Effect of perfect fluid dark matter on particle motion around a static black hole immersed in an external magnetic field
EEffect of perfect fluid dark matter on particle motionaround a static black hole immersed in an external magnetic field
Sanjar Shaymatov,
1, 2, 3, ∗ Daniele Malafarina, † and Bobomurat Ahmedov
2, 3, ‡ Institute for Theoretical Physics and Cosmology,Zheijiang University of Technology, Hangzhou 310023, China Ulugh Beg Astronomical Institute, Astronomicheskaya 33, Tashkent 100052, Uzbekistan Tashkent Institute of Irrigation and Agricultural Mechanization Engineers,Kori Niyoziy 39, Tashkent 100000, Uzbekistan Department of Physics, Nazarbayev University, Kabanbay Batyr 53, 010000 Nur-Sultan, Kazakhstan (Dated: April 16, 2020)We investigate particle and photon motion in the vicinity of a static and spherically symmetricblack hole surrounded by perfect fluid dark matter in the presence of an external asymptoticallyuniform magnetic field. We determine the radius of the innermost stable circular orbit (ISCO) forcharged test particles and the radius for unstable circular photon orbits and show that the effect ofthe presence of dark matter shrinks the values of ISCO and photon sphere radii. Finally, we considerthe effect of the presence of dark matter on the center of mass energy of colliding particles in theblack hole vicinity. We show that the center of mass energy grows as the value of the dark matterparameter increases. This result, in conjunction with the fact that, in the presence of an externalmagnetic field, the ISCO radius can become arbitrarily close to the horizon leads to arbitrarilyhigh energy that can be extracted by the collision process, similarly to what is observed in thesuper-spinning Kerr case.
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I. INTRODUCTION
The existence of black holes is a direct consequenceof Einstein’s general theory of relativity. However, un-til recently, black holes had been considered as poten-tial explanation for some observed phenomena, like someX-ray sources, but not directly detected. This changedwith the discovery of gravitational waves produced by themerger of pairs of stellar mass black holes by the LIGOand Virgo collaborations [1, 2] and as well as with thefirst direct imaging of the supermassive black hole in thegalaxy M87 by the Event Horizon Telescope (EHT) col-laboration [3, 4]. These observations allow for the firsttime to test the nature of the geometry in the vicinityof the black hole’s event horizon. The possibility thatthe geometry of such astrophysical compact objects mayexhibit departures from the Kerr line element has beeninvestigated thoroughly in the literature (see for example[5, 6]).In this context, particle motion around black holes andexotic compact objects has been a productive field ofstudy for several years. It is well known that the mo-tion of test particles and photons is strongly affectedby the geometry in the strongly relativistic regime, i.e.in the vicinity of the black hole candidate. However,other elements may affect the geometry and the parti-cles’ geodesics. For example deformations of the source ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] (see for example [7–10]), the presence of external mag-netic fields (see for example [11, 12]) and the presenceof other matter fields (see for example [13, 14] ) may allcontribute to altering the motion of test particles.A large amount of work has been devoted to the studyof charged particle motion in the geometry of a black holeimmersed in an external magnetic field [15–23]. Simi-larly, one can consider the motion of test particles in thespacetime describing a black hole immersed in an exter-nal matter field. Among the matter fields that may sur-round a supermassive black hole candidate the presenceof dark matter is particularly important. Although therestill exist no direct detection of dark matter, its presencecan be inferred from galactic rotation curves [24]. Thus,by extrapolation of known dark matter profiles for outerregions of a galaxy [25] to the inner regions, one can ex-pect the dark matter contribution to be relevant near thegalactic center (see for example [26]). There are severalways to include dark matter fields in the background ofa black hole geometry (see for example [27–29]).In the present article we consider a static black holesurrounded by perfect fluid dark matter, as described bythe line element proposed in [27]. For this geometry, weinvestigate photon orbits and the motion of charged testparticles in the presence of an external asymptoticallyuniform magnetic field.The relevance of the motion of charged particles stemsfrom recent astronomical observations of particle out-flows from active galactic nuclei (AGN) such as windsand jets. These outflows have been observed in X -ray, γ -ray and Very Long Baseline Interferometry (VLBI) ob-servations and can have energies of the order of E ≈ − erg / s [30–32]. A proposed explanation for a r X i v : . [ g r- q c ] A p r these observations comes from considering high energycollisions of particles near the black hole horizon. Theprocess was first theoretically described by Banados, Silkand West (BSW) in [33] and it has since been consideredin a large variety of contexts [34–52]. Theoretical investi-gation of high energy phenomena around compact objectshas been considered also in alternate theories of gravity[53], in the vicinity of naked singularities [54–56], andnear regular black hole solutions [57]. The well-knownPenrose process was addressed in [58] while the effect ofspinning test particles on the Penrose collision processwas studied in [59]. The presence of a magnetic fieldplays an important role also in understanding the mech-anisms of energy extraction from black holes [60–64] andcoupling between accretion disks and jets in AGNs [65].Since black holes are not endowed with their own mag-netic fields [66, 67] one needs to consider the presence ofan external magnetic field induced by nearby objects suchthe accretion discs around rotating black holes [11] ormagnetars [68, 69] and neutron stars [66, 70–72]. Thus,the magnetic field can be regarded as a test field, whichdoes not modify the background geometry [73–79].The paper is organized as follows: In Sec. II we brieflydiscuss the metric for a black hole immersed in a staticanisotropic perfect fluid, which can be used to model adark matter distribution. In Sec. III we study the mo-tion of photons and charged particles in the spacetimewhen an external magnetic field is present. The discus-sion of the effects of dark matter on the collision energyof particles in the spacetime is presented in Sec. IV andconcluding remarks with the relevance of the work forastrophysical black holes are in the Sec. V.Throughout the paper we use a system of units inwhich G = c = 1. Greek indices are taken to run from 0to 3, Latin indices from 1 to 3. II. THE METRIC
The metric describing a static and spherically sym-metric black hole immersed in perfect fluid dark matterin Schwarzschild coordinates ( t, r, θ, ϕ ) is given by [27] ds = − F ( r ) dt + F ( r ) − dr + r d Ω , (1)where d Ω is the line element on the unit 2-sphere andwhere we have defined F ( r ) = (cid:18) − Mr + λr log r | λ | (cid:19) , (2)with M being black hole mass and λ related to the darkmatter density and pressure. In the case of vanishing λ ,the spacetime metric (1) reduces to the Schwarzschildmetric, while for λ (cid:54) = 0 the stress energy-momentumtensor of the dark matter distribution is that of ananisotropic perfect fluid T µν = diag( − ρ, p r , p θ , p φ ) where λ / M r / M FIG. 1: The dependence of the horizon radius r h (thickline), photon sphere r ph (dashed line) and ISCO (dot-dashedline) on the perfect fluid dark matter parameter λ . For smallvalues of λ all three radii have maximum value in the absenceof dark matter. density, radial and tangential pressures are given by ρ = − p r = λ πr and p θ = p φ = λ πr . (3)In order to model a dark matter distribution we shallrestrict ourselves to the case λ > r − M + λ log r | λ | = 0 , (4)and for small values of parameter λ (cid:28) M it can be ap-proximated by the analytical expression r h = M + (cid:115) M − M λ log 2 M | λ | . (5)We notice that the radius of the horizon is r h = 2 M for two values of λ , i.e. λ = 0 and λ = 2 M . Therefore r h decreases as the dark matter profile is introduced and itreaches a minimum for λ = λ < M .We move now to the study of particle and photon mo-tion in the above spacetime. From the usual relation forthe 4-momentum p µ p µ = k we have that for massive par-ticles one has to set k = − m (with m the mass of the testparticle) while for photons one has to set k = 0. Fromthe Hamilton-Jacobi formalism we obtain the action S inthe form S = − Et + Lϕ + S r ( r ) + S θ ( θ ) , (6)where E and L are the usual conserved quantities associ-ated with the time translations and spatial rotations anddescribe the energy E and angular momentum L of theparticle or photon, respectively and S r and S θ are func-tions of only r and θ , respectively. Now it is straight-forward to obtain the Hamilton-Jacobi equation in thefollowing form k = − E F ( r ) + F ( r ) (cid:18) ∂S r ∂r (cid:19) + 1 r (cid:18) ∂S θ ∂θ (cid:19) + L r sin θ . (7)Due to the spherical symmetry of the spacetime we canrestrict the analysis to the equatorial plane θ = π/ r = E + kF ( r ) − L r F ( r ) = E − ˜ V eff ( r ) , (8)where, for simplicity, we shall now set k = − E and L describe energy and angularmomentum per unit mass. We can use the effective po-tential to find the radii of circular orbits for given valuesof E and L by solving simultaneously ˙ r = ¨ r = 0. This isequivalent to solving simultaneously V eff ( r, E, L ) = 0 , ∂V eff ( r, E, L ) ∂r = 0 , (9)for the function V eff ( r, E, L ) defined by V eff ( r, E, L ) = E r + kr F ( r ) − L F ( r ) . (10)In the case of massive particles the radius of the ISCO r i is obtained from the minimum value of the angularmomentum for which circular orbits are allowed. There-fore one determines L from V (cid:48) eff = 0 and then r i from V (cid:48)(cid:48) eff = 0. In our case this gives r i implicitly from thecondition0 = 2 λ (cid:0) r − L ( r ) (cid:1) log rλ − (4 M + 3 λ ) r ++ (24 M − r + 7 λ ) L ( r ) , (11)with L = L ( r ) determined by V (cid:48) eff = 0. On the otherhand, for photons it is sufficient to use V (cid:48) eff = 0 tofind the following condition for the radius of the photonsphere 6 M − r + λ (cid:16) − rλ (cid:17) = 0 . (12)In the limit of λ (cid:28) r i and the photon orbit r ph as r i ≈ M + (cid:20) − (cid:18) Mλ (cid:19)(cid:21) λ + O ( λ ) , (13) r ph ≈ M + 12 (cid:20) − log (cid:18) (cid:19)(cid:21) λ + O ( λ ) . (14) This clearly shows r i = 6 M and r ph = 3 M in the limit λ →
0, which corresponds to the ISCO and photon orbitin the case of the Schwarzschild black hole.The behaviour of the black hole horizon r h , photon or-bit r ph and ISCO r i is shown in Fig. 1. It is clear thateach radius decreases as a consequence of the presenceof λ in the limit of small λ . However, due to the repul-sive nature of the radial pressures the effect of the darkmatter profile ρ can turn repulsive for larger values of λ thus resulting in increasing values for the three radiihere considered as λ grows. In the following we will re-strict our attention to more realistic case where the blackhole mass dominates over the dark matter distribution,therefore considering λ < M . III. CHARGED PARTICLE MOTION
We now consider the motion of charged particles inthe geometry described by the line element (1) once anexternal magnetic field is present. For simplicity we willconsider the magnetic field to be uniform at large dis-tances. Also we assume that the presence of the electro-magnetic field does not affect the background geometry.However, the presence of the magnetic field alters themotion of charged particles depending on its strength.We can introduce the following dimensionless parameterto characterize the magnetic field strength [73]: b ≡ qBM Gmc , (15)where q is the test particle’s charge, B is the modulus ofthe uniform external magnetic field and we have reintro-duced G and c in order to be able to make quantitativeestimates. Notice that depending on the signs of q and B we may have b positive, when charge and magneticfield are ‘aligned’, or negative otherwise. This parameteris of the order of b ∼ for a proton around a stellarmass black hole of mass M ∼ M (cid:12) and of the order of b ∼ for a proton around a supermassive black holeof mass M ∼ M (cid:12) , where M (cid:12) is the mass of the Sun.This quantity can be larger for an electron due to the factof its mass is smaller as compared to the mass of proton.Thus, these estimates clearly show that the effect of themagnetic field on a charged particle motion can domi-nate over gravitational field for particles on orbits nearthe black hole [43, 73, 78, 79].In the following, we consider an external asymptoti-cally uniform magnetic field surrounding the black hole.Following Wald [11] we consider the timelike and space-like Killing vectors ξ α ( t ) = ( ∂/∂t ) α and ξ α ( ϕ ) = ( ∂/∂φ ) α associated to the time translational and rotational sym-metries of the spcetime (1). We can write the Killingequations as ξ α ; β + ξ β ; α = 0 , (16)from which we obtain the following equation (cid:3) ξ α = ξ α ; β ; β = R µδ ξ δ , (17)where R µν is the Ricci tensor. It is obvious that the right-hand side of expression (17) vanishes in the vacuum case,so that the equation takes the simpler form (cid:3) ξ α = 0.Then, in the vacuum case, the Maxwell equations forthe vector potential A α take the same form as in theLorentz gauge, i.e. (cid:3) A α = 0 . However, the line element(1) is not Ricci flat ( R αβ (cid:54) = 0), thus implying that thevector potential for the magnetic field surrounding thesource must be modified. In fact, in this case the vectorpotential of the electromagnetic filed can be defined inthe following way [52, 75] A α = C ξ α ( t ) + C ξ α ( φ ) + a α , (18)with C and C being integration constants and a α dueto the non vanishing Ricci tensor. The vector a α can beobtained from the following equation (cid:3) a α = a α ; β ; β = (cid:16) C ξ γ ( t ) + C ξ γ ( φ ) (cid:17) R αγ . (19)From the condition of the spacetime being static weimmediately obtain one of the integration constants as C = 0. Taking the second integration constant to be C = B/ A t = A r = A θ = 0 ,A ϕ = B r (cid:20) λr (cid:18) Mr (cid:19)(cid:21) sin θ . (20)The four-velocity of the zero angular momentum ob-servers (ZAMOs) in the spacetime under considerationis ( u α ) ZAMO = (cid:110) √ F , , , (cid:111) , (21)from which we obtain the non vanishing components ofthe Faraday tensor measured by the ZAMOs as F rφ = Br (cid:18) λ r log Mr (cid:19) sin θ , (22) F θφ = Br (cid:20) λr (cid:18) Mr (cid:19)(cid:21) sin θ cos θ . (23)From the above one easily obtains the expressions forthe orthonormal components of the electromagnetic fieldmeasured by the ZAMO observers as B ˆ r = − B (cid:20) λr (cid:18) Mr (cid:19)(cid:21) cos θ , (24) B ˆ θ = B √ F (cid:20) λ r log Mr (cid:21) sin θ . (25)As expected, the radial and polar components of theelectric field do not appear because of the vanishing ofdragging of inertial frames due to the spin parameter ofthe black hole being zero. In the limit of flat spacetime or in the limit of large distances, i.e. for M/r →
0, Eqs.(24) and (25) reduce to B ˆ r = − B cos θ , B ˆ θ = B sin θ , (26)which describe an homogeneous magnetic field in flatspacetime. The magnetic field lines are shown in Fig. 2.The Hamiltonian for the system of a charged parti-cle around a static and spherically symmetric black holeimmersed in an external magnetic field is H ≡ g αβ ( π α − qA α )( π β − qA β ) , (27)with π α being the canonical momentum of the chargedparticle and the four-vector potential of the electromag-netic field A α given by Eq. (20). The Hamiltonian is aconstant H = ˜ k/
2, where ˜ k = − m (with m the mass ofthe charged particle) [80].Accordingly Hamilton’s equations of motion in termsof x α and π α are dx α dτ = ∂H∂π α , (28) dπ α dτ = − ∂H∂x α , (29)where τ = ς/m is the affine parameter with proper time ς . As usual, the first equation is a constraint equationproviding the definition of the four-momentum of thecharged particle. Note that the action S correspondingto the Hamilton-Jacobi equation can be separated in thefollowing form S = −
12 ˜ kτ − Et + Lϕ + S r ( r ) + S θ ( θ ) , (30)where the quantities E ≡ − π t and L ≡ π ϕ , together withthe rest energy of the test particle m , are the constantsof motion, namely, energy and axial angular momentumof the charged particle. Using Eqs (27) and (30), one caneasily obtain the Hamilton-Jacobi equation in the form˜ k = − F ( r ) − ( E + qA t ) + F ( r ) (cid:18) ∂S r ∂r (cid:19) + 1 r (cid:18) ∂S θ ∂θ (cid:19) + ( L − qA ϕ ) r sin θ . (31)A fourth constant of motion can be obtained due to sepa-rability of the action. However, since the fourth constantof motion is related to the latitudinal motion of test parti-cles, we won’t need to specify it when considering motionin the equatorial plane.By virtue of Eq. (31), the radial equation of motionfor the charged test particle can be written in the usualform 12 ˙ r + V eff ( r ; L , λ, b ) = E , (32) x / M z / M x / M z / M x / M z / M FIG. 2: The configuration of magnetic field lines in the vicinity of a black hole surrounded by perfect fluid dark matter fordifferent values of λ and a given magnetic field with strength B = 0 . B as a dimensionless quantityon the basis of Eq. (15) having set G = c = 1). The values of the dark matter parameter used in the figures are λ = 0 .
05 (leftpanel), λ = 0 . λ = 0 . z corresponds to the symmetry axis θ = 0 alongwhich the magnetic field is oriented, the horizontal axis x corresponds to an arbitrary radial direction orthogonal to z . λ = λ = λ = . r / M V e ff b = = b = . r / M V e ff b = = b = . r / M V e ff FIG. 3: Radial dependence of the effective potential for massive particles around a black hole in perfect fluid dark matterimmersed in an external asymptotically uniform magnetic field. Left panel: V eff is plotted for different values of λ in the casewithout magnetic field, i.e. b = 0. Middle panel: V eff is plotted for different values of b in the case without dark matter, i.e. λ = 0. Right panel: V eff is plotted for different values of b in the case of fixed λ = 0 . where the effective potential V eff ( r ; L , λ, b ) which deter-mines the motion of the particle is given by V eff = F ( r ) (cid:32) (cid:2) L − bM (cid:0) λr (cid:0) Mr (cid:1)(cid:1) r (cid:3) r (cid:33) , (33)and where we have used the specific constants of motionper unit mass, namely E = E/m , L = L/m . The mag-netic parameter b measuring the effect of the magneticfield on the charged particle motion is given in Eq. (15).In the case of vanishing λ and b parameters, Eq. (33) re-covers the effective potential for the Schwarzschild space-time.In Fig. 3 we show the radial dependence of the effectivepotential (33) for different values of λ and b . We see thatthe presence of dark matter, i.e. λ > b >
0, interms of the strength of the potential, therefore suggest-ing the possibility that these two effects may cancel eachother at some radius for certain values of λ and b . Onthe other hand we notice that regardless of the sign ofthe magnetic field parameter, the ISCO radius is always smaller with respect to the Schwarzschild case, thus sug-gesting that the geometry could be distinguished fromthe Schwarzschild geometry, provided that one is able tohave an independent measurement of M .We shall now consider circular orbits and in particu-lar study the innermost stable circular orbit (ISCO) forcharged particles moving in the black hole spacetime withasymptotically uniform magnetic field and dark matter.The condition for circular orbits ˙ r = ¨ r = 0 is obtained bytaking the following conditions for the effective potentialand its first derivative V eff ( r, L , λ, b ) = E , ∂V eff ( r, L , λ, b ) ∂r = 0 . (34)We can then solve the above equations to find the cor-responding values of the specific energy E and angularmomentum L at the circular orbits. In Fig. 4 we showthe dependence on λ and b of the specific angular momen-tum L for particles in circular orbit. For small radii both b and λ have similar effect, thus reducing the value of L for the particle to be on circular orbit. Finally, to findthe ISCO we solve the equation for the second derivative b = = = r / M L λ = λ = λ = r / M L FIG. 4: The dependence of the specific angular momentum on the radial motion of the charged particles moving in the vicinityof the black hole in the perfect fluid dark matter. Left panel: For the different values of magnetic parameter b for dark matterparameter λ = 0. Right panel: for b = 0 for the different values of dark matter parameter λ . The case λ = 0, b = 0 correspondsto Schwarzschild case.TABLE I: The values of the ISCO radius r i for charged parti-cles moving around the black hole surrounded by perfect fluiddark matter for different values of λ and b . Notice that r i hasmaximum value in the Schwarzschild case. bλ .
000 0 .
001 0 .
005 0 .
010 0 .
050 0 . − . − . − . − . − . of the effective potential to vanish ∂ V eff ( r, L , λ, b ) ∂r = 0 . (35)In table I, we show the ISCO radius obtained by solvingEqs. (34) and (35) numerically for different values ofthe magnetic and dark matter parameters. It can beseen that the radius of the ISCO decreases due to thecombined effects of dark matter and magnetic field.We can compare the behaviour of test particles withthe Schwarzschild case by considering the difference be-tween the effective potential V Schw eff for Schwarzschildspacetime and the effective potential V eff ( r, L , λ, b ) for the black hole surrounded by dark matter and magneticfield. Their difference is given by R ( r ; L , λ, b ) = V Schw eff ( r ) − V eff ( r ; λ, b ) = (cid:18) − Mr (cid:19) (cid:18) L r (cid:19) − (cid:18) − Mr + λr log r | λ | (cid:19) × (cid:104) L − bM (cid:16) λr (cid:0) Mr (cid:1) (cid:17) r (cid:105) r . (36)For a given radius r there may exist non zero values ofboth λ and b for which the effective potential is the sameas Schwarzschild, i.e. for which R ( r ; L , λ, b ) = 0.We can evaluate the implicit function λ ( b ) for which R ( r, L , λ, b ) = 0 at a given r of circular orbit by imposingthe following conditions R ( r ; L , λ, b ) = 0 , ∂R ( r ; L , λ, b ) ∂r = 0 . (37)In Fig. 5, we show the relation between the dark mat-ter parameter λ and the magnetic field parameter b forwhich the effective potential mimics the Schwarzschildcase. Additionally from Fig. 5 one can see the combinedeffect of the magnetic field and dark matter around ablack hole. Namely, as one moves at larger radii, for anygiven value of b it is necessary a larger value of λ in orderto mimic the Schwarzschild behaviour. IV. PARTICLE COLLISIONS
In this section, we investigate the effect of perfect fluiddark matter on the center of mass energy for collisionsof two particles in the vicinity of a static black hole im-mersed in an external magnetic field. As it is usuallydone, we consider two particles with energies at infinityequal to the rest masses m i ( i = 1 , r = = = = b / M FIG. 5: The dependence of the dark matter parameter λ onthe magnetic field parameter b for a charged test particle oncircular orbit at a fixed radius r such that the orbit mimics acircular orbit in the Schwarzschild spacetime. the four-momentum and the total momentum of the twoparticles as π αi = m i u αi , (38) π αtot = π α + π α , (39)with u αi being the four velocity of the i -th particle ( i =1 , E cm as E m m = m + m m m − g αβ u α u β . (40)Our aim is to understand the effect of dark matter andexternal magnetic field on the amount of energy extractedfrom the collision of two particles. In the following, wewill focus on a specific collision scenario that was orig-inally developed by Frolov [43]. More specifically, weconsider the collision between a neutral particle in freefall from spatial infinity with a charged particle revolvingat a circular orbit, and more specifically we will considerthe ISCO orbit. In [43] it was found that for a static andspherically symmetric black hole surrounded by an ex-ternal magnetic field the scenario leads to arbitrary highcenter of mass energy.For convenience, let us denote with the subscript n the freely falling neutral particle with mass m = m n and with the subscript q the charged particle on circularorbit with mass m = m q . Then their four momenta shallbe π n and π q and the total momentum π αtot = π αn + π αq for which the center of mass energy of these two particlesis written as E = − π αtot π totα = m n + m q − g αβ π αn π βq . (41) The four momentum of a charged particle moving atan arbitrary circular orbit with radius r is π αq = m q γ (cid:32) rr − M + λ log r | λ | (cid:33) / δ αt + 1 r υδ αϕ , (42)with υ being the particle’s velocity for the rest frame withLorentz factor γ = 1 / √ − υ . For the sake of clarity, wecan rewrite our quantities in terms of ˜ L q = L q / (2 m q M ),˜ r = r/ M and ˜ λ = λ/ M . Then the expression for theangular momentum together with dϕ/dς = υγ/r leads to υγ = ˜ L q ˜ r − b (cid:34) λ ˜ r (cid:18) r (cid:19)(cid:35) ˜ r , (43)with the Lorentz factor γ given by γ = 1 + (cid:32) ˜ L q ˜ r − b (cid:34) λ ˜ r (cid:18) r (cid:19)(cid:35) ˜ r (cid:33) , (44)so that we find the velocity as υ = ˜ L q − b (cid:104) ˜ r + ˜ λ ˜ r (cid:0) r (cid:1)(cid:105)(cid:114) ˜ r + (cid:16) ˜ L q − b (cid:104) ˜ r + ˜ λ ˜ r (cid:0) r (cid:1)(cid:105)(cid:17) . (45)Hence, recalling Eq. (41) and employing Eq. (42) wederive the center of mass energy for the collision of aneutral particle in free fall from spatial infinity with acharged particle orbiting at a circular orbit with radius r as E = m n + m q + 2 m q γ × E n (cid:32) rr − M + λ log r | λ | (cid:33) / − υr L n . (46)Notice that the above expression diverges for r = r h . Ofcourse, stable circular orbits in the vicinity of the horizonare generally not allowed as r i > r h . However, we haveseen that the value of the ISCO decreases as we increasethe absolute value of the magnetic field (i.e. b ) and thecontribution due to the dark matter part (i.e. λ ). Thisallows for the extraction of larger collision energies froma black hole immersed in an external magnetic field andsurrounded by a dark matter distribution as comparedto the Schwarzschild case. Considering almost radial fallfrom infinity for the neutral particle one can neglect thesecond term in the bracket of Eq. (46) due to the factthat L n is small enough as compared to the first term.Hence, Eq. (46) yields E ≈ m q γE n √ r (cid:113) r − M + λ log r | λ | . (47)To evaluate the value of the center of mass energy forthe collision between a freely falling neutral particle with E n = m n and a charged particle on circular orbit we needto determine the angular momentum ˜ L q of the chargedparticle as a function of the radius of the circular orbitfrom V (cid:48) eff (˜ r, ˜ L, ˜ λ, b ) = 0 similarly to what was done inEq.(11) with the addition of the external magnetic field.From the effective potential given in Eq. (33) we get˜ L q = b ˜ r (3˜ r − / (3 − ˜ r ) / (cid:40) − ˜ λ b (3˜ r − r + − ˜ λ (cid:16) (cid:0) (3 − ˜ r )(3˜ r − (cid:1) / (cid:17) r − r −
1) ++ 9˜ λ ˜ r − λ ˜ r log(˜ r/ | ˜ λ | )2(˜ r − r − (cid:41) . (48)We can then express the center of mass energy E cm ( r )for the collision as a function of the radius of circular orbit r for any given set of values of ˜ λ and b . In Fig. 6 one cansee the effect of perfect fluid dark matter and the externalmagnetic field on the extracted energy from the collision.The center of mass energy increases when increasing thevalues of λ and b . However, one should keep in mindthat E cm is evaluated for charged particles on circularorbits and therefore it is valid only in the regime wherestable circular orbits are allowed, that is for r > r i . It isthen useful to evaluate the center of mass energy at theISCO orbit, which is the maximum possible extractedenergy for any given λ and b . By substituting Eq. (48)into V (cid:48)(cid:48) eff (˜ r, ˜ L, ˜ λ, b ) = 0, we derive the following conditionthat implicitly determines the value of the ISCO0 = (cid:0) r − r + 3 (cid:1) + (cid:112) (3˜ r − − ˜ r ) ++ ˜ λ (cid:16) r − log ˜ r | ˜ λ | (cid:17)(cid:0) (3 − ˜ r )(3˜ r − (cid:1) / + − ˜ λ (cid:16) (cid:0) ˜ r + 2˜ r − (cid:1) log ˜ r | ˜ λ | − (cid:0) r − r + 6 (cid:1)(cid:17) ˜ r − − ˜ λ (cid:16) (cid:0) r − r + 3 (cid:1) log ˜ r | ˜ λ | + 6˜ r − r + 3 (cid:17) b (3˜ r − r + − ˜ λ (3 − ˜ r ) / b (3˜ r − / ˜ r − (˜ λ + 1)(3 − ˜ r )2 b ˜ r . (49)The equation above allows us to determine r i in the casein which λ (cid:28) b (cid:29)
1. In this limit we get r i M ≈ √ b (cid:20) λ M (cid:18) λM (cid:19) + O (cid:0) λ (cid:1)(cid:21) + O (cid:0) b − (cid:1) , (50)which shows how the ISCO radius approaches the blackhole horizon for large values of the magnetic parameter b . In the vacuum case, i.e. for λ = 0, the approximateexpression for the ISCO radius becomes r i M ≈ √ b + O (cid:0) b − (cid:1) , (51)which corresponds to the result obtained by Frolov andShoom [73]. Also it is worth noting that the ISCO ra-dius decreases also in case the dark matter parameter λ increases.Considering the limiting case of a charged particle or-biting at the ISCO, from Eq. (47) we get E cm m ≈ α ( λ ) b / , (52)with m q = m n = m and α ( λ ) = (2 γ ) / (cid:34) √
32 + (1 + log λ ) λ (cid:35) / . (53)In the limiting case of small λ , with b (cid:29) γ at the ISCO we can rewrite Eq. (53) as α ( λ ) = √ (cid:20) √ (cid:18) √ λ √ (cid:19) λ (cid:21) / × (cid:32) √
32 + (1 + log λ ) λ (cid:33) / . (54)It is not difficult to show that α is an increasing func-tion of λ as it can be seen from Fig. 7, which helps tounderstand the dependence of E cm on the dark matterparameter as seen in Fig. 6.Also, the above expression becomes α (0) = 2 / (3 / ) inthe vacuum case. Thus, the expression (52) representsthe center of mass energy for collision of the charged andneutral particles at the ISCO. In the limit λ →
0, onecan easily recover the result of Ref. [43] E cm m ≈ . b / . (55)In table II, we show the collision energy E cm and theradius of the ISCO r i for different values of the darkmatter parameter λ . It is immediately clear that, whilethe value of r i decreases with λ , the center of mass energyextracted via the collision increases with increasing valueof the dark matter parameter. Therefore in the presenceof dark matter one always obtains a larger value for E cm with respect to vacuum case. Also, as b grows, E cm in-creases, diverging in the limit of r i → r h obtained for b → + ∞ . V. CONCLUSIONS
Accretion disks around super-massive black hole candi-dates are the primary source of information about gravity λ =
0, b = λ = = λ = = r / M E C M / m λ =
0, b = λ = = λ = = r / M E C M / m λ =
0, b = λ = = λ = = r / M E C M / m FIG. 6: The center of mass energy E cm for the collision between a neutral particle in free fall from spatial infinity with acharged particle on a circular orbit in the geometry of a black hole immersed in perfect fluid dark matter and an externalmagnetic field as a function of the radius of the circular orbit r . One must keep in mind that E cm ( r ) here is defined only whenstable circular orbits are allowed and therefore, for given values of λ and b the above plot will be valid only for r > r i . It iseasy to notice that at any given radius E cm increases when λ and/or b increase. λ M α ( λ ) FIG. 7: The dependence of the function α ( λ ) on the darkmatter parameter λ for the collision of neutral particles withcharged particles orbiting at the ISCO radius. See text fordetails.TABLE II: The values of the ISCO radius r i and the centerof mass energy E cm /m for the collision of charged particlesorbiting at the ISCO with neutral particles or radial fall fordifferent values of the dark matter parameter λ at fixed valuesof the magnetic field parameter b . λ r i E cm /m . b − . b / . b − . b / . b − . b / . b − . b / . b − . b / . b − . b / in the strong field regime and the geometry surroundingsuch black hole candidates [81]. Energetic particles areproduced by collisions in the accretion disk and the disk’sluminosity depends on the underlying geometry. How-ever, in a realistic scenario the object can not be con-sidered to be in vacuum, as we know that dark matter distributions exist at the center of galaxies. Also mag-netic fields play an important role in the dynamics ofcharged particles around black holes, especially close tothe black hole’s horizon. Therefore, in order to have con-fidence in the conclusions drawn from the observations ofaccretion disks, it is important to study the effects thatthe presence of external matter fields and magnetic fieldshave on the particles in the disks.In this paper, we studied particle motion in the vicinityof a static, spherically symmetric black hole surroundedby perfect fluid dark matter and immersed in an exter-nal asymptotically uniform magnetic field. We showedthat the radius of the innermost stable circular orbit forcharged particles decreases under the effects of dark mat-ter and external magnetic field. A similar behaviour oc-curs for the unstable photon orbits..We showed that the combined effects of dark matterand magnetic field can cancel each other out only at aspecific radius depending on the values of the dark matterand magnetic parameters. This suggests that the emittedspectrum of an accretion disk in the geometry consideredhere could be distinguished from the spectrum of the diskaround a static black hole.Also, as a consequence of the decrease in value for theISCO, we showed that the center of mass energy for thecollision of neutral and charged particles increases underthe effects of both dark matter and external magneticfield. In this work, following [43], we considered the sim-ple collision scenario of a freely falling neutral particlecolliding with a charged particle revolving at a circularorbit. We showed that the center of mass energy becomesmaximum for charged particles at the ISCO. And sincethe ISCO can become arbitrarily close to the horizon asthe magnetic field strength increases, we showed that thecenter of mass energy can become arbitrarily large.These theoretical studies can help constraint the va-lidity of alternative models to black holes in explainingastrophysical observations.0 Acknowledgments
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