# Accretion onto some well-known regular black holes

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## Abstract

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.

### Keywords

Black Hole Dark Energy Radial Velocity Critical Velocity Stiff Matter## 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 fluid with the equation of state (EoS) \(\rho = \omega p\). The candidates of dark energy are a phantom-like fluid \((\omega < -1)\), quintessence \((-1 < \omega < -1/3)\), and the cosmological constant \((\omega = -1)\) [7]. Other models are also proposed as an explanation of dark energy, like k-essence, DBI-essence, Hessence, dilation, tachyons, Chaplygin gas, etc. [8, 9, 10, 11, 12, 13, 14, 15, 16].

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 Einstein’s field equations. To avoid these singularities, regular BHs (RBHs) have been developed. These BHs are solutions of Einstein’s 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 fluid from its surroundings, then the mass of condensed object has been effected. This process is known as accretion of fluid 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 first 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 Madrid and Gonzalez-Diaz [30] studied accretion of dark energy on a static BH and a Kerr–Newman 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 profile, energy density, and the rate of change of the RBH mass. In the end, we conclude our results.

## 2 General formalism for accretion

*r*only. The energy-momentum tensor is considered in terms of a perfect fluid which is isotropic and inhomogeneous and defined as follows:

*p*is the pressure, \(\rho \) is the energy density, and \(u^{\mu }\) is the four-velocity, which is given by

*r*. The normalization condition of the four-velocity must satisfy \(u^{\mu } u_{\mu } = -1\), and we get

*r*. By integrating the last equation, we obtain

## 3 Spherically symmetric metrics with charged RBHs

### 3.1 Charged RBH using Fermi–Dirac distribution

The velocity profile for different values of \(\omega \) is shown in Fig. 1. Here \(\omega =1,0,-1\) refer to the stiff, dust, and cosmological constant cases, respectively, and \(-1 < \omega < -1/3\) and \(\omega < -1\) refer to quintessence and phantom energy. It can be seen that for \(\omega =-1.5,-2\) the radial velocity of the fluid is negative and it is positive for \(\omega =-0.5,0,0.5,1\). If the flow is outward then \(u < 0\) is not allowed and vice versa. In the case of \(\omega =-1.5,-0.5\) the fluid is at rest at \(x = 10\). Figure 2 represents the behavior of energy density of fluids in the surrounding area of the RBH. Obviously the WEC and DEC satisfied by dust, stiff, and quintessence fluids. When the phantom fluid (\(\omega =-1.5,-2\)) moves toward the RBH then the energy density decreases and the reverse will happen for dust, stiff, and quintessence fluids (\(\omega =-0.5,0,0.5,1\)). Asymptotically \(\rho \rightarrow 0\) at infinity for \(\omega =-1.5,-0.5\), while it approaches the maximum at \(x=1.2,1.3,1.8\) and near the RBH.

Charged RBH using the Fermi–Dirac distribution

\(\omega \) | \(r_c\) | \(u(r_c)\) | \(c^{2}_{s}\) |
---|---|---|---|

\(-\)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 |

### 3.2 Charged RBH using logistic distribution

The velocity profile for different values of \(\omega \) is shown in Fig. 4. It can be observed that for \(\omega =-1.5,-2\) the radial velocity of the fluid is negative and it is positive for \(\omega =-0.5,0,1\). If the flow is inward then \(u > 0\) is not allowed and vice versa. In the case of \(\omega =-2,0\) the fluid is at rest at \(x\approx 5\). Figure 5 represents the behavior of energy density of fluids in the surrounding area of the RBH. Obviously the WEC and DEC are satisfied by dust, stiff, and quintessence fluids. When a phantom-like fluid (\(\omega =-1.5,-2\)) moves toward a RBH the energy density decreases and the reverse will happen for dust, stiff, and quintessence fluids (\(\omega =-0.5,0,0.5,1\)).

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 fluids and it decreases due to phantom fluids.

Charged RBH using logistics distribution

\(\omega \) | \(r_c\) | \(u(r_c)\) | \(c^{2}_{s}\) |
---|---|---|---|

\(-\)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 |

### 3.3 Charged RBH from nonlinear electrodynamics

*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

The absolute value of the velocity profile for different values of \(\omega \) is shown in Fig. 7. It can be observed that for \(\omega = -2\) the radial velocity of the fluid is negative and it is positive for \(\omega =0.5,0,1\). If the flow is inward then \(u > 0\) is not allowed and vice versa. In the case of \(\omega = -2,0\) the fluid is at rest at \(x \approx 5\). Figure 8 represents the energy density of fluids in the region of the RBH. It is apparent that the WEC and DEC is satisfied by phantom fluids. When the phantom fluids moves toward the RBH the energy density increases; on the other hand it decreases for dust and stiff matter.

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 fluids. It is also noted that the maximum rate of the RBH mass increases due to \(\omega =1\) followed by \(\omega =0.5,0,-2\).

Charged RBH from nonlinear electrodynamics

\(\omega \) | \(r_c\) | \(u(r_c)\) | \(c^{2}_{s}\) |
---|---|---|---|

\(-\)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 |

### 3.4 Kehagias–Sftesos asymptotically flat BH

*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],

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

*x*. We see that the RBH mass will increase for \(\omega =-0.35, 0,0.5,1\), and it will decrease for \(\omega =-2\).

Kehagias–Sftesos asymptotically flat BH

\(\omega \) | \(r_c\) | \(u(r_c)\) | \(c^{2}_{s}\) |
---|---|---|---|

\(-\)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 |

## 4 Concluding remarks

In this work, we have investigated the accretion onto various RBHs (such as an RBH using the Fermi–Dirac distribution, a RBH using the logistic distribution, an RBH using nonlinear electrodynamics, and a Kehagias–Sftesos asymptotically flat RBH) which asymptotically leads to Schwarzschild and Reissner–Nordstrom 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 fluid is positive for stiff, dust, and quintessence matter and it is negative for phantom-like fluids. If the flow is inward then \(u < 0\) is not allowed and \(u > 0\) is not allowed for outward flow. Also, we have seen that the energy density remains positive for quintessence, dust, and stiff matter, while it becomes negative for a phantom-like fluid 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 fluids which accrete onto it. Therefore, we have considered the various possibilities of accreting fluids, 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 fluids, since these fluids do not have enough repulsive force. However, the mass of all RBHs decreases in the presence of a phantom-like fluid (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, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. 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 fluid, this may be extended for a non-static fluid without assuming any EoS and thus can be obtained more interesting results. This is left for future considerations.

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