Strong deflection limit of black hole gravitational lensing with arbitrary source distances

Strong deflection limit of black hole gravitational lensing with arbitrary source distances

V. Bozza, G. Scarpetta Dipartimento di Fisica “E.R. Caianiello”, Università di Salerno, via Allende, I-84081 Baronissi (SA), Italy.
Istituto Nazionale di Fisica Nucleare, Sezione di Napoli.
International Institute for Advanced Scientific Studies, Vietri sul Mare (SA), Italy
July 23, 2019
Abstract

The gravitational field of supermassive black holes is able to strongly bend light rays emitted by nearby sources. When the deflection angle exceeds , gravitational lensing can be analytically approximated by the so-called strong deflection limit. In this paper we remove the conventional assumption of sources very far from the black hole, considering the distance of the source as an additional parameter in the lensing problem to be treated exactly. We find expressions for critical curves, caustics and all lensing observables valid for any position of the source up to the horizon. After analyzing the spherically symmetric case we focus on the Kerr black hole, for which we present an analytical 3-dimensional description of the higher order caustic tubes.

Relativity and gravitation; Classical black holes; Gravitational lensing
pacs:
95.30.Sf, 04.70.Bw, 98.62.Sb

I Introduction

Black holes have always attracted the interest of all physicists who hope to see General Relativity at work in a completely non-perturbative regime, outside any Post-Newtonian expansion. Since most of the information we receive from black holes and their surroundings is in the form of electromagnetic waves, one of the fundamental problems to be faced is the propagation of such waves in a black hole spacetime. The situation can be possibly complicated by an accretion disk formed by neutral plasma BroBla (). For wavelengths at which observations are typically lead, the geometrical optics approximation provides a very robust description for the propagation of the light rays, defined as the lines orthogonal to the wavefronts. In all situations in which the plasma physics has little effect on the rays trajectories, the light rays simply follow null geodesics. Then all questions that involve the propagation of an electromagnetic signal require integration of the null geodesics equation. In the case of Kerr metric, the null geodesics have been expressed by Carter in terms of first integrals through the separation of the Hamilton-Jacobi equation Car (). Then, the integrals involved in these geodesics can be solved in terms of elliptic functions (see e.g. RauBla (); CFC ()). Although these analytical solutions are not particularly illuminating by themselves, they can be successfully employed to build fast and accurate numerical codes Vie (); RauBla (); CFC (); BecDon (), by which one can get particular answers, such as the shape of the iron K-line or the appearance of the accretion flow into the black hole in a future hypothetical high-resolution image Lum (); Accret (); BroLoe ().

The problem of finding the null geodesics connecting a source to an observer in a curved background is usually referred to as gravitational lensing. It has been pointed out by many authors that in a black hole spacetime there are infinitely many possible trajectories for the photons emitted by a point-source to reach the observer Dar (); Lum (); Oha (); Nem (); VirEll (); BCIS (); HasPer (). For each of these trajectories the observer will detect an independent image of the original source. The images can be classified according to the number of loops performed by the corresponding photons around the black hole. One starts from the primary and secondary image, which are formed by photons performing no loops. These are already present in the classical weak deflection limit of gravitational lensing. Besides these, there are two infinite sequences of higher order images, formed by photons performing one or more loops around the black hole before reaching the observer. These images are progressively fainter and appear closer and closer to the apparent shadow cast by the black hole on the sky.

The higher order images contribute much less than the primary and secondary images to the total flux and are often completely masked in situations in which they are not separable from the main images Pet (). Therefore, the best chances to observe higher order images are offered by a black hole as massive and close to us as possible, so that its apparent angular size is the largest. The natural candidate is the black hole at the center of our Galaxy, identified with the radio-source Sgr A*. This is believed to be a supermassive black hole slowly accreting material from the surrounding environment MelFal (); Eis (). Its distance is kpc, so that the Schwarzschild radius of this black hole spans an angle of roughly 9 as in our sky. A resolution of this order of magnitude is needed to detect the higher order images, besides the requirement of negligible absorption in the wavelength of the emitted signal by the surrounding material. In spite of these difficulties, the detection of higher order images of sources around Sgr A* should be at hand of future interferometers operating in the sub-mm range, where one expects to detect higher order images of bright spots on the accretion disk BroLoe (), or in the X-ray band, where Low Mass X-Ray Binaries and other sources are active Muno (); RauBla (); KerObs (); KerGen (). Such images would be invaluable witnesses of the strong gravitational field around the supermassive black hole at the center of our Galaxy; their direct observation would thus be of striking importance in the confirmation of our gravitational theory.

In addition to the considerations about their importance, higher order images can boast two advantages with respect to lower order ones: if the black hole has non-negligible spin, they can easily form large arcs and additional images because their caustics have a larger angular extension compared to the first caustic; secondly, they enjoy a relatively much simpler analytical description.

The treatment of higher order images can take advantage of the fact that the deflection diverges logarithmically when the impact parameter reaches a minimum value. Then, the higher order images can be obtained by a simplified lens equation where the deflection terms are replaced by the first terms of their expansions in terms of the impact parameter. This procedure is analogous to the weak deflection limit but sets its starting point in the opposite regime and is thus conventionally called strong deflection limit. It was firstly used by Darwin in 1959 for the Schwarzschild black hole Dar () and then revived several times Lum (); Oha (); Nem (); BCIS () until it was generalized to all spherically symmetric black holes Boz1 (). This method was then applied to several interesting black hole metrics, also motivated by string and alternative theories Spheric (); BorInf (); BHdS (). The time delay calculation for higher order images was done in Ref. TimDel (). The method was recently extended to the presence of external shear fields in a setup analogous to the Chang & Refsdal lens BozMau (). The extension to Kerr metric has required several steps, starting from the purely equatorial case BozEq () to the case of generic trajectories with equatorial observers KerObs (), and finally to the general case KerGen () (in the latter two works, the treatment is however limited to the second order in the black hole spin). An application to the Kerr-Sen dilaton-axion black hole has also been performed KerSen (). Recently Iyer and Petters have found an alternative expansion parameter that significantly reduces the discrepancy between the strong deflection limit and the exact deflection formula IyePet (). They have also explored the next to leading order terms in the strong deflection expansion.

The strong deflection limit allows a simple analytic investigation of the gravitational lensing properties of any black hole metric in a well-defined limit, in which the results are easily comparable from one metric to another. For the Kerr metric it has been able to provide the first analytical formulae for the caustics and the critical curves involving higher order images and for lensing of sources near caustics. However, in the formulation used up to now, it has just been applied to sources very far from the black hole, so that the limit has been taken in all equations (here is the distance of the source from the black hole and is the Schwarzschild radius of the black hole).

The purpose of this paper is to remove the limitation to very far sources, enlarging the investigation of gravitational lensing in the strong deflection limit to sources placed at arbitrary distances from the black hole. We will thus be able to discuss the mathematical structure of the lensing problem and all the lensing observables taking as an additional parameter. We will show that the strong deflection limit is well-defined even for sources inside the photon sphere, so that our discussion can be safely pushed up to sources lying just outside the horizon of the black hole. Similarly, in the Kerr metric we will be able to describe the caustic hypersurface from infinite radial distance up to the horizon.

This paper is structured as follows. Sect. II contains the new outline of the strong deflection limit for spherically symmetric black holes with arbitrary source position; it analyzes the lens equation and observables and discusses the Schwarzschild metric as a simple example. Sect. III contains the extension of Kerr black hole lensing to arbitrary source distances, with some details moved to the appendix. Section IV contains the conclusions.

Ii Spherically symmetric black holes

In this section we shall present an updated version of the method outlined in Ref. Boz1 (). Besides including the finiteness of source and observer distances from the black hole, we also make some more slight refinements that allow further generalization of the method and more physical insight. We stress the importance of the study of spherically symmetric black holes as propaedeutic to the investigation of the Kerr metric, to be tackled in the next section.

As in Ref. Boz1 (), we start from three basic assumptions on the spacetime metric:

a) The spacetime is stationary and spherically symmetric, so that the line element can be written in the form

(1)

b) We assume that our metric is asymptotically flat, so that for we have

(2)

c) Furthermore, we assume that the function has one minimum at , corresponding to the radius of the photon sphere CVE ().

In some gravitational theories, the photons do not follow geodesics of the background geometry, but the self-interaction makes them follow geodesics with respect to some effective metric BorInf (). These particular cases can fit into our treatment, provided that one uses the effective metric felt by the photons. Assumption (b) can be generalized to spacetimes that are asymptotically conformal to flat, thanks to the conformal invariance of null geodesics. In this way one can include e.g. black holes with a cosmological constant BHdS ().

Figure 1: The function plotted for the Schwarzschild metric (, ). In this case the static limit is and the photon sphere radius is . The light-gray region is forbidden because it corresponds to imaginary . The dark-gray region lies below the horizon of the black hole and is uninteresting for gravitational lensing.

Let us give a closer look at assumption (c) and specifically at the function . We can note that asymptotic flatness drives the function to approach at very large . Moreover, in any metric admitting a static limit , such that , the function diverges at . So, by continuity it must have at least one minimum greater than . As a useful reference, in Fig. 1, we plot the function for the Schwarschild spacetime, where it assumes the form . Following Ref. CVE (), all stationary points of are technically photon spheres. However, maxima are not significant for gravitational lensing, since they are only accessible to locally emitted photons. We are actually interested in minima of , since they give rise to logarithmic divergences in the deflection angle, as it will be clear later. Although it is relatively easy to find metrics which also develop a maximum in (e.g. Reissner-Nordström with superextremal charge), it seems difficult to imagine a metric developing a second minimum. Such a metric would require a quite exotic source to be sustained as a viable solution of gravitational equations. Through assumption (c), we are discarding these too problematic spacetimes and stick to more reasonable metrics with only one minimum for .

Now let us start the calculation of the photon deflection. The spherical symmetry allows us to choose the equatorial plane as the plane where the entire motion of the photon takes place, so that and , where the dot denotes derivative with respect to the affine parameter.

The dynamics of the photon can be derived from the Lagrangian (see e.g. Cha () for a complete discussion of null geodesics in Schwarzschild and Kerr spacetimes)

(3)

The coordinates and are cyclic so that their conjugate momenta are constants of motion. They can be identified with the specific energy and angular momentum

(4)
(5)

We can choose the orientation of the polar axis so that .

The last constant of motion comes from the fact that the photon moves along null geodesics of the metric (1) so that . From this equation, we derive the expression of :

(6)

The angle formed by the spatial components of the photon momentum with a normalized vector tangent to a sphere centered on the black hole is

(7)

So, in any point of the photon trajectory, the knowledge of the value of the combination

(8)

allows us to calculate the angles formed by the photon with respect to the radial and tangent directions. The photon moves radially if and tangentially in points such that . It is also easy to prove that for those photons reaching the asymptotic flat region, this quantity equals the impact parameter, defined as the distance between the black hole and the asymptotic trajectory followed by the photon. On the basis of its immediate connection with the observed direction of the photon, we will eliminate in favor of , where possible.

Inversion of the radial motion can occur only at the points that make the argument of the square root vanish in Eq. (6), which correspond to points of instantaneous tangential motion by virtue of Eq. (7). However, assumption (c) states that the function has a single minimum at . So, a quick look at Fig. 1 convinces that the equation

(9)

has real roots only if , with

(10)

where we have introduced the short notation and similarly for and .

At this point it is convenient to distinguish the case (source outside the photon sphere) from the case (source inside the photon sphere). We shall finally find that they both lead to the same expression for the deflection of a photon in the strong deflection limit, given by Eq. (51). The next two subsections deal with the details of the calculations in the two mentioned cases.

ii.1 Source outside the photon sphere

Let us analyze all possibilities for the radial motion of a photon emitted by a source outside the photon sphere.

Some photons will leave the source with positive . Since the photons are emitted at , these photons never meet inversion points and run towards the asymptotic region without experiencing any effective deflection by the black hole. If some of them reach the observer and give rise to the primary image, which is not the subject of our analysis anyway.

Some other photons leave the source with negative . If , there is still the possibility that some of them can reach the observer without inverting their motion and form the primary image. But some other photons do not meet the observer and run towards the black hole. The photons with will inexorably sink into the black hole, since Eq. (9) will admit no real roots. If , the photons invert their motion at the largest root of Eq. (9), which we indicate by and identify with the closest approach distance. After the inversion in the radial motion, these photons go back towards the asymptotic region and eventually reach the observer, giving rise to the secondary and higher order images.

All these considerations can be summarized by saying that light rays shot at too small impact parameters are swallowed by the black hole, whereas those shot at larger impact parameters are just deflected, the limiting value of being . Our objective is to quantify the deflection of these photons as a function of .

The azimuthal shift of the photon is

(11)

where we have separated the motion of the photon into approach phase (with running from to the inversion point ) and departure phase (with running from to ). In the first integral we use the expression for with the minus sign, in the second integral we use the expression with the plus sign. Using Eqs. (5) and (6), and reversing the extrema in the first integral, we have

(12)
(13)

with the short notation and .

Note that the integrand diverges at , which is defined as the largest root of the last factor. In order to study the character of the divergence, it is opportune to perform a detailed analysis of the function

(14)

which governs the divergence of the integrand in the lower extremum. From the previous discussion, we know that has a minimum at for any fixed value of ; it vanishes at by the definition of ; it vanishes at by the definition of . These properties can be formalized by the equations

(15)
(16)
(17)

Since we are interested into those trajectories whose inversion point is very close to the minimum , we define a parameter by the equation

(18)

Correspondingly, also the impact parameter must be very close to the minimum. We thus define the parameter by the equation

(19)

Then, inserting (18) and (19) into Eq. (17) and expanding to the lowest order in and , we have

(20)

The first two terms vanish because of Eqs. (15) and (16) and we are left with a simple relation between and , which tells us how much the inversion point differs from the photon sphere radius , when we increase the impact parameter of the photon from the minimum value to . Given the form of it simply reads

(21)

with

(22)

where the prime denotes derivative with respect to the argument and the subscript means that the result must be evaluated at as usual. Thus we obtain that is of the same order as .

Let us analyze the behavior of when is very close to . We introduce the parametrization

(23)

with , and study for small values of . Expanding to the second order in , , and first order in , we find

(24)

Again, the first two terms vanish because of Eqs. (15) and (16). Moreover, the -term cancels with the remaining -term because of Eq. (20). The behavior of for small is thus

(25)

Now that we have found the dominant terms in for close to , which corresponds to close to , we can return to the integral in Eq. (13). Changing the integration variable from to , the integration ranges become , where . Each integral assumes the form

(26)

Now we add and subtract a term containing the divergence of the integrand at small values of . We then separate the integral into two parts:

(27)
(28)
(29)

Of course, the sum of the two integrals and is just the original integral (26), but now the first integral is elementary and reads

(30)

The divergence for appears explicitly in , while is the integral of a regular function and does not diverge any more for . Fig. 2 illustrates an interesting comparison between the original integrand of Eq. (26) and the integrand of in Eq. (28) taking the special case of Schwarzschild metric as an example. Although the integrand of is a drastically simplified form of the original one which approximates it for very close to zero, it turns out to be a very good approximation in the whole range of . Only for close to 1 we see a sensible difference. Such difference is stored in the integrand of . As tends to zero, the contribution by becomes more and more dominant with respect to that of as the divergence of the integrand becomes stiffer and stiffer. These considerations are a good premise to the strong deflection limit.

Figure 2: The plot illustrates the integrand of in Eq. (26) as a function of for () in the case of Schwarzschild black hole; the dashed line is the integrand of in Eq. (28), the difference between the two curves being the integrand of in Eq. (29).

Until now we have done no approximation. We have just added and subtracted some terms and made some changes of variables. The expression (27) is still exact. The strong deflection limit amounts to save the first dominant terms in the expressions for and as . We have a logarithmic divergent term in and then some terms converging to constant values. We also note that the parametrization (23) tends to

(31)

when . Consequently, the integration limits tend to .

After the truncation of the expansion in , we have

(32)

where the coefficients and are given by

(33)
(34)
(35)

where is just the integrand of in the limit for , also implying . The function has been introduced only for uniforming the expression of to the corresponding terms we shall derive in section II.2 in the case of a source inside the photon sphere. For a source outside the photon sphere, the variable is always positive. Finally, the deflection suffered by the photon is quantified by the full azimuthal shift , given by

(36)

When the source and the observer are very far from the black hole, it makes sense to define a deflection angle as the difference between the azimuthal shift suffered by the photon minus , which represents the total azimuthal shift of a photon travelling in a flat space without the black hole on a rectilinear trajectory. This concept becomes ill-defined for sources and observers that are not in the asymptotic flat region.

The fact that source and observer are at finite distances is encoded in the presence of and . Setting them to 1, the deflection angle so derived coincides with the expression originally given in Ref. Boz1 ().

With the definition of given by Eq. (22) and Eq. (14), we can formulate an explicit expression for the coefficient of the logarithmic term in terms of the metric functions

(37)

This coefficient is independent of the source and observer positions.

ii.2 Source inside the photon sphere

Now let us consider the case in which the source is inside the photon sphere, but still outside the horizon. The photons leaving the source with negative sink into the black hole. As for the photons leaving with positive , we have two possibilities: those starting with meet an inversion point before reaching the photon sphere radius . Therefore, they fall back into the black hole. The photons with meet no inversion point and escape towards the asymptotic region. The observer will thus see the deformed images of a source inside the photon sphere.

Recalling the relation between and the angle formed by the photon momentum with the tangential direction (Eq. (7)), we can re-interpret this discussion noting that only photons shot along the radial direction () or very close to the radial direction () will be able to escape to infinity. Photons shot at larger angles with respect to the radial direction invert their motion before crossing the photon sphere. It is interesting to note that the angle formed with the tangent direction by photons emitted at decreases until they cross the photon sphere. After that moment, they align more and more with the radial direction as they move farther and farther from the black hole.

At first sight, one may think that this situation is very different from the one described in the previous subsection. Actually, even in this case it is possible to define a strong deflection limit, corresponding to photons with just slightly smaller than . Let us see this in detail.

The azimuthal shift of the photon is

(38)

with . Even if the function never vanishes, it becomes minimum at . Correspondingly, the integrand has a maximum at this point and is largely dominated by this peak at if is very close to . So it is convenient to revisit the analysis of the function .

Now we have to keep in mind that , so that the parametrization (19) yields . As pointed before, the function has no real roots when , and in fact, Eq. (21) with gives an imaginary value for , so that the inversion point is no longer a real number. Moreover, it is again convenient to introduce the parametrization (31), but this time extended to corresponding to . Thus now the variable ranges in the interval .

We can now study the function for small values of as in the previous subsection and find

(39)

As usual, the first two terms vanish because of Eqs. (15) and (16). As for the -term, we can replace it by the corresponding -term through Eq. (20). We just have to keep in mind that now . The behavior of for small is thus

(40)

Returning to the integral (38), we can change the integration variable from to using Eq. (31) to get

(41)

where the integration extrema are already in the form . Note that, since the source is inside the photon sphere, we have .

As before, we add and subtract a term that contains the main structure of the integrand, that is the peak at , corresponding to . We have

(42)
(43)
(44)

The first integral is again elementary and reads

(45)

where we have made the sign of explicit. The second integral contains an integrand that is regular everywhere for . Fig. 3 illustrates a comparison between the integrand of Eq. (41) and the integrand of in the Schwarzschild case. We see that even if there is no divergence for , the integrand has a very pronounced peak that dominates the integral. The peak structure is catched by , while the wings are corrected by the contribution of . As , the peak grows larger and larger, dominating the wings.

Figure 3: The integrand of in Eq. (41) as a function of for (), compared to the integrand of in Eq. (43) in dashed style.

Now we make the strong deflection limit approximation, by requiring that and consequently is small. Saving the logarithmically divergent term and the constant terms, we have

(46)

where the coefficient is still given by Eq. (33) and is

(47)

with still given by Eq. (35). It is interesting to note that the argument of the logarithm remains positive, since both and are negative.

We can split the integral in Eq. (47) in two parts

(48)
(49)
(50)

We observe that and have the same formal expression as and .

ii.3 Deflection and higher order images formulas

As a result of the previous two subsections, we have a unique expression for the photon deflection, which can be conveniently stated in terms of , which represents the impact parameter shift from the minimum value (see Eq. (19)). By Eq. (36) and (21) we finally get

(51)

where we have defined the coefficient

(52)

For quick reference, is defined in Eq. (19), is given by Eq. (10), is given by Eq. (22), is given by Eq. (33), ; and are given by the integrals (34), where is found in Eq. (35) and in Eq. (14), having changed the integration variable from to through the parametrization (31).

This expression for the total deflection of the photon is valid for any position of observer and source. Even if we have not explicitly considered it, time-reversal symmetry warrants that Eq. (51) is valid even in the unrealistic case of an observer inside the photon sphere (provided that one correctly relates to the sky coordinates of an observer in a strongly curved region). The only approximation performed is that the impact parameter is very close to the critical value .

The general lens equation for spherically symmetric black holes can be simply stated as

(53)

Fixing the origin of the azimuthal coordinate in such a way that and using the expression of the total deflection in the strong deflection limit (51), we can easily solve the lens equation and find the position of the images. In general we have

(54)

where denotes the number of loops done by the photons before reaching the observer and . Of course, the strong deflection limit becomes exact in the limit but is typically a very good approximation already for , as will be shown in the next subsection.

For an observer in the asymptotic region, which is the most physically interesting case, the angular separation between the direction of arrival of the photon and the direction of the black hole is simply . So, we have

(55)
(56)

is usually called the angular radius of the shadow of the black hole, since all images of sources outside the photon sphere reach the observer from angles and the region within the angular radius appears empty. However, when the source is inside the photon sphere, and then . We thus have and the sequence of images will appear within the shadow of the black hole, with the lowest order ones closer to the center and the higher order ones closer and closer to the shadow border.

The study of the Jacobian of the lens equation confirms that the critical curves are simply given by (55) and (54) with for standard lensing for retrolensing. They are circles outside the shadow for sources outside the photon sphere and inside the shadow for sources inside the photon sphere. The caustics are always pointlike and are located behind and in front of the source. Altogether, the caustics cover the whole optical axis as varies from to the static limit .

It is interesting to take the limit of the total azimuthal shift for and calculate the first order in . Recalling that , we have

(57)

can be calculated using the asymptotic limit of all metric functions evaluated in that appear in its expression and taking the limit for . It is simply

(58)

Summing up, we get

(59)

where is the deflection angle calculated on the asymptotic trajectories (), is defined in Eq. (56) and

(60)

is the angular size of the shadow of the black hole as measured by a distant source. The lens equation then becomes

(61)

The first correction to the lens equation with far source and observer is thus universal and simply takes into account the geometry of the lensing problem (compare with the discussion of the lens equation in Ref. S2 ()).

ii.4 Testing the formulas in the Schwarzschild case

In this subsection, we shall specify all our general formulae for black hole gravitational lensing in the case of the simplest possible metric. This will allow us to understand the sense, the validity and the power of the strong deflection approximation throughout the range of possible source positions.

The Schwarzschild metric reads (with )

(62)
(63)
(64)

The minimum of the function is at . Correspondingly the minimum impact parameter as well-known Dar (); Cha ().

Now let us calculate the coefficients of the deflection formula in the strong deflection limit. As already noted after Eq. (37), the coefficient of the logarithmic term is independent of the source and observer positions. So, Eq. (37) simply gives the already known value

(65)

This can be expected, since the logarithmic term is a characteristic of geodesics winding around the photon sphere which does not depend on the start and arrival point.

The constant coefficient in the deflection formula is

(66)
(67)

Putting everything together, we have

(68)

which reduces to the well-known formula Dar ()

(69)

in the limit . Eq. (68) is the generalization of Darwin’s formula (69) to sources and observers at finite distance from the black hole, the only approximation remaining (see Appendix A for a discussion of formulae expressed in terms of alternative perturbative parameters). In order to test our formula for the deflection of a photon, we can use it to calculate the radius of the critical curves.

Figure 4: (a) Angular radius of the first relativistic Einstein ring relative to the black hole shadow for a source behind the black hole as a function of the source distance measured in Schwarzschild radii (, ). (b) Difference between the value of calculated by the strong deflection limit formula and the one calculated by the exact formula.

The angular radius of the critical rings is given by Eq. (54) with or . Explicitly, for the Schwarzschild metric we have

(70)
(71)

where is an even number in the retro-lensing case (), and an odd number in the standard lensing case (). The first critical curve for is created by photons experiencing weak deflection and is beyond the range of validity of Eq. (70). The critical curve with is the first retro-lensing ring, while for we obtain the first higher order Einstein ring of standard lensing.

The displacement of the first relativistic Einstein ring from the black hole shadow is shown in Fig. 4a, where it can be appreciated how stays small throughout the range of source distances, validating the strong deflection limit approximation. For , the ring radius tends to its asymptotic value. Our analytic formula for the critical curve nicely joins the region within the photon sphere to that outside the photon sphere . The divergent term in the deflection formula at forces , in order to keep finite. Since this is true for any order , we can conclude that all higher order images of a source right at the photon sphere collapse into one degenerate image appearing right at the shadow border. As , becomes negative as expected. This means that the sequence of Einstein rings for a source inside the photon sphere is reversed: the brightest rings will appear closer to the center of the shadow and the fainter will be farther, approaching the shadow border as grows more and more.

The Schwarzschild metric is simple enough to allow an exact integration of the azimuthal motion in terms of elliptic integrals. It is thus very important and instructive to compare the critical curve radius calculated by the formula (51) obtained in the strong deflection limit to the exact Einstein ring position, calculated using the exact deflection angle. The difference between the two positions is plotted in Fig. 4b, where it can be appreciated that it stays of the order throughout the range of , testifying the accuracy of the strong deflection limit as a powerful approximation to describe higher order images.

The updated analysis of extreme gravitational lensing by Schwarzschild black holes presented here can be repeated for any kind of spherically symmetric black holes, using the formulae derived in this section. It is interesting to consider that some metrics could give the same lensing observables for sources at infinity, whereas they could be distinguished in gravitational lensing of sources at small distances from the black hole. Therefore, the introduction of as a new parameter enriches the arena for the comparison of different metrics.

Iii Kerr black hole

In this section, we shall describe gravitational lensing of sources placed at arbitrary distances from a spinning black hole. With respect to the analysis of Ref. KerGen (), the finiteness of and only intervenes in the calculation of the radial integrals. As a consequence, the first three sections of Ref. KerGen (), concerning the description of the unstable circular orbit and the shadow of the Kerr black hole, remain unaffected. Their content is briefly reviewed in the following subsection. The modifications in the radial integrals are reflected in the lens equation and its Jacobian. Consequently, also the critical curves and the caustics contain a dependence on the source position. They are described in Section III.2. Finally, lensing of sources near caustics is updated in Section III.3. Throughout this section, we shall preserve the spirit of Ref. KerGen () expanding all quantities to the second order in the black hole spin . The perturbative expansion is of key-importance to keep all calculations fully analytic up to the final results. At the same time, the second order approximation proves to be very reliable up to values of the black hole spin , as noted in the comparison with numerical results KerObs ().

iii.1 Derivation of the lens equation

The Kerr metric in Boyer-Lindquist coordinates BoyLin () is

(72)
(73)
(74)

Distances are measured in units of the Schwarzschild radius () and is the specific angular momentum of the black hole, running from (Schwarzschild black hole) to (extremal Kerr black hole) in our units.

The Kerr geodesics are described in integral form by the equations

(76)

with

(78)

is the component of the angular momentum of the photon along the spin axis and is the Carter integral Car (), related to the total angular momentum of the photon (we set the specific energy to 1 by a suitable choice of the affine parameter).

An observer in the position , defines angular coordinates in the sky, such that the black hole is in with its spin projected along . A photon travelling on a geodesic characterized by constants of motion and hits a distant observer from the direction

(79)
(80)

where as usual Cha ().

iii.1.1 Shadow of the Kerr black hole

Among all photon trajectories ending at the observer, there is a family of trajectories that approach an unstable circular orbit around the black hole when traced back asymptotically in the past. This family can be parameterized by the parameter ranging from to 1. The constants of motion identifying the geodesics of this family are then given by

(81)
(82)

The radius of the unstable circular orbit asymptotically approached in the past is

(83)

Correspondingly, a distant observer sees such photons from the directions

(84)
(85)

Eqs. (84) and (85) define a curve in the observer sky as varies between and . This curve represents the border of the shadow of the black hole, in the sense that all photons emitted by a source outside the unstable circular orbits reach the observer from directions outside this border. In the Schwarzschild limit, the shadow border is simply a circle of radius , whereas for generic values of the spin, the shadow satisfies the ellipse equation

(86)

with the origin shifted rightward by

(87)

and semiaxes given by

(88)
(89)

with ellipticity

(90)

For more details on the derivation of the parameterization (83) and of the shadow border, see the deep and detailed discussion in Ref. KerGen ().

iii.1.2 Strongly deflected photons

We now introduce the following parametrization of the observer sky by

(91)

One half of the sky is covered as varies from to and varies from to . The double sign in selects which half of the sky we are covering. We are interested into strongly deflected photons, which correspond to very small values of . One may regard as the relative displacement of the photon direction from the shadow border. As , the photon spends more and more time close to the unstable circular orbit, and performs more and more loops around the black hole before emerging.

Using Eqs. (79) and (80) we can calculate the values of and identifying the geodesics of such photons. For each value of and , we can calculate the inversion point in the radial motion examining the roots of the function , defined by Eq. (78). It is immediate to see that satisfies the properties