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Web End = Eur. Phys. J. C (2016) 76:123DOI 10.1140/epjc/s10052-016-3967-2

Regular Article - Theoretical Physics

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Web End = Accretion onto some well-known regular black holes

Abdul Jawad1,a, M. Umair Shahzad1,b

1 Department of Mathematics, COMSATS Institute of Information Technology, Lahore 54000, Pakistan

Received: 19 January 2016 / Accepted: 17 February 2016 / Published online: 7 March 2016 The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract In this work, we discuss the accretion onto static spherically symmetric regular black holes for specic choices of the equation of state parameter. The underlying regular black holes are charged regular black holes using the FermiDirac distribution, logistic distribution, nonlinear electrodynamics, respectively, and KehagiasSftesos asymptotically at regular black holes. We obtain the critical radius, critical speed, and squared sound speed during the accretion process near the regular black holes. We also study the behavior of radial velocity, energy density, and the rate of change of the mass for each of the regular black holes.

1 Introduction

At present, the type 1a supernova [1], cosmic microwave background (CMB) radiation [2], and the large scale structure [3,4] have shown that our universe is currently in an accelerating expansion period. Dark energy is responsible for this acceleration and it has the strange property that it violates the null energy condition (NEC) and the weak energy condition (WEC) [5,6] and produces strong repulsive gravitational effects. Recent observations suggests that approximately 74 % of our universe is occupied by dark energy and the rest 22 and 4 % is of dark matter and ordinary matter, respectively. Nowadays dark energy is the most challenging problem in astrophysics. Many theories have been proposed to handle this important problem in last two decades. Dark energy is modeled using the relationship between energy density and pressure by a perfect uid with the equation of state (EoS) = p. The candidates of dark energy are a phantom-

like uid ( < 1), quintessence (1 < < 1/3),

and the cosmological constant ( = 1) [7]. Other mod

els are also proposed as an explanation of dark energy, like

a e-mails: mailto:[email protected]

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

Web End [email protected]

b e-mail: mailto:[email protected]

Web End [email protected]

k-essence, DBI-essence, Hessence, dilation, tachyons, Chap-

lygin gas, etc. [816].

On the other hand, the existence of essential singularities [which leads to various black holes (BHs)] is one of the major problems in general relativity (GR) and it seems to be a common property in most of the solutions of Einsteins eld equations. To avoid these singularities, regular BHs (RBHs) have been developed. These BHs are solutions of Einsteins equation with no essential singularity; hence their metric is regular everywhere. The strong energy condition (SEC) is violated by these RBHs somewhere in space-time [17,18], while some of these satisfy the WEC. However, it is necessary for those RBHs to satisfy the WEC having a de Sitter center. The study of an RBHs solutions is very important for understanding the gravitational collapse. Since the Penrose cosmic censorship conjecture claims that singularities predicted by GR [19,20] occur, they must be explained by event horizons. Bardeen [21] has done pioneering work in this way by presenting the RBH known as the Bardeen black hole, satisfying the WEC.

The discussion as regards the properties of the BHs have led to many interesting phenomena. Accretion onto the BHs is one of them. When massive condensed objects (e.g. black holes, neutron stars, stars etc.) try to capture a particle of the uid from its surroundings, then the mass of condensed object has been effected. This process is known as accretion of uid by condensed object. Due to accretion the planets and star form inhomogeneous regions of dust and gas. Supermassive BHs exist at the center of giant galaxies, which suggests that they could have formed through an accretion process. It is not necessary that the mass of the BH increases due to the accretion process, sometimes in-falling matter is thrown away like cosmic rays [22]. For a rst time, the problem of accretion on a compact object was investigated by Bondi using the Newtonian theory of gravity [23]. After that many researchers such as Michel [24], Babichev et al. [25,26], Jamil [27] and Debnath [31] have discussed the accretion on Schwarzschild BHs under different aspects. Kim and Kang [29] and Jimenez

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123 Page 2 of 11 Eur. Phys. J. C (2016) 76 :123

Madrid and Gonzalez-Diaz [30] studied accretion of dark energy on a static BH and a KerrNewman BH. Sharif and Abbas [28] discussed the accretion on stringy charged BHs due to phantom energy.

Recently, the framework of accretion on general static spherical symmetric BHs has been presented by Bahamonde and Jamil [22]. We have extended this general formalism for some RBHs. We analyze the effect of the mass of a RBH by choosing different values of the EoS parameter. This paper is organized as follows: In Sect. 2, we derive a general formalism for a spherically static accretion process. In Sect. 3, we discuss some RBHs and for each case, we explain the critical radius, critical points, speed of sound, radial velocities prole, energy density, and the rate of change of the RBH mass.In the end, we conclude our results.

2 General formalism for accretion

The generalized static spherical symmetry is characterized by the following line element:

ds2 = X(r)dt2 +

energy or any kind of dark matter. For a spherically symmetric BH, the proper dark energy model could be obtained by generalizing Michels theory. In dark energy accretion, Babichev et al. [25] have introduced the above generalization of the Schwarzschild black hole. Similarly, some authors [22,31] have extended this procedure for a generalized static spherically symmetric BH. In these works, the equation of continuity plays an important role, which turns out to be

( + p)u

X(r)

Y (r)

u2 + Y (r)Z(r) = A0, (5)

where A0 is the constant of integration. Using uT = 0,

we obtain the continuity (or relativistic energy ux) equation

u, + ( + p)u; = 0. (6)

Furthermore, we assume p = p(), a certain EoS in this

case. After some calculations, the above equation becomes

+ p +

u u +

X 2X +

Y 2Y +

1Y (r)dr2

+Z(r)(d2 + sin 2d2), (1) where X(r) > 0, Y (r) > 0, and Z(r) > 0 are functions of r only. The energy-momentum tensor is considered in terms of a perfect uid which is isotropic and inhomogeneous and dened as follows:

T = ( + p)uu + pg, (2)

where p is the pressure, is the energy density, and u is the four-velocity, which is given by

u =

Z = 0, (7)

where a prime represents the derivative with respect to r. By integrating the last equation, we obtain

uZ(r)

X(r)

Y (r) e d+p() = A1, (8)

where A1 is the constant of integration. By equating Eqs. (5) and (8), we get

( + p)

u2 + Y e d+p() =

A0A1 = A3, (9)

where A3 is another constant, depending upon A0 and A1. Moreover, the equation of mass ux yields

X(r) Y (r)

dxd = (ut, ur, 0, 0), (3)

where is the proper time. u and u both are equal to zero due to spherical symmetry restrictions. Here the pressure, the energy density, and the four-velocity components are only functions of r. The normalization condition of the four-velocity must satisfy uu = 1, and we get

ut :=

dtd =

u2 + YXY , (4)

where u = dr/d = ur [22], ut can be negative or positive

due to the square root which represents the backward or forward in time conditions. However, u < 0 is required for the accretion process, otherwise for any outward ows u > 0. Both inward and outward ows are very important in astrophysics. One can assume that the uid is determined by dark

X(r)

Y (r) Z(r) = A2, (10)

where A2 is the constant of integration. By using Eqs. (5) and (10), we obtain the following important relation:

( + p)

u2 + Y =

A1A2 A4, (11)

where A4 is arbitrary constant which depends on A1 and A2. Taking differentials of Eqs. (10) and (11) and some manipulation lead to

X(r) Y (r)

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Eur. Phys. J. C (2016) 76 :123 Page 3 of 11 123

V 2 u2u2 + Y duu +

(V 2 1)

Y Y

Z V 2

Y 2(u2 + Y )

dr = 0. (12)

In addition, we have introduced the variable

V 2

d ln + p

d ln 1. (13)

If the bracketed terms in Eq. (12) vanish, we obtain the critical point (where the speed of sound equals the speed of the ow), which is located at r = rc. Hence at the critical point, we get

V 2c =

Z(rc)X (rc)

2Z (rc) ,

V 2c =

u2cu2c + Y (rc)

Z(rc)X (rc)

2X(rc)Z (rc) + Z(rc)X (rc)

, (14)

and Eq. (12) turns out to be

(V 2c 1)

. (20)

Although our focus is on charged RBHs metrics with event horizons, the present analysis is forbidden for a horizon space-time. In many cases, we are concerned with critical values (critical radius), critical velocities, speed of sound in uid, behavior of energy density of uid, radial velocity, and the rate of change of the mass of the accreting objects. So the horizon is not involved anywhere [22].

3.1 Charged RBH using FermiDirac distribution

The said RBH solution has the following metric functions [32]:

X(r) = 1

X (rc)

X(rc)

Y (rc)

Z (rc)

Z(rc) V 2c

Y (rc)

= 2(u2c + Y (rc))

. (15)

Also, uc is the critical speed of the ow evaluated at the critical value r = rc. We can decouple the above two equations

and obtain

u2c =

Y (rc)Z(rc)X (rc)

2X(rc)Z (rc) ,

V 2c =

Z(rc)X (rc)

2X(rc)Z (rc) + Z(rc)X (rc)

2M r

(x)(r)

. (16)

The speed of sound is evaluated at r = rc as follows:

c2s =

= Y (r), (21)

where the FermiDirac distribution function is

(x) =

1ex + 1

d pd |r=rc = A4

Y (rc)

X(rc)(u2c + Y (rc))

. (22)

By replacing x =

q2Mr , we can obtain the distribution func-

tion as

1. (17)

Obviously, u2c and V 2c can never be negative and hence

X (rc)

Z (rc) > 0. (18)

Moreover, the rate of change of the BH mass can be dened as follows [31]:

Macc = 4 A3M2( + p). (19)

Here a dot is for a derivative with respect to time. We can observe that the mass of the BH will increase for the uid, + p > 0, and hence the accretion occurs outside the BH.

Otherwise, for + p < 0 like a uid, the mass of the BH will

decrease. The mass of the BH cannot remain xed because it will decrease due to Hawking radiation, while it will increase

due to accretion. If we consider the time dependence of the BH mass, then we rst assume that it will not change the geometry and symmetry of space-time. Hence the space-time metric remains static spherically symmetric [22].

3 Spherically symmetric metrics with charged RBHs

In this section, we discuss the spherically symmetric metrics with charged RBHs in which X(r) = Y (r). For this assump

tion, Eq. (16) gives

u2c =

(r) =

1 , (23)

with normalization factor =

2 . Also the distribution

function satises

q2 Mr

+ 1

(r)

1, (24)

where r . Hence the metric functions turn out to be

X(r) = Y (r) = 1

2M r

q2 Mr

+ 1

, Z(r) = r2.

(25)

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Fig. 1 Velocity prole against x =

rM for = 1, q = 1.055M,

M = 1, and A4 = 0.45 of the charged RBH using the FermiDirac

distribution

If we set 0 and , we obtain

X(r) = Y (r) = 1

rM for = 1, q = 1.055M, M = 1,

A2 = 1, and A4 = 0.45 of the charged RBH using the FermiDirac

distribution

1/3 and < 1 refer to quintessence and phantom energy.

It can be seen that for = 1.5, 2 the radial velocity of the

uid is negative and it is positive for = 0.5, 0, 0.5, 1. If

the ow is outward then u < 0 is not allowed and vice versa. In the case of = 1.5, 0.5 the uid is at rest at x = 10.

Figure 2 represents the behavior of energy density of uids in the surrounding area of the RBH. Obviously the WEC and DEC satised by dust, stiff, and quintessence uids. When the phantom uid ( = 1.5, 2) moves toward the RBH

then the energy density decreases and the reverse will happen for dust, stiff, and quintessence uids ( = 0.5, 0, 0.5, 1).

Asymptotically 0 at innity for = 1.5, 0.5, while

it approaches the maximum at x = 1.2, 1.3, 1.8 and near the

RBH.

Using this metric, Eqs. (19) and (29), the rate of change of the mass of the RBH due to accretion becomes

M =

Fig. 2 Energy density against x =

Mr , (26)

2Mr e

X(r) = Y (r) = 1

2Mr e

q22Mr . (27)

In both equations, the difference of the factor 2 must be noted [32].

It is possible to integrate the conversation laws and obtain analytical expressions of the physical parameters. For simplicity, we will study the barotropic case where the uid has the equation p(r) = (r). Using (5) and (11), we obtain u(r)

12 eq2Mr+ 12 r

( + 1)2 + A24r 1/2

( + 1)r

(28)

4 A22 A4( + 1)

r3/2

12 eq2Mr+ 12 r

( + 1)2 + A24r

(r)

A2( + 1)

r3/2

12 eq2Mr+ 12 r

( + 1)2 + A24r

(29)

(30)

Figure 3 represents the change in RBH mass for different values of . The mass of the RBH will increase near it and at x = 1.2, 1.3, 1.7 for = 1, 0.5, 0, respectively. On the

other hand, the mass of the RBH decreases near it and at x = 1.7 for = 2. Hence the mass of the RBH increases

due to the accretion of quintessence, dust, and stiff matter,

The velocity prole for different values of is shown in Fig. 1. Here = 1, 0, 1 refer to the stiff, dust, and

cosmological constant cases, respectively, and 1 < <

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Eur. Phys. J. C (2016) 76 :123 Page 5 of 11 123

Mr q2

q2 Mr

+Mr

V 2c =

2r2

eq2 Mr

q2 Mr

3Mr +q2

+3Mr

(32)

Also, the condition (18) yields

Mr q2

q2 Mr

+ Mr

> 0. (33)

3.2 Charged RBH using logistic distribution

The logistic distribution function is [32]

(x) =

eq2 Mr+ 1

(ex + 1)2

, (34)

rM for = 1,

q = 1.055M, M = 1, A2 = 1, and A4 = 0.45 of a charged RBH using

the FermiDirac distribution

Table 1 Charged RBH using the FermiDirac distribution

rc u(rc) c2s

2 1.37495 0.3138832070 0.0000002580

1.5 7.5044 0.2382908936 0.4999997476

0.5 7.5044 0.2382908936 0.4999997476

0 1.3749 0.3138832070 0.0000002580

0.5 1.092 0.2476468259 0.503110174

1 0.999 0.1998921298 1.002986469

while it decreases due to the accretion of phantom-like uids.

The critical values, critical velocities, and speed of sound are obtained for different values of the EoS parameter in Table 1. The critical radius is shifting to the left when

0 increases. Thus, the in-falling uid acquires supersonic speeds closer to RBH. The same critical radius is obtained for = 2, 0 and = 1.5, 0.5 with the same critical

velocities but in an opposite direction. We get a negative speed of sound at x = 7.5044 and a positive speed of sound

for the remaining critical radius. Also, the speed of sound increases near the RBH. For this metric, we nd that

Fig. 3 Rate of change of the mass of a RBH against x =

in which we replace x =

2q2Mr ; then we obtain the distribution

function

2q2 Mr

(r) =

2 , (35)

with normalization factor =

14 . Also the distribution

function satises

e 2q2Mr+ 1

(r)

1, (36)

where r . The horizons can be obtained for = 1

where q = 1.055M. The metric function can be written as

X(r) = Y (r) = 1

2q2 Mr

2M r

2q2 Mr

+ 1

Z(r) = r2. (37) If we set 0, then we obtain the Schwarzschild BH, and

if we set we get

X(r) = Y (r) = 1

2Mr e

q22Mr . (38)

It is noteworthy that this metric function corresponds to an Ayon-Beato and Garca BH [32].

The radial velocity and energy density for the metric (37) using Eqs. (5) and (10) is given by

q2 Mr

Mr q2

+ Mr

u2c =

, (31)

eq2 Mr+ 1

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123 Page 6 of 11 Eur. Phys. J. C (2016) 76 :123

rM for = 1, q = 1.055M,

M = 1, and A4 = 0.4 of a charged RBH using the logistic distribution

u(r) =

1( + 1)r

Fig. 4 Velocity prole against x =

rM for = 1, q = 1.055M,

M = 1, A2 = 1, and A4 = 0.4 of a charged RBH using logistic

distribution

The M of an RBH for distinct EoS parameters is obtained

by using (19),

4 A22 A4( + 1)

1/2

2q2 Mr

(+1)2 A24r

2q2 Mr

1/2 .

(39)

2q2 Mr

r3/2

(+1)

2 +A24r

2q2 Mr

(r) =

A2( + 1)

1/2 .

(41)

Figure 6 represents the change in the RBH mass against x. It is evident that the mass of the RBH increases due to quintessence, dust, and stiff uids and it decreases due to phantom uids.

The critical radius, the critical velocity, and the speed of sound are obtained for different values of EoS parameter in Table 2. The critical radius is shifting to the right when 0 increases. Thus the in-falling uid acquires super-

sonic speeds closer to the RBH. For a phantom-like uid, quintessence, dust, and stiff matter the critical radius and critical velocities are explained in Table 2. Same critical radius is obtained for = 2, 0 and = 1.5, 0.5 with the

same critical velocities but different in sign. We obtained a negative speed of sound at x = 1.36375, 3.777412 and pos

itive speed of sound at x = 1.12974, 1.1850. Near the RBH

the speed of sound will increase. For this metric we nd that

2q2 Mr

r3/2

(+1)2+ A24r

2q2 Mr

(40)

The velocity prole for different values of is shown in Fig. 4. It can be observed that for = 1.5, 2 the

radial velocity of the uid is negative and it is positive for = 0.5, 0, 1. If the ow is inward then u > 0 is

not allowed and vice versa. In the case of = 2, 0 the

uid is at rest at x 5. Figure 5 represents the behavior

of energy density of uids in the surrounding area of the RBH. Obviously the WEC and DEC are satised by dust, stiff, and quintessence uids. When a phantom-like uid ( = 1.5, 2) moves toward a RBH the energy density

decreases and the reverse will happen for dust, stiff, and quintessence uids ( = 0.5, 0, 0.5, 1).

Fig. 5 The energy density against x =

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Eur. Phys. J. C (2016) 76 :123 Page 7 of 11 123

Table 2 Charged RBH using logistics distribution

rc u(rc) c2s

2 1.36375 0.3998729763 0.1018620364

1.5 3.777412 0.3138895411 0.5007414180

0.5 3.77412 0.3138895411 0.5007414180

0 1.36375 0.3998724197 0.1018620364

0.5 1.1850 0.4018068205 0.116622918

1 1.12974 0.4014558621 0.231766770

Here the function

M(r) = M

1 tan h q2 2Mr

, (46)

and its associated electric eld source is

E =

1 tan h2 q2 2Mr

rM for = 1,

q = 1.055M, M = 1, A2 = 1, and A4 = 0.4 of a charged RBH using

the logistic distribution

Fig. 6 Rate of change of the mass of a RBH against x =

1 q24Mr tan h

q2 2Mr

, (47)

22+2Me

2q2 Mr

2q + 2 Mr +

2q + 2 Mr

2q2 Mr

u2c =

, (42)

r3/2

2q2 Mr

V 2c =

2 Me

2q2 Mr

2q + 2

2q2 Mr

Mr3/2

2q2 Mr

2Mq + 6

r M3/2

2q2 Mr

. (43)

+ 2Mq + 6

r M3/2

Also, the condition (18) yields

21+2Me

2q2 Mr

2q + 2 Mr

2q2 Mr

+ 2q + 2 Mr

> 0. (44)

r7/2

2q2 Mr

3.3 Charged RBH from nonlinear electrodynamics

We use the line element

X(r) = Y (r) = 1

where q and M represent the electric charge and the mass, respectively [33]. The solution elaborates RBH and its global structure is like R-N BH. The asymptotic behavior of the solution is

X(r) = 1

2M(r)r . (45)

2Mr +

r2 + O

1 r4

. (48)

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Fig. 7 Velocity prole against x =

rM for q = 1.055M, M = 1, and

A4 = 0.7 of a charged RBH from nonlinear electrodynamics

So we have the metric function

X(r) = Y (r)

= 1

rM for q = 1.055M, M = 1, A2 =

1, and A4 = 0.7 of a charged RBH from nonlinear electrodynamics

The rate of change of the mass is given by

M =

4 A22 A4(+1)

1 tan h(q22Mr ) , Z(r) = r2.

(49)

The radial velocity and energy density for this metric are given by

u(r) =

2M r

(52)

The rate of change of in the RBH mass against x is plotted in Fig. 9. Due to accretion of dust and stiff matter the mass of the RBH will increase for small values of x and vice versa for phantom uids. It is also noted that the maximum rate of the RBH mass increases due to = 1 followed by =

0.5, 0, 2.

The critical values, critical velocities, and speed of sound are obtained for different values of the EoS parameter in Table 3. The critical radius is shifting to the right when 0 increases. The speed of sound is

negative at x = 3.685523529 and near the BH the

speed of sound will increase. For this RBH we nd

that

u2c =

r3/2

2M tan h

2Mr

+2M r

(+1)2+ A24r

2M tan h

2Mr

2M r

( + 1)2 + A24r

( + 1)r

(50)

(r) =

( + 1)A2

(51)

The absolute value of the velocity prole for different values of is shown in Fig. 7. It can be observed that for = 2 the radial velocity of the uid is negative and it is

positive for = 0.5, 0, 1. If the ow is inward then u > 0 is

not allowed and vice versa. In the case of = 2, 0 the uid

is at rest at x 5. Figure 8 represents the energy density of

uids in the region of the RBH. It is apparent that the WEC and DEC is satised by phantom uids. When the phantom uids moves toward the RBH the energy density increases; on the other hand it decreases for dust and stiff matter.

Fig. 8 Energy density against x =

r3/2

2M tan h( q

2 +2M r)

(+1)2+ A24r

tan h2 q2 2Mr

1 + 2Mr

tan h q2 2Mr

4r2 ,(53)

tan h2 q2 2Mr

V 2c =

+2Mr

tan h q2 2Mr

4r2+q2

tan h2

q2 2Mr

+6+Mr

tan h q2 2Mr

(54)

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Eur. Phys. J. C (2016) 76 :123 Page 9 of 11 123

rM for

q = 1.055M, M = 1, A2 = 1, and A4 = 0.7 of charged RBH from

nonlinear electrodynamics

Table 3 Charged RBH from nonlinear electrodynamics

rc u(rc) c2s

2 3.685523529 0.2993097288 0.125

0 3.685523529 0.2993097288 0.125

0.5 1.506050868 0.2844719573 0.312500584

1 1.106971797 0.1633564212 0.750003072

Fig. 9 Rate of change of the mass of an RBH against x =

Fig. 10 Velocity prole against x =

rM for M = 1, b = 0.9 and

A4 = 0.9 of a KehagiasSftesos asymptotically at BH

for r (rb )1/3. The KS metric has two horizons at

r = M

2bM2

, (58)

with 2bM2 1 [34].

The radial velocity and energy density are given by

u(r) =

A24 + b2r4 + 4Mbr br2 1 ( + 1)2 1/2

+ 1

(59)

Also, the condition (18) yields

tan h2

q2 2Mr

1 q2 + 2Mr

tan h

q2 2Mr

2r4 > 0.(55)

3.4 KehagiasSftesos asymptotically at BH

KS studied the following BH metric:

X(r) = Y (r) = 1 + br2 b2r4 + 4Mbr, Z(r) = r2.(56)

In the frame work of Horava theory, where m is the mass, b is the positive constant related to the coupling constant of the theory. The metric asymptotically behaves like the usual Schwarzschild BH [34],

X(r) = Y (r) 1

(r) =

A2( + 1)

A24+ b2r4+4Mbr br21 ( + 1)2 1/2

(60)

The radial velocity for different values of is shown in Fig. 10. The radial velocity is negative for a phantom-like uid and positive for quintessence, dust, and stiff matter. The evolution of the energy density of the uids in the surrounding area of an RBH is plotted in Fig. 11. The energy density for phantom uids is negative, while the energy density for stiff, dust, and quintessence uids is positive.

For this RBH, rate of change of the mass becomes

M =

4 A22 A4( + 1)

2Mr + O

1 r4

, (57)

A24 + b2r4 + 4Mbr br2 1 ( + 1)2 1/2

(61)

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Table 4 KehagiasSftesos asymptotically at BH

rc u(rc) c2s

2 7.8946 0.2505321935 0.0000008330

0.34 30267.74 0.6327458490 0.0513167023

0 7.8946 0.2505321736 0.0000008330

0.5 2.3185 0.3961993774 0.500013404

1 1.8183 0.3888079314 0.500013404

for dust and phantom-like uids and the same critical velocities but different in sign. If we increase the EoS parameter then the critical radius is shifted near RBH. It is evident that the critical velocity is negative for a phantom-like uid and positive for quintessence, dust, and stiff matter. The speed of sound is negative at x = 30267.74 and positive for the

remaining critical radius. For this metric, we nd that

u2c =

2br

Fig. 11 Energy density against x =

rM for M = 1, A2 = 1, b = 0.9,

and A4 = 0.9 of the KehagiasSftesos asymptotically at BH

Fig. 12 Rate of change of the mass of an RBH against x =

r 4

2b2r3 + 2Mb

b2r3 + 4Mb

, (62)

2br 2b2r3+2Mb r(b2r3+4Mb)

b2r3+4Mb

V 2c =

2br 2b2r3+2Mb r(b2r3+4Mb)

+4r

1+br2 r

(63)

The condition (18) becomes

2br 2b2r3+2Mb r(b2r3+4Mb)

2r > 0. (64)

4 Concluding remarks

In this work, we have investigated the accretion onto various RBHs (such as an RBH using the FermiDirac distribution, a RBH using the logistic distribution, an RBH using nonlinear electrodynamics, and a KehagiasSftesos asymptotically at RBH) which asymptotically leads to Schwarzschild and ReissnerNordstrom BHs (most of them satisfy the WEC). We have followed the procedure of Bahamonde and Jamil [22] and obtained the critical points, critical velocities, and the behavior of the speed of sound for the chosen RBHs. Moreover, we have analyzed the behavior of the radial velocity, the energy density, and the rate of change of the mass for RBHs for various EoS parameters. For calculating these quantities, we have assumed the barotropic EoS and found the relationship between the conservation law and the barotropic EoS. We have found that the radial velocity (u) of the uid is positive for stiff, dust, and quintessence matter and it is

rM for

M = 1, A2 = 1, b = 0.9, and A4 = 0.9 of a KehagiasSftesos

asymptotically at BH

Figure 12 represents the rate of change in an RBH mass against x. We see that the RBH mass will increase for =

0.35, 0, 0.5, 1, and it will decrease for = 2.

The critical values, critical velocities, and speeds of sound for different values of are presented in Table 4. For quintessence matter, we obtain a very large critical radius. Similarly to the case before, we obtain the same critical radius

123

Eur. Phys. J. C (2016) 76 :123 Page 11 of 11 123

negative for phantom-like uids. If the ow is inward then u < 0 is not allowed and u > 0 is not allowed for outward ow. Also, we have seen that the energy density remains positive for quintessence, dust, and stiff matter, while it becomes negative for a phantom-like uid near RBHs.

In addition, the rate of change of the mass of the BH is a dynamical quantity, so the analysis of the nature of its mass in the presence of various dark energy models may become very interesting in the present scenario. Also, the sensitivity (increasing or decreasing) of the BHs mass depends upon the nature of the uids which accrete onto it. Therefore, we have considered the various possibilities of accreting uids, such as dust and stiff matter, quintessence, and phantom. We have found that the rate of change of the mass of all RBHs increases for dust and stiff matter, and quintessence-like uids, since these uids do not have enough repulsive force.However, the mass of all RBHs decreases in the presence of a phantom-like uid (and the corresponding energy density and radial velocity become negative) because it has a strong negative pressure. This result shows the consistency with several works [22,31,3547]. Also, this result favors the phenomenon that the universe undergoes the big rip singularity, where all the gravitationally bounded objects are dispersed due to the phantom dark energy.

Although we have assumed the presence of a static uid, this may be extended for a non-static uid without assuming any EoS and thus can be obtained more interesting results.This is left for future considerations.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/

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Web End =ons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP3.

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In this work, we discuss the accretion onto static spherically symmetric regular black holes for specific choices of the equation of state parameter. The underlying regular black holes are charged regular black holes using the Fermi-Dirac distribution, logistic distribution, nonlinear electrodynamics, respectively, and Kehagias-Sftesos asymptotically flat regular black holes. We obtain the critical radius, critical speed, and squared sound speed during the accretion process near the regular black holes. We also study the behavior of radial velocity, energy density, and the rate of change of the mass for each of the regular black holes.

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Title

Accretion onto some well-known regular black holes

Author

Jawad, Abdul; Shahzad, M Umair

Pages

1-11

Publication year

2016

Publication date

Mar 2016

Publisher

Springer Nature B.V.

ISSN

14346044

e-ISSN

14346052

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1140/epjc/s10052-016-3967-2

ProQuest document ID

1771071483

SIF and Springer-Verlag Berlin Heidelberg 2016

Accretion onto some well-known regular black holes

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