Curious case of gravitational lensing by binary black holes: a tale of two photon spheres, new relativistic images and caustics.

# Curious case of gravitational lensing by binary black holes: a tale of two photon spheres, new relativistic images and caustics.

## Abstract

Binary black holes have been in limelight off late due to the detection of a gravitational waves from coalescing compact binaries in the events GW150914 and GW151226. In this paper we study gravitational lensing by the binary black holes modelled as an equal mass Majumdar-Papapetrou di-hole metric and show that this system displays features that are quite unprecedented and absent in any other lensing configuration investigated so far in the literature. We restrict our attention to the light rays which move on the plane midway between the two identical black holes, which allows us to employ various techniques developed for the equatorial lensing in the spherically symmetric spacetimes. If distance between the two black holes is below a certain threshold value, then the system admits two photon spheres. As in the case of single black hole, infinitely many relativistic images are formed due to the light rays which turn back from the region outside the outer (unstable) photon sphere, all of which lie beyond a critical angular radius with respect to the lens. However in the presence of the inner (stable) photon sphere, the effective potential after admitting minimum turns upwards and blows up for the smaller values of radii and the light rays that enter the outer photon sphere can turn back, leading to the formation of a new set of infinitely many relativistic images, all of which lie below the critical radius from the lens mentioned above. As the distance between the two black hole is increased, two photon spheres approach one another, merge and eventually disappear. In the absence of the photon sphere, apart from the formation of a finite number of discrete relativistic images, the system remarkably admits a radial caustic, which has never been observed in the context of relativistic lensing before. Thus the system of binary black hole admits novel features both in the presence and absence of photon spheres. We discuss possible observational signatures and implications of the binary black hole lensing.

## I Introduction

The first detection of gravitational waves was made recently by Advanced LIGO on September 14, 2015 validating the Einstein‘s general theory of relativity det1 (). It was soon followed by the second detection on December 26, 2015. These events are referred to as GW150914 and GW151226 det2 (). In both the events the gravitational waves were generated by a pair of black holes which orbited one another with decaying orbits and eventually merged to form a single black hole. A gigantic amount of energy worth solar masses was emitted in the first event GW150914 within the timescale of few milli-seconds exceeding the integrated intensity of all stars in the observable universe by two orders of magnitude. These and many other observations related to binary black holes in electromagnetic band mac (),oj287 () suggest that the system consisting of two black holes in the close vicinity can host a wide variety of remarkable phenomenon that would be interesting both from the point of view of astrophysics and fundamental physics. In this paper we explore binary black holes from a different perspective. We study gravitational lensing by a pair of black holes and show that this system can exhibit novel features which are quite unprecedented and absent in any other gravitational lensing configuration studied so far. The analysis of the gravitational lensing is also a timely from the point of view of Event Horizon Telescope and Gravity collaborations which would in near future take snapshots of supermassive black holes at the center of our and nearby galaxies Bambi (),Johansen (),Doeleman (),Lu (),Bartko (),Cunha (),Cunha1 ().

The two body problem in general relativity is extremely difficult, in contrast with Newtonian gravity. This is essentially consequence of the fact that Einstein equations are complicated coupled non-liner partial differential equations that are difficult to deal with. There are no exact solutions to Einstein equations depicting binary black holes that could be applicable in the realistic astrophysical context. In this paper we work with the Majumdar-Papapetrou solution depicting multiple black holes in the equilibrium, which could be thought of as a simple toy model that might capture some of the features in a realistic scenario. Majumdar-Papapetrou metric is perhaps the simplest multi-black hole solution known so far. It was discovered independently by Majumdar and Papapetrou majumdar (),papa () and later demonstrated to represent spacetime with multiple black holes hartle (). There were many interesting investigations that were carried out in the context of Majumdar-Papapetrou spacetime Dettmann (),Cornish (),Dettmann1 (),shadow1 (),2phs (),patil (),shadow2 (). We also note that the gravitational lensing in the context of the full-fledged numerical relativity simulation of binary black hole merger was studied in Bohn ().

We focus on the case where Majumdar-Papapetrou metric represents two identical equal mass non-rotating black holes at rest with respect to one another at certain distance. The metric is static, has a rotational symmetry around the line joining two black holes and reflection symmetry about the plane that is midway between the two black holes, which intersects line joining two black holes orthogonally. We assume that the light source and observer are located on the symmetry plane and focus on light rays which are confined to move on this plane. This allows us to employ various techniques developed to study gravitational lensing on the equatorial plane of spherically symmetric spacetimes narasimha (),virbhadra (),bozza (),bozza2 (),sahu (),sahu2 (). The central point of the symmetry plane which is also the point exactly midway between two black holes on the axis essentially acts as a gravitational lens. The photon sphere plays an important role in the relativistic gravitational lensing. Recently it was demonstrated 2phs () that the number of photon spheres on the symmetry plane depends on the distance between the two black holes. If the distance between the two black holes is small, then the symmetry place exhibits two photon spheres, one of which is unstable and the other one is stable. As the distance is increased the two photon spheres approach each other, merge and disappear. So when the distance is larger than the certain critical threshold value, there are no photon spheres on the equatorial plane. The properties of relativistic images and caustics are radically different depending on whether two photon spheres are present and absent. The existence of the stable photon sphere in the context of the axially symmetric electro-vacuum spacetimes was studied in Dolan (). We would like to mention that what we refer to photon sphere is in fact a ring rather than being a sphere since we restrict out attention on the plane midway between the black holes.

The structure of images formed due to the photons that turn back from the region above the outer unstable photon sphere is same as in the case of Schwarzschild black hole virbhadra (),bozza (). All the images lie beyond certain critical angular radius with respect to the lens. There are infinite images clubbed together close to this critical radius. In case of Schwarzschild black hole there is a dark region below the critical radius with no images, which is consequence of the fact that the light rays which enter the single unstable photon sphere never turn back and enter black hole. The situation is drastically different in case of the di-hole in the presence of second stable inner photon sphere. As we will show, the effective potential turns upward again and the photons that enter outer photon sphere can now turn back. This leads to the formation of new infinite set of relativistic images, all of which lie inside the critical radius mentioned above. The region which would have been dark in the case of a single black hole, is not dark in the presence of the two photon spheres. Again most of the images are crowded together close to the critical radius and few discrete images lie close to the lens. In the absence of the photon sphere a finite number of discrete relativistic images are formed. Interestingly the radial caustic is present in this case. This is the location where the map between image plane and source plane is degenerate. Relativistic radial caustic has never been reported before in any other investigation so far. Thus we demonstrate that new novel features arise both in the presence and absence of twin photon spheres.

We note that the system similar to the one we investigated in this paper where there are two black holes and source which could be bright spot in the accretion disk or maybe in the gas cloud falling towards black holes may not be so hard to realize in the nature. Based on our analysis and expectation that we might capture some of the features in the realistic scenario, we anticipate that the gravitational lensing signature of such a system will be characteristic and peculiar. Thus gravitational lensing signal will allow us to locate a binary black hole system in the sky and thus reduce the number of parameters in the search for gravitational waves from binary black holes in the interferometric detectors such as LIGO, VIRGO and KAGRA using matched filtering. This suggests that our investigation could possibly have serious implications for gravitational wave data analysis.

## Ii Majumdar-Papapetrou di-hole metric and photon spheres

In this section we describe the Majumdar-Papapetrou di-hole metric and null geodesics. In classical mechanics a system of particles at rest with charge equal to mass is in the equilibrium for arbitrary distance between the particles. Interestingly same is true in general relativity as well. A system of maximally charged non-rotating black holes are in equilibrium irrespective of the distance between them. The static electro-vacuum spacetime depicting this scenario is Majumdar-Papapetrou metric which is the simplest multi-black hole exact solution in general relativity and thus it has served as a toy-model for numerous analytical investigations as quoted in the introductory section.

In this paper we focus on the special case of the Majumdar-Papapetrou metric which represents two equal mass maximally charged non-rotating black holes in equilibrium at fixed distance between them. We call it a di-hole metric. The spacetime is static and asymptotically flat. It admits a rotational symmetry about the line joining two black holes and reflection symmetry about the plane that is midway between the black holes. In the cylindrical coordinate system , which is well-adapted to the symmetries, the di-hole metric is given by

 ds2=−dt2U2+U2(dρ2+ρ2dϕ2+dz2), (1)

where the metric function is given by

 U(ρ,z)=1+M√ρ2+(z−a)2+M√ρ2+(z+a)2. (2)

Each of the black holes has mass . The black holes are located at and on axis about which there is a rotational symmetry and the symmetry plane is located midway between the two black holes. In this coordinates black holes appear as points. We find it to convenient to work in the units where is set to unity and thus all dimensional quantities are appropriately expressed in units of which is half of the coordinate distance between the two black holes.

We focus our attention on the null geodesics that are restricted to move on the symmetry plane . Let be the four-velocity of the light ray. Using the standard techniques of dealing with null geodesics with the help of conserved quantities associated with symmetries of metric and normalization of velocity sahu () we obtain the non-zero components of four-velocity as

 Vt=U2b, Vϕ=±1U2ρ2, Vρ=±√1b2−1U4ρ2, (3)

where is the impact parameter, in the expression for stands for the photons moving clockwise and anti-clockwise respectively and in the expression for stands for the photons which move in radially outwards and inwards respectively. The function in the expression above is now

 U(ρ)=1+2M√ρ2+1, (4)

where is twice ratio of mass and distance between the black holes. The equation describing the radial motion of the light ray can be recast in the form

 Vρ2+Veff(ρ)=1b2  ;  Veff(ρ)=1U4ρ2. (5)

is the effective potential for radial motion. It is quite useful to use the analogy of a particle moving in potential well in classical mechanics while dealing with the radial motion of photon.

The location of the circular photon orbit i.e. photon sphere can be obtained by solving the equation which is given by

 (ρ2+1)32=2M(ρ2−1). (6)

With the substitution , the above equation can be cast into a cubic equation

 η3−2Mη2+4M=0. (7)

The existence or not of the photon spheres is determined by the discriminant of the cubic . The critical value of parameter for which discriminant is zero is given by

 M∗=√278. (8)

When , the discriminant of cubic is positive and two photon spheres are present. We obtain location of the photon spheres by solving the cubic equation with Trigonometric method Dickson (). The location of the outer photon sphere denoted by is given by

 ρout=√4M29(1+2cos[13cos−1(1−274M2)])2−1. (9)

The effective potential admits a maximum at and hence the outer photon sphere is ”unstable”. The location of inner photon sphere denoted by is given by

 ρin=√4M29(1−2sin[π6−13cos−1(1−274M2)])2−1. (10)

Since the effective potential admits a minimum at , the inner photon sphere is ”stable”. The location of photon spheres as a function of is depicted in Fig.1, which is same as Fig.4 in 2phs ().

When , two photon spheres coincide and we have a single degenerate photon sphere located at

 ρdeg=√5. (11)

Whereas in the case , the discriminant of a cubic is negative and circular photon orbits are absent.

As mentioned earlier is ratio of mass to the distance between the black holes. So the results obtained in this section imply that for a fixed mass, if the distance between the black holes is below certain critical value, two photon spheres are present on the symmetry plane midway between the black holes. As the black holes recede from another, two photon spheres come together, merge and eventually disappear above the critical distance. As we demonstrate in this paper, the gravitational lensing signature of binary black holes is radically different depending on whether the photon spheres are present. Thus the separation between the black holes for fixed mass dictates the existence or otherwise of twin photon spheres and the gravitational lensing signature of the binary black holes.

## Iii Gravitational lensing by the Majumdar-Papapetrou di-hole

In this section we describe the basics of the gravitational lensing formalism that we employ in our investigation of Majumdar-Papapetrou di-hole. We assume that the source of light and observer are located on the symmetry plane placed midway between the black holes. We restrict our attention to the light rays that are allowed to move on the plane . This allows us to employ the formalism developed for the lensing of light on the equatorial plane of spherically symmetric spacetimes. We note that the central point with on the plane , which is also the point midway between the two identical black holes, essentially acts as a gravitational lens in our investigation for all practical purposes.

We assume that both source and observer are located faraway from the central point and also from the two black holes in the asymptotic region which is approximately flat. Thus the light ray starts from infinity, falls toward the center as it gets bent, admits a turning point at and returns to infinity. The total amount of deflection suffered by the light ray in its journey is given by

 ^α(ρ0)=2∫∞ρ0|Vϕ||Vρ|dρ−π=2∫∞ρ01ρ√Veff(ρ)√Veff(ρ0)−Veff(ρ)dρ−π. (12)

While obtaining the equation above from (3) we have used the fact that radial component of velocity is zero at the turning point, i.e., and thus the impact parameter can be related to via

 b=1√Veff(ρ0). (13)

Later in the paper we employ clever techniques proposed by Bozza bozza2 () and develop their generalizations in order to get an approximate expression for deflection angle in various situations.

We further assume that the source is located approximately behind the lens with respect to the observer. This allows us to use Virbhadra-Ellis lens equation virbhadra (). We note that there are other lens equations in the literature which also work equally well in this approximation such as Virbhadra-Narasimha-Chitre lens equation proposed earlier in narasimha () and so on.

The lens diagram is depicted in Fig.2. in the diagram is the lens which is central point of plane in this context. The observer is located at . The line joining observer and lens is the optic axis. is the location of source which is almost exactly behind the lens with respect to observer. The source is located at an angle with respect to optic axis as seen by the observer. The light ray which initially travels along line gets bent in the vicinity of the lens and arrives at the observer moving along . Thus to the observer the light seems to originate from which is the perceived image. The image is located at angle with respect to the optic axis. The deflection suffered by the light ray on its journey from source to the observer is . The light can go around the lens multiple times. Hence the deflection angle can be very large. The distances between lens-image, lens-observer and source-observer are given by , and respectively.

The Virbhadra-Ellis lens equation allows us to relate the source location , image location and the deflection angle and is given by

 tanβ=tanθ−DdsDs(tanθ+tan(^α0−θ)). (14)

Another relation that we can write down from the lens diagram is

 sinθ=bDd. (15)

From Eqs.(12),(13),(14) we can write deflection angle in terms of and then use lens equation Eq.(14) to solve for the image locations for a given source location . Solving lens equation is not so easy as it is a complicated transcendental equation. One has to often resort to numerical techniques. Analytical way of solving lens equation in certain situations was proposed by Bozza in bozza2 () which we generalize and develop further in this paper. For this we use the fact that the source is almost exactly behind the source and thus angles and are small and deviation of deflection angle from multiple of is very small, i.e.,

 ^α0=2πn+δαn  ;  |δαn|<<1. (16)

This allows us to simplify lens equation Eq.(14) and Eq.(15) as

 β=θ−DdsDsδαn, (17)

and

 θ=bDd. (18)

We will employ these equations later in the paper.

The lens equation allows us to relate image location to the source location and set up a map from image plane to source plane. A radial caustic is admitted if this map is degenerate, i.e.

 dβdθ=0. (19)

We now derive an expression for . From Eq.(14), we get

 dβdθ=cos2βcos2θ [1−DdsDs{1+cos2θcos2(^α0−θ)(d^α0dρ0dρ0dθ−1)}], (20)

where and are as given below. From Eqs.(13),(15) we get

 dρ0dθ=−2Veff(ρ0)V′eff(ρ0)√D2dVeff(ρ0)−1, (21)

and from Eq.(12) after implementing few clever tricks we get

 d^α0dρ0=2∫∞ρ0⎡⎢ ⎢⎣1√Veff(ρ0)−Veff(ρ)ddρ(√Veff(ρ)ρ)−12√Veff(ρ)(V′eff(ρ0)−V′eff(ρ))ρ(Veff(ρ0)−Veff(ρ))32⎤⎥ ⎥⎦dρ, (22)

where prime denotes the derivative with respect to . Radial caustic occurs when two images for a given source location that are separated from one another along the radial sense with respect to the lens coincide, has never been reported so far in the context of relativistic gravitational lensing. We will show that the radial caustic is admitted in the absence of the photon sphere, when the distance between two black holes is larger than a critical threshold value in Majumdar-Papapetrou di-hole spacetime.

Whenever required, for numerical computation we choose the following set of convenient parameters. We assume that the source and observer are equidistant from the lens. The mass of each of the black hole in binary is taken to be same as the mass of the galactic central supermassive black hole in Milky way which is and also we take distance of observer from lens to be same as the distance of earth from central supermassive black hole, i.e. . The angular location of source is taken to be .

Three cases , and when two photon spheres are present, when a single degenerate photon sphere is present and when no photon spheres are present respectively require separate consideration and analysis and thus they are dealt with independently in the subsequent sections. Our aim is to analyze the structure of images and existence of caustic in the near-aligned configuration of source, lens and observer.

## Iv Two photon spheres

In this section we focus our attention on the case when two photon spheres are present on the symmetry plane centered at . The locations of the outer unstable photon sphere and inner stable photon sphere are given by Eqs.(9),(10). The effective potential admits maximum at and minimum at as shown in Fig.3. For lower values of , the effective potential rises upwards below the inner photon sphere for the lower radii. We note that we talk about the stability with respect to the radial perturbations restricting ourselves in plane. The situation can be different as far the stability in the two dimensional plane is concerned as analyzed in Dolan (). is a point such that the effective potential at this location is same as the effective potential at maximum, i.e. . The light rays for which turn back from . Whereas the light rays for which , enter outer photon sphere and admit turning point below , i.e. . This situation is quite different in case of the black holes where the effective potential does not rise again for lower values of radial coordinates and light rays which enter the photon sphere necessarily enter the event horizon. Thus new relativistic images are formed due to the lights rays which turn back in the region below outer photon sphere in dihole spacetime which are absent in black hole case.

### iv.1 Images due to the light rays that turn back outside outer photon sphere.

Initially we focus on the light rays that admit turning point at outside the outer photon sphere and calculate the location of the images formed. Further we assume that the deflection point is in fact very close to the photon sphere. This greatly simplifies the discussion and allows us to do calculations analytically.

We find it convenient to introduce new radial coordinate following the discussion in bozza2 () which is related to the old coordinate by

 y=(1U(ρ)−1U(ρ0))(1−1U(ρ0)). (23)

is a monotonically decreasing function. Thus it follows that increases monotonically from to as varies from to . It is also useful to invert the relation above and write in terms of as

 ρ(y)=U−1(U(ρ0)U(ρ0)+1−y), (24)

where is an inverse function of . We define a function as

 T(ρ0,ρ)=dρdy=−U(ρ)2U′(ρ)(1−1U(ρ0)). (25)

The deflection angle can be written as

 ^α0=I−π. (26)

in the expression above is

 I=∫∞ρ02ρ√Veff(ρ)√Veff(ρ0)−Veff(ρ)dρ=∫10F(ρ0,ρ(y))√Veff(ρ0)−Veff(ρ(y))dy, (27)

where is a function given by

 F(ρ0,ρ(y))=2ρ√Veff(ρ(y))T(ρ0,ρ(z)). (28)

The function is finite and well-behaved for all values of . Whereas diverges at . Taylor expanding around , we get

 Veff(ρ0)−Veff(ρ(y))=α1(ρ0)y+β2(ρ0)y2+O(y3), (29)

where and are given by

 β1(ρ0)=−12(T2(ρ0,ρ=ρ0)V′′eff(ρ0)+T(ρ0,ρ=ρ0)T′(ρ0,ρ=ρ0)V′eff(ρ0)), α1(ρ0)=−F(ρ0,ρ=ρ0)V′eff(ρ0). (30)

When turning point is away from the outer photon sphere , is a non-zero positive finite number. Thus the integral Eq(27) and the deflection suffered by the light ray is finite.

The situation is quite different when the turning point is close to the outer photon sphere. We expand and around and get

 β1(ρ0)=−12T2(ρout,ρ=ρout)V′′eff(ρout)+O(ρ0−ρout), α1(ρ0)=2β1(ρout)T(ρout,ρ=ρout)(ρ0−ρout)+O((ρ0−ρout)2). (31)

To the leading order is vanishingly small and thus the integral Eq(27) and the deflection angle show divergence. We isolate the divergence piece in the integral Eq(27) as

 I1D(ρ0)=F(ρout,ρ=ρout)∫101√α1(ρ0)y+β2(ρ0)y2dy=−A1log(ρ0ρout−1)+~B1+O(ρ0−ρout), (32)

where

 A1=F(ρout,ρ=ρout)√β1(ρout), ~B1=F(ρout,ρ=ρout)√β1(ρout)log(2T(ρout,ρ=ρout)ρout), (33)

and the regular piece in the integral can be written as

 I1R(ρ0)=∫10(F(ρ0,ρ(y))√Veff(ρ0)−Veff(ρ(y))−F(ρout,ρ=ρout)√α1(ρ0)y+β2(ρ0)y2)dy=I1R(ρout)+O(ρ0−ρout). (34)

Combining divergent part and regular parts of the integral , we can write down the deflection angle as

 ^α0=−A1log[B1(ρ0ρout−1)]−π+O(ρ0−ρout), (35)

where is related to , and by

 B1=exp(−~B1+I1R(ρout)A1). (36)

Thus the deflection angle shows logarithmic divergence as the reflection point approaches the outer photon sphere.

The impact parameter upon expansion around the outer photon sphere can be written as

 b=C1+D1(ρ0ρout−1)2+O((ρ0−ρout)3), (37)

where and are given by

 C1=1√Veff(ρout), D1=−12V′′eff(ρout)V32eff(ρout)ρ2out. (38)

Using Eqs.(18),(35),(37) we can relate to the deflection angle as

 θ=C1Dd+D1Dd1B21exp(−2A1(^α0+π)), (39)

and using Eqs.(16),(17),(39) we can compute locations of images for a given source location in the near-aligned configuration, which are given by

 θ1,n=C1Dd+D1Dd1B21% exp(−2A1(2n+1)π)(1+2A1DsDdsβ). (40)

Here stands for the number of times light ray goes around the lens during its journey from source to observer. It turns out that all the images lie beyond a certain critical angle given by

 ¯θ1=C1Dd. (41)

As increases images get closer and closer to the critical angle and asymptotically approach it from right. So far we dealt with the light rays which go around the lens in the clockwise sense. For the light rays which go around the lens in counter-clockwise sense the images occur on the opposite side of the optic axis and their locations are given by

 θ′1,n=−C1Dd+D1Dd1B21exp(−2A1(2n+1)π)(−1+2A1DsDdsβ). (42)

The pattern of the images formed due to the single black hole is qualitatively similar to the images formed in case of the di-hole due to the light rays that are reflected back outside the outer photon sphere. In case of the black hole there is a dark region below the critical radius as there are no images formed in this region. As we show in this paper that is not the case for di-hole as the light rays which turn back from the region inside photon sphere form new images filling in the void.

### iv.2 Images formed due to the light rays that turn back inside the photon sphere.

We now compute the pattern of images formed due to the light rays that enter the outer photon sphere and turn back at the radial coordinate and reach infinity again. As mentioned earlier is a location such that

 Veff(ρ1)=Veff(ρout). (43)

We use the coordinate introduced earlier Eq.(23). The deflection angle can be written as before in Eqs.(26),(27)

 ^α0=I−π=∫10F(ρ0,ρ(y))√Veff(ρ0)−Veff(ρ(y))dy−π. (44)

is finite and well-behaved everywhere . On the other hand diverges when i.e. at and at i.e. when is close to . Here is given by .

We first focus on and expand around . To the leading order we obtain

 Veff(ρ0)−Veff(ρ(y))=α2(ρ0)y+O(y2), (45)

where is given by

 α2(ρ0)=−T′(ρ0,ρ=ρ0)V′eff(ρ0). (46)

Since is a finite, positive non-zero number, divergent behavior of integrand can be tamed when we compute the integral and it does not result in the divergent behavior of deflection angle.

We now focus on and expand around . We get

 Veff(ρ0)−Veff(ρ(y))=α3(ρout)+β3(ρout)(y−yout)2+O((y−yout)3), (47)

where and are given by

 α3(ρout)=Veff(ρ0)−Veff(ρout)=−V′eff(ρ1)(ρ1−ρ0)+O((ρ1−ρ0)2), β3(ρout)=−12T2(ρ1,ρ=ρout)V′′eff(ρout)+O(ρ0−ρ1). (48)

We used Eq.(43) to derive the expression above. When is very close to , integral is divergent and so is the deflection angle .

We identify the divergent piece in the integral as

 I2D=F(ρ1,ρ=ρout)∫101√α3+β3(y−yout)2dy=−A2log(1−ρ0ρ1)+~B2+O(ρ0−ρ1), (49)

where

 A2=F(ρ1,ρ=ρout)√β3, ~B2=F(ρ1,ρ=ρout)√β3log⎛⎝4yout(1−yout)β3−V′eff(ρ1)ρ1⎞⎠, (50)

and the regular piece in the integral can be written as

 I2R(ρ0)=∫10(F(ρ0,ρ(y))√Veff(ρ0)−Veff(ρ(y))−F(ρ1,ρ=ρout)√α3+β3(y−yout)2)dy=I2R(ρ1)+O(ρ1−ρ0). (51)

While writing the equations above we set in the expression for since and hence the error involved is at higher order. Combining the divergent piece and regular piece in the integral we can write the deflection angle as

 ^α0=−A2log(B2(1−ρ0ρ1))−π+O(ρ0−ρ1), (52)

where

 B2=exp(−~B2+I2R(ρ1)A2). (53)

The deflection angle again shows the logarithmic divergence as the reflection point approaches . We now write the impact parameter , expanding it around . We get

 b=C2−D2(1−ρ1ρ0)+O((ρ0−ρ1)2), (54)

where and are given by

 C2=1√Veff(ρ1) D2=−12V′eff(ρ1)V32eff(ρ1)ρ1. (55)

Using Eqs.(18),(52),(54) we can relate to the deflection angle as

 θ=C2Dd+D2Dd1B2exp(−1A2(^α0+π)), (56)

and using Eqs.(16),(17),(56) we can compute locations of images for a given source location in the near-aligned configuration, which are given by

 θ2,n=C2Dd−D2Dd1B22% exp(−1A2(2n+1)π)(1+1A2DsDdsβ). (57)

All the images lie below the critical angle which is given by

 ¯θ2=C2Dd. (58)

As we increase , images get closer and closer to critical angle and asymptotically approach it from left. From Eqs.(38),(41),(43),(55),(58), we see that

 ¯θ1=¯θ2. (59)

This implies that the second set of images formed due to the light rays which enter photon sphere and turn back fill in what would have been a void and dark region in case of the single black hole at an angle less that the critical angle towards optic axis.

Here denote the location of images formed on the right side of the optic axis due to the light rays that move around the lens in the clockwise sense. The location of images formed on the left side of the optic axis due to the light rays which move in anti-clockwise sense are given by

 θ′2,n=−C2Dd+D2Dd1B22exp(−1A2(2n+1)π)(1−1A2DsDdsβ). (60)

We note that and must be computed numerically since the integrals Eqs.(34),(51) are difficult to evaluate analytically. It might also be necessary to compute numerically since one needs to solve transcendental equation . All other calculations presented above can be carried out analytically.

The pattern of images formed is shown in Fig.(4). The images on the right side of the optic axis i.e. lens are associated with the light rays that move clockwise and images on the left side are associated with the light rays which move in the counter-clockwise sense. In case of a single black hole light rays which enter the photon sphere are doomed to enter the black hole. Thus images are formed due to the light rays that turn back outside the photon sphere. All images occur above the critical angle and infinitely many images are clubbed together just above the critical angle as shown in the figure. No images are formed below the critical angle towards the lens resulting in the dark void. In case of the binary black hole due to the presence of inner stable photon sphere inside the outer unstable photon sphere effective potential turns upwards again and light rays which enter the outer photon sphere can turn back giving rise to another set of new relativistic images. The new images lie below the critical angle as shown in the figure. Infinitely many images clubbed just below . Thus the region which would have been dark void is filled up with images and turns bright again. Thus the pattern of images is qualitatively different in the binary black holes spacetime.

## V A single degenerate photon sphere

We now deal with the case where the parameter assumes the critical value . The two photon spheres we encountered in the earlier section when now coincide and we have a single degenerate photon sphere present at , where both the first and second derivative of the effective potential vanish, i.e., . The value of the effective potential at the location of the degenerate photon sphere is given by . The light rays admit turning point at when the impact parameter satisfies and at when the impact parameter is . We present here the unified treatment of the light rays that turn back both above and below the photon sphere. Such a treatment was not possible for since the light rays admitted turning points either at or where and were distinct points with qualitatively different behavior of effective potential .

Once again we use the new radial coordinate with the finite range introduced in Eq.(23). The deflection angle can be written as

 ^α0=I−π=∫10F(ρ,ρ(y))√Veff(ρ0)−Veff(ρ(y))dy−π. (61)

We request readers to refer to section IV-A for the definition of various quantities that we use here. The function is finite and well-behaved everywhere as earlier whereas is divergent at , i.e., at .

We expand around as

 Veff(ρ0)−Veff(ρ(y))=α4(ρ0)y+β4(ρ0)y2+γ4(ρ0)y3+O(y4), (62)

where

 α4(ρ0)= −T(ρ0,ρ=ρ0)V′eff(ρ0), (63) β4(ρ0)= −12(T(ρ0,<