Spherical Accretion Flow Onto General Parameterized Spherically Symmetric Black Hole Spacetimes*

Chinese Physics C Vol. 45, No. 1 (2021) 015102

Spherical accretion flow onto general parameterized spherically symmetric black hole spacetimes*

1,2† 2,3‡ 2,3§ 1♮ 2,3♯ Sen Yang(杨森) Cheng Liu(刘诚) Tao Zhu(朱涛) Li Zhao(赵力) Qiang Wu(武强) 4¶ 2,3,5♭ Ke Yang(杨科) Mubasher Jamil 1Institute of theoretical physics, Lanzhou University, Lanzhou 730000, China 2Institute for theoretical physics and Cosmology, Zhejiang University of Technology, Hangzhou 310032, China 3United center for gravitational wave physics (UCGWP), Zhejiang University of Technology, Hangzhou 310032, China 4School of Physical Science and Technology, Southwest University, Chongqing 400715, China 5School of Natural Sciences, National University of Sciences and Technology, Islamabad 44000, Pakistan

Abstract: The transonic phenomenon of black hole accretion and the existence of the photon sphere characterize strong gravitational fields near a black hole horizon. Here, we study the spherical accretion flow onto general parametrized spherically symmetric black hole spacetimes. We analyze the accretion process for various perfect fluids, such as the isothermal fluids of ultra-stiff, ultra-relativistic, and sub-relativistic types, and the polytropic fluid. The influences of additional parameters, beyond the Schwarzschild black hole in the framework of general parameterized spherically symmetric black holes, on the flow behavior of the above-mentioned test fluids are studied in detail. In addition, by studying the accretion of the ideal photon gas, we further discuss the correspondence between the sonic radius of the accreting photon gas and the photon sphere for general parameterized spherically symmetric black holes. Possible extensions of our analysis are also discussed. Keywords: spherical accretion, black hole, RZ parametrization, photon sphere DOI: 10.1088/1674-1137/abc066

I. INTRODUCTION Bondi [4], where an infinitely large homogeneous gas cloud, steadily accreting onto a central gravitational ob- Accretion around a massive gravitational object is a ject, was considered. Bondi's treatment was formulated in basic phenomenon in astrophysics and has been essential the framework of Newtonian gravity. Later, in the frame- to the understanding of various astrophysical processes work of general relativity (GR), the steady-state spheric- and observations, including the growth of stars, the form- ally symmetric flow of a test fluid onto a Schwarzschild ation of supermassive black holes, quasar luminosity, and black hole was investigated by Michel [5]. Since then, X-ray emission from compact star binaries [1-3]. The ac- spherical accretion has been considered for various spher- cretion of matter in a realistic astrophysical process is ically symmetric black holes in GR and modified gravit- rather complicated, since it involves many challenging ies; see [6-30] and references therein for examples. aspects of general relativistic magnetohydrodynamics, in- One important feature of spherical accretion onto cluding turbulence, radiation processes, and nuclear burn- black holes is the phenomenon of transonic accretion and ing. To understand these accretion processes, it is useful the existence of a sonic point (or a critical point). At the to simplify the problem by making assumptions and/or sonic point, the accretion flow transits from the subsonic considering simple scenarios. to the supersonic state. Normally, the locations of the The simplest accretion scenario describes a stationary, sonic points in a given black hole spacetime are not far spherically symmetric solution, as first discussed by from its horizon. What is important and intriguing is that

Received 2 August 2020; Published online 25 September 2020 * C.L., T.Z., and Q.W. are supported by National Natural Science Foundation of China (11675143), the Zhejiang Provincial Natural Science Foundation of China (LY20A050002), and the Fundamental Research Funds for the Provincial Universities of Zhejiang in China (RF- A2019015) † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected], corresponding author ♮ E-mail: [email protected] ♯ E-mail: [email protected] ¶ E-mail: [email protected] ♭ E-mail: [email protected] ©2021 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

015102-1 Sen Yang, Cheng Liu, Tao Zhu et al. Chin. Phys. C 45, 015102 (2021) the narrow region around the sonic point is closely re- phenomena not only for specific theories of gravity but lated to some ongoing observations about the spectra of also for analysis in a unified way by exploring the influ- electromagnetic and gravitational waves. Therefore, ence of different black hole parameters on the spherical studying the spherical accretion problem can not only accretion process [40]. Specifically, we focus our atten- help us to understand accretion processes in different tion on the perfect fluid accretion onto general parameter- black holes but, importantly, also provide us with an al- ized spherically symmetric black hole spacetimes and fur- ternative approach to explore the nature of the black hole ther investigate transonic phenomena for different fluids, spacetimes in the regime of strong gravity. including isothermal fluids and polytropic fluids. By On the other hand, the EHT collaboration recently re- studying the accretion of the ideal photon gas, we further ported their first image concerning the detection of the reveal the correspondence between the sonic points of the shadow of a supermassive black hole at the center of a accreting photon gas and the photon sphere, for general neighboring elliptical M87 galaxy [31-36]. This image re- parameterized spherically symmetric black holes. vealed that the diameter of the center black hole shadow Our paper is organized as follows. In Sec. 2, we is (42 3) μ, leading to the measured center mass present a very brief introduction to general parameter- M = (6.5 0.7) × 109 M⊙ [31]. The outer edge of the shad- ized spherically symmetric black holes. Then, in Sec. 3, ow image, if one considers a Schwarschild black hole, we derive the basic equations for subsequent discussions forms a photon sphere near the black hole horizon, at on the spherical accretion of various fluids and present which the trajectories of photons create a closed circular several useful quantities. Sec. 4 is devoted to performing orbit. Within astrophysical observations, the existence of a dynamical systems analysis of the accretion process and a photon sphere is related to the electromagnetic observa- finding the critical points of the system. In Sec. 5, we ap- tions of black holes via the background electromagnetic ply these results to several known fluids and further in- emission and the frequencies of quasi-normal modes. The vestigate the transonic phenomena of the accretion of latter is determined by the parameters of null geodesic these fluids onto general parameterized spherically sym- motions on and near the photon sphere of a given black metric black holes. In Sec. 6, by studying the spherical hole spacetime. accretion of the ideal photon gas and photon sphere of Recently, it was shown that there is a correspondence general parameterized spherically symmetric black holes, between the sonic points of an accreting ideal photon gas we establish the correspondence between the sonic points and the photon sphere, for static spherically symmetric of the ideal photon gas and its photon sphere. The conclu- spacetimes [37]. This important result is valid not only sion of this paper is presented in Sec. 7. for spherical accretion of the ideal photon gas but also for rotating accretion in static spherically symmetric space- II. PARAMETERIZED SPHERICALLY SYMMET- times [38, 39]. In an observational viewpoint, as men- RIC BLACK HOLE SPACETIME tioned in [39], this correspondence connects two independent observations, the observation of light coming In this section, we present a brief introduction to the from sources behind a black hole and the observation of parameterization by L. Rezzolla and A. Zhidenko's (RZ) emission from the accreted radiation fluid onto the black [40], for generic spherically symmetric black hole space- hole. This is because the size of the hole's shadow is de- times. First, let us consider the line element of any spher- termined by the radius of the photon sphere, and the ac- ically symmetric stationary configuration in the spherical creted fluid can signal the sonic point. polar coordinate system (t,r,θ,ϕ), which can be written as In light of the above studies, it is interesting to explore spherical accretion flows in different black hole B2(r) spacetimes. The additional parameters, beyond the ds2 = −N2(r)dt2 + dr2 + r2(dθ2 + sin2 θdϕ2), (1) N2(r) Schwarzschild black hole in these spacetimes, may affect the accretion flow behavior; thus, we have a potentially important approach for studying the strong gravity beha- where N(r) and B(r) are two functions of the radial co- vior of black holes in many alternative theories of grav- ordinate r alone. In the RZ parameterization, N(r) is ex- ity. Instead of finding the exact solution and studying the pressed as spherical accretion case by case for each given theory, a reasonable strategy is to consider a model-independent N2(x) = xA(x), (2) framework that parameterizes the most generic black- hole geometry through a finite number of adjustable where A(x) > 0 for 0 < x < 1 with x = 1 − r0/r. It is obvi- quantities. For this purpose, in this paper, we consider ous that x = 0 represents the location of the event horizon spherical accretion flows in general parameterized spher- of the black hole, and x = 1 is the spatial infinity. Then, ically symmetric black hole spacetimes [40]. This para- the functions A(x) and B(x) can be further parameterized meterized description allows one to consider accretion in terms of the parameters ϵ, ai , and bi as

015102-2 Spherical accretion flow onto general parameterized spherically symmetric black hole spacetimes Chin. Phys. C 45, 015102 (2021)

radial coordinate r only. For the sake of simplicity, we set A(x) = 1 − ϵ(1 − x) + (a − ϵ)(1 − x)2 + A˜(x)(1 − x)3, (3) 0 the radial velocity as ur = u < 0 for the accreting case. Then, using the normalization condition, it is easy to in- = + − + ˜ − 2, B(x) 1 b0(1 x) B(x)(1 x) (4) fer that

˜ ˜ 2 + 2 2 where the functions A and B are introduced to describe t 2 N (r) B (r)u ≃ (u ) = . (8) the metric near the horizon (i.e., x 0) and at the spatial N4(r) infinity (i.e., x = 1). The coefficients a0 and b0 can be seen as combinations of the PPN parameters. The func- There are two basic conservation laws that govern the ˜ ˜ tions A and B can be expanded using the continuous Padé evolution of a fluid in a black hole spacetime. One is the approximation as conservation law of the particle number, and another one is the conservation law of the energy momentum. The as- ˜ = a1 , ˜ = b1 , A(x) a x B(x) (5) sumption of the conservation of the particle number im- + 2 b2 x 1 a x + plies there is no particle creation and/or annihilation dur- + 3 1 b x 1 1+··· + 3 1 1+··· ing the accreting process. Defining the proper particle number density n and number current Jµ = nuµ in the loc- where a1,a2,··· ,an and b1,b2,··· ,bn are dimensionless al inertial rest frame of the fluid, the conservation of the constants that can be determined by matching the above particle number gives parameterization to a specific metric. In addition, the ϵ µ µ parameter in the RZ parameterization measures the de- ∇µ J = ∇µ(nu ) = 0, (9) viation of the position of the event horizon in the general metric from the corresponding location in the Schwarz- where ∇µ denotes the covariant derivative with respect to schild spacetime, i.e., the coordinate. For the RZ parameterization of a generic spherically symmetric black hole spacetime, Eq. (9) can 2M − r0 ϵ = . (6) be rewritten as r0 1 d The RZ parameterization can be matched to many (r2Bnu) = 0. (10) r2B dr black hole solutions which differ from GR. These in- clude the Reissner-Nordström (RN) black hole in GR and Integrating this equation, we obtain black holes in the Brans-Dicke gravity (BD), f (R) gravity, the Einstein-Maxwell axion dilaton theory (EMAD), 2 = , and the Einstein-Aether theory [40, 41]. Recently, the RZ r Bnu C1 (11) parameterization has also been extended to the rotating case [42]. where C1 is the integration constant. The conservation law of the energy momentum is ex- pressed as III. BASIC EQUATIONS FOR SPHERICAL AC- CRETION FLOWS µν ∇µT = 0. (12) In this section, we consider the steady-state spherical accretion flow of matter near an RZ-parameterized black It is also convenient to introduce the first law of the hole. For this purpose, the accreting matter is approxim- thermodynamics of the perfect fluid, which is given by ated as a relativistic perfect fluid, by neglecting effects [43] related to viscosity or heat transport. Thus, the energy momentum tensor of the fluid can be described by dp = n(dh − Tds), dρ = hdn + nTds, (13) µν = ρ + µ ν + µν, T ( p)u u pg (7) where T is the temperature, s is the specific entropy, and h is the specific enthalpy, defined as where ρ and p are the proper energy density and the pressure of the perfect fluid. The four-velocity uµ obeys the ρ + p µ h ≡ . (14) normalization condition uµu = −1. We assume that the n fluid is radially flowing into the black hole; therefore, we have uθ = 0 = uϕ . For the same reason, the physical quant- Then, projecting the conservation law of the energy- ities (ρ, p) and others introduced later are functions of the momentum (12) along uµ , one obtains

015102-3 Sen Yang, Cheng Liu, Tao Zhu et al. Chin. Phys. C 45, 015102 (2021) [ ] µν µ ν µν uν∇µT = uν∇µ nhu u + pg On the other hand, by considering radial accretion θ = ϕ = = − µ∇ + µ∇ . flows, i.e., d d 0, the black hole metric can be de- nu µh u µ p (15) composed as [44]

( )2 In the above, we have used the conservation of the 2 2 B ∇ µ = µ∇ = ∇ µ = ds = −(Ndt) + dr , (22) particle number, i.e., µ(nu ) 0 and u νuµ uµ νu N 1 ∇ µ = 2 ν(u uµ) 0. Noticing that the first law of thermodynamics (13) can be rewritten as ∇µ p = n∇µh − nT∇µ s, from which one can define an ordinary three-dimension- from the above projection one arrives at al velocity v measured by a static observer as

− µ∇ = , B dr nTu µ s 0 (16) v ≡ . (23) N2 dt implying that there is no heat transfer between the differ- Considering ur = u = dr/dτ and ut = dt/dτ with τ be- ent fluid elements, and the specific entropy is conserved ing the proper time of the fluid, one finds along the evolution lines of the fluid. For a parameterized spherically symmetric black hole, the conservation ( ) B2 u 2 B2u2 of the specific entropy reduces to ∂ s = 0, i.e., s = con- v2 = = . (24) r 4 t 2 + 2 2 stant. For this reason, the fluid is isentropic and Eq. (13) N u N B u reduces to 2 2 2 Then, one can express u and ut in terms of v as = , ρ = . dp ndh d hdn (17) N2v2 u2 = , (25) B2(1 − v2) With the above thermodynamical properties of the perfect fluid, the conservation law of the energy-momentum (12) can be written as 2 2 = N . ut (26) µ µ µ 1 − v2 ∇µTν = ∇µ(hnu uν) + ∇µ(δν p) = µ∇ + ∇ nu µ(huν) n νh These quantities will be used in the following dynam- µ µ λ = nu ∂µ(huν) − nu Γµνhuλ + n∇νh = 0. (18) ical systems analysis for the radial, steady-state perfect fluid flow in a parametrized spherically symmetric black hole. Then, the time component ν = t of the above equation yields IV. SONIC POINTS AND DYNAMICAL SYSTEMS

∂r(hut) = 0. (19) ANALYSIS The two basic equations (11) and (20) constitute a dy- Integrating it for the parameterized spherically sym- namical system for the radial accretion process. In this metric black hole we consider in this paper, one arrives at section, we use these equations to study the accretion pro- √ cess in a parametrized spherically symmetric black hole. 2 2 2 h N + B u = C2, (20) A. Sonic points where C2 is the integration constant. This equation, to- In the trajectories of an accretion flow into a black gether with Eq. (11), constitutes the two basic equations hole, there exists a specific point called the sonic point, at describing a radial, steady-state perfect fluid flow in the which the four-velocity of the moving fluid becomes parameterized spherically symmetric black hole. equal to the local speed of sound, and the accretion flow To proceed further, let us introduce several useful attains the maximal accretion rate. To determine the son- quantities for describing the accretion flow, which will be ic point, let us first take the derivative of the two basic used in the subsequent analysis. The first quantity is the equations (11) and (20) with respect to r, which leads to sound speed of the perfect fluid, which is defined by [ ( ) ] ( ) − 2 2 − 2 dlnv = 1 v 2 + dln B − − 2 dN . dp n dh dlnh v cs cs NB 2 r B(1 cs )r c2 ≡ = = . (21) dr BNr dr dr s dρ h dn dlnn (27)

015102-4 Spherical accretion flow onto general parameterized spherically symmetric black hole spacetimes Chin. Phys. C 45, 015102 (2021)

2 = 2 At the sonic point r∗ (cs (r∗) v (r∗)), one has 2 [ ] h 2 2 2 2 v˙ ≡ g(r,v) = − rN, (1 − c ) − 4N c . (34) ( ) r(1 − v2) r s s

2 + dln B − − 2 dN = , cs∗N∗B∗ 2 r∗ B∗(1 cs∗)r∗ 0 (28) dr ∗ dr ∗ These equations constitute an autonomous, Hamilto- nian two-dimensional dynamical system. Its orbits are where ∗ denotes the values evaluated at the sonic point. composed of the solutions of the two basic Eqs. (11) and This equation allows us to determine the sonic point once (20). In the construction of the above dynamical system, 2 ≡ / ρ the speed of sound cs dp d is known. The above equa- we considered the two quantities (r,v) as the two dynam- tion can be rewritten as ical variables of the system. It is worth mentioning that there are actually different ways to select dynamical vari- dN ables; for example, one may choose the dynamical vari- N∗r∗ 2 = ( dr ∗ ). ables to be (r,h), (r, p), or (r,u) [45]. u∗ (29) 2 dln B At critical points, the right-hand sides of Eqs. (33) B∗ 2 + r∗ dr ∗ and (34) vanish, and the following equations provide a set of critical points that are solutions to r˙ = 0 and v˙ = 0, Therefore, once r∗ is determined, one can use this ex- 2 = 2, pression to find the value of u at the sonic point. The ex- v∗ cs (35) istence of the sonic point in the black hole spacetime 2 physically exhibits a very interesting accreting phe- r∗N∗, c2 = r∗ . (36) nomenon; it highlights transonic solutions that are super- s 2 2 r∗N∗, + 4N∗, sonic near and subsonic far from the black hole. In the r∗ r∗ following sections, we are going to find sonic points by using the equations obtained in this subsection and dis- It is easy to see that sonic points are the critical points cuss the transonic phenomenon in detail for different flu- of this dynamical system. Hereafter, we use (r∗,v∗) to de- ids. note the critical points of the dynamical system. For a dynamical system, critical points can be divided into sever- B. Dynamical system and critical points al different types. To observe which critical points could From the two basic equations (11) and (20), we ob- arise from the black hole accretion processes, let us per- serve that there are two integration constants C1 and C2 . form the following linearization of the dynamical system For this system, we may treat the square of the left-hand by Taylor-expanding Eqs. (33) and (34) around the critic- side of Eq. (20) as a Hamiltonian H of this system, al points, i.e., ( ) ( ) 2 2 2 2 δr˙ δr H = h (N + B u ), (30) = X , (37) δv˙ δv so C2 of every orbit in the phase space of this system is kept fixed. Inserting Eq. (25) into the Hamiltonian H one where δr, δv denote the small perturbations of r, v about finds the critical points, and X is the Jacobian matrix of the dynamical system at the critical point (r∗,v∗), which is h2(r,v)N2 defined as H(r,v) = . (31) 1 − v2    ∂ f ∂ f     ∂ ∂  Then, the dynamical system associated with this X =  r v  . (38)  ∂g ∂g  Hamiltonian reads (r∗,v∗) ∂r ∂v

r˙ = H,v, v˙ = −H,r, (32) Depending on the determinant ∆ = det(X) of X and its trace χ = Tr(X), the types of the critical points (r∗,v∗) of where the dot denotes the derivative with respect to t¯ (the the dynamcial system can be summarized as follows: time variable of the Hamiltonian dynamical system). ● Saddle points if ∆ < 0. Then, inserting the Hamiltonian, one finds ● Attracting nodes if ∆ > 0, χ < 0, and χ2 − 4∆ > 0. ● Attracting spirals if ∆ > 0, χ < 0, and χ2 − 4∆ < 0. 2h2N2 ∆ > χ > χ2 − ∆ > r˙ ≡ f (r,v) = (v2 − c2), (33) ● Repelling nodes if 0, 0, and 4 0. v(1 − v2)2 s ● Repelling spirals if ∆ > 0, χ > 0, and χ2 − 4∆ < 0. ● Degenerate nodes if ∆ > 0, and χ2 − 4∆ = 0.

015102-5 Sen Yang, Cheng Liu, Tao Zhu et al. Chin. Phys. C 45, 015102 (2021)

● Centers if ∆ > 0, χ = 0. and ∆ = ● Line or plane critical points if 0. ( ) When the critical points and their types are determ- (w + 1)ρ n w h = 0 , (44) ined, the constant C1 in Eq. (11) can be rewritten in terms n0 n0 of the quantities evaluated at the critical point (r∗,v∗) as ρ ρ 4 2 2 2 5 2 2 where n0 and 0 denote the values of n and evaluated at r∗ n∗v∗ N∗ r∗ n∗ N∗, C2 = = r∗ . (39) some reference point. Using Eq. (40), we arrive at 1 2 1 − v∗ 4 ( ) 1 − v2 w h2 = K , (45) This equation is satisfied not only at the critical point r4N2v2 but also at any point in the same streamline in the phase portrait, so one can easily get where K is a constant. Through the transformation t¯→ Kt¯ ( ) 2 5 2 and H → H/K, the constant K is absorbed into the re- n r∗ N∗, 1 − v2 = r∗ . (40) defined time t¯. Then, the new Hamiltonian becomes n∗ 4 r4N2v2 N2(1−w) H(r,v) = . (46) If there is no solution to Eqs. (33) and (34) at the crit- (1 − v2)1−wv2wr4w ical point, one can introduce any reference point (r0,v0) from the phase portrait, obtaining [46] Considering the first-order RZ parameterization (tak- ( ) ing only the first three items of Eq. (3)), one can approx- n 2 r4N2v2 1 − v2 = 0 0 0 . (41) imately write n − 2 r4N2v2 0 1 v0 ( )[ ] 2M 4M2(a − ϵ) 2Mϵ The above expressions will be used later to analyze N2 ≃ 1 − 1 + 0 − . (47) + ϵ 2 + ϵ 2 + ϵ the spherical accretion processes for some test fluids. r(1 ) r (1 ) r(1 )

Then, Eq. (46) can be approximately rewritten as V. APPLICATIONS TO TEST FLUIDS

In this section, we consider the accretion processes of ( ) − [ ] − 2M 1 w 4M2(a − ϵ) 2Mϵ 1 w several test fluids, using the equations derived in the 1 − 1 + 0 − r(1 + ϵ) 2 + ϵ 2 r(1 + ϵ) above sections for a parameterized spherically symmetric H ≃ r (1 ) . black hole. Specifically, we consider the isothermal and (1 − v2)1−wv2wr4w polytropic fluids in the following subsections. (48)

A. Isothermal test fluid 2 = At the sonic point, with cs w, Eq. (34) reduces to

In this subsection, we consider the accretion pro- 2 rN, cesses for isothermal (constant-temperature) fluids. The w = r . (49) 2 rN, + 4N2 corresponding system can be viewed as an adiabatic one, r r=r∗ owing to the fast movement of the fluid. For such a system, we define its equation of state (EoS) w as With Eq. (47), Eq. (49) can be approximately rewritten as w ≡ p/ρ. (42) 2 2 3 M[−12M ϵ + r∗ (1 + ϵ) + 4a M(3M − r∗(1 + ϵ))] w = 0 , (50) where ρ and p represent the energy density and pressure LT1 of the fluid, respectively. It is worth noting that 0 < w ⩽ 1 for isothermal fluids [20]. In addition, the adiabatic speed where dp of sound is given by c2 ≡ = w. = 3ϵ − 2 + ϵ 3 + 3 + ϵ 3 s dρ LT1 4M 3Mr∗ (1 ) 2r∗ (1 ) According to h = (ρ + p)/n = (1 + w)ρ/n and c2 = 2 s + 4a0 M (−M + r∗ + r∗ϵ). (51) dlnh/dlnn = w, we have

( ) + n 1 w 1.. Solution for an ultra-stiff fluid (w = 1) ρ = ρ0 , (43) n0 Let us first consider an ultra-stiff fluid, whose energy density is equal to its pressure. In this case, the equation

015102-6 Spherical accretion flow onto general parameterized spherically symmetric black hole spacetimes Chin. Phys. C 45, 015102 (2021) of state is w = p/ρ = 1. The Hamiltonian (48) for the ul- and red curves represent the flows with minimal Hamilto- tra-stiff fluid becomes nian Hmin for the accretion outflow. All of the flows in Fig. 1 between the red and green curves are physical and 1 have Hamiltonian H > H . H = . (52) min v2r4 The values of the minimal Hamiltonian Hmin depend on the RZ parameterization parameters. In Table 1, the H ϵ For physical flows, one has |v| < 1. Therefore, the values of r0 and for the different values of parameter Hamiltonian (52) for the ultra-stiff fluid has a minimal for the ultra-stiff fluid are presented. Since the horizon ra- −4 dius decreases with respect to ϵ, it is shown clearly that value Hmin = r . With Eq. (52), the two-dimensional dy- 0 the minimal Hamiltonian H = 1/r4 increases with in- namical system (33), (34) is min 0 creasing ϵ. 2 r˙ = − , (53) 2.. Solution for an ultra-relativistic fluid (w = 1/2) r4v3 Let us now consider an ultra-relativistic fluid, for which the equation of state is w = 1/2, i.e., p = ρ/2. In 4 this case, the fluid's isotropic pressure is less than its en- v˙ = . (54) w = / 5 2 ergy density. With 1 2, the Hamiltonian (48) be- r v comes It is easy to see that this dynamical system has no crit- ( ) / [ ]1/2 ical points. The phase space portrait of this dynamical 2M 1 2 4M2(a − ϵ) 2Mϵ 1 − 1 + 0 − system for the ultra-stiff fluid with M = 1, a0 = 0.001, and r(1 + ϵ) r2(1 + ϵ)2 r(1 + ϵ) ϵ = . H = , (55) 0 1 for a general parameterized black hole is depicted r2|v|(1 − v2)1/2 in Fig. 1, in which the physical flow of the ultra-stiff fluid in the general parameterized black hole is represented and then, the two-dimensional dynamical system (33), by several curves with arrows. It is shown that the curves (34) is with v < 0 have arrows directed toward the black hole, representing the accreting flow of the ultra-stiff fluid, ( ) [ ] / 1/2 2 1 2 while the curves with v > 0 have arrows directed toward 2M 4M (a0 − ϵ) 2Mϵ 1 − 1 + − the outside, representing the outflow fluids. The green r(1 + ϵ) r2(1 + ϵ)2 r(1 + ϵ) r˙ = r2(1 − v2)3/2 ( ) / [ ] / 2M 1 2 4M2(a − ϵ) 2Mϵ 1 2 1 − 1 + 0 − r(1 + ϵ) r2(1 + ϵ)2 r(1 + ϵ) − , (56) r2v2(1 − v2)1/2

( ) / [ ] / 2M 1 2 4M2(a − ϵ) 2Mϵ 1 2 2 1 − 1 + 0 − r(1 + ϵ) r2(1 + ϵ)2 r(1 + ϵ) v˙ = r3|v|(1 − v2)1/2 ( )[ ] 2M 8M2(a − ϵ) 2Mϵ 1 − − 0 − r(1 + ϵ) 3 + ϵ 2 2 + ϵ − r (1 ) r (1 ) LT ( 2 ) 2 4M (a0 − ϵ) 2Mϵ 2M 1 + − r2(1 + ϵ)2 r(1 + ϵ) Fig. 1. (color online) Phase space portrait of the dynamical − , (57) 2 system (33), (34) for the ultra-stiff fluid (w = 1) with the black r (1 + ϵ)LT2

hole parameters M = 1, ϵ = 0.1, and a0 = 0.0001.

Table 1. Values of r0 and Hmin for different values of the black hole parameter ϵ for the ultra-stiff fluid with w = 1. In the calculation, −4 we set M = 1 and a0 = 10 .

ϵ 0 0.1 0.2 0.3 0.4 0.5

r0 2 1.81818 1.66667 1.53846 1.42857 1.33333 H min 0.0625 0.0915063 0.1296 0.178506 0.2401 0.316406

015102-7 Sen Yang, Cheng Liu, Tao Zhu et al. Chin. Phys. C 45, 015102 (2021)

where black hole parameters ϵ and a0 . ( ) 1/2 The phase space portrait of this dynamical system for 2 2 1/2 2M the ultra-relativistic fluid with M = 1, a = 0.0001, and LT2 =2r |v|(1 − v ) 1 − 0 r(1 + ϵ) ϵ = 0.1 for a general parameterized black hole is depicted [ ] / 4M2(a − ϵ) 2Mϵ 1 2 in Fig. 3, in which the physical flow of the ultra-relativ- × 1 + 0 − . (58) r2(1 + ϵ)2 r(1 + ϵ) istic fluid for a general parameterized black hole is represented by several curves. Clearly, both the critical , ,− For some given value of H , one can obtain v2 from Eq. (55), points in Fig. 3, (r∗ v∗) and (r∗ v∗), are saddle points of √ the dynamical system. The five curves in Fig. 3 corres- 4F(r) pond to the different values of the Hamiltonian 1 1 + 4H2 H0 = {H∗ − 0.05, H∗ − 0.02, H∗, H∗ + 0.03, H∗ + 0.08} . r 0 v2 = , (59) This plot shows several different types of fluid motion. 2 The magenta (with H = H∗ + 0.08) and blue (with H = H∗ + 0.03) curves correspond to the purely superson- where ic accretion (v < −v∗ branches), purely supersonic out- 2M 8M3 8a M3 flow (v > v∗ branches), or purely subsonic accretion fol- F(r) = − 1 + + + 0 r r3(1 + ϵ)3 r3(1 + ϵ)3 lowed by the subsonic outflow (−v∗ < v < v∗ branches). 3 2 The red (with H = H∗ − 0.02) and green (with 8M 4a0 M − − . (60) H = H∗ − 0.05) curves correspond to the non-physical be- r3(1 + ϵ)2 r2(1 + ϵ)2 havior of the fluid. The most interesting solution of the fluid motion is In addition, one can obtain the critical points of the depicted by the black curves in Fig. 3, revealing the tran- accretion process for the ultra-relativistic fluid by solv- sonic behavior of the fluid outside the black hole horizon. ing the two-dimensional dynamical system, when both For v < 0, there are two black hole curves that go through right hand sides of Eq. (56) and Eq. (57) vanish. With the the sonic point (r∗,−v∗). One solution starts at the spatial black hole parameters set to M = 1, ϵ = 0.1, and infinity with a sub-sonic flow followed by a supersonic a = 0.0001, one obtains the physical critical points 0 flow after it crosses the sonic point, which corresponds to (r∗,v∗), i.e., (2.30139,−0.707107) and (2.30139, the standard nonrelativistic accretion considered by 0.707107) for the outflow and the accreting flow, respect- Bondi in [4]. Another solution, which starts at the spatial ively. Inserting these critical point values (r∗,v∗) into infinity with a supersonic flow but becomes sub-sonic Eq. (55), one finds the critical Hamiltonian H∗ = 0.160335. The values of r∗ , v∗ , and H∗ at the sonic point with different values of the black hole parameters are summarized in Table 2 for w = 1/2, M = 1, and a0 = 0.0001. Clearly, as the value of the black hole parameter ϵ increases, the following occurs: (1) the value of r∗ at the sonic point decreases, while the distance from horizon r0 to the critical point increases; (2) the values of velocity v∗ at the sonic points are two constants, because they are equal to the fluid's speed of sound; and (3) the value of the Hamiltonian for the fluid at the critical

points increases. We also show the behavior of the critical radius r∗ with respect to the black hole parameter ϵ for Fig. 2. (color online) Relation between r∗ and ϵ for differ- different values of parameter a0 in Fig. 2, which shows ent a0 in the spherical accretion process for the ultra-relativist- that the critical radius r∗ decreases with increase in the ic fluid (w = 1/2).

Table 2. Values of r∗ , v∗ , and H∗ at the sonic point, for different values of the black hole parameter ϵ for the ultra-relativistic fluid

with w = 1/2. We use M = 1 and a0 = 0.0001 in the calculation.

ϵ 0 0.1 0.2 0.3 0.4 0.5

r0 2 1.81818 1.66667 1.53846 1.42857 1.33333

r∗ 2.49998 2.30139 2.14917 2.04111 1.97876 1.96016

v∗ 0.70711 0.70711 0.70711 0.70711 0.70711 0.70711

H∗ 0.14311 0.16034 0.17465 0.18507 0.19098 0.19271

015102-8 Spherical accretion flow onto general parameterized spherically symmetric black hole spacetimes Chin. Phys. C 45, 015102 (2021)

3.. Solution for a radiation fluid (w = 1/3) For a radiation fluid, the equation of state is w = 1/3. In this case, the Hamiltonian (48) becomes

( ) / [ ] / 2M 2 3 4M2(a − ϵ) 2Mϵ 2 3 1 − 1 + 0 − r(1 + ϵ) r2(1 + ϵ)2 r(1 + ϵ) H = , (61) r4/3|v|2/3(1 − v2)2/3

and then, the two-dimensional dynamical system (33), (34) is

( ) [ ] / 2/3 2 − ϵ ϵ 2 3

2M 4M (a ) 2M 4v1/3 1 − 1 + 0 − Fig. 3. (color online) Phase space portrait of the dynamical r(1 + ϵ) r2(1 + ϵ)2 r(1 + ϵ) r˙ = system (33), (34) for the ultra-relativistic fluid (w = 1/2), for 3r4/3(1 − v2)5/3 = ϵ = . = . ( ) [ ] / the black hole parameters M 1, 0 1, and a0 0 0001. The 2/3 2 2 3 2M 4M (a0 − ϵ) 2Mϵ critical (sonic) points (r∗,v∗) of this dynamical system are 2 1 − 1 + − r(1 + ϵ) 2 + ϵ 2 r(1 + ϵ) presented by the black spots in the figure. The five colored − r (1 ) , / / / curves (black, red, green, magenta, and blue) correspond to 3r4 3|v|5 3(1 − v2)5 3 the Hamiltonian values H = H∗, H∗ − 0.02,H∗ − 0.05,H + 0.03, (62) and H∗ + 0.08. ( ) / [ ] / 2M 2 3 4M2(a − ϵ) 2Mϵ 2 3 4 1 − 1 + 0 − after it crosses the sonic point, is unstable, according to r(1 + ϵ) r2(1 + ϵ)2 r(1 + ϵ) v˙ = the analysis presented in [46]; such a behavior is very dif- 3r7/3|v|2/3(1 − v2)2/3 > ( )[ ] ficult to achieve. For v 0, there are two solutions as 2M 8M2(a − ϵ) 2Mϵ 2 1 − − 0 + well. One solution, which starts at the horizon with a su- r(1 + ϵ) r3(1 + ϵ)2 r2(1 + ϵ) personic flow followed by a sub-sonic flow after it − LT crosses the sonic point, corresponds to the transoinc solu- ( 3 ) 4M2(a − ϵ) 2Mϵ tion of the stellar wind, as discussed in [4] for the non-re- 4M 1 + 0 − < r2(1 + ϵ)2 r(1 + ϵ) lativistic accretion. Another solution, similar to the v 0 − , (63) 2 + ϵ case, is unstable and too difficult to achieve [46]. r (1 )LT3 Here, we would like to add several remarks about the physical explanations of the flows in Fig. 3 for different where ( ) values of Hamiltonian H . In general, different values of 1/3 2M the Hamiltonian represent different initial states of the LT =3r4/3|v|2/3(1 − v2)2/3 1 − 3 r(1 + ϵ) dynamical system. For the transonic solution of the ultra- [ ] / 2 1 3 relativistic fluid, its Hamiltonian can be evaluated at the 4M (a0 − ϵ) 2Mϵ × 1 + − . (64) sonic point. The Hamiltonian with values different from r2(1 + ϵ)2 r(1 + ϵ) the transonic one does not represent any transonic solutions of the flow. For example, the green curve shows the The sonic points can be found by solving the above subcritical fluid flow since such a flow does not pass two-dimensional dynamical system, when both right hand through the critical point and fails to reach the critical sides of Eq. (62) and Eq. (63) vanish. The values of the point. In fact, such solutions have a turning or bouncing critical radius r∗ , sound speed v∗ , and critical H∗ for point, which is the nearest point reachable by such fluids, different values of ϵ are summarized in Table 3 for beyond which they are bounced back or turned around to w = 1/3, M = 1, and a0 = 0.0001. Similar to the ultra-re- infinity. A similar explanation holds for the red curves. lativistic fluid, the critical radius decreases with increas- The curves shown in blue and magenta can be termed su- ing ϵ, while the critical Hamiltonian H∗ increases. We per-critical flows. Although such fluids do not go through also illustrate the behavior of the critical radius r∗ for the the critical point either, they already possess velocities radiation fluid with respect to ϵ for different values of a0 above the allowed critical value. Such flows end up enter- in Fig. 4. ing the black horizon. It is also worth mentioning that a The phase space portrait of this dynamical system for similar analysis also applies to other fluids, including ra- the radiation fluid with M = 1, a0 = 0.001, and ϵ = 0.1 is diation, sub-relativistic, and polytropic fluids. displayed in Fig. 5, in which the physical flow of the radi-

015102-9 Sen Yang, Cheng Liu, Tao Zhu et al. Chin. Phys. C 45, 015102 (2021)

Table 3. Values of r∗ , v∗ , and H∗ at the sonic point, for different values of the black hole parameter ϵ for the radiation fluid with

w = 1/3. We use M = 1 and a0 = 0.0001 in the calculation.

ϵ 0 0.1 0.2 0.3 0.4 0.5

r0 2 1.81818 1.66667 1.53846 1.42857 1.33333

r∗ 2.99996 2.8096 2.67692 2.59413 2.55246 2.54108

v∗ 0.57735 0.57735 0.57735 0.57735 0.57735 0.57735

H∗ 0.20999 0.22082 0.22845 0.23316 0.2355 0.23613

blue curves represent the supersonic flows for v < −v∗ or v > v∗ , while they correspond to sub-sonic flows if −v∗ < v < v∗ . The transonic solutions are presented by the black curves. For v < 0, one of the black curves, which starts at the spatial infinity with a sub-sonic flow and then becomes supersonic after it crosses the sonic point (r∗,−v∗), corresponds to the standard transonic accretion, and another black curve represents an unstable solution. For v > 0, one black curve corresponds to the transonic outflow of wind, and another one represents an unstable flow, similar to the case of the ultra-relativistic fluid. The Fig. 4. (color online) Relation between r∗ and ϵ for differ- green and red curves are non-physical solutions. ent a0 in the spherical accretion process for the radiation fluid (w = 1/3). 4.. Solution for a sub-relativistic fluid (w = 1/4) Let us now consider a sub-relativistic fluid, whose energy density exceeds its isotropic pressure; the equation of state for such a fluid is w = 1/4. In this case, the Hamiltonian (48) takes the form

( ) / [ ] / 2M 3 4 4M2(a − ϵ) 2Mϵ 3 4 1 − 1 + 0 − r(1 + ϵ) r2(1 + ϵ)2 r(1 + ϵ) H = √ , (65) r |v|(1 − v2)3/4

and then the two-dimensional dynamical system is

( ) / [ ] / √ 2M 3 4 4M2(a − ϵ) 2Mϵ 3 4 3 |v| 1 − 1 + 0 − r(1 + ϵ) r2(1 + ϵ)2 r(1 + ϵ) r˙ = − 2 7/4 2r(1 v ) ( ) [ ] / Fig. 5. (color online) Phase space portrait of the dynamical 3/4 2 3 4 2M 4M (a0 − ϵ) 2Mϵ system (33), (34) for the radiation fluid (w = 1/3) for black 1 − 1 + − r(1 + ϵ) r2(1 + ϵ)2 r(1 + ϵ) hole parameters M = 1, ϵ = 0.1, and a0 = 0.0001. The critical − , r|v|3/2(1 − v2)3/4 (66) (sonic) points (r∗,v∗) of this dynamical system are presented by the black spots in the figure. The five colored curves (black, red, green, magenta, and blue) correspond to the val- ( ) [ ] / 3/4 2 3 4 2M 4M (a0 − ϵ) 2Mϵ ues of Hamiltonian H = H∗, H∗ − 0.03,H∗ − 0.05,H + 0.05, 1 − 1 + − H + . r(1 + ϵ) r2(1 + ϵ)2 r(1 + ϵ) and ∗ 0 1, respectively. v˙ = √ 2 | | − 2 3/4 ( r)[ v (1 v ) ] 2 ation fluid for a general parameterized black hole is rep- 2M 8M (a0 − ϵ) 2Mϵ 3 1 − − + r(1 + ϵ) 3 + ϵ 2 2 + ϵ resented by several curves. One can see that both the crit- − r (1 ) r (1 ) , ,− ical points in Fig. 5, (r∗ v∗) and (r∗ v∗), are saddle points ( LT4 ) of the dynamical system. From Fig. 5, one also observes 6M2(a − ϵ) 2Mϵ 4M 1 + 0 − that the radiation fluid shares the same types of the fluid r2(1 + ϵ)2 r(1 + ϵ) − , (67) = / = / 2 motion (w 1 3) as the ultra-relativistic fluid (w 1 2), r (1 + ϵ)LT4 as shown in Fig. 3. Similar to Fig. 3, the magenta and

015102-10 Spherical accretion flow onto general parameterized spherically symmetric black hole spacetimes Chin. Phys. C 45, 015102 (2021)

where creasing ϵ and a0 . The phase space portrait of the dynamical system for ( ) / √ 2M 1 4 the sub-relativistic fluid is shown in Fig. 7. From this fig- LT =4r |v|(1 − v2)3/4 1 − 4 r(1 + ϵ) ure, we observe that the type of the fluid motion for the [ ] / sub-relativistic fluid (w = 1/4) is same as those for the ul- 4M2(a − ϵ) 2Mϵ 1 4 × 1 + 0 − . (68) tra-relativistic fluid (w = 1/2) and the radiation fluid 2 + ϵ 2 + ϵ r (1 ) r(1 ) (w = 1/3). For v > v∗ , the magenta and blue curves are purely supersonic outflows, while for v < −v∗ , they rep- For this dynamical system, similar to the above two resent supersonic accretions. For −v∗ < v < v∗ , these cases, we present the values of r∗ , v∗ , and H∗ for differ- curves are sub-sonic flows. The black curves shown in ent values of ϵ in Table. 4 for w = 1/4, M = 1, and Fig. 7 are more interesting since they represent the tran- a0 = 0.0001. We also plot the behavior of the critical radi- sonic solution of the spherical accretion for v < 0 and us r∗ for the sub-relativistic fluid with respect to the black spherical outflow for v > 0 around the black hole. Similar hole parameter ϵ for different values of a0 ; this is shown to the results for the ultra-relativistic and radiation fluids, in Fig. 6. Clearly, the critical radius r∗ decreases with in- the red and green curves represent non-physical solutions.

Table 4. Values of r∗ , v∗ , and H∗ at the sonic point, for different values of the black hole parameter ϵ for the sub-relativistic fluid

w = 1/4. The black hole parameters M and a0 are set to M = 1 and a0 = 0.0001.

ϵ 0 0.1 0.2 0.3 0.4 0.5

r0 2 1.81818 1.66667 1.53846 1.42857 1.33333

r∗ 3.49993 3.32303 3.20741 3.13980 3.10743 3.09881

v∗ 0.5 0.5 0.5 0.5 0.5 0.5

H∗ 0.26557 0.27281 0.27750 0.28022 0.2815 0.28184

Fig. 6. (color online) Relation between r∗ and ϵ for differ-

ent a0 in the spherical accretion process for the sub-relativistic fluid (w = 1/4).

Fig. 7. (color online) Phase space portrait of the dynamical B. Polytropic test fluid system (33), (34) for the sub-relativistic fluid (w = 1/4), for the

The state of a polytropic test fluid can be described by black hole parameters M = 1, ϵ = 0.1, and a0 = 0.0001. The

parameters are r0 ≃ 1.81818, r∗ ≃ 3.32303, v∗ ≃ 0.5. Black plot: γ p = κn , (69) the solution curve through the saddle CPs (r∗,v∗ ) and (r∗,−v∗ ) for which H = H∗ ≃ 0.272806. Red plot: the solution curve for where κ and γ are constants. For ordinary matter, one which H = H∗ − 0.03. Green plot: the solution curve for which generally works with the constraint γ > 1. Following [45], H = H∗ − 0.05. Magenta plot: the solution curve for which we obtain the following expressions for the specific en- H = H∗ + 0.03. Blue plot: the solution curve for which thalpy: H = H∗ + 0.1.

κγnγ−1 h = m + , (70) where the constant of integration has been identified with γ − 1 the baryonic mass m. The three-dimensional speed of sound is given by

015102-11 Sen Yang, Cheng Liu, Tao Zhu et al. Chin. Phys. C 45, 015102 (2021)

γ − ( )(γ−1)/2 2 = ( 1)Y ≡ κγ γ−1 . 1 − v2 cs γ − + (Y n ) (71) Y = m(γ − 1)Z (78) m( 1) Y r4N2v2

Using Eq. (41) in Eq. (71), we obtain into Eq. (71), we obtain  ( ) γ− /   2 ( 1) 2 ( )  1 − v  (γ−1)/2 h = m1 + Z , (72) 1 − v2 r4N2v2 c2 = Z(γ − 1 − c2) . (79) s s r4N2v2 where This, along with Eq. (36), takes the form of the following 2 2 2 κγ expressions at the critical points (c (r∗) = v (r∗) = v∗ ): ≡ | |γ−1 = . > , s Z γ − C1 const 0 (73) m( 1) ( ) γ− / − 2 ( 1) 2 2 2 1 v∗ c r∗ = Z γ − − v , (80) s ( ) ( 1 ∗) 4 2 2 and Z is a positive constant. If critical points exist, Z takes r∗ N∗ v∗ the special form

  γ− / γ−1 5 2 ( 1) 2 κγ r∗ N∗,  2 2 3 n∗  r∗  M − M ϵ + r + ϵ + a M M − r∗ + ϵ Z ≡   = const. > 0. (74) 2 = [ 12 ∗ (1 ) 4 0 (3 (1 ))]. m(γ − 1) 4 v∗ (81) LT1

The constant Z depends on the black hole parameters Here, we have used Eq. (36) to obtain the right-hand side and the test fluid. From Eq. (74), it is clear that Z is of Eq. (81). If there are critical points, the solution of this roughly proportional to κn∗/m for a given black hole system of equations in (r∗,v∗) provides all the critical solution and certain test fluids. points, with a given value of the positive constant Z. Inserting Eq. (72) into Eq. (31), we evaluate the Then, one can use the values of critical points to reduce Hamiltonian by n∗ from Eq. (74).   Numerical solutions to the dynamical system of Eqns. ( )(γ−1)/2 2 N2  1 − v2  (80) and (81) are shown in Fig. 8. Clearly, there is only H = 1 + Z  , (75) 1 − v2 r4N2v2 one critical point, a saddle point, in accretion (−1 < v < 0) where m2 has been absorbed into a redefinition of (t¯,H). 2 > 2 > Obviously, N (r) 0 and N,r 0 for all r. This means that the constant Z > 0 (recall that γ > 1). It is easy to see that there are no global solutions, since the Hamiltonian re- mains constant along the solution curves. Notice that since γ > 1, the solution curves do not cross the r axis at points where v = 0 and r , r0 ; other- wise, the Hamiltonian (75) would diverge there. The point on the r axis which the solution curves may cross is only (r0,0). The horizon r = r0 is a single root to N2(r) = 0, in the vicinity of which v behaves as

2−γ | | ∝ | − | 2(γ−1) .

v r r0 (76) Fig. 8. (color online) Accretion of a polytropic test fluid. We see that only solutions with 1 < γ < 2 may cross Contour plots of the Hamiltonian (75) for γ = 5/3 and Z = 5, = ϵ = . = . the r axis. Here, H(r0,0) is the limit of H(r,v) as (r,v) → for the black hole parameters M 1, 0 1, and a0 0 0001. γ ≃ . ≃ . ≃ . (r0,0). When 1 < γ < 2, the pressure p = κn diverges at The parameters are r0 1 81818, r∗ 2 30998, v∗ 0 704098. the horizon as Black plot: the solution curve through the saddle CPs (r∗,v∗ ) and (r∗,−v∗ ) for which H = H∗ ≃ 5.5208. Red plot: the solution −γ curve for which H = H∗ − 0.3. Green plot: the solution curve 2(γ−1) p ∝ |r − r0| . (77) for which H = H∗ − 1.0. Magenta plot: the solution curve for which H = H∗ + 0.5. Blue plot: the solution curve for which Then, inserting H = H∗ + 1.0

015102-12 Spherical accretion flow onto general parameterized spherically symmetric black hole spacetimes Chin. Phys. C 45, 015102 (2021) of a polytropic test fluid. The motion types for the poly- time is spherically symmetric, we can perform the calcu- tropic test fluids, as shown in Fig. 8, are the same as the lations in the equatorial plane θ = π/2. To find the null motion types for the isothermal test fluids with w = 1/2 geodesics around the black hole we can use the Hamilton- (c.f. Fig. 3), w = 1/3 (c.f. Fig. 5), and w = 1/4 (c.f. Fig. 7). Jacobi equation, given as follows:

∂ ∂ ∂ VI. CORRESPONDENCE BETWEEN SONIC S = −1 µν S S , ∂λ g ∂ µ ∂ ν (87) POINTS OF PHOTON GAS AND PHOTON 2 x x SPHERE where λ is the affine parameter of the null geodesic, and Recently, a correspondence was shown between the S denotes the Jacobi action of the photon. The Jacobi ac- sonic points of the ideal photon gas and the photon sphere tion S can be separated in the following form: in static spherically symmetric spacetimes [37]. This important result is valid not only for spherical accretion of S = −Et + Lϕ + S r(r), (88) the ideal photon gas but also for rotating accretion in static spherically symmetric spacetimes [38, 39, 47]. In this where E and L represent the energy and the angular mo- section, we establish this correspondence for parameter- mentum of the photon, respectively. The function S (r) ized spherically symmetric black holes. r Let us first consider the spherical accretion of the depends only on r. ideal photon gas and derive the corresponding sonic Substituting the Jacobi action into the Hamilton-Jac- points. The equation of state for the ideal photon gas in d- obi equation, we obtain dimensional space is ∫ √ r B2(r) R(r) S r(r) = dr, (89) kγ γ− r2N2(r) h = n 1, (82) γ − 1 where with 2 2 2 4 2 = −r N (r)L + r E . d + 1 R(r) (90) γ = , (83) B2(r) B2(r) d The variation of the Jacobi action gives the following where k is a constant of the entropy [37]. The speed of equations of motion for the evolution of the photon: sound for the ideal photon gas is constant dt E dlnh = , (91) c2 ≡ = γ − 1. (84) dλ N2(r) s dlnn

For general parameterized spherically symmetric dϕ L spacetimes (d = 3), the equation of state of the ideal = , (92) photon gas becomes dλ r2

h = 4kn1/3, (85) √ dr R(r) = . (93) λ 2 2 = d r and the speed of sound for the ideal photon gas is cs 2. For the accretion of the ideal photon gas in parameter- To determine the radius of the photon sphere of the ized spherically symmetric spacetimes, the radius r∗ of the sonic point is specified by black hole, we need to find the critical circular orbit for the photon, which can be derived from the unstable con- ( ) d N dition = 0. (86) dr r dR(r) R(r) = 0, = 0. (94) To proceed, let us derive the photon sphere by analyz- dr ing the evolution of a photon in a parameterized spherically symmetric black hole. The photon follows the null For a parameterized spherically symmetric black hole, geodesics in a given black hole spacetime. As the space- from the above conditions, one finds

015102-13 Sen Yang, Cheng Liu, Tao Zhu et al. Chin. Phys. C 45, 015102 (2021)

( ) d N = 0. (95) dr r

This is the condition for determining the radius of the photon sphere. Clearly, Eq. (86) is actually the same as Eq. (95), which means that the critical radius r∗ of a sonic point, for the accretion of the ideal photon gas in parameterized spherically symmetric spacetimes, is equal to the radius of the photon sphere. With Eq. (95), one obtains the critical radius r∗ and the radius of the photon sphere in parameterized spherically symmetric spacetimes ( ) −1 1 dN r∗ = . (96) Fig. 9. (color online) Schematic of the spherical accretion of N dr = r r∗ the ideal photon gas onto a spherically symmetric black hole and its photon sphere (the red circle). The red circle has a two- By substituting Eq. (47) into the above equation, we fold meaning, since it represents both the photon sphere and obtain the expression for r∗ , the sonic radius of the spherical accretion of the ideal photon √ gas. 2 2 (1 + i 3)M (1 + ϵ)[−8a0 + 3(1 + ϵ) ] r∗ =M + √ 2LT5 process and presented the general formulas for determin- (1 − i 3)LT ing the sonic points (or critical points). These two equa- + 5 , (97) 6(1 + ϵ)3 tions were derived from the conservation laws of energy and particle number of the fluid. Using these two equa- where tions, we analyzed the accretion processes of various perfect fluids, such as the isothermal fluids of the ultra-stiff, ultra-relativistic, and sub-relativistic types, and polytrop- 3 6 2 3 LT5 =[−27M (1 + ϵ) (1 + a0(6 − 4ϵ) − 7ϵ + 3ϵ + ϵ ) ic fluids. The flow behaviors of these test fluids around a √ √ (98) 1/3 general parameterized spherically symmetric black hole + 6 3 LT6] , were studied in detail and are shown graphically in Figs. 1, 3, 5, 7. For the isothermal fluid, it is interesting to men- and tion that the sonic point does not exist for the ultra-stiff fluid with w = 1 alone; thus, transonic solutions exist for 6 12 3 2 2 = / LT6 =M (1 + ϵ) [128a − 9a (−11 + 68ϵ + 4ϵ ) the ultra-relativistic fluid with w 1 2, for the radiation 0 0 = / 2 3 2 3 fluid with w 1 3, and for the sub-relativistic fluid − 135ϵ(1 − 2ϵ + 3ϵ + ϵ ) + 135a0(1 − 3ϵ + 7ϵ + ϵ )]. w = 1/4. The value of ϵ affects r0, r∗, and H∗ but not v∗ , (99) which indicates the effect of the position of the event horizon. For the polytropic fluid, Fig. 8 shows that it exhib- It is easy to verify that when the additional paramet- its a similar flow behavior as the isothermal fluids with ϵ ers (a0 and ) are set to zero, the critical radius reduces to w = 1/2, w = 1/3, and w = 1/4. Here, we would like to r∗ = 3M. In Fig. 9, we schematically show the spherical mention that the results presented in this paper can also accretion of the ideal photon gas onto a spherically sym- be reduced to specific cases in several modified theories metric black hole and its photon sphere (represented by of gravity. For example, one can map the results here to the red circle). The red circle in Fig. 9 thus has a two-fold the first-type Einstein-Aether black hole in [48, 49] by meaning, since it represents both the photon sphere and setting the sonic radius of the spherical accretion of the ideal √ photon gas. M − M2 − æ2 ϵ = √ , (100) M + M2 − æ2 VII. CONCLUSIONS AND DISCUSSION In this paper, we studied the spherical accretion flow æ2 of a perfect fluid onto a general parameterized spheric- a0 = √ , b0 = 0, (101) ally symmetric black hole. For this purpose, we first for- (M + M2 − æ2)2 mulated two basic equations for describing the accretion

015102-14 Spherical accretion flow onto general parameterized spherically symmetric black hole spacetimes Chin. Phys. C 45, 015102 (2021)

ai = 0, bi = 0, (i > 0), (102) creting behaviors of various types of matter when the spherical symmetry approximation is relaxed by consid- where ering a non-zero relative velocity between the black hole and the accreting matter. This scenario is also known as 2c − c wind accretion or Bondi–Hoyle–Lyttleton accretion [51- æ2 = − 13 14 M2, (103) 2(1 − c13) 53] (see [54] for a review). We will consider the more complicated accretion disk model, which is more related to real observations, in our future work. with c13 and c14 being the coupling constants in the Ein- stein-Aether theory. It is interesting to mention that flow Second, it is also interesting to extend our analysis to behaviors for different test fluids in this paper are qualit- rotating black holes. In a rotating background, one may atively consistent with those studied in [50] for the spher- consider rotating fluids accreting onto a rotating black ical accretion in the Einstein-Aether theory. hole. The rotation of the fluids can lead to the formation We further considered the spherical accretion of the of a disc-like structure around the black hole, and such ideal photon gas and derived the radius of its sonic point. accretion discs are the most commonly studied engines Comparing the radius with that of the photon sphere for a for explaining astrophysical phenomena such as active general parameterized spherically symmetric black hole, galactic nuclei, X-ray binaries, and gamma-ray bursts. we studied the correspondence between the sonic points However, considering rotation introduces complications of the accreting photon gas and the photon sphere for a into the accretion problem, in which case, the study heav- general parameterized spherically symmetric black hole. ily relies on numerical calculations. With the above main results, we would like to men- Finally, when one considers a rotating black hole, its tion several directions that can be pursued for extending shadow does not correspond to a photon sphere but a our analysis. First, spherical accretion is the simplest ac- photon region. An immediate question now arises as to cretion scenario, in which the accreting matter falls stead- what structure in the rotating accretion of the ideal photon ily and radially into a black hole. This is an extreme gas corresponds to the photon region of a rotating black simple case. Therefore, it is interesting to explore the ac- hole. This is still an open issue.

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