Horizon structure of rotating Bardeen black hole and particle acceleration

Abstract

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We investigate the horizon structure and ergosphere in a rotating Bardeen regular black hole, which has an additional parameter (g) due to the magnetic charge, apart from the mass (M) and the rotation parameter (a). Interestingly, for each value of the parameter g, there exists a critical rotation parameter (......), which corresponds to an extremal black hole with degenerate horizons, while for ...... it describes a non-extremal black hole with two horizons, and no black hole for ....... We find that the extremal value ...... is also influenced by the parameter g, and so is the ergosphere. While the value of ...... remarkably decreases when compared with the Kerr black hole, the ergosphere becomes thicker with the increase in g. We also study the collision of two equal mass particles near the horizon of this black hole, and explicitly show the effect of the parameter g. The center-of-mass energy (......) not only depend on the rotation parameter a, but also on the parameter g. It is demonstrated that the ...... could be arbitrarily high in the extremal cases when one of the colliding particles has a critical angular momentum, thereby suggesting that the rotating Bardeen regular black hole can act as a particle accelerator.

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Web End = Horizon structure of rotating Bardeen black hole and particle acceleration

Sushant G. Ghosh1,2,a, Muhammed Amir1,b

1 Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India

2 Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa

Received: 6 October 2015 / Accepted: 7 November 2015 / Published online: 25 November 2015 The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract We investigate the horizon structure and ergo-sphere in a rotating Bardeen regular black hole, which has an additional parameter (g) due to the magnetic charge, apart from the mass (M) and the rotation parameter (a). Interestingly, for each value of the parameter g, there exists a critical rotation parameter (a = aE), which corresponds to

an extremal black hole with degenerate horizons, while for a < aE it describes a non-extremal black hole with two horizons, and no black hole for a > aE. We nd that the extremal value aE is also inuenced by the parameter g, and so is the ergosphere. While the value of aE remarkably decreases when compared with the Kerr black hole, the ergo-sphere becomes thicker with the increase in g. We also study the collision of two equal mass particles near the horizon of this black hole, and explicitly show the effect of the parameter g. The center-of-mass energy (ECM) not only depend on the rotation parameter a, but also on the parameter g. It is demonstrated that the ECM could be arbitrarily high in the extremal cases when one of the colliding particles has a critical angular momentum, thereby suggesting that the rotating Bardeen regular black hole can act as a particle accelerator.

1 Introduction

The spherically symmetric ReissnerNordstrm metric [1] is given by

ds2 = g dx dx, (, = 0, 1, 2, 3), (1)

with g = diag( f (r), f (r)1, r2, r2 sin2 ) and

f (r) = 1

2mr +

q2

r2 ,

ae-mail: mailto:[email protected]

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ae-mails: mailto:[email protected]

Web End [email protected] ; mailto:[email protected]

Web End [email protected]

where m and q, respectively, denote the mass and charge. The solution represents a black hole shielding a singularity, the black hole is being formed as an end state of the collapse of a star. That the gravitational collapse of a sufciently massive star (3.5M ) inevitably leads to a spacetime singularity is

a fact established by an elegant theorem due to Hawking and Penrose [2] (see also Hawking and Ellis [3]).

However, it is widely believed that these singularities do not exist in Nature, but that they are the creation or an artifact of classical general relativity. The existence of a singularity, by its very denition, means spacetime fails to exist, signaling a breakdown of the laws of physics. Thus, in order for these laws to exist, singularities must be substituted by some other objects in a more suitable theory. The extreme condition, in any form, that may exist at the singularity implies that one should rely on quantum gravity, expanded to resolve this singularity [4]. While we do not yet have any denite quantum gravity allowing us to understand the inside of the black hole and resolve it separately, we must turn our attention to regular models, which are motivated by quantum arguments. The earliest idea, in mid-1960s, due to Sakharov [5] and Gliner [6], suggests that singularities could be avoided by matter, i.e., with a de Sitter core, with the equation of state p = . This equation of state is obeyed by the cosmolog

ical vacuum and hence, T takes on a false vacuum of the form T = g; is the cosmological constant.

Thus spacetime lled with a vacuum could provide a proper discrimination at the nal stage of gravitational collapse, replacing the future singularity [6]. The rst regular black hole solution, based on this idea, was proposed by Bardeen [7], according to whom there are horizons but there is no singularity. The matter eld is a kind of magnetic eld. The solution yields a modication of the Reisnner Nordstrm black hole solution, with the metric function f (r) being

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553 Page 2 of 12 Eur. Phys. J. C (2015) 75 :553

f (r) = 1

2mr2 (r2 + g2)3/2

= 1 [parenleftbigg]

m g

[parenrightbigg]

2(r/g)2(1 + (r/g)2)3/2

, and r 0.

Numerical analysis of f (r) = 0 reveals a critical value

such that f (r) has a double root if = , two roots if

< and no root if > , with = m/g. These cases

illustrate, respectively, an extreme black hole with degenerate horizons, a black hole with Cauchy and event horizons, and no black hole.

The Bardeen solution is regular everywhere, which can be realized from the scalar invariants

Rab Rab =

6mg2(4g2 r2) (r2 + g2)7/2

,

Rabcd Rabcd =

12m2(r2 + g2)7 [

8g8 4g6r2 + 47g4r4

12g2r6 + 48r8], (2) which are well behaved everywhere including at r = 0. How

ever, for the ReissnerNordstrm case (g = 0), they diverge

at r = 0 indicating a scalar polynomial singularity [3]. The

Bardeen black hole is asymptotically at, and near the origin it behaves as de Sitter, since

f (r) 1

2mg r2, r 0+,

whereas for large r, it is according to Schwarzschild.

Thus the black hole interior does not result in a singularity but develops a de Sitter like region, eventually settling with a regular center, thus its maximal extension is the one of the ReissnerNordstrm spacetime but with a regular center [8,9]. There has been an enormous advance in the analysis and application of regular black holes [1014], however, all subsequent regular black holes were based on the Bardeen idea. The detailed study of circular geodesics of photons of a non-rotating regular black hole can be found in [15]. The deection of light rays and gravitational lensing in regular Bardeen spacetime was also studied [16]. The ghost images of Keplerian discs, generated by photons with low impact parameters for a regular Bardeen black hole was also discussed [16].

The no-hair theorem suggests that astrophysical black hole candidates are Kerr black holes, but there still lacks direct evidence and its actual nature has not yet been veried. This opens the arena for investigating the properties for black holes that differ from Kerr black holes. Lately, the rotating (spinning) counterpart of Bardeens black hole has been proposed, which can be written in Kerr-like form in BoyerLindquist coordinates [17]. The rotating Bardeen regular metric has been tested with a black hole candidate in Cygnus X-1 [18], and thereafter more rotating regular black holes were proposed [1922]. Interestingly the 3-bound

a > 0.95 [23] and a > 0.983 [24] for the Kerr black hole

changes, respectively, to a > 0.78 and | |< 0.41, and

a > 0.89 and | |< 0.28 for the rotating Bardeen regular

black hole. Further, the measurement of the Kerr spin parameter of rotating Bardeen regular black holes from the shape of the shadow of a black holes was also explored [25]. The rotating Bardeen black holes accommodate the Kerr black holes in the special case when the deviation parameter, g = 0,

may be regarded as a well-suited framework for exploring astrophysical black holes.

In this paper, we investigate the horizon structure and ergosphere of the rotating Bardeen regular black hole and explicitly show the effect of the Bardeen charge g. The paper is organized as follows. In Sect. 2, we review the rotating Bardeen regular black hole and analyze the horizon structure and ergosphere, with respect to the charge g. We analyze the equatorial equations of motion of the particles and the effective potential in Sect. 3. Section 4 is devoted to the collision of two equal masses particles against the background of a rotating Bardeen regular black hole, and numerically we calculate ECM in a near horizon particle collision, and nally we summarize our results and evoke some perspectives to end the paper in Sect. 5.

2 Rotating Bardeen regular black hole

We do not have a denite quantum theory of gravity, hence an important direction of research is to consider regular rotating models to solve the singularity problem of the standard Kerr black hole. These are called non-Kerr black holes with different spin. Bambi [17], starting from a regular Bardeen metric, via the NewmanJanis algorithm [26] constructed a Kerr-like regular black hole solution, which in BoyerLindquist coordinates reads

ds2 = [parenleftbigg]1

2mr

dt2 4amr sin2 dtd + dr2

+ d2 + [parenleftbigg]r2 + a2 +

2a2mr sin2

sin2 d2,

(3)

where

= r2 + a2 cos2 , = r2 2mr + a2, (4) and

m m,(r, ) = M [parenleftbigg]

r2+ /2

r2+ /2 + g2r /2 [parenrightbigg]3/2

,

(5)

where m = m,(r, ) is a function of both r and ; ,

are two real numbers, and a is the rotation parameter. The

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Eur. Phys. J. C (2015) 75 :553 Page 3 of 12 553

0.8

0.6

a 0.70

aE 0.95145807

a 0.80

a aE

a 1.00

g

rHE =1.02899

a 0.60

aE 0.80802796

a 0.70

a aE

a 0.85

g

rHE =1.08549

0.6

0.4

0.4

0.2

0.2

0.0

0.0

0.2

0.4

0.2

0.5 1.0 1.5

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

r

r

Fig. 1 Plot showing the behavior of vs. r for xed values of = 2, = 3, = /4, and M = 1 by varying a. The case a = aE corresponds to

an extremal black hole

0.5 1.0 1.5

1.0

a 0.70

a 0.80

a aE

a 1.00

aE 0.942439535325

g

rHE =1.04769

a 0.60

aE 0.7851573623591

a 0.70

a aE

a 0.85

g

rHE =1.11879

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

0.2

0.4

0.2

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

r

r

0.8

0.8

g 0.20

g 0.30

g 0.39

g 0.50

g 0.30

g 0.40

g 0.49

g 0.60

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

0.2

0.2

0.0 0.5 1.0 1.5

0.0 0.5 1.0 1.5

r

r

Fig. 2 Plot showing the behavior of vs. r for xed values of = /2 and M = 1. The case a = aE corresponds to an extremal black hole

parameter g is the magnetic charge of non-linear electrodynamics, which measures the deviation from the Kerr black hole, and when we switch off non-electrodynamics (g = 0),

one recovers the Kerr metric. It turns out that the curvature invariants are regular everywhere, when g = 0, including at

the origin [17]. For deniteness, throughout this paper we shall call the metric (3) a rotating Bardeen regular black hole metric.

The BoyerLindquist coordinates are most widely used by astrophysicists, as in these coordinates the rotating black hole has just one off-diagonal term. The rotating Bardeen metric, like the Kerr metric, in BoyerLindquist coordinates does not depend on t, , which means the Killing symmetries and the Killing vectors are given by = t and = ,

with a the Kronecker delta. The existence of the two Killing

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553 Page 4 of 12 Eur. Phys. J. C (2015) 75 :553

of a rotating Bardeen regular black hole. The behavior of the event horizon is shown in Figs. 1, 2, and 3 for different values of a and g. Figure 3 implies that, for a < aE, there exists a set of values of the parameters for which one gets two horizons, and when a = aE the two horizons coincide, i.e., we have an

extremal black hole with degenerate horizons (cf. Table 1).

An extremal black holes occurs when = 0 has a double

root, i.e., when the two horizons coincide. When = 0 has

no root, i.e., no horizon exists, which means there is no black hole (cf. Figs. 1 and 2). There exists an upper bound on the spin parameter, aE 0.998, for astrophysical black holes,

which is called Thornes bound [27]. In the equatorial plane ( = /2), the mass function m,(r, ) does not depend

on the parameters and , and so we have the same for Eq. (7). This is also the case when both = = 0. Thus,

the two cases = /2, and = = 0 (for any ) will

have the same horizon structure. The non-trivial effect of the parameters and ( = /2) on the horizon structure

is shown in Fig. 1. Note that the horizon is called extremal when r = r EH is a double root of = 0 when a = aE. It

is seen that, for = 2, = 3, = /4, and g = 0.2, we

have aE = 0.95145807 and r EH = 1.02899, and for g = 0.4

one gets aE = 0.80802796 and r EH = 1.08549. Thus, the

extremal value of the rotation parameter aE decreases due to the presence of the magnetic charge g when compared with the Kerr black hole. Thus, for each g, there exists an extremal rotating Bardeen black hole.

The static limit surface or innite redshift surface of a black hole is a surface where the time-translation Killing vector becomes null, = 0. An innite redshift surface

is referred to as a null Killing surface to distinguish it from the null surface corresponding to the horizons. To nd the null Killing surfaces, one sets the gtt component of the metric tensor in Eq. (3) equal to zero. For the rotating Bardeen regular black hole, we nd that gtt = 0 gives 2(r2+ /2 + g2r /2)3

4M2r8+3 3/2 = 0. (8)

In Fig. 4, we depict the possible roots of the equation gtt = 0

with different combinations of the parameters a and g and dif-

Table 1 The values of the horizons of a rotating Bardeen regular black hole with the parameters M = 1 and = /2 (gh = r+H rH)

a g = 0 g = 0.2 g = 0.3 g = 0.4rH r+H gh rH r+H gh rH r+H gh rH r+H gh

0.5 0.13397 1.86603 1.73206 0.29652 1.82784 1.53132 0.40030 1.77671 1.37641 0.52235 1.69683 1.17448

0.6 0.20000 1.80000 1.60000 0.36555 1.75683 1.39128 0.48388 1.69783 1.21395 0.62877 1.60175 0.97298

0.7 0.28585 1.71414 1.45829 0.45416 1.66265 1.20849 0.59211 1.58940 0.99729 0.77907 1.45731 0.67824

0.8 0.40000 1.60000 1.20000 0.57655 1.53228 0.95573 0.75247 1.42484 0.67237

aE 1.00000 1.00000 0.00000 1.04769 1.04769 0.00000 1.08604 1.08604 0.00000 1.11879 1.11879 0.00000

aE = 1, 0.9488364581472, 0.875075019710, and 0.785157362359, which, respectively, correspond to g = 0, 0.2, 0.3, and 0.4

0.8

0.6

g

0.4

Black Hole

Region

No Black Hole Region

0.2

0.0

0.0 0.2 0.4 0.6 0.8 1.0

a

Fig. 3 Plot showing the behavior of the spin parameter (a) and the magnetic charge parameter (g) plane of the rotating Bardeen regular metric (contour plot of = 0). The blue dashed line is the boundary

which separates the black hole region from the no black hole region

vectors and implies that the corresponding momenta of a test particle, pt and p, are constants of the motion.

2.1 Horizons and ergosphere

The metric (3) is singular at = 0, which corresponds to an

event horizon of a black hole. The horizons of the rotating Bardeen regular black hole are solutions of

2 ()() = 0, (6)

which leads to

(r2 + a2)2(r2+ /2 + g2r /2)3

4M2r8+3 3/2 = 0. (7)

This depends on a, g, and , and it is different from the Kerr black hole where it is independent. The numerical analysis of (7) suggests the possibility of two roots for a set of values of parameters which corresponds to two horizons

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Eur. Phys. J. C (2015) 75 :553 Page 5 of 12 553

1.0

1.0

a 0.80

a 0.90

a 1.00

a 1.12

a 1.20

g

a 0.90

a 1.00

a 1.20

a 1.37

a 1.50

g

0.8

0.6

0.5

0.4

g tt

g tt

0.2

0.0

0.0

0.2

0.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

r

r

1.0

1.0

a 0.70

a 0.80

a 0.90

a 1.05

a 1.20

g

a 0.90

a 1.00

a 1.10

a 1.29

a 1.50

g

0.8

0.8

0.6

0.6

0.4

0.4

g tt

g tt

0.2

0.2

0.0

0.0

0.2

0.2

0.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

r

r

1.0

1.0

a 0.60

a 0.70

a 0.80

a 0.96

a 1.20

g

a 0.80

a 0.90

a 1.00

a 1.18

a 1.50

g

0.8

0.8

0.6

0.6

0.4

g tt

g tt

0.4

0.2

0.2

0.0

0.0

0.2

0.2

0.4 0.0 0.5 1.0 1.5 2.0

0.0 0.5 1.0 1.5 2.0

r

r

Fig. 4 Plot showing the variation of the innite redshift surface with the parameters a, g, and

ferent values of . The event horizon of the rotating Bardeen regular metric (3) is located at r = r+H, the larger root of

= 0, and the static limit surface is located at r = r+sls,

which is zero of gtt = 0. Observing the outer event hori

zon and the stationary limit surface of the rotating Bardeen regular black hole, it is veried that the stationary limit surface always lies outside the event horizon for all values of g. Hence, as in the Kerr black hole, we call the region between two surfaces the ergosphere, which lies outside of the black hole. The ergosphere is given by r+H < r < r+sls, and an important feature of the ergosphere is that the time-like

Killing vector becomes space-like after crossing the static limit surface, i.e., in the ergosphere = gtt > 0. The

existence of the ergosphere allows various kinds of energy extraction mechanisms for a rotating black hole. It has been suggested that the ergosphere can be used to extract energy from rotating black holes through the Penrose process [28]. Further, an observer moving along the time-like geodesics is always stationary in the ergosphere due to the frame-dragging effect [29]. Its shape is that of an oblate spheroidbulging at the equator, and attened at the poles of the rotating black hole. We have studied how the parameters a, g affect the

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a 0.7, g 0 a 0.8, g 0

a 0.998, g 0 a 1, g 0

a 0.7, g 0.2 a 0.8, g 0.2

a 0.941, g 0.2 a 1, g 0.2

a 0.6, g 0.3 a 0.7, g 0.3

a 0.873, g 0.3 a 0.9, g 0.3

a 0.6, g 0.4 a 0.7, g 0.4

a 0.783, g 0.4 a 0.8, g 0.4

a 0.5, g 0.5 a 0.6, g 0.5

a 0.669, g 0.5 a 0.7, g 0.5

Fig. 5 Plot showing the variation of the shape of ergosphere in xz-plane with parameter g, for different values of a, of the rotating Bardeen regular black hole. The blue and the red lines correspond, respectively,

to static limit surfaces and horizons. The outer blue line corresponds to the static limit surface, whereas the two red lines correspond to the two horizons

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Eur. Phys. J. C (2015) 75 :553 Page 7 of 12 553

a 0.9997, g 0 a 0.9421, g 0.2 a 0.8748, g 0.3 a 0.7847, g 0.4

Fig. 6 Plot showing the variation of the shape of ergosphere for a aE (extremal black hole) in xz-plane with parameter g, for different values

of a, of the rotating Bardeen regular black hole

Table 2 The values of the outer static limit surface and event horizon of a rotating Bardeen regular black hole with parameters = 2, = 3,

M = 1, and = /4 (ge = r+sls r+H)

a g = 0 g = 0.2 g = 0.3 g = 0.4r+H r+sls ge r+H r+sls ge r+H r+sls ge r+H r+sls ge

0.5 1.86603 1.93541 0.06938 1.82855 1.90211 0.07356 1.77855 1.85831 0.07976 1.70092 1.79209 0.09117

0.6 1.80000 1.90554 0.10554 1.75807 1.87081 0.11274 1.70115 1.82492 0.12377 1.60973 1.75498 0.14525

0.7 1.71414 1.86891 0.15477 1.66490 1.83227 0.16737 1.59578 1.78355 0.18777 1.47589 1.70843 0.23254

0.8 1.60000 1.82462 0.22462 1.53680 1.78541 0.24861 1.44038 1.73279 0.29241 1.19504 1.65023 0.45519

0.9 1.43589 1.77136 0.33547 1.33282 1.72860 0.39578 1.67041 1.57651

shape of the ergosphere, and the behavior of ergosphere for a < aE is depicted in Fig. 5 and for a aE in Fig. 6. It can

be seen that the ergosphere is sensitive to the parameter g, the ergosphere area enlarges with an increase in the parameter g, as well as with a (cf. Table 2). In the extremal case a

aE (Fig. 6), the inner and outer horizons coincide, and the thickness of the ergosphere still increases with the increase in the parameter g.

3 Equations of motion and the effective potential

In this section, we would like to study the equations of motion of a particle with rest mass m0 falling in the background of the rotating Bardeen regular black hole. Henceforth, we shall restrict our discussion to the case of the equatorial plane ( = /2), which simplies the mass function (5) to

m = M [parenleftbigg]

r2r2 + g2 [parenrightbigg]3/2

[bracketleftbigg]a(aE L) + (r2 + a2)

T

[bracketrightbigg] , (12)

u =

1 r2

[bracketleftbigg](aE L) +

aT

[bracketrightbigg] , (13)

ur =

. (9)

It is easy to see that the mass function (9) can also be obtained from (5) for = = 0. There are two Killing vectors, time-

like (t) and axial () Killing vectors. So there must be two conserved quantities corresponding to these Killing vectors. The conserved quantities corresponding to these time-like and axial Killing vectors are the energy (E) and angular

momentum (L), respectively. The conserved quantities at the equatorial plane are dened by the following equations:

E = gtu = gttut + gtu, (10)

L = gu = gtut + gu, (11)

where u is the four-velocity of the particle. To calculate ECM for the colliding particles, rst of all we need to calculate the four-velocities of the particles. The four-velocities are calculated by solving Eqs. (10) and (11) simultaneously and using the condition uu = m20. Hence the four-velocities

of the falling particles have the following form:

ut =

1 r2

1 r2

T 2 [bracketleftbig]m20r2 + (L aE)2[bracketrightbig],

(14)

where T = E(r2 + a2) La. In Eq. (14) the + sign corre

sponds to the outgoing geodesic and the sign corresponds

to the incoming geodesics. The effective potential is calculated as

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2.0

2.0

L 0.1

L 2.10714

L 3.0

Event horizon

L 0.1

L 2.37934

L 3.0

Event horizon

1.5

1.5

r

1.0

r

1.0

0.5

0.5

0.0

0.0

1 2 3 4 5 6

1 2 3 4 5 6

r

r

Fig. 7 The behavior of r vs. r for an extremal black hole. a The magnetic charge g = 0.2 and a = 0.942439535325. b The magnetic charge

g = 0.4 and a = 0.7851573623591

1.0

1.0

a 0.9; g 0.2

L 5

L 6

L 7

a 0.7; g 0.4

L 5

L 6

L 7

0.5

0.5

V eff

V eff

0.0

0.0

0.5 1 2 3 4 5 6

0.5 1 2 3 4 5 6

r

r

Fig. 8 Plot showing the behavior of Veff vs. r for different angular momenta L

1

2(ur)2 + Veff = 0. (15)

Hence, the form of the effective potential is like

Veff = [

E(r2 + a2) La]2 [m20r2 + (L aE)2]

2r4 .

(16)

L =

aa2 + (r EH)2

The range of the angular momentum for the falling particles is calculated by the following equations:

Veff = 0 and

dVeff

dr = 0. (17)

The plots of r vs. r can be seen from Fig. 7 for different

values of L, a, and g. We can see from this gure that if the angular momentum of the particle L > Lc, then the geodesics never fall into the black hole. On the other hand if the angular momentum L < Lc, then the geodesics always fall into the black hole and if L = Lc, then the geodesics fall

into the black hole exactly at the event horizon. The behavior

of the effective potential (Veff) with radius (r) can be seen from Fig. 8.

For the time-like particles, ut 0, from Eq. (12) the

condition

E[r3 + 2ma2 + a2r] 2mLa, (18) must be satised; as r r EH, this condition reduces to

E

a 2mr EH

L = H L, (19)

where H is the angular velocity at the event horizon.

4 Near horizon collision in the rotating Bardeen regular black hole

Recently, Bandos, Silk, and West (BSW) [30] analyzed the possibility that a Kerr black hole can act as a particle accelerator by studying the collision of two particles near the event horizon of the Kerr black hole and found that the center-of-mass energy (ECM) of the colliding particles in the equato-

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Eur. Phys. J. C (2015) 75 :553 Page 9 of 12 553

Table 3 The limiting values of the angular momentum for different extremal cases of a rotating Bardeen regular black hole

g aE r EH L2 L1

0 1.0 1.00000 4.82843 2.00000

0.2 0.942439535325 1.04769 4.77988 2.10714

0.3 0.875075019710 1.08604 4.72090 2.22294

0.4 0.785157362359 1.11879 4.63854 2.37934

0.5 0.671964660485 1.13979 4.52854 2.60418

0.6 0.529498982374 1.14404 4.38025 3.00134

rial plane can be arbitrarily high in the limiting case of an extremal black hole. Thus the extremal Kerr black hole can act as a Planck energy scale particle accelerator. This gave an opening to explore ultrahigh energy collisions and astrophysical phenomena, such as gamma ray bursts and active galactic nuclei. Hence, the BSW mechanism received significant attention in the study of the collision of two particles near a rotating black hole [20,27,3137] (see also [38] for a

review). In this section, we want to study the properties of ECM asr tends to the event horizonr+H in the case of a rotating

Bardeen regular black hole. Let us nd ECM for two colliding particles with the same rest mass, m1 = m2 = m0, coming

from rest at innity. The collision energy in the center-of-mass frame is dened as

ECM = m02[radicalBig]

1 gu(1)u(2), (20)

where u(1) and u(2) are the four-velocities of the colliding particles. By using Eq. (20) and substituting the values of four-velocities we can get ECM for the black hole (3):

E2CM

2m20 =

1

r(r2 2mr + a2)[bracketleftBig]

2a2(m + r)

2am(L1 + L2)

L1L2(2m + r) + 2(m + r)r2

30

30

2000

1500

L1 2.0L2 4.82843

1000

500

0 0.98 1.00 1.02 1.04 1.06

Event horizon

2000

1500

L1 2.10714 L2 4.77988

1000

500

0

1.040 1.045 1.050 1.055 1.060

Event horizon

25

25

20

20

E CM

15

E CM

15

10

10

5

5

0.8 1.0 1.2 1.4 1.6 1.8

1.0 1.1 1.2 1.3 1.4

r

r

30

30

2000

1500

L1 2.22294 L2 4.72090

1000

500

0

1.080 1.085 1.090 1.095 1.100

Event horizon

2000

1500

L1 2.37934 L2 4.63854

1000

500

0 1.10 1.11 1.12 1.13 1.14 1.15 1.16

Event horizon

25

25

20

20

E CM

E CM

15

15

10

10

5

5

1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40

1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40

r

r

Fig. 9 Plot showing the behavior of ECM vs. r for an extremal black hole. Top For the magnetic charge g = 0, spin aE = 1, angular momen

tum L1 = 2.0 (blue), 1.8 (green), 1.5 (red), and L2 = 4.82843 (left).

For g = 0.2, aE = 0.942439535325, L1 = 2.10714 (blue), 2.02

(green), 1.90 (red), and L2 = 4.77988 (right). Bottom For g = 0.3,

aE = 0.875075019710, L1 = 2.22294 (blue), 2.12 (green), 1.95 (red),

and L2 = 4.72090 (left). For g = 0.4, aE = 0.7851573623591,

L1 = 2.37934 (blue), 2.27 (green), 2.10 (red), and L2 = 4.63854

(right)

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553 Page 10 of 12 Eur. Phys. J. C (2015) 75 :553

Table 4 The limiting values of the angular momentum for different non-extremal cases of a rotating Bardeen regular black hole

g a r+H rH L4 L3

0 0.95 1.31225 0.68775 4.79285 2.44721

0.2 0.90 1.31866 0.78085 4.74908 2.49499

0.3 0.80 1.42484 0.75247 4.66471 2.70926

0.4 0.70 1.45731 0.77907 4.57216 2.84820

0.5 0.60 1.43116 0.83941 4.46958 2.94361

0.6 0.50 1.31840 0.96243 4.38474 3.18263

[radicalBig]

2m(a L1)2 L21r + 2mr2

[radicalBig]

0.81L2i (i = 1, 2). Equation (22) suggests that an unlimited

ECM is possible if one of the particles has the critical angular momentum Lc. Therefore, ECM is divergent at the horizon r = rE, if one of the particles satises the critical condi

tion E H Lc = 0. The restriction on the Lc is shown in

Table 3. The numerical value of the critical angular momentum is Lc = E/ H = 2.10714, which is exactly the same

as L1. Hence, we can say that ECM is innite for the extremal rotating Bardeen regular black hole. We plot in Fig. 9 ECM vs. r for various values of the angular momentum.

4.1 Near horizon collision in non-extremal rotating Bardeen regular black hole

Finally, we study the properties of ECM in the limitr r+H of

a non-extremal rotating Bardeen regular black hole. Again as r r+H, both numerator and denominator of Eq. (21) vanish.

Hence, by application of lHospitals rule, we nd that ECM, for the near horizon case for a non-extremal rotating Bardeen black hole, reads

E2CM

2m20

(r r+H) =

2m(a L2)2 L22r + 2mr2[bracketrightBig]

. (21)

Here m = (Mr3)/(r2 + g2)3/2. Thus the above equation is g

dependent, and when g = 0, it will look exactly the same with

m replaced by M, which is also exactly the same as obtained for the Kerr black hole [30]. Obviously as r r EH, Eq. (21)

has an indeterminate form when we choose numerical values of r EH, a, M, and g. We apply lHospitals rule twice, then the value of the ECM, as r r EH, becomes

E2CM

2m20

1(L3 L c)(L4 L c)

[12.14

+1.19(L23 + L24)

4.28(L3 + L4) 0.86L3L4] . (23)

In the above calculation we have xed a = 0.9, r = r+H =

1.31866, and g = 0.2. Equation (23) is the formula for ECM

of two colliding particles for a non-extremal rotating Bardeen black hole. ECM will be innite if either L3 or L4 is equal to

L c = E/ H, where L4 < L < L3 is the range for the angu

lar momentum with which a particle can reach the horizon. The critical value of the angular momentum is calculated as L c = 2.83213. Hence, L c is not in the acceptable range (cf.

Table 4). Therefore, we can say that ECM is nite in the case of a non-extremal black hole (cf. Figs. 10 and 11).

(r r EH) = 8.08 0.51L1L2 + 0.48(L1 + L2)

A1(L2 Lc) 2(L1 Lc)

A2(L1 Lc) 2(L2 Lc)

B1B2

3(L1 Lc)(L2 Lc) +

B21(L2Lc) 6(L1Lc)3

+

B22(L1 Lc) 6(L2 Lc)3

, (22)

with xed a = aE = 0.942439535325, r = r EH = 1.04769,

M = 1, and g = 0.2. The constants Ai and Bi correspond to

Ai = 3.56 + 0.91Li 0.51L2i and Bi = 4.35 0.35Li

30

30

6.0

5.8

5.6

5.4

5.2

1.38 1.39 1.40 1.41 1.42

Event horizon

4.9

4.8

4.7

4.6

4.5

4.4 1.68 1.69 1.70 1.71 1.72

Event horizon

25

25

20

20

E CM

15

E CM

15

10

10

5

5

1.0 1.1 1.2 1.3 1.4 1.5 1.6

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

r

r

Fig. 10 Plot showing the behavior of ECM vs. r. Left For aE = 0.942439535325, L1 = 2.10714, L2 = 4.77988, and g = 0.2 (blue), 0.24

(green), 0.28 (red). Right For a = 0.7851573623591, L1 = 2.37934, L2 = 4.63854, and g = 0.4 (blue), 0.44 (green), 0.48 (red)

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8

8

L 2.40, L 4.79285

L 2.10, L 4.79285

L 1.60, L 4.79285

Event horizon

L 2.47, L 4.74908

L 2.20, L 4.74908

L 1.70, L 4.74908

Event horizon

7

7

6

6

E CM

E CM

5

5

4

4

3

3

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

r

r

8

8

L 2.67, L 4.66471

L 2.30, L 4.66471

L 1.80, L 4.66471

Event horizon

L 2.82, L 4.57216

L 2.40, L 4.57216

L 1.90, L 4.57216

Event horizon

7

7

6

6

E CM

E CM

5

5

4

4

3

3

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

r

r

Fig. 11 Plot showing the behavior of ECM vs. r for non-extremal black hole. Top For a = 0.95, g = 0 (left). For a = 0.9, g = 0.2 (right). Bottom

For a = 0.8, g = 0.3 (left). For a = 0.7, g = 0.4 (right)

5 Conclusion

There have been made many efforts to push Einstein gravity as much as possible to its limit, trying to avoid the central singularity. Following some very early ideas of Sakharov [5], Gliner [6], and Bardeen [7], solutions possessing a global structure like the one of black hole spacetimes, but in which the central singularity is absent, have been found. The rst regular black hole solution having an event horizon was obtained by Bardeen [7], which is a solution of Einstein equations in the presence of an electromagnetic eld. Recently, the rotating Bardeen regular black hole was also found [17].Further, astrophysical black hole candidates are thought to be the Kerr black holes of general relativity, however, the actual nature of these objects is still to be veried. In this paper, we have investigated in detail the horizons and ergosphere for the rotating Bardeen regular black hole and also analyzed the possibility that it can act as a particle accelerator by studying the collision of two particles falling freely from rest at innity. The horizon structure of the rotating Bardeen regular black hole is complicated as compared to the Kerr black hole.It has been observed that, for each g and suitable choice of the parameters, we can nd the critical value a = aE, which

corresponds to an extremal black hole with degenerate horizons; i.e., when a = aE, the two horizons coincide (cf. Figs. 1

and 2). Interestingly, the value aE is sensitive to the parameter g: aE decreases with the increase in g. However, when a < aE, we have a regular black hole with Cauchy and event horizon, and for a > aE, no horizon exists. Furthermore, the ergosphere area increases with the increase in a. On the other hand, when the value of g increases, the ergosphere area enlarges. It has been suggested that the ergosphere can be used to extract energy from rotating black holes through the Penrose process [28]. However, Penrose himself said that the method is inefcient [39], although later [29] showed that the theoretical efciency could reach 20% extra energy up to 60%. In the case of a rotating black hole the efciency of the collisional process is not high, if a magnetic eld effect is not involved, rather it is efcient in the case of a Kerr naked singularity [40,41]. Hence, the parameter g may play a signicant role in the energy extraction process from rotating Bardeen regular black holes, which is being investigated separately.

It has been shown by BSW [30] that ECM of two colliding particles may occur at an arbitrarily high energy for the case of extremal Kerr black holes. The BSW analysis when

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extended to the rotating Bardeen regular black hole, with some restriction on the parameters, shows an arbitrary high ECM can be achieved when the collision takes place near the horizon of an extremal rotating Bardeen regular black hole; one of the colliding particles must have a critical angular momentum. We have also calculated the range of the angular momentum for which a particle may reach the horizon; i.e., if the angular momentum lies in the range, the collision is possible near the horizon of the rotating Bardeen regular black holes. The calculation of ECM was performed in the case of a rotating Bardeen regular black hole for various values of g, and we found that they are innite if one of the colliding particles has the critical angular momentum in the required range.On the other hand, ECM has always a nite upper limit for the non-extremal rotating Bardeen regular black hole. Thus, the BSW mechanism, for the rotating Bardeen regular black hole, depends both on the rotation parameter a as well as on the deviation parameter g. For the non-extremal black hole, we have also seen the effect of g on ECM, demonstrating an increase in the value of the ECM with an increase in the value of g.

According to the no-hair theorem, all astrophysical black holes are expected to be like Kerr black holes, but the actual nature of this has still to be veried [18]. The impact of the parameter g on the horizon structure, ergoregion, and particle acceleration presents a good theoretical opportunity to distinguish the rotating Bardeen regular black hole from the Kerr black and to test whether astrophysical black hole candidates are the black holes as predicted by Einsteins general relativity.

Acknowledgments M.A. acknowledges the University Grant Commission, India, for nancial 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, also would like to thank IUCAA, Pune, for hospitality while part of this work was done.

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