Gravitational Lensing by Rotating Naked Singularities

# Gravitational Lensing by Rotating Naked Singularities

## Abstract

We model massive compact objects in galactic nuclei as stationary, axially-symmetric naked singularities in the Einstein-massless scalar field theory and study the resulting gravitational lensing. In the weak deflection limit we study analytically the position of the two weak field images, the corresponding signed and absolute magnifications as well as the centroid up to post-Newtonian order. We show that there are a static post-Newtonian corrections to the signed magnification and their sum as well as to the critical curves, which are function of the scalar charge. The shift of the critical curves as a function of the lens angular momentum is found, and it is shown that they decrease slightingly for the weakly naked and vastly for the strongly naked singularities with the increase of the scalar charge. The point-like caustics drift away from the optical axis and do not depend on the scalar charge. In the strong deflection limit approximation we compute numerically the position of the relativistic images and their separability for weakly naked singularities. All of the lensing quantities are compared to particular cases as Schwarzschild and Kerr black holes as well as Janis–Newman–Winicour naked singularities.

Relativity and gravitation; Classical black holes; Naked singularities; Gravitational lensing
###### pacs:
95.30.Sf, 04.20.Dw, 04.70.Bw, 98.62.Sb

## I Introduction

Is there a naked singularity exist or not, it is probably one of the most important unsolved problems in general relativity. According to the standpoint of the cosmic censorship conjecture the spacetime singularities of gravitational collapse are hidden within black holes RP1 (); RMWald () and therefore can not be observed. In this work, we model the massive, dark, radio object Sgr A in the center of the Galaxy as a rotating generalization of the Janis–Newman–Winicour naked singularity. Using the predictions of the gravitational lensing caused by the space-time under consideration we obtain the physical parameters of the rotating gravitational lens, which uncover the rotating naked singularities concealed in the black holes.

In the last decades the gravitational lensing attracted ever more and more the interests of the science community. As result the theory have developed in two frames. The former examine the problem for the distribution of a photon on a radial distance much larger than the gravitational radius of the lens. In this case the light ray has a small deflection angle, therefore we are speaking about the gravitational lensing in the weak deflection limit. Then two weak field images on the each side of the lens appeared. The letter discuss the photons winding many times closely around the lens before reaching into the observer. This process is also known as a gravitational lensing in the strong deflection limit. I this case, an infinite series of highly demagnified relativistic images on both sides of the lens shadow should be appeared.

It is known that the lensing observables might be successfully described by a perturbation theory applied to general relativity. The theory of gravitational lensing in the weak deflection limit approximation has been developed for a Schwarzschild point-mass lens SEF (); PettersLevine (). The rotational case has been studied for the first time up to a post-Newtonian expansion by Epstein and Shapiro EpsteinShapiro () and after that by Richter and Matzner RichterMatzner (). Later on, Bray Bray () investigated the multi-imaging aspect of Kerr black hole lensing as resolved the equations of motion for a light ray up to and including second order terms in scaled black hole mass and angular momentum , where is the distance of closest approach. Gravitational lensing by rotating stars has been considered by Glinstein Glinstein () and later on by Sereno Sereno1 (). The weak field Reissner-Nordstrom black hole lensing has been done by Sereno Sereno2 (). The gravitatonal-magnetic effect in the propagation of light in the field of self-gravitating bodies has been investigated by Kopeikin at al. Kopeikin (). Asada and Kasai AsadaKasai () have found that up to the first order in the gravitational constant , a rotating lens is not distinguishable from a not-rotating one. They have found that because of the global translation of the center of lens mass the Kerr lens is observationally equivalent to the Schwarzschild one at linear order in mass and the specific lens angular momentum . Later on, Asada, Kasai, and Yamamoto have shown AsadaKasaiYamamoto (), that the nonlinear coupling break the degeneracy so that the rotational effect becomes in principle separable for multiple images or a single source. After that, Sereno Sereno3 () considered a gravitational lensing in metric theories of gravity in post-post-Newtonian order with gravitomagnetic field. Recently, Keeton and Petters KeetonPetters () have developed a general formalism for lensing by spherically symmetric lenses up to post-post-Newtonian order. Sereno and de Luca SerenoLuca () extended their approach to the case of the Kerr black hole. Finally, Werner and Petters WernerPetters () based on the analysis by Asada, Kasai, and Yamamoto, applied a simpler method to derive image positions and magnifications up to post-Newtonian order. Using the degeneracy in the case of the Kerr black hole lensing in the weak deflection limit, they presented lensing observables for the two weak field images in post-Newtonian terms with scaled lens angular momentum .

In 1959 Darwin Dar () examined a photon trajectories passing in the vicinity of the black hole and showed the larger deflection, which suffer a light ray without falling into the event horizon. This effect was considered again also in LON (). After that Fritteli, Kling and Newman FKN () as well as Virbhadra and Ellis Vir1 () developed this idea and worked out a definition of an exact lens equation. Virbhadra, Nirasimha and Chitre Vir3 () as well as Virbhadra and Ellis Vir2 () studied later numerically the lensing by static and spherically symmetric naked singularity and showed the influence of the scalar field to the lensing observables. In contrast to this Perlick Per () has considered lensing in a spherically symmetric and static spacetime, based on the lightlike geodesic equation without approximations. Bozza Bozza1 () developed an analytical technique based on the larger deflection of the light ray and showed that the deflection angle diverges logarithmically as light rays approach the photon sphere of a Schwarzschild black hole. The expectation that the relativistic images should be test the gravity in the strong deflection limit has leaded to application of this method to various metrics from the general relativity, string and alternative theories Group1 (); Group2 (); Group3 (); Eiroa (); MukherjeeMajumdar (). Analytically was studied and the Kerr black hole lensing in Refs. Group5 () and extended to Kerr-Sen black hole lensing GyulchevYazadjiev (). The time delay calculation for higher order images have done by Bozza and Mancini BozzaMancini1 () and Bozza and Sereno BozzaSereno (). Recently, Vibhadra and Keeton VirbhadraKeeton () have examined numerically the time delay and magnification centroid due to gravitational lensisng by black holes and naked singularities. Out of the assumption for the very far source positions Bozza and Scarpetta BozzaScarpetta () have considered black hole gravitational lensing with arbitrary source distances with respect to the black hole.

In our desire to bridge the gap between the weak and the strong deflection limit analysis of gravitational lensing we can refer to Amore and Arceo AmoreArceo (). They present a method which can be used to obtain arbitrarily, accurate, analytical expressions for the deflection angle of light propagating in a given metric. An effective analytical formalism for the Schwarzschild deflection angle which describe with satisfactorily accuracy both weak and strong deflection series have also been developed by Iyer and Petters IyerPetters ().

The purpose of this paper is to consider the gravitational lensing by rotating naked singularities and to explore how it differs from the Kerr black hole lensing. The weak gravitational lensing allow us to describe the light ray trajectory, where the closest approach distance and the impact parameter both lie outside the gravitational radius . Besides, the strong gravitational lensing could provide profound examination of the space–time around different kinds of black holes and naked singularities. Therefore, following Bozza1 (); WernerPetters () in the present work we wish to study gravitational lensing in the weak and strong deflection limit due to a stationary, axially-symmetric weakly naked singularities and to compare the results to rotating strongly naked singularities with the aim of investigating the influence of the scalar field on the behavior of the bending angle, on the position of the images and on their magnification as well as in the critical curves and caustics.

The outline of this paper is as follows. The second section contains a description of a Kerr-like solution of the Einstein-massless scalar field equations. In Sec. III we discuss the full lens equation. In Sec. IV a gravitational lensing in weak deflection limit by Kerr black hole, weakly and strongly naked singularity is investigated and the critical curves and the caustic structure are considered. In Sec. V the deflection angle is numerically computed in the equatorial plane and its dependence from the scalar charge and the lens angular momentum is shown. In Sec. VI we discuss the equatorial lensing by Kerr black hole, weakly naked singularity and marginally strongly naked singularity in the strong deflection limit and compute the positions of the relativistic images and their separability. A discussion of the results is given in Sec. VII.

## Ii Rotating singularity spacetime

We consider a Kerr-like solution KB () to the Einstein-massless scalar equations ( with , where is the Ricci tensor and is the massless scalar field). This solution is rotating generalization of the Janis–Newman–Winicour (see for example VirJNW ()) solution and is given by the line element

 ds2 = (1−2Mrγρ)γ(dt−wdϕ)2 (1) − (1−2Mrγρ)1−γρ(dr2Δ+dϑ2+sin2ϑdϕ2) + 2w(dt−wdϕ)dϕ,

and the scalar field

 φ=√1−γ24ln(1−2Mrγρ), (2)

where

 γ=M√M2+q2,w=asin2ϑ,ρ=r2+a2cos2ϑ,Δ=r2+a2−2Mrγ. (3)

and , the Arnowitt-Deser-Misner (ADM) mass and scalar charge, are constant real parameters in this solution. is the angular momentum of the rotating object in units of the mass. Hereafter we will not consider the massless scalar field case (e. i. or ). For and this solution reduces to the Janis–Newman–Winicour solution, for and to the Kerr black hole, while in the particular case and it reconstructs the Schwarzschild solution. The above solution is in fact the Einstein-frame version of the original Jordan-frame solution to the Brans-Dicke equations found in KB ().

In the case () when the Kerr black hole solution is recovered, there is an event horizon with spherical topology, which is the biggest root of the equation and is given by

 rH=M+√M2−a2 (4)

for . Beyond this critical value of the spin there is no event horizon and causality violations are present in the whole space-time, with the appearance of a naked singularity. The ergosphere is defined as a surface on which the Killing vector is isotropic, i.e . So the ergosphere lies at

 res=M+√M2−a2cos2ϑ. (5)

For the metric (1) describes rotating naked singularities with mass and angular momentum . In order to see that the solution is singular for we calculate the Ricci scalar curvature and find

 R=2(γ2−1)M2γ2ρ5(1−2Mrγρ)γ−3[Δ(r2−a2cos2ϑ)2+(ra2sin22ϑ)2]. (6)

As it is seen, the scalar curvature diverges where which shows the presence of a curvature singularity at

 rcs=1γ[M+√M2−γ2a2cos2ϑ], (7)

as the domain of variation of depends on the ratio . One can show that there is at least one equatorial null geodesic with one end on the singularity and the other on the future null infinity i.e. the singularity is indeed naked. The global naked singularity nature of the static JNW solution for was first shown in VJJ () (see also the Seifert conjecture for naked singularities VirS ()).

## Iii Equation of the gravitational lens

The equation of the gravitational lens, which allows small as well as large bending of the light ray is Vir1 ()

 tanB=tanΘ−DLSDOS[tanΘ+tan(~α−Θ)], (8)

where and respectively are the lens-source and the observer-source angular diameter distances. The optical axis of the gravitational lens systems connects the origins of Cartesian angular coordinates , and , respectively of the source plane and the lens plane. Given a source position , the values of , that solve this equation, give the pos ition of the observed images measured from the optical axis. The projections of the deflection angles into the lens plane are denoted by , . For the impact parameter of the light ray we have , where is the observer-lens angular diameter distance. For small angles, Eq. (8) reduces to the well known in the literature weak field lens equation, as well as to the strong deflection limit lens equation BozQuazi (). For spherically symmetric lenses we refer to BS (), where most general lens equation is constructed.

As we shall see below, we will solve the lens equation in the weak deflection limit in the cases of presence and absence of the photon sphere in order to explore the differences in the gravitational lensing by various of naked singularities. Moreover, since the strong deflection limit allows a simple analytical investigation of the gravitational lensing properties, after calculating of for the rotating singularity space-time in this approximation, we will solve the lens equation in order to derive the relativistic images when a photon sphere exists BozQuazi ().

## Iv Gravitational lensing by Rotating Naked Singularities in the weak deflection limit.

### iv.1 Image positions

In this part we will study the gravitational lensing by rotating naked singularities in the weak field regime up to the post-Newtonian order. We will distribute the source and the observer in the asymptotically flat region in such a way that the source is situated beyond the lens plane. Following the scheme exposed in WerPet () up to the post-Newtonian limit, we can express the angular coordinates of the source and the image in Einstein ring scale in the following way

 B = θEβ=θE(β(0)+β(1)ϵ+O(ϵ2)), (9) Θ = θEθ=θE(θ(0)+θ(1)ϵ+O(ϵ2)), (10)

where angular radius of the Einstein ring and the expansion parameter are

 θE=√4MDLSDOLDOS, ϵ=θEDOS4DLS. (11)

The Janis–Newman–Winicour deflection angle is

 ~α=4MJ+4M2J2[1−116γ2]π+O(M3J3). (12)

We refer to Vir3 (); VirbhadraKeeton () for different representations of the deflection angle.

Then up to the post-Newtonian order the weak field Janis–Newman–Winicour lens equation gets the form

 β=θ−1θ−[1−116γ2]πϵθ2. (13)

Hence, we can advance to the image positions. Let us orient lens coordinates such that the axis is along the projected lens angular momentum and the axis is perpendicular to the optical axis. Following AsadaKasai (); AsadaKasaiYamamoto (); SerenoLuca (), one can show that up to the post-Newtonian order lensing by rotating naked singularities under consideration is equivalent to lensing by Janis–Newman–Winicour lens but shifted by

 δθ=θE(δθ1,0)=θE(δθ1(1)ϵ,0), δθ1(1)=asinϑOM. (14)

Then we can use the lens equation (13) to describe the image properties WerPet (). Because of this shift (14) the image position of a direct photon () translates to position . By analogy the retrograde image position () is . Based on this, allowing positive and negative angular momenta, via the substitution for the two kinds of photons, one can compute the deflection angle again and rewrite the lens equation. Thereby, the rotating singularity lens equations are

 β1 = θ1−θ1−δθ1(θ1−δθ1)2+θ22−[1−116γ2]πθ1−δθ1((θ1−δθ1)2+θ22)3/2ϵ+O(ϵ2), β2 = θ2−θ2(θ1−δθ1)2+θ22−[1−116γ2]πθ2((θ1−δθ1)2+θ22)3/2ϵ+O(ϵ2). (15)

At Newtonian order (, ) Eq. (IV.1) reduces to the Janis–Newman–Winicour lens equation (13). The solution can be expanded in a series by ,

 θ1 = θ1(0)+θ1(1)ϵ+O(ϵ2), θ2 = θ2(0)+θ2(1)ϵ+O(ϵ2). (16)

The square of the solution of Newtonian order lens equation is , where

 θ±1(0) = β12(1±√1+4β2), θ±2(0) = β22(1±√1+4β2). (17)

The square of the scaled angular source position is . After deriving the post-Newtonian order correction of the lens equations (IV.1) we find the second terms of the image positions

 θ1(1) = [1−116γ2]πθ1(0)(1+θ2(0))θ(0)+(1−θ21(0)+θ22(0))δθ1(1)1−θ4(0), θ2(1) = [1−116γ2]πθ2(0)(1+θ2(0))θ(0)−2θ1(0)θ2(0)δθ1(1)1−θ4(0), (18)

which reduce to the correction terms expected for rotating lenses WerPet () in the case of Kerr black hole (). The post-Newtonian corrections of the positive and negative parity images can be found since we already know , .

### iv.2 Critical curves and caustics.

The critical curves separate the regions in the lens plane where the Jacobian determinant of the lens map has opposite sign. For a point lens at these curves, the magnification factor of the images diverges. Formally looking, the critical curves are solution to the equation

 ~J=∂β1∂θ1∂β2∂θ2−∂β1∂θ2∂β2∂θ1=0. (19)

According to the lens Eqs. (IV.1) and (19) up to the post-Newtonian order the Jacobian is

 ~J=1−1θ4(0)+⎛⎝[1−116γ2]π(1−θ2(0))2(1+θ2(0))θ5(0)−4θ1(0)δθ1(1)(1+θ2(0))θ4(0)⎞⎠ϵ+O(ϵ2), (20)

which reduces to the correction terms expected for Kerr black hole SerenoLuca () up to post-Newtonian order.

We look for a parametric solution up to the post-Newtonian order in the form

 θcr1 = Θcr1θE=cosφ{1+δθE(φ)ϵ+O(ϵ2)}, θcr2 = Θcr2θE=sinφ{1+δθE(φ)ϵ+O(ϵ2)}, (21)

where is an angle in a polar coordinate system taken in the lens plane with origin at the lens. In that system . The first term of (IV.2) gives the Schwarzschild black hole Einstein ring with radius . Solving Eq. (19) we obtain the deviation coefficient

 δθE=[1−116γ2]π2+asinϑOMcosφ+O(ϵ2). (22)

For Kerr black hole lensing (), (22) reduces to the result of Sereno and Luca SerenoLuca (). Then, for an observer set in position and fixed values of and a critical curve in the plane exists. The positions of the equatorial cross sections of the critical curves are shifted with respect to the static case by

 tanδΘcr≃asinϑODOL+O(ϵ2). (23)

In order to calculate the caustics we have to find the corresponding source positions. Through lens equations (IV.1) and (IV.2), up to order the caustic are point-like and are positioned in

 {tanBcau1,tanBcau2}≃{asinϑODOL+O(ϵ3),0}. (24)

Our results for the critical curves shift (23) and the caustic positions (24) coincides with those found in SerenoLuca () up to the post-Newtonian order.

Critical curves are plotted in Fig. 1 for some values of the scalar charge, the lens angular momentum and the observer’s positions. We model the massive dark object in the center of our galaxy as a Kerr black hole and as a rotating generalization of the JNW naked singularities. We assume a point source and set the lens between the source and the observer as assuming that kpc. Studying the influence of the lens parameters over the critical curves, as an illustration we made all graphics for the source position pc. According to Eis () the lens has a mass . In this situation the expansion parameter .

### iv.3 Magnification.

In the approximation of geometrical optics gravitational lensing causes a change in the cross section of a bundle of light rays, such that the surface brightness is conserved. Therefore, the ratio between the angular area element of the image in the celestial sky, , and the angular area element of the source in absence of the lens, , gives the signed magnification

 μ=~J−1=[∂β1∂θ1∂β2∂θ2−∂β1∂θ2∂β2∂θ1]−1. (25)

Disposing with the Eq. (IV.1) the calculations yield

 μ=θ4(0)θ4(0)−1−⎛⎝[1−116γ2]πθ3(0)(1+θ2(0))3−4θ4(0)θ1(0)δθ1(1)(1−θ2(0))2(1+θ2(0))3⎞⎠ϵ+O(ϵ2), (26)

which describes the signed magnification for both images. Eq. (26) reduces to the signed magnification of the Kerr black hole images WerPet (), when .

The individual magnifications of the positive and the negative parity image up to post-Newtonian order can be calculated using (IV.1) and (26). They are respectively

 μ+ = (β+√β2+4)4(β+√β2+4)4−16−1(4+β2)3/2([1−116γ2]π−4β1δθ1(1)β3)ϵ+O(ϵ2), μ− = (β−√β2+4)4(β−√β2+4)4−16−1(4+β2)3/2([1−116γ2]π+4β1δθ1(1)β3)ϵ+O(ϵ2). (27)

Than the sum of the signed magnification for the rotating singularity lens, which has the form

 μ++μ−=1−2π(4+β2)3/2[1−116γ2]ϵ+O(ϵ2), (28)

does not depend on the specific angular momentum and is equivalent to the result for Janis–Newman–Winicour lens to post-Newtonian order. At order the deviations from the magnification invariant for the Janis–Newman–Winicour lens and the lens under consideration are the same.

In the vicinity of the caustics (or the critical curves) the power series (26) does not work properly and describe the image magnifications exactly for scaled source angular positions (or ). Therefore, in order to disentangle the influence of the scalar charge and the lens angular momentum over the signed magnifications, in Figs. 2, 3 and 4 we have plotted the reciprocal values of the Jacobian corresponded to and according to the current estimates for the massive dark object in the center of our Galaxy Eis () as we assume that the observer is equatorial.

In the static case of the Janis–Newman–Winicour lens for all values of the scaled source angular positions and the magnification of the positive parity and the value of that of the negative parity image increases with the increase of the scalar charge. Let us set the source on the equatorial plane (i.e. ) and move it to the optical axis. Then two weak field images will appear, one on each side of the optical axis. The signed image magnifications start to grow from and respectively for the positive and negative parity images when the absolute value of the source scaled angular coordinate is at infinity. In the case when the source is on the optical axis () the signed magnifications diverge and infinitely bright Einstein rings appear. Removing the source from the optical axis (i.e. ) and keeping we will see two weak field images situated respectively on each side of the optical axis in the directions perpendicular to the equatorial plane. Their magnifications decrease with the increase of and approach the limit values and respectively for the positive and negative parity image when the source scaled angular coordinate goes to infinity. In all of these cases the presence of the scalar charge leads to an increment of the signed magnifications.

When the lens is rotational and the source is equatorial the angular momentum of the lens decreases the signed magnification of the negative parity image and increases the value of that of the positive parity image for every value of and a scalar charge . For the source position the negative parity image is outside the critical curve while the positive parity image is inside. Moving the source from infinity to the left hand side of the optical axis the negative parity image magnification starts to grow from while the positive parity image magnification start to decrease from . Passing through the optical axis the source reaches the point-like caustic , then the value of the signed image magnifications diverges and infinitely bright critical curves appear. Drawing back the equatorial source from the left hand side of the caustic point the negative signed magnification increases, while the positive signed magnification decreases with the increase of and approaches respectively and when . When we remove the source from the equatorial plane for every and the lens angular momentum leads to an increase in the negative parity magnification and to a decrease in the positive parity magnification in comparison to the magnification of the static case. When the negative parity magnification decreases, while the positive parity magnification increases. The signed magnifications coincide for a source in the point () where the static signed magnifications have a maximum. When non-equatorial motion of the source leads to occurrence of the positive parity image outside of the critical curve and to raising of the negative parity image inside the critical curve. For fixed value of scaled angular coordinate and lens angular momentum as well as all values of the values of the positive and the negative parity image magnifications increase with the increase of the scalar charge .

For big enough values of the scalar charge and the lens angular momentum the graphics do not describe reliable authenticity of the pictured quantities in the post-Newtonian order, which works sufficiently well for small values of the source scaled angular coordinates and . Therefore, we do not show this situations.

### iv.4 Total magnification and centroid.

When the two weak field images are not resolved and are packed together the main observables become the total magnification and magnification-weighted centroid position. Taking into account that the image parities give for to the absolute magnifications and , then the total absolute magnification for the rotating singularities space-time is

 μtot=|μ+|+|μ−|=2+β2β√4+β2+8β1β3(4+β2)3/2asinϑOMϵ+O(ϵ2), (29)

and up to terms do not differ from the result for the Kerr lensing WerPet (). When the observer is on the rotational axis or in the particular case of the circularly symmetric lens (), the term vanishes.

The magnification centroid position is defined by GauPett ()

 ΘCent=θ+|μ+|+θ−|μ−||μ+|+|μ−|. (30)

Using (IV.1), (IV.1) and (IV.3) we obtain the same expressions as those for Kerr black hole lensing up to post-Newtonian order WerPet ()

 ΘCent1 = θE[(3+β2)β12+β2+(β21−β22−2)(2+β2)2asinϑOMϵ+O(ϵ2)], ΘCent2 = θE[(3+β2)β22+β2+2β1β2(2+β2)2asinϑOMϵ+O(ϵ2)]. (31)

For the case or , the magnification centroid for the rotating singularity space-time coincides with the result by Keeton and Petters KeePett ().

Since the total magnification is the difference of the reciprocal values of the Jacobi determinant related to the positive and the negative parity images, we have plotted it in Fig. 5 as a function of the source scaled angular coordinates and for different values of the lens angular momentum and the scalar charge. The magnification centroid is plotted also in Figs. 7 and 7. The total magnification has a similar behavior as the positive parity magnification, with the difference that for a fixed with the increase of the scalar charge increases for and decreases for . For the equatorial source (i.e. ) the magnification centroid grows with an increase of . When the source is near the optical axis the lens angular momentum decreases more the magnification centroid and less for bigger values of the source scaled angular coordinate , where the graphics coincide and become straight lines. When the source drifts away from the equatorial plane (i.e. ) the dependence of on decreases for bigger values of . The second magnification centroid coordinate increases when grows and has a maximum for . As lens the angular momentum increases first decreases for the source on the left hand side of the optical axis and then increases for a source on the opposite side of the axis. When the source is at the point () the magnification centroid curves coincide with those for the static case. The magnification centroid is plotted in the Fig. 7 over the image plane . For different values of the lens parameters and and also for the scaled angular coordinates of the source and the graphics resemble the behavior of .

## V Exact numerical investigation of the deflection angle

Let us impose the condition and set the light ray on the equatorial plane. In this case, if we substitute as a new radial coordinate and measure all distances in units we obtain the reduced metric in the form

 ds2=A(x)dt2−B(x)dx2−C(x)dϕ2+D(x)dtdϕ. (32)

The metric components are:

 A(x)=(1−1γx)γ, (33) B(x)=x2[x2+a2−xγ](1−1γx)1−γ, (34) C(x)=a2[2−(1−1γx)γ]+x2(1−1γx)1−γ, (35) D(x)=2a[1−(1−1γx)γ]. (36)

The relation between the impact parameter (the perpendicular distance from the lens to the tangent to the null geodesic of the source) and the distance of closest approach of the light ray can be obtained from the conservation of the angular momentum of the scattering process, and it is given by

 J(x0)=a(1−1γx0)γ−a+√x20+a2−x0γ(1−1γx0)γ. (37)

The sign in front of the square root is chosen to be positive when the light ray is winding counterclockwise. For the black hole rotates counterclockwise, while for the black hole and the photons rotate in converse direction.

A starting point of the strong field limit expansion is the photon sphere, which has been defined by Virbhadra and Ellis Vir1 () (and has subsequently been investigated by Claudel, Virbhadra and Ellis PSphere ()) as a time-like hypersurface on which the light bending angle becomes unboundedly large when the closest distance of approach tends to . The photon sphere equation for a stationary, axially-symmetric metric is

 A0C′0−A′0C0+J(A′0D0−A0D′0)=0, (38)

for which we require to admit at least one positive solution. For the Kerr-like metric the photon sphere equation takes the form

 4γ3x4−8(1+γ)γ2x3+γ(1+2γ)(5+2γ)x2−[(1+2γ)2+8a2γ3]x+4a2γ2(1+γ)=0. (39)

The biggest real root of this equation external to the outer Kerr black hole horizon or to the curvature singularity defines the radius of the photon sphere . The radius of the photon sphere is computed numerically and is plotted in Fig. 8 for different values of the ratio . The analytical investigation of the fourth order algebraic Eq. (39) shows the kind of the naked singularity. Depending on whether or not the naked singularity is covered within a photon sphere Virbhadra and Ellis Vir2 () classify the naked singularity as weakly naked (WNS) (those contained within at least one photon sphere) and strongly naked (SNS) (those not contained within any photon sphere). In the particular case of when the Kerr black hole is recovered, the photon sphere exists for angular momentum . In the case of rotating naked singularities when the photon sphere exists for angular momentum , while for the naked singularity allows a photon sphere for angular momentum . Here is defined by

 am=√2γ(9γ−√45γ2−192γ6+144γ4+3)12γ2. (40)

An intermediate position takes the static case and , when the so called marginally strongly naked singularity (MSNS) is realized. In this case there is no photon sphere, but the gravitational lensing leads to the appearing of relativistic images. We do not show the numerically investigation to this situation, which is qualitatively similar to the static WNS.

The Kerr black hole and WNS photon spheres decrease with the increase in and have a similar behavior for different scalar charges. In the case of the Kerr black hole () when reaches its extremal value, the photon sphere coincides with the black hole horizon. With the increase of the photon sphere and the curvature singularity increase as for when the singularities are weakly naked and meet at and . Moreover, for the bigger values of the scalar charge the photon sphere in the static case and therefor in this cases the singularity is strongly naked. The different values of the upper limit of WNS angular momentum, for which a photon sphere exists and the limits of the photon sphere corresponding to them can be seen on Fig. 8 and are also written in Table 1.

The bending angle of a light ray in a stationary, axially-symmetric space-time, described by the line element (32) is given by

where is the total azimuthal angle. With the decrease of the distance of closest approach the deflection angle increases, and for a certain value of the deflection angle becomes , so that the light makes a complete loop around the lens. Let decreases further, then the light ray will wind several times around the lens before reaching the observer and finally when becomes equal to the radius of the photon sphere the deflection angle will become unboundedly large and the photon will be captured by the lens object.

Let us discuss the behavior of the bending angle for the Kerr black hole, WNS and SNS. One has for all values of and the angular momentum , for , (Kerr black hole) and , and , (weakly naked singularities) and for , and , (strongly naked singularities). is the minimum of the deflection angle for SNS, which is a function of the angular momentum and the scalar charge and is given by

 limx0→xcs~α(x0) = −π+2aγ√1−4a2γ2ln∣∣∣1+√1−4a2γ21−√1−4a2γ2∣∣∣,4a2γ2<1, (42) = −π+4aγ,4a2γ2=1, (43) = −π+2aγ√4a2γ2−1[π−2arctan1√4a2γ2−1],4a2γ2>1. (44)

In Fig. 9 we plot the behavior of the bending angle against the impact parameter (scaled in terms of ) for different values of the scalar charges and the lens angular momentum . In the top left, top middle and middle left figures we plot the curves respectively for and a scalar charge (WNS) with a step , and a scalar charge (WNS) with a step and for and a scalar charge (WNS) with a step . It is obvious that they have similar qualitative behavior for the different scalar charges; the bending angle strictly increases with the decreases of the impact parameter and becomes unboundedly large as the impact parameter approaches the respective value it has on the photon spheres (i. e ). In Fig. 9 (bottom left) we plot the curves for and scalar charge (WNS) with a step ; for all curves but the uppermost () the deflection angle changes its sign as it first increases up to its maximum with a decrease in the impact parameter, after that becomes null with the further decreases of the impact parameter and finally, after reaching its minimum becomes unboundedly large as . In Fig. 9 (top right) we plot curves for positive angular momentum and a scalar charge with a step and without the uppermost (WNS) these are for SNS. first increases with the decrease of the impact parameter and further decreases to the minimum value as the impact parameter approaches the impact parameter for their respective (i. e. ). As there are no photon spheres for SNS the deflection angle for these cases are never unboundedly large. Therefore the gravitational lensing by SNS would not give rise to relativistic images. In Fig. 9 (center, middle right, bottom middle, bottom right) we plot for some other values of the angular momentum and the scalar charge for SNS. translates its maximum to the right at the bigger impact parameters and decreases with the increase of the angular momentum or of the scalar charge .

## Vi Gravitational lensing by Kerr black hole and rotating weakly naked singularities in strong deflection limit

In this section we study the gravitational lensing by Kerr black hole and weakly naked singularities which have a photon sphere. We find the logarithmic behavior of the deflection angle around the photon sphere in the space-time under consideration and discuss the lensing observables.

### vi.1 Deflection angle in the stong deflection limit

Considering equatorial light ray trajectory we can find the behavior of the deflection angle very close to the photon sphere following the evaluation technique for the integral (V) developed by Bozza Bozza1 (). The divergent integral is first split into two parts one of which contains the divergence and the other is regular. Both pieces are expanded around and with sufficient accuracy are approximated with the leading terms. At first, we express the integrand of (V) as a function of two the new variables and which are defined by

 y=A(x), (45) z=y−y01−y0, (46)

where .

The whole azimuthal angle then takes the form

 ϕf(x0)=∫10R(z,x0)f(z,x0)dz, (47)

where the functions are defined as follows