Black holes in asymptotic safety with higher derivatives: accretion and stability analysis

The event horizon of a black hole is defined in general relativity as the frontier of a region of space-time where gravity is so strong that nothing, not even light, that enters that region can ever escape. The characteristic features of black holes, including event horizons, accretion and outflow processes, singularities, Hawking radiation, etc., and the possibility of observation of their immediate environment with angular resolution comparable to the event horizon using facilities such as the Event Horizon Telescope 1 (1) and the GRAVITY instrument 2 (2), make black holes useful laboratories to test general relativity in the strong field regime and, hopefully, to test ideas for quantum gravity.

Accretion process onto compact objects, on the other hand, is relevant since it is believed to provide the energy of Quasars, accreting supermassive black holes, X-ray binaries and Gamma-ray bursts. The simplest situation, which has reached a paradigmatic status in studies on accretion astrophysics, is spherically symmetric accretion onto a static compact object of inviscid gas which is at rest at infinity. In the framework of Newtonian gravity the problem was analyzed in a seminal paper by Bondi 3-Bondi (3). The relativistic generalization of the Bondi accretion problem was done in 4-Michel (4), which was followed by a large number of studies analyzing spherical accretion onto a wide variety of astrophysical objects including black holes 5-various (5).

The problem of the stability of spherically symmetric accretion under linearized perturbations has extensively been studied in the literature both in the context of Newtonian gravity 6-PETT (6, 7, 8, 9) and in the classical general relativistic realm 10-Moncrief (10, 11, 12, 13, 14, 15, 16, 17). In both of these regimes no evidence has been found of the development of instabilities. In 6-PETT (6) it has been shown that in the Newtonian regime and neglecting viscosity, the perturbation has a constant amplitude for the conserved flow so that stability is ensured. In general relativity the background solution is also stable with the amplitude of the perturbation being damped in time, that is, general relativistic effects enhance the stability 10-Moncrief (10), 17-Naskar (17). A mechanism that favors stability in this case is the coupling of the infalling flow with the geometry of space-time, which acts in the manner of a dissipative effect.

In this work we consider quantum gravity corrections to the accretion, and to its stability, onto an improved Schwarzschild black hole in the framework of the asymptotic safety scenario for quantum gravity. In this scenario it is conjectured that gravity constitutes a fundamental theory at the non-perturbative level 18-Weinberg (18). A basic ingredient in this construction is a non-Gaussian fixed point (NGFP) of the gravitational renormalization group (RG) flow which controls the behavior of the theory at trans-Planckian energies, where the physical degrees of freedom interact predominantly antiscreening, and that renders physical quantities safe from unphysical divergences at all scales 19-Reuter (19). This means that the gravitational field itself is the principal carrier of the relevant classical and quantum degrees of freedom, with the renormalization flow relating its infrared and ultraviolet physics. On condition that the set of RG trajectories approaching the fixed point in the UV are parameterized by a finite number of relevant (that is, physically motivated) running couplings, asymptotic safety is as predictive as the standard perturbatively renormalizable quantum field theory. Notwithstanding asymptotic safety defines a consistent and predictive quantum theory for gravity within the framework of quantum field theory, it remains a prediction in the sense that a rigorous existence proof for the NGFP is still lacking. There exist, however, substantial evidence for the existence of such a fixed point suitable for the Weinberg’s asymptotic safety scenario 20-various2 (20).

Quantum corrections to the structure of black hole spacetimes within the asymptotic safety program was initiated in 21-BoRe (21) by constructing the so-called “RG-improved” solution to the Schwarzschild metric. Following up on these initial work, the RG-improved Kerr metric has been discussed in 22-ReTu (22), and the improved Schwarzschild-(A)dS and Reissner-Nordström-(A)dS metrics have been studied in 23-Ko-Sa (23) and 24-Go-Ko (24), respectively. The inclusion of a cosmological constant is relevant because it has been shown in 23-Ko-Sa (23) that is this constant which determines the short-distance structure of the RG-improved black holes. Further aspects of these quantum compact objects including thermodynamical properties, evaporation processes, mini-black hole production, etc, have been analyzed in 25-various3 (25), while quantum corrections to the accretion process onto a Schwarzschild black hole have been discussed in 26-Yang (26), 27-FAYJ (27).

In this paper we review the results by the authors in 27-FAYJ (27) where the study of the accretion onto an improved asymptotically safe Schwarzschild metric with high-derivative terms was done, and we also discuss the stability of the accretion process and contrast with the result coming from GR. The interest in the improved Schwarzschild space-time with higher-derivatives in the infrared limit comes from the fact that, as shown in 28-Cai-Easson (28), describes a black hole that never completely evaporate and, consequently, makes an excellent dark matter candidate. Since our interest will be to analyze the most general aspects of the problem, we will neglect viscous effects, heat transport, fluid self-gravity, and effects associated to the back-reaction of the fluid on the geometry. As done in 27-FAYJ (27), we describe the accretion process of isothermal fluids by using the Hamiltonian dynamical formalism and we show that, contrary to what claimed in that work, there exist physical solutions for the accretion of an ultra-relavitivistic fluid in AS. Regarding the analysis of the stability, instead of adopting the approach of perturbing a scalar potential whose gradient is prescribed to be the velocity of the ideal fluid 10-Moncrief (10), we use a perturbation scheme based on the continuity equation 6-PETT (6), 7-TD (7), 17-Naskar (17). The stationary solution of the continuity equation gives a first integral, which, within a constant geometric factor, is actually the matter accretion rate. Our perturbative procedure entails the perturbation of this constant of motion. This quantity, being a flux of mass can be observed and, in principle, can be measured using the present day and the projected observational instruments 1 (1), 2 (2).

This paper is organized as follows. In the next section we present the RG-improved Schwarzschild metric. In Sec. III we review the mathematics of the accretion process and present the Hamiltonian dynamical system formalism. In Sec. IV we apply the formalism to the spherically symmetric accretion of isothermal fluids and analyze the quantum gravity effects by comparing with the GR description. In Sec. V we discuss the implementation of the perturbation scheme based on the continuity equation, and analyze the quantum gravity corrections to the stability of the accretion by studying the behavior of standing wave and travelling wave perturbations to the mass accretion rate. Finally, in the last section, we present our conclusions.

We start recalling that the classical Schwarzschild line element for a black hole with mass $M$, in units $c=G_{0}=1$, is written as

$$ds^{2}=-f_{0}(r)dt^{2}+f_{0}(r)^{-1}dr^{2}+r^{2}\Big{(}d\theta^{2}+\sin^{2}\theta d\varphi^{2}\Big{)},$$

(1)

with the metric coefficient $g^{tt}$ given by: $f_{0}(r)=1-2M/r$. The event horizon or Schwarzschild radius comes from solving $f_{0}(r)=0$, which gives $r_{hS}=2M$.

On the other hand, in the high-derivative gravity theory, which includes the Ricci scalar square, the Ricci tensor square, the Kretschmann scalar and running gravitational couplings in the effective action 28-Cai-Easson (28), the quantum gravity corrections to the metric are accounted for by promoting the classical gravitational Newton constant through a running coupling that evolve under the equations of the gravitational RG-flow. Thus, in the low energy limit the RG-improvement to the metric in Eq. (1) is obtained by doing

$$f_{0}(r)\rightarrow f(r)=1-\frac{2M}{r}\left(1-\frac{\xi}{r^{2}}\right),$$

(2)

where $\xi$ is a parameter with dimensions of length squared associated to the scale identification between the momentum scale $p$ and the radial coordinate $r$ which, in the IR regime, takes the form $p\sim 1/r$. Note that the quantum effects on the space-time geometry are all encoded in $\xi$, such that for $\xi=0$ the classical metric coefficient $f_{0}(r)$ is recovered. The condition for the radius of the new event horizon: $f(r)=0$, takes the form of a generic cubic equation

$$r^{3}-2Mr^{2}+2M^{3}\tilde{\xi}=0,$$

(3)

where the dimensionless parameter $\tilde{\xi}=\xi/M^{2}$ has been introduced for convenience. The only real solution $r_{hIR}$ to Eq.(3) is

$$\frac{r_{hIR}}{r_{hS}}=\frac{1}{6}\left(2+\frac{4}{\sqrt[3]{8-27\tilde{\xi}+3\sqrt{3}\sqrt{\tilde{\xi}(27\tilde{\xi}-16)}}}+\sqrt[3]{8-27\tilde{\xi}+3\sqrt{3}\sqrt{\tilde{\xi}(27\tilde{\xi}-16)}}\right).$$

(4)

The left panel in Fig. 1 shows the quotient $q\equiv r_{hIR}/r_{hS}$ as a function of $\tilde{\xi}$. For $\tilde{\xi}=0$ we have $r_{hIR}=r_{hS}$ as expected, while for $\tilde{\xi}$ greater than the critical value $\tilde{\xi}_{c}=16/27$, there is no horizon at all and a naked singularity arises. This implies that each value of $\tilde{\xi}$ in the range $0\leq\tilde{\xi}\leq\tilde{\xi}_{c}$ picks out a critical mass $M_{c}=\sqrt{27\xi/16}$ in such a way that for $M>M_{c}$ there are two horizons: one inner Cauchy horizon and one outer event horizon, for $M<M_{c}$ a naked singularity develops, while when $M=M_{c}$ the two horizons merge. The right panel in Fig. 1 illustrates this situation for $\xi=0.5$ that is, for $M_{c}=0.918$. Since our aim is to discuss quantum gravity effects on the accretion onto a Schwarzschild black hole, naked space-time singularities will be ignored.

For future reference we note that for $M>M_{\mathrm{c}}$, the outer horizon of the improved solution, upon expanding to the leading order in $\xi$, acquires the form: $r_{hIR}=r_{hS}-\xi/(2M)$. This result exhibits the typical shifting of the horizon of RG-improved black hole solutions towards smaller values with respect to its classical counterpart.

Here we describe the accretion flow by adopting the Hamiltonian procedure. The formulation of the problem of accretion onto compact objects as a Hamiltonian dynamical system was proposed for the first time in 29-CHA-SA (30).

We first recall that the phenomenon of accretion is based on two conservation laws: the continuity equation which expresses the conservation of the particle number, and the energy-momentum conservation. These requirements are given, respectively, by the covariant derivatives

$$\nabla_{\mu}\left(\eta u^{\mu}\right)=0\qquad\mbox{and}\qquad\nabla_{\mu}T^{\mu\nu}=0,$$

(5)

with $\eta$ being the particle number density of the fluid, and $u^{\mu}$ the four-velocity of the fluid normalized as $u^{\mu}u_{\mu}=-1$.

Neglecting effects related to viscosity or heat transport, and assuming that the fluid’s energy density is sufficiently small so that its self-gravity can also be ignored, the accreting matter can be approximated as a perfect fluid described by the energy-momentum tensor

$$T_{\mu\nu}=\left(\epsilon+p\right)u_{\mu}u_{\nu}+pg_{\mu\nu},$$

(6)

where $\epsilon$ and $p$ are the proper energy density and the proper pressure of the fluid, respectively.

For purely radial flow in the equatorial plane the normalization condition $u^{\mu}u_{\mu}=-1$ produces

$$u_{t}=-fu^{t}=\sqrt{f+(u^{r})^{2}}$$

(7)

where $u^{r}$ denotes the radial flow velocity and $f=f(r)$ is the metric coefficient in the spherically symmetric line element.

We assume as infalling matter a test fluid which is both isothermal and isentropic, that is a test fluid with barotropic equation of state $h=h(\eta)$ where $h$ is the specific enthalpy

$$h=\frac{\epsilon+p}{\eta}.$$

(8)

This fluid is described by the thermodynamical equations

$\displaystyle dp=\eta dh,\qquad d\epsilon=hd\eta,$

(9)

so that the square of the local speed of sound is written as

$$a^{2}=\frac{\partial p}{\partial\epsilon}\Bigg{|}_{ds=0}=\frac{dp}{d\epsilon}=\frac{\eta}{h}\frac{dh}{d\eta}=\frac{d\ln h}{d\ln\eta}.$$

(10)

Then, as it is well known, for spherically symmetric stationary accretion the conservation laws in Eq. (3) yield

$$\eta u^{r}r^{2}=C_{1},$$

(11)

and

$$hu_{t}=C_{2},$$

(12)

where $C_{1}$ and $C_{2}$ are arbitrary integration constants.

In order to describe the accretion process as a two-dimensional Hamiltonian dynamical system, it is useful to define the three-velocity $v$ of a fluid element as measured by a locally static observer as $v=dl/d\tau$, where $dl=dr/\sqrt{f}$ and $d\tau=\sqrt{f}dt$ are the proper radial distance and the proper time, respectively. It is important to remark that $v$ is defined outside the horizon, that is for an observer for which the time coordinate is timelike 30-AHMED (31). Using $u^{r}=dr/d\tau$, $u^{t}=dt/d\tau$ and Eq. (7) we get

$$v^{2}=\frac{(u^{r})^{2}}{f+(u^{r})^{2}}=\frac{(u^{r})^{2}}{(u_{t})^{2}},$$

(13)

from which

$$(u^{r})^{2}=\frac{fv^{2}}{1-v^{2}},\quad\mathrm{and}\quad(u_{t})^{2}=\frac{f}{1-v^{2}}.$$

(14)

As the Hamiltonian for the flow fluid we can choose either of the two integrals of motion $C_{1},C_{2}$, or even any combination of them. We choose $C_{2}^{2}$ as our Hamiltonian $H$ and fix the dynamical variables of the system to be $r$ and $v$. Using Eq. (14) we then have

$$H(r,v)=\frac{h^{2}(r,v)f(r)}{1-v^{2}},$$

(15)

and the dynamical system is defined by

$$\dot{r}=\frac{\partial H}{\partial v},\qquad\dot{v}=-\frac{\partial H}{\partial r},$$

(16)

where the dot denotes time derivative. After using Eq. (10) these two equations become

$$\dot{r}=\frac{2fh^{2}}{v(1-v^{2})^{2}}(v^{2}-a^{2}),,$$

(17)

$$\dot{v}=-\frac{h^{2}}{r(1-v^{2})}\left[(1-a^{2})r\frac{df}{dr}-4fa^{2}\right]$$

(18)

The critical points of the dynamical system, which coincides with the sonic points of the fluid flow, are the points $(r_{c},v_{c})$ where both $\dot{r}$ and $\dot{v}$ are zero. Assuming that $h\neq 0$ for all values of $r>r_{hIR}$ where also $f\neq 0$, it follows that at the critical points

$$v_{c}^{2}=a_{c}^{2},\qquad r_{c}(1-a_{c}^{2})f_{c,r_{c}}=4f_{c}a_{c}^{2},$$

(19)

where $f_{c}=f(r)|_{r_{c}}$ and $f_{c,r_{c}}=(df/dr)|_{r_{c}}$.

In this section, following 30-AHMED (31), we will obtain an adequate expression for $h(r,v)$ in Eq. (15) when the accreted fluid is isothermal. This in turn will provide a neat expression for the Hamiltonian $H(r,v)$.

The barotropic equation of state for an isentropic fluid can be expressed as $\epsilon=\epsilon(\eta)$. Moreover, for an isothermal fluid the equation of state reads $p=k\epsilon$ where the constant $k$ is the parameter state: $0<k<1$. Note that the definition of the speed of sound in Eq. (10) leads to $a^{2}=k$ and so the speed of sound remains constant through the accretion process. Now, from Eqs. (9) we have

$$h=\frac{d\epsilon}{d\eta},$$

(20)

and

$$\frac{dp}{d\eta}=\eta\frac{dh}{d\eta}=\frac{d^{2}\epsilon}{d\eta^{2}},$$

(21)

which, upon integration, produces

$$\eta\frac{d\epsilon}{d\eta}-\epsilon(\eta)=k\epsilon(\eta),$$

(22)

where we have used $p(\eta)=k\epsilon(\eta)$. Integration of this last equation yields

$$\epsilon(\eta)=C\eta^{k+1}=\frac{\epsilon_{c}}{\eta_{c}^{k+1}}\eta^{k+1},$$

(23)

where the integration constant $C$ has been chosen so that Eqs. (8) and (20) give rise to the same expression for $h$

$$h(r,\eta)=\frac{(k+1)\epsilon_{c}}{\eta_{c}}\left(\frac{\eta}{\eta_{c}}\right)^{k},$$

(24)

An expression for $\eta/\eta_{c}$ can be obtained from the constant of integration $C_{1}$ in Eq. (11). In fact, using Eqs. (14) and (19) we can write

$$C_{1}^{2}=\frac{r^{4}\eta^{2}fv^{2}}{1-v^{2}}=\frac{r_{c}^{4}\eta_{c}^{2}f_{c}v_{c}^{2}}{1-v_{c}^{2}}=\frac{r_{c}^{5}\eta_{c}^{2}f_{c,r_{c}}}{4},$$

(25)

so that

$$\left(\frac{\eta}{\eta_{c}}\right)^{k}=\left(\frac{r_{c}^{5}f_{c,r_{c}}}{4}\frac{1-v^{2}}{r^{4}fv^{2}}\right)^{k/2}.$$

(26)

Replacing into Eq. (24) we obtain

$$h^{2}=K\left(\frac{1-v^{2}}{r^{4}fv^{2}}\right)^{k},$$

(27)

with the constant $K$ given by $K=(r_{c}^{5}f_{c,r_{c}}/4)^{k}[(k+1)\epsilon_{c}/\eta_{c}]^{2}$.

Redefining the Hamiltonian as $\mathcal{H}=H/K$ and using Eq. (26), $\mathcal{H}$ acquires finally the form

$$\mathcal{H}(r,v)=\frac{[f(r)]^{1-k}}{r^{4k}v^{2k}(1-v^{2})^{1-k}}.$$

(28)

It must be stressed that, due to the definition of the three-velocity $v$, this expression for $\mathcal{H}$ is valid for an observer outside the horizon. This applies both for the classical Schwarzschild metric and for the improved AS metric.

We now focus on four types of isothermal accretion: ultra-stiff fluid ($k=1$), ultra-relativistic fluid ($k=1/2$), radiation fluid ($k=1/3$), and sub-relativistic fluid ($k=1/4$). For the purpose of comparison with the analysis in 27-FAYJ (27), we will take $\xi=0.5$ and $M=1>M_{c}=0.918$ in the expressions for the quantum $\mathcal{H}^{(AS)}$ and the classical $\mathcal{H}^{(GR)}$ Hamiltonians. Notwithstanding this amounts to consider a black hole solution with two horizons, the quantum effects on the accretion we will describe below are present for all the values of the parameter $\xi$ in the range $0\leq\xi\leq(16/27)M^{2}$, with the orbits in the contour plot of the quantum Hamiltonian going continuously towards the orbits of its classical counterpart in the limit $\xi\rightarrow 0$ as we will show ahead.

Let us start, following 17-Naskar (17), by writing the equations describing the accretion process in a more appropriate way for the stability analysis.

First, the explicit form of the continuity equation reads

$$\partial_{t}\left(\eta u^{t}\right)+r^{-2}\partial_{r}\left(\eta u^{r}r^{2}\right)=0.$$

(36)

To solve for $\eta$ and $u^{r}$, we use the equation for energy-momentum conservation and the thermodynamic equation for mass-energy conservation $d(\epsilon/\eta)+Pd(1/\eta)=Tds$. Along with the condition of constant entropy $ds=0$, and with the square of the speed of sound given by Eq. (10), the condition of energy-momentum conservation reads

	$\displaystyle u^{t}\partial_{t}u^{t}$	$\displaystyle+$	$\displaystyle u^{r}\partial_{r}u^{t}+f\left(\partial_{r}f\right)u^{r}u^{t}$		(37)
		$\displaystyle+$	$\displaystyle\frac{a^{2}}{\eta}\left[\left((u^{t})^{2}-f\right)\partial_{t}\eta+u^{r}u^{t}\partial_{r}\eta\right]=0$

Once the expression for the speed of sound $a$ of the particular test fluid is known and after fixing the time derivatives to zero, the solution to the system of coupled equations (36) and (37) provides us with the stationary fields $u^{r}(r)$, $\eta(r)$ and $a(r)$ and allow, in principle, the quantitative study of the spherical accretion onto a static spherically symmetric black hole. Assuming that the flow is smooth at all points of space-time, the stationary solutions of Eqs. (36) and (37) are written as

$$4\pi\bar{\mu}\eta u^{r}r^{2}=\dot{m},$$

(38)

and

$$\frac{d}{dr}\left[\ln\left(fu^{t}\right)\right]+\frac{a^{2}}{\eta}\frac{d\eta}{dr}=0,$$

(39)

respectively, where the constant of motion $\dot{m}$ is to be identified as the matter flow rate.

The use of the perturbation scheme based on the continuity equation starts by considering linear perturbations to the stationary velocity and particle number density fields: $u^{r}(r,t)=u^{r}(r)+u^{r\prime}(r,t)$ and $\eta(r,t)=\eta(r)+\eta^{\prime}(r,t)$, where primed quantities stand for small time-dependent perturbations. We now define the variable $\psi=\eta u^{r}r^{2}$, whose stationary value $\psi(r)$ coincides, except for the constant $4\pi\bar{\mu}$, with the steady mass accretion rate given by Eq. (38).

The first order perturbation to $\psi$ around stationary values is

$$\psi^{\prime}(r,t)=[u^{r}(r)\eta^{\prime}(r,t)+\eta(r)u^{r\prime}(r,t)]r^{2},$$

(40)

and Eq. (36) acquires the form

$$u^{t}\partial_{t}\eta^{\prime}+\frac{\eta u^{r}}{f^{2}u^{t}}\partial_{t}u^{r\prime}=-\frac{1}{r^{2}}\partial_{r}\psi^{\prime}.$$

(41)

In general, the first order perturbation to the square of the speed of sound is given by

$$a^{\prime 2}=a^{2}+\frac{da^{2}}{d\eta}\eta^{\prime},$$

(42)

The time evolution of $\eta^{\prime}$ and $u^{r\prime}$ follows from Eqs. (40) and (41) and are given, respectively, by

$$\partial_{t}\eta^{\prime}=-\frac{1}{r^{2}}\left(\frac{u^{r}}{f}\partial_{t}\psi^{\prime}+fu^{t}\partial_{r}\psi^{\prime}\right),$$

(43)

$$\partial_{t}u^{r\prime}=\frac{fu^{t}}{\eta r^{2}}\left(u^{t}\partial_{t}\psi^{\prime}+u^{r}\partial_{r}\psi^{\prime}\right).$$

(44)

On the other hand, Eq. (37) is written in terms of perturbed quantities in the form

	$\displaystyle u^{t}\left[u^{r}\frac{a^{2}}{\eta}\partial_{t}u^{r\prime}+\partial_{t}\eta^{\prime}\right]+\partial_{r}\left(u^{r}u^{r\prime}\right)+2u^{r}u^{r\prime}\frac{a^{2}}{\eta}\partial_{r}\eta$
	$\displaystyle+(fu^{t})^{2}\partial_{r}\left(\frac{a^{2}}{\eta}\eta^{\prime}\right)=0.$
			(45)

Taking the time derivative of this last equation and substituting for the time derivatives of $\eta^{\prime}$ and $u^{r\prime}$ from Eqs. (43) and (44), we obtain the differential equation obeyed by the perturbation to the mass accretion rate $\psi^{\prime}$

		$\displaystyle\partial_{t}$	$\displaystyle\left(\eta h^{tt}\partial_{t}\psi^{\prime}+\eta h^{tr}\partial_{r}\psi^{\prime}\right)$
		$\displaystyle+$	$\displaystyle\partial_{r}\left(\eta h^{rt}\partial_{t}\psi^{\prime}+\eta h^{rr}\partial_{r}\psi^{\prime}\right)$
		$\displaystyle=$	$\displaystyle(1-2a^{2})\left(h^{rt}\partial_{t}\psi^{\prime}+h^{rr}\partial_{r}\psi^{\prime}\right)\frac{d\eta}{dr},$

where the coefficients $h^{\alpha\beta}$ are given by

	$\displaystyle h^{tt}$	$\displaystyle=$	$\displaystyle\frac{u^{r}(fu^{t})}{f^{2}}\left[(fu^{t})^{2}+u^{r}-(u^{r})^{2}a^{2}\right],$
	$\displaystyle h^{tr}$	$\displaystyle=$	$\displaystyle h^{rt}=\frac{(u^{r})^{2}(fu^{t})^{2}}{f}(1-a^{2}),$		(47)
	$\displaystyle h^{rr}$	$\displaystyle=$	$\displaystyle u^{r}(fu^{t})\left[(u^{r})^{2}-(fu^{t})^{2}a^{2}\right].$

In this work we have reviewed the problem, previously analyzed in Ref. 27-FAYJ (27), of the steady accretion onto a R-G improved Schwarzschild black hole in the AS theory with higher derivatives. In this sense the accretion process is considered as a quantum correction to the known spherically symmetric accretion in general relativity. As in 27-FAYJ (27), we have used an isothermal fluid as our test fluid and we have specialized to the cases: ultra-stiff fluid ($k=1$), ultra-relativistic fluid ($k=1/2$), radiation fluid ($k=1/3$), and sub-relativistic fluid ($k=1/4$). Since our interest has been to describe the most general aspects of the problem, we have neglected viscous effects, heat transport, fluid self-gravity, and effects associated to the back-reaction of the fluid on the geometry. By using the Hamiltonian dynamical system procedure introduced in 29-CHA-SA (30), we have recovered the results by the authors in 27-FAYJ (27) for the accretion of ultra-stiff, radiation and sub-relativistic fluids but, at difference to what established in that work, we have found that accretion of ultra-relativistic fluids are a physical possibility in AS. In fact, for this type of fluid, as for the others considered here, we have found not only a shifting of the orbits of the Hamiltonian in the $(v,r)$ plane towards the central object and an enhancement of the flow velocity near the black hole as compared to what happens in GR, but also subsonic, supersonic and transonic regimes. This is an important result because there seems to be no physical reason for the no physical significance of the accretion of an ultra-relativistic fluid in AS. As expected, the quantum effects on the accretion are fully governed by the parameter $\xi$ in the RG-improved metric coefficient $f(r)$ in Eq. (2) with the classical behavior recovered in the limit $\xi\rightarrow 0$.

We have also analyzed the issue of the stability of the accretion and we have contrasted the results with the known stability of the accretion flow in classical general relativity. Using a perturbative procedure based on the continuity equation, the mass accretion rate of the infalling matter, considered as an isothermal fluid, has been subjected to small linear perturbations in the form of a standing wave and in the form of a travelling wave. For the standing wave perturbation we can conclude that, as in the general relativistic realm, the disturbance is damped in time so that the accretion is stable. As for the travelling wave perturbation a simple criterion, based on the behavior of the coefficient $k_{0}$ in the series expansion of the spatial part of the perturbation, has allowed us to compare between the quantum and the classical frameworks, and to reach to the conclusion that, where quantum gravity effects will have to be accounted for, the amplitude of the travelling wave perturbation for ultra-stiff fluid becomes enhanced as compared with the classical one. This indicates that the coupling of this type of fluid with the space-time curvature is weaker in AS than in GR. For ultra-relativistic, radiation and sub-relativistic fluids, the amplitude of the perturbation is reduced or enhanced depending on whether the local observer is located in the immediate neighborhood of the black hole horizon or not, respectively. The value of the radial coordinate $r_{\mathrm{cross}}$ at which this transition takes place can be easily calculated by solving for $r$ the equation that equals the classical amplitude with the quantum one. This means that the coupling of the accreting fluid with the geometry of the space-time, which is responsible for the damping of the perturbation, is stronger in the quantum case than in the classical one for $r<r_{\mathrm{cross}}$ and weaker for $r>r_{\mathrm{cross}}$. Since the disturbed quantity it has been the mass accretion rate, a physical quantity which is in principle measurable, it would be very interesting to study the possible observable effects that could be used to test asymptotic safety in the future.

We acknowledge financial support from COLCIENCIAS through the project with code 50754 associated to the CONVOCATORIA 757 PARA DOCTORADOS NACIONALES 2016. We also acknowledge financial support from Universidad Nacional de Colombia through the project with code 36031 within the program for the institutional strengthening of scientific research.