √ 2] and the NUT charge |n| ≤ 1, the particle with critical angular momentumĽ H can reach the horizon. Thus, for the fixed a ∈ [1, √ 2], one can find that the CM energy will be unlimited if L 1 =Ľ H and L 2 is in a proper range. In Fig.1, we plot the effective potential V eff(L→Ľ H ) and CM energy E cm of collision for the different values of a and n. From Fig. 1 (a), one can find that for 1 ≤ a ≤ √ 2 the effective potential V eff(L→Ľ H ) is negative when r ≥ r H = 1 so that the particle can reach the horizon. However, the effective potential V eff(L→Ľ H ) for a = 1.8 is positive near the horizon, which implies that the particle can not reach the horizon in this case. From Fig.1 (b), we also find that for the case 1 ≤ a ≤ √ 2 the CM energy at the horizon is be unlimited, which can be explained by a fact that the effective potential V eff(L→Ľ H ) is negative for the two colliding particles with angular momenta L 1 =Ľ H and For the case a = 1.8, it is obvious that the CM energy is limited and the particle cannot reach the horizon. Moreover, we also find that with the increase of the NUT charge , we obtain that the maximal CM energy for the collision particles can be approximated as  in the near-extremal KTN spacetime. This means that the maximal CM energy of the collision particles increases proportionally to a 3/4 . We assume that the rest masses of the colliding particles m 1 , m 2 are of about 1 GeV, just like the mass of a neutron. In order to obtain the Planck-scale energy E cm ∼ 10 19 GeV, we need δξ ∼ 10 −76 , which is similar to that in the Kerr-Newman black hole [12]. This implies that it is very hard for a near-extremal case to be a particle accelerator of Planck-scale energy.

IV. SUMMARY
In this paper, we studied the collision of two particles with the different rest masses moving in the equatorial plane of the KTN spacetime and get the center-of-mass (CM) energy for the particles. Our result shows that the CM energy depends not only on the rotation parameter of the KTN spacetime, a, but also on the NUT charge of the KTN spacetime, n. For the extremal KTN spacetime, the presence of the NUT charge modified the restrict conditions for the spin a when arbitrarily high CM energy appears, i.e., 1 ≤ a ≤ √ 2, which is a significant difference from the Kerr [1] and Kerr-Newman [12] black holes. For the nearextremal case, we also found that the CM energy E cm decreases with the increase of the NUT charge n when one particle has the maximum angular momentum L max and the other has the minimum angular momentum L min . We also estimated the maximal value of the CM energy of particles for both the extremal and the near-extremal KTN spacetime when one particle has the critical angular momentum and the other has zero angular momentum and discussed the change of the maximal CM energy with the parameters a and n.