Maximal efficiency of the collisional Penrose process with spinning particles in Kerr-Sen black hole

We study the collision of two uncharged spinning particles around an extreme Kerr-Sen black hole and calculate the maximal efficiency of the energy extraction from the Kerr-Sen black hole via super Penrose process. We consider the collision of two massive particles as well as collision of a massless particle with a massive particle. For all the cases, we find that the maximum efficiency decreases as the Kerr-Sen black hole's parameter($b=1-a$) increases.


A. Equations of motion of a spinning particle
The equations of motion for spin particle in the curved spacetime can be described by Mathission-Papapetrou-Dixon(MPD) equations [23][24][25] is the tangent vector of the center-of-mass world line, D Dτ is the covariant derivative along worldline, and p a = mu a is the canonical 4-momentum of the spinning particles which satisfy p a p a = −m 2 .
(2.4) Moreover S ab is the particle's antisymmetric spin tensor, and its square turns out to be the spin of the particle as follows, S ab S ab = 2S 2 = 2m 2 s 2 (2. 5) where s and m are the spin and mass of the given particle respectively. In the following, for the convenient of the calculation, we add a supplementary conditions between S ab and P a as follows S ab p a = 0 (2.6) Furthermore, we also normalize the affine paramenter τ through u a v a = −1 (2.7) A detailed calculation shows a relation between v a and u a as v a − u a = S ab R bcde u c S de 2(m 2 + 1 4 R bcde S bc S de ) The Eq. (2.8) means that the 4-velocity and 4-momentum are not always parallel. In addition, we can obtain the conserved quantities for spin particles with Killing vector fields ξ a as follows: Q ξ = p a ξ a + 1 2 S ab ∇ a ξ b (2.9)

C. Equations of motion on the equatorial plane
When the particle's spin is aligned with the spin of the black hole, the spin s (a) can be show as follow: Furthermore, we consider that the particle was confined in the equatorial plane(θ = π/2) [31]. The non-zero components of spin tensor read Combining Eqs (2.14), (2.15), and (2.18), the equations of momentum can be written as there are a normalization condition of the 4-momentum as [7] p (a) p (a) = k (2.22) where k = −m 2 for the massive and k = 0 for massless particles. As for massive particles, we defined a specific 4-momentum u (a) , by u (a) = p (a) /m. Hence, with Eq.(2.22) in hand, for the massive particles, we have Here σ = ±1 denote the outgoing and ingoing motions respectively. Moreover, combining Eqs.(2.8), (2.18) and (2.10), the expressions of the 4-velocity read (2.28) By employing the tetrad basis (2.12), 4-velocity can be rewritten as By plugging (2.25) to Eq.(2.31), the radial equation of motion for spin particle gives rise to In order to facilitate the numerical calculation and without loss generality, we simply set the variables to the dimensionless variables as This is equivalent to discuss the energy and other quantity with unity mass. In the following, we omit the for simplicity. For example, E in the following text actually meansẼ.

D. Constraints on the orbits
In this part, we devoted to find the admissible trajectory of the spin particle which can apporoch to the horizon Hence the condition that the orbit can reach the horizon equal to B r ≤ B cr . On the other hand, we know that for a massive particle, the 4-velocity along the admissible trajectory must be timelike as Along the same line of [10], the above timelike condition is equivalent to the following constraint where C a = r(2b + r) 3 − s 2 (b + r), F 1 is at Eq.(2.28) , and the detailed expression for U can be found in APPENDIX.
Since we consider the maximal energy contraction efficiency from black hole, in the following we only focus on the extreme Kerr-Sen black hole(b = 1 − a). In this situation, if one of the collision particle possess the critical angular momentum, it is easy to see that B cr = 2 from Eq.(2.37). Then from Eq. (2.39), we have which gives us a constraint on energy E for different values of spin s and b and is showed in the Fig. 1(a). The figure shows that when b increase, the admissible range of spin s shrinks for a given energy E. If the particle falling from infinity, that is E ≥ 1, combining this fact with Eq. (2.40), the spin s will be restricted to s min < s < s max for a given value of b. For example, when b = 0.1, we can obtain s min ≈ −0.285 and s max ≈ 0.471. More information of s min and s max for different value of b can be found in Fig. 2. Moreover, it worth to note that, the authors of Ref [28] point out that when the particle process critical , the timelike condition is violated. We show in Fig. 2, our admissible spin s corresponding to the maximum of efficiency is always bigger than critical value (s > −s c ), and therefore the timelike condition is satisfied in our case.
If the particle's angular momentum is deviate from critical value, we set B r = 2(1 + ξ) with ξ being a negative number. From Eq.(2.39), the energy E now is a function of the s, b and ξ and is showed as the Fig. 1(b). This figure shows that the allowed range of ξ increase when b increases.
We assume the particles are freely falling from infinity. If B r > B rc , such a particle falling from infinity will find a turning point away from horizon, and then bounce back to infinity. So if B r = B rc + δ(δ → 0 + ), the turning point of the particle can very close to the horizon. Then, these particles will moving outward. Therefore this situation should also need to be taken into account.

III. COLLISION OF SPINNING PARTICLES
In this section, we consider the collision of two spin particles that are freely falling from infinity, and find the formula of the efficiency of the energy extraction from the extreme Kerr-Sen black hole.
We denotes that the 4-momentum of particle 1 and particle 2 are p 1 µ and p 2 µ . Our picture is the following: The particles collide outside the horizon. After collision, particle 3, whose 4-momenta is p 3 µ , will move to infinity, while the particle 4 with p 4 µ falls into the Kerr-Sen black hole. We assume that the sum of initial spins and 4-momenta are conserved throughout the collision process. That is, Since the Kerr-Sen spacetime exists two Killing vectors, contracting these two Killing vector with the above equation gives the conservation of the energy E and angular momentum J as follows From the Eq. (3.1) and (3.2), we can also obtain the conservation of particle's spin and the radial components of 4-momentum throughout the collision Now we assume that particle 1 and particle 2 collide near the horizon of extreme Kerr-Sen black hole, the radial position of collision point r c is very close to extreme Kerr-Sen black hole's horizon r H (r H = a = 1 − b), so that we can assume (r c = a/(1 − ǫ)) with ǫ → 0 + . Then, we expand the particles' radial 4-momentum in terms of ǫ as follows: In the following analysis, without loss of generality, along the same line of [10], we doing calculation in case that particle 1 is critical (J 1 = 2E 1 ), while particle 3 is near-critical(J 3 = 2E 3 + O(ǫ)) and particle 2 is non-critical(J 2 < 2E 2 ) [10].
Then the total angular momentum of the particle can relate to the energy as follows: where α 3 and β 3 are expansion parameters of O (ǫ 0 ). For particle 2, since it is non-critical, we assume that: where ξ < 0 and ξ = O(ǫ 0 ) From the conservation law (3.3) and (3.4), we get the following equations which give us: Since we consider the collision of the particle 1 and particle 2, the particle 2 must be ingoing (σ 2 = −1) because the particle 2 is noncritical [10]. Combine Eq.(3.7) with the conservation of 4-momentum (3.6), we can get the equation as follows: , we find that σ 4 = σ 2 and s 4 = s 2 . Then Eq.(3.5) further forces us to impose s 3 = s 1 .
In the following section, we will consider three different types of collision. The first case is the collision of two massive particles(MMM). The second type is the collision of one massless particle with another massive particle, which is called as compton scattering(PMP) [10] and third type is the inverse compton scattering(MPM) [10], which is the inverse process of type two case. Now, we come to calculate E 2 and E 3 for the cases The radial component of the 4-momentum of massive particle can be calculated from the Eqs.
where k 1 (E 1 , s 1 , b, 0), k 2 (s 2 , b, ξ), h 1 (s 1 , b), h 2 (s 2 , b) h 71 (s 1 , b), g 3 (b, s 2 , ξ) and so on are the functions of different parameters and we will show them in the appendix. From the Eq.(3.22), with the detailed expressions given by the above, we obtain the equation of E 3 as follow From Eqs.(3.32) and (3.33), we find that σ 3 is decoupled. So the sign of σ 3 will not affect the value of E 3 . Since the quadratic equation of E 3 (3.32) has two solutions. The larger solution of E 3 = E 3,+ gives larger efficiency because the efficiency depends on the value of E 3 that will became explicit in following parts. Therefore, it is sufficient to consider the case of σ 3 = −1 with the larger solution of E 3 = E 3,+ . In conclusion, we can get the expression of E 3 and E 2 from the Eqs. (3.22) and (3.23).
With all those ingredients, the efficiency can be calculated through the following expression:

Efficiency
With the detailed expressions of E 3 and E 2 above. We have three different types of parameters involved in the calculation of the efficiency η. The first type is the charge of extreme Kerr-Sen black hole(b = 1 − a). Second type is particles spins(s 1 and s 2 ), the third type is orbit parameters of the particles such as (α 3 , β 3 and ξ) and direction of the particles' motion (σ 1 , σ 2 , σ 3 and σ 4 ).
Then, for a given value of E 1 , the maximal efficiency η max would be reached with the minimum value of E 2 and the maximal value of E 3 . Without loss of generality, we just normalize the ingoing energy E 1 as E 1 = 1.
From the Eq. (3.34), we find that the expression of E 3 decoupled with the parameters ξ and β 3 . So we analyze the maximal value of E 3 with the remaining parameters for different values of b. Note that Fig. 1(a) shows that the spin magnitude s 1 close to zero for larger value of E 3 . So we first assume s 1 = 0 in order to find the relation of E 3 and α 3 . The contour maps of E 3 in terms of α 3 and s 2 showed in Fig. 3. From the Fig. 3, we know that the largest efficiency can found with α 3 → 0 + . Therefore, we set α 3 = 0 + to calculate the corresponding maximal efficiency.
In Fig. 4, the contour map of E 3 in terms of s 1 and s 2 is showed. The maximal value of E 3 is labeled with the red point.
Note that E 2 ≥ 1 if the particle 2 falling from infinity, if E 2 = 1 is possible, we find that the maximal value of E 3 gives the maximal efficiency. Note that E 3 is decoupled with parameters β 3 and ξ. So our target is equivalent to find E 2 = 1 with some admissible values of β 3 and ξ. In Fig. 1(b), we already have the constraint on ξ, that is, 0 > ξ ≥ −0.5 > ξ min for different values of s and b. For such constrained ξ, the relation between β 3 and ξ which gives E 2 = 1 can be found in Fig. 5(a).
Hence the maximum efficiency is given by η max = E 3max /2. The Fig. 5(b) shows the maximum efficiency η max with different b. We found that the efficiency η max decreases with the increase of b. While when b = 0 which corresponds the Kerr case, our results is the same as the previous results [10].   So the expression of f 22 , f 23 , f 42 , and f 43 can write in an explicit way: Note that the radial component of the 4-momentum of massive particle do not change through the collision process. As in case [A], we finally get the detail expression for E 3 and E 2 respectively. (3.45) where A 1 , B 1 , C 1 and P 1 are given by Eqs. (3.33) and (3.36) with s 2 = 0.

Efficiency
On the one hand, when the value of E 1 is given, the maximal efficiency η max would be reached with minimum value of E 2 and maximal value of E 3 . On the other hand, we consider particle 1 and particle 2 falling from infinity, we obtain the constrains of E 1 ≥ 1 and E 2 ≥ 0. Without loss generality, we again normalize E 1 to unity (E 1 = 1) as last subsection and then analyze E 3 and E 2 which are directly associated to the maximal efficiency.  If E 2 → 0 can be achieved, it certainly gives the minimal value of E 2 and thus the maximal efficiency can be simply given by η max = E 3max . Hence it is important to analyze whether E 2 → 0 is possible or not. From Eq. (3.36), we obtain the asymptotic expression of P as Eq. (3.47) tells us that E 2 → 0 + , if β 3 ξ → +∞. For example, the value of parameters at red point in Fig. 6(b) are b = 0.1, σ 1 = 1, σ 3 = −1, α 3 = 0, s 1 = 0.03513, s 2 = 0, E 1 = 1, E 3 = 12.2977. So the detail expression of E 2 can be rewritten as: which means E 2 → 0 + can be realized β 3 ξ → ∞ for the case of b = 0.1. Hence by employing formula η max = E 3max , we found that the efficiency η max decreases with the increase of b. While b = 0 which corresponds the Kerr case, our results is again the same as the previous results [10].

C. Maximal Efficiency in Case [C] PMP
Now we come to the last case, which is the Compton scattering. The radial components of 4-momenta of massless particles have already been given in Eq. (3.40). So we can write the coefficients f 12 , f 13 , f 32 and f 33 in terms of energy as: From the conservation of the radial components of the 4-momenta, we find where the amplification factor S is given by and where P 3 keeps the same form of P 1 given by Eq. (3.36) by replaceing f 13 and f 33 with Eqs.

Efficiency
It is easy to see in Compton scattering, the efficiency η is defined as: again, we consider massless particle 1 and massive particle 2 falling from infinity, we assume the constrains of E 1 ≥ 0 and E 2 ≥ 1 and obtain that the maximal value of S and the minimal value of E 2 /E 1 gives the maximal efficiency. First, we can easily find that the ratio E 2 /E 1 doesn't depends on the E 1 and E 2 , but rather depends on the parameters α 3 , β 3 , ξ, s 2 and b. From Eq. (3.55), the asymptotic expression of E 2 /E 1 behaves Note that the particle 2 is massive and can reach the horizon, therefore the constraint on ξ keeps the same form as in previous section, namely, ξ min < ξ < 0. With this parameter space, a direct calculation shows that S = 0. From Eq. (3.57), we can see that if denominator of the equation is not equal to zero, the condition E 2 /E 1 → 0 can be archived when β 3 ξ → ∞. Thus the maximal energy contraction efficiency is η max = S max .
The Fig. 7 shows the maximum value of E 3 with the red point in the contour map of S in terms of α 3 and s 2 for different values of b. The figure shows the maximum efficiency η max = S max decreases when b increases.

IV. CONCLUSIONS
In this paper, we study the collision of two uncharged spinning particles around an extreme Kerr-Sen black hole and calculate the maximal efficiency of the energy extraction from the black hole. We consider the particles freely falling from infinity to the Kerr-Sen black hole. The Kerr-Sen spacetime is determined by three parameters, which are mass M, angular momentum a, and charge Q(b = Q 2 /2M). It reduces to a Kerr black hole when the parameter b = 0 and all our results coming back to the Kerr case [10] when b = 0. We viewed this as a consistent check.
In this paper, we consider three types of collision, the first one is the MMM case[A], we obtain that the maximum efficiency is given by η max = E 3max /2 and decreases monotonously with the increase of b. Then, in the MPM case[B], we obtain the maximum efficiency η max = E 3max and decreases monotonously with the increase of b. Finally, in the PMP case[C], we get the maximum efficiency η max = S max which decreases when the b increases. All our results can reduce to the Kerr situation [10] when b = 0. The Compton scatting and inverse Compton scatting of spinless particle in Kerr background is discussed in [6], and our results shows when the spin take into account, the maximum efficiency can be greatly improved.
In summarize, for extreme Kerr-Sen black hole, decrease the charge parameter b = Q 2 /2M always increase the maximum efficiency of energy extraction.