Investigation of circular geodesics in a rotating charged black hole in the presence of perfect fluid dark matter
aa r X i v : . [ g r- q c ] S e p Prepared for submission to JCAP
Investigation of the circulargeodesics in a rotating chargedblack hole in presence of perfectfluid dark matter
Anish Das, a Ashis Saha, b Sunandan Gangopadhyay a a Department of Theoretical Sciences, S.N. Bose National Centre for Basic Sciences,JD Block, Sector-III, Salt Lake, Kolkata 700106, India b Department of Physics, University of Kalyani, Kalyani 741235, IndiaE-mail: [email protected], [email protected],[email protected]
Abstract.
We give a charged black hole solution in perfect fluid dark matter (PFDM). Themetric corresponding to the rotating avatar of the black hole solution is obtained by incorpo-rating the Newman-Janis algorithm. We then compute two type of circular geodesics, namely,the null geodesics and time-like geodesics for the mentioned spacetime geometry. For the caseof time-like geodesics we consider massive particles characterized by charge with values q = 0 and q = 0 . The effective potentials of the corresponding circular geodesics has also beenstudied briefly. We then continue the subsequent analysis by graphically representing thecollective effects of the black hole parameters, namely, the charge of the black hole ( Q ), spinparameter ( a ) and the PFDM parameter ( α ) on the energy ( E ), angular momentum ( L ) andeffective potential ( V eff ) of the concerned particle. We then discuss the Penrose process in or-der to study the negative energy particles which have possible existence within the ergosphereand which in turn leads to the energy gain of the emitting particle. ontents The study of the black holes, one of the most fascinating objects in the universe has beenplaying the key role in the study about the observational aspects of the geometric theory ofrelativity. Although the general theory of relativity is still not able to grasp the whole storyof the black holes, it turns out that upto a great extent the results of general relativity matchwith observations which urges us to work following the same. The supermassive black holesare believed to be the central core of all the galaxies and are responsible for holding the entiregalaxy together. Recently the shadow images of the M87 [1] provided the first direct evidencefor the existence of black holes and this motivates us to study some interesting aspects ofblack holes. It is difficult to study black holes directly and measure any of it’s propertieslike Hawking temperature, etc. So the study of spacetime structure and thereby the geodesicmotion of various types of particles in it’s vicinity, helps to gain knowledge regarding thefeatures of black holes. A vast amount of study has been done on the geodesic motion aroundblack holes and it is very hard to mention them all, yet some of them has to be lookedupon due to their novelty. In [2, 3], the study of geodesics around Schwarzchild and Kerrblack holes can be found in detail. Besides that, geodesic motion of massive particles havebeen studied around conformal Schwarzchild black hole in [4]. The study of geodesics inregular Hayward BH [5], Kerr-Sen BH [6, 7], Kerr-Newman-Taub-NUT BH [8], generic blackhole coupled to non-linear electrodynamics [9] and Schwarzchild black hole in quintessence[10] are worth mentioning. Since the observer in reality is far away from the black hole sotheoretically one places the observer at infinity from which all planes are identical and hencefor convenience and effectiveness the geodesics in the equatorial plane of the black holes areable to capture the whole story. The geodesics in equatorial plane for dyonic Kerr-Newmanblack hole has been studied in [11] and that for quintessential rotating black hole with andwithout cosmological constant ( Λ ) has been studied in [12] and [13] respectively. Also the– 1 –tudy of geodesics for distorted static BH can be found in [14] along with particle motion inKerr spacetime in [15] and Kerr pierced by cosmic string in [16]. Besides massive test particles,the study of null geodesics demands more attention since they consume an extensive area ofresearch now-a-days after the discovery of black hole shadow image recently by The EventHorizon Telescope collaboration [1]. The motivation to study the null geodesics is related tothe complete understanding about the photon sphere which in turn leads to the formationof the black hole shadow. There are many studies on the null geodesics and thereby onthe shadow. Some of them can be found in [7],[17] − [24]. Apart from the study of photonsand massive particles, researchers have shown considerable interest in the study of chargedparticles due to the presence of the Maxwell fields which leads to additional interacting terms.The study of charged particles around Kerr Newman black hole has been carried out in [25]-[28]. Also the motion of the charged particles in case of black holes immersed in externalelectromagnetic fields has been studied in [29] − [33].Another interesting thing in this context is dark matter and dark energy which seeks ourattention by its own right. Dark matter is supposed to fill about 27% of the universe alongwith dark energy (68%) and ordinary matter(5%). Due to the indirect observations of darkmatter, it is assumed to be present everywhere within and outside the observable universeand the amount increases as we move away from the galactic center ( from the observationof galaxy rotation curves ). There are different models of dark matter which are very usefulto explain large and small scale structure of the universe [34] − [36]. Besides, dark matter inmost theories is assumed to be present around black holes as halos and black hole shadowshas been calculated in those cases as in [37, 38]. Also quintessential dark matter solutionswere studied by Kiselev [40]. In recent times dark matter has been studied using perfectfluid model as in [42]. This model has gained attention recently and helps to identify theimpact of the dark matter on various observables related to the black holes. There has beenstudy of shadows in perfect fluid dark matter in case of rotating black holes with and withoutcosmological constant [43, 44] along with non-rotating charged black holes [45]. Dark matteris composed of non-baryonic matter and being present around black holes can contribute tothe effctive mass of the black hole system. The geodesics around black hole in PFDM (perfectfluid dark matter) should in principle have an appreciable influence due to the presence ofextra matter and hence can led to interesting observations. Besides, black holes are sources ofextreme energy which can be gained as theorised by Penrose [46]. The process has maximumefficiency at the event horizon of the black hole and requires particles of negative energyand angular momentum to be absorbed by the black hole which requires the presence of ergosphere , the place where the process takes place. There are different studies of Penroseprocess and thereby the extraction of energy from the black hole as in [47] − [50] where energyextraction for spinning particles have been studied and also it is found that the efficiencyof the Penrose process increases in case of higher dimensions. In this paper we study thePenrose process for Kerr-Newman black holes in asymptotically flat spacetime in presence ofthe PFDM. We can briefly describe the plan for this paper in the following points below.In this work, we give a static, charged black hole solution surrounded by perfect fluid darkmatter (PFDM). We obtain the rotating version of the black hole solution by incorporatingthe Newman-Janis algorithm. The variation of the event horizon with respect to the PFDMparameter α has been observed. We study the geodesics corresponding to both massless andmassive particles in the vicinity of the rotating charged black hole with PFDM. We then studythe circular geodesics of photons and use them to determine the radius of unstable photonorbits. Furthermore, we look into how the dark matter parameter α as well as the spin ( a )– 2 –nd charge ( Q ) of the black hole affect the photon radius. In case of massive particles, wecompute the energy ( E ) and the angular momentum ( L ) of the particle in addition with theeffective potential ( V eff ). The effects of the PFDM parameter α , spin ( a ) and charge ( Q ) onthese computed entities, has been observed briefly. We then study the Penrose process. Wewish to study how PFDM parameter α affects the size of the ergosphere, the negative energyof the particle and thereby the efficiency of the Penrose process.The paper is organised as follows. In section 2 we discuss the charged black hole spacetimein perfect fluid dark matter. In section 3, we introduce rotation in the black hole by usingthe Newman-Janis algorithm. In section 4, we then discuss the circular geodesics of variousparticles (photons, uncharged and charged massive particles) and then we show the natureof potential of the black hole in section 5 . Later we discuss the Penrose process in detailin section 6 and conclude by summarising the results in section 7. In this paper we haveassumed ~ = c = G = 1 . We consider a ( )-dimensional gravity theory minimally coupled with a U(1) gauge field,in presence of a perfect fluid dark matter (PFDM). The action can be written in the followingform [36, 39–41, 45] S = Z d x √− g (cid:16) R πG + 14 F µν F µν + L DM (cid:17) (2.1)where, R is the Ricci scalar and G is the Newton’s gravitational constant. F µν is the Maxwellfield strength which is related with the electromagnetic potential A µ as F µν = ∂ µ A ν − ∂ ν A µ and L DM gives the Lagrangian density for the perfect fluid dark matter. Extremizing theaction we get the Einstein field equations as R µν − g µν R = 8 πG ( T Mµν + T DMµν ) . (2.2)In the above equation, T Mµν represents the energy-momentum tensor corresponding to theordinary matter (in this case the Maxwell field). This can be denoted as [55] ( T µµ ) M ≡ g µν T Mµν = diag (cid:18) − Q πGr , − Q πGr , Q πGr , Q πGr (cid:19) (2.3)where Q is the electric charge. On the other hand T DMµν corresponds to the energy-momentumtensor of the perfect fluid dark matter [51]. It is specified as ( T µµ ) DM ≡ g µν T DMµν = diag ( − ρ, P, P, P ) (2.4)where ρ and P correspond to density and pressure of the perfect fluid dark matter. Followingthe approach given in [41, 51], we further consider ( T θθ ) DM = ( T tt ) DM (1 − ǫ ) (2.5)with ǫ being a constant. By substituting the components (from eq.(2.4)) in eq.(2.5), we obtainthe equation of state for PFDM to be [51] Pρ = ( ǫ − . (2.6)– 3 –n order to obtain a static, spherically symmetric solution, we assume an ansatz metric of theform ds = − e ν dt + e λ dr + r ( dθ + sin θdφ ) (2.7)with ν = − λ and ν , λ being functions of r only. Now, the Einstein field equations read e − λ (cid:16) r − λ ′ r (cid:17) − r = 8 πG (cid:16) − ρ − Q πGr (cid:17) e − λ (cid:16) r + ν ′ r (cid:17) − r = 8 πG (cid:16) P − Q πGr (cid:17) e − λ (cid:16) ν ′ + ν ′ ν ′ − λ ′ r − ν ′ λ ′ (cid:17) = 8 πG (cid:16) P + Q πGr (cid:17) e − λ (cid:16) ν ′ + ν ′ ν ′ − λ ′ r − ν ′ λ ′ (cid:17) = 8 πG (cid:16) P + Q πGr (cid:17) (2.8)where the prime ( ′ ) denotes derivative with respect to radial coordinates ( r ). The first andthird equations can be rearranged to the form e − λ (cid:16) r − λ ′ r (cid:17) − r + Q r = − πGρe − λ (cid:16) ν ′ + ν ′ ν ′ − λ ′ r − ν ′ ′ λ ′ (cid:17) − Q r = 8 πGP . (2.9)By using the equation of state for the PFDM (given in eq.(2.6)) in the above eq.(s) and takingtheir ratio we get e − λ (cid:16) ν ′ + ν ′ ν ′ − λ ′ r − ν ′ ′ λ ′ (cid:17) − Q r = (1 − ǫ ) " e − λ (cid:16) r − λ ′ r (cid:17) − r + Q r . (2.10)In order to solve the above equation, we set ν = − λ = ln(1 − U ) where U ≡ U ( r ) . This inturn simplifies the above equation to the following form U ′′ ǫ Ur + ( ǫ − Ur + (2 − ǫ ) Q r = 0 . (2.11)Eq.(2.11) can be solved for different values of ǫ [41]. However we are particularly interestedin the solution for ǫ = [39, 41]. For ǫ = , eq.(2.11) reduces to the following form r U ′′ + 3 rU ′ + U + Q r = 0 . (2.12)The solution of the above equation is obtained to be U ( r ) = r s r − Q r − αr ln (cid:16) r | α | (cid:17) (2.13)where r s and α are integration constants. In order to evaluate r s , we set Q = 0 and α = 0 .In this limit, by utilizing the weak field approximation, r s is obtained to be GM . Thus thelapse function becomes f ( r ) = e ν = e − λ = e ln (1 − U ) = 1 − U = 1 − GMr + Q r + αr ln (cid:16) r | α | (cid:17) (2.14)corresponding to the following metric of a static, spherically symmetric, charged black holein PFDM ds = − f ( r ) dt + 1 f ( r ) dr + r dθ + r sin θdφ . (2.15)– 4 – Rotating charged black hole in perfect fluid dark matter: Newman-Janisalgorithm
From the point of view of a more realistic set up, inclusion of spin parameter in the black holemetric is necessary. In order to incorporate spin ( a ), we shall use the Newman-Janis algorithm[52] which is by far the easiest and most effective technique to include the spin parameter toany metric without cosmological constant. Before that we would like to consider a generalmetric and follow the approach accordingly as in [53]. Let us assume a metric of the form ds = − f ( r ) dt + 1 g ( r ) dr + h ( r ) (cid:16) dθ + sin θdφ (cid:17) . (3.1)Then the metric can be rewritten in the Eddington-Finkelstein coordinates ( u ) by the trans-formation dt = du + dr p f ( r ) g ( r ) (3.2)which modifies the metric as ds = − f ( r ) du − s f ( r ) g ( r ) dudr + h ( r ) (cid:16) dθ + sin θdφ (cid:17) . (3.3)We now introduce the null tetrads Z µ = (cid:16) l µ , n µ , m µ , m µ (cid:17) in terms of which the metric tensorcan be written as g µν = − l µ n ν − l ν n µ + m µ m ν + m ν m µ (3.4)with the relation between the tetrad components as l µ n µ = − m µ m µ = 1 and all others arezero. Now in order to get the inverse metric, we need to represent the tetrad components interms of ( u, r, θ, φ ). This reads l µ = δ µr n µ = s g ( r ) f ( r ) δ µu − g ( r )2 δ µr m µ = 1 p h ( r ) (cid:16) δ µθ + i sin θ δ µφ (cid:17) . (3.5)To incorporate spin to the metric, we make the transformation of the form u → u ′ = u − ia cos θ and r → r ′ = r + ia cos θ which transforms the null tetrads, and the transformationrelations look like Z ′ µβ = ∂x ′ µ ∂x ν Z νβ . (3.6)Here µ denotes the components of the tetrads along the directions u, r, θ, φ and β denotes thetetrads ( l, n, m, ¯ m ). So the transformation results in a new set of tetrads which read l ′ µ = δ µr ; n ′ µ = s g ( r ) f ( r ) δ µu − f ( r )2 δ µr ; m ′ µ = 1 p h ( r ) (cid:16) ia sin θ (cid:16) δ µu − δ µr (cid:17) + δ µθ + i sin θ δ µφ (cid:17) . – 5 –ince r and u got transformed, hence the components of the metric tensor will also change. Sothe modified functions are f ( r ) → F ( r, θ ) , g ( r ) → G ( r, θ ) and h ( r ) → H ( r, θ ) . The non-zerocomponents of the inverse metric tensor are as follows g uu = a sin θH ; g rr = G + a sin θH ; g ur = g ru = − r GF − a sin θHg θθ = 1 H ; g φφ = 1 H sin θ ; g uφ = g φu = aH ; g rφ = g φr = − aH . We now obtain the non-zero components of the metric g uu = − F ; g ur = g ru = − r FG ; g uφ = g φu = (cid:16) − a r FG + aF (cid:17) sin θg θθ = H ; g φφ = sin θ (cid:16) H + 2 a sin θ r FG − a F sin θ (cid:17) ; g rφ = g φr = a sin θ r FG .
The resulting metric reads ds = − F du − r FG dudr + 2 a sin θ (cid:16) F − r FG (cid:17) dudφ + 2 a sin θ r FG drdφ + Hdθ + sin θ h H + a sin θ (cid:16) r FG − F (cid:17)i dφ . (3.7)Now we need to remove u and express the metric in the Boyer-Lindquist coordinates for whichthe necessary transformation is du = dt + ξ ( r ) dr and dφ = dφ + ξ ( r ) dr . Only the diagonalelements and dtdφ component of the metric survives, rest are zero. This leads to the valuesof ξ and ξ as ξ ( r ) = − (cid:16)q GF H + a sin θ (cid:17)(cid:16) GH + a sin θ (cid:17) ; ξ ( r ) = − a (cid:16) GH + a sin θ (cid:17) and the metric becomes ds = − F dt + (cid:16) HGH + a sin θ (cid:17) dr + Hdθ + 2 a sin θ (cid:16) F − r FG (cid:17) dtdφ + sin θ h H + a sin θ (cid:16) r FG − F (cid:17)i dφ . (3.8)Now in order to keep the terms in the metric real, we use the transformation r p → r p +2 r + a cos θ = r p +2 ρ ; p ≥ following [54]. This approach of incorporating spin ( a ) into the terms, leads tothe modified functions of the form f ( r ) → F ( r ) = 1 − M rρ + Q ρ + αrρ ln (cid:16) r | α | (cid:17) g ( r ) → G ( r ) = 1 − M rρ + Q ρ + αrρ ln (cid:16) r | α | (cid:17) – 6 – ( r ) → H ( r ) = ρ . Using the above relations, we obtain the final expression for the metric of rotating chargedblack hole in PFDM to be ds = − ρ (cid:16) ∆ − a sin θ (cid:17) dt + ρ ∆ dr + ρ dθ − a sin θρ h M r − Q − αr ln (cid:16) r | α | (cid:17)i dtdφ + sin θ h r + a + a sin θρ (cid:16) M r − Q − αr ln (cid:16) r | α | (cid:17)(cid:17)i dφ (3.9)with ∆ = r + a − M r + Q + αr ln (cid:16) r | α | (cid:17) and ρ = r + a cos θ . As we have mentioned previously, we are interested for an observer far away (theoretically atinfinity) from the black hole. This motivates to confine ourselves only for the case of geodesicsin the equatorial plane ( θ = π ). The consideration of equatorial plane simplifies the followingfunctions as ∆ = r + a − M r + Q + αr ln (cid:16) r | α | (cid:17) ; ρ = r . The logarithmic term with parameter α corresponds to the contribution of the PFDM to thecharged black hole metric. This term can be positive or negative depending on the sign of α .In [51], the authors have discussed about relevance of both positive and negative values of α but here we are interested only in positive values ( α > ). The allowed maximum value for α is α max = 2 M [43, 51]. For the sake of simplicity, we will consider M = 1 in our subsequentanalysis. The analysis with negative values of α can also be done by following the subsequentanalysis. The PFDM term influences the structure of the spacetime and hence the trajectoriesof the geodesics. α r h + r h − Table 1 . Values of inner ( r h − ) and outer ( r h + ) horizon of the black hole for various values of α at a = 0 . , Q = 0 . . In Table 1, we show the variations in the values of the inner horizon ( r h − ) and outer horizon( r h + ) of the black hole for various values of α at constant spin parameter a = 0 . and charge Q = 0 . of the black hole. We find that the outer horizon ( r h + ) decreases with the increasein the value of α . This behavior can be observed upto certain critical value α c (which inthis case α c ≈ . , however, after this critical value ( α c ), the outer horizon ( r h + ) starts toincrease slowly. On the other hand with the increase in the value of α , the value of the innerhorizon ( r h − ) decreases. This critical value α c can be interpreted as the point of reflection.The point of reflection can be found by plotting ∆( r ) with r at fixed values of spin ( a ) and– 7 – Q α c a Q α c Table 2 . Critical values of PFDM parameter ( α c ) for valid combinations of spin ( a ) and charge ( Q )of the black hole. charge ( Q ) of the black hole. With increase in α , the outer event horizon ( r h + ) reduces andat some value of α starts to increase. That value of α gives the point of reflection ( α c ).This effect has also been observed for the shadow of the rotating black hole with PFDM [44].Due to such effect observed on event horizon of black hole, we are interested in analysingdifferent properties of the black hole spacetime in terms of the nature of particles in tworanges of α , namely, the lower range α < α c and the higher range α > α c . This apparentincrease in size of the black hole may be assigned to the fact that after a critical value ( α c )of the dark matter, it contributes to the effective mass of the black hole system. It can beexplained by the fact that dark matter acts as a point mass distribution. So as mentionedin [44], we consider that the total system consists of two parts - one is the original BH withmass M and the other effective black hole due to the dark matter with mass M ′ . Whenthe PFDM parameter α is less than the critical value α c , then the dark matter hinders theoriginal BH system, hence the effective horizon is less than M . But as α gradually increasesand becomes α > α c , the total system is dominated by the dark matter component. Thus,the event horizon effectively increases. Hence we observe such effects in the system concerned.In Table 2, we show the values of α c for various valid combinations of spin and charge whichhas been used in the subsequent analysis. In Table 2, we fix the spin a of the black hole at aparticular value, then we obtain the values of α c for various values of charge Q . We observethat the obtained values of α c lies within the range α c ∈ [0 . − . depending upon thevalues of spin ( a ) and charge ( Q ). On the basis of these values, we define the lower range ofvalues of α which are less than these values and higher range of values of α which are greaterthan these values.In order to continue the analysis for circular geodesics, we consider a particle with Lagrangian L = g µν ˙ x µ ˙ x ν where ˙ x µ = u µ = dx µ dλ is the four-velocity obtained by undertaking the deriva-tive of spacetime position ( x µ ) with respect to the affine parameter λ . The affine parametercorresponds to the proper time ( τ ) of the massive particles in case of timelike geodesics. TheLagrangian is expressed in terms of metric and we observe that the metric coefficients areindependent of t and φ , hence the metric is invariant along those directions (directions ofsymmetry) which results into conserved quantities E and L . These two quantities physicallyrepresent the specific energy (energy per unit mass) and angular momentum (angular mo-mentum per unit mass) of the particle respectively with respect to a stationary observer atrelatively infinite distance. In terms of these quantities, the geodesic equations of t and φ – 8 –akes the following form ˙ t = 1 r h r + a ∆ (cid:16) E ( r + a ) − aL (cid:17) + a ( L − aE ) i (4.1) ˙ φ = 1 r h a ∆ (cid:16) E ( r + a ) − aL (cid:17) + ( L − aE ) i . (4.2)The Hamiltonian H = p µ ˙ x µ − L therefore reads H = p ˙ x + p ˙ x + p ˙ x = − Et + L ˙ φ + r ∆ ˙ r = constant = ǫ . (4.3)with ǫ = − , , for timelike, null and spacelike geodesics respectively. We are mainlyinterested in the first two types of geodesics which are physically relevant. Substituting thevalues of ˙ t and ˙ φ , we obtain the geodesic equation for r . This reads ˙ r = 1 r h(cid:16) E ( r + a ) − aL (cid:17) − ∆( L − aE ) i + ∆ r ǫ . (4.4)The radial equation is very useful for the analysis of the circular geodesics and also for thecomputation of the effective potential. In case of null geodesics ǫ = 0 , hence the radial equation becomes ˙ r = 1 r h(cid:16) E ( r + a ) − aL (cid:17) − ∆( L − aE ) i = F ( r ) . (4.5)For the sake of convenience we define LE = D as the impact parameter which reduces twoconstants into one. In terms of the impact parameter, the above equation becomes ˙ r = E r h r + 2 Mr ( a − D ) − Q r ( a − D ) − αr ln ( rα ) + ( a − D ) i . (4.6)In general D = a , but for the trivial case considering D = a , we get the geodesic equationsto be dtdλ = r + a ∆ E ; dφdλ = a ∆ E ; drdλ = ± E . (4.7)For the general case ( D = a ), we aim to find the circular photon orbits subject to theconditions F ( r ) = F ′ ( r ) = 0 . These two conditions yield r p + 2 Mr p ( a − D ) − Q r p ( a − D ) − αr p ln ( r p | α | )( a − D ) + ( a − D ) = 0 (4.8) r p − Mr p ( a − D ) + 2 Q r p ( a − D ) + αr p ln ( r p | α | )( a − D ) − αr p ( a − D ) = 0 . (4.9)Solving for D from eq.(4.9), we get D = a ∓ s r p M r p − Q r p − αr p ln ( r p | α | ) + αr p . (4.10)– 9 – =0.5, Q =0.3 α r p Q =0.3, α =0.2, 1.0 α a r p a =0.5, α =0.2, 1.0 α Q r p Table 3 . Radius ( r p ) of the co-rotating (prograde) photon orbits. a =0.5, Q =0.3 α r p Q =0.3, α =0.2, 1.0 α a r p a =0.5, α =0.2, 1.0 α Q r p Table 4 . Radius ( r p ) of the counter-rotating (retrograde) photon orbits. Here the signs ∓ corresponds to the counter rotation (-) and the co-rotation (+) of the orbitsalong with the black hole. By substituting the expression for D in eq.(4.8), we obtain thefollowing equation which leads to the radius of the photon orbits M r p − Q r p − αr p ln (cid:16) r p | α | (cid:17) + αr p − r p ± a r r p (cid:16) M r p − Q − αr p ln (cid:16) r p | α | (cid:17) + αr p (cid:17) = 0 . (4.11)The ± signs denote the co-rotating and the counter rotating photon orbits. Tables (3) and(4) show the photon sphere radius ( r p ) with the variation in spin ( a ), charge ( Q ) and theparameter ( α ) which gives the weightage of the dark matter. Here r p and r p correspond tothe co-rotating and counter rotating photon sphere radius.Since the black hole horizon shows different behaviour in different range of PFDM parameter α , so we have analysed the different geodesics and the corresponding characteristics of theparticles in two separate ranges. One is the low range of α where α < α c and the other is therange α > α c . • From the above Table(s) (3),(4), it is quite clear that for the lower range of valuesof α < α c , the photon radius corresponding to both the co-rotating and the counter-rotating orbits decreases with increase in PFDM parameter. • In the higher range of values of α > α c , increase in α increases the photon radius as isevident from the nature of the outer event horizon radius ( r h + ).– 10 – With the increase in the value of spin parameter ( a ) of the black hole, the radius ofthe co-rotating orbits increases while that of the counter rotating orbits decreases forboth α > α c and α < α c . Since the spin parameter ( a ) of the black hole assists theco-rotation and opposes the counter rotation, thus we observe such feature of photonradius. • The presence of the charge ( Q ) also affects the radius of the photon sphere as can beobserved from the Tables (3), (4). With the increase in the value of charge ( Q ), thephoton sphere radius decreases both for prograde and retrograde orbits with α =0.2( α < α c ) and α =1.0 ( α > α c ). • Besides we also find that the radius of prograde orbits are larger than the retrogradeorbits of the photons moving around the black hole.
In this case, we consider massive particles and this consideration makes the geodesics time-likehence ǫ = − . This results in the modification of the radial equation (4.4) into the form ˙ r = h E + 2 Mr ( aE − L ) − Q r ( aE − L ) − αr ln (cid:16) r | α | (cid:17) + 1 r ( a E − L ) i − ∆ r = F ( r ) . (4.12)Again for the trivial case, L = aE which simplifies the radial equation to the form drdτ = (cid:16) E − ∆ r (cid:17) . (4.13)So the proper time can be evaluated as τ = Z (cid:16) E − ∆ r (cid:17) − dr . (4.14)We now confine ourselves for the general case ( L = aE ) . Our aim is to calculate and showthe variation of the energy ( E ) and the angular momentum ( L ) of the particle with variationin parameter α . So we proceed by assuming x = L − aE and rewrite the above equation interms of x and E . Upon imposing the conditions for circular orbits ( F ( r ) = F ′ ( r ) = 0 ), weobtain F ( r ) = x (cid:16) a − ∆ (cid:17) + r E − aEr x − ∆ r = 0 (4.15) F ′ ( r ) = 4 r E − aErx − r − ∆ ′ ( r + x ) = 0 . (4.16)Solving the above equations, we obtain the expression for x . Using the obtained expressionof x , we determine the energy ( E ) and angular momentum ( L ) of the particle.The expression for E is obtained from eq.(s)(4.15), (4.16) as E = 1 r ax h(cid:16) a − ∆ + r ∆ ′ (cid:17) x + (cid:16) ∆ ′ r − ∆2 r (cid:17)i . (4.17)On replacing E in eq(4.15), we get an equation in x as h (cid:16) ∆ − a − r ∆ ′ (cid:17) − a (cid:16) a − ∆ + r ∆ ′ (cid:17)i x + h(cid:16) a −
4∆ + r ∆ ′ (cid:17) × (cid:16) r ∆ ′ − r ∆ (cid:17) − r a ∆ ′ i x + h r ∆ − r ∆ ′ i = 0 . (4.18)– 11 –he equation is quadratic in x with the discriminant ∆ D = 16 a ∆ r h a − ∆ + r ∆ ′ i . (4.19)Now we can factorize the coefficient of x as (cid:16) ∆ − a − r ∆ ′ (cid:17) − a (cid:16) a − ∆ + r ∆ ′ (cid:17) = F + F − (4.20)where F ± = 2 (cid:16) ∆ − a − r ∆ ′ (cid:17) ± a r(cid:16) a − ∆ + r ∆ ′ (cid:17) . (4.21)Incorporating the above, the solution for x reads x = r (cid:16) ∆ F ± − F + F − (cid:17) F + F − = r (cid:16) ∆ − F ∓ (cid:17) F ∓ (4.22)where we consider F − ≡ F ∓ and F + ≡ F ± . The solution for x becomes α = α = α = α = r E ne r g y (a) a = 0 . , Q = 0 . α = α = α = α = r E ne r g y (b) a = 0 . , Q = 0 . a = = = r E ne r g y (c) α = 0 . , Q = 0 . Q = = = r E ne r g y (d) a = 0 . , α = 0 . Figure 1 . Plots for energy of massive particles co-rotating with the black hole with variation in a , Q and α . x = ± r √F ∓ " a ± r a − ∆ + r ∆ ′ . (4.23)– 12 –eplacing the values of x , the expression for energy becomes E = 1 √F ∓ r " ∆ − a a ± r a − ∆ + r ∆ ′ ! (4.24)and that of angular momentum becomes L = 1 √F ∓ r " a ∆ − a − r ! ∓ (cid:16) r + a (cid:17)r a − ∆ + r ∆ ′ . (4.25)The plots of the above expressions for energy ( E ) and angular momentum ( L ) with respectto the radial distance ( r ) gives a firm idea about the orbits and thereby motions around theblack hole which in effect gives an impression about the spacetime structure around the blackhole. Besides in our case it helps us to get an idea about the impact of the surrounding darkmatter on the geodesics. • In Fig.(1), we find that the energy ( E ) of the co-rotating particle falls with distance ( r )from the black hole. • While the particle is near the black hole, it is assisted by the spin ( a ) of the black holeand its energy is thereby increased but as it starts to move away, the energy followingeq.(5.1) behaves partially like the potential ( V eff ) where we showed only a portion ofthe plot where the energy falls in order to distinguish both (co and counter rotating)type of particle(s) energy and hence the motion. α = α = α = α = r E ne r g y (a) a = 0 . , Q = 0 . α = α = α = α = r E ne r g y (b) a = 0 . , Q = 0 . Figure 2 . Plots of energy of the massive particles counter-rotating with respect to the black holewith variation in α . • The effect of dark matter ( α ) is similar for both lower values ( α < α c ) and higher values( α > α c ). In both cases energy increases with increase in the effective amount of darkmatter in the system. • The effect of spin ( a ) both for low ( α = 0 . ) and high ( α = 1 . ) values of the parameter α results in the increment of the energy of the particle which means an assist by therotation of the black hole. – 13 – On the other hand, increase in charge ( Q ) results in decrease in the energy of theparticle. • The energy of the particle in all the cases can be found to be approaching towards unityas it is the energy of the particle at infinity observed by a stationary observer.In Fig.(2), we find that energy ( E ) of the counter-rotating particle increases with distance( r ) from the black hole and as it moves away it slowly approaches towards unity. As theparticle is closer it is opposed by the black hole rotation and hence it has less energy. a = = =
10 20 30 40 50 r E ne r g y Figure 3 . Plots of energy of counter-rotating mas-sive particles for α =0.2 and Q =0.3 with spin( a )variation. • As the particle moves away, the inten-sity of opposing falls and the particlehas positive energy due to rotation incircular orbit. • In this case, the effect of dark matter( α ) is similar both for low values ( α <α c ) and high values ( α > α c ). In bothcases (varying α ) energy increases withincrease in the effective amount of darkmatter in the system. • The increase in spin ( a ) of the blackhole both for low ( α =0.2) and high( α =1.0) values (not shown) of the parameter α results in decrement of energy of theparticle as is evident from Fig.(3), whereas the increase in charge ( Q ) results in increasein energy of the particle.The plots of the angular momentum present a completely different picture with respect tothe different range (higher and lower values) of the parameter α of PFDM. • Fig.(4) shows the angular momentum of the particles moving in prograde (co-rotating)orbits. The plots show that in the lower range values of α ( α < α c ), increasing thevalue of α decreases the angular momentum whereas the reverse is observed in case ofthe higher range values of α ( α > α c ). • The different behaviour of the angular momentum of the particle is due to the natureof dark matter which gets reflected on the particle trajectories. Also the variation ofthe angular momentum ( L ) with respect to the spin ( a ) shows that L increases withthe increment in the value of the spin of the black hole for α > α c whereas there is veryslight variation in case of α < α c . • The effect of the charge ( Q ) shows that increase in the charge of the black hole de-creases the angular momentum ( L ) of the particles with prograde orbits and the rateof reduction increases with further increment in the value of the charge ( Q ). • The angular momentum of the particles in the retrograde orbits are negative since theymove opposite to the direction of rotation of the black hole as shown in Fig.(5).– 14 – = α = α = α = L (a) a = 0 . , Q = 0 . α = α = α = α = L (b) a = 0 . , Q = 0 . Figure 4 . Plots of angular momentum of the massive particles co-rotating with the black hole withvariation in α with fixed spin( a ) and charge( Q ). α = α = α = α = - - - - - - - - L (a) a = 0 . , Q = 0 . α = α = α = α = - - - - - - L (b) a = 0 . , Q = 0 . Figure 5 . Plots of angular momentum of the massive particles counter-rotating with the black holewith variation in α with fixed values of spin( a ) and charge( Q ). • As the particle moves away ( r increases) from the black hole the angular momentumincreases (in the opposite direction which in turn increases the negative value) for α < α c and decreases for α > α c . • As can be seen from the plots that for α < α c , the increment in the value of α reducesthe angular momentum of the massive test particle. However, for α > α c , it increasesthe angular momentum. • Also with increase in the value of the spin ( a ) of the black hole, angular momentumincreases for α < α c and decreases for α > α c . • The increment in the charge ( Q ) of the black hole reduces the angular momentum ofthe particle for both α < α c and α > α c . After studying the null-geodesics and geodesics of the chargeless massive particles, we move onto study the geodesic motion of the massive charged particles. Incorporating the interactions– 15 –f the gauge fields, the Hamiltonian of the particle in this case gets modified to [28] H = g µν (cid:16) p µ + qA µ (cid:17)(cid:16) p ν + qA ν (cid:17) = ǫ = − (4.26)with the electromagnetic potential for charged spinning black hole coupled to PFDM givenby [33] A = A µ dx µ = Qrρ (cid:16) dt − a sin θdφ (cid:17) . (4.27)Using the Legendre transformation H = p µ ˙ x µ − L , we obtain the Lagrangian of the particleas L = 12 g µν ˙ x µ ˙ x ν − qA µ ˙ x µ (4.28)where q denotes the charge of the particle. Also we consider the particle to have unit mass( m = 1 ) for the sake of simplicity. Using the symmetry of the metric we compute theconserved quantities E and L which have same physical meaning as mentioned previously.Also since we are interested in the equatorial geodesics, we use θ = π and ˙ θ = 0 which leadsto following the geodesic equations ˙ t = 1 r " r + a ∆ (cid:16) E − qQr (cid:17)(cid:16) r + a (cid:17) − a (cid:16) L − qaQr !(cid:17) + a (cid:16) L − qaQr (cid:17) − a (cid:16) E − qQr (cid:17)! (4.29) q = = = = r E ne r g y q = = = = L Figure 6 . Plots for energy and angular momentum of charged particles co-rotating with the blackhole with varying q for α =0.2, Q =0.3 and a =0.5. ˙ φ = 1 r " a ∆ (cid:16) E − qQr (cid:17) ( r + a ) − a (cid:16) L − qaQr (cid:17)! + (cid:16) L − qaQr (cid:17) − a (cid:16) E − qQr (cid:17)! (4.30) ˙ r = − ∆ r + (cid:16) E − qQr (cid:17) r "(cid:16) r + a (cid:17) − a ∆ − ar (cid:16) r + a − ∆ (cid:17)(cid:16) E − qQr (cid:17)(cid:16) L − qaQr (cid:17) − r (cid:16) ∆ − a (cid:17)(cid:16) L − qaQr (cid:17) . (4.31)– 16 – = = = = r E ne r g y q = = = = - - - - L Figure 7 . Plots for energy and angular momentum of charged particles counter-rotating with theblack hole with varying q for α =0.2, Q =0.3 and a =0.5. In order to determine the circular orbits, we use the conditions F ( r ) = F ′ ( r ) = 0 where F ( r ) = − ∆ r + (cid:16) E − qQr (cid:17) r "(cid:16) r + a (cid:17) − a ∆ − ar (cid:16) r + a − ∆ (cid:17)(cid:16) E − qQr (cid:17)(cid:16) L − qaQr (cid:17) − r (cid:16) ∆ − a (cid:17)(cid:16) L − qaQr (cid:17) . (4.32) q = = = = r E ne r g y q = = = = r L Figure 8 . Plots for energy and angular momentum of charged particles co-rotating with the blackhole with varying q for α =1.0, Q =0.3 and a =0.5. It is very difficult to find an exact solution for E and L from these conditions. So we proceedby assuming qm ≪ (particle with small specific charge) and approximately write down thefollowing solutions by incorporating Taylor expansion about qm = 0 [26] E ( q ) = E (0) + qE ′ (0) + O ( q ) + ... (4.33) L ( q ) = L (0) + qL ′ (0) + O ( q ) + ... . (4.34)The approximate solutions of E and L satisfy the condition of circular orbits given as F ( r ) = F ′ ( r ) = 0 . We display the plots. • The plots of energy ( E ) show that for both α < α c and α > α c , the increase in thevalue of the charge of the particle q from − . to . , decreases the energy of co-rotating– 17 – = = = = r E ne r g y q = = = = - - - - r L Figure 9 . Plots for energy and angular momentum of charged particles counter-rotating with theblack hole with varying q for α =1.0, Q =0.3 and a =0.5. particles whereas there is an increase in case of counter rotating particles. In both casesthe energy tends towards unity. • In case of angular momentum, we observe that for α = 0 . the angular momentum ofboth co-rotating ( + ve increase) and counter rotating ( − ve increase) particles increaseswith the increase in the value of the charge q of the particle. • However, for α = 1 . we find that with increase in the value of charge q , angularmomentum ( L ) for co-rotating and counter rotating particles decreases. • The observations depict that the particle’s charge q responses differently depending onthe dark matter. Also it implies that the particle with more charge ( q ) is hindered moreif the intensity of dark matter increases. In this section we study the effective potential which results in both stable and unstable orbitsdepending upon the condition ∂ V eff ∂r > or ∂ V eff ∂r < respectively. The stable and unstableorbits correspond to the local minima and maxima of the potential which we obtain from theradial geodesic equations. The potential depends upon the following parameters, the chargeof the black hole ( Q ), spin parameter ( a ), the dark matter parameter α and on the chargeof the particle ( q ). The effective potential in case of the massive particles obtained from thecorresponding radial geodesic equation eq.(4.12) reads ˙ r + V eff = E (5.1)where the effective potential V eff is given by V eff = − Mr ( aE − L ) + Q r ( aE − L ) + αr ln (cid:16) r | α | (cid:17) − r ( a E − L ) + ∆ r . (5.2)For circular geodesics, the particle moves in a circular trajectory of constant radius r whichimplies ˙ r = 0 . In case of photons, the effective potential as obtained from eq.(4.5) takes theform V eff = − Mr ( aE − L ) + Q r ( aE − L ) + αr ln (cid:16) r | α | (cid:17) − r ( a E − L ) . (5.3)– 18 – = α = α = α = V e ff (a) a = 0 . , Q = 0 . , L = 3 . α = α = α = α = V e ff (b) a = 0 . , Q = 0 . , L = 3 . a = = = V e ff (c) α =0.2, Q = 0 . , L = 3 . Q = = = V e ff (d) a = 0 . , α =0.2, L = 3 . L = = = = V e ff (e) a = 0 . , α =0.2, Q = 0 . L = = = = V e ff (f) a = 0 . , α =1.0, Q = 0 . Figure 10 . Plots of effective potential for null geodesics with variation in a , Q , L and α . Also for massive particles with charge q , the effective potential takes the following formobtained using eq.(4.31) V eff = E + (cid:0) ∆ − a (cid:1) (cid:16) L − aqQr (cid:17) r + 2 a (cid:0) a − ∆ + r (cid:1) (cid:16) E − qQr (cid:17) (cid:16) L − aqQr (cid:17) r − (cid:16)(cid:0) a + r (cid:1) − a ∆ (cid:17) (cid:16) E − qQr (cid:17) r + ∆ r . (5.4)– 19 – = α = α = α = V e ff (a) a = 0 . , Q = 0 . , L = 3 . α = α = α = α = V e ff (b) a = 0 . , Q = 0 . , L = 3 . a = = = V e ff (c) α =0.2, Q = 0 . , L = 3 . Q = = = V e ff (d) a = 0 . , α =0.2, L = 3 . L = = = = V e ff (e) a = 0 . , α =0.2, Q = 0 . L = = = =
34 r V e ff (f) a = 0 . , α =1.0, Q = 0 . Figure 11 . Plots of effective potential for timelike geodesics with variation in a , Q , L and α . In the trivial case when L = aE , the radial equations and effective potentials become ˙ r massive = ± r E − ∆ r ; ˙ r null = ± E (5.5) V massive = ∆ r ; V null = 0 . (5.6)In Figures (10),(11), we show the plots of the effective potential for both null and timelikegeodesics and observe the effects of a , Q , L and α on it. The first set of plot Fig.(10) representsthe effective potential for null geodesic particles.– 20 – We find that for α < α c , the potential increases with increase in the amount of darkmatter around the black hole, whereas for α > α c the effective potential falls withincrease in the PFDM parameter α . • Also we find that the potential increases with increase in the spin ( a ) and charge ( Q )of the black hole both for α < α c and α > α c . • The maxima of the potential in both of these cases shifts towards smaller values of r .The plots shown for null particles has been computed by setting E = 1 (particles withunit energy). • Besides we observe that with increase in angular momentum ( L ) of the particle, theeffective potential rises in both cases ( α < α c and α > α c ) and the maxima shiftstowards larger radial distance r . • Fig.(11) shows the effective potential for massive particles. In this case we find thatwith the increase in parameter α , the effective potential rises sharply for α < α c but itdecreases in case of α > α c . • The spin parameter ( a ) of the black hole is suitable for the increment of the potentialand thereby V eff increases with increase in the value of a for both α < α c and α > α c . • Similar to the effect of the spin parameter, increase in the value of the charge of theblack hole ( Q ) increases V eff for both α < α c and α > α c . • Also the potential increases steadily with increase in the angular momentum ( L ) forboth cases of α < α c and α > α c . • Here also the plots are shown for the massive particle with unit energy ( E = 1 ). Thenature of the shift of maxima with a , Q and L is similar to the case of massless particles. • The effective potential in case of the null particles finally approaches zero whereas formassive particles approaches unity which can be observed from the plots.
Black hole is a vessel of extreme energy and there are many processes theorised which areresponsible to gain energy from the black hole. One of them is the Penrose process namedafter Roger Penrose who proposed the mechanism in [46]. In case of rotating black hole aregion gets created between the outer event horizon ( g rr = 0 ) and the stationary limit surface( g tt = 0 ). These two surfaces meet at the poles and have largest separation in the equatorialplane. This varying annular region is known as the ergosphere . In case of static black holethis region vanishes. The speciality of this region is that the Killing vector ∂∂t which hasa unit norm as observed by a stationary observer at infinity becomes spacelike within theregion. The symmetry of the metric with change in the said Killing vector results in energyconservation and hence the energy in this region can be negative. This fact can be utilised togain energy from the black hole.Let a particle (uncharged) with positive energy fall into this ergoregion and split into twoparticle, one with positive energy and the other with negative energy. The negative energyparticle is absorbed by the black hole and that with positive energy comes out of the black– 21 –ole having more energy than the particle that entered the black hole and hence resulting inenergy gain.The condition of negative energy of the particle can be found using the condition of circularorbits. The equation with ˙ r = 0 results in E h(cid:16) r + a (cid:17) − a ∆ i − E h aL (cid:16) r + a − ∆ (cid:17) + L (cid:16) a − ∆ (cid:17) + ∆ r ǫ i = 0 (6.1)which can be solved for both E and L as given by E = aL (cid:16) r + a − ∆ (cid:17) ± r r ∆ h r L − ǫ (cid:16)(cid:16) r + a (cid:17) − a ∆ (cid:17)i(cid:16) r + a (cid:17) − a ∆ (6.2) L = aE (cid:16) r + a − ∆ (cid:17) ± r r ∆ h r E + ǫ (cid:16) ∆ − a (cid:17)i(cid:16) a − ∆ (cid:17) . (6.3)If one assumes positive sign in eq.(6.2) along with the condition a L (cid:16) r + a − ∆ (cid:17) > ∆ r h r L − ǫ (cid:16)(cid:16) r + a (cid:17) − a ∆ (cid:17)i (6.4)and L < , then E < , i.e., particle with negative energy is possible. This gives the ideathat E < is possible for L < which is the case for counter rotating orbits. The negativeenergy particle following counter rotating orbits must lie within the ergoregion. The plots ofnegative energy with the variation in different parameters are shown below.In order to discuss the Penrose process in detail, we must start by considering an unchargedparticle of energy E entering the ergosphere and let, it breaks down into two photons withenergies E and E . Let the angular momentum of the particles be L (entering), L (leaving)and L (captured). Also let the energy of the particle entering the ergosphere be E = 1 .Hence the angular momentum of the particles are L = a (cid:16) r + a − ∆ (cid:17) + r r ∆ h r + (cid:16) ∆ − a (cid:17)i(cid:16) a − ∆ (cid:17) (6.5) L = aE (cid:16) r + a − ∆ (cid:17) + r r ∆ (cid:16) r E (cid:17)(cid:16) a − ∆ (cid:17) = b E (6.6) L = aE (cid:16) r + a − ∆ (cid:17) − r r ∆ (cid:16) r E (cid:17)(cid:16) a − ∆ (cid:17) = b E (6.7)where b = a (cid:16) r + a − ∆ (cid:17) + r √ ∆ (cid:16) a − ∆ (cid:17) ; b = a (cid:16) r + a − ∆ (cid:17) − r √ ∆ (cid:16) a − ∆ (cid:17) . (6.8)– 22 – = α = α = α = - - - - - r E ne r g y (a) a = 0 . , Q = 0 . , L = − . α = α = α = α = - - - - - r E ne r g y (b) a = 0 . , Q = 0 . , L = − . L =- =- =- =- - - - - - r E ne r g y (c) a = 0 . , α =0.2, Q = 0 . L =- =- =- =- - - - - - r E ne r g y (d) a = 0 . , α =1.0, Q = 0 . Figure 12 . Plots for negative energy particle with variation in L and α . By conservation of energy and angular momentum we get E = E + E = 1 ; L = b E + b E . (6.9)Solving for E and E we obtain E = 12 " r r + ∆ − a r (6.10) E = 12 " − r r + ∆ − a r (6.11)where E and E corresponds to the positive and negative energies of the two particles. Thusthe particle with energy E is captured by the black hole while that with energy E comesout of the black hole resulting in an energy gain of ∆ E = E − "r r + ∆ − a r − = − E . (6.12)In the limit a → , the ergosphere vanishes and the region of ergosphere corresponds to eventhorizon with radius r h + and hence ∆ = 0 and we get energy gain ∆ E = 0 , E = 1 and E = 0 and hence no particle with negative energy exists.– 23 – = α = α = α = r E ne r g y ga i n (a) a = 0 . , Q = 0 . α = α = α = α = r E ne r g y ga i n (b) a = 0 . , Q = 0 . a = = a = r E ne r g y ga i n (c) α =0.2, Q = 0 . Q = (cid:5) = (cid:6)(cid:7)(cid:8)(cid:9) = (cid:10) r E ne r g y ga i n (d) a = 0 . , α =0.2 Figure 13 . Plots showing energy gain from the black hole with variation in a , Q and α . The plots of negative energy and energy gain from the black hole are shown above. The plotsdepict how the negative energy states depend on the parameters characterising the black holespacetime. • From Fig. (12) we find that for α < α c , the negative energy increases with increase in α and similar is true for α > α c also. However the change is less prominent for highervalues. • Also it is noticeable that with the increase in spin ( a ), the negative energy increasesthough the effect is very feeble. • The influence of charge ( Q ) and angular momentum ( L ) are firmly observed, where forboth large and small constant values of α , the increase in the charge and the negativeangular momentum increases the negative energy of the particle quite impressively whichresults in the fact that the particle absorbed by the black hole will have higher negativeenergy and will lead to increased energy gain from the black hole.The energy gain from the black hole via., the Penrose process is astrophysically very importantand significant. Also indirectly more the negative energy absorbed by the black hole more isthe gain in positive energy via Penrose process. The plots of energy gain in Fig.(13) show usthe impact of different black hole parameters on the proportion of increment or decrement ofthe gain in the energy. – 24 – We find that energy gain increases with increase in the PFDM parameter α for both α < α c and α > α c . The increment is more significant for lower range values of PFDMparameter. • With increase in the value of the charge ( Q ), energy gain increases for both α < α c and α > α c . We now summarise our findings. We have made some interesting observations in this paper.First of all we give a static, charged black hole solution in PFDM. Furthermore, we haveincorporated the Newman-Janis algorithm in order to compute the metric corresponding toa rotating, charged black hole surrounded by perfect fluid dark matter. Our initial study onthe event horizon radius of the mentioned black hole reveals that the PFDM parameter α creates a noticeable influence on both outer and inner event horizons. We observe that thereexists a certain value α c (at a constant value of spin a and charge Q ) upto which the outerevent horizon radius ( r h + ) decreases with the increase in the value of α . However, after α c ,surprisingly r h + starts to increase with increasing value of α . On the basis of this criticalvalue, we define two range of values for α . We speculate that it might be due to the fact thatthe dark matter contributes to the effective mass of the black hole system. Then we look forthe radius of photon spheres for both prograde and retrograde orbits and found that with theincrease in α the radius decreases for α < α c and increases for α < α c . Besides we also foundthat increase in the value of the spin parameter ( a ) increases the radius of the photon orbits.Then we observed the energy ( E ) and angular momentum ( L ) for massive particles moving inprograde and retrograde orbits. The energy ( E ) of the particle in prograde orbits decreaseswhereas that of retrograde orbits increases with increase in the radial distance ( r ) from theblack hole and gets close to unity as the particle approaches infinity. The increment in thevalues of α and spin ( a ) increases the energy of the particle considerably for the progradeorbits. On the other hand in case of retrograde orbits, the energy ( E ) increases with theincrease in α but falls with the increasing value of the spin ( a ). This is because when theparticle spins along the black hole, the black hole helps its motion whereas in the reverse caseit opposes. The most important observation is in the case of angular momentum of the blackhole which decreases with the increase in value of α for α < α c and increases for α > α c forboth types of orbits. Also we observe that with increase in the spin ( a ) of the black hole, theangular momentum ( L ) of the particle increases for co-rotating particles ( + ve increase) forboth α < α c and α > α c . But for counter-rotating particles, the angular momentum ( L ) rises(-ve increase) for α < α c while it decreases for α > α c . We found that with the increase inthe charge ( Q ) of the black hole, the angular momentum ( L ) of the particle decreases.We have then studied charged particles. In case of charged particles we analyse both theenergy ( E ) and angular momentum ( L ) with the variation in the charge ( q ) of the particleand found that with the increase in the value of charge ( q ), the energy falls in prograde orbitsand increases in case of retrograde orbits. Besides we observe that angular momentum ( L )increases with increasing q for α < α c and falls for α > α c . It is observed that the effectivepotential of the black hole for photons and massive particles, increases with the increasingspin ( a ) and charge ( Q ) of the black hole as also with angular momentum ( L ) of the particle.The change is quite sharp with the change in the angular momentum ( L ). Also with increasein the value of α ( α < α c ), the potential increases whereas for α > α c , it decreases slightly.– 25 –he potential of the black hole for the charged particle is analysed with variation in q andwe found that potential increases with increase in the charge from − . to . .Finally, we studied the Penrose process. The negative energy particles are very importantwith respect to the idea of energy gain from black hole and we observed that negative energyconsiderably increases with increase in negative angular momentum (counter-rotating parti-cle) and also with increase in the charge ( Q ) of the black hole. The effect of dark matter onthe negative energy is less pronounced even though negative energy slightly increases. Morethe negative energy of the particle absorbed by the black hole, more is the gain, and we foundthat the energy gain via Penrose process increases due to the presence of dark matter in thesystem. We also observed that more the black hole charge ( Q ), more is the energy gain andhence more efficient is the Penrose process. Acknowledgments
A.D. would like to acknowledge the support of S.N. Bose National Centre for Basic Sciences forJunior Research Fellowship. A.S. acknowledges the financial support by Council of Scientificand Industrial Research (CSIR, Govt. of India).
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