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Light paths of normal and phantom Einstein-Maxwell-dilaton black holes

Null geodesics of normal and phantom Einstein-Maxwell-dilaton black holes are determined analytically by the Weierstrass elliptic functions. The black hole parameters other than the mass enter, with the appropriate signs, the formula for the angle of deflection to the second order in the inverse of the impact parameter allowing for the identification of the nature of matter (phantom or normal). Such identification is also possible via the time delay formula and observation of relativistic images. Scattering experiencesmay favor black holes of Einstein-anti-Maxwell-dilatonic theory for their high relative discrepancy with respect to the Schwarzschild value. For the cases we restrict ourselves to, phantom black holes are characterized by the absence of many-world and two-world null geodesics.


Introduction
Most experimental settings for testing gravitational theories are designed to evaluate trajectories of light rays. Accuracy in this field is a growing interest. From this point of view, the leading experimental settings are aiming to achieve high accuracies beyond the known first order level and to reach a sensitivity of 1 part in 10 9 in measuring the Eddington parameter γ [1], which is an important parameter in post-Newtonian formalism.
On the theoretical front, workers have been striving to evaluate exactly light paths using (hyper)elliptic functions mainly the Weierstrass elliptic functions denoted by ℘ [2,3]. On the one hand, this has provided answers to some open questions, for instance, whether the cosmological constant could be a cause of the Pioneer anomaly [4], has raised the question of whether lensing could be used as a test of the cosmic censorship (CC) [5] (much work on testing the CC has been done in [6][7][8]), and has lead to the discovery new light paths, the Pascal Limaçon trajectories for black holes with cosmological constant [9]. On the other hand, the analytical solutions derived so far, Refs. [10]- [26] to mention but a few, could be useful for any of the experimental settings aiming to test gravitational theories. Moreover, they provide new academic techniques for tackling the motion of massive and massless particles in the geometries of various gravitational fields, may serve as references for testing the accuracy of numerical methods [27] and provide unique benchmarks for testing and improving perturbation and decomposition methods [28]. For that purpose it is very helpful to have relatively simple solutions.
In case of spherical symmetry, one of the equations governing geodesic motion reduces to dr dφ 2 = P(r) (1.1) where P(r) is a polynomial function of the radial variable r, the parameters of the solution and the constants of motion. Depending on the dimension of the space-time, P(r) may be reduced, as described in [19], to a polynomial of degree 3 or 5. We are interested in the former case and we assume that (1.1) is brought to dρ dΘ 2 = 4ρ 3 − g 2 ρ − g 3 (1.2) by coordinate transformations. Here g 2 , g 3 depend on the parameters of the solution and the constants of motion. So far no special terminology has been introduced to simplify notations and expressions. We introduce the following terminology to describe (1.2) and the related polynomial and coordinates. We shall call (1.2) Weierstrass differential equation, w(ρ) = 4ρ 3 − g 2 ρ − g 3 Weierstrass polynomial and (ρ, Θ) Weierstrass coordinates. We bring (1.1) to (1.2) by a series of coordinate transformations relating r to Weierstrass radial coordinate ρ where ρ(r) is a nontrivial and nonlinear transformation; however, Θ(φ ) is a linear transformation and in many cases Θ = φ , where φ is the azimuthal angle.
Most workers prefer to use the effective potential approach by which they determine all planar trajectories [absorbed paths (captured photons), scattering paths, trapped or confined paths, (un)stable circular paths, spiral paths approaching the circular paths from above and/or below and some other special closed curves]. The method we shall apply is entirely based on the properties of the Weierstrass differential equation, and of its polynomial. We shall develop and use this method, which has been used in [12,22] (and partially used in [9,11]), leading to a systematic approach for all problems governed by (1.2). This will allow us to determine all types of trajectories.
None of the papers mentioned above has ever dealt with light paths of normal Einstein-Maxwelldilaton (EMD) black holes. One of the purposes of this paper is to address this question; the other one is to extend the analysis to that of light paths of phantom black holes of EMD and to draw a comparison between the systems of trajectories for a given ratio of charge to mass squared (a 2 = q 2 /M 2 ).
In a phantom gravitating field theory one or more of the matter fields appear in the action with an unusual sign of the kinetic term, so that they are coupled repulsively to gravity. In the case of "phantom EMD" theory, which is also a short term for the theory, we may have a number of ways the matter fields are coupled to gravity: Einstein-anti-Maxwell-anti-dilaton, Einstein-Maxwell-anti-dilaton, Einstein-Maxwell-dilaton and so on. The presence of phantom fields continues to receive support from both collected observational data [29] and theoretical models [30].
The static, spherically symmetric black hole solutions to EMD theory with phantom Maxwell and/or dilaton field were derived, and their causal structure was analyzed, among which one finds nine classes of asymptotically flat and two classes of nonasymptotically flat phantom black holes [31]. In a subsequent work [32], these solutions have been generalized to multicenter solutions of phantom EMD. Recently, their thermodynamic properties and stability were investigated too [33]. One of the remaining tasks is to investigate their null geodesics to see how phantom fields may affect the light paths, particularly the angles of deflection, the photon spheres and related lensing effects. Deviations of the angle of deflection from the Schwarzschild value are generally attributable to extensions in the theory (inclusion of Maxwell fields or scalar ones, cosmological constant and so on), departure from spherical symmetry or motion of the solution itself (mostly rotation). In this paper we examine the case due to the inclusion of (anti)dilatonic and/or (anti)-Maxwell fields.
In Section 2 we consider the cosh and sinh black hole solutions of the generalized phantom EMD, which depend on three parameters (M, q, γ). We evaluate, and discuss, the angle of deflection δ φ to the second order of approximation in the inverse of the impact parameter as a function of the black hole three parameters. Figures, relying on exact formulas, depicting δ φ for phantom and normal black holes are plotted against the Schwarzschild angle of deflection for different values of the parameters. The relative discrepancy is discussed and plotted showing high values from some set of the parameters. The time delay is also evaluated.
In Section 3 we introduce the Weierstrass elliptic functions and use and develop the method based on the Weierstrass polynomial to determine exactly all kinds of null geodesics to any spherically symmetric geometry, provided the equation of (planar) motion of light rays may be brought to (1.2). Applications are considered in Section 3.4 and in Sections 4 and 5. In Subsection 3.4 we consider the strong field limit and relativistic images and derive an analytic reference equation for the log-formula for the angle of deflection, which applies to any geometry provided the light motion or a plane projection of it is described by (1.2). In Section 4 we consider the case γ = 1 and show that the problem of determining the null geodesics of normal Reissner-Nordström black holes by the method based on the Weierstrass polynomial, which was initiated in [22], is tractable analytically and extend the analysis to phantom Reissner-Nordström black holes upon applying the results of Section 3, and in Section 5 we consider the case γ = 0 and determine all the null geodesics of the phantom cosh and normal sinh EMD black holes by mere comparison to the work done in Section 3. In Sections 3 to 5, we do not aim to go into the details of each null geodesic motion; rather, we present a general procedure (Section 3) by which we discuss some type of null geodesic motions and the nature of existing divergencies and present exact reference and standard formulas for specific geodesics, the angle of deflection, the time delay, and the log-formula. The paper ends with a conclusion section and an two appendix sections.

The deflection angle of light paths in the cosh-sinh solutions of EMD
The action for EMD theory with phantom Maxwell and/or dilaton field reads where λ is the real dilaton-Maxwell coupling constant, and η 1 = ±1, η 2 = ±1. Normal EMD corresponds to η 2 = η 1 = +1, while phantom couplings of the dilaton field ϕ or/and Maxwell field F = dA are obtained for η 1 = −1 or/and η 2 = −1.
The metrics of the so-called cosh and sinh solutions, derived in [31], take the form where we have introduced the parameter γ following the notation of [33] 1 . These are asymptotically flat spherically symmetric black holes of mass M, electric charge q and event horizon r + > 0 related by [31] where we have substituted, into the original formula of q, 1 + η 1 λ 2 = 2/(1 + γ). Since q is real, r − and η 2 (1 + γ) must have the same sign. Using this fact in (2.3), we have r − < 0 for the cosh solution and r − > 0 for the sinh one. As shown in Subsection 4.1 case 2. (d) of [31], r = 0 corresponds to a singularity for the cosh solution where geodesics terminate (a Penrose diagram is given in figure 1 of [31]). Similarly, in Subsection 4.3 case 1. (a) (ii) of [31] it is established that, for generic values of 1 + η 1 λ 2 as this is the case for γ = 0 (to which we restrict ourselves in Section 5), r = r − is a null singularity for the sinh solution (a Penrose diagram is given in Figure 3 of [31]). The curvature scalar of (2.2) diverges at these two points for γ = 0 Expressing (r + , r − ) in terms of M and a 2 = q 2 /M 2 , one obtains (2.7) [The limit γ → 0 in (2.7) leads to (2.6)].
Angle of deflection. The derivation of the angle of deflection is given in Appendix A by where u n = 1/r n is the point on the light scattering geodesic nearest the origin where du dφ (u n ) = 0. Since the values of γ depend on η 1 according to (2.4) and the sign of r − is that of η 2 (1 + γ) by (2.5), the deflection angle depends on the type of EMD under investigation. From (2.6, 2.7) one sees that, for both cases γ = 0 and γ = 0, the limit case q = 0 corresponds to r − = 0 and r + = 2M. Thus in the limit q → 0, δ φ approaches the value δ φ (r − = 0, r + = 2M), which is the Schwarzschild angle of deflection δ φ S : Using this along with (2.6, 2.7) in (2.8) we express δ φ in terms of the charges (M, q) and δ φ S where we have made use of M − M = η 2 |M − M | and (2.7). [The limit γ → 0 in (2.11) or in (2.12) leads to (2.10)]. Thus to the first order of approximation in 1/r n all normal and phantom black holes deflect light paths in the same way with δ φ = 4M/r n + · · · . To the second order of approximation in 1/r n , the added contribution to the Schwarzschild one [second terms in (2.10, 2.11)] does not depend on the sign of q but depends on the signs of (η 1 , η 2 ). First consider the special case γ = 0, which corresponds to η 1 = +1. In normal EMD (η 2 = +1) we have δ φ < δ φ S . In phantom EMD (η 2 = −1), which is E-anti-MD theory, we have δ φ > δ φ S provided we restrict ourselves to the physical case a 2 < 1 [4(3π − 8)/π ≃ 1.8]. Thus, in the presence of phantom   Figure 2: The relative discrepancy R δ φ = (δ φ − δ φ S )/δ φ S , which defines the relative difference of the actual deflection angle with respect to the Schwarzschild value, is sketched on its domain of definition against γ for fixed (M = 1, a 2 = 1/16, u n = 0.05, η 2 = 1), r n = 1/u n is the point on the null geodesic nearest the origin. This is the EM-(anti)-D case (η 1 depends on γ). R δ φ decreases on its domain of definition and changes sign for some γ 0 between −0.1 and −0.05. fields, light rays are more deflected than in the normal case. Phantom fields cause light rays to bend with an angle (3π − 8)M 2 a 2 /(2r 2 n ) larger than the angle of deflection caused by normal fields. This difference is independent of the sign of q but depends on the mass of the black hole through r n . Using (2.9) we obtain 3π − 8 which is 0.22% of the Schwarzschild value δ φ S if M = 1, a 2 = 1/4, u n = 0.05 and 0.89% of it (the exact value is 1.03%) if M = 1, a 2 = 1, u n = 0.05. The case γ = 1 is phantom or normal Reissner-Nordström black hole. With δ φ = δ φ S − η 2 3πM 2 a 2 4 1 r 2 n + · · · , if γ = 1 (2.14) we confirm the previous conclusions: δ φ < δ φ S for normal Reissner-Nordström black holes and δ φ > δ φ S for phantom ones. A phantom Reissner-Nordström black hole deflects light with an angle 3πq 2 /(2r larger than the deflection angle caused by a normal Reissner-Nordström black hole which is 1.47% of the Schwarzschild value δ φ S if M = 1, a 2 = 1/4, u n = 0.05 and 5.89% of it (the exact value is 6.56%) if M = 1, a 2 = 1, u n = 0.05. Now, consider the case γ = 0 and γ = 1 (γ = −1). Here again we confirm the previous conclusions: δ φ < δ φ S for normal black holes and δ φ > δ φ S for phantom ones provided |γ| is large enough. This is no longer true if γ is closer to −1 as the coefficient of 1/r 2 n becomes too large invalidating (2.12). The relative discrepancy function R δ φ = (δ φ − δ φ S )/δ φ S is shown in Figures 1 and 2, which have been plotted using the exact formula (A.4). The Figures illustrate the existence of a zero γ 0 beyond which R δ φ > 0 (δ φ > δ φ S ) for phantom black holes and R δ φ < 0 (δ φ < δ φ S ) for normal ones. Figures 1 and 2 have been plotted for fixed (M = 1, a 2 = 1/16) and a relatively large value of r n (u n = 0.05). Based on these and on some other figures (not shown in this paper) for small values of r n up to the photon sphere (u n = 0.3) and larger values of a 2 up to 1, we can draw a general conclusion: For fixed (M, a 2 , u n ), there is always a root γ 0 in the interval (−0.2, 0) to R δ φ (γ) = 0. Otherwise, for some values of γ in the interval (−0.2, 0), it seems there is always a critical value r n = r c , larger than the photon sphere, where R δ φ = 0.
As γ → +∞, R δ φ approaches the limit As Figures 3 to 6 reveal, the vertical spacing |δ φ − δ φ S |, whenever defined, depends slightly on γ, which itself depends on η 1 , and increases with a 2 . In the extreme case (a 2 = 1), the winding number for phantom black holes with η 2 = −1 and γ > γ 0 (regardless of the sign of η 1 ) diverges near u n ≃ 0.3, a value for which the angle of deflection for phantom (η 1 = −1) or normal (η 1 = +1) black holes with η 2 = +1 is less than a few radians. As we shall see later, this is a consequence of the fact that the photon δ φ ever increases and exceeds 2π, then diverges, as r n decreases from ∞ to r ps (the photon sphere). |δ φ − δ φ S | decreases with sphere for black holes with η 2 = −1 (black holes where the Maxwell field F is coupled repulsively to gravity) and γ > γ 0 is larger than 3M, which is the Schwarzschild limit, allowing photons to orbit the hole at larger, ever-decreasing, radii. The Schwarzschild limit 3M is larger than the photon sphere for black holes with η 2 = +1 and γ > γ 0 . If γ < γ 0 and η 2 γR δ φ < 0, all that is true for black holes with η 2 = −1 (respectively, η 2 = +1) applies to black holes with η 2 = +1 (respectively, η 2 = −1). It is useful to express δ φ in terms of the charges (M, q) and the impact parameter b = L/E. For that end we need to solve the equation E 2 = L 2 g(u n ) (see Appendix A) for u n = 1/r n to the second order in Using this in (2.8) we derive the desired equation In terms of the Schwarzschild value, , the expressions (2.10, 2.11, 2.14) become, respectively, Time delay. We evaluate the coordinate time T (U ) required for light to travel from a point U = 1/R to u n = 1/r n in the plane θ = π/2. Using Eqs (A.1, A.2) along with E 2 = L 2 g(u n ), we obtain where we have set u = u n x and X = U /u n = r n /R satisfies 0 < X < 1. An expansion in terms of powers of u n leads to where T S is the corresponding Schwarzschild value [34] Notice that the correction, which has been added to the Schwarzschild value in (2.21), vanishes for Reissner-Nordström black holes (γ = 1), and a second order correction is needed for these black holes as shown in (2.22). Since the sign of (γ − 1)/(γ + 1) is, by (2.4), that of −η 1 , we conclude from (2.21) The time delay ∆T or the excess in time is the difference between the time required for light to travel from the source located at the point U s = 1/R s through the point u n = 1/r n to the observer located at the point U o = 1/R o and the time needed for light to travel from U s to U o in the absence of gravity (and any field that may cause light to deflect). As shown in [8] the time delay may be positive, zero or negative.
Using the same notation as in Figures 1 of [35,36], the axis joining the observer O and the black hole (the lens or deflector L) is the optic axis. The angle LOS is β and the angle LOI is θ where I is the image of the source S on the same side as the latter. D s and D d are the distances of the source and the lens from the observer (measured along the optic axis) and D ds = D s − D d represents the projected distance on the optic axis from the lens to the source. With these notations we have Far away from the lens, for large values of the impact parameter, one may expand ∆T in powers of ε = θ E /(4D) where θ E is a good estimate of the angular radius of the Einstein ring of Schwarzschild where the terms independent of γ correspond to the Schwarzschild value ∆T S . For γ = 1, the last term in (2.23) vanishes and a term proportional to ε 3 is needed. Expanding where ∆T S is the Schwarzschild value up to ε 3 as derived in [34].

Determination of geodesics by Weierstrass elliptic functions: general procedure
Any differential equation of the form (1.2) has a unique solution in terms of the Weierstrass elliptic function of the form [2,3] where C is generally a complex constant.
To determine all types of geodesic motion in a given geometry, one has to consider all allowed possible locations of (ρ ∞ , ρ 0 , ρ + , ρ − , . . .) with respect to (e 3 , e 2 , e 1 ). Once this is done, any geodesic motion that can be brought to (1.2) is integrated by mere comparison with the work done in this section. We shall provide some examples in this section and in the next two we apply the procedure to phantom and normal Reissner-Nordström and EMD black holes. To illustrate the procedure, we shall envisage only some locations of (ρ ∞ , ρ 0 , ρ + , ρ − , . . .) with respect to (e 3 , e 2 , e 1 ), most of which are related to phantom and normal Reissner-Nordström and EMD black holes.

Scattering and trapped paths
We consider four possible situations.
As light scatters from r = +∞ → r n → r = +∞, the corresponding point on the ρ-axis moves from ρ ∞ → e 2 → ρ ∞ , if ρ(r) is a decreasing function of r, or from ρ ∞ → e 3 → ρ ∞ , if ρ(r) is an increasing function of r. We consider the former case throughout this section, which is also going to be the case for the next two sections. The solution to (1.2) is To fix C we may assume Θ = 0 at ρ n = e 2 , corresponding to r n , or assume Θ = 0 at ρ ∞ . The former case looks simpler leading to e 2 = ℘(C) and thus, by (3.1), C = ω 2 or C = −ω 2 (℘ is an even function!). We choose the latter solution so that by (3.2) C = ω 1 + ω 3 and The angles Θ and φ are related by a linear formula that may be put on the form Θ = κφ . The angle of deflection is then given by where we have used the second formula in (3.7). The other possible motion, the so-called trapped or terminating bound path, is in the region between e 1 and ρ 0 where w(ρ) ≥ 0. If the path starts from the singularity ρ 0 = ρ sing (r = 0), it reaches the farthest point ρ f = e 1 (r = r f ) and then returns to ρ 0 . Again choosing Θ = 0 at the farthest point, the solution is If ρ 0 is a singularity, then this case is identical to (a) with a scattering path from ρ ∞ → e 2 → ρ ∞ given by (3.9) and a trapped path between e 1 and ρ 0 given by (3.11). If ρ − is a singularity but ρ 0 is not, then there is a trapped path between e 1 and ρ − given by (3.11). If neither ρ 0 nor ρ − is a singularity, the path is a many-world periodic bound orbit [19] in that the path, after crossing the inner horizon at r = r − , emerges in another copy of the space-time then in another copy of it and so on. If we choose Θ = 0 at ρ = e 1 , then the solution will be given by (3.11). e 2 < ρ 0 < e 1 < ρ ∞ < ρ + < ρ − = +∞ e 2 < ρ 0 < e 1 < ρ ∞ < ρ + < ρ − = +∞ e 2 < ρ 0 < e 1 < ρ ∞ < ρ + < ρ − = +∞. Since w(ρ) < 0 for ρ ∈ (e 2 , e 1 ), there are no paths that can reach or emanate from the origin.
There is a path that extends from spatial infinity (ρ ∞ ) to the inner horizon (ρ − ). This is not a spiral path since the integral converges (Θ remains finite). If ρ − is not a singularity, the path is called a two-world scattering orbit in that the path emerges, after crossing the inner horizon at r = r − , in another copy of the space-time and back to spatial infinity. If ρ − is a singularity, we have an absorbed path from spatial infinity to the singularity. The solution is again ρ(r) = ℘(Θ(φ ) +C). Since there is no farthest or nearest point on the path, we choose Θ = 0 at ρ ∞ leading to ρ ∞ = ℘(C). Using the inverse function to ℘, C = ∞ ρ ∞ dρ/ w(ρ) and Here again no paths that can reach or emanate from the origin.
We have a scattering path from ρ ∞ → e 2 → ρ ∞ , and the solution is again given by (3.9) but with different p.
There is another path from r = r 1 (ρ = e 1 ) to r = r − (ρ = ρ − ). Since in this case r 1 is finite (ρ ∞ < e 1 ⇒ r 1 < +∞), the path is a many-world periodic bound orbit if ρ − is not a singularity or a trapped path if ρ − is a singularity. If we choose Θ = 0 at ρ = e 1 , then the solution will be given by (3.11).

Absorbed and circular paths
Absorbed paths extend from spatial infinity (ρ ∞ ) to the (nearest) singularity (ρ sing ). Such paths exist if both points ρ ∞ , ρ sing are in [e 3 , e 2 ] or in [e 1 , +∞). [There are no such paths in the Schwarzschild case when w(ρ) has three distinct real roots.] It is clear that there are no circular paths when w(ρ) has three distinct real roots.

Two distinct real roots
The Weierstrass polynomial w(ρ) will have two real roots if This happens when one of the local extreme values of w(ρ) is zero. The second condition in (3.13) splits into two cases.

One real root
The Weierstrass polynomial w(ρ) will have one real root with multiplicity 1 if The sign of the real root e r e r = 1 is related to that of g 3 by e r · g 3 > 0 and e r = 0 if g 3 = 0 .

(3.25)
Motion is possible for ρ ≥ e r . Absorbed paths exist if the range e r ≤ ρ < ∞ includes r sing < r < ∞. In that case, we will have e r ≤ ρ ∞ < ρ sing and the solution will be given by (3.12) where the upper limit of integration "∞" is replaced by "ρ sing ". If ρ ∞ < e r < ρ sing , the solution is a trapped path given by (3.11).
One can envisage other situations as ρ sing < e r and so on. However, we are giving examples that are more or less related to EMD and Reissner-Nordström black holes.
The case g 2 = g 3 = 0 implies e r = 0 [this is the only case where the three real roots of w(ρ) = 0 are equal]. This is no different from the two cases discussed above for the generic case. However, if ρ(r) were an increasing function of r and ρ ∞ > e r = 0, the angle Θ would diverge as for paths approaching e r = 0 from the right. This is not a logarithmic behavior as the one we have seen earlier. These spiral paths would approach, without reaching, the unstable circular path at ρ = e r = 0.

Application: Strong field limit and relativistic images
Because of the orbiting effect [23,39] geodesics may orbit many times around the deflector before escaping to infinity. In the case of light paths, images that are formed because of geodesic deflection by more than 3π/2 are called relativistic images [35]. As we have seen in Subsection 3.2, such an orbiting effect happens in the strong field region and takes place as the nearest point r n (to the origin) approaches (but remains larger than) the radius of the photon sphere r ps . The limit r n → r ps leads to spiral paths approaching endlessly the photon sphere. This provides the most common definition of the photon sphere [40] 6 .
In the context of Weierstrass elliptic functions, we have established that photon spheres are the unstable circular paths with two equal positive roots of the Weierstrass polynomial, e 1 = e 2 > 0, and a negative root, e 3 = −2e 1 .
In the strong field limit, all relations governing the light trajectory as well as the determination of the angular positions of images and related entities form a set of transcendental, analytically nontractable, equations and inequations. Numerical solutions to the case of a Schwarzschild black hole exit and have led to the following conclusions [42]: 1) Observations of relativistic images would mean high accuracy in the determinations of masses and distances of massive deflectors at the centers of galaxies, and 2) the ratio mass to differential time delay for the two outermost relativistic images is almost constant with respect to changes in (β , D d , D ds ).
In the hope that these numerical solutions would be extended to other massive deflectors, some authors resorted to analytic approximate solutions [23,24,[36][37][38]. In the case of relativistic images these methods are valid if the angular position θ of the image is small enough to allow for series expansions. This has been done in [36][37][38] for the Schwarzschild, Reissner-Nordström, and other black holes and has resulted, among other derivations, in the determination of a log-formula for the angle of deflection and the angular position of the image. Very recently, the method used in [38] has been applied to gravitational lensing by phantom black holes [43].
In the following we will derive an approximate analytic reference equation for the log-formula for the angle of deflection, which applies to any geometry provided the light motion or a plane projection of it is described by (1.2). We assume that a photon sphere r ps exits. The corresponding Weierstrass radial coordinate is denoted by ρ ps , and ρ n corresponds to r n . Since we are assuming ρ(r) to be a decreasing function of r, a scattering case corresponds to e 3 < ρ ∞ < e 2 = ρ n and e 2 < ρ ps < e 1 . In the strong field limit, as r n → r ps , both e 2 and e 1 approach ρ ps . If we set 7 ρ ps − ρ n =ε > 0, then to the relevant first order of approximation we have e 1 − ρ ps =ε + O[ε] 2 , whereε is assumed to be small in order to have relativistic images. With e 1 = ρ ps +ε + O[ε] 2 , e 2 = ρ n = ρ ps −ε, and e 3 = −2ρ ps + O[ε] 2 the first equation (3.10) takes the form δ φ = 1 |κ| We introduce the variable z = (ρ n − ρ)/(ρ n − ρ ∞ ). Following [38] we obtain The final step is to express δ φ in terms of ε: r n − r ps = 2Mε. We havē where ρ ′ (r ps ) = (dρ/dr) r=r ps . Substituting in (3.27), we arrive at which is valid whether ρ is increasing or decreasing. As we mentioned earlier, this formula applies to all relativistic images of light paths governed by (1.2). The formula looks much easier to handle analytically than that given in [38]. Using an expansion of b in powers of ε, one can determine the positions of the relativistic images if these are too small to allow for series expansions as shown in the following example. 7 Had we assumed ρ(r) increasing, a scattering case would correspond to ρ n = e 1 < ρ ∞ . In the strong field limit we set ρ n − ρ ps =ε > 0 leading to ρ ps − e 2 =ε + O[ε] 2 . Eq. (3.26) would read For the Schwarzschild black hole we have seen that [22] ρ = M/(2r) − 1/12, κ = 1, ρ ps = 1/12, and ρ ∞ = −1/12. We obtainε = M/(2r ps ) − M/(2r n ) = ε/9 + · · · , where we have used r ps = 3M, and finally which is the same as in [36,37]. For small angular positions of the source and its image, the impact parameter b and θ are related by b = D d θ . Using an expansion of b in powers of ε: as in [38]. The position of the relativistic image of order n in terms of β (the angular position of the source) is determined as in [36][37][38].

Phantom and Normal Reissner-Nordström black holes
It is difficult or impossible to reduce (A.3) to (1.2) for any γ . For γ = ±2 and probably for some other values, the problem can be tackled semianalytically in a similar way to what is done in [4,9,19]: Limits to the analytical treatment are 1) lack of "compact" solutions to the polynomial equation (or its polynomial reduced form) P(r) = 0 and/or 2) lack of solutions to the generally nonpolynomial equation ∆ ≡ g 3 2 − 27g 2 3 = 0. All that does not apply to the cases γ = 1 and γ = 0, which can be entirely analytically solved. In this section we investigate the former case and in the next one we tackle the latter case.
If u r is a root with multiplicity 1, following the general procedure, we introduce the radial coordinate y = u − u r followed by z = 1/y, and finally where we have used (4.4) to eliminate u r from the expressions of g 2 , g 3 . The new parameters (ℓ − , ℓ + ) are defined by In the physical case a 2 = q 2 /M 2 < 1, 0 < ℓ + < ℓ − for the phantom black hole and ℓ − < 0 < ℓ + for the normal one. The transformation (4.5) "splits" the point r = r r into r + r and r − r (corresponding to u − r and u + r , respectively). If C 1 < 0, then r + r and r − r are sent to ρ = ρ r = +∞ and ρ = −∞, respectively, and if C 1 > 0 the latter limits are reversed. As we shall see in Appendix B, it is always possible to choose the real root u r so that C 1 < 0: We choose u r to be the smallest root for phantom black holes and the largest root for normal ones. The points ρ ∞ , ρ 0 and ρ ± (corresponding to r = +∞, r = 0, and r = r ± ) on the ρ-axis are given by which depend on u r whose analytic expression in terms of (M, q 2 , ℓ) is sizable.
For the normal case, Eq. (B.6) of Appendix B reads The only possible paths are scattering ones from ρ ∞ to e 2 given by (3.9, 4.5) or many-world ones from any point e 1 ≤ ρ ≤ ρ + to ρ = +∞ (from any point r + ≤ r ≤ r 1 = 1/u 1 to r r = 1/u r where u r > u − is the largest root of ℓ − u 2 + 2Mu 3 − q 2 u 4 = 0 and u 1 < u + is the second largest root [ Figure 7 (b)]). If the many-world path starts from ρ = e 1 its equation is given by (3.11, 4.5). Since ρ 0 is a singularity for the phantom black hole, there is a trapped path for this hole from ρ + → ρ 0 given by (3.17, 3.18) with η 2 = −1.

Two distinct real roots for
There is a many-world periodic path for the normal black hole from ρ + → ρ − given by (3.16, 4.5).
then using the first equation But instead of (4.19) we would obtain
For the normal solution (η 2 = +1), this is the case where the point M 3 is above the u-axis [ Figure 7 (b)]. The two real roots of ℓ − u 2 + 2Mu 3 − q 2 u 4 = 0 satisfy u 3 < 0 and u r > u − > 0 is the largest one. (u 1 , u 2 ) are now complex roots, so (e 1 , e 2 ) no longer exist. Eqs. (B.5, B.6) become The only existing paths are two-world scattering paths from spatial infinity to r = r r = 1/u r > 0 given by (3.12) and (4.5).

Null geodesics of phantom and normal EMD
In this section we restrict ourselves to the case γ = 0, which corresponds to η 1 = +1, and then (2.3) implies η 2 = −1 for the cosh solution and η 2 = +1 for the sinh one. Thus we will be considering E-anti-MD for the cosh solution and normal EMD for the sinh one. Instead , we use f − as a radial coordinate. This way we reduce (A.3) for light paths (ε = 0) to where, using (2.6), In the physical case a 2 < 1, to which we restrict ourselves, α is constrained by for the phantom cosh and normal sinh solutions, respectively. The next step is to introduce the variable R = 1/( f − − f 0 ) where f 0 is a zero of the fourth order polynomial in f − on the r.h.s of (5.1). We choose f 0 = 0, leading to dR dφ 2 = α − (3α + 1)R + (3α + β + 2)R 2 − (α + 1)R 3 .

One real root for
This case corresponds to [Eq. (3.23)]: β > β 3 leading to ρ ∞ < e r < ρ + < ρ 0 < ρ − = +∞ (5.19) where e r is the real given by (3.24). For α > 2 it is not possible to have g 2 = 0 and g 3 = 0, so there is no solution e r = 0 with multiplicity 3. Since ρ 0 is a singularity for the cosh black hole, there is a trapped path from e r to ρ 0 for the generic case β > β 3 given by (3.11, 5.5, 5.8).
The case β > β 3 has been treated in Section 3.1.1 case (c). Since ρ − is a singularity, we have an absorbed path from spatial infinity to the singularity. The solution is given by (3.12, 5.5, 5.8).
The discussion of the case β = 0 for the normal sinh black hole is similar to that for the phantom cosh one. The information in the first paragraph following (5.16) applies to this case if we replace "ρ 0 " by "ρ − ", "r = 0" by "r = r − " and "cosh" by "sinh". Thus, there is a trapped path from ρ + to the singularity ρ − given by (3.16, 5.5, 5.8).

Conclusion
To the first order of approximation, all black holes of phantom and normal EMD deflect light in the same manner. If we restrict ourselves to physical conditions [a 2 ≤ 1 for η 2 = −1 and a 2 ≤ (1 + γ)/(2γ) for η 2 = +1], then 1) for γ larger than some γ 0 , which is likely in (−0.2, 0) and depending on the parameters of the black hole, black holes of E-anti-M-(anti)-D theory (regardless of the sign of η 1 ) cause light rays to deflect with larger angles than black holes of EM-(anti)-D. The difference in the angles and the relative discrepancy ever increase with a 2 for fixed (u n , γ). For fixed (a 2 , γ), they increase with 1/r n and diverge as r n approaches the photon sphere of E-anti-M-(anti)-D black holes. 2) For γ < γ 0 and η 2 γR δ φ < 0 light is more deflected by the black holes of EMD than by those of E-anti-MD; the relative discrepancy for larger values of the impact parameter is, however, much larger for the black holes of E-anti-MD.
Time delay and relativistic images are other ingredients, besides deflection, allowing for the determination of the nature of matter. From this point of view a very useful log-formula for the positions of the images has been determined.
The method based on the Weierstrass polynomial to integrate geodesic motion and determine exact solutions is equivalent to other methods using potential barriers and can be applied systematically. The advantage of using the method based on the Weierstrass polynomial is that motion is allowed in at most two regions: In between the smallest root of the polynomial and the intermediate one and/or for values greater than the largest root. This highly simplifies the problem. Some of the systematic resolutions consist as follows: 1) The angle of deflection and the log-formula have standard expressions for all problems that can be brought to Weierstrass differential equation. 2) If the smallest and intermediate roots of the Weierstrass polynomial are equal for some value of the vector of parameters, there should be a stable circular path for the corresponding radial coordinate r if the latter is within accessible limits to observers. 3) If the largest and intermediate roots are equal for some value of the vector of parameters, there should be an unstable circular path (photon sphere) for the corresponding radial coordinate r if the latter is within accessible limits to observers. 4) Existence of spiral paths, which approach endlessly the photon spheres, is a consequence of 3). 5) Existence and identification of divergencies for the angle of deflection: a logarithmic one if 3) holds or a power law one (to the power −1/2) if the three real roots are zero. 6) Ordering of the parameters expressing spatial infinity, singularity, horizons and so on on the Weierstrass axis is derived by circular rotation (from their given order relations on the r-axis) in the one or the other way depending on the coordinate transformation relating the Weierstrass radial coordinate to the spherical radial one (increasing or decreasing).
Phantom Reissner-Nordström black holes are characterized by the existence of trapped and absorbed null paths that do not exist for normal Reissner-Nordström black holes. Their other noncommon paths include many-world (periodic bound) and two-world paths that exist only for normal Reissner-Nordström black holes. Their common paths include scattering, spiral (existence of logarithmic divergencies), and unstable circular paths with radii approaching the Schwarzschild limit from above for phantom black holes and from below for normal ones.
Both phantom cosh and normal sinh black holes of EMD theory are characterized by the presence of scattering, trapped, and unstable circular paths, thus spiral paths and the existence of logarithmic divergencies. The photon spheres are larger or smaller than the Schwarzschild one, respectively, and approach it in the limit of no electric charge. The phantom solution has no absorbed path while the normal one does. massive, massless particles, respectively) dr dτ For scattering states E 2 − ε > 0. Eliminating τ in (A.1, A.2) and using u = 1/r we arrive at du dφ From now on we take ε = 0 so that the condition for light scattering is E 2 > 0. Now, let g(u) := u 2 f + f 2γ−1 − and u n = 1/r n be the point on the light scattering geodesic nearest the origin where du dφ (u n ) = 0. Since E 2 = L 2 g(u n ), the angle of deflection, which is twice the variation of φ minus π, takes the form where u = u n x. If u n ≪ 1, corresponding to scattering with large values of the impact parameter (b = L/E), + r 2 − [4γ 2 − 1 + 6(2γ − 1)x + (8γ 2 + 1)x 2 + 6x 3 + 3x 4 ] u 2 n + O[u n ] 3 where we have used (2.5): r + + γr − = 2M. Performing the integrations over x we obtain (2.8).