Cyclic and heteroclinic flows near general static spherically symmetric black holes: Semi-cyclic flows -- Addendum and corrigendum

We present new accretion solutions of a polytropic perfect fluid onto an f(R)-gravity de Sitter-like black hole. We consider two f(R)-gravity models and obtain finite-period cyclic flows oscillating between the event and cosmological horizons as well as semi-cyclic critical flows executing a two-way motion from and back to the same horizon. Besides the generalizations and new solutions presented in this work, a corrigendum to Eur. Phys. J. C (2016) 76:280 is provided.

The aim of this short work is two-fold: (1) Generalize the dynamical-system formalism for accretion of perfect fluids to metrics of the form ds 2 = −A(r)dt 2 + dr 2 B(r) + C(r)(dθ 2 + sin 2 θdφ 2 ). (1) This will allow us to generalize the properties of the accreting fluid intended for future use. (2) Obtain new interesting solutions not discussed so far in the scientistic literature.
In (1), (A, B, C) are any functions of the radial coordinate r assumed to be well-defined and positive-definite in the regions where the Killing vector ξ µ = (1, 0, 0, 0) is timelike and their ratio D ≡ A/B is positive-definite on the horizons too. For the metric (1) to describe a black hole solution, the equations B(r) = 0 and A(r) = 0 should have the same set of solutions with same multiplicities. In our applications we restrict ourselves to the cases where the global structure of the spacetime is well-determined by (A, B, C). To keep the analysis general, we do not assume asymptotic flatness of the metric as we intend to apply it to the de Sitter and anti-de Sitter-like black holes. This metric form generalizes the one used in Refs. [1][2][3]; In Ref. [1] we restricted ourselves to the case A = B = f (r) and C = r 2 . It is easy to show that equations (5), (7), (11), (23), (24), (25) of Ref. [1] generalize respectively to where (C 1 , C 2 ) are constants of motion, u ≡ u r , and −1 < v < 1 is the three-velocity of a fluid element as measured by a locally static observer. The second line in (4) expresses the law of particle conservation, ∇ µ (nu µ ) = 0, and C 2 is the constant of motion hu µ ξ µ [ξ µ = (1, 0, 0, 0) is a timelike Killing vector]. This constant is the inertial-equivalent generalization of the energy conservation equation mu µ ξ µ [4]. The constant C 2 1 in (4) can be written as where "0" denotes any reference point (r 0 , v 0 ) from the phase portrait; this could be a CP, if there is any, spatial infinity (r ∞ , v ∞ ), or any other reference point. We can thus write The equation of state (EoS) of the fluid may be given in the form e = F(n) or equivalently in the form p = G(n) [1]. It has been shown [1] that (F, G), the specific enthalpy, and the three-dimensional speed of sound satisfy where the prime denotes derivative with respect to n. Equations (5) and (7) imply that the specific enthalpy h depends explicitly on (A, C, v) only. There is no explicit dependence on B. As we shall see in the next subsection, this implies that the Hamiltonian H, which defines the dynamical system, will also depend explicitly on (A, C, v) only. This fact will have consequences on the location of the critical points (CP's). If the fluid had further properties, say, being isothermal or polytropic, h and H can be expressed explicitly in terms of (A, C), as this is done in the following section.
Horizons r h are defined by A(r h ) = 0 and B(r h ) = 0 or simply by B(r h ) = 0 since the equations A(r h ) = 0 and B(r h ) = 0 are assumed to have the same set of solutions with same multiplicities. The zeros of B(r h ) = 0 determine the regions in three-space where the fluid flow takes place: These are the regions where ξ µ = (1, 0, 0, 0) is timelike.
Note that the case C 1 ≡ 0, corresponding either to n = 0 (2) (no fluid) or to v = 0 (4) and any n (no flow), is not interesting. So, we assume C 1 = 0. As the fluid approaches, or emanates from, any horizon (r → r h ), A approaches 0. Three cases emerge from (5) 3. |v| assumes any value between 0 and 1 there. This yields a divergent n as r → r h . Since 0 < |v| < 1 there is no reason that the fluid cumulates near the horizon: The flow continues until all fluid particles have crossed the horizon. We rule out this case for it is not physical. This conclusion, due to the law of particle conservation, is general and it does not depend on the fluid characteristics.
Since a non-perfect simple fluid (containing a single particle species) also obeys the law of particle conservation (4), these conclusions remain valid for real fluids too.

B. Dynamical system -Critical points
If the fluid had a uniform pressure, that is, if the fluid were not subject to acceleration, the specific enthalpy h reduces to the particle mass m and the first equation in (3) reduces to mu µ ξ µ = C 2 along the fluidlines. This is the well know energy conservation law which stems from the fact that the fluid flow is in this case geodesic. Now, if the pressure throughout the fluid is not uniform, acceleration develops through the fluid and the fluid flow becomes non-geodesic; the energy conservation equation mu µ ξ µ = cst, which is no longer valid, generalizes to its relativistic equivalent [4] hu µ ξ µ = C 2 as expressed in the first equation in (3).
Let the Hamiltonian H of the dynamical system be proportional to C 2 2 (4), which is a constant of motion. Substituting the first equation in (4) into the first equation in (3) yields With H given by (9), the dynamical system readṡ (here the dot denotes thet derivative wheret is the time variable of the Hamiltonian dynamical system). In (10) it is understood that r is kept constant when performing the partial differentiation with respect to v in H ,v and that v is kept constant when performing the partial differentiation with respect to r in H ,r . The critical points (CPs) of the dynamical system are the points (r c , v c ) where the rhs's in (10) are zero. To take advantage of the calculations made in [1] we introduce the radial coordinate ρ and the notation f defined by The Hamiltonian takes the form The derivative H(ρ, v) ,ρ has been evaluated in Ref. [ Using Introducing the notation g c = g(r)| r=r c and g c,r c = g ,r | r=r c where g is any function of r, the following equations provide a set of CPs that are solutions toṙ = 0 anḋ v = 0: where a c is the three-dimensional speed of sound evaluated at the CP. The first equation states that at a CP the three-velocity of the fluid equals the speed of sound. The second equation determines r c once the EoS, e = F(n) or p = G(n) (6), is known. From the set of equations (16) we see that the metric function B does not enter explicitly in the determination of the CP's; however, it does that implicitly via the successive dependence of h on n, of n on v, and of v on D.
Other sets of CPs, solutions toṙ = 0 andv = 0, may exist too. For instance, we may have (1) A c = 0 and A c,r c = 0 which, by (14) and (15), yieldṙ = 0 andv = 0 without having to impose the constraint v 2 c = a 2 c at the CP. This corresponds to a double-root horizon of an extremal black hole. When this is the case, the accretion becomes transonic well before the fluid reaches the CP, which is the horizon itself (recall that the three-velocity v, as the fluid approaches the horizon, tends to −1). However, extremal black holes are unstable and whatever accretes onto the hole modifies its mass making it non-extremal so that A c,r c = 0 no longer holds. We may also have (2) h = 0 at some point where ξ µ = (1, 0, 0, 0) is timelike, which may hold only for non-ordinary (dark, phantom, or else) accreting matter.

A. Hamiltonian system for test isothermal perfect fluids
The isothermal EoS is of the form p = ke = kF(n) with G(n) = kF(n) where k, the so-called state parameter, obeys the constraints 0 < k ≤ 1. The differential equation (6) reads where we have used (7). Using this and (5) in (9) we obtain where all the constant factors have been absorbed into the redefinition of the timet and the Hamiltonian H.

B. Hamiltonian system for test polytropic perfect fluids
The polytropic equation of state is where K and γ > 1 are constants. Inserting (20) into the differential equation (6), it is easy to determine the specific enthalpy by integration [1] where we have introduced the baryonic mass m. Introducing the constant then using (5), h takes the form Finally, the Hamiltonian (9) reduces to where m 2 has been absorbed into a re-definition of (t, H). The three-dimensional speed of sound is obtained from (8) Since γ > 1, this implies a 2 < γ − 1 and, particularly, v 2 c < γ − 1 if there are CPs to the Hamiltonian system. Equation (25) bears a striking similarity with Eq. (2.249) on page 119 of Ref. [4].

Corrigendum
Using the second equation in (4) and (22), we rewrite X (25) as Substituting into (25), we arrive at This equation along with the second line in (16) take the following expressions at the CPs For a given value of the positive constant Y, the resolution of this system of equations in (r c , v c ) provides all the CPs, if there are any. In Ref. [1] we worked with A = B = f and C = r 2 reducing (28) to which is the correct expression of Eq. (112) of Ref. [1]. In both equations (111) and (112) (2) to construct new solutions.

A. f(R)-gravity model of Ref. [5]
In Ref. [1], we considered three models of f(R) gravity [5][6][7]. For the model of Ref. [5] the black hole solution is of the form A = B = f and C = r 2 with (31) Following the notation of Ref. [1], we employ in this work the symbol " f " for the metric component, −g tt , and the symbol "f" for the function f(R) defining the f(R)-gravity model. The solutions shown in Fig. 5 of Ref. [1], which depict the accretion of a polytropic perfect fluid onto an antide Sitter-like f(R) black hole (31), are re-derived using the same values of the parameters: M = 1, β = 0.05, Λ = −0.04, γ = 1/2, and Y = −1/8. The re-derived solutions using the correct expressions (28) and (29) are plotted in Fig. 1 of this work. The first thing we note is that the re-derived solution corresponding to γ = 5.5/3, right panel of Fig. 1, has no CPs. Apart from this, the rederived solutions have the same characteristics as those shown in Fig. 5 of Ref. [1].
The solutions depicting the accretion of a polytropic perfect fluid onto a de Sitter-like f(R) black hole (31), which are shown in Fig. 2 of this work, have been constructed using the same values of the parameters used in Fig. 6 of Ref. [1]: M = 1, β = 0.05, Λ = 0.04, γ = 1.7, and Y = 1/8. The magenta and blue solutions were discovered in Ref. [1]. The new solutions are the semicyclic critical black plots of Fig. 2. The first semi-cyclic solution represents a supersonic accretion from the cosmological horizon, where the initial three-velocity is almost −1, then it becomes subsonic passing the CP, and it vanishes on the event horizon. The accretion is followed by a flowout back to the cosmological horizon reversing all the details. The second semi-cyclic solution is a flowout from the event horizon with an initial threevelocity in the vicinity of 1, which decreases gradually until it is sonic at the CP then zero at the cosmological horizon. This flowout is then followed by an accretion back to the event horizon.
Notice that on the horizons, r h = r eh (event horizon) or r h = r ch (cosmological horizon), the pressure of the fluid diverges as [1] if there v = 0. This explains why the fluid, once it reaches any horizon with vanishing three-velocity, it is repulsed backward under the effect of its own pressure.
As the value of the Hamiltonian exceeds the critical value H c ≡ H(r c , v c ), cyclic flows between the two horizons form. These flows are sandwiched by two separate branches corresponding to supersonic accretion and flowout, as depicted by the magenta and blue plots of Fig. 2. The separation between these supersonic branches increases with the value of the Hamiltonian resulting in faster accretion and flowout while the cyclic flow tends to become more and more nonrelativistic.

IV. CONCLUSION
In this addendum we have first generalized the dynamical-system procedure describing the accretion/flowout of perfect fluids to all black holes endowed with spherical symmetry. This is needed for many future investigations [8]. In our dynamical-system procedure we took the radial coordinate and the threevelocity as dynamical variables of the Hamiltonian, which is proportional to the square of the constant of motion hu µ ξ µ [ξ µ = (1, 0, 0, 0) is a timelike Killing vector]. This constant is the relativisitc equivalent general-ization of the energy conservation equation mu µ ξ µ [4].
We have shown that the de Sitter-like black holes, of different f(R)-gravity models, present cyclic non-critical flows and semi-cyclic critical flows all characterized by a vanishing three-velocity on either horizon or a luminal three-velocity there: no situations where the fluid reaches, or emanates from, either horizon with intermediate three-velocity occur. This is due to the law of particle conservation and it is not related to the nature of the fluid. This conclusion remains valid for real fluids too as they approach any horizon from within a region where ξ µ = (1, 0, 0, 0) is timelike.