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

## Abstract

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. We calculate the maximum efficiency for all the cases, and found that the efficiency increases as the Kerr-Sen black hole’s parameter (\(b=1-a\)) decreases.

## 1 Introduction

Penrose process, a mechanism to extract rotational energy from black hole, was first discovered by Penrose in 1969 with Kerr black hole [1]. The original version of Penrose process happened in the ergosphere, an object splits into two parts while it falls toward Kerr black hole. The one falls into the black hole with negative energy, while the other escapes to infinity. The energy of escaped part is larger than the original one. Therefore, rotational energy can be extracted from Kerr black hole. For such a process, Wald obtain the maximum efficiency \(\eta _{max} = (output energy)/(input energy) \approx 1.21\) [2]. After that, Piran et al. consider a different type of collision Penrose process which two particles collide inside the ergosphere, and found to be have similar energy contracting efficiency with the original Penrose process [3].

In 2009, Bañados, Silk and West(BSW) proposed that a rotating black hole can act as accelerators for non-spin particles [4]. They show that the collision center-of-mass energy can be arbitrarily high for extremal Kerr black hole [4]. Inspired by this work, some authors suggest to construct Penrose process based on the BSW mechanism [5, 6, 7]. These collision process are called as super Penrose process since it usually has far more higher energy contraction efficiency. For example, Schnittman obtain the maximal efficiency is about 13.92 when a massless and a massive particle collide near the horizon [7]. Along this line, the super Penrose process have been extended to various black holes [8, 9, 10, 11].

Recently, the BSW mechanism has been generalized to include the spinning particles [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. It has been shown [23, 24, 25, 26, 27] that the trajectory of a spinning test particle is no longer a geodesic and therefore is more close to the real particle. The corresponding super Penrose process also have been investigated in many cases [8, 9, 10, 11]. It worth to note that in [28], the authors obtain some general result on the energy in the center of mass frame for BSW mechanism. However, this paper is devoted to the efficiency of Penrose process that was not studied in [28].

On the other hand, the Kerr-Sen black hole is a rotating and charged solution of the low-energy effective field theory for heterotic string theory [29]. After it proposed, many aspects of Kerr-Sen solution has been investigated [30]. This black hole solution characterized by three parameters, which are mass *M*, angular momentum *a*, and charge *Q*(\(b=Q^2/2M\)). It reduces to the Kerr black hole when the parameter \(b=0\). As a grand unified theory, string theory is the most promising candidate of unified all the interactions, to this sense, the expected rotating and charged black hole solution would be the Kerr-Sen black hole rather than the Kerr-Newman one. Therefore in this paper we investigate the issue of the maximal energy contraction efficiency of the super Penrose process for spin particles in Kerr-Sen background. We provide a deep analysis of the super Penrose process for spinning particles and investigate the dependence of the maximal energy contraction efficiency with the Kerr-Sen black hole’s parameter.

This paper is organized as follows: after an introduction, we discuss the equations of motion for spinning particles in Kerr-Sen black hole in Sect. 2. While in Sect. 3, we study the super Penrose collision of spinning particles in extreme Kerr-Sen background. This section is divided into three cases and calculate the the maximal efficiency with different parameters of extreme Kerr-Sen black hole. The summary and conclusion was given in Sect. 4. Through out the paper, we adopt the geometrical unit \(( c = G =1)\).

## 2 Basic equation

### 2.1 Equations of motion of a spinning particle

*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

### 2.2 Conserved quantities in the Kerr-Sen black hole

*E*and

*J*are the energy and angular momentum of the particle respectively.

### 2.3 Equations of motion on the equatorial plane

*E*in the following text actually means \(\tilde{E} \).

### 2.4 Constraints on the orbits

*E*for different values of spin

*s*and

*b*and is showed in the Fig. 1a. 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 \(E\ge 1\), 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 \(s_{min}\approx -0.285\) and \(s_{max}\approx 0.471\). 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 spin \(s=-s_c=-a^2\left( \frac{2}{a}-1\right) ^{\frac{3}{2}}\), 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 \(B_r=2(1+\xi )\) with \(\xi \) being a negative number. From Eq. (2.39), the energy *E* now is a function of the *s*, *b* and \(\xi \) and is showed as the Fig. 1b. This figure shows that the allowed range of \(\xi \) increase when *b* increases.

## 3 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.

*E*and angular momentum

*J*as follows

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 [A] (MMM), [B] (MPM), and [C] (PMP).

### 3.1 Maximal efficiency in case [A] MMM

*m*, i.e. \(m_1=m_2=m_3=m_4=m\). With this in hand, the equations of conservation laws (3.3)–(3.6) can be simplified as

#### 3.1.1 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 \(\eta \). 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 (\(\alpha _3\), \(\beta _3\) and \(\xi \)) and direction of the particles’ motion (\(\sigma _1\), \(\sigma _2\), \(\sigma _3\) and \(\sigma _4\)).

Note that we already fix the value of \(\sigma _2,\sigma _3,\sigma _4\) as \(\sigma _2=\sigma _4=-1\) and \(\sigma _3=-1\) in the last section. So the only remaining parameter for the direction of the particles’ motion is \(\sigma _1\). However, a good efficiency can’t be found for \(\sigma _1=-1\) [10], so we set that \(\sigma _1=1\).

Then, for a given value of \(E_1\), the maximal efficiency \(\eta _{max}\) 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 \(E_1=1\).

*b*. Note that Fig. 1a 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 \(\alpha _3\). The contour maps of \(E_3\) in terms of \(\alpha _3\) and \(s_2\) showed in Fig. 3. From the Fig. 3, we know that the largest efficiency can found with \(\alpha _3 \rightarrow 0^+\). Therefore, we set \(\alpha _3=0^+\) 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 \(E_2 \ge 1\) 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 \(\beta _3\) and \(\xi \). So our target is equivalent to find \(E_2=1\) with some admissible values of \(\beta _3\) and \(\xi \). In Fig. 1b, we already have the constraint on \(\xi \), that is, \(0> \xi \ge -0.5 > \xi _{min}\) for different values of *s* and *b*. For such constrained \(\xi \), the relation between \(\beta _3\) and \(\xi \) which gives \(E_2=1\) can be found in Fig. 5a.

Hence the maximum efficiency is given by \(\eta _{max}=E_{3max}/2\). Figure 5b shows the maximum efficiency \(\eta _{max}\) with different *b*. We found that the efficiency \(\eta _{max}\) 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].

### 3.2 Maximal efficiency in case [B] MPM

*m*and the massless particles are nonspinning. The equations of conservation law (3.5) and (3.6) reduce to

#### 3.2.1 Efficiency

On the one hand, when the value of \(E_1\) is given, the maximal efficiency \(\eta _{max}\) 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 \(E_1 \ge 1\) and \(E_2 \ge 0\). 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.

Figure 6 shows the contour map of \(E_3\) in terms of \(\alpha _3\) and \(s_1\). The maximum value of \(E_3\) is given at the red point.

*b*. While \(b=0\) which corresponds the Kerr case, our results is again the same as the previous results [10].

### 3.3 Maximal efficiency in case [C] PMP

#### 3.3.1 Efficiency

*b*. From Eq. (3.55), the asymptotic expression of \(E_2/E_1\) behaves

Figure 7 shows the maximum value of \(E_3\) with the red point in the contour map of \(\mathcal {S}\) in terms of \(\alpha _3\) and \(s_2\) for different values of *b*. The figure shows the maximum efficiency \(\eta _{max} = \mathcal {S}_{max}\) decreases when *b* increases.

## 4 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 \(\eta _{max}=E_{3max}/2\) and decreases monotonously with the increase of *b*. Then, in the MPM case[B], we obtain the maximum efficiency \(\eta _{max} = E_{3max}\) and decreases monotonously with the increase of *b*. Finally, in the PMP case[C], we get the maximum efficiency \(\eta _{max} = \mathcal {S}_{max}\) 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.

## Notes

### Acknowledgements

This work is supported by NSFC with No. 11775082. The authors could like to thank prof. Kazumasa Okabayashi for helpful discussion.

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