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Recently, Cai and Easson [27] found black hole solution in ASG scenario considering higher derivative terms in their investigation. They discuss how the inclusion of quantum corrections modifies the Schwarzschild black hole solution. According to Ref. [27], the geometry of a static spherically symmetric Schwarzschild (anti)-de Sitter black hole in ASG in the IR limit is given by
ds2=−(1−2GMr+2G2Mξr3)dt2+(1−2GMr+2G2Mξr3)−1dr2+r2dθ2+r2sin2θdϕ2,
(1) where G and M denotes the gravitational constant and mass of the black hole, respectively. The outer horizon, which is merely the null hypersurface of the modified version Eq. (1) of the Schwarzschild black hole taking quantum corrections into account can be written in approximate form as
rh=2GM3[1−2cosh(13cosh−1β)],
(2) where
β=27ξ8GM2−1 .rh given in Eq. (2) is the only real root of1−2GMr+2G2Mξr3=0 , which can be calculated using the Weierstrass Polynomialr=z+2GM3 . By expanding Eq. (2) to the leading order ofξ , we can approximate it as [27]rIR≃2GM−ξ2M.
(3) If we insert the running coupling parameter
ξ=0 into Eqs. (1) and (3), we can retrieve the classical Schwrazschild black hole metric and the corresponding event horizon, respectively. Here, we review some important equations describing the steady state Michel flow on a Schwrazschild (anti) de Sitter black hole in ASG. Further detail and generalization to an even more general static spherically symmetric black hole background is provided in Refs. [16, 28, 29].As described in the introduction, we model the flow of a perfect relativistic fluid, neglecting the effects related to viscosity or heat transport, and further assume that the fluid's energy density is sufficiently small, such that its self-gravity can be neglected. We assume that the flow of the perfect fluid onto the improved Schwrazschild black hole is steady-state flowing in the radial direction, described by the particle density n (also called baryonic number density), pressure p, and the energy density e by an observer moving along the fluid four-velocity
uαuα=−1 . To investigate the accretion process onto high derivative black hole, as described above, we need to review the fundamental equations of accretion for the underlying geometry of the spacetime.The accretion dynamics of a perfect matter is governed by the following conservation laws
∇αJα=0,
(4) ∇αTαβ=0,
(5) where
Jα=nuα is the particle current density, andTαβ=nhuαuβ+pgαβ is the stress energy tensor.∇ refers to the covariant derivative with respect to the spacetime metric. Here and onwards, we assume that h denotes the enthalpy per particle defined byh=p+en [30] whereh=h(n) is a function of the particle density n only. In the spherical symmetry stationary case, the above Eqs. (4) and (5) reduce tor2nu=const=K,
(6) h(1−2GMr+2G2Mξr3+u2)1/2=const=L,
(7) which expresses the conservation of particle and energy flux through a sphere of constant areal radius r. We stress here that to analyze the perfect fluid flow, Eqs. (6) and (7) will play the main role in the background of improved Schawrzschild black hole, as they will be helpful to convert the present problem into a Hamiltonian dynamical system.
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Physically, a critical point
r=rc describes the transition of the flow's radial velocity measured by the static observer from subsonic to supersonic. If we consider the barotropic fluid for which there is a constant pressure throughout (i.e.h=h(n) ), then its equation of state can be expressed as [30]dhh=a2dnn,
(8) where a denotes the local speed of sound.
Differentiating Eqs. (6) and (7) with respect to r, we obtain
dudr=2ur.c2s(1−2GMr+2G2Mξr3+u2)−GM2r−3G2Mξ2r3u2−c2s(1−2GMr+2G2Mξr3+u2),
(9) where
c2s=k is the square of the speed of sound, and k is a state parameter for the isothermal equation of state (EoS)p=ke . The above Eq. (9) can be converted into a two-dimensional autonomous Hamiltonian dynamical system:f1(r,u)=drdl=r{u2−c2s(1−2GMr+2G2Mξr3+u2)},
(10) f2(r,u)=dudl=2u{c2s(1−2GMr+2G2Mξr3+u2)−GM2r−3G2Mξ2r3},
(11) with an arbitrary parameter l, whose phase portraits consist of r versus s, indicate solutions of Eqs. (6) and (7). To obtain critical points, we set the right hand side of Eqs. (10) and (11) equal to zero, which after solving yield
u2c=GM2rc+3G2Mξ2r3c,
(12) c2s=GM2rc+3G2Mξ2r3c1−3GM2r2c+7G2Mξ2r3c.
(13) By using Eqs. (12) and (13), we can obtain the sonic points, which refer to the critical points of the dynamical systems of Eqs. (10) and (11). From Eqs. (6) and (7), after performing several intermediate steps, one can arrive at very important equation, which is helpful to describe the critical flows of the fluid under consideration
√1−2GMr+2G2Mξr3+(u)2=Ar2kuk.
(14) For the standard equation of state, the critical point
(rc,uc) is the saddle point, and thus the solution must pass through it. The detailed discussion of this critical point will be presented in the forthcoming sections. -
To analyze the perfect matter flow, it is useful to employ a dynamical system whose orbits consist of graphs of solutions of the system Eqs. (6) and (7). Such a system can be defined conveniently in terms of r and v (where v is the three-velocity of the fluid). We formulate our problem in terms of the Hamiltonian dynamical system on the phase space
(r,n) , where the vector field describing the dynamics is the Hamiltonian vector field associated with the functionF(r,n) . By assumption, F is constant along the trajectories of phase flow and thus meets the definition of level curves. The main usefulness of converting the accretion problem into a dynamical system is that the fluid behavior near the critical point of F can be analyzed using standard tools of the theory of dynamical systems [31, 32].In Eqs. (6) and (7), we used two integrals of motion K and L. We stress here that any one of them , or any combination of these integrals can be utilized as Hamiltonian system for the fluid flow. Assuming that the Hamiltonian system is a function of two variables r and v and square of the left hand side of Eq. (7), i.e.,
H(r,v)=h2(1−2GMr+2G2Mξr3+u2),
(15) which in a more general form can be written as
H(r,v)=h2(r,v)(f(r)+u2)1−v2.
(16) We introduce the following pair of a dynamical system
˙r=H,v˙v=−H,r,
(17) where the dots denote the
˜t derivatives, and theH,r andH,v denote partial derivatives ofH with respect to r and v, respectively. By solving the right-hand-side of the above equation and subsequently equating to zero results in the desired critical point(rc,uc) , we obtain the following fundamental pair of equationsv2c=a2c,rc(1−a2c)fc,rc=4fca2c,
(18) which are thus helpful to derive the following important equations
(a2c=k)[14rcfc,rc+f∗]=[14rcfc,rc],
(19) (urc)2=14rcfc,rc.
(20) We point out here that the above pair of Eqs. (19) and (20) is equivalent to Eqs. (12) and (13) and provide the critical radius and critical speed of the moving fluid. Thus, we shall use these equations to locate the position of the critical point
(r∗,ur∗) . We know that one of the most appreciative tools for energy conservation is the Hamiltonian. In the current study, the precise form of the general Hamiltonian Eq. (15) in terms of the variables r and v for the isothermal test fluid can be expressed asH(r,v)=(1−2GMr+2G2Mξr3)1−k(1−v2)1−kv2kr4k,
(21) where k is the state parameter, and v is the ordinary three-dimensional speed of the fluid, which is given by
v2=u21−2GMr+2G2Mξr3+u2.
(22) We remark here that u is well-defined everywhere, and the velocity v is defined outside the horizon. Complete derivation of these fundamental equations is provided in Refs. [33-35].
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Expressing accreting matter by the isothermal equation of state (EoS)
P=ke (where k is a state parameter), we present a complete description of the fluid flow behavior near the black hole.1) The fluid at which isotropic pressure and the energy density of the fluid particles is same is referred to as ultra stiff fluid. In this case, the state parameter has the value
k=1 . This value of the state parameter reduces Eq. (19) tofc=0 , which yieldsrc=rh , i.e., the critical radius and event horizon coincide. In this case, the Hamiltonian in Eq. (21) reduces toH=1v2r4.
(23) As above, the Hamiltonian shows constant of motion i.e.,
H=H0 , and we observe that v behaves as1r . To explain the physical behavior of the fluid flow, we need to sketch contour plots ofH(rc,vc)=Hc . From the Fig. 1 on top left, the black curve indicates the solution forH=Hc , the red curve indicates the solution forH=Hc+0.005 , the green curve shows the solution forH=Hc+0.02999 , the magenta curve depictsH= Hc−0.0001 , and the blue curve depictsH=Hc−0.09 . In summary, we observe that forv>0 , there is particle emission, andv<0 depicts the fluid accretion.Figure 1. (color online) Contour plot of Hamiltonian
H(21) for ultra-stiff(k=1) , ultra-relativistic(k=1/2) , radiation(k=1/3) and sub-relativistic(k=1/4) fluids whereM=1,G,ξ=0.5 . The black curve in these graphs depicts the curve that passes through the critical saddle point i.e.H=H∗. 2) If the isotropic pressure is less than the energy density, it has characteristics of an ultra-relativistic fluid. In this type of fluid, the EoS takes the form
p=e/2 . After settingk=1/2 in Eq. (19), we obtain the following expression for the critical radiusrc≃52GM−1425ξM.
(24) The Hamiltonian in Eq. (21) in this case reduces to
H=√(1−2GMr+2G2Mξr3)r2v√1−v2.
(25) We can observe that
H in Eq. (25) is not defined for(r,v2)=(rh,1) . However, for some constant values ofH=H0 , one can solve it forv2 . The five trajectories of solutions to Eq. (25) in the phase space are shown in top right diagram of Fig. 1. Here, the black curve indicates the solution forH=Hc , the red curve depicts the solution forH=Hc−0.01 , the green curve forH= Hc−0.005 , the magenta curve forH=Hc+0.01 , and the blue curve forH=Hc+0.005087 . From the contour plots, we see that they are doubly-valued and show unphysical behavior, such we can say that there is no physical significance of such fluid in ASG.3) For the radiation fluid, we have the state parameter
k=1/3 . This fluid has the property to absorb the radiations emitted by the black hole. The insertion ofk=1/3 in Eq. (19) results in the following real approximation of the critical radiusrc≃3GM−59ξM,
(26) while the Hamiltonian in Eq. (21) takes the form
H=(1−2GMr+2G2Mξr3)2/3r4/3v2/3(1−v2)2/3.
(27) From above Hamiltonian, we see that the point
(r,v2)=(rh,1) is not a critical point of the dynamical Hamiltonian system. However, the expression forv2 can be obtained by fixingH=H0 . The characteristics of solution curves are depicted in the left lower picture, where the black curve shows the solution forH=Hc , the red curve forH=Hc+0.00099 , the green curve forH=Hc+0.0009 , the magenta curve forH=Hc−0.04 , and the blue curve forH=Hc−0.09 .Here, for the radiation fluid, we find some surprising characteristics as it gets closer to the black hole. The black, magenta, and blue curves exhibit unphysical behavior, however, the green curves describe highly interesting behavior of the transonic type. The fluid has supersonic velocity before the critical point, but as soon as it approaches the critical point, the speed becomes subsonic.
4) In sub-relativistic fluids, energy density exceeds the isotropic pressure, and the assigned value to the state parameter is
k=1/4 . Repeating previous steps, we obtain an approximation of the critical radius asrc≃72GM−2649ξM.
(28) The insertion of Eq. (28) into Eq. (20) provides the desired critical point
(rc,uc) .In this case of a sub-relativistic fluid, the Hamiltonian Eq. (24) takes the following form
H=(1−2GMr+2G2Mξr3)3/4rv1/2(1−v2)3/4.
(29) From above equation, it is evident that the point
(r,v2)=(rh,1) is not a critical point of the dynamical system. Now, we draw contour plots ofH in the(r,v) plane by fixingH=Hc which describes the following behavior of the moving fluid. The black curve shows the solution forH=Hc , the red curve forH=Hc+0.03 , the green curve forH=Hc−0.0399 , the magenta curve forH=Hc+0.0009 , and the blue curve forH=Hc−0.0317999 . The curves shown in the lower right figure describe the behavior of the moving fluid as follows: in the blue and green curves, the fluid reaches near the critical point to show transonic behavior, but surprisingly fails to do so. Hence, in this scenario, we define this motion as unphysical behavior of the fluid (as they show velocity as a double valued function). However, blue and magenta curves show the supersonic accretion motion in the regionv>vc and subsonic motion in the region wherev<vc . -
Unlike the Schawrzschild black hole, the quantum gravity affects the accreting fluid near the improved version of Schawrzschild black hole. Furthermore, if we do not entertain the quantum gravity effects, the above presented results are easily reducible to what already been published in [2]. We discuss the asymptotic behavior of isothermal fluids with EoS
p=ke , such that0<k<1 . Eq. (13) can be written in the following formGM2rc+3G2Mξ2r3c=k(1−3GM2r2c+7G2Mξ2r3c),
(30) which can again be reduced to a depressed cubic equation by introducing the Wierstrass polynomial
r=t+7GM6 , which is equivalently expressed ast3c−ptc−q=0,
(31) where
p=49G2M212,q=343G3M3108.
(32) Here, Eq. (31) has three roots: one real root and the other two will be complex conjugates of each other. This follows directly from the Cardano formula
rc=3√−q+ι√W+3√−q−ι√W,
(33) where
W=√p3+q2 . We can perform a detailed analysis by computing the Jacobian matrix for the Eqs. (10) and (11), for instanceJ=(∂f1∂r∂f1∂u∂f2∂r∂f1∂u).
(34) With the help of the above Jacobian matrix, one can determine that either the critical values are center, saddle, or spiral. If both eigenvalues are real and have different signs, we have a saddle point. If the real part of the complex eigenvalues is negative, then we have a spiral and if the real part of the complex number is zero, we have a center.
Hence, using Eqs. (19) and (20), one can obtain
rc andurc , respectively (velocity of the fluid at sonic point). Then, after putting(rc,urc) in Eq. (14), we find the constantA to obtain u in an explicit form. Moreover, from the normalization of the four-velocity vector, one can also derive an expression forut(r) . Thus, after finding the explicit forms of u andut , one can sketch(uut)2 along ther -axis to see whether for each case k = 1, 1/2, 1/3, 1/4, the fluid passes through the sonic point, as sketched in Fig. 2. In Fig. 1, we have discussed non-transonic solutions, but we have also plotted the transonic solutions for the isothermal fluid in Fig. 2. The transonic solutions yield a maximum accretion rate, because they pass through the critical point. In Fig. 2, we see that the fluid trajectories may form an orbit fork=1/4 near the Cauchy and event horizon.
Accretion on high derivative asymptotically safe black holes
- Received Date: 2019-11-11
- Accepted Date: 2020-01-29
- Available Online: 2020-06-01
Abstract: Asymptotically safe gravity is an effective approach to quantum gravity. It is important to differentiate modified gravity, which is inspired by asymptotically safe gravity. In this study, we examine particle dynamics near the improved version of a Schwarzschild black hole. We assume that in the context of an asymptotically safe gravity scenario, the ambient matter surrounding the black hole is of isothermal nature, and we investigate the spherical accretion of matter by deriving solutions at critical points. The analysis of various values of the state parameter for isothermal test fluids, viz., k = 1, 1/2, 1/3, 1/4 show the possibility of accretion onto an asymptotically safe black hole. We formulate the accretion problem as Hamiltonian dynamical system and explain its phase flow in detail, which reveals interesting results in the asymptotically safe gravity theory.