Dependence of Acoustic Surface Gravity on Geometric Configuration of Matter for Axially Symmetric Background Flows in the Schwarzschild Metric
In black hole evaporation process, the mass of the hole anti-correlates with the Hawking temperature. This indicates that the smaller holes have higher surface gravity. For analogue Hawking effects, however, the acoustic surface gravity is determined by the local values of the dynamical velocity of the stationary background fluid flow and the speed of propagation of the characteristic perturbation embedded in the background fluid, as well as by their space derivatives evaluated along the direction normal to the acoustic horizon, respectively. The mass of the analogue system - whether classical or quantum - does not directly contribute to extremise the value of the associated acoustic surface gravity. For general relativistic axially symmetric background fluid flow in the Schwarzschild metric, we show that the initial boundary conditions describing such accretion influence the maximization scheme of the acoustic surface gravity and associated analogue temperature. Aforementioned background flow onto black holes can assume three distinct geometric configurations. Identical set of initial boundary conditions can lead to entirely different phase-space behavior of the stationary flow solutions, as well as the salient features of the associated relativistic acoustic geometry. This implies that it is imperative to investigate how the measure of the acoustic surface gravity corresponding to the accreting black holes gets influenced by the geometric configuration of the inflow described by various thermodynamic equations of state. Such investigation is useful to study the effect of Einstenian gravity on the non-conventional classical features as observed in Hawking like effect in a dispersive medium in the limit of a strong dispersion relation.
PACS numbers: 04.40.Dg, 04.70.Dy, 95.30.Sf
Black hole analogues are fluid dynamical analogue of the black hole space time as perceived in the general theory of relativity [1, 2, 3, 4, 5, 6, 7, 8]. Such analogue systems may be realized by studying the propagation of small amplitude linear perturbation through a dissipationless, irrotational, barotropic transonic fluid. Contemporary research in the field of analogue gravity phenomena has gained widespread currency since it opens up the possibility of understanding the salient features of the horizon related effects through experimentally realizable physical configurations within the laboratory set up.
Conventional works in this direction, however, concentrate on systems not directly subjected to the gravitational force. Gravity like effects are manifested as emergent phenomena. In such cases, only the Hawking like effects can be studied and no direct connection can be made to such effects with the general relativistic Hawking effects since such non gravitating systems do not include any source of strong gravity capable of producing the Hawking radiation.
To explore whether (and how) the emergent gravity phenomena may be observed in a physical system which itself is under the influence of a strong gravitational field, a series of recent works describe how the acoustic geometry may be realized for stationary, spherically and axially symmetric hydrodynamic flow onto astrophysical black holes [9, 10, 11, 12, 13, 14, 15, 16].
Accreting black holes represent systems which simultaneously contain gravitational as well as acoustic horizons and are shown to be natural examples of large scale classical analogue systems found in the universe. This allows us to study the influence of the original background black hole space time metric on the embedded perturbative acoustic metric.
Axisymmetric, general relativistic, low angular momentum, inviscid hydrodynamic accretion onto non-rotating astrophysical black holes can be studied for three different geometric configurations of matter – disc accretion with constant flow thickness (hereafter constant height flow), quasi-spherical accretion in conical configuration (hereafter conical flow), and for axisymmetric flow maintained in the hydrostatic equilibrium along the vertical direction (hereafter vertical equilibrium flow). Details about such geometric configurations can further be found in  and in section 4 of . For these three geometric configurations, the nature of the sonic geometry embedded within the infalling material has recently been studied for accretion processes under the influence of the generalized post-Newtonian pseudo-Schwarzschild potentials .
In our present work, we would like to extend such calculations on a more formal foundation. We shall study the properties of the sonic geometry for general relativistic axisymmetric accretion (resulting in the existence of a curved background geometry for the stationary fluid configuration) onto a Schwarzschild black hole for three different geometric configurations of non self gravitating matter and for each configuration, two different thermodynamic equations of state. We would like to understand how crucial is the role of the relativistic gravitation as well as the geometric configuration of the background stationary flow (subjected to that gravitational field) in determining the essential features of the analogue gravity phenomena. We thus intend to demonstrate how the estimation of the acoustic surface gravity gets influenced by the geometric configuration of matter for general relativistic background matter flow onto a Schwarzschild black hole.
For canonical Hawking effect in connection to a Schwarzschild black hole, the Hawking temperature ( being the mass of the Hawking radiating black hole). This indicates that one requires a black hole of reasonably small mass – e.g., a primordial black hole of cosmological origin – to maximize the observable Hawking effect. The extremisation of the observable Hawking effect can thus be parameterized by the mass of the black hole only. One can not have such a straight forward (anti) correlation available for the analogue temperature with the mass parameter of the system to conclude that the acoustic black hole of microscopic dimension will indeed produce a larger analogue temperature. Extremisation of such temperature as well as depends on various initial boundary conditions determining the background stationary states of the system under consideration.
2 Acoustic surface gravity () for accreting black hole systems
For a stationary flow configuration, the acoustic horizon is the surface defined by the equation
where is position dependent sound speed (the speed of propagation of the perturbation under consideration in general), the bulk flow velocity is measured along the direction normal to the acoustic horizon.
where space gradient is taken along the normal to the acoustic horizon. The subscript indicates that the quantities under consideration have been evaluated on acoustic horizon.
where is the Killing field which is null on the corresponding acoustic horizon. In subsequent sections, we will show that the explicit expression for the norm of can be evaluated in terms of the values of the background metric elements evaluated on the acoustic horizon and on certain flow parameters.
One needs to calculate the location of the acoustic horizon for a stationary configuration, as well as to evaluate the expression for the normal bulk flow velocity and the speed of the propagation of the acoustic perturbation along with their space gradients normal to the acoustic horizon to compute the value of .
For the background stationary accretion solutions considered in the present work, and (evaluated on ) is determined using the initial boundary conditions defined by the triad for the polytropic accretion and the diad for the isothermal accretion, where and are the specific total conserved energy, specific conserved angular momentum, the adiabatic index (, where and are specific heats at constant pressure and volume, respectively), and the bulk ion temperature of the accreting matter, respectively. Extremisation of , as will be shown in the subsequent sections, nonlinearly depends on and on for the adiabatic and the isothermal flows, respectively. One thus needs to explore the three dimensional parameter space spanned by and the two dimensional parameter space spanned by to apprehend what values of the initial boundary conditions are favoured for the extremisation of . This might help to enhance the possibility of obtaining the observable signature of the analogue radiation. One also needs to understand which one out of these three flow configurations favours the production of reasonably large value of .
This enables one to provide a ‘calibration space’ spanned by various astrophysically relevant parameters governing the flow, for which the extremisation of can be performed. This also provides a comprehensive idea about the influence of the geometric configuration of the black hole accretion flow on the extremisation process of .
At this point, it is important to clarify that the present work does not make any attempt to understand the thermal properties of the Hawking like effects. We do not intend to analyze the analogue radiation – it’s origin, propagation and observational manifestation. We rather concentrate to explore the underlying sonic geometry through the detailed study of the dependence of on various factors governing the stationary background configuration. This does not involve the detailed analysis of quantum acoustic Hawking process at least at this stage. We do not deal with the quantization process of the associated phonon field. To accomplish that task, one needs to demonstrate that the effective action for the acoustic perturbation is equivalent to a field theoretic action in curved space, and the associated commutation relation as well as the dispersion relation will directly follow [2, 8]. Such considerations are rather involved and is clearly beyond the scope of our present work. Our main motivation is rather to employ the analogy to describe the classical perturbation of the fluid flow in terms of a field satisfying the wave equation in an effective geometry and to study the consequences relevant to a large-scale gravitating system. We believe that itself is a rather important entity to understand the flow structure as well as the associated sonic metric, irrespective of the existence of quantum Hawking like phenomena characterized by their feeble temperature too difficult to detect experimentally.
The significant role of in influencing the non negligible classical effects associated with the emergence of the stimulated Hawking effects through the modified dispersion relations at the sonic horizon has recently been emphasized from the theoretical front [23, 24], as well as within the experimental framework in the laboratory set up [19, 20, 21, 22]. The deviation of the Hawking like effects in a dispersive medium from the original Hawking effect is sensitive to the spatial velocity gradient corresponding to the stationary solutions of the background fluid flow [23, 24]. For relativistic accretion onto a Schwarzschild black hole, the expression for is found to be an analytical function of the space gradient of the steady state bulk velocity of the background fluid. Such velocity gradient influences the universality of the Hawking like radiation (as well as the departure from it), and various other properties of the anomalous scattering of the acoustic mode due to the modified dispersion relation at the acoustic horizon.
One of the main importances of our work is to identify a natural large-scale gravitating relativistic system where there is a probability to estimate the aforementioned deviation. For such a system the exact value of the space gradient of the flow velocity can explicitly be computed in terms of realistic, observationally measurable, astrophysically relevant physical entities. It is surely a step ahead of some abstract theoretical calculation as we believe. Existing works which study the anomalous dispersion relation, consider the gradient of the bulk velocity only and the space gradient of the sound speed is not taken into account in any such literature. For adiabatic flow, the speed of propagation of the linear perturbation embedded within the fluid is a position dependent quantity, the role of the gradient of the sonic velocity in influencing the estimation of the deviation of the Hawking like effect from universality cannot be underestimated. In our work we calculate the space gradient of the sonic velocity in terms of the observationally obtainable physical quantities and include such factors in the calculation of . Our work can contribute to enrich the formalism as presented in [23, 24] in a more realistic way.
In what follows, we describe our overall scheme for the computation of in terms of various flow geometries and for different thermodynamic equations of state.
Hereafter, any relevant distance will be scaled in units of and any velocity will be scaled by the velocity of light in vacuum, , where represents the mass of the black hole and represents the universal gravitational constant.
For adiabatic accretion, the equation of state of the form
is considered to describe the flow. is assumed to be constant throughout the flow in the steady state. A more realistic flow model, however, perhaps requires the implementation of a non constant polytropic index having a functional dependence on the radial distance of the form [41, 42, 43, 44, 45]. We, nevertheless, have performed our calculations for a reasonably wide spectrum of and thus believe that the whole astrophysically relevant range of polytropic indices is covered in our analysis. The proportionality constant in eq. (4) is a measure of the specific entropy of the accreting fluid provided no additional entropy generation takes place.
Isothermal accretion is assumed to be described by the following equation of state
and are the universal gas constant, the Boltzmann constant, the isothermal flow temperature, the reduced mass and the mass of the Hydrogen atom, respectively. in the above equation represents the position independent isothermal sound speed which implies that identically. For isothermal accretion, only the space gradient of the bulk advective velocity, and not that of the speed of propagation of the acoustic perturbation, contributes to the estimation of the acoustic surface gravity.
For energy momentum tensor corresponding to an ideal fluid considered in a Boyer-Lindquist  line element for a non rotating black hole, we demonstrate (see subsequent sections for detailed derivation) that can be expressed as
In subsequent sections, we provide the expressions for for three different flow configurations for the adiabatic as well as the isothermal (for which will vanish everywhere, including at the acoustic horizon, for obvious reason) equation of state. We use those values to study the dependence of on various flow parameters as well as on various flow geometries.
3 Overall solution scheme
We consider low angular momentum axially symmetric accretion and the viscous transport of angular momentum has not been taken into account. Such low angular momentum inviscid flow is not a theoretical abstraction. For astrophysical systems, such sub-Keplerian weakly rotating flows are exhibited in various physical situations, such as detached binary systems fed by accretion from OB stellar winds [25, 27], semi-detached low-mass non-magnetic binaries , and super-massive black holes fed by accretion from slowly rotating central stellar clusters ([29, 30] and references therein). Even for a standard Keplerian accretion disc, turbulence may produce such low angular momentum flow (see, e.g., , and references therein). Reasonably large radial advective velocity for the slowly rotating sub-Keplerian flow implies that the infall time scale is considerably small compared to the viscous time scale for the flow profile considered in this work. Large radial velocities even at larger distances are due to the fact that the angular momentum content of the accreting fluid is relatively low [32, 33, 34]. The assumption of inviscid flow for the accretion profile under consideration is thus justified from an astrophysical point of view. Such inviscid configuration has also been addressed by other authors using detailed numerical simulation works [34, 35]. The general relativistic Euler and the continuity equations are thus obtained using the vanishing of the four divergence of the energy momentum tensor of an ideal fluid.
Considering the flow to be steady and the steady state to be a stable state111One can perform a linear stability analysis to ensure that the steady axially symmetric inviscid flow is stable, see ., the time independent Euler and the continuity equations will then be integrated to obtain the respective integrals of motion, since the time independent Euler and the continuity equations are examples of first order ordinary homogeneous differential equations in advective velocity. The integral solution of the Euler equation provides the conserved total specific energy (denoted by in this work) as the first integral of motion for polytropic accretion. For isothermal flow, the corresponding first integral of motion (denoted by in this work) cannot be identified with the specific energy of the flow since energy exchange with the surrounding is required to maintain the space invariance of the bulk temperature. The first integral of motion obtained from the Euler equation does not depend on the geometric configuration of the flow.
The equation of continuity implies the conservation of mass and hence its integral solution will provide the mass accretion rate (denoted by in this work) as another first integral of the motion. Explicit expression for the mass accretion rate may not depend on the equation of state used and is found to be a function of the flow thickness. explicitly depends on the flow geometry. For polytropic accretion, we will have three different expressions for for different flow geometries and for the isothermal accretion will have same set of expressions for for flow with constant thickness and conical flow, but the explicit expression will be different for accretion in the vertical equilibrium. This is due to the fact that for accretion in vertical equilibrium the expression for the flow thickness comes out to be a function of the corresponding sound speed. We solve six different cases in this work, three different flow models for a particular energy first integral for the polytropic flow as well as for the first integral corresponding to the isothermal flow.
Once the first integrals of motion are obtained, we find the space gradient of the dynamical velocity and that of the sonic velocity (for isothermal flow is position independent) and perform the critical point analysis to find out the critical point(s) of the flow. For all flow models other than the flow in vertical equilibrium, the critical points coincide with the sonic points. For flow in vertical equilibrium, critical surfaces are not isomorphic with the sonic surfaces and the integral flow solutions are to be used to find the sonic point (location of the acoustic horizon) by integrating the flow starting from the corresponding saddle type critical points. We study the dependence of on and on for the polytropic as well as for the isothermal accretion, respectively. We compare such dependence for three different flow profiles.
Hereafter, the subscripts CH, CF and VE will indicate that the quantities are evaluated/expressions are formulated for flow with constant height (CH), for quasi spherical conical flow (CF), and for flow in hydrostatic equilibrium along the vertical direction (VE), respectively.
4 Configuration of the background fluid flow
We consider a (3+1) stationary axisymmetric space-time endowed with two commuting Killing fields, within which the dynamics of the background fluid will be studied. For the energy momentum tensor of any ideal fluid with certain equation of state, the combined equation of motion in such a configuration can be expressed as
being the velocity vector field defined on the manifold constructed by the family of streamlines. The normalization condition for such velocity field yields , and is the speed of propagation of the acoustic perturbation embedded inside the bulk flow. is the local rest mass energy density. The local timelike Killing fields and are the generators of the stationarity (constant specific flow energy is the outcome) and axial symmetry, respectively.
In general, the acoustic ergosphere and the acoustic event horizon do not co-incide. However, for a radial flow onto a sink placed at the origin of a stationary axisymmetric geometry they do (see, e.g., [4, 11] for detail discussion), since only the radial component of the flow velocity remains non zero everywhere. In this work we consider accretion flow with radial advective velocity confined on the equatorial plane. The flow will be assumed to have finite radial spatial velocity (the advective flow velocity as designated in usual astrophysics literature [37, 38]) defined on the equatorial plane of the axisymmetric matter configuration. We focus on stationary solutions of the fluid dynamic equations (to determine the stationary background geometry) and hence consider only the spatial part of such advective velocity. Considering to be the magnitude of the three velocity, is the component of three velocity perpendicular to the set of timelike hypersurfaces defined by .
The local radial Mach number of the accreting fluid is defined as the ratio of the radial component of the local dynamical flow velocity to that of the propagation of the acoustic perturbation embedded inside the accreting matter – . The flow will be locally subsonic or supersonic according to or . The flow is transonic if at any moment it crosses the hypersurface. This happens when a subsonic to supersonic or supersonic to subsonic transition takes place either continuously or discontinuously. Such a point where such crossing takes place continuously is called a sonic point, and where such transition takes place discontinuously is called a shock or a discontinuity. The particular value of the radial distance for which , is referred as the transonic point or the sonic point, and will be denoted by hereafter. and is thus identical for a transonic system. For , infalling matter becomes supersonic. Any acoustic perturbation created in such a region is destined to be dragged towards the black hole, and can not escape to the domain . In other words, any co-moving observer from region can not communicate with any observer (co-moving or stationary) located in the sub-domain by sending any signal which travels with velocity , where is defined as the velocity of propagation of the acoustic perturbation (the sound speed) embedded in the moving fluid. Hence the hypersurface through is generated by the acoustic null geodesics, i.e., by the phonon trajectories, and is actually an acoustic horizon for stationary configuration, which is produced when accreting fluid makes a transition from subsonic () to the supersonic () state.
At a distance far away from the black hole, accreting material almost always remains subsonic (except possibly for the supersonic stellar wind fed accretion) since it possesses negligible dynamical flow velocity. On the other hand, the flow velocity will approach the velocity of light while crossing the event horizon, while the maximum possible value of sound speed, even for the steepest possible equation of state, would be [37, 38], resulting close to the event horizon. In order to satisfy such inner boundary condition imposed by the event horizon, accretion onto black holes exhibit transonic properties in general .
For a transonic flow as perceived within the aforementioned configuration, the collection of the sonic points (where the radial Mach number, the ratio of the advective velocity and the speed of propagation of the acoustic perturbation in the radial direction, becomes unity) at a specified radial distance forms the acoustic horizon, the generators of which are the phonon trajectories. An axially symmetric transonic black hole accretion can thus be considered as a natural example of the classical analogue gravity model which contains two different horizons, the gravitational (corresponding to the accreting black hole) as well as the acoustic (corresponding to the transonic fluid flow).
To describe the flow structure in further detail, the energy momentum tensor of an ideal fluid of the form
being the Kerr parameter related to the black holes spin angular momentum. The required metric elements are:
The specific angular momentum (angular momentum per unit mass) and the angular velocity can thus be expressed as
We also define
which will be used in the subsequent sections to calculate the value of the acoustic surface gravity.
For flow onto a Schwarzschild black hole, one can obtain the respective metric elements (and hence, the expression for and thereof) by substituting in eq. (10 - 13). We construct a Killing vector where the Killing vectors and are the two generators of the temporal and axial isometry groups, respectively. Once is computed at the acoustic horizon , becomes null on the transonic surface. The norm of the Killing vector may be computed as
Hence the explicit form of the acoustic surface gravity for relativistic flow onto a Schwarzschild black hole looks like
5 The first integrals of motion
Vanishing of the four divergence of the energy momentum tensor provides the general relativistic version of the Euler equation
whereas the corresponding continuity equation is obtained from
The time independent part of the linear momentum conservation equation (Euler equation) is a first order homogeneous differential equation. Its integral solution will provide a constant of motion (first integral of motion) for whatever equation of state is used to describe the accreting matter. Such first integral of motion, however, cannot formally be identified with the total energy of the background fluid flow for any equation of state other than the polytropic one.
5.1 Integral solution of the linear momentum conservation equation
5.1.1 Polytropic accretion
For polytropic accretion, the specific enthalpy is formulated as
where the energy density includes the rest mass density and internal energy and is defined as
The adiabatic sound speed is defined as
At constant entropy, the enthalpy can be expressed as
Contracting eq. (16) with one obtains (since , and )
Since , the angular momentum per baryon for the axisymmetry flow is conserved. Contraction of eq. (16) with provides
from where the quantity comes out as one of the first integrals of motion of the system. is actually the relativistic version of the Bernouli’s constant  and can be identified with the total specific energy of the general relativistic ideal fluid (see, e.g.,  and references therein) scaled in units of the rest mass energy. Hence
From the normalization condition one obtains
We thus obtain
The exact form for will depend on the space time structure appearing in the expression for through the metric elements. It will not depend on the matter geometry since the accretion is assumed to be non self gravitating. For isothermal flow, the total specific flow energy does not remain constant, rather the first integral of motion obtained by integrating the relativistic Euler equation has a different algebraic form which can not be identified with the total energy of the system.
5.1.2 Isothermal flow
For isothermal flow, the system has to dissipate energy to keep the temperature constant. The isotropic pressure is proportional to the energy density through
From the time part of eq. (16), one obtains
Using the definition of enthalpy, the above equation may be re-written as
Since the isothermal sound speed can be defined as (see, e.g.,  and references therein)
Which further implies that
Hence is the first integral of motion for the isothermal flow, which is not to be confused with the total conserved specific energy .
space invariance of temperature requires the sound speed to be position independent for isothermal accretion, and hence the space gradient of the speed of propagation of the isothermal perturbation does not contribute to the estimation of the acoustic surface gravity. The expression for the first integral of motion obtained from the integral solution of the Euler equation is independent of the geometrical configuration of matter as already discussed, and is found to be
5.2 Integral solution of the mass conservation equation
For , the mass conservation equation (17) implies
which further leads to
being the co-variant volume element. We assume that there is no convection current along any non equatorial direction, and hence no non-zero terms involving (for spherical polar co-ordinate) or (for flow studied within the framework of cylindrical co-ordinate) should become significant. This assumption leads to the condition
for the stationary background flow studied using the spherical polar co-ordinate and
for such flow studied using the cylindrical co-ordinate .
We integrate eq. (37) for and ; being the value of the polar co-ordinates above and below the equatorial plane, respectively, for a local flow half thickness , to obtain the conserved mass accretion rate in the equatorial plane. The integral solution of the mass conservation equation – the mass accretion rate – comes out to be another first integral of motion for our stationary background fluid configuration. For conical wedge shaped flow studied in the spherical polar co ordinate, remains constant. Flow with such geometric configuration was first studied by  and followed by  for pseudo-Schwarzschild flow geometry under the influence of the Paczyński & Wiita  pseudo-Schwarzschild Newtonian like black hole potential. The relativistic version for such flow has further been studied by [53, 54, 55, 56, 57, 58, 59, 49, 60, 61].
In a similar fashion, eq. (38) can be integrated for (where is the local half thickness of the flow) symmetrically over and below the equatorial plane for axisymmetric accretion studied using the cylindrical polar co-ordinate to obtain the corresponding mass accretion rate on the equatorial plane. Contrary to the first integral of motion obtained by integrating the Euler equation for a particular thermodynamic equation of state, the expression for the mass accretion rate does not explicitly depend on the equation of state, but is different for different geometric configuration of the matter distribution. The general expression for the mass accretion rate can be provided as
being the two dimensional surface area having surface topology or through which the inward mass flux is estimated in the steady state. For (and for not so large value of ), , and for (axisymmetric accretion studied using the cylindrical co ordinate), .
In subsequent sections, we provide the explicit expressions for the conserved mass accretion rate and related quantities for three different flow geometries
6 Stationary transonic accretion solutions
6.1 polytropic accretion
We substitute the values of the corresponding metric elements and of in eq. (26) and obtain
In what follows, we shall illustrate the procedure to obtain the stationary transonic flow solutions for flow with constant thickness. We shall then provide the corresponding similar expressions for other flow geometries.
6.1.1 Flow with constant thickness
As stated in the paragraphs preceding eq. (39), we integrate the continuity equation to obtain the conserved mass accretion rate to be
being the constant disc height.
Equations (40 – 41) can not directly be solved simultaneously since it contains three unknown variables and , all of which are functions of the radial distance . Any accretion variable from the triad has to be eliminated in terms of the other two. We are, however, interested to study the radial Mach number profile to identify the location of the acoustic horizon (the radial distance at which becomes unity), and hence the study of the radial variation of and are of prime interest in this case. We would thus like to express in terms of and other related constant quantities. To accomplish the aforementioned task, we make a transformation . Employing the definition of the sound speed as well as the equation of state used to describe the flow, the expression for can further be elaborated as
The entropy per particle is related to and as 
where the constant depends on the chemical composition of the accreting material. The above equation implies that is a measure of the specific entropy of the accreting matter. We thus interpret as the measure of the total inward entropy flux associated with the accreting material and label to be the stationary entropy accretion rate. The concept of the entropy accretion rate was first introduced in [50, 51] to obtain the stationary transonic solutions of the low angular momentum non relativistic axisymmetric accretion under the influence of the Paczyński and Wiita  pseudo-Schwarzschild potential onto a non rotating black hole.
The conservation equations for and may simultaneously be solved to obtain the complete accretion profile on the radial Mach number vs radial distance phase space, see, e.g., [12, 48] for the depiction of several such phase portraits.
The relationship between the space gradient of the acoustic velocity and that of the advective velocity can now be established by differentiating eq. (42)
Differentiation of eq. (40) with respect to the radial distance provides another relation between and . We substitute as obtained from eq. (43) into that relation and finally obtain the expression for the space gradient of the advective velocity as
|We define another quantity which shall be used later,|
Eq. (43 – 44) can now be identified with a set of non-linear first order differential equations representing autonomous dynamical systems , and their integral solutions provide phase trajectories on the radial Mach number vs the radial distance plane. The ‘regular’ critical point conditions for these integral solutions are obtained by simultaneously making the numerator and the denominator of eq. (44) vanish. The aforementioned critical point conditions may thus be expressed as
Since in this work we deal with a transonic fluid in real space for which the flow is continuous along the entire real line, only the ‘regular’ or ’smooth’ critical point is considered, for which as well as their space derivatives remain regular and do not diverge. Such a critical point may be of saddle type allowing a transonic solution to pass through it, or may be of centre type through which no physical transonic solution can be constructed. Other categories of critical point include a ‘singular’ one for which are continuous but their derivatives diverge. All such classifications have been discussed in detail and the criteria for a critical point to qualify as a ‘regular’ one which is associated with a physical acoustic horizon has been found out in .
Equation (46) provides the critical point condition but not the location of the critical point(s). It is necessary to solve eq. (40) under the critical point condition for a set of initial boundary conditions as defined by . The value of and , as obtained from eq. (46), may be substituted in eq. (40) to obtain the following 11 degree algebraic polynomial equation for , being the location of the critical point
where, the coefficients are functions of . The explicit form of such co-efficients are derived, and the results are presented in the appendices.
A particular set of values of will then provide the numerical solution for the algebraic expression to obtain the exact value of . Astrophysically relevant domain for such initial boundary conditions are  defined by
For accretion with constant height, the critical point condition reveals that the advective velocity and the sound velocity are same at the critical point. Hence the critical surface at and the acoustic horizon coincide for this flow geometry.
For an astrophysically relevant set of the critical point(s) of the phase trajectory can be identified, and a linearisation study in the neighbourhood of these critical points(s) may be performed  to develop a classification scheme to identify the nature of the critical point(s). Since viscous transport of angular momentum has not been taken into account in the present work, such critical points are either of saddle type through which a stationary transonic flow solution can be constructed, or of a centre type which does not allow any transonic solution on phase portrait to pass through it. A complete understanding of the background stationary transonic flow topologies on the phase portrait will require a numerical integration of the non analytically solvable non linearly coupled differential equations describing the space gradient of the advective velocity as well as that of the speed of propagation of the acoustic perturbation embedded within the stationary axisymmetric background spacetime.
For a particular set of , solution of eq. (47) provides either no real positive root lying outside the gravitational black hole horizon implying that no acoustic horizon forms outside the black hole event horizon (non availability of the transonic solution) for that value of , or provides one, two or three (at most) real positive roots lying outside the black hole event horizon. Typically, if only one root is found, the critical point is of saddle type and a mono-transonic flow profile is obtained with a single acoustic horizon for obvious reason. Solutions containing two critical saddle points imply the presence of a homoclinic orbit222A homoclinic orbit or a homoclinic connection is a bi-asymptotic trajectory converging to a saddle like orbit as time goes to positive or negative infinity. For our stationary systems, a homoclinic orbit on a phase portrait is realized as an integral solution that re-connects a saddle type critical point to itself and embarrasses the corresponding centre type critical point. For a detailed description of such phase trajectories from a dynamical systems point of view, see, e.g., [69, 70, 71]. on the phase plot and hence such solutions are excluded.
Stationary configuration with three critical points requires a somewhat detailed understanding. Although a full description is available in , we provide a brief account over here for the sake of completeness. One out of the aforementioned three critical points is of centre type which is circumscribed by two saddle type critical points, see, e.g.,  for details about the characteristic features of the saddle and the center type critical points. With reference to the gravitational horizon, one of these saddle points forms sufficiently close to it, even closer than the innermost circular stable orbit. i.e., ISCO (see, e.g.  and  for details about ISCO) in general, and is termed as the inner type critical point. The other saddle type point, termed as the outer saddle type critical point, is usually formed at a fairly large distance away from the gravitational horizon. The inner critical point thus forms in a region of substantially strong gravitational field whereas the outer type critical point, in many cases, is formed in a region of asymptotically flat spacetime. This is because, depending on the choice of , such a critical point can be located at a distance (or even more) away from the gravitational horizon. The centre type critical point, termed as the middle critical point because of the fact that , forms usually at a length scale ranging from 10 to 10 in units of , depending on the value of used. thus provides the multi-critical behaviour of stationary transonic solution. The parameter space spanned by can further be classified into two different subspaces for which the representative phase portraits are topologically different. Such subspaces are characterized by the relative values of the stationary entropy accretion rate evaluated at the inner and the outer critical points, respectively. For , accretion can have three allowed critical points, and a homoclinic orbit is generated through the inner saddle type critical point, whereas for transonic accretion can have only one saddle type (inner) critical point and the homoclinic orbit forms through the outer saddle type critical point. Hence only provides the multi-critical accretion configuration for which two saddle type and one center type (delimited between two such saddle types) critical points are available. As already mentioned, a physically acceptable transonic solution for inviscid accretion cannot be constructed through a centre type critical point. A multi-critical flow with three critical points is thus a theoretical abstraction.
On the other hand, a bi-transonic accretion is a practically realizable configuration where the stationary transonic accretion solution passes through one inner and one outer saddle type sonic points. For flow geometries providing the isomorphism between the critical and the sonic points, such sonic points define the acoustic horizons. For flow configuration which does not allow such isomorphism, a sonic point can be identified on the integral stationary flow solutions corresponding to every saddle type critical point. For a bi-transonic solution, however, it should indeed be realized that a smooth stationary solution can not encounter more than one regular sonic point since once it crosses the outer type sonic point (for accretion) it becomes supersonic and only a subsonic solution can have access to pass through the inner sonic point. No continuous transonic solution can accommodate more than one acoustic horizons. Multi transonicity could only be realized as a specific flow configuration where the combination of two different otherwise smooth solutions passing through two different saddle type critical (and hence sonic) points are connected to each other through a discontinuous shock transition. Such a shock has to be stationary and will be located between two sonic points. For certain where ‘nss’ stands for no shock solution, three critical points (two saddle embracing a centre one) are routinely obtained but no stationary shock forms for the stationary transonic accretion. Hence no multi transonicity is observed even if the flow is multi-critical, and real physical accretion solution can have access to only one saddle type critical point (the outer one) out of the two. Thus multi critical accretion and multi transonic accretion are not topologically isomorphic in general. A true multi-transonic flow can only be realized for where ‘ss’ stands for ‘shock solution’, if the criteria for the energy preserving relativistic Rankine-Hugoniot shock [72, 73, 74, 75, 76] for the adiabatic accretion and temperature preserving relativistic shock [77, 49] for the isothermal accretion are met. In this work, however, we will not be interested to deal with the shock solutions and would mainly concentrate on the mono-transonic flow to study the accretion model dependence of the acoustic surface gravity . Further details will be provided in subsequent paragraphs where we describe the methodology of constructing the Mach number vs radial distance (measured from the gravitational horizon in units of ) phase portrait, see, e.g., figure 2 of  and related discussions may be found therein for details of such multi-transonic shocked accretion flow configurations.
Space gradient of the advective velocity at the critical points for various flow configurations can be obtained by evaluating the limiting values of at such points using the l’Hôpital’s rule . The expressions for the critical values of and have been provided in the appendix.
The critical acoustic velocity gradient can also be computed by substituting the value of in eq. (43) and by evaluating other quantities in eq. (43) at . Both and can be reduced to an algebraic expression in with real coefficients that are complicated functions of . Once is known for a set of values of , the critical slope for the advective velocity, i.e., the space gradient for at can be computed as a pure number, which may either be a real number providing a saddle type point (for stationary transonic accretion solution to exist) or an imaginary number for providing a centre type point (no transonic solution can be found).
To obtain the Mach number vs radial distance phase plot for the stationary transonic accretion flow, one needs to simultaneously integrate the set of coupled differential equations (43 – 44) for a specific set of initial boundary conditions determined by . The initial value of the space gradient of the advective velocity, i.e., the critical velocity gradient evaluated at the critical point and provided in eq. (89) and the critical space gradient of the sound speed can be numerically iterated using the fourth order Runge - Kutta method  to obtain the integral solutions for the mono-transonic as well as for the multi-transonic flow. Details of such numerical integration scheme, along with the representative phase plots are available in [48, 12, 16]. Acoustic surface gravity is not relevant for a centre type critical point since no stationary transonic solution can be constructed through such points. For flow with constant thickness and for conical flow – both for the adiabatic as well as the isothermal accretion – saddle type critical points and the sonic points are isomorphic. Since the critical surface and the acoustic horizon is identical, numerical construction of the integral stationary solution is not required to calculate the corresponding acoustic surface gravity for these flow geometries, and value of is all we need to calculate the value of for the respective flow configuration. In such cases, for axisymmetric flow with constant height for example, the surface gravity can be computed as
For accretion in hydrostatic equilibrium along the vertical direction, critical points and the sonic points are not isomorphic. As a result, the acoustic horizon does not form on the critical surface. The location of the sonic point will always be located at a radial distance (hereafter we will designate a sonic point as instead of ). Such is to be found out by integrating the expression of and and by locating the radial co ordinate on the equatorial plane for which the Mach number becomes exactly equal to unity. Since the condition is satisfied at and not at , is to be used to calculate the corresponding value of the acoustic surface gravity instead of for those particular flow geometries.
It is relevant to note that the absolute value of the (constant) disc thickness does not enter anywhere in the expression of the acoustic surface gravity (and hence, into the calculation of the Hawking like temperature). Similar result is to be obtained for the conical flow where the geometrical factor (the solid angle) representing the angular opening of the conical flow does not show up in the expression for or as well. This implies that it is only the geometrical configuration of the matter (non self gravitating) and not the absolute measure of the flow thickness (for constant height flow) or the ratio of the local height to the local radial distance (for conical flow) which influences the computation of the acoustic surface gravity. This may not be the situation where the radius dependent flow thickness itself is found to be a function of the speed of propagation of acoustic perturbation.
6.1.2 Conical flow model
The mass accretion rate is calculated as
is the geometric factor determining the exact shape of the flow, over which integration of the continuity equation is performed.
The corresponding entropy accretion rate is
The relationship between the corresponding space gradient of the speed of propagation of the acoustic perturbation (the space gradient of the adiabatic sound speed) and that of the stationary advective velocity can be found as
The explicit expression for the velocity space gradient comes out to be
The critical point conditions are calculated as
To compute the numerical value(s) of the critical point(s), one needs to substitute the critical point condition into the expression for to obtain an algebraic polynomial of the form , which is to be solved for for initial boundary conditions described by the astrophysically relevant values of . For conical flow, provides a polynomial in of eleventh degree. The explicit expression for such polynomial is provided in the appendix.
Polynomials of degree higher than four can not be solved analytically. However, the number of roots of such equations lying between infinity and the event horizons can be estimated analytically using the generalized Sturm sequence algorithm .
The expressions for the critical values of and have been provided in the appendix.
The expression for the acoustic surface gravity can be obtained as
6.1.3 Flow in hydrostatic equilibrium along the vertical direction
The radius dependent disc height is calculated as
and the corresponding mass accretion rate comes out to be
The entropy accretion rate is
The relationship between and is
The explicit expression for thus comes out to be
The critical point condition
indicates that critical surfaces are not the acoustic horizons for flow in vertical equilibrium since the value of the Mach number at the critical point is found to be
Unlike other flow models, to locate the radius of the acoustic horizon for flow in vertical equilibrium, one needs to integrate the flow equations from the critical point upto the radial distance where Mach number becomes unity.
The location of the critical point can be obtained by solving an degree polynomial equation, explicit expression of which is provided in the appendix. The expression for critical velocity gradients are also provided in the appendix.
The acoustic surface gravity can be calculated as
where is the sonic point which is to be obtained by integrating the flow equations from the corresponding critical points, and
6.2 Isothermal accretion
As is understood, the expression for the mass accretion rate for constant height and conical flow model for isothermal accretion will exactly be the same as those obtained for the adiabatic flow since the flow thickness does not depend on . For accretion in vertical equilibrium, however, the flow thickness will be different for two different equation of states and hence the corresponding mass accretion rate for the isothermal flow will also be different from its polytropic counterpart
The corresponding (advective) velocity (space) gradients are obtained as