Null geodesic of Schwarzschild AdS with Gaussian matter distribution
Abolhassan Mohammadi, Tayeb Golanbari, Behrooz Malakolkalami, Shahram Jalalzadeh
aa r X i v : . [ g r- q c ] F e b Null geodesic of Schwarzschild AdS with Gaussian matter distribution
Abolhassan Mohammadi , ∗ Tayeb Golanbari , † Behrooz Malakolkalami , ‡ and Shahram Jalalzadeh § Department of Physics, Faculty of Science, University of Kurdistan, Sanandaj, Iran. Departmento de F´ısica, Universidade Federal de Pernambuco, Pernambuco, PE 52171-900, Brazil. (Dated: February 11, 2021)One of the best ways to understand the gravitation of a massive object is by studying the photon’smotion around it. We study the null geodesic of a regular black hole in anti-de Sitter spacetime,including a Gaussian matter distribution. Obtaining the effective potential and possible motions ofthe photon are discussed for different energy levels. The nature of the effective potential impliesthat the photon is prevented from reaching the black hole’s center. Different types of possible orbitsare considered. A photon with negative energy is trapped in a potential hole and has a back andforth motion between two horizons of the metric. However, for specific values of positive energy,the trapped photon still has a back and forth motion; however, it crosses the horizons in everydirection. The effective potential has an unstable point outside the horizons, which indicates thepossible circular motion of the photon. The closest approach of the photon and the bending angleare also investigated.
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I. INTRODUCTION
Black holes are known as a region of spacetime wheregravity is so strong that nothing, even light, can escapefrom it. It is a natural prediction of general relativitythat sufficient mass could change spacetime and forma black hole. The black hole boundary is specified bythe event horizon in which nothing inside the horizonhas a chance to escape. Then, black holes are entirelyblack, and we could see nothing inside. Nevertheless,the situation is different near the event horizon. Thequantum field theory in curved spacetime indicates thatit is possible to emit radiation near the event horizon.The point was first pointed out by Steven Hawking [1],who worked out the mathematical argument, and theradiation is known as Hawking radiation.The discovery of Hawking radiation is the first windowfor quantum gravity and has raised hope for construct-ing a unifying theory [2]. The quantization of generalrelativity, so that gravitons carry the gravitational force,ended up in infinities. Then, gravity was consideredentirely classical. Hawking could find a semi-classicalargument to support his conjecture about the blackhole evaporation [3–5]. However, it was valid onlywhen emitted particles’ energy was small compared tothe black hole’s mass [2].The black hole evaporates,and its mass decreases, and eventually, at a certainpoint, the conjecture was failed to be hole because thesemi-classical description is broken down. As the decaycontinues, a quantum theory of gravity is required toexplain the process. So far, there are two candidates forquantum gravity: string theory and loop quantum grav- ∗ Electronic address: [email protected] ; [email protected] † Electronic address: [email protected] ; [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ity in which both have their own merit and drawbacks[6].Regarding our understanding of quantum gravity, blackholes are assumed to be our best chance [2]. Althougheverything seems to work for understanding the theoryof quantum gravity, the problem is that the candidatetheories do not explain the final stage of the blackhole. A complete quantum description of black holeevaporation is unlikely since the scattering amplitudesand cross sections by perturbative techniques are notcomputed. The new approach for solving the problem isthe Noncommutative Geometry arguments [7–15]. It iswidely believed that there should be an uncertainty inthe theory of quantum theory to prevent the measuringof position to an accuracy better than Planck length [2].The statement comes to this conclusion that the theoryshould satisfy a new form of uncertainty indicating thatthe positions do not commute, i.e. [ x i , x j ] = 0 (referto [2] for more information). The interesting pointabout including the noncommutativity to the theory isthat the spacetime coordinates become noncommutativeoperators by means of [ x µ , x ν ] = iα µν [52], where α µν is an anti-symmetric matrix with the length squareddimension.Noncommunity modification has appealing resultsin both quantum field theory and general relativity[2, 16–22]. It has been concluded that, under suchmodification, the final stage of black hole evaporation sothat the temperature no longer diverges and the curva-ture reaches a finite value [2]. The modification altersEinstein’s action, which brings out new field equations[23–31]. Another viewpoint is that the noncommutativeeffects leaves the Einstein tensor unchanged, and theeffects only appear in the matter source [2, 32–35]. Oneconsequence of the noncommunity effect is that, in flatspacetime, it removes the point-like structures in favorof smeared objects [34]. In the mathematical view,the work is done by replacing everywhere the positionDirac-delta function with a Gaussian distribution ofminimal width √ α . The coordinate commutator encodesan intrinsic uncertainty, which implies that instead ofhaving a particle mass M localized in a point, it spreadsout through a region of linear size √ α . The resultingmass density of a static, spherically symmetric, smeared,the point-like gravitational source is presented by Eq.(1).Replacing the vacuum with Gaussian matter distributionleads to a new regular metric. The metric has beenconsidered for different aspects [2, 34–37].The present work’s primary goal is to study the geometryof the null geodesics of AdS Schwarzschild with Gaussianmatter distribution. The main motivation of the presentwork is two folds. First, we have the non-singular natureof the metric. Assuming that general relativity is validup to arbitrarily high energy and curvature, it wasproved by Hawking and Penrose that spacetime with ablack hole is not geodesically complete, and there willbe a singularity. This result was obtained under thedominant energy condition. The singularity, infiniteenergy, and curvature are nonphysical and undesirablefor us. It is widely believed that formulating a quantumtheory of gravity will release gravitation from singular-ities. One approach to formulating quantum gravityis the noncommutative geometry effects, which leadsto some uncertainties and minimal length. The secondmotivation stands on the importance of null-geodesics.The motion of the massless particle is recognized as wayto understand the gravitational field around a blackhole. Due to this fact, the null geodesics has been thetopic of many research works, where different typesof black holes have been considered. In some cases,the black holes are assumed to be surrounded by darkenergy candidates such as quintessence. [38–47].The paper is organized as follows: In Sec. II, thespacetime we are working on is introduced. We workwith the Lagrangian of the black hole in Sec.III. Then, inSec.IV, the radial motions are studied where the angularmomentum is ignored. Null geodesic with angularmomentum is considered in Sec. V, where the effectivepotential, energy levels, classical force, and circularorbits are discussed. The light passes for different energylevels of the photon are investigated in Sec. VI. Finally,in Sec.VII, the closest approach is considered, and wecalculate the bending angle of the photon. The resultsare concluded in Sec.VIII. II. THE SPACETIME
Assuming that the spacetime is filled with a negativecosmological constant. Solving the Einstein equation un-der this condition leads to the regular Schwarzschild anti-de Sitter (AdS) metric, which is static and possesses aspherical symmetry. Adding a matter with Gaussian dis- tribution, as follow ρ ( r ) = M (cid:0) πα (cid:1) / e − r / α , (1)where the parameter α indicates the variance of the dis-tribution (i.e., √ α ), will change the geometry of space-time. The presence of such energy density is due to thenon-commutative effect, which states that the coordi-nates do not commute and there is a minimal length; asexplained in the introduction. Then, we have no longera point-like particle, instead, the mass ( M ) of the parti-cle is spread in a region with size √ α . In mathematicallanguage, it means that the Dirac delta function shouldbe replaced by the Gaussian distribution function. Then,as α →
0, the non-commutative effects vanish and coor-dinates commute. In this limit, the Gaussian distribu-tion function in Eq.(1) comes to the Dirac delta function[2, 34, 48, 49].The matter distribution results an energy-momentumtensor T µν which under the condition ▽ µ T µν = 0 and g = − g − gets the following form T = T rr = − ρ ( r ) , T θθ = T φφ = − ρ ( r ) − r ∂ρ ( r ) ∂r . (2)The energy-momentum tensor describes an anisotropicfluid. Including this source of energy to the Einsteinfield equations come to a spherically symmetric solutiondescribed by ds = − g ( r ) dt + g − ( r ) dr + r d Ω , (3)where d Ω = dθ + sin θdφ , and the metric coefficient g ( r ) is read as g ( r ) = 1 + r b − G N Mr √ π γ (cid:18) , r α (cid:19) , (4)where G N is the Newton’s constant, which from now onit is taken as G N = 1, and b is the curvature radius ofthe AdS space. The function γ (cid:16) , r α (cid:17) is known as thelower incomplete gamma function described as γ ( n, x ) = Z x dt t n − e − t . Fig.1 describes the behavior of the function g ( r ). It isrealized that depending on the values of the constants α and b , the metric could have two, one, and no horizon,which is plotted in the figure. The behavior of the metriccoefficient g ( r ) versus the radius r is illustrated in Fig.1for different values of b . It is realized that as r approacheszero, the coefficient g ( r ) tends to one. The main differ-ence with the case of no cosmological constant is that,as the radius goes to infinity, the coefficient gets biggerand bigger. However, for the case Λ = 0, as r → ∞ , thecoefficient asymptotically approaches to one; Fig.2. b = = = - r ˜ g ( r ˜ ) r in r out FIG. 1: The metric coefficient g ( r ) versus the radius ˜ r ≡ r/ √ α is plotted for different values of b . The other constantsare taken as ˜ M ≡ M/α = 2 .
5, and α = 0 . M ˜= M ˜= M ˜= - r ˜ g ( r ˜ ) FIG. 2: The metric coefficient g ( r ) (for Λ = 0) versus theradius ˜ r ≡ r/ √ α is plotted for different values of b , where α is taken as α = 0 . III. NULL GEODESICS
The Lagrangian of the described black holes, as statedby L = − (cid:16) − g ( r ) ˙ t + g − ( r ) ˙ r + r ˙ θ + r sin θ ˙ φ (cid:17) . (5)The metric has two killing vector, i.e. ∂ t and ∂ φ , leadingto the two constants of motions g ( r ) ˙ t = E n , (6) r sin θ ˙ φ = L. (7)As the initial conditions, it is assumed that θ = π/ θ = 0. It results in ¨ θ = 0, which means that θ stands in π/ π/
2. By substituting Eqs.(6)and (7) into the Lagrangian ((5)), one arrives at˙ r + g ( r ) (cid:18) L r + ε (cid:19) = E n , (8) in which ε = 2 L and ε = +1 and 0 respectively corre-spond to the massive and massless particle. ComparingEq.(8) with the relation ˙ r + V eff = E n , one could de-fined an effective potential V eff = g ( r ) (cid:18) L r + ε (cid:19) . (9)To consider the circular orbit of the particle, it is requiredto realize the relation between φ and r . From Eq.(7), onefinds dφdr = Lr p E n − V e . (10)For the rest of the work, we concentrate on the nullgeodesic so that ε = 0. In this case, the effective po-tential is given by V eff = L g ( r ) /r . IV. RADIAL NULL GEODESICS
There is no angular momentum for the radialgeodesics, i.e., L = 0. Consequently, the effective po-tential is null, V eff = 0. Besides, from Eqs.(6) and (8),one arrives at ˙ r = ± E n , ˙ t = E n g ( r ) . (11)Combining the relation, the time t is extracted as a func-tion of the radial t , so that dtdr = ± g ( r ) , (12)and the time is obtained by taking integration, however,analytically solving the integration faces difficulties de tothe presence of incomplete gamma function.The proper time s is derived from the relation ˙ r = dr/ds = ± E n by integration as s = ± rE n + c, (13)where c is a constant of integration. It is realized thatas the particle approaches the horizons, r → r in,out , theproper time is always a finite value, s → ± r in,out /E n . V. ANGULAR MOMENTUM FOR NULLGEODESICS
In the case of L = 0 the situation will be different. Thepotential is no longer zero, and it is V eff = L g ( r ) r . (14)Fig.3 displays the potential of the model versus the radius r for different values of angular momentum. L = = = - - - - r ˜ V e ff ( r ˜ ) ( × - ) r in r out FIG. 3: The behavior of the potential versus the radius ˜ r fordifferent values of the angular momentum. The other con-stants are taken as ˜ M ≡ M/α = 2 . α = 0 .
1, and b = 10.FIG. 4: The figure shows the effective potential of the particleand some energy levels. The constants are taken as ˜ M ≡ M/α = 2 . α = 0 . b = 10, and L = 0 . The energy relation ˙ r + V eff = E n is utilized to describethe motion of the particle based on its energy level. As aninstance, Fig.4 illustrate the effective potential of particleand also there are some energy level. Based on the energylevel of the particle, there are different types of motionwhich are explained as follows: • E = E : At this energy level, the particle stays inthe bottom of the potential hole r = r , and has acircular orbit. It could not have a radial motion.The shape of the potential implies that the r = r point is a stable point. Note that, since r in < r A
The force on the photon is obtained by taking radialdifferentiation from the effective potential, as F = − dV eff dr (15)= L (cid:20) r − M √ π r γ (cid:18) , r α (cid:19) + M α √ π α r e − r / α (cid:21) . The first term on the right-hand side (RHS) is alwayspositive, which implies a positive force that pushes theparticle away from the center. This force is high for small r and rapidly decrease by enhancement of r . The cosmo-logical constant has no role in the classical force. Thecosmological constant’s central role is in the potential,which brings an asymptotical flat (non-zero) potentialfor large r . The second and third terms on the RHS aredue to the presence of Gaussian matter distribution. Thetotal force that a photon experience is depicted in Fig.6versus the radius for different values of angular momen-tum. The force changes sign two times in r and r .For small r < r , the force is positive and it tends to L = = = - - - - r ˜ F ( r ˜ ) ( × - ) r in r out r r FIG. 6: The total imposed on a photon during its motionaround the black hole versus the radius ˜ r for different valuesof angular momentum. The other constants are taken as ˜ M ≡ M/α = 2 . α = 0 .
1, and b = 10. push out the photon. The point r is inside the hori-zons. The photon feels this preventing force when it isinside the horizons and wants to reach the black hole’scenter. The force gets larger as the photon comes closerto the center and prevents it from reaching the center.In r = r , the force changes sign and become negativefor r < r < r . The photon is pulled back to the center(note that the outer horizon is in between). The force iszero again for r = r , and it is positive for r > r . Then,for r > r , the photon feels a positive force that drivesit away from the black hole. The plot at the left side ofFig.7 enlarges the force’s shape at r = r which clearlyillustrates that the force becomes zero and changes sign.Then, the force is positive and gets larger in magnitude for a while and again gets weaker and tends to zero, de-picted in Fig.7. Zero force for large r was expected fromthe shape of the potential where it was mentioned thatthe potential becomes flat for large r . Therefore, whenthe photon is at a large distance from the black hole, italmost feels no force. FIG. 7: The plot displays the behavior of the total force inthe second and third critical point of the radius. It is realizedthat the total force is zero in the point and it changes sign.
B. Circular orbits
When the particle has a constant energy, i.e. E n = E c ,the particle’s radius remains unchanged, ˙ r = 0. At thispoint, the radial derivative of the effective potential iszero dV eff /dr = 0, and the point is an unstable point.Then, the potential and energy get equal, and the radialcoordinate of the particle remains unchanged, r = r c .Suppose that the constants of the model have been jus-tified in a way that there are two horizons. Then, therewill be two r c describing circular orbits.The energy E c and the angular momentum L c at the cir-cular orbits are related through the impact parameter D c at the circular stage. The relation is read as E c L c = g ( r c ) r c = 1 D c . (16)From Fig.4, it is realized that there might be two circularorbits occurring in r = r and r respectively related tothe two energy levels E c = E , and E . The first circularorbit stays between the two horizons, which is happeningat the stable point of the potential. The second circularorbit is outside of the horizon, which stands on the un-stable point. C. Time period of circular orbit
The circular orbit’s period is derived via the Eq.(6) interms of proper time and physical time. Since the radial r c for the circular motion is constant, the proper time isobtained as T s = 2 πr c L c , (17)and in terms of the physical time, the period is obtainedas T t = 2 πr c p g ( r c ) . (18)Both periods are depicted in two plots regarding the twocircular orbits in r = r and r in Fig.8. At r = r , (a)(b) FIG. 8: The proper and physical time period versus angularmomentum are plotted for r c and r c circular orbits. To plotthe quantities, the constants are taken as ˜ M ≡ M/α = 2 . α = 0 .
1, and b = 10. the circular orbit is inside the two horizons and is hiddenfrom the outside observer. Then there is no access to theparticle for the outsider observer to measure the physicaltime. Due to this fact, Fig.8a only exhibits the properperiod of the first circular orbit versus the angular mo-mentum. The proper time period increases by decreasingthe angular momentum.For the second circular orbit, which stays outside of thehorizons, the two time period is displayed in Fig.8b. Forlarger angular momentum values, the physical period ishigher. However, the proper period is higher for smallerangular momentum. Note that T t is independent of theangular momentum. D. Unstable circular orbit
The Lyapunov exponent λ [50], which is defined as λ = s − V ′′ eff ( r c )2 ˙ t ( r c ) , (19)is measure the instability of the unstable circular nullgeodesics. For the present case, the parameter λ is ob-tained as λ = s − V ′′ eff ( r c ) r c g ( r c )2 L . (20)Then, from Eq.(9), it is realized that the instability pa-rameter λ does not depend on the angular momentum.There are two circular orbits, one stable and the otherunstable, which stays outside the horizon. The instabilityparameter for the outsider orbit is about λ = 0 . VI. LIGHT PATH
The effective potential has been considered, and wecould describe the possible path of light based on its en-ergy level. Then, the photon’s force was investigated,and it was specified whether the force is attractive or re-pulsive. Now, we are going to consider the path of lightand its orbit in more detail.First, we consider an unbounded photon, namely a highenergy photon that crosses two horizons and encountersthe potential bond in the turning point r F . The pho-ton then comes back and recedes from the black hole.The path is depicted in Fig.9. The blue line describeshe incoming photon with start point ( r, φ ) = (20 , φ = 3 π which is illustrated inFig.9(b).A photon path with energy E = E c is portrayed inFig.10. The incoming photon is plotted in Fig.10(a)where the start point is assumed to be ( r, φ ) = (15 , r = r , the effec-tive potential of the photon equals to the its energy, i.e. V eff ( r = r ) = E c . The radial point r is an equilib-rium point, specified in Fig.4, and the particle continuesits movement as a circular orbit. However, it should benoted that the point is an unstable equilibrium point.Fig.10(b) describes an outgoing photon with the sameenergy level, E = E c . The photon recedes from the cen-ter of black hole, crosses the two horizons and reach thepoint r = r . At this point, the potential energy of thephoton reaches to its energy and the photon has no radialmovement any more. Then, it continues its motion as a (a)(b) FIG. 9: The unbounded orbit of photon with high energy ispresented. The incoming and outgoing photon are respec-tively illustrated by blue and red lines. The figure is a polarplot, i.e. r − φ , which is drawn in x − y coordinate (the θ = π/ r, φ ) = (20 , circular orbit.For a photon with energy lower the energy E c , there aretwo situations. First, suppose that the photon is awayfrom the outer horizon of the black hole. In this case, foran incoming photo, it approaches to the black hole, en-counters to the turning point (like the r E point in Fig.4)where V eff = E . Then, the photon changes its directionand recedes from the black hole. The light path for thiscase is presented in Fig.11. The blue line in Fig.11(a) de-scribes the incoming photon which is assumed to start itsmotion at ( r, φ ) = (20 , π . The blue line in Fig.11(a) describes theincoming photon which is assumed to start its motion (a)(b) FIG. 10: The blue (red) line present the orbit of incoming(outgoing) photon. The photon approaches (recedes from)black hole. The effective potential increases and at the point r = r the effective potential and photon energy are equal.The point is an extremum point of potential (an unstablepoint). Then, the radial distant of the photon remains un-changed and photon start a circular orbit. at ( r, φ ) = (20 , π .The last path that is considered is for a photon in thepotential hole. A photon with this energy could not es-cape from the hole and it has a back and forth motion,like a photon with energy E in Fig.4 that has two turn-ing point, r c and r D . The photon start point is assumedto be ( r, φ ) = (5 , (a)(b) FIG. 11: The plot represents the orbit of photon with mediumenergy, L /b < E < E c , away from the outer horizon. Theblue (red) line presents the incoming (outgoing) photon. Thephoton comes to the black hole, encounters the turning pointout of the r out horizon. Then, it turns back and its radialdistance r rapidly increases; as presented in (b). VII. GRAVITATIONAL LENSING
One of the main application of null geodesics are grav-itational lensing and finding the bending of light. Thefirst step to this matter is to take an unbounded orbitwith energy E = E . The light approached the blackhole, and it reaches the minimum distance, r = r , andthen recedes from the black hole. We need to find howphotons deviate from their original path due to the blackhole’s gravity. In this regard, it is required first to findthe closest approach r . A. Closest approach r An unbounded orbit of light has been picked out. So,the photon approaches the black hole, reaches a minimumdistance r (close approach), and then keeps out. Theorbit occurs in θ = π/ r of light changes with the angle φ . Therefore, to findthe closest approach, and one needs to solve dr/dφ = 0. FIG. 12: The figure illustrates the orbit of a photon trappedin potential hole. The energy of photon is not enough to es-cape from the hole and it only has a back and forth motion.The polar plot of the orbit is drawn in figure, which is picturedin x − y coordinate. The blue (red) line presents the incoming(outgoing) photon. The photon starts at ( r, φ ) = (5 , From Eqs.(6), (9) and (8), it is realized that (cid:18) r drdφ (cid:19) = E n L − r − b + 4 M √ π r γ (cid:18) , r α (cid:19) ≡ χ ( r ) . (21)The energy level has a vital role in determining theclosest distance r . From the potential, Eq.(9) andFig.3, it is realized that there is no unbounded orbitfor negative energy. As a matter of fact, the potentialasymptotically approaches to a constant value L /b forlarge radial distance. Then, the minimum energy whichshould be picked is L /b . Fig.13 illustrates the function χ ( r ) versus the radius r for different values of energy. Itis exhibited that there are one, two, or three values of r leading to χ = 0 depending of the magnitude on energy.For larger energy, The photon approaches the blackhole for more considerable energy, crosses two hori-zons, gets to the closest distance r , and then recedesfrom the black hole. Therefore, there is only one clos-est distance r for the photon with energy E > V eff ( r ) . B. Bending angle
The unbounded photon could be divided into twotypes. The first corresponds to the photons that haveonly one turning point, which stays inside the hori-zons. The second type includes the photon with energy L /b < E < E c which has three turning points, and FIG. 13: The plot displays the behavior of the function χ ( r )versus the radius r for different values of energy. the point χ ( r ) = 0 is clear from the figure. It is realized that for se-lected energy, there could be found one, two, or three r whichmainly depends on the magnitude of energy. The constantsare taken as ˜ M ≡ M/α = 2 . α = 0 . b = 10, and L = 0 . only one of them is out of horizons. The other two turn-ing points stand on the potential hole wall, which is re-lated to the trapped photon. Here, we are only interestedin unbounded photons. At the turning points, we have χ ( r ) = 0.For static, spherically symmetric metric, with generalform of line element ds = A ( r ) dt − B ( r ) dr − D ( r ) r (cid:16) dθ + sin θdφ (cid:17) , the bending angle is obtained as [51]∆ φ = 2 Z ∞ r q B ( r ) D ( r ) r(cid:16) rr (cid:17) D ( r ) D ( r ) A ( r ) A ( r ) − drr − π, (22)where r indicates the closest approach [51]. Comparingthe above line element with Eq.(3), it is realized thatthere are A ( r ) = − g ( r ), B ( r ) = − g − ( r ), and D ( r ) = −
1, leading to bending angle∆ φ = 2 Z ∞ r p g ( r ) r(cid:16) rr (cid:17) g ( r ) g ( r ) − drr − π. (23)The integral could not be solved analytically due to theincomplete gamma function. However, it could be solvednumerically. Fig.14 presents the bending angle versus theclosest approach r , in which the smaller r correspondsto the photon’s higher energy. The bending angle is pos-itive for smaller r and decreases by enhancement of r .The bending angle asymptotically approaches a constantvalue that is not zero. The main reason for this resultis that the metric is not asymptotically flat. For largeradial distance, the AdS term in the metric dominatesand causes such behavior. FIG. 14: The plots describe the bending angle (∆ φ ) versus theclosest approach r . The higher energy of photon indicatesby smaller r . The constants are taken as ˜ M ≡ M/α = 2 . α = 0 .
1, and b = 10. VIII. CONCLUSION
Including the Noncommutative Geometry argumentsin general relativity has been the topic of many researchprojects. The effect of noncommunity could be consid-ered in two aspects: it could modify the Einstein fieldequation or it leads to a modified energy density so thatremoving the point-like structure is one of the noncom-munity effects.Solving the Einstein field equations in AdS spacetime,including a Gaussian matter distribution, leads to a met-ric that describes a regular black hole. In the presentedwork, the null geodesics, as a tool for understanding amassive source’s gravity, were discussed. Through theLagrangian of the black hole, the effective potential wasextracted, which indicates that based on the metric’s con-stants, the described black hole could have two, one, orno horizon. The work was dedicated to the case of pos-sessing two horizons. Based on the effective potential, ina general view, photon’s possible motions were consid-ered based on the energy levels.The classical force on the photon and also the possiblecircular orbits were discussed. There would be two cir-cular orbits, one at the bottom of the effective potentialhole, which is a stable point of the effective potential.This orbit would occur between the horizons. Moreover,the second circular orbit is at r = r which is an unstablepoint of the effective potential. The physical and properperiod was estimated. There is no physical time for thefirst circular orbit between the horizons, and only theproper time is measurable. However, for the second cir-cular orbit, both periods could be found. It was realizedthat the proper time is smaller than the physical periodfor small angular momentum. As the angular momentumincreases, the proper period will enhance as well, and itcould be even larger than the physical period.Next, the path of the photon was discussed in more de-tail. It was explained that for negative energy, the photonis trapped in the potential hole, and it has a back andforth motion between two horizons, in which the turn-0ing points are inside the horizons. For positive energies, E n > L /b , there are more options. For intermediateenergies, i.e., L /b < E n < E c , the photon could ei-ther be still trapped in the potential hole or have anunbounded orbit to infinity. If the photon is trapped inthe potential hole, it has a back and forth motion andcrosses the horizons in each direction, and changes thedirection as it reaches the turning points. A photon withenergy E c has a circular orbit at r which is an unsta-ble point of the effective potential. The higher energy ofphoton leads to an unbounded orbit. However, no mat-ter how much we increase the photon’s energy, it neverreaches the center of the black hole due to the nature ofthe potential. The effective potential rapidly grows upas r approaches zero.The closest approach of photon and the bending anglewas the last topic considered. The bending angle was ob- tained positive for small r and reduced by growing r ,and asymptotically approaches to a non-zero constantvalue. Due to the AdS nature of the metric, the AdSterm is the dominant term of the metric for large radialdistance. Acknowledgments
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