# Collision of two general particles around a rotating regular Hayward’s black holes

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## Abstract

The rotating regular Hayward’s spacetime, apart from mass (*M*) and angular momentum (*a*), has an additional deviation parameter (*g*) due to the magnetic charge, which generalizes the Kerr black hole when \(g\ne 0\); for \(g=0\) it goes over to the Kerr black hole. We analyze how the ergoregion is affected by the parameter *g* to show that the area of the ergoregion increases with increasing values of *g*. Further, for each *g*, there exists a critical \(a_E\), which corresponds to a regular extremal black hole with degenerate horizons \(r=r^E_H\). \(a_E\) decreases whereas \(r^E_H\) increases with an increase in the parameter *g*. Banãdos, Silk, and West (BSW) demonstrated that the extremal Kerr black hole can act as a particle accelerator with arbitrarily high center-of-mass energy (\(E_{\mathrm{CM}}\)) when the collision of two particles takes place near the horizon. We study the BSW process for two particles with different rest masses, \(m_1\) and \(m_2\), moving in the equatorial plane of the extremal Hayward’s black hole for different values of *g*, to show that \(E_{\mathrm{CM}}\) is arbitrarily high when one of the particles takes a critical value of the angular momentum. Our result, in the limit \(g \rightarrow 0\), reduces to that of the Kerr black hole.

## 1 Introduction

Recently, Banãdos, Silk, and West (BSW) [1] demonstrated that a Kerr black hole can act as a particle accelerator, i.e., when two particles collide arbitrarily close to the horizon of an extremal Kerr black hole, the center-of-mass energy (\(E_{\mathrm{CM}}\)) can grow infinitely in the limiting case of maximal rotation. The \(E_{\mathrm{CM}}\) becomes higher for an extremal black hole when one of the colliding particles has a critical angular momentum. This aroused much attention [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] in a series of subsequent papers by generalizing the BSW mechanism to different spacetimes, also opening a window into new physics with astrophysical applications, e.g., to understand the phenomenon like gamma ray bursts and AGNs in the Galaxy. Jacabson and Sotiriou [2] elucidated the BSW mechanism to discuss some practical limitations for the Kerr black hole to act as a particle accelerator. It was pointed out that infinite \(E_{\mathrm{CM}}\) for the colliding particles can only be attained when the black hole is exactly extremal and only in infinite time and on the horizon of the black hole. On the other hand, Lake [3] studied the BSW mechanism near the Cauchy horizon to show that \(E_{\mathrm{CM}}\) of the colliding particles is infinite. With the BSW mechanism extended to the Kerr–Newman black hole case [4], an unlimited \(E_{\mathrm{CM}}\) requires an additional restriction on the value of the spin parameter *a*. Zaslavskii [5] argued that a similar effect exists for a nonrotating but charged black hole even for the simplest case of radial motion of particles in a Reissner–Nordström background and also demonstrated that the BSW mechanism is a universal property of the rotating black holes [6, 7]. The BSW mechanism has been extended to an Einstein–Maxwell–Dilaton black hole [8], rotating stringy black hole [9], a Kerr–(anti-) de Sitter black hole [10], BTZ [11], a rotating black hole in Horava–Lifshitz gravity [12] and around the four-dimensional Kaluza–Klein extremal black hole in [8], and there resulted an infinitely large \(E_{\mathrm{CM}}\) near the horizon.

Later, Gao and Zhong showed that the BSW mechanism is possible for the nonextremal black holes, showing that for a critical angular momentum, the \(E_{\mathrm{CM}}\) diverges at the inner horizon. The interesting work of Grib and Pavlov [13, 14, 15] shows that the scattering energy of the particles in the center-of-mass frame can be arbitrarily large, not only for extremal black holes but also for nonextremal ones, when we take into account the multiple scattering. It turns out that the divergence of the \(E_{\mathrm{CM}}\) of the colliding particles is a phenomenon not only associated with black holes but also with naked singularities [16, 17, 18, 19]. The BSW mechanism was further extended to the case of two different massive colliding particles near the Kerr black hole [20] and also in the case of Kerr–Newman black hole [21], and for the Kerr–Taub–NUT spacetime [22], suggesting that \(E_{\mathrm{CM}}\) depends not only on the rotation parameter *a* but also on the NUT charge *n*. These works on two different massive colliding particles are generalizations of previous studies (see also Harada and Kimura 2014 for a review of the BSW mechanism [23]).

While we are far away from any robust quantum theory of gravity that tell us how the singularities of classical black holes are solved, there are some models of black hole solutions without the central singularity. These are regular black holes first proposed by Bardeen metric [24], which is a solution of Einsteins gravity coupled to a nonlinear electrodynamics field [25]. Another interesting model is the Hayward’s black hole metric [26] and Ayón-Beato–García [25]. The rotating or Kerr-like solutions of the Bardeen and the Hayward’s metrics have also been obtained [27]. These rotating black holes violate even the weak energy condition is violated, but such a violation can be made very small [27]. Astrophysical black holes are more interested in the study of rotating regular black hole, but the actual nature of these objects has still to be tested [28, 29]. These regular black holes have an additional deviation parameter (say *g*) apart from the mass (*M*) and angular momentum (*a*), which provides a deviation from the Kerr black hole. It turns out that for these black holes, for each nonzero *g*, there exists a critical \(a_{E}\), which corresponds to a regular extremal black hole with degenerate horizons [30]. The BSW mechanism for these rotating regular black holes has also been analyzed in [31, 32, 33], suggesting that a rotating regular black hole can also act as a particle accelerator, which in turn provides a suitable framework for Planck-scale physics.

## 2 Rotating regular Hayward’s black holes

*m*is the mass and

*l*is a constant. The metric (1) asymptotically behaves as

*M*is replaced by

*m*(

*r*). In this section, we shall discuss the horizon structure and the ergoregion of a rotating regular Hayward’s black hole metric. The rotating regular Hayward’s black hole in Boyer–Lindquist coordinates reads [27]

*a*is a rotation parameter, and

*m*(

*r*) is a function related to mass of the black hole via

*M*represents the black hole mass,

*g*is the magnetic charge which provides a deviation from the standard Kerr black hole, and \(\alpha \), \(\beta \) are two real numbers. In the equatorial plane (\(\theta = \pi /2\)), the mass function (4) takes the form

Table for different values of *a* and *g* for a rotating regular Hayward’s black hole. \(\delta \) is the region between the static limit surface and event horizon (\(\delta ^{a}=r^{+}_{\mathrm{sls}}-r^{+}_{H}\))

| \(a=0.6\) | \(a=0.7\) | \(a=0.8\) | \(a=0.9\) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

\(r^{+}_{H}\) | \(r^{+}_{\mathrm{sls}}\) | \(\delta ^{a}\) | \(r^{+}_{H}\) | \(r^{+}_{\mathrm{sls}}\) | \(\delta ^{a}\) | \(r^{+}_{H}\) | \(r^{+}_{\mathrm{sls}}\) | \(\delta ^{a}\) | \(r^{+}_{H}\) | \(r^{+}_{\mathrm{sls}}\) | \(\delta ^{a}\) | |

0 | 1.80000 | 1.90554 | 0.10554 | 1.71414 | 1.86891 | 0.15477 | 1.60000 | 1.82462 | 0.22462 | 1.43589 | 1.77136 | 0.33547 |

0.1 | 1.79961 | 1.90523 | 0.10562 | 1.71367 | 1.86858 | 0.15491 | 1.59935 | 1.82426 | 0.22491 | 1.43477 | 1.77095 | 0.33618 |

0.3 | 1.78944 | 1.89725 | 0.10781 | 1.70103 | 1.85992 | 0.15889 | 1.58186 | 1.81466 | 0.23280 | 1.40355 | 1.76004 | 0.35649 |

0.5 | 1.74839 | 1.86576 | 0.11737 | 1.64846 | 1.82553 | 0.17707 | 1.50329 | 1.77620 | 0.27291 | – | 1.71562 | – |

### 2.1 Ergoregion of rotating regular Hayward’s black hole

*g*. There is another important surface of a black hole, particularly known as a static limit surface. The main property of the static limit surface is that the nature of particle geodesics changes after crossing the static limit surface, i.e., a timelike geodesic becomes spacelike and vice versa. The static limit surface satisfies \(g_{tt}=0\), i.e.,

*g*, and \(\theta \). It turns out that for each

*g*, there exist critical \(a_E\) and \(r_H^E\), which corresponds to a regular extremal black hole with degenerate horizons, and \(a_E\) decreases whereas \(r^E_H\) increases with increase in

*g*. The region between the event horizon and static limit surface is known as the ergoregion; from this region energy can be extracted via the Penrose process [36]. We show that as we increase the value of

*g*for corresponding

*a*, then the region between the event horizon and the static limit surface increases. In Figs. 1 and 2, we plot the contours of the horizon surfaces in the

*xz*-plane of the rotating regular Hayward’s black hole to show the ergoregion for various values of

*g*. The ergoregion for the rotating regular Hayward’s black hole is shown in Fig. 2, whereas the ergoregion of the Kerr black hole is shown in Fig. 1. The ergoregion has two boundaries: the event horizon and the outer static limit surface. The observer in the ergoregion cannot remain static. The ergoregion is sensitive to the deviation parameter

*g*, and its area increases with

*g*. This can be seen by the event horizon \(r_{H}^{+}\) and the static limit surface (\(r^+_{\mathrm{sls}}\)), which are depicted in Table 1, for various values of

*a*and

*g*. Interestingly, \(\delta ^a=r^+_{\mathrm{sls}}-r^+_H\) increases with

*g*as well as

*a*, thereby suggesting that the ergoregion is enlarged. Further, we find that for each value of rotation parameter

*a*, there exists a critical value of \(g=g^*\), such that for \(g<g^*\), we have two horizons suggesting a nonextremal Hayward’s black hole, whereas for \(g=g^*\), one gets an extremal black hole where the two horizons coincide (cf. Fig. 2); for \(g>g^*\), we obtain no horizons or no black hole.

## 3 Equations of motion of the particle

*E*and the angular momentum

*L*. Furthermore, we can write the geodesic equations for a particle in terms of these conserved quantities

*m*is given by

*m*corresponds to the mass of a particle, and

*g*, and in the limit \(g \rightarrow 0\), we set the corresponding equations for the Kerr black hole [1]. The radial equation for the timelike particles moving along the geodesics is described by

## 4 Center-of-mass energy of the colliding particles in Hayward’s black hole

*i*th particle is \(p^{\mu }_{i} = m_{i} u^{\mu }_{i}\), where \(m_{i}\) and \( u^{\mu }_{i}\) correspond, respectively, to the mass and the 4-velocity of the

*i*th particle (\(i=1,2\)), and the total 4-momenta of the particles is given by

*m*(

*r*) is given by Eq. (5). Obviously, the result Eq. (27) confirms that the parameter

*g*has influence on \(E_{\mathrm{CM}}\). Thus, \(E_{\mathrm{CM}}\) depends on the parameters

*g*and

*a*. When \(g \rightarrow 0\), in Eq. (27), we obtain the \(E_{\mathrm{CM}}\) of two different mass particles of the Kerr black hole [20],

### 4.1 Near horizon collision in near-extremal Hayward’s black hole

*i*th particle and \(\Omega _{H}=a/(r_H^2+a^2)\) is the angular velocity at the event horizon of the black hole. It is clear from Eq. (31) that if either \(L_{C_1}=L_1\) or \(L_{C_2}=L_2\), then the \(E_{\mathrm{CM}}\) becomes infinite, i.e., the necessary condition for getting \(E_{\mathrm{CM}}\) infinite is \(L=L_C\) or \(\Omega _H L =E\), for each particle. One can also see the behavior of \(E_{\mathrm{CM}}\) vs.

*r*for an extremal rotating regular Hayward’s black hole from Fig. 3; it can be seen that \(E_{\mathrm{CM}}\) is infinite for the critical values of angular momentum \(L_1=2.0, 2.0305, 2.34136, 2.52987\), corresponding to \(g=0, 0.2, 0.5, 0.6\), and it remains finite for other values.

### 4.2 Particle collision for nonextremal black hole

*r*for a nonextremal black hole, which indicates that \(E_{\mathrm{CM}}\) remains finite for each value of \(L_{1}\) and \(L_{2}\). Furthermore, the effect of

*g*can also be seen from Fig. 4, which indicates that \(E_{\mathrm{CM}}\) increases with

*g*. The limit of nonextremal \(E_{\mathrm{CM}}\) for two different mass particles, in the case of \(g \rightarrow 0\), has the following form:

## 5 Conclusion

The singularity theorems under fairly general conditions imply that a sufficiently massive collapsing object will undergo continual gravitational collapse, resulting in the formation of a singularity. However, it is widely believed that singularities do not exist in Nature, but that they are an artifact of general relativity. Hence, especially in the absence of well-defined quantum gravity, the regular models without singularities received much attention. Also, the astrophysical black holes may be different from the Kerr black holes predicted in general relativity [28], but the actual nature of these objects has still to be verified. The rotating regular Hayward’s black hole can be seen as one of the non-Kerr black hole metrics, which in Boyer–Lindquist coordinates is the same as the Kerr black hole with *M* replaced by a mass function *m*(*r*) , which has an additional parameter due to the magnetic charge *g*; Hayward’s black hole reduces to the Kerr black hole in the absence of charge (\(g=0\)). Interestingly, for each nonzero value of *g*, the Hayward’s black hole is extremal with critical spin parameter \(a^*\) with degenerate horizons [31]. In this paper, we have performed a detailed analysis of the ergoregion in the rotating regular Hayward’s black hole [27], and we discuss the effect of the deviation parameter *g* on the ergoregion. Our analysis reveals (cf. Fig. 2) that for each *a* there exists a critical \(g=g^*\) such that for \(g >g^*\), the rotating regular Hayward’s black hole has a disconnected horizon, and the horizons coincide when \(g=g^*\), and we have two horizons when \(g<g^*\). When the ergoregion of a rotating regular Hayward’s black hole is compared with the Kerr black hole (cf. Fig. 1), we find that the ergoregion is sensitive to the deviation parameter *g*, and the ergoregion enlarges as the value of the deviation parameter increases.

We have also studied the collision of two particles with different rest masses moving in the equatorial plane of the rotating regular Hayward’s black hole, and we calculated the \(E_{\mathrm{CM}}\) for these colliding particles. We obtain a general expression of \(E_{\mathrm{CM}}\) for the rotating regular Hayward’s black hole, the \(E_{\mathrm{CM}}\) is calculated when the collision takes place near the horizon for both extremal and nonextremal black holes. It is demonstrated that \(E_{\mathrm{CM}}\) not only depends on the rotation parameter *a* of the rotating regular Hayward’s black hole, but also on the deviation parameter *g*. Further, for an extremal black hole, we address the case of arbitrarily high \(E_{\mathrm{CM}}\) when the collision occurs near the horizon and one of the particles has critical angular momentum. We also calculate \(E_{\mathrm{CM}}\) for the nonextremal rotating regular Hayward’s black hole, and we found that \(E_{\mathrm{CM}}\) is finite with an upper bound which increases with an increase in the parameter *g*.

## Notes

### Acknowledgments

M.A. acknowledges the University Grant Commission, India, for financial support through the Maulana Azad National Fellowship For Minority Students scheme (Grant No. F1-17.1/2012-13/MANF-2012-13-MUS-RAJ-8679). S.G.G. would like to thank SERB-DST for Research Project Grant NO SB/S2/HEP-008/2014. We also thank IUCAA for hospitality while a part of this work was done.

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