# Accretion processes for general spherically symmetric compact objects

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## Abstract

We investigate the accretion process for different spherically symmetric space-time geometries for a static fluid. We analyze this procedure using the most general black hole metric ansatz. After that, we examine the accretion process for specific spherically symmetric metrics obtaining the velocity of the sound during the process and the critical speed of the flow of the fluid around the black hole. In addition, we study the behavior of the rate of change of the mass for each chosen metric for a barotropic fluid.

## Keywords

Black Hole Dark Energy Critical Velocity Naked Singularity Charged Black Hole## 1 Introduction

About two decades ago, astronomical observations such as CMB radiation [1], Supernova Type Ia [2] and large scale structure data [3, 4] showed that the Universe is undergoing an accelerating expansion period. This discovery was a revolution in cosmology and the agent responsible for this effect was named dark energy. This energy has the strange property that produces repulsive gravitational effects and it violates the null and weak energy condition [5, 6]. Surprisingly, observations suggests that around two-thirds of the total energy of the Universe belongs to the dark energy [1, 2]. However, the nature of this energy is not well understood and nowadays is one of the most challenging problems in theoretical physics. Over the last two decades, a number of theorists have been trying to tackle this important problem. Several ideas have been proposed like cosmological constant, phantom energy, quintessence, k-essence, dynamic scalar fields, and others. Usually, dark energy is modeled using a perfect fluid such as the pressure and energy density are related by a perfect fluid with a barotropic equation \(p=w\rho \) where *w* is the state parameter with \(w=-1\) a cosmological constant, \(-1<w<-1/3\) for quintessence and \(\omega <-1\) for phantom models [7].

Accretion is the process by which a massive astrophysical object such as a black hole or a star can take particles from a fluid from its vicinity which leads to increase in mass (and possibly angular momentum) of the accreting body [8, 9, 10, 11]. It is one of the most ubiquitous processes in the Universe. Indeed the stars and planets form only as a result of accretion in some inhomogeneous regions of gas and dust. The existence of supermassive black holes at the centers of giant elliptical and spiral galaxies suggests that such black holes could have evolved via accretion process. Other processes of formation of giant black holes such as merger of several small mass black holes (or compact objects) or stellar collapse of several stars in a small domain leading to merger seem very remote possibilities. The most likely model of the formation of massive giant black holes is the accretion of dust or matter from nearby regions for sufficiently long times. The formation of giant elongated astrophysical jets from active galaxies or small compact objects indicates the existence of large amounts of hot dust (likely to occur in the form of an accretion disk) around the regions of jet formation. However, an accretion process not always increases the mass of the compact source; sometimes the in-falling matter is thrown away in the form of jets or cosmic rays. It is more probable that the accretion process may not be static and the velocity of free-fall and the energy density of the fluid change with time and position. The accretion of normal matter and dust onto compact objects is a well-studied problem, however, the accretion of more exotic types of energy matter is not so commonly probed, including dark energy and stiff fluids. Since the Universe is dominated by dark energy, it is more pertinent to study the accretion of various forms of dark energy onto black holes.

Bondi investigated accretion for compact objects using Newtonian gravity [12]. After the arrival of general relativity, it became possible to investigate the accretion onto more compact objects such as neutron stars and black holes. Michel was the first who studied the relativistic accretion process for the Schwarzschild black hole [13]. Babichev et al. derived the fate of a stationary uncharged black hole in the phantom energy dominated universe [14]. They concluded that the phantom energy falling onto a black hole will decrease its mass. We use the same formalism in order to study the accretion process for different spherically symmetric space-times. In [15], they studied spherical, steady-state accretion of cosmological fluids onto an intermediate-mass black hole at the center of a globular cluster. More recently, Debnath [16] presented a framework of static accretion onto general static spherically symmetric black holes, thereby generalizing the work of Babichev et al. [14]. We extend these previous studies using a more general ansatz for a static spherically symmetric space-time, which covers all possible static black hole (and non-black hole) solutions such as Schwarzschild-monopole, string cloud, Jannis-Newman–Winicour, and dilaton (stringy charged) black hole solutions. In particular, we investigate the parameters of the fluid and the effect on the black hole mass.

This paper uses the geometrized units \(c=G=\hbar =1\) and the signature \((-,+,+,+)\) for the metrics. This paper is organized as follows: In the second section, we derive a general formalism for the relativistic spherical accretion process. In the third section, we study some specific metrics such as black holes with topological defects, string cloud, JNW metric, and a static charged black hole. For each case, we derive the velocity profiles, the speed of sound, the critical speed of the flow, and the rate of change of the black hole mass for a barotropic fluid. We explain these parameters with the help of figures. Finally, we discuss the results of our paper.

## 2 General formalism

*r*only. This metric ansatz with \(C(r)=r^2\) was used by Debnath [16]. We are going to perform a general accretion formalism for those space-times geometries.

*p*the pressure and \(u^{\mu }\) is the 4-velocity, which in general will be

*r*only. Since the 4-velocity must satisfy the normalization condition \(u_{\mu }u^{\mu }=-1\) we find

*r*.

## 3 Spherically symmetric metrics with horizons

### 3.1 Schwarzschild–de Sitter black hole with a topological defect (monopole)

*x*, near the black hole, \(\dot{M}>0\) showing that mass of black hole increases for \(w=0\) and \(w=-0.5\), being for matter and quintessence, respectively. In other words, the mass of the black hole will increase for matter and quintessence accretions. However, for stiff matter accretion, \(\dot{M}>0\) for both small and large values of

*x*, thereby increasing in mass of black hole. It should be noted that quintessence, dust, and stiff matter satisfy the relativistic energy conditions for energy matter which ensure the increase in BH mass and satisfy the second law of black hole thermodynamics. It is also apparent that \(\dot{M}<0\) for phantom-like equations of state such as \(w=-1.5,-2\), thus mass of the BH will decrease by the accretion of phantom like fluids.

It is important to remark that we have another possibility to study the accretion process for this metric, which is by taking \(A_{2}<0\) and using (32) and (33) with a minus of difference. In that case we have \(\rho >0\) for \(w>-1\) as we required, but \(u(r)<0\) for \(w<-1\). However, the case \(A_{2}<0\) seems to be an unnatural choice, since we may notice that we will need a barotropic case such as \(p=-w \rho \) to have identical solutions to solving (5), (9), and (11) directly and setting \(A_{0}=1+w\) and \(A_{2}=-1\). In other words, it is the same to have \(A_{0}=1+w\) and \(A_{2}=-1\) and solving the system of equations (5), (9), and (11) as solving the system of Eqs. (5), (11), and \(p=-w \rho \). Therefore, we will have the same behavior as we discussed above, since for \(A_{2}=-1\), the state parameter *w* will be minus the state parameter in the case of \(A_{2}=1\). Thus, for all the following subsections, we are going to find the solutions using (5), (11), and \(p=w \rho \), and we will take \(A_{2}=1\).

### 3.2 Schwarzschild black hole in a string cloud background

*M*is the mass of black hole and it appears as a constant of integration during the solution of field equations [22]. The limit \(\alpha =0\) gives the Schwarzschild solution. The radial velocity and the energy density for a barotropic fluid are given by

*u*(

*r*) and \(\rho (r)\). For this case, the critical values are given by

*w*, its value will be different for every fluid.

*x*and find that the mass of the BH will increase due to the fluids satisfying energy conditions while it decreases for fluids violating the same energy conditions.

### 3.3 Janis–Newman–Winicour space-time

The physical existence of naked singularities is still questionable, however, there are few exact solutions of Einstein field equations that represent naked singularity. These include the JNW, JMN and gamma metrics [23]. The Janis–Newman–Winicour (JNW) solution is obtained as an extension of the Schwarzschild space-time when a massless scalar field (with vanishing potential) is introduced. Thus this solution is not a vacuum solution [24]. Here the coordinate singularity in the Schwarzschild space-time becomes a naked singularity in JNW space-time. Joshi et al. [23] (see also references therein) have investigated several astrophysical features distinguishing a black hole from a naked singularity. The gravitational lensing features due to a naked singularity have also been explored [25, 26].

*a*and \(\mu \) are related by

*x*, however, interestingly we do observe some symmetry in the profile of the velocity curves and the location of critical points. After passing the critical point, the fluid flow becomes supersonic or trans-sonic. In Fig. 8, the graph of the energy density of different fluids in the neighborhood of a naked singularity. The fluid energy density for dust and quintessence rises in the near vicinity of singularity and goes to zero asymptotically. Similarly, we also see that the energy density of dust and quintessence remains positive, while for phantom fluids it is negative.

### 3.4 Charged black hole in string theory

*w*. We write down the stringy charged metric in the following form [31]:

*Q*is the charge parameter and \({\varPhi }_{0}\) the asymptotic value of the dilatonic field which can be set to zero under certain cases [32]. In this case, we have

*x*. We also represent the location of critical velocity by thick dots whose presence in the fluid flow profiles show that critical flow is a generic property of all fluids in the strong gravitational regimes. All velocity profiles \(u(x)>0\) are physically forbidden, since nothing can escape from the black hole. However, flows of dust, stiff matter, and quintessence are permissible under our assumptions of inward flows, \(u<0\) for all

*x*.

*x*for different values of state parameter. In all cases, it is observed that the BH mass will increase by the accretion of quintessence, dust or the stiff fluid when

*x*is small. Strikingly, the maximum rate of increase in BH mass arises due to quintessence followed by dust and stiff fluid.

## 4 Discussion

We have proposed a most general framework for the study of spherical accretion onto spherically symmetric compact objects. We explored various features of the fluid flow near black holes. Although we assumed all the fluids to satisfy a certain linear equation of state with distinct state parameters, the above analysis can alternatively be performed without specifying the equation of state, since the number of equations are sufficient to close the system of equations. Interestingly we found a link between the barotropic equation of state and the conservation laws. However, the equation of state helps us identify which kind of fluid is falling onto the black hole. In other words there is no multiple accretion scenario into play, though it can be done. In particular, we focused on dust, stiff matter, quintessence, and phantom dark energy accretion onto black holes unlike previous studies in the literature, which focused on only one kind of test fluid. We did not consider the case of the cosmological constant or vacuum energy since its accretion does not alter the evolution of black holes. It is found that different fluids with distinct state parameters have different evolutions in the black hole backgrounds. Certain fluids acquire positive or negative energy density near the black hole, while some fluids cause black hole masses to increase or decrease. Although we plotted all the fluid behaviors via single graphs, it is assumed that a single test fluid is accreted at one time. For a future work it will be interesting to perform a similar analysis with a dynamically spherically symmetric geometry or with a non-static fluid.

## Notes

### Acknowledgments

S. B. is supported by the Comisión Nacional de Investigación Científica y Tecnológica (Becas Chile Grant No. 72150066).

## References

- 1.D.N. Spergel et al., WMAP Collaboration. Astrophys. J. Suppl.
**170**, 377 (2007). arXiv:astro-ph/0603449 - 2.S. Perlmutter et al., Supernova Cosmology Project Collaboration. Astrophys. J.
**517**, 565 (1999). arXiv:astro-ph/9812133 - 3.D.J. Eisenstein et al., SDSS Collaboration. Astrophys. J.
**633**, 560 (2005). arXiv:astro-ph/0501171 - 4.A.G. Riess et al., Supernova Search Team Collaboration. Astron. J.
**116**, 1009 (1998). arXiv:astro-ph/9805201 - 5.V.B. Johri, Phys. Rev. D
**70**, 041303 (2004). arXiv:astro-ph/0311293 CrossRefADSMathSciNetGoogle Scholar - 6.F.S.N. Lobo, Phys. Rev. D
**71**, 084011 (2005). arXiv:gr-qc/0502099 CrossRefADSMathSciNetGoogle Scholar - 7.S. Nojiri, S. Odintsov, Phys. Rep.
**505**, 59–144 (2011)CrossRefADSMathSciNetGoogle Scholar - 8.I.G. Martnez-Pas, T. Shahbaz, J.C. Velzquez,
*Accretion Processes in Astrophysics*(Cambridge University Press, Cambridge, 2014)Google Scholar - 9.P. Mach, E. Malec, J. Karkowski, Phys. Rev. D.
**88**(8), 084056 (2013). arXiv:1309.1252 [gr-qc] - 10.J. Karkowski, B. Kinasiewicz, P. Mach, E. Malec, Z. Swierczynski, Phys. Rev. D.
**73**, 021503 (2006). arXiv:gr-qc/0509079 - 11.P. Mach, E. Malec, Phys. Rev. D.
**91**(12), 124053 (2015). arXiv:1501.04539 [gr-qc] - 12.H. Bondi, Mon. Not. R. Astron. Soc.
**112**, 195 (1952)CrossRefADSMathSciNetGoogle Scholar - 13.F.C. Michel, Astrophys. Space Sci.
**15**, 153–160 (1972)CrossRefADSGoogle Scholar - 14.E. Babichev, V. Dokuchaev, Y. Eroshenko, Phys. Rev. Lett.
**93**, 021102 (2004). arXiv:gr-qc/0402089 - 15.C. Pepe, L.J. Pellizza, G.E. Romero, Mon. Not. Roy. Astron. Soc.
**420**, 3298 (2012). arXiv:1111.5605 [astro-ph.HE] - 16.U. Debnath, Eur. Phys. J. C
**75**, 3, 129 (2015). arXiv:1503.01645 [gr-qc] - 17.N. Dadhich, K. Narayan, U.A. Yajnik, Pramana
**50**, 307 (1998)CrossRefADSGoogle Scholar - 18.Y.W. Han, S.Z. Yang, Commun. Theor. Phys.
**47**, 1145 (2007)CrossRefADSMathSciNetGoogle Scholar - 19.A.R. Amani, H. Farahani, Int. J. Theor. Phys.
**51**, 2943 (2012)CrossRefzbMATHGoogle Scholar - 20.E.O. Babichev, V.I. Dokuchaev, Y.N. Eroshenko, Phys. Usp.
**56**, 1155 (2013). arXiv:1406.0841 [gr-qc] [Usp. Fiz. Nauk**189**, 12, 1257 (2013)] - 21.A. Ganguly, S.G. Ghosh, S.D. Maharaj, Phys. Rev. D
**90**, 6, 064037 (2014). arXiv:1409.7872 [gr-qc] - 22.P. Letelier, Phys. Rev. D
**20**, 1294 (1979)CrossRefADSMathSciNetGoogle Scholar - 23.S. Sahu, M. Patil, D. Narasimha, P.S. Joshi, Phys. Rev. D
**86**, 063010 (2012)CrossRefADSGoogle Scholar - 24.S. Zhou, R. Zhang, J. Chen, Y. Wang, Int. J. Theor. Phys.
**54**(8), 2905 (2015). arXiv:1408.6041 [gr-qc] - 25.K.S. Virbhadra, G.F.R. Ellis, Phys. Rev. D
**65**, 103004 (2002)CrossRefADSMathSciNetGoogle Scholar - 26.K.S. Virbhadra, D. Narasimha, S.M. Chitre, Astron. Astrophys.
**337**, 1 (1998)ADSGoogle Scholar - 27.A.N. Chowdhury, M. Patil, D. Malafarina, P.S. Joshi, Phys. Rev. D
**85**, 104031 (2012).arXiv:1112.2522 [gr-qc]CrossRefADSGoogle Scholar - 28.G.W. Gibbons, K. Maeda, Nucl. Phys. B
**298**, 741 (1998)CrossRefADSMathSciNetGoogle Scholar - 29.D. Garfinkle, G.T. Horowitz, A. Strominger, Phys. Rev. D
**43**, 3140 (1991)CrossRefADSMathSciNetGoogle Scholar - 30.M. Sharif, G. Abbas, Chin. Phys. Lett.
**29**, 010401 (2012)CrossRefADSGoogle Scholar - 31.
- 32.R. Li, J. Zhao, Eur. Phys. J. C
**74**, 3051 (2014)CrossRefADSGoogle Scholar

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