Lyapunov exponent ISCO and Kolmogorov Senai entropy for Kerr Kiselev black hole
EEur. Phys. J. C (2021) 81:84 https://doi.org/10.1140/epjc/s10052-021-08888-1
Regular Article - Theoretical Physics
Lyapunov exponent, ISCO and Kolmogorov–Senai entropy forKerr–Kiselev black hole
Monimala Mondal , Farook Rahaman , Ksh. Newton Singh Department of Mathematics, Jadavpur University, Kolkata, West Bengal 700032, India Department of Physics, National Defence Academy, Khadakwasla, Pune 411023, IndiaReceived: 16 July 2020 / Accepted: 16 January 2021© The Author(s) 2021
Abstract
Geodesic motion has significant characteristicsof space-time. We calculate the principle Lyapunov expo-nent (LE), which is the inverse of the instability timescaleassociated with this geodesics and Kolmogorov–Senai (KS)entropy for our rotating Kerr–Kiselev (KK) black hole. Wehave investigate the existence of stable/unstable equatorialcircular orbits via LE and KS entropy for time-like and nullcircular geodesics. We have shown that both LE and KSentropy can be written in terms of the radial equation of inner-most stable circular orbit (ISCO) for time-like circular orbit.Also, we computed the equation marginally bound circularorbit, which gives the radius (smallest real root) of marginallybound circular orbit (MBCO). We found that the null cir-cular geodesics has larger angular frequency than time-likecircular geodesics ( Q o > Q σ ). Thus, null-circular geodesicsprovides the fastest way to circulate KK black holes. Further,it is also to be noted that null circular geodesics has short-est orbital period ( T photon < T I SC O ) among the all possi-ble circular geodesics. Even null circular geodesics traversesfastest than any stable time-like circular geodesics other thanthe ISCO. The first detection of black hole (BH) merger GW150914strongly supports the existence of black holes [1,2] byobserving the gravitational waves generated during the pro-cess of coalescence. Further, the first observation of theshadow of a super massive black hole in the giant ellipti-cal galaxy M87 strengthen this claim for the existence ofBHs [3]. BHs are usually surrounded by diffused matter inorbital motion named as “accretion disk”. An accretion disk a e-mail: [email protected] b e-mail: [email protected] (corresponding author) c e-mail: [email protected] can be influenced by cosmic repulsion and magnetic field. Adetail analysis was presented by Stuchlk et al. [4] by consid-ering thin and thick disks. Stuchlk [5] have also discussed themotion of test particle with non-zero cosmological constant,further he also presented equation of motions for electricallycharge and magnetic monopoles. There exist many stable andunstable circular orbits. Another type of orbit also exist whichlie between dynamically unstable and stable orbits knownas “homoclinic orbits” [6–8]. These orbits started asymptoti-cally closed to unstable circular orbits and ends with spiralingin and out about the central BH. During the in-spiraling phasea significant amount of angular momentum is lost into grav-itational waves thereby circularizing the orbit. In addition tothese orbits, “chaotic orbits” have been discovered recentlyfor fast spinning BHs [9–11]. The intermost stable circularorbits (ISCO) [12] is of more interest as it will identify theonset of dynamical instability.The instability of an orbit can be identified by a positivevalue of Lyapunov exponent [13,14]. It is also well-knownthat maximal Lyapunov exponents (MLE) can distinguisha chaotic dynamics from others, which may initiate unsta-ble/homoclinic orbits by a perturbation. For a spinning blackhole the number of unstable orbits increase very rapidly andcrowded into the corresponding phase [15–17]. However,determining LE in general relativistic regime is very diffi-cult as each orbits has different LE and it is impractical toscan the overall nature of the orbits. Also, since LE deter-mines the separation to neighboring orbits in time and timeitself is relative which will eventually lead to relative LE[18]. Lyapunov coefficient delineates the measure of insta-bility of the circular orbits and not a measure of chaos. Onthe other hand, if we consider magnetized black holes i.e. ablack holes immersed in an external magnetic field, the equa-tions of motion become chaotic [19–21]. P ´ a nis et al. [22] dis-cussed about Keplerian disk orbiting a Schwarzschild blackhole embedded in an asymptotically uniform magnetic fieldwhere they have found three possible scenarios i.e. (1) regular
84 Page 2 of 16 Eur. Phys. J. C (2021) 81:84 oscillatory motion, (2) destruction due to capture by the mag-netized black hole and (3) chaotic motion. They have used atime series of the solution of equations of motion for differentconditions to analyzed the transition from regular to chaoticmotion. Extending this work, Kolo ˇ s et al. [23] explainedobservational data for few microquasars using magnetizedstandard geodesic models of QPOs. Further, Tursunov et al.[24] presents the possible source of ultrahigh-energy cos-mic rays from supermassive black holes (SMBH), where thecharged particles are accelerated by magnetic Penrose pro-cess from the rotating SMBH.Wu et al. [25] developed a method of determining fast Lya-punov indicator (FLI) using two-nearby-orbits method with-out projection operations and with proper time as the inde-pendent variable, which can quickly identify chaos. SinceLE is not invariant under coordinates transformations in GR,Wu and Huang [26] proposed a new relativistic LE whichis invariant in a curve manifold. In general relativity, chaostakes an important role in describing more realistic physi-cal systems. The amplification of chaos restricted relativisticthree-body systems [27,28]. It has also been reported thatchaos also exists in two relativistic systems like two fixedblack holes [29,30], Schwarzschild’s black hole and a dipolarshell [31–33] etc. Chaotic behavior must be well understoodin binary gravitational waves sources or otherwise its detec-tion will be highly uncertain. Levin et. al. [7] shown that in theabsence of radiative back reaction a chaotic motion is devel-oped in rapidly rotating compact stars where the relativisticprecession of apastron is supplemented by chaotic precessionof the orbital plane. Burd and Tavakol [34] used Bianchi type-IX cosmology to describe chaos as guage invariant model bycalculation spectrum of LEs. The invariant formulation ofLE was developed by Motter [35] showing that chaos is rep-resented by positive LE and coordinate invariant.The organization of the article is as follows: In Sect. 2:we begin with a basic definition of Lyapunov exponent andprovide a simple formula for finding Lyapunov exponent ( λ )in terms of the second derivative of the of effective poten-tial in radial motion ˙ r . In Sect. 3: We encapsulate the rela-tion between Lyapunov exponent and KS-entropy. In Sect. 4:we analyze the rotating black hole and described completelythe equatorial circular geodesics. We showed that Lyapunovexponent can be demonstrated in terms of ISCO equation anddeliberated stability of time-like geodesics. We derived angu-lar velocity and reciprocal of Critical exponent for both thecases of time-like and null-circular geodesics of the space-time. In Sect. 5: we constructed the equation of marginallybound circular orbit for finding the radius of marginallybound circular orbit to the black holes. We calculated theratio of angular velocity of null-circular geodesics to time-like geodesics in Sect. 6. In Sect. 7: we determined the ratio oftime period of null-circular geodesics to time-like geodesics.In Sect. 8: we summarize our work. In a dynamical system the LE is a quantity that classified therate of separation of extremely close trajectories. This rateof separation depends on orientations of initial separationvector. Among all LE, a positive Maximal Lyapunov expo-nent (MLE) is taken to indicate that the system is chaotic.If the system is conservative then the sum of all Lyapunovexponents must be zero and if negative then the system isdissipative. The Lyapunov spectrum can be utilized to givean evaluate of the rate of entropy production. According toPesin’s theorem [36], the sum of all positive LEs provide anestimate of the KS entropy. Also, a positive LE and a nega-tive LE designates a divergence and a convergence betweento nearby trajectories, respectively.In classical physics, the study of autonomous smoothdynamical system analyzes differential equations of the form d xdt = F ( x ), (1)where t represents the time parameter. If the following fourconditions satisfied by chaos, we may quantify chaos in termsof LEs. The condition are as follows:(i) the system is autonomous;(ii) the relevant part of the phase space is bounded;(iii) the invariant measure is normalizable;(iv) the domain of the time parameter is infinite.This types of characterization is advantageous for space-timebecause LE does not change under space diffeomorphismsof the form z = ψ( x ) . That’s why we may say that chaosis a property of the physical system and independent on thecoordinate which is used to describe the system.There is no absolute time in general relativity. Therefore,the time parameter in-force us to consider the Eq. (1) underthe diffeomorphism: z = ψ( x ), d τ = η( x , t ) dt in thespace-time. Since the classical indicator of chaos i.e. LE andKS entropy depends on the choice of the time parameter, thereis a conceptual problem. This problem was first arise in themix-master cosmological model [37–39], where the largestLE was positive or zero depends on different choice of thecoordinate. In general relativity (GR), this non-invariancefeatures implies that chaos is not a property of physical sys-tem, it is a property of the coordinate system.Following the work by Motter [35], we find that the chaoscan be characterized by positive LE and KS entropy. The Lya-punov exponent and KS entropy transform under the space-time according to λ τ j = λ tj (cid:2) ξ (cid:3) t ( j = , . . . , N ), (2) ur. Phys. J. C (2021) 81:84 Page 3 of 16 84 and h τ ks = h tks (cid:2) ξ (cid:3) t (3)respectively. Where N is phase-space dimension and 0 < (cid:2) ξ (cid:3) t < ∞ is time average of ξ = d τ/ dt over typical tra-jectory. For natural measure of the Eqs. (2) and (3) to bewell defined, 0 < (cid:2) ξ (cid:3) t < ∞ is the basic requirement condi-tion. Finally we can say that, the coordinate transformation isalways transformed into the time independent transformationwhich is given by z = ψ( x ), d τ = ξ( x , t ) dt (4)where ξ is a strictly positive, continuously differentiablefunction and ψ is a diffeomorphism. Then it is clear thatthe LE and KS entropy are invariant under space diffeomor-phism.Now we will calculate the LE using proper time from λ = ± (cid:2) ( ˙ r ) (cid:5)(cid:5) . (5)where ˙ r represents the radial potential. Here we will ignorethe ± sign and we shall take only positive Lyapunov expo-nent. The circular orbit is stable when λ is imaginary andcircular orbit is unstable when λ is real and when λ =
0, thecircular orbit is marginally stable or saddle point.Following Pretorius and Khurana [40], we can define crit-ical exponent γ = (cid:9) πλ = T λ T (cid:9) (6)where, T λ represents the Lyapunov time scale, T (cid:9) representsthe orbital time scale and (cid:9) represent the angular velocity.Also here T λ = λ and T (cid:9) = π(cid:9) . Now the critical exponentcan be written , in terms of second order derivative of thesquare of radial velocity ( ˙ r ) , as γ = π (cid:2) (cid:9) ( ˙ r ) (cid:5)(cid:5) . (7)Also the reciprocal of critical exponent is given by1 γ = T (cid:9) T λ = π (cid:2) ( ˙ r ) (cid:5)(cid:5) (cid:9) . (8) Kolmogorov–Senai ( h ks ) entropy [41,42] is an importantquantity which is connected to the Lyapunov exponents.When the chaotic orbit evolves, this entropy gives a mea-surement about information lost or gained. Another way itcan be explored that when h ks > h ks = ( h ks ) entropyis the sum of all positive Lyapunov exponent, that is, h ks = (cid:3) λ j > λ j . (9)There are two Lyapunov exponent in 2-dimensional phase-space, so in terms of effective radial potential the Kol-mogorov -senai entropy can be written as h ks = (cid:2) ( ˙ r ) (cid:5)(cid:5) There are many choices of rotation axis and a multitude ofangular momentum parameter in higher dimension whereas,only possible rotation axis with only one angular momen-tum for an axisymmetric in four dimension. Here we con-sider the metric of a Rotating black hole surrounding withquintessence [44] is, ds = − (cid:4) − ( Mr + cr − w )(cid:11) (cid:5) dt + (cid:11) dr (cid:12) − a sin θ (cid:4) Mr + cr − w (cid:11) (cid:5) dt d φ + (cid:11) d θ + sin θ (cid:4) r + a + a sin θ (cid:6) Mr + cr − w (cid:11) (cid:7)(cid:5) d φ . (11)where c is a new parameter, which describes the hair of theblack hole and w represents a quintessential equation of stateparameter and (cid:12) = r + a − Mr − c ≡ ( r − r + )( r − r − ),(cid:11) = r + a cos θ. (12)Also this rotating black hole is always bounded from aboveby w = , which was first considered by Schee and Stuchlik[45]. Slan ´ y a and Stuchl ´ i k [46] have also considered test par-ticle in equatorial circular orbits around Kerr–Newman–deSitter BH and naked singularity.The metric is similar to Kerr–Newman black hole when w = , c = − Q , Reissener-Nordstr ¨ o m black hole when w = , c = − Q , a =
0, Kerr black hole w = , c = w = , c = , a =
0. Thehorizon take place at g rr = ∞ or (cid:12) =
84 Page 4 of 16 Eur. Phys. J. C (2021) 81:84 r ± = M ± (cid:8) M − a + c , (13)here r + is called event horizon and r − is called Cauchy hori-zon.The BH solution considered in this work is the rotationalversion of the Kiselev static quintessence BH. Originally,Kiselev used quintessence surrounding the BH characterizedby the EMT of the quintessence. Here, the scalar field couplesto gravity through a Lagrangian of the form L = − g μν ∂ μ φ ∂ ν φ − V (φ) (14)and the stress-energy tensor T μν for the matter field took asperfect fluid. The complete action and the field equations canbe given as S = (cid:9) d x √− g (cid:4) R − g μν ∂ μ φ ∂ ν φ − V (φ) (cid:5) , (15) R μν − g μν R = − π T μν − π (cid:4) ∂ μ φ ∂ ν φ − g μν (cid:10) g μν ∂ μ φ ∂ ν φ + V (φ) (cid:11)(cid:5) , (16)1 √− g ∂ μ (cid:12) √− g g μν ∂ ν φ (cid:13) − ∂ V (φ)∂φ = . (17)The pressure and energy density of the scalar field aregiven by p φ = ˙ φ − V (φ) , ρ φ = ˙ φ + V (φ). (18)The equation of state parameter ω φ = p φ /ρ φ is found to be ω φ = ˙ φ − V (φ) ˙ φ + V (φ) (19)which lies between − < ω φ < ˙ φ < V (φ) then ω φ < − /
3. The rotating Kiselev BH was obtained by usingNewman–Janis algorithm from Kiselev static quintessenceBH, for details see [44].4.1 Circular geodesics in the equatorial planeTo calculate the geodesic equation in the equatorial plane forthis space-time we follow Chandrasekhar [47]. To computethe geodesics motions of the orbit in the equatorial planewe set ˙ θ = θ = constant = π/ L = − (cid:6) − Mr − cr − ( + w) (cid:7) ˙ t − (cid:4) Mar + acr − ( + w) (cid:5) ˙ t ˙ φ + (cid:4) r + (cid:6) + Mr + cr − ( + w) (cid:7) a (cid:5) ˙ φ + r (cid:12) ˙ r , (20) where φ is an angular coordinate. Then the generalizedmomenta can be derived from it as follows p t = − (cid:4) − Mr − cr − ( + w) (cid:5) ˙ t − (cid:4) Mar + acr − ( + w) (cid:5) ˙ φ = − E = const . (21) p φ = − (cid:4) Mar + acr − ( + w) (cid:5) ˙ t + (cid:4) r + (cid:6) + Mr + cr − ( + w) (cid:7) a (cid:5) ˙ φ = L = const . (22) p r = r (cid:12) ˙ r . (23)Since Lagrangian is unhampered by both of t and φ , so p t and p φ preserve quantities. Solving the Eqs. (21) and (22)for ˙ φ and ˙ t we get ˙ φ = (cid:12) (cid:4)(cid:6) − Mr − cr − ( + w) (cid:7) L + (cid:6) Mar + acr − ( + w) (cid:7) E (cid:5) , (24) ˙ t = (cid:12) (cid:4)(cid:6) r + (cid:6) + Mr + cr − ( + w) (cid:7) a (cid:7) E − (cid:6) Mar + acr − ( + w) (cid:7) L (cid:5) . (25)The Hamiltonian in terms of the metric is given by2 H = p t ˙ t + p φ ˙ φ + p r ˙ r − L = − (cid:6) − Mr − cr − ( + w) (cid:7) ˙ t − (cid:6) Mar + acr − ( + w) (cid:7) ˙ t ˙ φ + (cid:4) r + (cid:6) + Mr + cr − ( + w) (cid:7) a (cid:5) ˙ φ + r (cid:12) ˙ r (26)Since the Hamiltonian does not depends on ‘ t ’, then we canwrite it as follows:2 H = − (cid:4)(cid:6) − Mr − cr − ( + w) (cid:7) ˙ t + (cid:6) Mar + acr − ( + w) (cid:7) ˙ φ (cid:5) ˙ t + r (cid:12) ˙ r + (cid:4) − (cid:6) Mar + acr − ( + w) (cid:7) ˙ t + (cid:10) r + (cid:6) + Mr + cr − ( + w) (cid:7) a (cid:11) ˙ φ (cid:5) (27) ur. Phys. J. C (2021) 81:84 Page 5 of 16 84 which is equivalent to − E ˙ t + L ˙ φ + r (cid:12) ˙ r = ξ = constant (28)Here ξ = − , , r ˙ r = r E + (cid:6) Mr + cr − ( + w) (cid:7) ( a E − L ) + ( a E − L ) + ξ (cid:12) (29) For circular null geodesics (ξ = ) , the radial Eq. (29) gives r ˙ r = r E + (cid:6) Mr + cr − ( + w) (cid:7) ( a E − L ) + ( a E − L ), (30)where, E is the energy per unit mass and L is the angularmomentum per unit mass of the particle express the trajec-tory.At the points r = r o , E = E o and L = L o the circulargeodesics condition ˙ r = ( ˙ r ) (cid:5) = r o E o + (cid:6) Mr o + cr − ( + w) o (cid:7) ( a E o − L o ) + ( a E o − L o ) = , (31)2 r o E o − (cid:6) Mr o + c ( + w) r − ( + w) o (cid:7) ( a E o − L o ) = . (32)We can write the above Eqs. (31) and (32) in simplified formby introducing the impact parameter D o = L o E o as r o + (cid:6) Mr o + cr − ( + w) o (cid:7) ( a − D o ) + ( a − D o ) = , (33) r o − (cid:6) Mr o + c ( + w) r − ( + w) o (cid:7) ( a − D o ) = . (34)From the Eq. (34) we get D o = a ∓ r o (cid:14) Mr o + c ( + w) r ( − w) o . (35)Notice that the Eq. (33) is valid if and only if | D o | > a . Inthe case of counter rotating orbits, | D o − a | = − ( D o − a ) ,which correlates to upper sign in the above Eq. (35) while inthe case of co-rotating orbits, | D o − a | = ( D o − a ) whichcorrelates to the lower sign in the above Eq. (35). Putting the Eqs. (35) into (33), we obtain an equation forthe radius of null circular geodesics r o − Mr o ± a (cid:15) Mr o + c ( + w) r ( − w) o − c ( + w) r ( − w) o = . (36)When c =
0, we find the well known result [47]. By usingEqs. (33) and (35) we can find an another important relationfor null circular orbits as follows D o = a + r o (cid:6) Mr o + c ( + w) r ( − w) o Mr o + c ( + w) r ( − w) o (cid:7) . (37)To analyze the null circular geodesics we will derived animportant quantity is the angular frequency ((cid:9) σ ) which isgiven by (cid:9) o = (cid:6) − Mr − cr − ( + w) o D o (cid:7) + a (cid:6) Mr + cr − ( + w) o (cid:7)(cid:6) r + a + Ma r + a cr − ( + w) o D o (cid:7) − a (cid:6) Mr + cr − ( + w) o D o (cid:7) = D o (38)With the help of the Eqs. (35) and (33) we show that the angu-lar frequency ((cid:9) o ) of the equatorial null circular geodesicsis inverse of the impact parameter D o , which generalize thefour dimensional result [47]. Also it is a general property ofany stationary space-time. For time-like circular geodesics (ξ = − ) , the radial Eq.(29) gives r ˙ r = r E + (cid:6) Mr + cr − ( + w) (cid:7) ( a E − L ) + ( a E − L ) − (cid:12). (39)Now we shall focus on radial equation of ISCO which gov-erning the circular time like geodesics in terms of reciprocalradius u = r , can be written as V = u − ˙ u = E + Mu ( a E − L ) + cu ( + w) ( a E − L ) + ( a E − L ) u − u ( a − cu w − ) + Mu − , (40)where V is the effective potential.The condition for the existence of the circular orbits areat r = r σ or u = u σ is given by V = , (41)and d V du = . (42)
84 Page 6 of 16 Eur. Phys. J. C (2021) 81:84 V = E + Mu σ ( a E − L ) + cu ( + w)σ ( a E − L ) + ( a E − L ) u σ − u σ ( a − cu w − σ ) + Mu σ − = , (43)and d V du = Mu σ ( a E − L ) + c ( + w) u + wσ ( a E − L ) + ( a E − L ) u σ − u σ ( a − cu w − σ ) + c ( w − ) u wσ + M = . (44)For circular orbits , let L σ and E σ be the value of energyand angular momentum at the radius r σ = u σ , respectively.Now putting z = L σ − a E σ , in the Eqs. (43) and (44) weobtain the followings equations cu ( + w)σ z + Mu σ z − ( z + az E σ ) u σ − u σ ( a − cu w − σ ) + Mu σ − + E σ = , (45)and3 c ( + w) u ( + w)σ z + Mu σ z − ( z + az E σ ) u σ − u σ ( a − cu w − σ ) + c ( w − ) u wσ + M = . (46)With the help of the Eq. (46) we get an equation for E σ fromEq. (45) as E σ = − u σ M + M z u σ + c ( w − ) u w + σ + c ( w + ) u ( + w)σ z . (47)Again from the Eq. (46) we get2 az E σ u σ = z (cid:6) Mu σ + c ( + w) u + wσ − u σ (cid:7) − (cid:6) ( a − cu w − σ ) u σ − c ( w − ) u wσ − M (cid:7) . (48)After eliminating E σ from the Eqs. (47) and (48), we can getan quadratic equation for z as P z + Q z + S = . (49)where P = u σ (cid:4)(cid:6) Mu σ − + c ( + w) u w + σ (cid:7) − a (cid:6) Mu σ + c ( + w) u ( + w)σ (cid:7)(cid:5) Q = − u σ (cid:4)(cid:6) Mu σ − + c ( + w) u w + σ (cid:7) × (cid:6)(cid:6) ( a − cu w − σ ) u σ − c ( w − ) u wσ − M (cid:7) + a u σ (cid:6) − Mu σ + c ( w − ) u w + σ (cid:7)(cid:5) S = (cid:4)(cid:6) ( a − cu w − σ ) u σ − c ( w − ) u wσ − M (cid:7)(cid:5) Now, the solution of the quadratic Eq. (49) is z = − Q ± D P , (50)where D denotes the discriminant of the Eq. (49) which canbe found as D = au σ (cid:12) u σ (cid:15) Mu σ + c ( w + ) u w + σ , (51)and (cid:12) u σ = ( a − cu w − σ ) u σ − Mu σ + . (52)In order to write the solution of the Eq. (49) in simple form,we consider the following expression X + X − = (cid:6) Mu σ − + c ( + w) u w + σ (cid:7) − a (cid:6) Mu σ + c ( + w) u ( + w)σ (cid:7) , (53)where X ± = (cid:6) − Mu σ − c ( + w) u w + σ ± a (cid:15) Mu σ + c ( w + ) u (w + )σ (cid:7) . (54)Thus the solution reduces to z u σ = (cid:12) u σ − X ∓ X ∓ (55)Again we can utilize the identity (cid:12) u σ − X ∓ = u σ (cid:4) a √ u σ ± (cid:15) M + c ( w + ) u wσ (cid:5) (56)Hence the solution for z can be written in the following simpleform as z = − (cid:4) a √ u σ ± (cid:14) M + c ( w + ) u wσ (cid:5) √ u σ x ± . (57)Here the lower sign in the foregoing equations indicates tothe co-rotating orbit whereas the upper sign indicates to thecounter-rotating orbit. Inserting expression of (57) in the Eq.(47) we obtain following expression for E σ as energy E σ = √ z ± (cid:4) − Mu σ ∓ a (cid:15) Mu σ + c ( w + ) u w + σ (cid:5) (58) ur. Phys. J. C (2021) 81:84 Page 7 of 16 84 Again with the help of Eqs. (57), (58) and the relation L σ = a E σ + z , we obtain the angular momentum for thetime circular geodesics as L σ = ∓ √ u σ z ± (cid:4)(cid:15) M + c ( w + ) u wσ × (cid:6) + a u σ ± au σ (cid:15) Mu σ + c ( w + ) u w + σ (cid:7) ∓ ac (cid:14) u σ (cid:5) (59)In order to find the minimum radius for a stable circular orbit,we will obtained the second derivative of V with respect to r for the value of E σ and L σ specific to circular orbit. Herewe use the Eqs. (41, 42) with further equation d Vdu (cid:16)(cid:16)(cid:16)(cid:16) u = u σ = , (60)and we found that d Vdu = u σ (cid:4) M z u σ + c (w + )( w + ) u w + σ z + c ( w − ) u wσ − M (cid:5) (61)By the Eq. (55) we get d Vdu (cid:16)(cid:16)(cid:16)(cid:16) u = u σ = u σ X ± (cid:4)(cid:6) M + c (w + )( w + ) u wσ (cid:7) (cid:12) u σ − (cid:6) c (w + )( w + ) u wσ + c ( w − ) u wσ + M (cid:7) X ± (cid:5) (62)Therefore the equation of ISCO at the reciprocal radius u σ X ± (cid:4)(cid:6) M + c (w + )( w + ) u wσ (cid:7) (cid:12) u σ − (cid:6) c (w + )( w + ) u wσ + c ( w − ) u wσ + M (cid:7) X ± (cid:5) = (63)or this is similar to (cid:6) Mu σ + c (w + )( w + ) u w + σ (cid:7)(cid:6) a − cu w − σ (cid:7) + c ( w + w + )(w + ) u w + σ + cM ( w + w + ) u w + σ ± a (cid:6) M + c ( w + w + ) u wσ (cid:7) × (cid:15) Mu σ + c ( w + ) u (w + )σ + M u σ − M = r σ , we get the equation of innermoststable circular orbit(ISCO) for non extremal BH is identifiedby Mr w + σ − M r wσ − Ma r w − σ − cM ( w + w + ) r wσ ± a (cid:6) Mr wσ + c ( w + w + ) (cid:7)(cid:15) Mr w − σ + c ( w + ) r w − σ − c (w + ) (cid:4) a ( w + ) r w − σ + c ( w + w − ) (cid:5) = r σ = r I SC O which will be innermost stable circular orbit(ISCO) of theblack hole. Here ( − ) sign determines for direct orbit and (+)sign determines for retrograde orbit.Special cases:(i) When w = and c = − Q , we can get the equation ofISCO for non-extremal Kerr–Newman black hole whichis given by Mr σ − M r σ − Ma r σ + M Q r σ ∓ a ( Mr σ − Q ) + Q ( a − Q ) = . (66)The smallest root ( real ) of the Eq. (66) is the radius ofISCO.(ii) When w = and c =
0, the Eq. (65) is similar to theequation of ISCO for Kerr black hole which is as follows: r σ − Mr σ ∓ a (cid:8) Mr σ − a = . (67)The radius of the ISCO is equal to the real root of theabove equation.(iii) When w = and a =
0, we obtained the equation ofISCO for Reissner-Nordstr ¨ o m black hole as follows Mr σ − M r σ + M Q r σ − Q = . (68)The radius of the ISCO for the Reissner-Nordstr ¨ o m blackhole can be found by obtaining the smallest real root ofthe previous equation.(iv) When w = and c = a =
0, we obtain the radius ofISCO for Schwarzschild black hole is given by r σ − M = . (69)
84 Page 8 of 16 Eur. Phys. J. C (2021) 81:84
Now we will calculate the Lyapunov exponent and KS entryfor time-circular orbit. Using the Eq. (5), we get the Lyapunovexponent and KS entry in terms of the radial equation of ISCOas follows λ time = h ks = (cid:17)(cid:18)(cid:18)(cid:18)(cid:18)(cid:18)(cid:19) − (cid:6) Mr w + σ − M r wσ − Ma r w − σ − cM Gr wσ ± a H − cT (cid:7) r σ (cid:6) r w + σ − Mr wσ − c ( + w) ± a (cid:14) Mr wσ + c ( w + ) (cid:7) , (70)where G = ( w + w + ), H = (cid:6) Mr wσ + c ( w + w + ) (cid:7) × (cid:15) Mr w − σ + c ( w + ) r w − σ , T = (w + ) (cid:4) a ( w + ) r w − σ + c ( w + w − ) (cid:5) . (71)The condition for existing the time-circular geodesics motionof the test particle are energy( E σ ) and angular momentum( L σ )must be real and finite. For these we must have r w + σ − Mr wσ − c ( + w) ± a (cid:14) Mr wσ + c ( w + ) > r wσ > − c ( w + ) M From the above Eq. (70) we can conclude that the time-circular geodesics of non extremal black hole is stable when Mr w + σ − M r wσ − Ma r w − σ − cM Gr wσ ± a H − cT > , (72)that is, λ time or h ks is imaginary, the time-circular geodesicsis unstable when Mr w + σ − M r wσ − Ma r w − σ − cM Gr wσ ± a H − cT < , (73)such that λ time or h ks is real and circular geodesics ismarginally stable when Mr w + σ − M r wσ − Ma r w − σ − cM Gr wσ ± a H − cT = . (74)that is, λ time or h ks is zero.Special cases:(i) When w = , c = − Q , we can get the Lyapunov expo-nent and KS entry for Kerr–Newman black hole in termsof ISCO equation are given by λ K N = h ks = (cid:17)(cid:18)(cid:18)(cid:18)(cid:18)(cid:18)(cid:19) − (cid:6) Mr σ − M r σ − Ma r σ + M Q r σ ∓ a ( Mr σ − Q ) + Q ( a − Q ) (cid:7) r σ (cid:6) r σ − Mr σ ∓ a (cid:8) Mr σ − Q + Q (cid:7) (75)(ii) For Kerr black hole w = , c =
0, the Lyapunov expo-nent and KS entry for circular time-like geodesics are λ K err = h ks Fig. 1
Dimensionless instability exponent λ σ (cid:9) σ as a function of rotation a for real value of Lyapunov eponent. We use units such that M = , c =
2. Solid lines refer to corotating orbits and dashed lines refer tocounterrotating orbits ur. Phys. J. C (2021) 81:84 Page 9 of 16 84 = (cid:17)(cid:18)(cid:18)(cid:18)(cid:18)(cid:18)(cid:19) − M (cid:6) r σ − M r σ ∓ a √ Mr σ − a (cid:7) r σ (cid:6) r σ − Mr σ ∓ a √ Mr σ (cid:7) (76)(iii) For the Reissner-Nordstr ¨ o m black hole w = , c =− Q , a =
0, the Lyapunov exponent and KS entry fortime-like geodesics are λ RN = h ks = (cid:17)(cid:18)(cid:18)(cid:18)(cid:19) − ( Mr σ − M r σ + M Q r σ − Q ) r σ (cid:6) r σ − Mr σ + Q (cid:7) (77)(iv) For Schwarzschild black hole w = , c = − Q = , a =
0, the Lyapunov exponent and KS entry for cir-cular time-like geodesics in terms of ISCO equation aregiven by λ Sch = h ks = (cid:2) − M ( r σ − M ) r σ ( r σ − M ) (78) Lyapunov exponent and KS entry for circular null geodesicsare λ Null = h ks = (cid:17)(cid:18)(cid:18)(cid:18)(cid:19) ( L o − a E o ) (cid:6) Mr o + c ( + w) r ( − w) o (cid:7) r o (79)Since ( L o − a E o ) ≥ r w o > − c (w + ) M then λ Null isreal, so the circular null geodesics are unstable. Special cases:(i) When w = , c = − Q , we can get the Lyapunov expo-nent and KS entry for Kerr–Newman black hole which isgiven by λ K N = h ks = (cid:2) ( L o − a E o ) ( Mr o − Q ) r o . (80)(ii) When w = , c =
0, the Eq. ( 79 ) is similar to theequation Kerr black hole which is as follows : λ K err = h ks = (cid:2) M ( L o − a E o ) r o . (81)(iii) For the Reissner-Nordstr ¨ o m black hole w = , c =− Q , a =
0, the Lyapunov exponent and KS entry forcircular null geodesics are λ RN = h ks = (cid:2) L o ( Mr o − Q ) r o . (82)When r o > Q M , λ RN is real which implies that null-circular geodesics for Reissner- Nordstr ö m black hole isunstable.(iv) For Schwarzschild black hole w = , c = , a = λ Sch = h ks = (cid:2) M L o r σ . (83)It can be verify that when r o = M , λ Sch is real. Hencenull-circular geodesics For Schwarzschild photon sphereare unstable.4.3 Angular velocity of time-like circular geodesicFor time-like circular geodesic, the angular velocity at r = r σ is given by (cid:9) σ = ˙ φ ˙ t = (cid:6) − Mr σ − cr − ( + w)σ (cid:7) L σ + a (cid:6) Mr σ + cr − ( + w)σ (cid:7) a E σ (cid:6) r σ + a + Ma r σ + a cr − ( + w)σ (cid:7) E σ − a (cid:6) Mr σ + cr − ( + w)σ (cid:7) a L σ (84)The Eq. (84) can be written as
84 Page 10 of 16 Eur. Phys. J. C (2021) 81:84
Fig. 2
The plots shows the time-like orbital frequency (cid:9) σ to the radialcoordinate r σ of corotating orbits (left panel) and counterrotating orbits(lower panel) for different values of w . We use units M = , a = . c = Fig. 3
The plot shows the orbital time period of time-like circulargeodesic T σ to the radius r σ . We use units such that M = , a = . c =
2. Solid lines refer to corotating orbits and dashed lines referto counterrotating orbits (cid:9) σ = [ L σ − Mu σ z − cu ( + w)σ z ] u σ ( + a u σ ) E σ − Mau σ z − acu ( + )σ z . (85)The numerator of the Eq. (85) can be written in the simplifiedform as L σ − Mu σ z − cu ( + w)σ z = ∓ (cid:14) M + c ( w + ) u wσ √ u σ z ± (cid:12) u σ . (86)Similarly, the denominator can be written as ( + a u σ ) E σ − Mau σ z − acu ( + )σ z = (cid:12) u σ √ z ± (cid:6) ∓ a (cid:15) Mu σ + c ( w + ) u (w + )σ (cid:7) . (87)Putting Eqs. (86) and (87) into Eq. (85) we obtain the angularvelocity for time circular orbit as (cid:9) σ = ∓ (cid:14) Mu σ + c ( w + ) u (w + )σ ∓ a (cid:14) Mu σ + c ( w + ) u (w + )σ (88)Now the angular velocity in terms of r σ for time circulargeodesics is (cid:9) σ = ∓ (cid:14) Mr wσ + c ( w + ) r (w + ) σ ∓ a (cid:14) Mr wσ + c ( w + ) (89)Now the time period for circular time-like orbit is givenby T σ = π(cid:9) σ = ∓ π · r (w + ) σ ∓ a (cid:14) Mr wσ + c ( w + ) (cid:14) Mr wσ + c ( w + ) (90)At the limit w = , c = , a =
0, the above equationcan be written in the form T σ ∝ r σ . This shows that theEq. (90) satisfies relativistic Kepler’s law for Schwarzschildblack hole.4.4 Critical exponent Now we will calculate the critical exponent for equatorialtime-like circular geodesics. Thus the reciprocal of criticalexponent is followed by the Eq. (8) ur. Phys. J. C (2021) 81:84 Page 11 of 16 84 γ = π (cid:17)(cid:18)(cid:18)(cid:18)(cid:18)(cid:18)(cid:18)(cid:19) − (cid:6) Mr w + σ − M r wσ − Ma r w − σ − cM Gr wσ ± a H − cT (cid:7)(cid:6) r (w + ) σ ∓ a (cid:14) Mr wσ + c ( w + ) (cid:7) r σ (cid:6) Mr wσ + c ( w + ) (cid:7)(cid:6) r w + σ − Mr wσ − c ( + w) ± a (cid:14) Mr wσ + c ( w + ) (cid:7) (91)Since (cid:6) r (w + ) σ ∓ a (cid:14) Mr wσ + c ( w + ) (cid:7) ≥ (cid:6) r w + σ − Mr wσ − c ( + w) ± a (cid:14) Mr wσ + c ( w + ) (cid:7) ≥ r wσ > − c ( w + ) M and Mr w + σ − M r wσ − Ma r w − σ − cM Gr wσ ± a H − cT <
0, so γ is real which shows that equatorialtime-like circular geodesics is unstable.Special cases:(i) When w = , c = − Q , we can get the equation ofcritical exponent for Kerr–Newman black hole in termsof ISCO which is given by1 γ = π (cid:17)(cid:18)(cid:18)(cid:19) − ( Mr σ − M r σ − Ma r σ + M Q r σ ∓ a ( Mr σ − Q ) + Q ( a − Q )( r σ ∓ a (cid:8) Mr σ − Q ) r σ ( Mr σ − Q )( r σ − Mr σ ∓ a (cid:8) Mr σ − Q + Q ) (92)(ii) When w = and c =
0, the Eq. (91) is similar to theequation of Critical exponent for Kerr black hole in termsof ISCO which is as follows :1 γ = π (cid:17)(cid:18)(cid:18)(cid:18)(cid:18)(cid:18)(cid:19) − (cid:6) r σ − M r σ ∓ a √ Mr σ − a (cid:7) ( r σ √ r σ ∓ a √ M ) r σ (cid:6) r σ − Mr σ ∓ a √ Mr σ (cid:7) (93)(iii) For the Reissner-Nordstr ¨ o m black hole w = , c = − Q and a =
0, the reciprocal of Critical exponent for time-like geodesic is1 γ = π (cid:2) − ( Mr σ − M r σ + M Q r σ − Q )( Mr σ − Q )( r σ − Mr σ + Q ) (94)(iv) When w = and c = a =
0, we obtained the recipro-cal of Critical exponent for Schwarzschild black hole interms of ISCO equation1 γ = π (cid:2) − ( r σ − M ) r σ − M (95) The reciprocal of critical exponent associated to null circulargeodesics is given by (cid:6) γ (cid:7) Null = π (cid:6) r (w + ) o ∓ a (cid:15) Mr w o + c ( w + ) (cid:7) × (cid:17)(cid:18)(cid:18)(cid:18)(cid:18)(cid:18)(cid:19) ( L o − a E o ) (cid:6) Mr o + c ( + w) r ( − w) o (cid:7) r o (cid:6) Mr w o + c ( w + ) (cid:7) (96)Special cases:We take following limits in the above Eq. (96):(i) When w = and c = − Q , we can get reciprocal ofCritical exponent for Kerr–Newman black hole which isgiven by
84 Page 12 of 16 Eur. Phys. J. C (2021) 81:84 (cid:6) γ (cid:7) Null = π (cid:2) ( L o − a E o ) ( Mr o − Q )( r o ∓ a (cid:8) Mr o − Q ) r o ( Mr o − Q ) (97)(ii) When w = and c =
0, the Eq. (96) is similar to thereciprocal of Critical exponent of null-circular geodesicsfor Kerr black hole which is as follows : (cid:6) γ (cid:7) null = π (cid:2) ( L o − a E o ) ( r o √ r o ∓ a √ M ) r (98)(iii) For the Reissner-Nordstr ¨ o m black hole w = , c = − Q and a =
0, the reciprocal of Critical exponent for null-circular geodesic is (cid:6) γ (cid:7) null = π (cid:2) L o ( Mr o − Q ) r σ ( Mr o − Q ) (99)(iv) When w = and c = a =
0, we obtained the recipro-cal of Critical exponent for Schwarzschild black hole interms of null-circular geodesics is (cid:6) γ (cid:7) null = π (cid:2) L o r o (100) It is known that for stable circular orbits the effective potentialhas a the local minima. Thus the condition for the existence ofstable circular orbit exists is d V ef f / dr > d V ef f / dr = E σ = E σ = z u σ = Mu σ − c ( w + ) u w + σ Mu σ + c ( w + ) u w + σ , (101)Again from Eq. (57) we have z u σ = (cid:4) au σ ± (cid:14) Mu σ + c ( w + ) u w + σ (cid:5) x ∓ . (102)Hence from (101) and (102) we get x ∓ = (cid:4) Mu σ + c ( w + ) u w + σ Mu σ − c ( w − ) u w + σ (cid:5) × (cid:4) au σ ± (cid:15) Mu σ + c ( w + ) u w + σ (cid:5) , (103)or the above equation can be written as (cid:6) Mu σ − c ( w − ) u w + σ (cid:7)(cid:6) − Mu σ − c ( + w) u w + σ ± a (cid:15) Mu σ + c ( w + ) u (w + )σ (cid:7) = (cid:6) Mu σ + c ( w + ) u w + σ (cid:7) × (cid:4) au σ ± (cid:15) Mu σ + c ( w + ) u w + σ (cid:5) (104)After simplification we get the equation of marginally boundcircular orbit in terms of u σ as follows: (cid:6) a + c ( w + ) u w − σ + ( w − )(w + ) ( w + ) u w − σ (cid:7) − M (cid:6) c ( w + ) u w − σ + a − c ( w − ) u w − σ (cid:7) u σ − c ( w + ) u w + σ − (cid:6) M u σ + c ( w − ) u wσ (cid:7) ∓ (cid:6) a Mu σ + ac ( w + ) u w − σ (cid:7) × (cid:15) Mu σ + c ( w + ) u ( w + )σ + M = r σ we can get the above equation as Mr w + σ − (cid:6) M r w σ + c ( w − ) r w + σ (cid:7) − M (cid:6) c ( w + ) r wσ + a r w − σ − c ( w − ) r wσ (cid:7) ∓ (cid:6) a Mr wσ + ac ( w + ) (cid:7)(cid:15) Mr w − σ + c ( w + ) r ( w − )σ − (cid:6) a r w − σ + c ( w + ) + ( w − )(w + ) ( w + ) (cid:7) = . (106) ur. Phys. J. C (2021) 81:84 Page 13 of 16 84 Let r σ = r mb be the smallest(real) root of the Eq. (106).The root will be the closest bound circular orbit to the blackhole.Special cases:(i) When w = and c = − Q , we can get the equation ofmarginally bound circular orbit for Kerr–Newman blackhole which is given by Mr σ − M r σ − Ma r σ + M Q r σ ∓ ( a Mr σ − a Q ) (cid:8) Mr σ − Q + Q ( a − Q ) = . (107)The smallest root (real) of the Eq. (107) is the radius ofmarginally bound circular orbit of the black hole.(ii) When w = and c =
0, the Eq. (106) is similar tothe equation of Marginally bound circular orbit for Kerrblack hole which is as follows : r σ − Mr σ ∓ a (cid:8) Mr σ − a = . (108)The radius say r σ = r mb can be obtained by finding realsmallest root of the above equation for the marginallybound circular orbit of the black hole.(iii) For the Reissner-Nordstr ¨ o m black hole w = , c = − Q and a =
0, we can get the equation of marginally boundcircular orbit which is as follows Mr σ − M r σ + M Q r σ − Q = w = and c = a =
0, we can obtained the radiusof marginally bound circular orbit for Schwarzschildback hole as follows r σ − M = . (110)More specifically, we can say that r σ = M is the radiusof marginally bound circular orbit for Schwarzschildback hole.The static radius for the circular orbits of test particles isgiven by the condition d V ef f dr = We have already computed the angular velocity for time cir-cular geodesics [50] in the Eq. (89) which is given by (cid:9) σ = ∓ (cid:14) Mr wσ + c ( w + ) r (w + ) σ ∓ a (cid:14) Mr wσ + c ( w + ) . (111)Again, the similar expression of angular velocity (cid:9) o = D o for null circular geodesics can be obtain from the above equa-tion as (cid:9) o = ∓ (cid:14) Mr w o + c ( w + ) r (w + ) o ∓ a (cid:14) Mr w o + c ( w + ) . (112)Now the ratio of angular velocity between null-circulargeodesics and time-circular geodesics is (cid:9) o (cid:9) σ = ⎛⎝ (cid:14) Mr w o + c ( w + ) (cid:14) Mr wσ + c ( w + ) ⎞⎠ × ⎛⎜⎝ r (w + ) σ ∓ a (cid:14) Mr wσ + c ( w + ) r (w + ) o ∓ a (cid:14) Mr w o + c ( w + ) ⎞⎟⎠ . (113)(i) When the radius of time-circular geodesics is equal to theradius of null circular geodesic, the corresponding angu-lar velocities are also equal, that is, when r σ = r o , Q σ = Q o which shows that the intriguing physical phenom-ena could occurs in the curve space-time. For instance,it would increase the interesting possibility of excitingquasinormal frequencies of the black hole by orbitingparticle, possibly leading to instabilities of the space-time[51,52].(ii) When the radius of time-circular geodesics is greaterthan the radius of null-circular geodesics,that is, r σ > r o then Q o > Q σ which implies that null-circular geodesicsis characterized by the largest angular frequency than theangular frequency of time-circular geodesics as measuredby asymptotic observer. Therefore, such type of space-time are characterized by (cid:9) null > (cid:9) timelike . (114)This type of characteristic of null-circular geodesic hasbeen graphed in the Fig. 4 for different values of radius r .Thus from the above equation we may conclude that thenull-circular geodesic provide the fastest way to circle ofblack hole [53,54]. This is satisfied for the case of Spher-ically symmetry Schwarzschild black hole, Kerr black
84 Page 14 of 16 Eur. Phys. J. C (2021) 81:84
Fig. 4
The plot shows the orbital angular frequency (cid:9)
Versus a fordifferent values of radius r in the range r o ≤ r ≤ r σ , where r σ and r o arethe radius of time-like geodesic and null-circular geodesic, respectively.Green (solid) color indicates for the value where the radius of time-likeand null-circular geodesics are coincide, Red(Dashed) color indicatesfor the value of radius of null-circular geodesic and blue (Dot-Dashed)color indicates the value of radius of time-like geodesic. We use unitssuch that M = , c = hole and is still much general and can be applied for thecase of Stationary, Kerr–Newman, axisymmetry space-time. From the Eq. (90), the time period for time-like geodesics isgiven by T σ = ∓ π (cid:6) r (w + ) σ ∓ a (cid:14) Mr wσ + c ( w + ) (cid:7)(cid:14) Mu wσ + c ( w + ) . (115)Substituting r σ = r o in the above equation, we can deducethe time period for null-circular geodesics as T o = ∓ π (cid:6) r (w + ) o ∓ a (cid:14) Mr w o + c ( w + ) (cid:7)(cid:14) Mu w o + c ( w + ) . (116)Now the ratio of time period between null-circular geodesicsand time-circular geodesics is given by T o T σ = ⎛⎝ (cid:14) Mr wσ + c ( w + ) (cid:14) Mr w o + c ( w + ) ⎞⎠ × ⎛⎜⎝ r (w + ) o ∓ a (cid:14) Mr w o + c ( w + ) r (w + ) σ ∓ a (cid:14) Mr wσ + c ( w + ) ⎞⎟⎠ . (117) Fig. 5
The plot shows the orbital time period T versus a for differentvalues of radius r in the range r o ≤ r ≤ r σ , where r σ and r o are theradius of time-like geodesic and null-circular geodesic, respectively.Blue color indicates for the value where the radius of time-like andnull-circular geodesics are coincide, green color indicates for the valueof radius of time-like geodesic and red color indicates the value of radiusof null-circular geodesic. We use units such that M = , c = (i) When r σ = r o , the Eq. (115) and the Eq. (116) becameidentical, that is, the time period for time-like geodesicsis similar to the time period of null-circular geodesicswhich leading to excitations of quasinormal modes.(ii) When the radius of null- circular geodesics is smaller thanthe radius of time-like geodesics, orbital time period ofnull-circular geodesics is smaller than the orbital timeperiod of time-circular geodesics, that is, for r o < r σ , T o < T σ . Let r o = r photon , r σ = r I SC O , the ratiobetween the orbital time period for photon sphere andthe orbital time period for ISCO be T photon T I SC O = ⎛⎝ (cid:14) Mr wσ + c ( w + ) (cid:14) Mr w o + c ( w + ) ⎞⎠ × ⎛⎜⎝ r (w + ) o ∓ a (cid:14) Mr w o + c ( w + ) r (w + ) σ ∓ a (cid:14) Mr wσ + c ( w + ) ⎞⎟⎠ . (118)which leads to T photon < T I SC O , (119)this implies that the time-like circular geodesics providesthe slowest possible orbital time period. This type of char-acteristic of time-like circular geodesic can be easily seenfrom Fig. 5. Therefore we conclude that among all thecircular geodesics, the null-circular geodesics is charac-terized by the fastest way to circle of black hole. ur. Phys. J. C (2021) 81:84 Page 15 of 16 84 In this paper we have clarify some aspect about principleLyapunov exponent, KS entropy and unstable null-circulargeodesics. We have presented that the principle Lyapunovexponent and KS entropy can be express in terms of theequation of ISCO (innermost stable circular orbit). Alsowe have highlighted that Lyapunov exponent can be utilizeto determine the instability of equatorial circular geodesicsfor both time-like and null-circular geodesics. We haveexplored the relation for null-circular geodesics that the angu-lar frequency ( Q o ) is equal to inverse of impact parameter ( D o ) , which is general characteristic of any stationary space-time. Also we computed the equation of ISCO in which thesmallest real root give the radius of ISCO for rotating blackhole and for each of the special cases black holes like Kerr–Newman, Kerr black hole, Reissner-Nordstr ¨ o m black holeand Schwarzschild black hole. We showed that r σ = M isthe radius of ISCO for Schwarzschild black hole.We have computed the equations of Lyapunov exponentand reciprocal of critical exponent for both the cases of time-like and null-circular geodesics for rotating black hole andfor special cases. We constructed the equation of marginallybound circular orbit(MBCO) and showed that r σ = M isthe radius of MBCO for Schwarzschild black hole . We havealso determined the ratio of angular frequency of null-circulargeodesics to time-circular geodesics. In this ratio we clarifythat null circular geodesics have largest angular frequencythan the time circular geodesics, that is, Q o > Q σ . Thischaracteristic of null-circular geodesic can be check by theFig. 4 above, which is graphed by taking M = c = T timelike > T photon . This scenario has been plottedin the Fig. 5, which is graphed by taking different valuesof radius r . In fact, we may conclude that any stable time-circular geodesics other than the ISCO traverses slowly thanthe null-circular geodesics. Acknowledgements
FR is grateful to the Inter-University Centre forAstronomy and Astrophysics (IUCAA), Pune, India for providing Asso-ciateship programme under which a part of this work was carried out.We are thankful to the referee for his valuable suggestions.
Data Availability Statement
This manuscript has no associated dataor the data will not be deposited. [Authors’ comment: It’s a theoreticalwork where the graphs were generated analytically.]
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