Study of relativistic accretion flow in Kerr-Taub-NUT spacetime
Indu K. Dihingia, Debaprasad Maity, Sayan Chakrabarti, Santabrata Das
aa r X i v : . [ a s t r o - ph . H E ] J un Study of relativistic accretion flow in Kerr-Taub-NUT spacetime
Indu K. Dihingia , , ∗ Debaprasad Maity , † Sayan Chakrabarti , ‡ and Santabrata Das § Department of Physics, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, India Discipline of Astronomy, Astrophysics and Space Engineering,Indian Institute of Technology Indore, Indore 453552, India (Dated: June 17, 2020)We study the properties of the relativistic, steady, axisymmetric, low angular momentum, invis-cid, advective, geometrically thin accretion flow in a Kerr-Taub-NUT (KTN) spacetime which ischaracterized by the Kerr parameter ( a k ) and NUT parameter ( n ). Depending on a k and n values,KTN spacetime represents either a black or a naked singularity. We solve the governing equationsthat describe the relativistic accretion flow in KTN spacetime and obtain all possible global tran-sonic accretion solutions around KTN black hole in terms of the energy ( E ) and angular momentum( λ ) of the flow. We identify the region of the parameter space in λ − E plane that admits the flow topossess multiple critical points for KTN black hole. We examine the modification of the parameterspace due to a k and n and find that the role of a k and n in determining the parameter space isopposite to each other. This clearly indicates that the NUT parameter n effectively mitigate theeffect of black hole rotation in deciding the accretion flow structure. Further, we calculate the discluminosity ( L ) corresponding to the accretion solutions around the KTN black hole and for a givenset of a k and n , we obtain the maximum luminosity ( L max ) by freely varying λ and E . We observethat L max decreases with the increase of n irrespective of a k . In addition, we also investigate allpossible flow topologies around the naked singularity and find that there exists a region around thenaked singularity which remains inaccessible to the flow. We study the critical point properties fornaked singularities and find that the flow possesses maximum of four critical points. Finally, weobtain the parameter space for multiple critical points for naked singularity and find that param-eter space is shrunk and shifted to lower λ and higher E side as a k is increased which ultimatelydisappears. PACS numbers: —————–
I. INTRODUCTION
The accretion process around the compact stars re-mains the subject of intense interest for last severaldecades in the astrophysical community. Understand-ing the electromagnetic properties of a large class of as-trophysical observations, particularly for the sources likequasars, active galactic nuclei and black hole X-ray bina-ries, the accretion of matter has been proved to be thepotentially possible physical mechanism till date [1–5].Generically, black holes are considered to be the cen-tral object which are essentially a very special class ofsolutions of the well known Einstein’s equation. Oneof the defining properties of black holes is the existenceof the horizons which is a surface that encompasses thecurvature singularity in a black hole spacetime and thehorizon behaves like a one way membrane through whichanything can enter but nothing can come out. This in-teresting property helps one to invoke unique boundarycondition for the accretion flow dynamics near the hori-zon. The underlying framework of studying the accretionof matter is based upon the principles of relativistic hy-drodynamics in gravitational background. Once the flow ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] properties, such as velocity, temperature, density etc. areunderstood, the relevant radiative processes can be com-puted and compared with the observation. Therefore, inprinciple, one can put constraints on the underlying the-oretical parameters, such as the mass accretion rate aswell as the mass and spin of the black hole.Vast amount of literature exists on the topic of accre-tion flow which are based on different physical conditionsin the hydrodynamic regime (see [5, 6] and the refer-ences therein). However, limited works involving hydro-dynamical aspects of accretion flow have been performedin the realm of modified gravitational backgrounds aswell as around the exotic compact objects in general rel-ativity. For instance, the accretion flows around brane-world black holes [7, 8], slowly rotating black holes in dy-namical Chern-Simons modified gravity [9], black holesin Hoˇrava gravity [10, 11], boson stars [12, 13], wormholes[14], gravastars [15], and quark stars [16], etc., were stud-ied considering particle dynamics. In addition, accretiondisc properties have been studied around naked singular-ities [17–19] as well.It may be noted that the astrophysical observationsare generally explained considering various backgroundgravitational systems which may comprise of black holeor any other exotic compact objects. Therefore, giventhe advent of high precession observations these days,it would be viable to probe the nature of gravitationalbackground through the study of the accretion flow dy-namics, and this is the main motivation of our presentstudy. Towards this, for the first time to the best of ourknowledge, we study the properties of the general rela-tivistic accretion flow around the general class of gravi-tational backgrounds which are the solutions of vacuumEinstein’s equation. Here, we emphasize that depend-ing upon the choice of parameters of the theory, thoseclasses of gravitational backgrounds do represent eitherblack holes or more exotic spacetimes with naked singu-larity.Historically, for the first time, Taub [20] reported agravitational background which is presently known asthe Taub-NUT spacetime. The initial motivation to con-struct such a spacetime was based on the assumption ofthe existence of a four-dimensional group of isometriessuch that the spacetime can be interpreted as a possi-ble vacuum homogeneous cosmological model. There-after, the solution was rediscovered by Newman, Untiand Tamburino (NUT) [21] as a simple generalization ofthe Schwarzschild spacetime. To include rotation, theKerr-Taub-NUT (KTN) spacetime was formulated thatgeneralizes the well known Kerr metric by introducinga new parameter called the NUT charge. Needless tomention that Kerr spacetime is described by mass andKerr parameter ( a k , the spin angular momentum per unitmass) of the black hole whereas three parameters are re-quired to uniquely specify the KTN spacetime. Theseparameters are the mass, the Kerr parameter ( a k ) andthe NUT parameter ( n , also called NUT charge). In thelimit n →
0, the KTN spacetime reduces to the Kerrspacetime and if a k → n ) as a gravitomagnetic charge determines the gravito-magnetic properties of the Taub-NUT spacetime. Thenon-diagonal term implies a singularity on the θ = π axis, which is known as the Misner string. This type ofsingularity is completely different from the ordinary coor-dinate singularity. According to Misner [35], this singu-larity can be avoided by introducing a periodic time co-ordinate and two different coordinate patches. All theseinterpretations of the metric were abstract and theoreti-cal in nature. However, because of the availability of thestate-of-the-art astrophysical observation facilities now adays, nothing could be more appropriate than taking apractical approach towards understanding the nature ofthose kind of non-standard gravitational backgrounds.Motivated by this, in the present work, we study theaccretion phenomena in a special class of gravitationalbackground, namely the KTN spacetime. Since the ac-cretion of matter around the compact object is believedto be the driving mechanism behind most of the ener-getic phenomena in the universe [36], the study of theeffect of the NUT parameter on the accretion phenomenais expected to shed some light not only on the observa-tional aspects but also on the theoretical understandingof the Einstein’s theory itself. Keeping this goal in mind,in the present work, we consider an optically and geo-metrically thin accretion disc in the KTN background.In order to investigate the flow properties, we considerthe full general relativistic hydrodynamic framework [37]with relativistic equation of state (EoS) [38–41] . Withthese considerations, we perform the critical point anal-ysis and obtain the solutions in terms of energy ( E ) andangular momentum ( λ ) of the flow. Here, we choose a k and n values in such a way that the combination rep-resents KTN spacetime. We then study the role of theNUT parameter in deciding the nature of the accretionsolutions. Farther, we identify the parameter space inthe λ − E plane for multiple critical points and examinehow parameter space is modified with the increase of a k and n , respectively. Moreover, we calculate the disc lumi-nosities corresponding to the flow solutions characterizedwith a given set of ( a k , n ) and find the maximum lumi-nosity ( L max ) in KTN spacetime. Finally, we extend ourwork for naked singularity as well, where we obtain theparameter space for multiple critical points and examinehow the nature of the flow solution changes with a k and n values.We arrange the paper in the subsequent sections as fol-lows. In section II & III, we formulate the mathematicalbuilding blocks to study the accretion flow and discussthe critical point analysis. In section IV, we describethe methodology to calculate the flow solutions and ob-tain the parameter space for multiple critical points. Insections V, we present the results obtained for naked sin-gularity. Finally, in section VI, we present the discussionand conclusion of this work. II. KERR-TAUB-NUT (KTN) BACKGROUND
We consider a special class of axisymmetric vacuumsolution of Einstein’s theory, which is the KTN metricexpressed in Boyer-Lindquist coordinates as [23], ds = g µν dx µ dx ν , = g tt dt + 2 g tφ dtdφ + g rr dr + g θθ dθ + g φφ dφ , (1)where x µ ( ≡ t, r, θ, φ ) denote coordinates and g tt =( a sin θ − ∆) / Σ, g tφ = ( A ∆ − a k B sin θ ) / Σ, g rr = Σ / ∆, g θθ = Σ and g φφ = ( B sin θ − A ∆) / Σ are the non-zero metric components. Here, A = a k sin θ − n cos θ ,Σ = ( a k cos θ + n ) + r , B = a + n + r and ∆ = r − r + a − n . Throughout this paper, we adopt thesign convention as ( − , + , + , +). We set the source mass M S = 1, and work in units where the universal gravi-tational constant G = 1 and the speed of light c = 1is used. In this unit system, we express length, angularmomentum, and time in terms of GM S /c , GM S /c and GM S /c , respectively.The event horizon ( r H ) of the metric is defined as ∆ =0, which gives r H = 1 + q − a + n . (2)We have taken only the outer horizon as the region of ourinterest. Equation (2) clearly suggests that depending onthe values of a k and n , KTN spacetime represents eitherblack hole with (1 − a + n ) > − a + n ) <
0. For n = 0, KTN spacetime boils downto the usual Kerr spacetime. One of the most importantphysical implications of this background, contrary to theconventional wisdom, is that the spin parameter a k cannow be larger than unity for black holes. This partic-ular fact specifically makes KTN black hole spacetimea fertile ground for accretion study having much richerphenomenology as compared to usual Kerr black holes. III. ASSUMPTIONS AND MODEL EQUATIONS
In this paper, we carry out the hydrodynamical anal-ysis of accretion flow based on some simple set of as-sumptions. We consider the flow to be axisymmetric inaccordance with the KTN background. For simplicity,we also consider non-dissipative, optically, and geometri-cally thin accretion flow in the steady state [46, 52, 60].
A. Equations of the fluid
In the framework of the relativistic accretion processes,the non-dissipative energy-momentum tensor for the fluid composed of ions and electrons can generally be ex-pressed as, T µν = ( e + p ) u µ u ν + pg µν , (3)where e , p , and u µ represent the energy density, pressure,and the four velocities of any fluid element, respectively.The time-like velocity field satisfies the following localnormalization condition u µ u µ = −
1. Here, µ and ν arethe spacetime indices running from 0 →
3, and g µν arethe components of the metric under consideration. Theconservation of energy-momentum tensor and the massflux give all the hydrodynamical equations required todescribe the flow, and are given by, T µν ; ν = 0 , ( ρu ν ) ; ν = 0 , (4)where ρ is the local mass density of the flow. In relativis-tic hydrodynamics, we employ the projection operator h iµ = δ iµ + u i u µ , where ‘ i ’ takes only the spatial coordi-nates, which satisfies h iµ u µ = 0. By taking the projectionof conservation equation on the spatial hypersurface, oneobtains the relativistic Euler equation, h iµ T µν ; ν = ( e + p ) u ν u i ; ν + ( g iν + u i u ν ) p ,ν = 0 , (5)and, projecting it along u µ , we have the first law of ther-modynamics as, u µ T µν ; ν = u µ (cid:20) (cid:18) e + pρ (cid:19) ρ ,µ − e ,µ (cid:21) = 0 . (6)To describe the flow completely, we need to know theequation of state (EoS) of the fluid under consideration,which relates the density ( ρ ), pressure ( p ), and the inter-nal energy ( e ) of the flow. Usually, the temperature ofthe accretion flow can go up to ∼ − K [42], partic-ularly, within a few Schwarzschild radius. Therefore, weconsider the relativistic EoS given by Chattopadhyay &Ryu [43], e = n e m e f = ρτ f. (7)Here, n e and m e are the number density and the massof the electrons, ρ = n e m e τ , τ = [2 − ζ (1 − /χ )], ζ = n p /n e , and χ = m e /m p , respectively, where n p and m p are the number density and the mass of the ions. Weconsider the flow to be composed of solely by ions andelectrons. Hence, throughout our study, we set ζ = 1,until otherwise stated. Finally, the extended form of f interms of the dimensionless temperature Θ (= k B T /m e c )is given by, f = (2 − ζ ) (cid:20) (cid:18)
9Θ + 33Θ + 2 (cid:19) (cid:21) + ζ (cid:20) χ + Θ (cid:18)
9Θ + 3 /χ
3Θ + 2 /χ (cid:19) (cid:21) . (8)According to the relativistic EoS, the explicit expressionsof the polytropic index ( N ), adiabatic index (Γ) and thesound speed ( a s ) are given as, N = 12 dfd Θ ; Γ = 1 + 1 N ; and a s = Γ pe + p = 2ΓΘ f + 2Θ . (9)In equation (9), N and Γ are expressed as a function of Θ,and therefore, these quantities would be determined selfconsistently while obtaining the flow properties acrossthe length-scale of the accretion disk. B. Governing Equations for Accretion Disc
In our analysis, we assume geometrically thin accre-tion disc around black hole in the steady state. There-fore, given the background axisymmetry, one can generi-cally consider the disc to be lying on the equatorial planewith θ = π/
2, and consequently u θ ∼
0. Under this as-sumption, the radial component of the relativistic Eulerequation (equation (5)) takes the following form, u r u r,r + 12 g rr g tt,r g tt + 12 u r u r (cid:18) g tt,r g tt + g rr g rr,r (cid:19) + u φ u t g rr (cid:18) g tφ g tt g tt,r − g tφ,r (cid:19) + 12 u φ u φ g rr (cid:18) g φφ g tt,r g tt − g φφ,r (cid:19) + ( g rr + u r u r ) e + p p ,r = 0 . (10)Subsequently, the second part of the equations (4), i.e .,the continuity equation can be expressed in terms of themass accretion rate ( ˙ M ), which is a constant of motionand is given by, ˙ M = − πru r ρH, (11)where H is the local half-thickness of the accretion disc.The functional form of H is obtained by following Riffert& Herold [44], and Peitz & Appl [45], in the form, H = pr ρ F , (12)with F = γ φ ( r + a k ) + 2∆ a k ( r + a k ) − a k , where γ φ = 1 / (1 − v φ ) and v φ = u φ u φ / ( − u t u t ), respec-tively. It has been shown to be convenient and physicallytransparent to study the dynamics in terms of all the flowvariable defined in the co-rotating frame [45, 46]. In theco-rotating frame, the radial three velocity is defined as v = γ φ v r and thus the associated radial Lorentz factoris given by γ v = 1 / (1 − v ), where v r = u r u r / ( − u t u t ).Employing these definitions of the velocities and usingthe expressions g µν for KTN metric in equation (10), weobtain vγ v dvdr + 1 hρ dpdr + d Φ eff e dr = 0 , (13)where h [= ( e + p ) /ρ ] is the specific enthalpy, Φ eff e denotesthe effective potential at the disc equatorial plane, and is FIG. 1: Plot of effective potential Φ effe with radial distancefor angular momentum λ = 3 .
5. In the left panel, we fixNUT parameter n = 0 .
5, and solid (black), dotted (blue),small-dash (red) and long-dash (green) curves are for Kerrparameter a k = 0 . , . , . .
7, respectively. In the rightpanel, we fix a k = 0 .
5, and solid (black), dotted (blue), small-dash (red) and long-dash (green) curves are for n = 0 , . , . .
75, respectively. given by [47],Φ eff e = 1 + 12 ln (cid:0) n + r (cid:1) ∆( a − a k λ + n + r ) − ( a k − λ ) ∆ ! . (14)In Fig. 1, we illustrate the variation of effective po-tential (Φ eff e ) with radial coordinate ( r ) for a given an-gular momentum λ = 3 .
5. In the left panel of the fig-ure, we demonstrate how Φ eff e varies with black hole spin( a k ). Here, we choose NUT parameter n = 0 .
5, and solid(black), dotted (blue), small-dash (red) and long-dash(green) curves are for Kerr parameter a k = 0 . , . , . .
7, respectively. Similarly, in the right panel of thefigure, we show the dependencies of Φ eff e on n , where a k = 0 . n = 0 , . , . .
75, respectively. Figure 1 clearlyindicates that for KTN black hole spacetime, the roleof a k and n are opposite to each other in deciding thefeatures of Φ eff e .Because the KTN spacetime is stationary and axisym-metric, there exists two mutually perpendicular Killingvectors, namely ∂ t and ∂ φ . These two Killing vectorshelps to construct two conserve quantities along the di-rection of the motion, and are given by, hu φ = L (constant); − hu t = E (constant) , (15)where E is the Bernoulli constant (equivalently specificenergy) of the flow. Here, u t = − γ v γ φ / p λg tφ − g tt ,where λ = − u φ /u t is the specific angular momentumof the flow which is also a constant of motion obviousfrom equation (15).Integrating equation (6) with the help of equation (7-8), we obtain the expression of density ( ρ ) in terms oftemperature (Θ) as, ρ = K exp( k )Θ / (3Θ + 2) k (3Θ + 2 /χ ) k , (16)where K is the entropy constant and k = 3(2 − ζ ) / k =3 ζ/
4, and k = ( f − τ ) / (2Θ). Following Chattopadhyayand Kumar [48], Kumar and Chattopadhyay [49]), wedefine the entropy accretion rate ( ˙ M ) as˙ M = ˙ M π K = exp( k )Θ / (3Θ + 2) k (3Θ + 2 /χ ) k Hru r . (17)It maybe noted that ˙ M is also a constant of motion. C. Wind Equation
To obtain the wind equation, we make use of equations(6), (7), (10), and (11). In fact, it is customary to expressthe wind equation as follows, dvdr = ND , (18)where denominator D is given by, D = γ v (cid:20) v − a s v (Γ + 1) (cid:21) , (19)and numerator N is given by, N = 2 a s Γ + 1 (cid:20) d ∆ dr + 12 η dηdr + 32 r − F d F dr (cid:21) − d Φ eff e dr . (20)Here, we write η = r / ( r + n ).Similarly, the gradient of the temperature is obtain byrewriting equation (5) using equations (9) and(11) as, d Θ dr = − N + 1 (cid:20) γ v v dvdr − F d F dr + 12∆ d ∆ dr + 12 η dηdr + 32 r (cid:21) . (21) D. Critical Point analysis
In order to obtain the accretion solution, one requiresto solve equations (18) and (21) simultaneously by us-ing the initial condition of the flow. In this work, theinitial condition of the flow is characterized by a set ofinput parameters, namely the radial velocity v ( r ), tem-perature Θ( r ) and the angular momentum ( λ ) of the flowalong with the spacetime parameters ( a k , n ). Interest-ingly, because of the nature of the black hole spacetime, the accretion flow around a black hole must be transonic,which essentially means that while accreting towards theblack hole, the flow must make smooth transition fromsub-sonic to supersonic velocity at some point before en-tering the black hole. Such a special point where flowchanges its sonic character is called as critical point ( r c ).At the critical point, the numerator and the denomi-nator of the wind equation (18) vanish simultaneously( i.e., dv/dr = 0 /
0) where we have the critical point con-ditions as N = D = 0. To calculate the radial veloc-ity gradient ( dv/dr ) c at r c , we apply the l ′ Hospital rule.In general, ( dv/dr ) c possesses two distinct values; one ofthem is for accretion and the other one is for wind. Whenboth values of ( dv/dr ) c are real and of opposite sign, thecritical point is called as saddle type critical point; when( dv/dr ) c are real and same sign it is called as nodal typecritical point (or N-type) and when ( dv/dr ) c are com-plex, it is the spiral type (or O-type) critical point [50,and references therein]. It may be noted that saddle typecritical points have special importance as the global tran-sonic accretion flow can only pass through it. In reality,depending on the input parameters, the flow may possesssingle or multiple critical points within the length scaleof the accretion disc [51, 52]. When critical points formclose to the horizon, it is referred as the inner criticalpoints ( r in ) and when they form far away from the blackhole, it is termed as the outer critical points ( r out ). IV. ACCRETION AROUND KTN BLACK HOLE
In this section, we intend to focus on the KTN blackhole background keeping the naked singularity case asidefor discussion in Section V. In reality, the behavior ofthe accretion flow depends on both the parameters de-scribing the KTN black hole, namely Kerr ( a k ) and NUTparameters ( n ). However, since the role of a k in study-ing the accretion solution around black hole is alreadywell explored, we plan is to concentrate on the NUT pa-rameter ( n ) only and investigate its impact on the flowproperties. A. Properties of the critical points
Since the accretion solutions around the black holescan only pass through the saddle type critical points, itis useful to examine how the nature of the critical pointsdepends on the NUT parameters ( n ). For that we choosea set of ( λ, a k , n ) values to calculate the flow energy ( E )at the critical points by using the critical point condi-tions. The obtained results are shown in Fig. 2 wherewe plot the variation of E with the critical point location( r c ) for different values of n . Here, we choose a k = 0 . λ = 2 .
04. In the figure, the left to right curves areobtained for n = 0, 0 .
25 and 0 . n valuesare marked. In a given curve, we denote the saddle, nodaland O-type critical points by solid, dotted and dashed FIG. 2: Plot of energy as a function of critical point location( r c ) for three different NUT parameter, namely n = 0, n =0 .
25, and n = 0 .
5. Here, we choose a k = 0 .
99, and λ = 2 . line styles, respectively. For n = 0, it is observed that thenature of the critical points changes in systematic orderas saddle − nodal − spiral − nodal − saddle with the shift ofthe location of the critical points from the black hole. Wealso observe that there exists an energy range that allowsthe flow to possess maximum of three critical points. Outof the three critical points, one is O-type, and the othertwo are either saddle type or combination of spiral andnodal types. For E <
1, flow contains two critical points,and between them one is saddle or nodal type and otheris O-type. When energy is above a critical value, flowonly owns a single saddle type critical point located closeto the horizon. For n = 0 .
25, we find similar results as inthe case of n = 0, however, the energy range for multiplecritical points is reduced and the locations of the criti-cal points are shifted outwards for a given energy. WhenNUT parameter is increased further to n = 0 .
5, we ob-serve that multiple critical points completely disappear.This clearly indicates that there exists a critical NUT pa-rameter (say, n cri ) beyond which multiple critical pointsdo not exist. Needless to mention that n cri does not haveany universal value, instead it strongly depends on theother input parameters. Overall, the above analysis sug-gests that NUT parameter ( n ) plays an important rolein deciding the properties of the accretion flow in KTNblack hole background. B. Global Transonic Accretion solution
In order to solve the hydrodynamic equations, theaforementioned criticality condition plays very impor-tant role in identifying the appropriate boundary con-ditions for the flow. Setting D = 0 and N = 0, andusing E = − hu t (see equation (15)), we obtain the radialvelocity ( v c ) and temperature (Θ c ) at the critical point( r c ) for a given set of ( E , λ, a k , n ) values. In other words,these two critical point conditions enable us to reduce thenumber of input parameters from six to four and there-fore, we can start the integration of equations (18) and(21) from the critical point itself to obtain the globalaccretion solutions. Accordingly, using the same set of( E , λ, a k , n ) values as the input parameters, we integrateequation (18) and (21) from the critical point ( r c ) first upto horizon and then up to a large distance (equivalentlythe outer edge of the disc). Finally, we join these twoparts of the solution to obtain a global transonic accre-tion solution around black hole [41, 47, 53, and referencestherein].Following the above procedure, we calculate the globalflow solutions for different n values and plot them in Fig.3. While obtaining the solution, we fix E = 1 . λ = 2 .
04, and a k = 0 .
99, and vary NUT parameter asin panel (a) n = 0, (b) 0 .
25, and (c) 0 .
5. In each panel,the Mach number ( M = v/a s ) is plotted as function ofradial coordinate where solid (black) and dotted (blue)curves denote accretion and wind branches, respectivelyand filled circles represent the critical points. Figureclearly shows how the nature of the flow solutions changeswith n for a given set of ( E , λ, a k ) values. In panel (a), theflow possesses multiple critical points, and the solutionpassing through the inner critical point ( r in = 1 . r out = 1021 . n = 0 .
25 (see panel (b)), flow continues to possessmultiple critical points ( r in = 2 . , r out = 1022 . n = 0 . r out = 1023 . FIG. 3: Plot of Mach number ( M = v/a s ) as a function of radial distance ( r ). Solid (black) curve represents accretion solutionand dotted (blue) curve denotes winds. Here, we choose E = 0 . λ = 2 .
04, and a k = 0 .
99 for all panels. Results presentedin panel (a), (b) and (c) are for n = 0, 0 .
25 and 0 .
5, respectively. See text for details.FIG. 4: Region of the parameter space in λ −E plane accordingto the nature of the flow solutions. At the insets, all possibleflow solutions (O, A, W, I) are presented. See text for details. C. Parameter space with NUT charge
In this section, we begin with the study of parameterspace in λ − E plane according to the nature of flow so-lutions. To do that we fix a k = 2 .
23 and n = 2 andvary both λ and E freely to calculate various flow solu-tions. Here, we restrict our investigation for E ≥ M = v/a s ) with radial coordinate is shown andindividual panels are also marked. In each panel, solidcurves denote the accretion solutions, dotted curves in-dicate the wind solutions and filled circles represent thecritical points. Arrows indicate the overall direction offlow motion towards the black hole.Next, we examine the range of flow parameters thatprovides the flow solutions containing multiple criticalpoints around the black hole having spin a k = 0 . E and λ freely. This al-lows us to obtain the parameter space spanned by E and λ which is depicted in Fig. 5. Effective region of the param-eter space separated by solid (black), dotted (blue) anddashed (red) boundaries are obtained for n = 0, 0 . a k is varied. Here,we fix the NUT parameter as n = 0 . a k = 0 . , .
99 and 1 . a k is increased keeping n fixed, the parameter space for multiple critical points isshifted toward higher energy and lower angular momen-tum domain. By comparing Fig. 5 and Fig. 6, we alsofind that n and a k play opposite role in determining theparameter space as expected.More interesting and rich phenomenology comes intoplay when the fast spinning KTN black holes are taken FIG. 5: Modification of the parameter space for multiple crit-ical points with the increase of NUT parameter ( n ). Regionbounded with solid (black), dotted (blue) and dashed (red)are for n = 0 . , .
5, and 1 .
0, respectively. Here, we fixed theKerr parameter a k = 0 .
99. See text for details.FIG. 6: Same as Fig. 5, but NUT parameter is fixed as n = 0 . a k is varied as marked in the figure. into considerations. It has already been emphasized thatKTN black hole can accommodate spin parameter a k larger than unity as opposed to the usual Kerr black holeprovided the chosen NUT parameter satisfies the condi-tion 1 − a + n >
0. Therefore, KTN spacetime opensup new opportunities to explore large class of observa-
FIG. 7: Comparison of parameter space for multiple criti-cal points. Effective regions of the parameter space boundedwith dotted (blue) and solid (black) curves are obtained fortwo different event horizon locations r H = 1 .
14 and r H = 2,respectively. The Kerr parameter and NUT parameter ( a k , n )are marked in the figure. see text for details. tions which may not be possible in usual Kerr black holespacetime (for a recent work see [55]). Keeping this inmind, we study the accretion flow dynamics around therotating black holes with Kerr parameter a k >
1, andcompare the results with that of the usual Kerr blackhole ( n = 0). In order to do so, we choose different com-bination of Kerr and NUT parameters keeping the eventhorizon location ( r H ) fixed and identify the ranges of E and λ that admit accretion solutions containing multi-ple critical points. The obtained results are depicted inFig. 7, where we identify the region of parameter spacein λ − E plane that render multiple critical points. Here,solid and dotted boundaries refer to the two event hori-zon locations as r H = 2 (solid) and r H = 1 .
14 (dotted).The chosen set of ( a k , n ) values are marked in the fig-ure. We notice that for a given n , as a k is increased,accretion flow generally possesses multiple critical pointsat lower angular momentum and higher energy ranges.Furthermore, n and a k play competing role in decidingthe black hole horizon (see equation (2)) for KTN space-time. Therefore, when n ≫ r H tends to be insensitiveto a k causing the parameter space for multiple criticalpoints indistinguishable as seen in Fig. 7. D. Radiative properties in KTN spacetime
During the course of accretion, inflowing matter expe-riences compression that causes the flow to become hotand dense. Hence, the flow is expected to emit high en-
FIG. 8: Plot of maximum luminosity ( L max ) as a function ofNUT parameter ( n ). Solid (black), dotted (blue) and dashed(red) curves denote the results corresponding to a k = 0 . , . .
0, respectively. Here, we choose black hole mass M S =10 M ⊙ , accretion rate ˙ m = 0 . M Edd , and inclination angle i = π/ ergy radiation. Since accretion flow is composed of bothions and electrons, free-free emission is viable and there-fore, we consider the bremsstrahlung radiation from theaccretion disc. Usually, the bremsstrahlung emission rateper unit volume, per unit time, per unit frequency is es-timated as [56], ǫ ( ν ) = 32 πe m e c (cid:18) π k B m e T e (cid:19) / n e e − hν/k B T e g br , (22)where e is the charge of the electron, h is the Planck’sconstant, T e is the electron temperature, ν is the fre-quency and g br is the Gaunt factor. Note that g br isa dimensionless quantity that varies between 0 . . g br = 1 for simplic-ity. Further, following the work of Chattopadhyay andChakrabarti [58], we estimate the electron temperature( T e ) as T e = ( m e /m p ) / T , where m e and m p denote theelectron and ions masses, and T refers the flow temper-ature. The total luminosity emitted from the accretiondisc is obtained upon integrating ǫ ( ν ) over the total vol-ume and is given by L = 2 Z ∞ Z r edge r H Z π Hrǫ ( ν e ) dν e drdφ. (23)Here, ν e refers the emitted frequency which is related tothe observed frequency ( ν o ) as ν e = (1 + z ) ν o , where z denotes the red-shift factor. Following Luminet [59], weobtain z as 1 + z = u t (1 + r Ω sin φ sin i ) , (24) where Ω = u φ /u t is the angular velocity and i is theinclination angle of the black hole. In this work, we con-sider i = π/ M S = 10 M ⊙ , where M ⊙ is the so-lar mass and accretion rate ˙ m = 0 . M Edd , ˙ M Edd beingEddington accretion rate. With this set up, we calculatethe maximum disc luminosity ( L max ). While doing so,we choose a set of ( a k , n ) values and freely vary the en-ergy ( E ) and angular momentum ( λ ) of the flow. Thisprovides the swarm of transonic accretion solutions [eachsolution is obtained for a particular set of ( a k , n, E , λ )]that are used in equation (23) to calculate the disc lu-minosity ( L ). Upon comparing various L values, we findthe maximum disc luminosity ( L max ). It is noteworthyto mention that during integration, we truncate the ac-cretion disc at the outer edge ( r edge ) where H/r → . L max is plotted as a function of NUT parameter( n ) for different a k values. In the figure, solid, dottedand dashed curves denote the results obtained for a k = 0(black), a k = 0 .
99 (blue), and a k = 2 . n = 0), L max increases with the increase of a k . On the other hand, as n gradually increases, L max decreases. Moreover, we findthat for Schwarzschild type KTN black hole ( a k = 0), therate of decrease of L max with n is relatively smaller com-pared to that of the rotating KTN black hole. Overall,we observe that L max generally decreases with increasing n irrespective of a k values and appears to merge for largeNUT parameter. V. HYDRODYNAMICAL FLOW AROUND KTNNAKED SINGULARITY
In this section, we study the properties of the hydro-dynamic flow around the KTN naked singularity. Themain motivation here is to explore the role of a k and n in deciding the nature of the critical points and the flowsolutions. To do that we follow the same methodologiesas discussed in Section IV. A. Properties of critical points
In Fig. 9, we present the variation of flow energy ( E )as a function of critical point location ( r c ) for differentangular momentum ( λ ). In the figure, we choose NUTand Kerr parameters as ( n, a k ) = (1 . , .
55) in panel(a), (1 . , .
60) in panel (b), (1 . , .
65) in panel (c), and(1 . , .
70) in panel (d). In panel (a), the energy varia-tion plotted using solid (black), dotted (blue) and dashed(red) are obtained for angular momenta λ = 2 .
90, 2 . .
70, respectively and they are marked. Here, weconsider n, a k in such a way that it yields the KTN blackhole spacetime. This results apparently help us to under-stand how the properties of the critical point alter as thespacetime geometry is changed from KTN black hole to0 FIG. 9: Plot of energy ( E ) as a function of critical point location ( r c ). Here we fix n = 1 .
24. In each panel (a-d), solid (black),dotted (blue) and dashed (red) curves denote results for different λ values which are marked. We choose a k = 1 .
55 for panel(a), a k = 1 .
60 for panel (b), a k = 1 .
65 for panel (c), and a k = 1 .
70 for panel (d), respectively. See text for details.
KTN naked singularity. From panel (a) it is clear that fora given set of λ and E , the flow may possess maximum ofthree critical points and minimum of one critical point.When multiple critical points are present, one of themis necessarily O-type in nature [51, 52]. In panel (b),we keep the NUT parameter same as in panel (a) ( i.e. , n = 1 .
24) and increase the Kerr parameter to a k = 1 . λ = 2 .
80, 2 .
70, and2 .
60, respectively. We find that near the origin, a newcritical point is appeared which was absent for black hole spacetime. In reality, this critical point is invisible forblack hole as it always remains inside the horizon [60].Hence, for a given set of λ and E , flow can have maximumof four critical points for KTN naked singularity. Amongthem, the innermost critical point is always O-type, andthe flow can have a maximum of two saddle type criticalpoints. More precisely, the flow contains critical pointsin a systematic order as O-type — saddle type — O-type— saddle type with the shift of the location of the crit-ical point away from the origin. In panel (c), we choose n = 1 .
24 and a k = 1 .
65 where spacetime represents KTNnaked singularity and the amount of spacetime deforma-1
FIG. 10: Modification of the parameter space for multiplecritical points as a k is increased. Here, we choose n = 1 . a k = 1 . , .
65, and 1 . tion is more compared to (b). In the plot, solid (black),dotted (blue) and dashed (red) curves are for λ = 2 . .
55, and 2 .
50, respectively. We find that there existsthe ranges of λ and E for multiple critical points which isreduced compared to the results presented in panel (b).Moreover, we observe that the locations of the criticalpoints are in general shifted outwards from the origin.Finally in panel (d), we fix n = 1 .
24 and a k = 1 . λ = 2 .
40, 2 . .
35, respectively. We find thatin this limit, flow possesses at most two critical pointswhere the inner one is O-type and the other is saddletype. We point out that possibly for the first time to thebest of our knowledge, this observation is explored in thepresent work which is not seen for black hole spacetime.
B. Parameter space for Multiple saddle typecritical points
From the discussion presented in § VA, it is clear thatflow may harbor maximum of four critical points depend-ing on the a k and n values. To quantify this, we identifythe region of the parameter space in the λ − E planefor a given set of ( a k , n ) that allows the flow to possessat least two saddle type critical points. For that we fix n = 1 .
24 and calculate the parameter space for multi-ple saddle type critical points for various a k values. Wepresent the results in Fig. 10, where the region identi- FIG. 11: Plot of Mach number ( M ) as function of radial co-ordinate ( r ). Here, we choose a k = 1 . n = 1 . E = 1 . λ = 2 .
80. Filled circles denote the critical points whichare marked. The contours are of constant entropy accretionrate ( ˙ M ) which are indicated by different line styles. See textfor details. fied with solid (black), dotted (blue) and dashed (red)boundaries are obtained for a k = 1 . , .
65, and 1 . λ and higher E sideas the a k is increased. This eventually indicates that fora given n , the possibility of having multiple saddle typecritical points in a flow is reduced when a k is increased. C. Flow solutions of different kind
In order to obtain flow solution around a naked singu-larity, we first choose a k and n values such that 1 − a >n . Then, we calculate the critical points correspondingto flow energy E and angular momentum λ . Followingthe criteria to classify the nature of the critical points, weidentify the saddle type critical points and calculate theflow solutions passing through it. In the next subsections,for the purpose of representation, we choose a k = 1 . n = 1 .
24, and obtain different types (altogether fivetypes) of flow solutions for various sets of E and λ values. A-type solutions:
In Fig. 11, we present the variation of Mach number( M ) with radial coordinate ( r ). Here, we fix a k = 1 . n = 1 . E = 1 .
001 and λ = 2 .
80, and obtain fourcritical points. Among them two are saddle type criti-cal points located at r in = 2 . r out = 207 . r c1 = 1 . r c2 = 7 . r out ,we calculate the radial velocity gradient ( dv/dr ) c at r out which yields two real values; one is positive and otheris negative. Using the negative values of ( dv/dr ) c , firstwe integrate equation (18) and (21) inward towards thenaked singular point and then outward up to the outeredge of the disc ( r edge , usually the large distance). Fi-nally, we join this two parts to obtain a complete branchof solution. Here, we choose r edge = 1000. Consideringthe positive values of ( dv/dr ) c , we repeat the above pro-cedure to obtain the other branch of the solution. In thefigure, these two branches of the solution is plotted us-ing solid (black) curve. It is noteworthy that the entropyaccretion rate ( ˙ M ) of a given solution always remainsconstant. We calculate the entropy accretion rate of theabove flow solutions and obtain as ˙ M out = 2 . × .Next, we calculate the flow solutions passing through theinner critical point ( r in ) in the same way as in the case ofsolutions passing through the outer critical point ( r out ).The noticeable difference here is that flow does not ex-tend up to the outer edge of the disc, instead it becomesclosed in between r c1 and r out . These solutions are plot-ted using dot-big-dashed (magenta). For this solution wefind ˙ M in = 5 . × . Now, keeping all the remain-ing flow parameters unchanged, if we consider ˙ M otherthan ˙ M in or ˙ M out , flow solution does not possess anycritical point. In that case, one can start integration ofthe equations (18) and (21) from any radial coordinate ofinterest. For example, when ˙ M = 1 . × , we calcu-late the radial velocity ( v ) and flow temperature (Θ) at r edge = 1000 using equation (17) and employing them, weobtain the solution depicted by doted (blue) curve. Simi-larly, solutions plotted using dashed (red) and dot-small-dashed (green) curves are obtained for ˙ M = 3 . × and ˙ M = 6 . × , respectively. We also observethat there exists a region around the naked singularitywhich remains inaccessible to the flow. We conjecturethat during accretion, flow is expected to pile up thereand tended to rotate along a surface around the nakedsingularity which we call as the naked surface . Since theacceptable flow solution connects the central object andthe outer edge of the disc (in case of black hole, it is eventhorizon to the outer edge of the disc), solutions plottedwith solid (black) and dotted (blue) curves are physi-cally acceptable. However, since the entropy content ofthe dotted (blue) solution is lower than the solid (black)one, nature therefore favors the flow solutions passingthrough r out only. W-type solutions:
Here, we choose the input parameters as a k = 1 . n = 1 . E = 1 .
001 and λ = 2 .
90, and obtain the criti-cal points as r c1 = 1 . r in = 1 . r c2 = 9 . r out = 206 . FIG. 12: Plot of Mach number ( M ) as function of radial co-ordinate ( r ). Here, we choose a k = 1 . n = 1 . E = 1 . λ = 2 .
90. Filled circles denote the critical points whichare marked. The contours are of constant entropy accretionrate ( ˙ M ) which are indicated by different line styles. See textfor details. the obtained results ( M vs. r ) in Fig. 12. The en-tropy accretion rate ( ˙ M ) corresponding to the flow solu-tions drawn using dotted (blue), solid (black), dot-small-dashed (green), dot-big-dashed (magenta) and dashed(red) curves are calculated as 1 . × , 2 . × ,2 . × , 2 . × , and 3 . × , respectively.Note that the flow solution passing through r out (dot-big-dashed, magenta) fails to join with the naked sur-face , however, solution passing through r in (solid, black)smoothly connects the naked surface with the outer edgeof the disc. Moreover, this solution is preferred over othersolutions as it has high entropy content. I-type solutions:
We continue our study of finding flow solutions andchoose the input parameters as a k = 1 . n = 1 . E = 1 .
030 and λ = 2 .
80. Here, we find that only twocritical points exist: one of them is O-type ( r c1 ) and theother is saddle type ( r in ). We calculate the flow solutionsfollowing the procedure as in Fig. 11. We observe thatflow solution with ˙ M = 6 . × passes through r in =2 . naked surface to the outer edgeof the disc which is shown using solid (black) curve in Fig.13. Other solutions obtained for ˙ M = 6 . × and8 . × are plotted with dotted (blue) and dashed(red) curves as shown in the figure. As before solutionpassing through r in is preferred as it has higher ˙ M . O-type solutions:
In this case, we choose the input parameters as a k =3 FIG. 13: Plot of Mach number ( M ) as function of radial co-ordinate ( r ). Here, we choose a k = 1 . n = 1 . E = 1 . λ = 2 .
80. Filled circles denote the critical points whichare marked. The contours are of constant entropy accretionrate ( ˙ M ) which are indicated by different line styles. See textfor details.FIG. 14: Plot of Mach number ( M ) as function of radial co-ordinate ( r ). Here, we choose a k = 1 . n = 1 . E = 1 . λ = 2 .
22. Filled circles denote the critical points whichare marked. The contours are of constant entropy accretionrate ( ˙ M ) which are indicated by different line styles. See textfor details. FIG. 15: Plot of Mach number ( M ) as function of radialcoordinate ( r ). Here, the solutions are obtained for boundedenergy ( E <
1) where flow parameters are chosen as a k = 1 . n = 1 . E = 0 . λ = 2 .
80, respectively. Filled circlesdenote the critical points which are marked. The contours areof constant entropy accretion rate ( ˙ M ) which are indicatedby different line styles. See text for details. . n = 1 . E = 1 .
001 and λ = 2 .
22 and find twocritical points. Here, we find that the flow solutionsare very much similar in character as in Fig. 13, ex-cept the saddle-type critical point ( r out ) forms far awayfrom the naked surface and flow solution passing through r out = 216 . naked surface to theouter edge. In Fig. 14, the solutions plotted using dotted(blue), solid (black) and dashed (red) curves are obtainedfor ˙ M = 2 . × , 2 . × , and 3 . × ,respectively. I ∗ -type solutions: Here, we present the flow solutions for bounded ener-gies, i.e., E <
1. For that, we choose the input param-eters as a k = 1 . n = 1 . E = 0 .
999 and λ = 2 . r c1 and r c2 ) and the remaining one is saddletype ( r in ). As before, we calculate the flow solutionsfollowing the procedure mentioned above while gener-ating Fig. 11 and depict all the solutions in Fig. 15.For ˙ M = 5 . × , flow solution passes through r in = 2 . naked surface to the outer edge of thedisc. Other solutions which are not transonic in nature,are obtained for ˙ M = 4 . × and 6 . × and we plot them using dotted (blue) and dashed (red)curve as shown in the figure.4 VI. DISCUSSION AND CONCLUSIONS
In this work, we study the properties of the accretionflow in a general axisymmetric KTN spacetime. Thisspacetime either describe black hole or naked singular-ity depending on the choice of Kerr parameter ( a k ) andNUT parameter ( n ). We consider the relativistic hydro-dynamic equations that govern the flow motion and solvethem to obtain the flow solutions around the black holesor naked singularities in the steady state limit. We ex-amine the role of a k and n in deciding the nature of thecritical points as well as the flow solutions. We presentour finding point wise below.1. For KTN black hole with fixed a k , there exists arange of n that admits maximum of three criticalpoints. Among them, the critical point that formsclose to the horizon is always saddle type. Beyondthis range, flow is left with only one critical point(see Fig. 2).2. We calculate all possible transonic flow solutionsaround a KTN black hole and separate the param-eter space in λ − E plane according to the natureof the flow solutions (see Fig. 4). We also observethat the nature of the flow solutions changes as n is varied (see Fig. 3). Considering this, we studythe modification of λ − E parameter space for mul-tiple critical points and find that for a given a k ,as n is increased, the parameter space is shiftedtowards the higher angular momentum and lowerenergy domain (see Fig. 5). On the other hand,when a k is increased keeping n fixed, the shift ofthe parameter space happens in the lower angularmomentum and higher energy sides (see Fig. 6).These findings suggest that a k and n respond inopposite way in determining the parameter spacefor multiple critical points. Overall, it appears thatthe NUT parameter ( n ) effectively shields the blackhole rotation for flows accreting on to them.3. It may be noted that, for KTN spacetime, a k > a k <
1. We therefore study the λ − E parameter space for multiple critical points con-sidering KTN black hole having Kerr parameter a k >
1. Considering the various combination of a k and n values, we obtain a fixed event horizon r H (see equation (2)), and obtain the multiple critical point parameter space. We observe that the param-eter space is very much dependent on r H when a k and n values are small with respect to unity, how-ever, it tends to become independent on r H whenboth a k and n is very large (see Fig. 7).4. We compute the maximum luminosity ( L max ) tobe emitted by the accretion flow considering theBremsstrahlung radiative process active in the flow.We find that L max in general decreases with theincrease of n irrespective to the a k values (see Fig.8).5. We examine the critical point properties consider-ing the naked singularity and reveal that flow maypossess maximum of four critical points. When flowcontains four critical points, two of them must besaddle type critical points (see Fig. 9). We calcu-late λ − E parameter space for multiple saddle typecritical points and find that the parameter spaceshrinks and shifted towards lower λ and higher E side as a k is increased (see Fig. 10). We further ob-tain the all possible transonic flow solutions wherewe find that flow tends to reach an imaginary sur-face called as naked surface avoiding the origin ofthe naked singularity (Figs. 11-15).Finally, we argue that our formalism may be usedto predict the possible range of NUT parameter ( n ) inthe astrophysical context. In order to do that, one re-quires the knowledge of the source luminosity, sourcemass and source spin, respectively (see Fig. 8). Tokeep our discussion simple, in this work, we only con-sidered Bremsstrahlung emission process neglecting theother radiative processes, namely synchrotron emission,Compton emission, etc., although they are expected toplay a role in determining the accretion disc luminosity.Therefore, in order to constrain the range of n , a rig-orous study is indispensable, involving all the emissionprocesses, which we intend to consider as a future workand plan to report elsewhere. Acknowledgments
All authors thank Indian Institute of TechnologyGuwahati, India for providing infrastructural support tocarry out this work. ID thanks the Max Planck Partnergroup grant (MPG-01) for financial support. [1] N. I. Shakura and R. A. Sunyaev, Astron. Astrophys. ,337 (1973).[2] M. C. Begelman, in Astrophysics of Active Galaxies andQuasi-Stellar Objects , edited by J. S. Miller (1985), pp.411–452.[3] O. Blaes, ASP Conf. Ser. , 75 (2007), astro- ph/0703589.[4] H. Netzer,
The Physics and Evolution of Active GalacticNuclei (Cambridge University Press, 2013).[5] M. A. Abramowicz and P. C. Fragile, Living Rev. Rel. , 1 (2013), 1104.5499.[6] J. A. Font, Living Reviews in Rela- tivity (2000), ISSN 1433-8351, URL http://dx.doi.org/10.12942/lrr-2000-2 .[7] C. S. J. Pun, Z. Kov´acs, and T. Harko,Phys. Rev. D , 084015 (2008), URL https://link.aps.org/doi/10.1103/PhysRevD.78.084015 .[8] M. Heydari-Fard, Class. Quant. Grav. , 235004 (2010).[9] T. Harko, Z. Kov´acs, and F. S. N.Lobo, Classical and Quantum Gravity , 105010 (2010), ISSN 1361-6382, URL http://dx.doi.org/10.1088/0264-9381/27/10/105010 .[10] T. Harko, Z. Kov´acs, and F. S. N. Lobo, Class. Quant.Grav. , 165001 (2011), 1009.1958.[11] T. Harko, Z. Kov´acs, and F. S. N. Lobo,Phys. Rev. D , 044021 (2009), URL https://link.aps.org/doi/10.1103/PhysRevD.80.044021 .[12] D. F. Torres, Nuclear Physics B ,377?394 (2002), ISSN 0550-3213, URL http://dx.doi.org/10.1016/S0550-3213(02)00038-X .[13] F. S. Guzm´an, Phys. Rev. D , 021501 (2006), URL https://link.aps.org/doi/10.1103/PhysRevD.73.021501 .[14] T. Harko, Z. Kov´acs, and F. S. N. Lobo,Phys. Rev. D , 064001 (2009), URL https://link.aps.org/doi/10.1103/PhysRevD.79.064001 .[15] T. Harko, Z. Kov´acs, and F. S. N. Lobo, Class. Quant.Grav. , 215006 (2009), 0905.1355.[16] Z. Kov´acs, K. S. Cheng, and T. Harko, MonthlyNotices of the Royal Astronomical Society , 1632?1642 (2009), ISSN 1365-2966, URL http://dx.doi.org/10.1111/j.1365-2966.2009.15571.x .[17] C. Reina and A. Treves, Astrophys. J. , 596 (1979).[18] P. S. Joshi, D. Malafarina, and R. Narayan,Classical and Quantum Gravity ,015002 (2013), ISSN 1361-6382, URL http://dx.doi.org/10.1088/0264-9381/31/1/015002 .[19] Z. Kov´acs and T. Harko, Phys. Rev. D , 124047(2010), 1011.4127.[20] A. H. Taub, Annals of Mathematics , 472 (1951), ISSN 0003486X, URL .[21] E. Newman, L. Tamburino, and T. Unti,Journal of Mathematical Physics , 915(1963), https://doi.org/10.1063/1.1704018, URL https://doi.org/10.1063/1.1704018 .[22] M. Demianski and E. T. Newman, Bull. Acad. Pol. Sci.,Ser. Sci. Math. Astron. Phys. , 653 (1966).[23] B. Carter, Phys. Rev. , 1242 (1966).[24] W. Kinnersley, J. Math. Phys. , 1195 (1969).[25] D. Kramer and G. Neugebauer, Commun. Math. Phys. , 132 (1968).[26] I. Robinson, J. Robinson, and J. Zund,J. Math. Mech. , 881 (1969), URL https://ntrs.nasa.gov/search.jsp?R=19690054843 .[27] C. J. Talbot, Commun. Math. Phys. , 45 (1969).[28] D. Lynden-Bell and M. Nouri-Zonoz,Rev. Mod. Phys. , 427 (1998), URL https://link.aps.org/doi/10.1103/RevModPhys.70.427 .[29] V. Kagramanova, J. Kunz, E. Hack-mann, and C. Lmmerzahl, Physical Re-view D (2010), ISSN 1550-2368, URL http://dx.doi.org/10.1103/PhysRevD.81.124044 . [30] C. Chakraborty and S. Bhattacharyya, Jour-nal of Cosmology and Astroparticle Physics , 034 (2019), ISSN 1475-7516, URL http://dx.doi.org/10.1088/1475-7516/2019/05/034 .[31] W. B. Bonnor, Proc. Camb. Phil. Soc. , 145 (1969).[32] J. G. Miller, J. Math. Phys. , 486 (1973), URL https://doi.org/10.1063/1.1666343 .[33] J. S. Dowker, Gen. Rel. Grav. , 603 (1974).[34] S. Ramaswamy and A. Sen, J. Math. Phys. , 2612(1981), URL https://doi.org/10.1063/1.524839 .[35] C. W. Misner, J. Math. Phys. , 924 (1963).[36] J. Frank, A. King, and D. J. Raine, Accretion Power inAstrophysics: Third Edition (2002).[37] L. Rezzolla and O. Zanotti,
Relativistic Hydrodynamics (2013).[38] S. Chandrasekhar,
An introduction to the study of stellarstructure (Univ. Chicago Press, Chicago, IL, 1939).[39] J. L. Synge,
The relativistic gas , vol. 32 (North-HollandPublishing Co., Amsterdam, 1957).[40] J. P. Cox and R. T. Giuli,
Principles of stellar structure (Gordon and Breach, New York, 1968).[41] I. K. Dihingia, S. Das, and A. Nandi, Mon. Not. Roy.Astron. Soc. , 3209 (2019), 1901.04293.[42] I. K. Dihingia, S. Das, and S. Mandal, Mon. Not. Roy.Astron. Soc. , 2164 (2018), 1712.05534.[43] I. Chattopadhyay and D. Ryu, Astrophys. J. , 492(2009), 0812.2607.[44] H. Riffert and H. Herold, Astrophys. J. , 508 (1995).[45] J. Peitz and S. Appl, Mon. Not. Roy. Astron. Soc. ,681 (1997), astro-ph/9612205.[46] S. K. Chakrabarti, Mon. Not. Roy. Astron. Soc. , 325(1996), astro-ph/9611019.[47] I. K. Dihingia, S. Das, D. Maity, and S. Chakrabarti,Phys. Rev. D , 083004 (2018), 1806.08481.[48] I. Chattopadhyay and R. Kumar, Mon. Not. Roy. Astron.Soc. , 3792 (2016), 1605.00752.[49] R. Kumar and I. Chattopadhyay, Mon. Not. Roy. Astron.Soc. , 4221 (2017), 1705.01780.[50] S. K. Chakrabarti and S. Das, Mon. Not. Roy. Astron.Soc. , 649 (2004), astro-ph/0402561.[51] J. Fukue, Publ. Astron. Soc. Japan , 309 (1987).[52] S. K. Chakrabarti, Astrophys. J. , 365 (1989).[53] I. K. Dihingia, S. Das, D. Maity, and A. Nandi, Mon.Not. Roy. Astron. Soc. p. 1852 (2019), 1903.02856.[54] S. Das, Mon. Not. Roy. Astron. Soc. , 1659 (2007),astro-ph/0610651.[55] C. Chakraborty and S. Bhattacharyya,Phys. Rev. D , 043021 (2018), URL https://link.aps.org/doi/10.1103/PhysRevD.98.043021 .[56] M. Vietri, Foundations of High-Energy Astrophysics ,Theoretical Astrophysics (Chicago Univ. Press, Chicago,IL, 2008), URL http://cds.cern.ch/record/1109401 .[57] W. J. Karzas and R. Latter, Astrophys. J. Supple. Ser. , 167 (1961).[58] I. Chattopadhyay and S. K. Chakrabarti, Mon. Not. Roy.Astron. Soc. , 454 (2002), astro-ph/0202351.[59] J. P. Luminet, Astron. and Astrophys. , 228 (1979).[60] S. Das, I. Chattopadhyay, and S. i. K. Chakrabarti, As-trophys. J.557