# Astrophysical flows near \(f\,\,(T)\) gravity black holes

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## Abstract

In this paper, we study the accretion process for fluids flowing near a black hole in the context of *f*(*T*) teleparallel gravity. Specifically, by performing a dynamical analysis by a Hamiltonian system, we are able to find the sonic points. After that, we consider different isothermal test fluids in order to study the accretion process when they are falling onto the black hole. We find that these flows can be classified according to the equation of state and the black hole features. Results are compared in *f*(*T*) and *f*(*R*) gravity.

## Keywords

Black Hole Solution Curve Critical Flow Torsion Tensor Accretion Process## 1 Introduction

One of the most important problems in modern cosmology is the dark energy issue, which is responsible for the accelerated expansion of the observed Universe. Over the last few decades, several studies have been focused on trying to tackle this problem. It is well known that this form of energy is acting as a repulsive gravitational force so that in General Relativity (GR) one needs to consider a further non-standard fluid with a negative pressure to justify this accelerated scenario. The simplest approach is to consider a cosmological constant in order to explain it. However, from quantum considerations, the necessary expected value of it must be extremely much larger than the observed value [1]. Another approach to the cosmic accelerated behavior comes from modified theories of gravities where, instead of searching for new material ingredients, the philosophy is to address cosmic dynamics taking into account possible further degrees of freedom of the gravitational field. A very well-studied approach to modified gravity comes out from the “Teleparallel equivalent to General Relativity” (TEGR). This theory yields the same field equations as in General Relativity, so that TEGR is an alternative and equivalent theory. However, the geometrical interpretations of these theories are different. On the one hand, GR assumes a non-zero curvature and a vanishing torsion by choosing the symmetric Levi-Civita connection. On the other hand, TEGR considers an antisymmetric connection provided with a non-vanishing torsion and a zero curvature (Weitzenböck connection). In other words, one can say that GR uses the curvature to geometrize the space-time, meanwhile TEGR uses torsion to explain gravitational effects. In TEGR, we need to use tetrad fields as the dynamical variables in order to define the Weitzenböck connection (see [2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13], and also the review [14] for the basis in TEGR).

A natural generalization of TEGR is, instead of using the scalar torsion *T*, to consider an arbitrary and smooth function of the torsion *f*(*T*) in the gravitational action [15, 16, 17, 18]. This theory is the so-called “*f*(*T*) gravity”. The idea comes out naturally exactly as when GR is generalized to *f*(*R*) gravity [19, 20, 21, 22]. An important problem related to *f*(*T*) gravity is that it is no longer invariant under local Lorentz transformations so that different tetrads might give rise to different solutions. Therefore one needs to be very careful choosing the correct tetrad [23]. Although TEGR is equivalent to GR, it is important to mention that *f*(*R*) is no longer equivalent to *f*(*T*) gravity [24]. One needs to consider a more general theory of gravity, the so-called “*f*(*T*, *B*) gravity” to obtain the teleparallel equivalent to *f*(*R*) gravity [25]. In addition, it is important to remark that *f*(*T*) gravity contains only second order derivative terms; meanwhile *f*(*R*) gravity contains up to fourth order derivative terms in the metric formalism.

In the last few years, *f*(*T*) gravity acquired a lot of interest in cosmology due to the possibility to explain by it the accelerated expansion of the cosmic Hubble fluid (see [26, 27, 28, 29, 30, 31, 32, 33, 34, 35]). In addition, astrophysical studies related with compact objects as black holes has been considered among *f*(*T*) gravity such as in [36, 37, 38, 39, 40]. However, it is worth noticing that this is not the only solution that can be achieved by the Noether symmetry approach. As shown in [41] for *f*(*R*) gravity, the symmetries select the form of the function and several Noether vectors can exist. In the specific case of *f*(*T*) gravity, other solutions have been found as discussed in [42, 43]. A very well-studied process, known as accretion, occurs when a fluid is situated in the vicinity of a black hole or a massive astrophysical object (see [44, 45, 46, 47]). In this process, the compact object takes particles from the fluid and increases its mass. Accretion takes place regularly in the Universe, and it can be used to test gravitational theories using observational measurements [48, 49, 50]. The first study of accretion was performed using Newtonian gravity by Bondi [51]. He found transonic solutions for a gas accreting onto compact objects. Michel extended the later work considering GR for a Schwarzschild black hole [52]. An important work in this field has been pursued by Babichev et al., where they showed that the mass of the black hole decreases when a phantom fluid is in accretion onto it [53]. Later, Jamil and Qadir showed that primordial black holes decay earlier when the effect of accretion of phantom energy is considered [54]. In addition, Nayak and Jamil also found that primordial black holes accrete radiation, matter, and vacuum energy when they pass through radiation, matter, and vacuum dominated eras, respectively, with the result that they live longer during the radiation era [55]. After that, several works have been published on accretion onto compact objects (see [56, 57, 58, 59, 60]).

Recently, Ahmed et al. studied accretion for cyclic and heteroclinic flows near *f*(*R*) black holes [61]. In this paper, we will use a similar formalism in order to study the accretion process in a black hole in the context of *f*(*T*) gravity.

This paper is organized as follows: In Sect. 2, we briefly introduce the TEGR and *f*(*T*) gravity. In Sect. 3, we discuss the metric representation of black holes in *f*(*T*) gravity. Section 4 is devoted to finding the general equations for spherical accretion. In Sect. 5, we perform a dynamical system analysis using the Hamiltonian formalism and we study the system at the critical points (CPs). In Sect. 6, we obtain solutions for isothermal test fluids for different kind of fluids. In Sect. 7, we analyze the accretion process for a polytropic test fluid. Finally, in Sect. 8, we discuss our results and draw conclusions. Throughout the paper we will use the metric signature \((-,+,+,+)\) and the geometric units \(G=c=1\).

## 2 Teleparallel equivalent of general relativity and *f*(*T*) gravity

*f*(

*T*) gravity. We will adopt the notation used in [25]. In this theory, the dynamical variable is the tetrad field \(e_{a}^{\mu }\) (or vierbein), where Latin and Greek index indicate tangents space and space-time index, respectively. The construction of this theory relies on the relationship between the tetrad field and the metric \(g_{\mu \nu }\) in the following way:

*T*can be defined as

*R*and the torsion scalar

*T*are related by

*R*as in GR, the TEGR Lagrangian density is described by the torsion scalar

*T*

*B*is a boundary term, from (9), one can see that the TEGR action will arise to the same field equations as the Einstein–Hilbert action, making these two theories equivalent.

*f*(

*T*) gravity” and it has numerous and interesting applications, for example in cosmology (see [14] for a comprehensive review of those models). One important feature of this theory is that meanwhile TEGR is an equivalent theory to GR,

*f*(

*T*) does not produce the same field equations as

*f*(

*R*) gravity (due to Eq. (9)) and therefore one needs to consider a generalization of (11) from \(f(T)\rightarrow f(T,B)\) to find the teleparallel equivalent to

*f*(

*R*) gravity as discussed in [25]. Starting from the action (11), the field equations read

*f*(

*T*) gravity.

## 3 Black hole in *f*(*T*) gravity

*M*in

*f*(

*T*) gravity is given by [40]

*X*must have the same sign: \(C_{5}/X>0\). Since

*A*must be positive at spatial infinity, we must have \(X>0\) resulting in \(C_{5}>0\). Upon performing the coordinate transformation

*f*(

*R*) black holes were investigated, among which we find the solution

*f*(

*T*) black hole (14) with that onto the

*f*(

*R*) black hole (21).

The metric (14) being equivalent to (20), all general equations expressed in terms of \(\alpha \), which were derived in Ref. [61], are thus applicable to our present investigation upon replacing \(\alpha \) by \(c_{3}^2A\). However, because of their importance, we will outline their derivations below.

## 4 General equations for spherical accretion

*n*be the baryon number density in the fluid rest frame and \(u^{\mu }=\mathrm{d}x^{\mu }/\mathrm{d}\tau \) be the four velocity of the fluid where \(\tau \) is the proper time. We define the particle flux or current density by \(J^{\mu }=n u^{\mu }\) where

*n*is the particle density. From the particle conservation law, we see that the divergence of current density is zero, i.e.

*p*is the pressure. We assume that the fluid is radially flowing in the equatorial plane \((\theta =\pi /2)\), therefore \(u^{\theta }=0\) and \(u^{\phi }=0\). For the sake of simplicity, we set \(u^r=u\). Using the normalization condition \(u^{\mu }u_{\mu }=-1\) and (14), we obtain

*T*is the temperature,

*s*is the specific entropy, and

*s*is constant, this reduces the canonical form of the equation of state (EOS) of a simple fluid \(e=e(n,s)\) to the barotropic form

*n*. Now, the first equation (32) yields \(p'=nh'\), with \(h=F'\), we obtain

*F*and

*G*can be derived upon integrating the differential equation

*a*is defined by \(a^2=(\partial p/\partial \epsilon )_s\). Since the entropy

*s*is constant, this reduces to \(a^2=\mathrm{d}p/\mathrm{d}\epsilon \). Using (32), we derive

*v*is defined by \(v\equiv \frac{\mathrm{d}r/\sqrt{A}}{c_{3}\sqrt{A}\mathrm{d}t}\) and yields

## 5 Hamiltonian systems

*f*has been replaced by \(c_{3}^2A\). We can absorb the constant \(c_{3}^2\) into a redefinition of the Hamiltonian, however, we will do that in a further step of our derivation.

### 5.1 Sonic points

## 6 Isothermal test fluids

*k*is the state parameter such that \((0<k\le 1)\) [63]. The differential equation (37) reads

^{1}so that (28) and (34) lead to the same expression for

*h*:

*k*. For instance, we have \(k=1\) (ultra-stiff fluid), \(k=1/2\) (ultra-relativistic fluid), \(k=1/3\) (radiation fluid) and \(k=1/4\) (sub-relativistic fluid). In the case of the metric (14), Eq. (51) reduces to

*k*that yield a critical flow with the presence of a CP given by (61) and \(v_c^2=k\). In Ref. [61] we have shown that if the flow approaches the horizon with a vanishing three-dimensional speed, the pressure must diverge as

### 6.1 Solution for ultra-stiff fluid (\({\pmb k=1}\))

From (63) we see that, for physical flows (\(|v|<1\)), the lower value of \(\mathcal {H}\) is \(\mathcal {H}_{\text {min}}=1/r_h^4\): \(\mathcal {H}>\mathcal {H}_{\text {min}}\). As shown in Fig. 1, physical flows are represented by the curves sandwiched by the two black curves, which are contour plots of \(\mathcal {H}(r,v)=\mathcal {H}_{\text {min}}\). The upper curves, where \(v>0\), correspond to fluid outflow or particle emission and the lower curves, where \(v<0\), correspond to fluid accretion. From (63) we see that for the global solutions, shown in Fig. 1, which are the only existing solutions for \(k=1\), the speed *v* behaves asymptotically as \(v\sim 1/r^2\). Using this and the fact that \(A\sim r^2\) in (42), we obtain \(n\sim 1/r\).

### 6.2 Solution for ultra-relativistic fluid (\({\pmb k=1/2}\))

*v*has two different asymptotic behaviors. Since \(\mathcal {H}\) retains the same constant value and \(A\sim r^2\), we have either (a) \(v\rightarrow 0\) as \(v\sim cst/r\) or (b) \(v\rightarrow 1\) such that \(r^2(1-v^2)\sim cst\) yielding \(v\sim 1-cst/(2r^2)\). Using these in (42), we obtain (a) \(n\sim 1/r^2\) and (b) \(n\sim 1/r^4\).

- 1.
Purely supersonic accretion (\(v<-v_c\)), which ends inside the horizon, or purely supersonic outflow (\(v>v_c\)).

- 2.
Purely subsonic accretion followed by subsonic flowout; this is the case of the branches of the blue and magenta solution curves corresponding to \(-v_c<v<v_c\). Notice that for this motion the fluid reaches the horizon, \(A(r_h)=0\), with a vanishing speed, ensuring that the Hamiltonian (65) remains constant. The critical black solution curve reveals two types of motions: if we assume that \(\mathrm{d}v/\mathrm{d}r\) is continuous at the CPs.

- 3.
- a.
Supersonic accretion until (\(r_c,-v_c\)), followed by a subsonic accretion until (\(r_h,0\)), where the speed vanishes, then a subsonic flowout until (\(r_c,v_c\)), followed by a supersonic flowout.

- b.
Subsonic accretion followed by a supersonic accretion which ends inside the horizon. In the upper plot, we have a supersonic outflow followed by a subsonic motion.

- a.

*f*(

*T*) black hole. It is thus assumed that the accretion does not modify the mass of the black hole nor its other intrinsic properties. The flow, being non-geodesic, however, still obeys the simple rule that if

*r*increases,

*v*must be positive, and if

*r*decreases,

*v*must be negative. For instance, for \(v>0\), we see from Fig. 2 that the red plot has two branches. Consider the branch on the right of the vertical line \(r=r_c\). The flow along the segment of that branch along which

*v*increases and

*r*decreases is unphysical, for this is neither an accretion nor a flowout.

### 6.3 Solutions for radiation fluid (\({\pmb k=1/3}\)) and sub-relativistic fluid (\({\pmb k=1/4}\))

*v*behaves asymptotically as

*r*may go to infinity, the Hamiltonian (59) behaves in the limit \(r\rightarrow \infty \) as

*r*goes to infinity is not valid. For \(k=1/3\) global flow is possible, as we shall justify below; however, non-global flow is also realizable. Figure 3 depicts typical non-global fluid flows for this class of fluids where \(k\le 1/3\). Let \(r_{\text {rm}}\) be the

*r*coordinate of the rightmost point on the solution curve. We observe:

- 1.
(Generally supersonic) accretion from \(r_{\text {rm}}\) that crosses the horizon with the speed of light. Such a flow is possible if a fluid source is available at \(r_{\text {rm}}\) that injects fluid particles with a non-vanishing speed.

- 2.
(Almost subsonic) accretion from \(r_{\text {rm}}\) that reaches the horizon with a vanishing speed, followed by a (almost subsonic) flowout back to \(r_{\text {rm}}\). Such a flow could be made possible if a source-sink system is available at \(r_{\text {rm}}\).

- 3.
(Generally supersonic) flowout that emanates from the horizon with the speed of light and reaches \(r_{\text {rm}}\) with a non-vanishing speed. Such a flow is possible if a sink is available at \(r_{\text {rm}}\).

- 1.
(Supersonic) accretion with an initial velocity \(-v_{\infty +}\) that crosses the horizon with the speed of light.

- 2.
(Subsonic) accretion with an initial velocity \(-v_{\infty -}\) that reaches the horizon with a vanishing speed, followed by a (subsonic) flowout that reaches spatial infinity with the same speed \(v_{\infty -}\).

- 3.
(Supersonic) flowout that emanates from the horizon with the speed of light and reaches spatial infinity with a speed \(v_{\infty +}\).

*n*as follows. Equation (70) with \(\mathcal {H}\) given by the right-hand side of (66) yields

### 6.4 Accretions in *f*(*T*) and *f*(*R*) gravities

*f*(

*T*) and

*f*(

*R*) gravities. For that end we select from

*f*(

*R*) gravity black holes a similar solution (21) to the one considered here (14), that is, an anti-de Sitter-like

*f*(

*R*) black hole [61]. The following enumeration shows similarities and differences.

- 1.
The accretion of an isothermal perfect fluid with \(k=1\) is characterized by the presence of global solutions, which are the only existing solutions with no CPs. The speed

*v*and the particle density*n*behave asymptotically as \(v\sim 0\) and \(n\sim 1/r\) for both gravities. - 2.
If the isothermal perfect fluid has \(k=1/2\), the accretion is characterized by the presence of two CPs and critical flow for both gravities. For the global solutions we have either \(v\sim 0\) and \(n\sim 1/r^2\) or \(v\sim 1\) and \(n\sim 1/r^4\).

- 3.
- (a)
For

*f*(*T*) gravity the accretion of an isothermal perfect fluid with \(k=1/3\) has no CP nor critical flow while for*f*(*R*) gravity the fluid flow has two CPs. For the global solutions of both gravities \(v\sim cst\), where*cst*may assume any value between 0 and 1, and \(n\sim 1/r^3\). - (b)
For \(k<1/3\), the accretion onto an

*f*(*T*) gravity black hole is again noncritical, with no CP, while that onto an*f*(*R*) gravity black hole may have four CPs, as was shown in Ref. [61] for the isothermal perfect fluid with \(k=1/4\). For both gravities there are no global solutions.

- (a)

*f*(

*R*) gravity reduces to that of GR and the theory itself reduces to GR, \(f(R)=R+\Lambda \), if the

*f*(

*R*)-parameter \(\beta =0\). This is not the case with the black hole (14) of the

*f*(

*T*) gravity which does not reduce to any of the known GR black holes no matter how the

*f*(

*T*)-parameters (\(X,C_5\)) are chosen.

A deeper investigation should focus on the evaluation of the rates of accretion and efficiencies of the outgoing spectra for different black holes and different gravity theories.

The efficiency of the conversion of gravitational (potential) energy into radiation is one of the open problems of radial accretion onto a black hole; this is if one assumes, as most workers concluded, that the infall velocity scales almost as the free fall velocity (the case of Fig. 1 or the case of the critical subsonic accretion followed by a supersonic accretion of Fig. 2). This efficiency problem becomes more involved if we consider the critical accretion of Fig. 2 along the branch where *v* vanishes as \(r\rightarrow r_h\) or accretions along the blue and magenta branches of the same figure. Here the three velocity has a deceleration phase from \(r_c\) to \(r_h\) and it does not scale as a free fall velocity. This is the main discovery in this work and in [61]. The deceleration of the fluid increases by far the conversion efficiency; moreover, the efficiency is roughly proportional to \(n^2\) [64], which diverges by (54) as \(r\rightarrow r_h\).

All that is out of the scope of this work and could be the aim and task of subsequent works. In a first step one may consider the simplest cases of the \(f(T)=T\) [\(f(R)=R\) or GR] gravity theory. We believe that, when all these tasks are performed (most likely numerically), the result that will be at hand will confirm the equivalence of these gravity theories.

## 7 Polytropic test fluids

*K*and \(\gamma \) are constants. For ordinary matter, one generally works with the constraint \(\gamma >1\). Inserting (74) into the differential equation (37), it is easy to establish [61] the following expressions of the specific enthalpy:

*m*. Since \(\gamma >1\), this implies \(a^2<\gamma -1\) and, particularly, \(v_c^2<\gamma -1\).

*Z*takes the special form

The constraint \(X>0\), in (14), yields \(A_{,r}>0\) for all *r*, and this implies that the constant \(Z>0\) (recall that \(\gamma >1\)). Thus, the sum of the terms inside the square parentheses in (80) is positive, while the coefficient \(A/(1-v^2)\) diverges as \(r\rightarrow \infty \) (\(0\le 1-v^2<1\)). So, the Hamiltonian too diverges as *r* approaches spatial infinity. Since the Hamiltonian has to remain constant on a solution curve, we conclude that there are no global solutions (solutions that extend to, or emanate from, spatial infinity). This conclusion is general and it extends to all anti-de Sitter-like solutions [61].

*r*axis at points where \(v=0\) and \(r\ne r_h\), for otherwise the Hamiltonian (80) would diverge there. The curves may cross the

*r*axis at \(r=r_h\) only. The horizon (18) being a single root to \(A(r)=0\), if we assume \(v\propto |r-r_h|^{\delta }\) and \(\delta >0\) near the horizon, it is easy to show that

*valid*for \(\delta >0\), we see that only physical solutions with \(1<\gamma <2\) may cross the

*r*axis. For these values of \(\gamma \), the pressure \(p=Kn^{\gamma }\) diverges at the horizon as

*Z*, the resolution of this system of equations in (\(r_c,v_c\)) provides all the CPs, if there are any; the values of these are then used to determine \(n_c\) from (79).

Numerical solutions to the system of Eqs. (84) and (85) are shown in Figs. 5 and 6. The constant *Z* is a collection of parameters depending on the black hole and the barotropic fluid. For a given black hole solution, *Z* is roughly proportional to \(Kn_c/m\). For the physical case one is generally interested in in astrophysics, \(1<\gamma <2\), the solution curve has two CPs of the same sign of *v* for large values of *Z* (in total four CPs as in the left plot of Fig. 5). As *Z* reaches some critical value, \(Z_0\), each couple of CPs of the same sign of *v* merge as in the middle plot of Fig. 5. Below that critical value of *Z* there are no CPs as in the right plot of Fig. 5. For \(Z\ge Z_0\), we have heteroclinic flow between two CPs of same value of \(r_c\) and opposite values of \(v_c\).

The critical flow in the left plot of Fig. 5 is no difference of that of Fig. 2 (black plot). The only different feature is that the former flow is non-global while the latter flow is global. Similarly, the magenta and blue curves (corresponding to \(\mathcal {H}>\mathcal {H}_c\)) of the left and middle plots of Fig. 5 have branches which are subsonic for the whole process of accretion-flowout as is the case of the curves of Fig. 2 corresponding to \(\mathcal {H}>\mathcal {H}_c\). Another similarity emerges upon comparing the solutions with no CPs corresponding to \(Z<Z_0\) (right plot of Fig. 5) with those of Fig. 3 where no CPs occur too.

A common conclusion we can draw upon comparing the solutions of this section with those of the previous one is that low pressure fluids (*k* and *K* small) do not develop critical flows (no CPs) and high pressure fluids develop critical flows but they may maintain purely subsonic, even non-relativistic, flows.

Barotropic fluids with \(\gamma >2\), if there are any, may have CPs but no critical flow and their accretion velocity never vanishes as depicted in Fig. 6. The accretion make take place along two different paths starting from rightmost point of the lower branch of Fig. 6. For large values of the Hamiltonian (this would be the case if *Z* is large, *n*, or *K*), the accretion along one of these two paths is almost non-relativistic for \(r>r_c\), then the velocity jumps to supersonic and relativistic values as *r* approaches \(r_h\). For lower values of the Hamiltonian, the accretion takes place near the CP and the polytropic fluid never reaches the horizon.

*test*fluids neglecting all back-reaction effects. This rules out any homoclinic flow and motion along closed paths, as those shown in Fig. 6, where

*v*conserves the same sign but

*r*increases and decreases.

## 8 Conclusions

In this paper, we discussed in detail the accretion process of a spherically symmetric black hole in the context of *f*(*T*) gravity. In order to select the form of *f*(*T*) model, we adopted the Noether symmetry approach, following [40]. In particular, we discussed spherically symmetric solutions coming from \(f(T)=T^m\) models (and, in general, analytic *f*(*T*) models) that give rise to metrics of the form (20) and related gravitational potentials of the form (21); see [40] for details.

We have analyzed the motion of isothermal relativistic and ultra-relativistic fluids by means of a Hamiltonian dynamical system capable of representing hydrodynamics around the black hole. The thermodynamical properties of the fluids have been discussed according to the suitable EOS. Furthermore, conserved quantities and CPs have been selected for any fluid. Roughly, the accretion mechanism can be classified as subsonic and supersonic according to the features of the black hole and the EOS. In particular, the three-dimensional velocity flow strictly depends on the EOS, the radius, and the CPs on the phase space. Finally, the results have been compared to the analog results in *f*(*R*) gravity putting in evidence similarities and differences.

Clearly, the accretion process of the fluids flowing the black holes strictly depends on the conserved quantities (Noether’s symmetries) and the structure of CPs, as shown above. If conserved quantities are not identified, it could become extremely difficult to define the phase space structure of the dynamical problem and consequently the features of CPs. In conclusion, identifying the Noether symmetries allows one to fix the model (i.e. the form of *f*(*T*)), to derive the metric and the gravitational potentials, thanks to the reduction of the dynamical system, to define the form of the space phase. Models without these features are very difficult to handle.

*f*(

*T*) black holes. In particular, the possibility to investigate

*f*(

*T*) vs.

*f*(

*R*) black holes could be a powerful tool to discriminate between the curvature (GR) and torsional (TEGR) formulation of theories of gravity (see [14] for a detailed discussion). Specifically, the accretion process onto a black hole could be the feature capable of discriminating among competing models and, in general, between a curvature or a torsional formulation. A main role in this discussion is played by the stability conditions. For example, as discussed in [65] for the case of

*f*(

*R*) gravity, the stability conditions for any self gravitating object strictly depend on the theory. There it is demonstrated that the Jeans stability criterion is different if one considers

*f*(

*R*) instead of GR because effective mass, stability radius, Jeans wave length, and the other parameters characterizing any astrophysical object slightly change according to the underlying model. In general, if the accretor has a mass

*M*and a radius \(\mathcal R\), the gravitational energy release is

*M*, the yields depend on the accretor radius. Considering alternative theories of gravity, the above relation can be written as

*f*(

*T*) and

*f*(

*R*) pictures can be put in evidence by a detailed study of the accretion process. In particular, the number of CPs, the state parameter

*k*and other features, besides the effective potential, can discriminate among competing models. From a genuine observational point of view, luminous phenomena powered by black holes could contain features capable of discriminating among theories as soon as the parameters

*G*,

*M*, and \(\mathcal R\) are combined into a gravitational potential. For example, the accretion luminosity,

## Footnotes

## Notes

### Acknowledgments

S.B. is supported by the Comisión Nacional de Investigación Científica y Tecnológica (Becas Chile Grant No. 72150066). S.C. is supported by INFN (*iniziativa specifica* TEONGRAV) and acknowledges the COST Action CA15117 (CANTATA).

## References

- 1.J. Martin, Comptes Rendus Physique
**13**, 566 (2012)ADSCrossRefGoogle Scholar - 2.Y.N. Obukhov, J.G. Pereira, Phys. Rev. D
**67**, 044016 (2003)ADSMathSciNetCrossRefGoogle Scholar - 3.H.I. Arcos, J.G. Pereira, Int. J. Mod. Phys. D
**13**, 2193 (2004)ADSMathSciNetCrossRefGoogle Scholar - 4.R. Weitzenböck,
*Invarianten Theorie*(Nordhoff, Groningen, 1923)zbMATHGoogle Scholar - 5.Y.M. Cho, Phys. Rev. D
**14**, 2521 (1976)ADSMathSciNetCrossRefGoogle Scholar - 6.Y.M. Cho, Phys. Rev. D
**14**, 3335 (1976)ADSCrossRefGoogle Scholar - 7.K. Hayashi, Phys. Lett. B
**69**, 441 (1977)ADSCrossRefGoogle Scholar - 8.K. Hayashi, T. Shirafuji, Phys. Rev. D
**19**, 3524 (1979)Google Scholar - 9.K. Hayashi, T. Shirafuji, Phys. Rev. D
**24**, 3312 (1981)Google Scholar - 10.W. Kopczyński, J. Phys. A
**15**, 493 (1982)ADSMathSciNetCrossRefGoogle Scholar - 11.R.T. Hammond, Rept. Prog. Phys.
**65**, 599 (2002)ADSMathSciNetCrossRefGoogle Scholar - 12.J.W. Maluf, Annalen Phys.
**525**, 339 (2013)ADSMathSciNetCrossRefGoogle Scholar - 13.R. Aldrovandi and J. G. Pereira,
*Teleparallel Gravity : An Introduction*. Fundamental Theories of Physics, Vol. 173. Springer Dodrecht, Heidelberg (2013)Google Scholar - 14.Y.F. Cai, S. Capozziello, M. De Laurentis, E. N. Saridakis. arXiv:1511.07586 [gr-qc]
- 15.R. Ferraro, F. Fiorini, Phys. Rev. D
**75**, 084031 (2007)ADSMathSciNetCrossRefGoogle Scholar - 16.G.R. Bengochea, R. Ferraro, Phys. Rev. D
**79**, 124019 (2009)ADSCrossRefGoogle Scholar - 17.B. Li, T.P. Sotiriou, J.D. Barrow, Phys. Rev. D
**83**, 064035 (2011)ADSCrossRefGoogle Scholar - 18.T.P. Sotiriou, B. Li, J.D. Barrow, Phys. Rev. D
**83**, 104030 (2011)ADSCrossRefGoogle Scholar - 19.T.P. Sotiriou, V. Faraoni, Rev. Mod. Phys.
**82**, 451 (2010)ADSMathSciNetCrossRefGoogle Scholar - 20.S. Capozziello, Int. J. Mod. Phys. D
**11**, 483 (2002)ADSMathSciNetCrossRefGoogle Scholar - 21.S. Capozziello, M. De Laurentis, Phys. Rept.
**509**, 167 (2011)ADSCrossRefGoogle Scholar - 22.S. Nojiri, S.D. Odintsov, Phys. Rept.
**505**, 59 (2011)ADSMathSciNetCrossRefGoogle Scholar - 23.N. Tamanini, C.G. Böhmer, Phys. Rev. D
**86**, 044009 (2012)ADSCrossRefGoogle Scholar - 24.K. Bamba, S. Capozziello, M. De Laurentis, S. Nojiri, D. Saez-Gomez, Phys. Lett. B
**727**, 194 (2013)ADSCrossRefGoogle Scholar - 25.S. Bahamonde, C.G. Böhmer, M. Wright, Phys. Rev. D
**92**(10), 104042 (2015)ADSMathSciNetCrossRefGoogle Scholar - 26.R. Myrzakulov, Eur. Phys. J. C
**71**, 1752 (2011)ADSCrossRefGoogle Scholar - 27.S.H. Chen, J.B. Dent, S. Dutta, E.N. Saridakis, Phys. Rev. D
**83**, 023508 (2011)ADSCrossRefGoogle Scholar - 28.P. Wu, H.W. Yu, Phys. Lett. B
**693**, 415 (2010)ADSMathSciNetCrossRefGoogle Scholar - 29.K. Bamba, C.Q. Geng, C.C. Lee, L.W. Luo, JCAP
**1101**, 021 (2011)ADSCrossRefGoogle Scholar - 30.H. Farajollahi, A. Ravanpak, P. Wu, Astrophys. Space Sci.
**338**, 23 (2012)CrossRefGoogle Scholar - 31.G.R. Bengochea, Phys. Lett. B
**695**, 405 (2011)ADSCrossRefGoogle Scholar - 32.M. Jamil, D. Momeni, R. Myrzakulov, Eur. Phys. J. C
**72**, 1959 (2012)ADSCrossRefGoogle Scholar - 33.M. Jamil, D. Momeni, R. Myrzakulov, Eur. Phys. J. C
**72**, 2075 (2012)ADSCrossRefGoogle Scholar - 34.M. Jamil, D. Momeni, R. Myrzakulov, Eur. Phys. J. C
**72**, 2137 (2012)ADSCrossRefGoogle Scholar - 35.K. Bamba, M. Jamil, D. Momeni, R. Myrzakulov, Astrophys. Space Sci.
**344**, 259 (2013)ADSCrossRefGoogle Scholar - 36.R. Ferraro, F. Fiorini, Phys. Rev. D
**84**, 083518 (2011)ADSCrossRefGoogle Scholar - 37.T. Wang, Phys. Rev. D
**84**, 024042 (2011)ADSCrossRefGoogle Scholar - 38.C.G. Böhmer, A. Mussa, N. Tamanini, Class. Quant. Grav.
**28**, 245020 (2011)ADSCrossRefGoogle Scholar - 39.M. Hamani Daouda, M.E. Rodrigues, M.J.S. Houndjo, Eur. Phys. J. C
**71**, 1817 (2011)Google Scholar - 40.A. Paliathanasis, S. Basilakos, E.N. Saridakis, S. Capozziello, K. Atazadeh, F. Darabi, M. Tsamparlis, Phys. Rev. D
**89**, 104042 (2014)ADSCrossRefGoogle Scholar - 41.S. Capozziello, A. De Felice, JCAP
**0808**, 016 (2008)ADSCrossRefGoogle Scholar - 42.P.A. Gonzalez, E.N. Saridakis, Y. Vasquez, J. High Ener. Phys.
**53**(2012)Google Scholar - 43.S. Capozziello, P.A. Gonzalez, E.N. Saridakis, Y. Vasquez, J. High Energy Phys.
**39**, 1302 (2013)Google Scholar - 44.E. Shima, T. Matsuda, T. Hidenori, K. Sawada, MNRAS
**217**, 367 (1985)ADSCrossRefGoogle Scholar - 45.T. Matsuda, M. Inoue, K. Sawada, MNRAS
**226**, 785 (1987)ADSCrossRefGoogle Scholar - 46.S.K. Chakrabarti, Phys. Rept.
**266**, 229 (1996)ADSCrossRefGoogle Scholar - 47.R. Taam, B. Fryxall, Astrophys. J.
**331**, L117 (1988)ADSCrossRefGoogle Scholar - 48.T. Harko, Z. Kovacs, F.S.N. Lobo, Phys. Rev. D
**80**, 044021 (2009)ADSCrossRefGoogle Scholar - 49.C.S.J. Pun, Z. Kovacs, T. Harko, Phys. Rev. D
**78**, 024043 (2008)ADSCrossRefGoogle Scholar - 50.D. Perez, G.E. Romero, S.E.P. Bergliaffa, Astron. Astrophys.
**551**, A4 (2013)ADSCrossRefGoogle Scholar - 51.H. Bondi, MNRAS
**112**, 195 (1952)ADSMathSciNetCrossRefGoogle Scholar - 52.F.C. Michel, Astrophys. Space Sci.
**15**, 153 (1972)ADSCrossRefGoogle Scholar - 53.E. Babichev, V. Dokuchaev, Y. Eroshenko, Phys. Rev. Lett.
**93**, 021102 (2004)ADSCrossRefGoogle Scholar - 54.M. Jamil, A. Qadir, Gen. Rel. Grav.
**43**, 1069 (2011)ADSMathSciNetCrossRefGoogle Scholar - 55.B. Nayak, M. Jamil, Phys. Lett. B
**709**, 118 (2012)ADSCrossRefGoogle Scholar - 56.T.K. Das, A. Sarkar, Astron. Astrophys.
**374**, 1150 (2001)Google Scholar - 57.S.K. Chakrabarti, S.A. Sahu, Astron. Astrophys.
**323**, 382 (1997)ADSGoogle Scholar - 58.U. Debnath, Eur. Phys. J. C
**75**, 129 (2015)ADSCrossRefGoogle Scholar - 59.S. Bahamonde, M. Jamil, Eur. Phys. J. C
**75**, 508 (2015)ADSCrossRefGoogle Scholar - 60.G.Z. Babar, M. Jamil, Y.K. Lim, Int. J. Mod. Phys. D
**25**, 1650024 (2016)ADSMathSciNetCrossRefGoogle Scholar - 61.A.K. Ahmed, M. Azreg-Aïnou, M. Faizal, M. Jamil, Eur. Phys. J. C (2016). doi: 10.1140/epjc/s10052-016-4112-y. arXiv:1512.02065 [gr-qc]
- 62.L. Rezzolla, O. Zanotti,
*Relativistic Hydrodynamics*(Oxford University Press, NY, 2013)CrossRefzbMATHGoogle Scholar - 63.P. Mach, E. Malec, Phys. Rev. D
**88**, 084055 (2013)ADSCrossRefGoogle Scholar - 64.A. Treves, L. Maraschi, M. Abramowicz, Astron. Soc. Pacif.
**100**, 427 (1988)ADSCrossRefGoogle Scholar - 65.S. Capozziello, M. De Laurentis, I. De Martino, M. Formisano, S.D. Odintsov, Phys. Rev. D
**85**, 044022 (2012)ADSCrossRefGoogle Scholar - 66.S. Capozziello, G. Lambiase, Phys. Lett. B
**750**, 344 (2015)ADSCrossRefGoogle Scholar - 67.E.E. Salpeter, Ap. J.
**140**, 796 (1964)ADSCrossRefGoogle Scholar - 68.G.B. Rybicki, A.P. Lightman,
*Radiative Processes in Astrophysics*(Wiley, New York, 1979)Google Scholar

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