Can a light ray distinguish charge of a black hole in nonlinear electrodynamics?
aa r X i v : . [ g r- q c ] J a n Can a light ray distinguish charge of a black hole in nonlinear electrodynamics?
Bobir Toshmatov, ∗ Bobomurat Ahmedov, † and Daniele Malafarina ‡ Ulugh Beg Astronomical Institute, Astronomy 33, Tashkent 100052, Uzbekistan Webster University in Tashkent, Alisher Navoiy 13, Tashkent 100011, Uzbekistan Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, Kori Niyoziy 39, Tashkent 100000, Uzbekistan Department of Physics, Nazarbayev University, 53 Kabanbay Batyr, 010000 Nur-Sultan, Kazakhstan
It is a well-known fact that light rays do not follow the null geodesics of the space-time in nonlinear electro-dynamics; instead, they follow the null geodesics of the so-called effective space-time. Taking this into account,in this paper, we aim to discuss the possibility of distinguishing the type of charge with which the black hole isendowed, via the motion of light rays. The results show that, for any black hole being a charged solution of thefield equations of general relativity coupled to the nonlinear electrodynamics, one cannot distinguish the twotypes of charge (magnetic or electric) through the motion of light rays around it.
I. INTRODUCTION
Despite the fact that Maxwell (linear) electrodynamics andgeneral relativity are completely different classical theories,they both endure the so-called singularity problem, i.e. thefact that physically viable solutions of the theory genericallyexhibit singularities. The problem is more pronounced in gen-eral relativity, where the occurrence of curvature singularitiesdisrupts the causal structure of the space-time. However, ide-ally, it would be preferable to develop classical theories forgravitation and electromagnetism that do not present singu-larities. It is clear that the solution of the singularity problemrequires us to extend beyond classical general relativity andMaxwell electrodynamics, since these theories cannot avoidor solve the problem on their own.In electrodynamics, experiments direct us to consider theLagrangian density of the field to be approximately linear. Inthe simplest case, it is exactly linear so that if the field equa-tions are solved, together with the Lorentz gauge, one will endup with the well-known Maxwell equations [1]. The validityof Maxwell’s equations at the classical level have been widelydemonstrated from experimental physics. However, theoret-ically, if these equations are solved for point charges, onewould obtain diverging field quantities at the location of thepoint charge, a fact that is already quite inexplicable, whichin turn results in infinite total energy for the electric field of apoint charge, which clearly is physically undesirable.General relativity also allows for the existence of space-time singularities. Of course, not all singularities appearing insolutions of Einstein’s equations are physical: For example,coordinate singularities that are defined by the divergence ofone of the metric functions, may be merely mathematical, i.e.they can be eliminated with an appropriate coordinate trans-formation. However, curvature singularities, defined by thedivergence of curvature invariants such as the Kretschmannscalar, cannot be eliminated by any change of coordinates andthey are an intrinsic feature of the geometry. The singular-ity theorems show that such curvature singularities are an in- ∗ [email protected] † [email protected] ‡ [email protected] evitable outcome of physically viable scenarios such as, forexample, the dynamical collapse that leads to the formation ofa black hole [2], thus making the problem of their resolutionan important piece of the hunt for a better theory of gravityand of our understanding of extreme astrphysical phenomena[3]. The existence of curvature singularities is still one of theunsolved problems of the theory. Many approaches have beentaken in the attempt to avoid this ‘space-time pathology’ andone of the most promising worked-out methods is based oncoupling general relativity to nonlinear electrodynamics [4–7]. These solutions can be of electrically, magnetically ordyonically (i.e. simultaneously electrically and magnetically)charged black holes [8–13, 31], like in the case of general rel-ativity coupled to the linear electrodynamics that results theReissner-Nordstr¨om solution.Other new interesting phenomena appear in nonlinear elec-trodynamics. One of the such phenomena is associated withthe propagation of light rays. It is a well-known fact thatelectromagnetic waves propagate along null geodesics of thespace-time in vacuum and linear electrodynamics. How-ever, this is not the case if the electromagnetic field is self-interacting as in the case of nonlinear electrodynamics. Thenlight rays do not follow the null geodesics of a given space-time metric, instead the paths of light can be described interms of an effective space-time metric which represents amodification of the original space-time [14–25]. This phe-nomenon can also be shown from perturbations theory, as inthe high energy limit, the effective potential of the electro-magnetic perturbations of the black hole in nonlinear electro-dynamics coincides with the one governing the photon motionin the field of a central object [26–30].Motivated by the peculiar phenomena of nonlinear electro-dynamics discussed above, i.e. that light rays do not follownull geodesics, in this paper we aim to determine whether itis possible to distinguish the type of charge of the black holefrom the motion of light rays in the given geometry. To do so,we consider a given spherically symmetric, static space-timethat is either of electrically charged or magnetically chargedblack hole. By constructing the effective metrics for the ge-ometry and studying the motion of light rays in these twospace-times, we establish the criteria necessary to distinguishthe two types of charges from the motion of photons. Weshow that it is not possible to distinguish the two types ofcharges only from the motion of light rays around the blackhole. The paper is organized as follows: In Sec. II we brieflyreview the main equations to construct black hole solutionsin general relativity coupled to nonlinear electrodynamics. InSec. III we derive the effective metrics of the electrically andmagnetically charged black holes and in Sec. IV we study themotion of light rays in these space-times. Finally, in Sec. Vwe discuss and summarize results. Throughout the paper, weadopt the following signature convention ( − , + , + , +) for thespace-time metric and make use of natural units, thus setting c = ~ = G = 1 . II. BASIC EQUATIONS
A generic theory of general relativity coupled to nonlinearelectrodynamics is characterized by the action S = Z d x √− g L , (1)with the Lagrangian density given by L = 116 π [ R − L ( F )] , (2)where g and R are the determinant of the metric tensor and theRicci scalar, respectively while L is the Lagrangian densitydescribing the nonlinear electrodynamics theory, which is afunction of the Faraday tensor F µν through F ≡ F µν F µν .The electromagnetic field tensor satisfies F µν = ∂ µ A ν − ∂ ν A µ , with A µ being the 4-potential. Since the Faraday ten-sor, F µν , is antisymmetric, it has only six nonvanishing com-ponents: i.e. three for the electric field and three for the mag-netic field.To construct a solution of the theory described by (1), oneneeds to solve Einstein’s field equations given by G µν = T µν , (3)where G µν is the Einstein tensor and T µν is the energy-momentum tensor of the nonlinear electrodynamics field.Note that, for simplicity, in equation (3) the coefficient π hasbeen absorbed in the energy-momentum tensor. The energy-momentum tensor of nonlinear electrodynamics is given by T µν = 2 (cid:18) L F F αµ F να − g µν L (cid:19) , (4)where L F = ∂ F L . At the same time, the electromagneticfield is governed by Maxwell’s equations for nonlinear elec-trodynamics, which can be written as ∇ ν ( L F F µν ) = 0 , ∇ ν ∗ F µν = 0 (5)Where ∗ F µν = ε µναβ F αβ / is the dual electromagneticstrength tensor. The electromagnetic 4-potential can be writ-ten in spherical coordinates { t, r, θ, φ } in the following form: A µ = ϕ ( r ) δ tµ − Q m cos θδ φµ , (6)where ϕ ( r ) and Q m are the electric potential and total mag-netic charge, respectively. The exterior of spherically sym-metric, static, electrically and magnetically charged compact objects is described by the same line elements which can bewritten in general as ds = − f ( r ) dt + dr f ( r ) + r d Ω , (7)where d Ω = dθ + sin θdφ is the metric on the unit two-sphere and the metric function f ( r ) is given in the parameter-ized form as f ( r ) = 1 − m ( r ) r , (8)with the mass function, m ( r ) determined by the Lagrangiandensity of the nonlinear electrodynamics. In the absence ofthe electromagnetic field, the mass function takes constantvalue m ( r ) = M , consistent with the description of a purelygravitational mass. For the sake of our further calculations,here below we will briefly review the main points of theformalism for deriving electrically and magnetically chargedblack hole solutions. A. Electrically charged solution
If the space-time is electrically charged, then the 4-potentialof the electromagnetic field is given solely by the first term inequation (6), as A t = ϕ ( r ) . The electromagnetic field 2-formcan be written as F = ϕ ′ ( r ) d r ∧ d t . (9)Note that F = − ϕ ′ . By solving the non linear Maxwell’sequations (5), we arrive at the expression for the total electriccharge inside a sphere with radius rQ e = r L F ϕ ′ . (10)At this point, to construct a solution, we need to solve Ein-stein’s equations (3), which, for this system, reduce to onlytwo independent equations. By solving them, we obtain thefollowing: L = 2 m ′′ r , (11) L F = 2 m ′ − rm ′′ r ϕ ′ . (12)By using (10) and (12), we find the expression for the scalarelectric potential as ϕ = 3 m − rm ′ Q e . (13)If the mass function is constant, then, we immediately recoverthe Schwarzschild solution. If the electromagnetic field is lin-ear, i.e., the Lagrangian density is linear function of F , bysolving eqs. (11), (12), and (13), one can find the Reissner-Nordstr¨om solution with mass function m ( r ) = M − Q e / r ,and the Coulomb potential ϕ ∼ Q e /r . B. Magnetically charged solution
If the black hole is magnetically charged, then the 4-potential of the electromagnetic field is given by the secondterm in equation (6), as A φ = − Q m cos θ . The electromag-netic field 2-form can then be written as F = Q m sin θ d θ ∧ d φ . (14)Note the electromagnetic field strength is F = 2 Q m /r . Bysolving Einstein’s equations (3), we obtain the following ex-pressions for the Lagrangian density: L = 4 m ′ r , (15) L F = r (2 m ′ − rm ′′ )2 Q m . (16)In the case that the electromagnetic field is linear, i.e. forMaxwell’s theory, we have that L = F and by solvingthe above equations, we arrive at the mass function m = M − Q m / r that again represents the Reissner-Nordstr¨omspace-time with a magnetic charge. We see that in the caseof linear electrodynamics the two charges are not distinguish-able in Reissner-Nordstr¨om’s solution. On the other hand, inthe nonlinear theory the two cases produce two different ef-fective geometries. We shall now investigate whether suchspace-times may be distinguished by looking at the trajecto-ries of light rays. III. EFFECTIVE METRICS
As we mentioned in the case where the line element (7) is asolution of the field equations for general relativity coupled tononlinear electrodynamics, light rays do not propagate alongthe null geodesics of the space-time metric, instead, they fol-low the null geodesics of the effective metric obtained fromthe metric tensor [8, 14] g µνeff = L F g µν − L F F F µλ F λν . (17) A. Electrically charged case
In the electrically charged case the effective metric is writ-ten in the following form: ds = − f ( r )Φ dt + 1Φ f ( r ) dr + r L F d Ω , (18)where Φ = L F +2 F L F F . When written in terms of the massfunction m ( r ) , then the above line element takes the followingform: ds = − r ( r − m ) (cid:16) rm ′′′ − m ′′ (cid:17) Q e dt + (19) + r (cid:16) rm ′′′ − m ′′ (cid:17) Q e ( r − m ) dr + r (2 m ′ − rm ′′ )2 Q e d Ω . B. Magnetically charged case
On the other hand, the effective metric for a magneticallycharged black hole is written in the following form: ds = − f ( r ) L F dt + 1 L F f ( r ) dr + r Φ d Ω , (20)where again Φ = L F + 2 F L F F . And again, when written interms of the mass function m ( r ) , it takes the following form: ds = − Q m ( r − m ) r (2 m ′ − rm ′′ ) dt + (21) + 2 Q m r ( r − m ) (2 m ′ − rm ′′ ) dr + 4 Q m r m ′′′ − rm ′′ d Ω . We see that for a generic mass function m ( r ) the effectivemetric of an electrically charged black hole (19) differs fromthat of a magnetically charged one (21). IV. LIGHT RINGS AND GRAVITATIONAL LENSING
Before finding the equations governing the motion of lightrays in the effective space-time metrics for the electrically andmagnetically charged cases, one may write the effective met-rics (19) and (21) in the following unified form: ds = − A ( r ) dt + B ( r ) dr + C ( r ) d Ω . (22)Taking into account the symmetry of the space-time, one caneasily notice that the momenta p µ corresponding to time, t ,and azimuthal angle, φ , are conserved. These are related tothe energy, E , and angular momentum, L , of test particles andphotons. Restricting the attention to motion in the equatorialplane, θ = π/ , the conserved quantities are given by E = A ˙ t , L = C ˙ φ . (23)Since for photon’s motion we have p µ p µ = 0 , the radial com-ponent of the 4-velocity of light rays can be written in termsof the conserved quantities as ˙ r = 1 AB (cid:0) E − V eff (cid:1) , with V eff = L AC . (24)Then circular null geodesics are obtained by imposing ˙ r =0 = ¨ r . Therefore, setting to zero the expression inside paren-thesis in (24), one can find that the energy for photons on cir-cular orbits, while the radius of circular null geodesics i.e.,the light ring, is determined by the radius for which ¨ r = 0 ,corresponding to the solution of the following equation: AC ′ − A ′ C = 0 . (25)Before turning to the further relativistic effects, let us con-sider the effective potentials in terms of the effective metricsof the electrically and magnetically charged black holes whichare given by equations (19) and (21).In the case of the Reissner-Nordstr¨om black hole with massfunction m = M − Q / r , we recover the effective poten-tial V eff = (1 − M/r + Q /r )L /r . As we have men-tioned in section II, for linear electrodynamics, the Reissner-Nordstr¨om solution describes both electrically as well as mag-netically charged black hole space-times via the same line el-ement (7). In other words, the mass functions for electricallyand magnetically charged black holes in linear electrodynam-ics coincide. From equations (19) and (21) it is easy to real-ize that even though the effective metrics differ, the effectivepotentials for the motion of light rays in the electrically andmagnetically charged cases are the same. In fact, the effectivepotentials from equation (24) in terms of the effective metricsof the electrically (19) and magnetically (21) charged blackholes take the form V eff = f ( m ′′ − rm ′′′ )2 r ( rm ′′ − m ′ ) L . (26)Consequently, the radii of the light rings (25) are identical andit is not possible to distinguish the electrically charged fromthe magnetically charged case solely from the location of thephoton sphere.Despite the above result that the expression inside theparenthesis in equation (24) is identical for both electricallyand magnetically charged cases, the product of the metricfunctions A and B differs in the two cases which suggeststhat there may be other ways of distinguishing the two chargesfrom the motion of photons. In fact in the electrically chargedcase we get AB = r (cid:16) m ′′ − rm ′′′ (cid:17) Q e , (27)while for the magnetically charged case we have AB = 4 Q m r ( rm ′′ − m ′ ) . (28)In principle, considering other relativistic effects where theseterms play a role may allow to distinguish the type of chargeof the black hole. With this objective in mind, below wewill consider gravitational lensing in the strong field regimeof both space-times. As it was mentioned before, even thoughelectrically and magnetically charged black holes in nonlinearelectrodynamics are described by the same line element, theireffective metrics differ and light rays follow the null geodesicsof the effective space-time. Let us then consider light rayspassing close to a compact, massive object and evaluate howlight rays deviate from a straight trajectory while followingthe geodesics of the space-time surrounding a massive com-pact object.We first derive the equation for the deflection angle of gravi-tational lensing in a generic spherically symmetric space-timegiven by equation (22). By using the equations of motion (23)and (24), we find the deflection angle of the light ray from dφdr = √ B √ C q A C CA − , (29) where we have denoted the light ring solution of equation (25)with r and consequently quantities X evaluated at r are in-dicated via the subscript X . One can easily notice that as theradius r tends to r ( r → r ), the deflection angle diverges .This fact shows that at r light rays move along circular orbits.Another interesting relativistic effect associated with grav-itational lensing is the apparent time delay in the propagationof light rays passing near a massive object, i.e. the delay intravel time of the light ray from the source to the receiver.The time delay is found by using the equations of motion (23)and (24) and it can be written as dtdr = √ B √ A q − C A AC , (30)From equation (30) we can see that when the turning point ofthe light ray reaches the light ring of the black hole at radius r , the delaying time diverges and the light ray never reachesthe observer.Now let us evaluate the deflection of light rays and the timedelay due to the gravitational lensing for a central object thatis either electrically or magnetically charged. A. Electrically charged case
Rewriting the expressions (29) and (30) in terms of the ef-fective metrics of the electrically charged black hole (19) weobtain dφdr = r z f q f z fz − , (31) dtdr = 1 f q − fzf z , (32)where z = m ′′ − rm ′′′ r ( rm ′′ − m ′ ) , and the zero subscript indicates the value of a function evalu-ated at the light ring r . B. Magnetically charged case
On the other hand, in terms of the effective metrics of themagnetically charged black hole (21), expressions (29) and The deflection angle of light rays is determined by solving the followingintegral: ∆ φ ( r ) = 2 Z ∞ r dr (cid:18) dφdr (cid:19) − π . (30) take the following form: dφdr = r z f q f z fz − , (33) dtdr = 1 f q − fzf z , (34)Interestingly, despite the fact that the equations of motiongoverning the propagation of light rays in the electrically andmagnetically charged black hole space-times are different, thetrajectories followed by light rays near a massive compact ob-ject coincide. Therefore, we conclude that it is not possibleto distinguish an electrically charged black hole from a mag-netically charged one just by measuring the deflection of lightrays. V. DISCUSSION
We have shown that, despite the fact that the effective met-rics of electrically and magnetically charged black holes ingeneral relativity coupled to nonlinear electrodynamics aredifferent, photons follow the same trajectories and therefore,the observation of light propagation alone can not distinguishthe two kinds of charges. Therefore, to gain some further in-sights it may be worth to return to the field equations presentedin section II and analyze their dependence on the non linearelectromagnetic Lagrangian term. In the electrically chargedcase, if L F in (12) is written in terms of the total electricalcharge of the black hole (10), and compared to the one (16)for the magnetically charged case, one would easily notice thefollowing relation: ( L F ) e = 1( L F ) m . (35)Moreover, it is easy to show that the following relation be-tween the electromagnetic field strengths in both cases holds: (cid:0) F L F (cid:1) e = − F m . (36) If the effective metrics in equations (18) and (20) for bothcases are rewritten taking into account the relations givenabove and considering the fact that the conformal factor doesnot affect the motion of light rays [32], one can easily obtainthe following relation: (cid:18) L F Φ (cid:19) e = (cid:18) Φ L F (cid:19) m , (37)and consequently, end up with the identical space-times.Based on the fact that the light ray does not follow nullgeodesics in general relativity coupled to nonlinear electrody-namics [14–16], instead they follow the null geodesics of theeffective metric, in this paper we have studied the possibilitythat the type of charge with which the black hole space-timein endowed may be distinguished from the motion of lightrays in the equatorial plane of a black hole geometry. To thisaim, we considered that a space-time metric (7) with metricfunction (8) that is a solution of the field equations of generalrelativity minimally coupled to the nonlinear electrodynamicsand describes a static and spherically symmetric black hole.Then, we considered separately the two cases in which theblack hole is endowed with an electrical or a magnetic chargeand studied the physical effects related to the propagation oflight rays. We have shown that light rays follow the same tra-jectory in both cases, despite of the fact that the correspond-ing effective metrics are different. Therefore the observationof photon trajectories alone is not able to distinguish the twokinds of charges. ACKNOWLEDGEMENTS
BT and BA acknowledge the support of Ministry of In-novative Development of the Republic of Uzbekistan GrantsNo. VA-FA-F-2-008 and No. MRB-AN2019-29. DM is sup-ported by the Ministry of Education of the Republic of Kaza-khstan’s target program IRN: BR05236454 and NazarbayevUniversity Faculty Development Competitive Research GrantNo. 090118FD5348. [1] L. D. Landau and E. M. Lifshitz,
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