Relativistic sonic geometry for isothermal accretion in the Kerr metric

# Relativistic sonic geometry for isothermal accretion in the Kerr metric

Md Arif Shaikh Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211019, India
###### Abstract

We linearly perturb advective isothermal transonic accretion onto rotating astrophysical black holes to study the emergence of the relativistic acoustic spacetime and to investigate how the salient features of such spacetime get influenced by the spin angular momentum of the black hole. We have perturbed three different quantities - the velocity potential, the mass accretion rate and the relativistic Bernoulli’s constant to show that the acoustic metric obtained for these three cases are same up to a conformal factor . By constructing the required causal structures, it has been demonstrated that the acoustic black holes are formed at the transonic points of the flow and acoustic white holes are formed at the shock location. The corresponding acoustic surface gravity has been computed in terms of the relevant accretion variables and the background metric elements. The linear stability analysis of the background stationary flow has been performed.

Keywords: accretion, accretion discs, analogue gravity, general relativity, fluid mechanics

###### pacs:
04.70.Dy, 95.30.Sf, 97.10.Gz, 97.60.Lf
: Class. Quantum Grav.

## 1 Introduction

Classical analogue systems are those where a curved acoustic geometry may be obtained by linear perturbing a transonic fluid. Such sonic geometries are embedded within the background fluid, and the propagation of the acoustic perturbation can be described by an acoustic metric which possesses sonic horizons [1, 2, 3, 4, 5, 6, 7]. Accreting astrophysical black holes provide an embedding for an acoustic spacetime, which is embedded within the fluid system accreting on the black hole, and thus such systems are very special examples of classical analogue gravity systems, since such configuration may be considered unique because both the gravitational as well as acoustic horizons are an integral part of such systems [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. In the majority of such works, characteristic features of the embedded sonic geometry have been obtained through perturbation of mass accretion rate in the astrophysical context.

Shaikh et al[15] have studied the perturbation of isothermal accretion in a more general context, where a generalized formalism was presented to show how one can obtain the corresponding sonic geometry through the linear perturbation of the velocity potential, mass accretion rate as well as the relativistic Bernoulli’s constant to manifest that there exist some general properties of the emergent gravity phenomena which are independent of the quantity that we linear perturb to obtain the spacetime geometry describing the emergence of the gravity like phenomena. In [15], the accretion dynamics was studied in the background Schwarzschild metric. Astronomers, however, believe that most of the astrophysical black holes possess non zero spin angular momentum, i.e, such black holes will be characterized by non zero Kerr parameter ‘’ (e.g., see [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38] which studies the influence of black hole spin and measurement of such spin for different astrophysical systems).

We would thus like to understand how the emergent gravity phenomena and various characteristic features of the sonic geometry get influenced by the black hole spin. In other words, in the present work, we would like to demonstrate how the property (spin) of the background metric influences the properties of the emergent acoustic metric. Saha et al[39] have studied the dependence of few of the properties of the acoustic metric on the spin parameter, however, such studies were done for accretion under the influence of generalized post-Newtonian pseudo-Kerr black hole potential and the studies in the case of isothermal flow was very limited. Whereas our work is in fully general relativistic Kerr background and concentrated on isothermal flow. Important discussion on shocked isothermal accretion in the Kerr geometry could be found in [40] but the corresponding acoustic geometry was not discussed in [40] which we study in the present work. Also, they discuss only the stationary solution, whereas here we study the time dependence through linear stability analysis, which has not been done before. In [41], it was shown that stationary shock formation is an integral part of accretion disc in the Kerr metric and in the present work we show how shock formation influences the properties of the acoustic spacetime, e.g., the causal structure of acoustic spacetime. We assume an inviscid flow as viscosity breaks the Lorentzian symmetry [4] and acoustic metric cannot be constructed. However, for relativistic viscous transonic accretion flow onto rotating black hole interested readers are referred to [42, 43].

Accretion of the surrounding matter onto astrophysical black holes is a crucial phenomenon to study in connection with the observational evidence of the black holes in the universe. While falling onto the black hole, accreting matter emits multi-wavelength radiation due to various radiative processes. The details of the characteristic features of such emitted photons may be understood by studying various dynamical properties of the accretion flow as well as by applying the knowledge of the radiative transfer. Such emitted photons form its characteristic spectra. Study of such spectra provides the information about the strong gravity spacetime close to the black hole event horizon, and, most importantly, helps ensure the presence of astrophysical black holes through observational means. Apart from the wind-fed accretion (where the supersonic stellar wind accretes onto black holes), the infalling material usually starts its journey toward the black hole subsonically. Close to the event horizon of the black hole, the dynamical velocity becomes extremely large, close to the speed of light in the vacuum (). Whereas, even for the steepest equation of state, the maximally allowed velocity of sound within the accreting matter never exceeds . This dictates that the accreting matter falls supersonically onto the black hole.

The black hole accretion is, thus, necessarily transonic, and hence matter makes a smooth (continuous) transition from subsonic to supersonic state. Such transonic property of the accreting material thus suggests that an acoustic geometry with the (at least one) acoustic horizon may be present within the accreting material, where the transonic surface may act as the corresponding acoustic horizon specified by the acoustic metric.

If the accretion is spherically symmetric, then the formation of more than one sonic horizon is not possible. For axially symmetric accretion flow, however, one may argue that two saddle type critical points may form and the accreting flow may make a smooth continuous transition from a subsonic to a supersonic state and hence two acoustic black hole horizons may form. In between these two sonic surfaces, a stationary shock forms which forces the flow to make a discontinuous transition from a supersonic state to a subsonic state. Such shock surfaces may be identified with an acoustic white hole. Shock transition is considered to be a crucial phenomenon in accretion astrophysics. The discontinuous changes of the values of various variables may affect the emergent spectra by introducing (possibly) a broken power law since the photon energy changes abruptly. The shock formation phenomena in black hole accretion disc thus explicitly manifest through the observation of the characteristic black hole spectra.

It is thus obvious that the accreting black hole system has a one to one correspondence with a classical analogue system naturally found in the universe. Hence it is important to study the accreting black hole system from the analogue gravity point of view. Usually, a steady state accretion flow is considered to understand various spectral features of the astrophysical black holes. It is imperative to ensure that such a task can be accomplished through the linear perturbation technique. The stationary integral solutions of the steady state accretion flow are linear perturbed to demonstrate that such perturbation does not diverge and hence the steady state remains stable.

In analogue gravity phenomena, a stationary state is linear perturbed and the propagation of the resultant acoustic perturbation is described by the acoustic metric. For astrophysical accretion, one has to show that the linear perturbation of the stationary accretion solutions leads to the emergence of the acoustic metric embedded within the accreting matter. Such acoustic metric will have acoustic horizons. By constructing the causal structures, one has to show that such horizons actually coincide with the physical transonic surfaces in the accretion flow, and the shock surface coincides with the acoustic white whole horizon. In this way, a formal correspondence can be made between an accreting astrophysical black hole and a classical analogue gravity system. Such task has been accomplished in the present work for general relativistic, axially symmetric, isothermal inviscid accretion onto rotating (Kerr type) astrophysical black holes.

We will consider general relativistic sub-Keplerian low angular momentum inviscid accretion of isothermal matter onto a Kerr black hole as the background steady flow. Such flow is considered to be characterized by the constant bulk temperature (ion temperature of single temperature flow), constant specific angular momentum , and the Kerr parameter . We consider the low angular momentum accretion to fulfill the inviscid ( and hence constant ) criteria as flow with large angular momentum may settle into a Keplerian orbit and in absence of viscous transport of angular momentum the accretion may not happen. However, an inviscid flow may not necessarily be sub-Keplerian as there may be other mechanisms (non-viscous) which may drive the accretion process. In the present accretion model, we do not include these mechanisms and hence restrict ourselves to low angular momentum flow. A large pool of works is available in the literature which deals with effectively inviscid flow [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. Also, see [61] for a general review of accretion disk. Such practically inviscid flow may be observed at our galactic center [62]. Hence the assumption of inviscid accretion is not unjustified from astrophysical context. The specific angular momentum of the axially symmetric rotating flow and the spin angular momentum (the Kerr parameter) of the astrophysical black holes are two effectively additive quantities in our work. This means, if one reduces the Kerr parameter (the algebraic value of the Kerr parameter), then the effective value of the maximum angular momentum for which a multi-transonic flow will form, will increase, and vice-versa. This is probably due to the fact that the frame-dragging effect does not explicitly influence our calculations, and hence the parameter effectively ‘adds up’ with .

The plan of the work is as follows. We shall first construct the general relativistic Euler and continuity equation corresponding to the ideal fluid described by the isothermal equation of state. We shall find the integral solutions of the time-independent part of the Euler and the continuity equation to obtain two first integrals of motion for background steady flow. Such constants of motion are found to be the mass accretion rate obtained as the integral solution of the continuity equation and the relativistic Bernoulli’s constant which is obtained through the integral solution of the Euler equation. We then introduce the irrotationality condition which provides the constant steady value of the velocity potential . We then linear perturb , and separately and find that the overall characteristics features of the embedded sonic geometries are more or less same. They might differ in the conformal factor (which may appear as the coefficient of the acoustic metric element) only.

Once the acoustic metric is found, the location of the acoustic horizon is identified. Considering the detailed phase portrait of the characteristic stationary integral solutions, we show that at most three acoustic horizons may be present in such configurations. Out of those three horizons, two are black hole horizons formed at the outer and the inner transonic points and one also obtains an acoustic white hole which is obtained at the location where a discontinuous stationary shock transition takes place. We then construct the causal structures at and around such acoustic horizons. We demonstrate how to calculate the acoustic surface gravity in terms of the background black hole metric elements and corresponding accretion variables, all computed at the location of the corresponding acoustic horizon on which the value of the acoustic surface gravity is being evaluated.

Finally, we perform the stability analysis of the stationary solutions used for the calculation of the causal structure. Such analysis ensures that the stationary solutions that we are using are stable under linear perturbation.

We shall set where is the universal gravitational constant, is the velocity of light and is the mass of the black hole. The radial distance will be scaled by and any velocity will be scaled by . We shall use the negative-time-positive-space metric convention.

## 2 Basic equations

### 2.1 The background metric

We Consider the following metric for a stationary rotating space time

 ds2=−gttdt2+grrdr2+gθθdθ2+2gϕtdϕdt+gϕϕdϕ2, (1)

where the metric elements are function of and . The metric elements in the Boyer-Lindquist coordinates are given by [63, 64]

 gtt=(1−2rA),grr=AB,gθθ=A,gϕt=−2arsin2θA,gϕϕ=CAsin2θ (2)

where

 A=r2+a2cos2θ,B=r2−2r+a2,C=(r2+a2)2−a2Bsin2θ. (3)

The event horizon of the Kerr black hole is located at .

### 2.2 Conservation equations

The energy momentum tensor for perfect fluid is given by

 Tμν=(p+ε)vμvν+pgμν (4)

where and are the pressure and the mass-energy density of the fluid respectively. is the four-velocity of the fluid. obeys the normalization condition . We assume the fluid to be inviscid and irrotational and is described by the isothermal equation of state, i.e., , where is the local rest mass energy density of the fluid. The mass-energy density includes the rest mass density and the internal energy (thermal energy), . The rest mass energy density could be related to the particle number density of fluid by the relation , where is the rest mass of the fluid particle. The continuity equation which ensures the conservation of mass is given by

 ∇μ(ρvμ)=0 (5)

where the covariant divergence is defined as . The energy-momentum conservation equation is given by

 ∇μTμν=0. (6)

Substitution of Eq. (4) in Eq. (6) provides the general relativistic Euler equation for barotropic ideal fluid as

 (p+ε)vμ∇μvν+(gμν+vμvν)∇μp=0. (7)

The enthalpy is defined as

 h=(p+ε)ρ. (8)

The isothermal sound speed could be defined as[40]

 c2s=1h∂p∂ρ. (9)

The relativistic Euler equation for isothermal fluid can thus be written as

 vμ∇μvν+c2sρ(vμvν+gμν)∂μρ=0. (10)

For general relativistic irrotational fluid we have the irrotationality condition given by [15]

 ∂μ(vνρc2s)−∂ν(vμρc2s)=0. (11)

## 3 The accretion disc structure and the reference frame

### 3.1 Disc structure

The accretion flow is described by the four-velocity ). We assume the flow to be axially symmetric and also to be symmetric about the equatorial plane and the velocity along the vertical direction to be negligible, i.e., . Now let us consider the continuity equation given by (5). Due to the axial symmetry, the term would vanish. It is a common practice for accretion flow with low angular momentum to do a vertical averaging of the flow [65]. The vertical averaging of a flow variable is approximated as

 ∫f(r,θ)dθ≈Hθf(r,θ=π2) (12)

where is the characteristic angular scale of the flow. Thus the continuity equation for vertically averaged axially symmetric accretion can be written as[65, 17, 15]

 ∂t(ρvt√−~gHθ)+∂r(ρvr√−~gHθ)=0 (13)

where is the value of , the determinant of the background metric , on the equatorial plane. For Kerr metric and thus . The term , as mentioned above, appears as a result of the vertical averaging. Vertical averaging of flow allows us to work fully in the equatorial plane by retaining the information of the vertical structure in the term. In other words, is the proper weight function to give correct value of the mass accretion rate obtained by spatially integrating Eq. (13) in stationary case. Thus can be related to the local flow thickness of the flow. In the present work, we consider the flow to be wedge-shaped conical flow where , where is the local radius. For a flow with conical geometry thus . does not depend on the accretion variables like density and velocities. Thus linear perturbation of these quantities (as defined by the Eq. (24),(25) (to be searched below)) will have no effect on it. For simplicity, therefore, we will write simply as . Further details on different flow geometries can be found in [13, 17, 15]. From now on all the equations will be derived by assuming the flow to be vertically averaged and the variables have values equal to that in the equatorial plane.

### 3.2 Reference frame

Apart from the Boyer-Lindquist coordinate frame (BLF) we shall use a second reference frame which is called the corotating frame (CRF)[65]. This frame is obtained by an azimuthal Lorentz boost from the locally nonrotating frame (LNRF) into a tetrad basis that corotates with the fluid. LNRF is an orthonormal tetrad basis who lives at , where on the equatorial plane (originally calculated by Bardeen et al[63]). Let be the radial velocity (referred as the ‘advective velocity’) of the fluid as measured in the CRF and be the specific angular momentum of the fluid. Then the four-velocity components in BLF is related to and in the following way [65]

 vr=u√grr(1−u2) (14)
 vt= ⎷(gϕϕ+λgϕt)2(gϕϕ+2λgϕt−λ2gtt)(gϕϕgtt+g2ϕt)(1−u2) (15)

and

 vt=− ⎷gttgϕϕ+g2ϕt(gϕϕ+2λgϕt−λ2gtt)(1−u2) (16)

and are the two velocity variables needed to describe the flow in the equatorial plane.

## 4 Linear perturbation of velocity potential, mass accretion rate and the relativistic Bernoulli’s constant

In the following sections, we introduce three quantities that we want to linear perturb to obtain the acoustic metric. These quantities are - the velocity potential which is defined through the irrotationality condition, the mass accretion rate which is the integral of the continuity equation for stationary flow and the relativistic Bernoulli’s constant which is the integral of the temporal component of the relativistic Euler equation for stationary flow introduced in Sec. 4.1,4.2 and 4.3 respectively. After that, we use the perturbation equations given by Eq. (24)-(25) in the continuity equation, irrotationality condition and the temporal component of the relativistic Euler equation and keep only the terms that are linear in the perturbed quantities. These equations give the wave equations describing the propagation of the perturbation of the three quantities mentioned above. From these wave equations, we identify the matrix , as discussed later, to obtain the acoustic metric. Thus the main aim of this section is to find the expressions for the symmetric matrix for three different quantities.

Before going to the details of the derivation of the acoustic metric by linear perturbation of different quantities, let us first write down some useful equations which will be essential in the following sections. The normalization condition gives

 gtt(vt)2=1+grr(vr)2+gϕϕ(vϕ)2+2gϕtvϕvt (17)

From irrotaionality condition given by Eq. (11) with and and with axial symmetry we have

 ∂t(vϕρc2s)=0 (18)

again with and and the axial symmetry the irrotationality condition gives

 ∂r(vϕρc2s)=0 (19)

So we get that is a constant of the motion. Eq. (18) gives

 ∂tvϕ=−vϕc2sρ∂tρ (20)

Substituting in the above equation provides

 ∂tvϕ=−gϕtgϕϕ∂tvt−vϕc2sgϕϕρ∂tρ (21)

Also differentiating the Eq. (17) with respect to gives

 ∂tvt=α1∂tvr+α2∂tvϕ (22)

where and . Substituting in Eq. (22) using Eq. (21) gives

 ∂tvt=(−α2vϕc2s/(ρgϕϕ)1+α2gϕt/gϕϕ)∂tρ+(α11+α2gϕt/gϕϕ)∂tvr (23)

We perturb the velocities and density around their steady background values as following

 vμ(r,t)=vμ0(r)+vμ1(r,t),μ=t,r,ϕ (24)
 ρ(r,t)=ρ0(r)+ρ1(r,t) (25)

where the subscript “0” denotes the stationary background part and the subscript “1” denotes the linear perturbations. Using Eq. (24)-(25) in Eq. (23) and retaining only the terms of first order in perturbed quantities we obtain

 ∂tvt1=η1∂tρ1+η2∂tvr1 (26)

where

 η1=−c2sΛvt0ρ0[Λ(vt0)2−1−grr(vr0)2],η2=grrvr0Λvt0andΛ=gtt+g2ϕtgϕϕ (27)

### 4.1 Linear perturbation of velocity potential

The irrotationality condition given by Eq. (11) can be used to introduce a potential field such that

 −∂μψ=vμρc2s (28)

It has been shown in the previous section that is a constant of motion and hence is a constant of the motion. Therefore perturbation of gives . Contracting both sides of Eq. (28) with and using the normalization condition gives

 ρc2s=vμ∂μψ (29)

We perform linear perturbation of the above Eq. (29) by substituting using Eq. (24),(25) respectively and by the following equation

 ψ=ψ0+ψ1 (30)

where is the stationary value of independent of time. Retaining only the terms that are linear in the perturbation quantities gives

 ρ1=1c2sρc2s−10[vr∂rψ1+vt∂tψ1] (31)

In obtaining the above equation we have used and . Eq. (28) can also be written as . Writing and using Eq. (25) and Eq. (30) in the equation keeping only terms which are linear in the perturbed terms gives

 vμ1ρc2s0=−gμν∂νψ1−vμ0vν0∂νψ1 (32)

Therefore we get

 vt1=1ρc2s[(gtt−(vt0)2)∂tψ1−vt0vr0∂rψ1] (33)

and

 vr1=1ρc2s0[(−grr−(vr0)2)∂rψ1−vr0vt0∂tψ1] (34)

Substituting the and in the continuity Eq. (13) using Eq. (24) and (25) respectively and retaining the terms which are of first order in perturbation terms gives

 ∂t[√−~gH0(ρ0vt1+vt0ρ1)]+∂r[√−~gH0(ρ0vr1+vr0ρ1)]=0 (35)

substituting and in the above equation using Eq. (31), (33) and (34) respectively gives

 ∂t[k1(r){−gtt+(vt0)2(1−1c2s)}∂tψ1]+∂t[k1(r){vr0vt0(1−1c2s)}∂rψ1] (36) +∂r[k1(r){vr0vt0(1−1c2s)}∂tψ1]+∂r[k1(r){grr+(vr0)2(1−1c2s)}∂rψ1]=0

where . The above equation can be written as , where is

 fμν1=k1(r)⎡⎢⎣−gtt+(vt0)2(1−1c2s)vr0vt0(1−1c2s)vr0vt0(1−1c2s)grr+(vr0)2(1−1c2s)⎤⎥⎦ (37)

### 4.2 Linear perturbation of mass accretion rate

For stationary flow the part of the equation of continuity, i.e., Eq. (13) vanishes and integration over spatial coordinate provides

 √−~gH0ρ0vr0=constant. (38)

Multiplying the quantity by the azimuthal component of volume element and integrating the final expression gives the mass accretion rate,

 Ψ0=Ω√−~gH0ρ0vr0. (39)

arises due to the integral over and is just a geometrical factor and therefore can be absorbed in the left hand side to redefine the mass accretion rate without any loss of generality as

 Ψ0=√−~gH0ρ0vr0. (40)

We now define a quantity which has the stationary value equal to . Using the perturbed quantities given by Eq. (24) and (25) we have

 Ψ(r,t)=Ψ0+Ψ1(r,t), (41)

where

 Ψ1(r,t)=√−~gH0(ρ0vr1+vr0ρ1). (42)

Using Eq. (24)-(26) and (41) in the continuity Eq. (13) gives

 a1∂tvr1+b1∂tρ1=∂rΨ1, (43)

where

 a1=−√−~gH0ρ0η2andb1=−√−~gH0(vt0+ρ0η1). (44)

Also differentiating Eq. (42) with respect to gives

 c1∂tvr1+d1∂tρ1=∂tΨ1, (45)

where

 c1=√−~gH0ρ0andd1=√−~gH0vr0. (46)

With these two equation given by Eq. (43) and (45) we can write and solely in terms of derivatives of as

 ∂tvr1=1Δ1[b1∂tΨ1−d1∂rΨ1],∂tρ1=1Δ1[−a1∂tΨ1+c1∂rΨ1] (47)

with where is given by

 ~Λ=grr(vr0)2Λvt0−vt0+c2s0Λvt0[Λ(vt0)2−1−grr(vr0)2]. (48)

Now let us go back to the irrotationality condition given by the Eq. (11). Using and and dividing by gives the following equation

 grrvt∂tvr+c2sgrrvrρvt∂tρ−∂r(ln(ρc2svt))=0 (49)

Let us substitute the density and velocities using Eq. (25), (24) and

 vt(r,t)=vt0(r)+vt1(r,t). (50)

Keeping only the terms that are linear in the perturbed quantities and differentiating with respect to time gives

 ∂t[grrvt0∂tvr1]+∂t[grrc2svr0ρ0vt0∂tρ1]−∂r[1vt0∂tvt1]−∂r[c2sρ0∂tρ1]=0 (51)

We can also write

 ∂tvt1=~η1∂tρ1+~η2∂tvr1 (52)

with

 ~η1=−[Λη1+gϕtvϕ0c2sgϕϕρ0],~η2=−Λη2 (53)

using Eq. (52) the Eq. (51) can be written as

 ∂t[grrvt0∂tvr1]+∂t[grrc2svr0ρ0vt0∂tρ1]−∂r[~η2vt0∂tvr1]−∂r[(~η1vt0+c2sρ0)∂tρ1]=0 (54)

Finally substituting and in Eq. (54) using Eq. (47) we get

 ∂t[k2(r){−gtt+(vt0)2(1−1c2s)}∂tΨ1]+∂t[k2(r){vr0vt0(1−1c2s)}∂rΨ1] (55) +∂r[k2(r){vr0vt0(1−1c2s)}∂tΨ1]+∂r[k2(r){grr+(vr0)2(1−1c2s)}∂rΨ1]=0

where

 k2(r)=grrvr0c2svt0vt0~Λandgtt=1Λ=1gtt+g2ϕt/gϕϕ (56)

Eq. (55) can be written as where is given by the symmetric matrix

 fμν2=grrvr0c2svt0vt0~Λ⎡⎢⎣−gtt+(vt0)2(1−1c2s)vr0vt0(1−1c2s)vr0vt0(1−1c2s)grr+(vr0)2(1−1c2s)⎤⎥⎦ (57)

### 4.3 Linear perturbation of relativistic Bernoulli’s constant

Energy momentum conservation equation is given by Eq. (6). The energy momentum conservation equation can also be written as . Thus using we have

 vμ∂μvν−Γλμνvλvμ+c2sρ(vμvν∂μρ+∂νρ)=0 (58)

Therefore the temporal component of the relativistic Euler equation is given by

 vt∂tvt+vr∂rvt−Γλμtvλvμ+c2sρ(vtvt∂tρ+vrvt∂rρ+∂tρ)=0. (59)

It can be shown that . Thus the Eq. (59) can be written as

 vt∂tvt+c2sρ(vtvt+1)∂tρ+vrvt∂r{ln(vtρc2s)}=0 (60)

For stationary case where all derivatives with respect to vanish, the solution of the above equation gives the relativistic Bernoulli’s constant as which is a constant of motion for stationary flow.

Let us define a quantity such that it has the stationary value equal to . Thus

 ξ(r,t)=ξ0+ξ1(r,t) (61)

Now we use the perturbation equations given by Eq. (25), (24), (50) along with Eq. (61) in the Eq. (60) and retain only the terms that are linear in the perturbed quantities. This provides

 ξ0grrvr0vt0c2sρ0∂tρ1+ξ0grrvt0∂tvr1=∂rξ1 (62)

In deriving the above equation we have also used Eq. (52), (53) and (27). is given by

 ξ1(r,t)=c2sξ0ρ0ρ1(r,t)+ξ0vt0vt1 (63)

Differentiating both sides of the above equation with respect to and using Eq. (52) we get

 ξ0[c2sρ0+~η1vt0]∂tρ1+ξ0~η2vt0∂tvr1=∂tξ1 (64)

where and are given by Eq. (53). Eq. (62) and (64) gives and solely in terms of derivatives of

 ∂tvr1=1Δ2[b2∂tξ1−d2∂rξ1],∂tρ1=1Δ2[−a2∂tξ1+c2∂rξ1] (65)

where

 a2=ξ0grrvt0,b2=ξ0grrvr0vt0c2sρ0,c2=ξ0~η2vt0,d2=ξ0[c2sρ0+~η1vt0] (66)

and

 Δ2=b2c2−a2d2=ξ20grrc2sρ0v2t0vt0 (67)

The continuity equation is given by Eq. (13). Using the perturbation equation given by Eq. (25), (24) and retaining only the terms that are linear in perturbed quantities gives

 ∂t[√−~gH0(ρ0vt1+vt0ρ1)]+∂r[√−~gH0(ρ0vr1+vr0ρ1)]=0 (68)

Differentiating the above equation with respect to and using Eq. (22) gives

 ∂t[√−~gH0{(ρ0η2)∂tvr1+(ρ0η1+vt0)∂tρ1}]+∂r[√−~gH0(ρ0∂tvr1+vr0∂tρ1)]=0 (69)

Substituting and in the above equation using Eq. (65) provides

 ∂t[k3(r){−gtt+(vt0)2(1−1c2s)}∂tξ1]+∂t[k3(r){vr0vt0(1−1c2s)}∂rξ1] (70) +∂r[k3(r){vr0vt0(1−1c2s)}∂tξ1]+∂r[k3(r){grr+(vr0)2(1−1c2s)}∂rξ1]=0

with . The above equation can be written as where is given by the symmetric matrix

 fμν3=√−~gH0ρc2s−10⎡⎢⎣−gtt+(vt0)2(1−1c2s)vr0vt0(1−1c2s)vr0vt0(1−1c2s)grr+(vr0)2(1−1c2s)⎤⎥⎦ (71)

## 5 The acoustic metric

The linear perturbation of the velocity potential, mass accretion rate and the relativistic Bernoulli constant, performed in Sec. (4.1),(4.2) and (4.3) respectively, provides equations

 ∂μ(fμνi∂ν~ψi)=0,i=1,2,3;~ψi=(ψ1,Ψ1,ξ1) (72)

These equations could be compared to the wave equation of a massless scalar field propagating in a curved spacetime given by [66]

 ∂μ(√−ggμν∂νφ)=0 (73)

Thus the acoustic metric can be obtained from the relation[4]

 √−GiGμνi=fμνi,i=1,2,3 (74)

where is the determinant of the acoustic metric . The ’s for are same except the factors ’s. Thus the acoustic will also be the same up to a conformal factor equal to . The acoustic metric obtained by linear perturbing three different quantities differ by only this conformal factor. Thus we can expect the acoustic curvature or the line element of the acoustic spacetime metric to be different for different acoustic spacetime obtained by perturbing different quantities. However, our goal is to investigate the features of the acoustic spacetime which are common to all the acoustic spacetime metric irrespective of the quantity that we have perturbed to obtain the same. The location of the event horizon, causal structure of the spacetime or the surface gravity do not depend on the conformal factor of the spacetime metric. Therefore we are interested in studying these conformally invariant features of the acoustic spacetime. Thus in order to investigate these properties of the acoustic spacetime we could take the acoustic metric to be the same by ignoring the conformal factors. Hence the acoustic metrics and , ignoring the conformal factor, are given by

 Gμν=⎡⎢⎣−gtt+(vt0)2(1−1c2s)vr0vt0(1−1c2s)vr