Geodesic of nonlinear electrodynamics and stable photon orbits
GGeodesic of nonlinear electrodynamics and stable photon orbits
A S. Habibina ∗ and H. S. Ramadhan † Departemen Fisika, FMIPA, Universitas Indonesia, Depok, 16424, Indonesia.
Abstract
We study the geodesics of charged black holes in polynomial Maxwell lagrangians, a subclassmodels within the nonlinear electrodynamics (NLED). Specifically, we consider black holes inKruglov, power-law, and Ayon-Beato-Garcia models. Our exploration on the corresponding nullbound states reveals that photon can orbit the extremal black holes in stable radii outside thecorresponding horizon, contrary to the case of Reissner-Nordstrom (RN) black holes. The reasonbehind this is the well-known theorem that photon in NLED background propagates along its own effective geometry. This nonlinearity is able to shift the local minimum of the effective potentialaway from its corresponding outer horizon. For the null scattering states we obtain corrections tothe weak deflection angle off the black holes. We rule out the power-law model to be physical sinceits deflection angle does not reduce to the Schwarzschild in the limit of vanishing charge. ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ g r- q c ] J u l . INTRODUCTION One of the many intriguing properties of black hole (BH) is the notion of photon sphere ,the path upon which null rays can orbit in constant radius. The recent profound observationof black hole is made possible by producing the ring-like image of photon sphere around thesupermassive BH [1]. This discovery relies on the rather realistic rotating BH whose (circularas well as spherical) photon orbits have extensively been investigated, for example in [2, 3]and the references therein. It is nevertheless also of high interest to study photon sphere instatic cases. The Schwarzschild is known to have (unstable) null orbit at r = 3 M in naturalunit. The RN black hole possesses two photon spheres r ± ps = M (cid:18) ± (cid:113) − (cid:0) QM (cid:1) (cid:19) , onlyone of which ( r + ) can be observed since it lies outside its outer horizon. This physical orbitis the local maximum of the corresponding effective null potential; thus also unstable. Aspointed out in [4, 5] the extremal RN can have stable photon orbit exactly on its (extreme)horizon, r EH . The meaning “ stable ” here, we argue, is still controversial since any smallperturbation around r EH shall collapse the photon inside the horizon. Naturally, one wouldexpect that this condition can be cured in the NLED case.Nonlinear electrodynamics (NLED) is not new in modern physics. Mie in 1912, and laterBorn and Infeld in 1934, proposed that electron is a nonsingular solution of field theory withfinite electromagnetic energy [6, 7]. With the development of quantum electrodynamics(QED) this classical nonlinear field theories were later abandoned. Ironically, it is preciselythe success of QED that resurrects the recent interest in NLED. Recent photon-photon scat-tering experimental results strongly indicate that in the vacuum electrodynamics might benonlinear [8–11]. Euler and Heisenberg predicted that vacuum magnetic birefringence mustoccur in QED [12]. This phenomenon is absent in Maxwell and Born-Infeld (BI) electrody-namics, but can be present in other NLED. The bound for the birefringence’s magnitudeis provided by the BMV and and PVLAS experiments [13–15], and improvements are stillbeing sought. It is then no wonder that in a recent decade there is abundant proposals forNLED. Kruglov proposed a generalization of BI electrodynamics as a model of fractionalelectrodynamics [16, 17] and a few nonlinear electrodynamics model with trigonometricterms [18, 19]. Euler-Heisenberg electrodynamics which features second order of Maxwellelectrodynamics was revisited in [20, 21]. Logarithmic electrodynamics was investigatedin [22, 23], while exponential electrodynamics along with its phenomenology were studied2n [24–26]. The first black hole solutions coupled with nonlinear charge was discussed byHoffmann and Infeld and by Peres [27, 28]. They presented exact solutions of Einstein-Born-Infeld (EBI) theory. Today several exact solutions of black holes charged with NLEDsources, both in general relativity (GR) as well as in modified gravity, have extensively beenexplored (see, for example, [29–38] and references therein). The cosmological effect of a formof NLED with one parameter was examined in [39, 40]. Some of the most studied models ofNLED are the conformally invariant power Maxwell electrodynamics which was analyzed ashigher-dimensional black holes in [41, 42] and the Ayon-Beato-Garcia electrodynamics whichwas developed using Hamiltonian formulation to construct an electrically-charged Bardeen’sregular black hole solution in [43] and the magnetically-charged one on [44], where it wasresurfaced in recent studies [45, 46].It is well-known that photon behaves differently in the NLED ambient compared to thelinear Maxwell electrodynamics. They do not propagate along the background geometry’snull geodesic. Rather, they follow the null geodesic of its effective geometry [47]. Thisbehavior gives shed on the photon sphere study on charged BH. Curiously, research on thistopic is rather rare . It is therefore of our interest to study this phenomenology in the vastliterature of NLED models. In this work we shall investigate the null geodesic of severalNLED models in the framework of GR. For simplicity, in this preliminary work we shallfocus on the polynomial Maxwell-type lagrangians. That is, we consider three models: theKruglov, the power-law Maxwell, and the Ayon-Beato-Garcia NLED models.This work is organized as follows. In Section II we give a brief overview of general NLEDmodel. Sections III-V are devoted to investigating the three different NLED models. In eachwe study their timelike and null geodesics, as well as the weak deflection angle of light. Wecompletely produce the photon orbit values for each model. Our conclusion is summarizedin Section VI. The relations between photon spheres in Einstein-BI gravity with its phase transitions are studied in [48,49] and the references therein I. OVERVIEW OF NLED
In general, all NLED models can be expressed as a functional of Maxwell’s Lagrangian, L = L [ F ], where F ≡ F µν F µν . By correspondence principle, in the low-energy /weak-coupling limit they all should reduce to Maxwell, L = −F .Any nonlinearization extension of an established theory must obey causality and uni-tary principles. In the context of electrodynamics they can be formulated as the followingconstraints [50, 51]: L F ≤ , L FF ≥ , L F + 2 F L FF ≤ , (1)where L F ≡ ∂ L /∂ F and L FF ≡ ∂ L /∂ F .The field equation is given by its corresponding Euler-Lagrange, ∇ µ ( L F F µν ) = 0 . (2)Alternatively, one can define, by means of Legendre transformation, the corresponding“Hamiltonian” [43, 52] H ≡ L F F − L . (3)The Lagrangian, in turn, can be written as L = 2 H P P − H , (4)where P ≡ P µν P µν and P µν ≡ L F F µν . It is easy to see that H = H [ P ]. The field equationis then given by ∇ µ P µν = 0 . (5)In any case, the field equations can be shown to be ∇ · D = 0 , ∂ D ∂t = ∇ × H , (6)with D ≡ ∂ L /∂ E the electric displacement field and H ≡ − ∂ L /∂ B the magnetic field. Thenonlinearity implies the relation between E and D as well as B and H not linear. In general, D = D ( E , B ) and H = H ( E , B ). One interesting phenomenological interpretation is thatNLED describes the electromagnetic wave propagation in nonlinear media. Throughout this work we shall not deal with
G ≡ F µν ˜ F µν . This can be done by setting its constantparameter to be zero. II. GENERALIZED BORN-INFELD
This model was proposed by Kruglov to generalize the BI Electrodynamics [17], L K = 1 β (cid:20) − (cid:18) βFq (cid:19) q (cid:21) . (7)Here β is a parameter with dimension of [ L ] and q is an arbitrary dimensionless parameter.For q = 1 the model reduces to Maxwell, while q = 1 / ∂ µ (cid:0) Γ q − F µν (cid:1) = 0 , Γ ≡ β F q . (8)This equation is equivalent to 6 with the following identifications D = ε E , H = µ − B , (9)where ε = µ − ≡ Γ q − . For electric point-charge source the displacement field D ( r ) issingular at the origin, but the electric field E ( r ) is not. It is regular at the core with finitevalue given by E (0) = (cid:113) qβ .The monopole black hole can be obtained from the Einstein-Kruglov model [35] S = (cid:90) d x √− g (cid:20) R κ + L K (cid:21) , (10)where κ ≡ πG . The ansatz employed here is magnetic monopole and spherical symme-try [32], A t = A r = A θ = 0 , A φ = Q (1 − cos θ ) , (11)and ds = − f ( r ) dt + f − ( r ) dr + r d Ω . (12)The solutions are [35] F θφ = Q sin θ, (13)and f ( r ) = 1 − Mr − κ r β (cid:20) F (cid:18) − , − q ; 14 ; − Q β qr (cid:19) − (cid:21) , (14)where Q is the magnetic charge and F ( a, b ; c ; d ) the hypergeometric function. It can beshown that in the limit of β → q → / =- / = = ( Maxwell ) q = / ( BI ) q = / - - f ( r ) β = q =- / = = ( Maxwell ) q = / ( BI ) q = / - - - f ( r ) β = FIG. 1: Typical plots of f ( r ) with M = Q = 1. β = β = β = β = f ( r ) q = /
4, Q = q = = ( Maxwell ) q = / ( BI ) q = / - f ( r ) β = = FIG. 2: The cases naked singularities of the metric function f ( r ). [Left] The no-horizon solutionwith fixed q for several values of β . [Right] The vice versa. Here we set M = 1. In Fig. 1 we show typical plots of the metric function for several values of q , both inthe strong and the weak coupling regimes. The behavior does not differ much from theRN solution; they all typically have two horizons. For values of q that does not reduce toMaxwell (for example q = − /
2) the metric stops being real. Since they generally possesstwo horizons, in principle the metric 14 can be extremal. While it is impossible to show theextremal condition for M and Q analytically, in Fig. 3 we show that the extremal conditionscan be satisfied for certain values of the parameters. The radius tangent to the minima of themetric is the extremal horizon r EH . As β goes stronger r EH shifts closer to the singularity.6 = = β = = β = = β = = - f ( r ) q = / q =
2, Q = =
1, Q = = /
2, Q = = /
4, Q = - f ( r ) β = FIG. 3: Typical of extremal cases of the metric function f ( r ). [Left] The extremal solution withfixed q for several values of β . [Right] The vice versa. Here we set M = 1. However, for q < β above which only one horizon exists. Thiscan be seen in Fig. 1 on the right. A. Timelike Geodesics
A test particle with mass µ and (electric/magnetic) charge (cid:15) around compact object canbe described by the geodesics equation [34] d x ν dτ + Γ ναβ dx α dτ dx β dτ = − (cid:15)µ F νσ dx σ dτ . (15)For our metric (12), the timelike geodesics on equatorial plane ( θ = π/
2) can be written as1 = f ˙ t − f − ˙ r − r ˙ φ . (16)The symmetry of the metric admits conserved quantities˙ t = E f , ˙ φ = L r . (17)where E and L are the energy-and angular momentum-per unit mass of the test chargedparticles, respectively. Eq. (16) can be rewritten as˙ r + f (cid:18) L r + 1 (cid:19) − E = 0 . (18)Comparing the equation to ˙ r + V eff ( r ) = 0, we can extract the effective potential as V eff ( r ) = 12 (cid:18) L r + 1 (cid:19)(cid:18) − Mr − κ r β (cid:20) F (cid:18) − , − q ; 14 ; − Q β qr (cid:19) − (cid:21)(cid:19) − E . (19)7 =- / = = ( Maxwell ) q = / ( BI ) q = / - - - - -
101 r V e ff β = q =- / = = ( Maxwell ) q = / ( BI ) q = / - - - -
101 r V e ff β = FIG. 4: The effective potential for massive particles (19) with M = Q = 1. It is interesting to note that for monopole black hole massive charged (either electrically ormagnetically) test particle behaves the same as the chargeless one.The plot of V eff ( r ) for several values of q are shown in Fig. 4. Here we set E = L = 1.The feature of V eff ( r ) is qualitatively the same as the Newtonian counterpart; there existsbounded orbits. The minimum of V eff ( r ) corresponds to the radius of stable circular orbit r SCO , while its local maximum represents the radius of unstable circular orbit r UCO . Theseclosed orbits is constrained by q and the nonlinear coupling β . For q ≤ − / V eff ( r )stops being real. B. Null Geodesics
Novello et al showed that in NLED photon follows the null geodesic of its effective geom-etry given by [47] g µνeff = L F g µν − L FF F µα F αν . (20)For our case, it is given by g µνeff = (cid:18) βFq (cid:19) g µν − β ( q − q F µα F να . (21)Defining a factor h ( r ) ≡ qr + βQ β (8 q − Q +2 qr , the conformally-rescaled effective line element can To be precise, here r SCO = r SCO ( M, Q, q, β ). The smallest r SCO corresponds to the minimum of hyper-surface r SCO is the (
M, Q, q, β )-hyperspace, and is called the Innermost Stable Circular Orbit , r ISCO . = = ( Maxwell ) q = / ( BI ) q = / - - - V e ff β = q = = ( Maxwell ) q = / ( BI ) q = / - - - V e ff β = FIG. 5: The effective potential for massless particles (25) in the non-extremal case. Here we set M = Q = 1. be written as ds eff = − f ( r ) dt + f ( r ) − dr + h ( r ) r d Ω . (22)The null rays in this line element follow the trajectories given by0 = f ˙ t − f − ˙ r − hr ˙ φ , (23)from which the V eff can be extracted out as V eff ( r ) = f L hr − E , (24)or, explicitly, V eff ( r ) = L r (cid:18) β (8 q − Q + 2 qr qr + βQ (cid:19)(cid:18) − Mr − κ r β (cid:20) F (cid:18) − , − q ; 14 ; − Q β qr (cid:19) − (cid:21)(cid:19) − E , (25)the stationary of which corresponds to the existence of photon orbits; i.e., V (cid:48) eff ( r ) = 0.Since the metric function is not in a simple closed-form function it is of little interest todetermine the r UCO and r SCO analytically. In Figs. 5-6 we plot V eff ( r ) for photon. In thenon-extremal case the situation is similar as in RN, except in the weak-coupling regime and q < r UCO < r
SCO . Both are still inside the corresponding outer horizon. For theextremal case, however, something interesting emerges. As can be seen from Fig. 6 thereare typically two r UCO and one r SCO , and the position of r SCO shifts farther away fromsingularity when β gets stronger, as opposed to the behavior of r EH discussed earlier. As a9 = = β = = β = = β = = - - - - - - V e ff q = / q =
2, Q = =
1, Q = = /
2, Q = = /
4, Q = - - - - - - - V e ff β = FIG. 6: The extremal case of effective potential for massless particles. [Left] V eff with a fixed q and several values of β parameter. [Right] The vice versa. We set M = 1. result, there is a window of parameter space where you can have stable photon orbits outside the extremal horizon; i.e., r SCO > r EH . This is clearly shown in Fig. 7. The consequence isnovel. Not only stable circular photon orbit is possible, but there exists a family of boundedorbits with r min ≤ r ≤ r max , as long as r min ≥ r EH .As we mentioned in the previous subsection, for q < β gets stronger; i.e., the black hole behaves Schwarzschild-like. The relevant thing iswhether r SCO is located inside the remaining single horizon r h . In Fig. 8 we plot the metric(left panel) as well as its corresponding V eff for timelike (center panel) and null (right panel)in the strong-coupling limit ( β = 2) and q = 1 /
4. It can be observed that timelike V eff possesses r UCO and r SCO with r UCO < r
SCO . On the other hand, for null geodesic only r SCO exists, and r h < r SCO . Thus for β = 2, and we infer that it is valid for strong-couplingregime β ≥
1, stable photon orbits also exist. In Table. I we show physical photon orbitswith r SCO > r EH for the case of q = 1 / β and the values of BH charge Q .This is a typical family of solutions with q <
1. For the specific β = 0 .
05 we found that the r SCO is rather metastable since it is the saddle point of V eff ; i.e., V (cid:48)(cid:48) ( r SCO ) = 0.
C. Null Geodesics in Born-infeld case ( q = 1 / ) In this section we examine the null behaviour of q = 1 / EH - - - - - - V e ff q = / β = FIG. 7: The zoomed-in version of V eff for q = 1 / r UCO coincides with r EH (as can be confirmed by Table I) while r SCO lies outside. - - f ( r ) q = / β = r EH - - - V e ff q = / β = r EH - - - -
101 r V e ff q = / β = FIG. 8: The metric function f ( r ) (left), effective potential for massive particles (center) andeffective potential for light particles (right) with M = 1 and Q = 2. there is only a few extensive studies that is worked on the magnetostatic scenario [54, 55].It is worth to note that the BI enjoys SO (2)-duality invariance (not necessarily presentin other NLED), where the spherically symmetric solution is exactly the same for electricand magnetic case. Breton has shown in her paper that for electric BI blackhole possesses r SCO outside the r EH [34]. Through our study, we want to take a look if the magnetic casealso performs the same result. Here we evaluate the solution in extremal case. The metricfunction f ( r ) and its potential V eff is shown in Fig. 9.11 Q r EH r UCO r SCO r EH ) with q = 1 / β , with its r SCO and r SCO . The starred value is a saddle point. β = = β = = β = = β = = - f ( r ) q = / β = = β = = β = = β = = - - - - - - V e ff q = / FIG. 9: [Left] The extremal case of metric function f ( r ) and; [Right] The extremal case of effectivepotential for massless particles with q = 1 / β parameter. Here we set M = 1. Here we see the metric behaviour is similar to the previous case ( q = 1 / r EH gets smaller as the value of β increases. The potential, on the otherhand, shows that the r SCO moves farther away from the center of the black holes as β rises.We analyze the numbers and we find that the r SCO lies outside the event horizon for almostall value of β , with the case of β = 0 . r EH and r SCO coincide. In Table. II we showphysical photon orbits with r SCO > r EH for the case of q = 1 / β and thevalues of BH charge Q . 12 EH - - - - - - V e ff q = / β = FIG. 10: The zoomed-in version of V eff for Born-Infeld case shown in Fig. 9. It can be observedthat the inner r UCO coincides with r EH (as can be confirmed by Table II) while r SCO lies outside. β Q r EH r UCO r SCO r EH ) in Born-Infeldcase for various number of β , with its r SCO . D. Deflection of Light
As the last analysis for this model, let us calculate the (weak) deflection angle of lightin the case other than BI. Consider the case of q = 1 /
4. Knowing the conserved quantities, E = f ˙ t , L = hr ˙ φ and defining impact parameter b = L / E , we rewrite the null geodesicsin term of u ≡ /r as d udφ + f hu = − u ddu ( f h ) + 12 b ddu ( h ) . (26)Assuming small β <<
1, we might expand the metric function f and conformal factor h using Taylor series for first order of β . Inserting the corresponding function, Eq. (26) up to13rst order in β is d udφ + u = 3 M u − κ Q u + β (cid:18) Q u b + 84 M Q u −
110 237 κ Q u − Q u (cid:19) . (27)Define (cid:15) ≡ M u and ξ ≡ u/u . This yields Eq. (27), up to second-order in (cid:15) , d ξdφ + ξ ≈ ξ (cid:15) + ξ (cid:15) (cid:18) βQ b0 M − κ Q M (cid:19) . (28)Now, expand ξ in power of (cid:15) , ξ = ξ + (cid:15)ξ + (cid:15) ξ + .... , then insert them into Eq. 28. We cansort the equation by collecting terms in different order of (cid:15) [38]: d ξ dφ + ξ = 0 ,d ξ dφ − ξ + ξ = 0 ,d ξ dφ + κ ξ Q M − βξ Q b M − ξ ξ + ξ = 0 . (29)Solving them, we find the approximation of inverse radial distance u as u (cid:39) u cos( φ ) + 12 M u (3 − cos(2 φ ))+ u b (cid:20) φ sin( φ ) (cid:0) b0 (cid:0) M − κ Q (cid:1) + 48 βQ (cid:1) + cos( φ ) (cid:0) b0 (cid:0) M − κ Q (cid:1) + 336 βQ (cid:1) + cos(3 φ ) (cid:0) b0 (cid:0) M + κ Q (cid:1) − βQ (cid:1) (cid:21) . (30)Asymptotically φ → π/ δ as u →
0. For the case of u ≈
0, we can solve δ keepingonly second order of u as δ ≈ M u + u (cid:18) πβQ b + 15 πM − πκ Q (cid:19) . (31)We want to see the contribution of first order β in the deflection angle. We use b ≈ /u ≡ r tp where r tp is the radius of turning point. The weak deflection angle can be obtained as∆ φ weak ≡ δ ≈ Mr tp + 15 πM r + Q (cid:18) πβr tp − πκ r (cid:19) . (32)Using the same method, the deflection angle for other cases of q is calculated. The behaviouris shown in Fig. (11).In Fig. (11) above we show the behaviour of deflection angles by setting the value of itsparameter with M = 10 M (cid:12) (solar mass), the charge is set (arbitrarily) to be Q = 0 . =- / = / = / ( BI ) q = ( Maxwell ) q = r tp [ r S ] Δ ϕ [ r ad ] β = FIG. 11: Deflection angles for various number of q . and the radius of turning point r tp = b in solar radius r (cid:12) which has been normalized bySchwarzchild radius r S . It can be seen that that the other cases beside Maxwell ( q (cid:54) = 1)lay on the same curves. They all asymptote to Maxwell for large r tp , as they should, butsignificantly differ in the short-length regime. The extra term of β contributes bigger valueof deflection angle near the Schwarzchild radius. Obviously we cannot trust this weakapproximation all the way to r tp = r S since it is the regime where the field gets strong andthus full strong deflection analysis is required [56–59]. IV. POWER-LAW NLED
This power-law electrodynamics was proposed by Hassaine and Martinez [30, 31] and isgiven by L = −F q . (33)Maxwell is recovered when q = 1. The corresponding “permittivity” and “permeability” areexpressed as (cid:15) m = µ − m = q F q − . (34)They found that the BH is given by f ( r ) = 1 − Mr − κ − q r − q (cid:18) Q r (cid:19) q . (35)15 = ( Maxwell ) q = = / = / = / - - - f ( r ) FIG. 12: Metric function f ( r ) with Q = M = 1. It can be seen at a glance that q (cid:54) = 3 /
4. Asymptotically, f ( r ) goes aslim r →∞ f ( r ) = , q > / , − √ , q = 1 / , −∞ , q < / . Thus, the black hole is asymptotically flat only for q > /
2. The case of q = 1 / A. Timelike Geodesics
The corresponding V eff for massive test particles is V eff ( r ) = 12 (cid:18) L r + 1 (cid:19) (cid:20) − Mr − κ − q r − q (cid:18) Q r (cid:19) q (cid:21) − E r →∞ V eff ( r ) = , q > / , − √ , q = 1 / , −∞ , q < / . The V eff behaves quite similarly with the metric, as shown in Fig. 13. No stable or unstableorbit exists outside the horizon. 16 = ( Maxwell ) q = = / = / = / - - - -
101 r V e ff FIG. 13: The effective potential for massive particles with M = 1, Q = 1, and E = L = 1. B. Null Geodesics
The effective geometry in this model is given by g µνeff = g µν − q − F F µα F να . (37)The line element can be written as Eq. (21) but with h ( r ) ≡ (8 q − − . The correspondingeffective potential is V eff ( r ) = (8 q − L r (cid:18) − Mr − κ − q r − q (cid:18) Q r (cid:19) q (cid:19) − E . (38)In Fig. 14 we show the uninteresting result of V eff , since no photon orbit (stable or unstable)exists either. In the following discussion we shall show that this power-law NLED model isproblematic phenomenologically, at least in the weak-deflection limit. C. Deflection of Light
For this model, it is easier to calculate the deflection angle through the first-order, ratherthan second-order as done previously, differential equation. Recalling the null geodesicsequation (23) and substituting u = 1 /r , the term ˙ r can be rewritten as˙ r = (cid:18) drdφ dφdτ (cid:19) = L (cid:18) dudφ (cid:19) . (39)17 = / = / = / = ( Maxwell ) q = - - - V e ff FIG. 14: The efective potential for ligt particles with M = 1, Q = 1, and E = L = 1. Eq.(23) then becomes (cid:18) dudφ (cid:19) = 1 b − f u h . (40)Defining σ ( u ) ≡ bu ( f /h ) / , we obtain dudφ = 1 b (1 − σ ) / . (41)As an example, let us take q = 2. In the weak-field limit we obtain bdu = (cid:18)
13 + 2
M σ b − κ Q σ b (cid:19) dσ. (42)The total change of angle φ w.r.t. u from infinity ( u = 0) to the minimun radius ( u = u =1 /b ) and back to infinity can be written as δφ = 2 (cid:90) u dφdu du = 2 (cid:90) u b (1 − σ ) − / du = 2 (cid:90) (1 − σ ) − / (cid:18)
13 + 2
M σ b − κ Q σ b (cid:19) dσ. (43)The total deflection angle is defined as ∆ φ ≡ δφ − π . Using this method we calculate the18ngles for several q as follows:∆ φ ( q = 1) = 4 Mb + 3 πκQ b , ∆ φ ( q = 2) = 4 M b + π (cid:18) − κ Q b − (cid:19) , ∆ φ ( q = 3) = 4 M b + π (cid:18) − κ Q √ b + 1 √ − (cid:19) . (44) q = ( Maxwell ) q = = - - r tp [ r S ] Δ ϕ [ r ad ] FIG. 15: Deflection angles for various number of q . Here we set M = 10 M (cid:12) and Q = 0 . As can be seen from the result above, unless q = 1 the deflection angle does not reduceto Schwarszchild even in the limit of Q →
0. What is more problematic is that as showni n Fig. 15 the angles are generically negative, and does not go to zero at large r tp ! Thiscorrespondence violation poses a doubt whether this result is physical or not. At best wecan say that the weak-field approximation seems to break down for this model. V. AYON-BEATO-GARCIA BLACK HOLE
In 1968 Bardeen [60] in his seminal proceeding paper published his famous regular blackhole solution, f ( r ) = 1 − M r ( r + Q ) / . (45)19 = = = - f ( r ) FIG. 16: Metric function for ABG black hole solution (45) with M = 1 and several values of Q . The metric is regular at the origin, as in Fig. 16. The black hole regularity is ensured bythe fact that the corresponding invariants are also regular everywhere. Ayon-Beato andGarcia [44] were the first to realize that such a solution can be interpreted as a black holecharged with NLED magnetic monopole whose Lagrangian is given by L = − sκ Q (cid:18) (cid:112) Q F (cid:112) Q F (cid:19) / , (46) s ≡ Q/ M . Its electrically-charged counterpart solution was later proposed by Rodriguesand Silva [46]. Being an NLED BH, we shall study the null geodesic structure of Ayon-Beato-Garcia (ABG) metric and investigate the stable photon orbits. A. Timelike and Null Geodesics
The explicit form of the ABG’s timelike geodesics can be written as V eff ( r ) = 12 (cid:18) L r + 1 (cid:19)(cid:18) − M r ( r + Q ) / (cid:19) − E . (47)This V eff allows (marginally) stable orbits for massive test particles, as can be seen in theright panel of Fig. 17.While null geodesics structure of the original Bardeen spacetime was investigated bynumerous authors (see, for example, [58, 61–64]), none assumes the NLED perspective.They thus neglected the photon’s effective geometry and considered photon and graviton to20 = = = - - - - - V e ff = = Q = = = - - V e ff = = FIG. 17: The effective potential for massive particles with M = 1 for various number of Q . follow the same null rays. Consequently no stable photon sphere is observed by an observeroutside the horizon. Here we follow the ABG’s perspective and found something novel.From Lagrangian 46 the effective metric can be written as g µνeff = g µν + 2 r (6 Q + 5 Q r − Q ( Q + r ) F µα F να . (48)Defining h ( r ) ≡ (cid:18) − ( Q +5 Q r − ) ( Q + r ) (cid:19) − , the effective potential can be obtained, V eff ( r ) = L r (cid:18) − Q + 5 Q r − Q + r ) (cid:19)(cid:18) − M r ( r + Q ) / (cid:19) − E . (49)The circular orbit radii satisfies0 = L (cid:18) r ( Q +10 Q r − ) ( Q + r ) − Q r ( Q + r ) (cid:19) (cid:16) − Mr ( Q + r ) / (cid:17) r + L (cid:16) − Q +10 Q r − Q + r ) (cid:17) (cid:16) Mr ( Q + r ) / − Mr ( Q + r ) / (cid:17) r − L (cid:16) − Q +10 Q r − Q + r ) (cid:17) (cid:16) − Mr ( Q + r ) / (cid:17) r . (50)Finding its analytical roots is not illuminating. We thus determine the r SCO by studyingFig. 18. It can be seen that for the case of extremal and two horizons there are minima,albeit of little depth, whose r SCO > r h . This means that ABG BH allows stable photonorbits. The detail numerical values of the corresponding radii is shown in Table. III. Thesestability is considered to be marginally stable, since the depth of minima is quite shallow.Nevertheless, it is not a problem classically.21 = = = - - - - V e ff = = - - - - - FIG. 18: The effective potential for light particles with M = 1 for various number of Q . Thecorresponding equation is (49) Q r h r UCO r SCO r EH ) Q , next to itscorresponding values of r SCO and r SCO of the null geodesics.
B. Deflection of Light
Finally, let us calculate the deflection angle of photon off the ABG black hole which, tothe best of our knowledge, has not been studied in the literature. In the limit of small Q the inverse radial distance is, approximately, u (cid:39) u α (cid:0) √ αφ sin (cid:0) √ αφ (cid:1) (cid:0) ( α − αb + 10 M (cid:1) + cos (cid:0) √ αφ (cid:1) (cid:0) α − αb + 54 M (cid:1) + cos (cid:0) √ αφ (cid:1) (cid:0) M − ( α − αb (cid:1) (cid:1) + M u α (cid:0) (cid:0) √ αφ (cid:1) + sin (cid:0) √ αφ (cid:1) sin (cid:0) √ αφ (cid:1) + 4 cos (cid:0) √ αφ (cid:1)(cid:1) + u cos (cid:0) √ αφ (cid:1) . (51)22here we define α ≡ − Q b . For α ≈ φ weak ≈ Mr tp + 15 πM r tp + O (cid:0) Q (cid:1) . (52)The zeroth-order is twice the Schwarzschild’s angle. This is because ABG describes a regular(Bardeen) black hole which is distinct from Schwarzschild. The dependence of ∆ φ on r tp isshown in Fig. 19. This might be used to distinguish ABG/Bardeen’s signature from otherordinary black hole. r tp [ r S ] Δ ϕ [ r ad ] FIG. 19: Deflection angles of ABG black holes.
VI. CONCLUSION
This work is intended to investigate null geodesic of several NLED black holes and its phe-nomenological aspects. In this work we specifically consider three polynomial-type NLEDmodels: the generalized BI model (we dubbed it the Kruglov-BI model), the power-lawmodel, and the Ayon-Beato-Garcia model used in Bardeen black hole. Each has been ex-tensively studied by a large number of authors. What is left uninvestigated, and this is ourmain result, is the behaviour of photon around them. Due to the field discontinuity, pho-23on follows the null geodesic that is different from graviton in NLED theories. Our simpleinvestigation shows novel results.In the first model, we show that in the extremal limit of some weak-coupling β the blackhole allows stable physical photon orbits. By physical we mean that the orbits lie outside thehorizon. For the strong-coupling β <
1, generically for q < r SCO coincides with r EH . It is not clear how stable this is since any smallperturbation might collapse the photon inside the horizon. Now, the possibility of having r SCO > r EH in NLED case evades such concern. Not only we now have stable circular orbits,but also there exists a family of bounded orbits parametrized by two radii, r + and r − , aslong as r − ≥ r EH . In Newtonian gravity these closed orbits would correspond to ellipse. InGR, however, to determine the orbits we must explicitly solve the null geodesic equation.This is left for our future investigation.For the second model, we found problematic phenomenological results since the weakdeflection angle does not coincide with Schwarzschild even in the chargeless limit ( Q → V eff possesses singularity [65];therefore it is futile to talk about photon orbit around Bardeen BH. But looking fromthe NLED perspective, the matter becomes non-trivial. A test photon follows its effectivegeometry, and our investigation reveals that it is non-singular. Similar to the Kruglov-BIcase the ABG black hole also allows photon sphere and other bounded orbits outside thehorizon. Our calculation on the weak deflection angle shows that up to zeroth-order it is24wice the Schwarzschild value. While we argue that this might be caused by the fact that thetwo have different nature regarding the singularity (thus, the effective null geodesic outsidethe horizon is influenced by the nature inside it), we also realize that the strong deflectionformalism is needed to have a conclusive hypothesis. Acknowledgments
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