arXiv:1206.3077v2 [gr-qc] 23 Aug 2012 al olpiga h n fislf yl) i.black viz. cycle), life its of contin- end star the massive at a as collapsing (such ually clouds matter collapse gravitational massive distin- of of can states lensing important end field various is between strong it how guish particular, and In whether study too field). to compact weak they various same (because of the Relativity testing have General the in weak for scenarios as same object well the it- as reproduce limit), in Rela- construction field Relativity General by of General should generalizations of tivity all test almost a regime. (because field as self in- strong important the the is in in rise This lensing great gravitational a in seen has terest however, approximation. decade, mostly field last weak reasons, The the to good confined for been have But these theoreti- observationally. been both has and lensing, there cally gravitational then of Since studies 1919. numerous predictions in verified In Relativity successfully first General gravity. the of of Newtonian one was observationally against it fact of in theory prediction helped Einstein has a testing which is Relativity lensing General gravitational of phenomenon § ‡ † ∗ lcrncades [email protected] address: Electronic [email protected] address: Electronic [email protected] address: Electronic lcrncades [email protected] address: Electronic eeto flgtb asv oisadteeoethe therefore and bodies massive by light of Deflection a toggaiainllnigdsigihnkdsingul naked distinguish lensing gravitational strong Can ASnmes 53.f 42.w 47.w 98.62.Sb 04.70.Bw, 04.20.Dw, 95.30.Sf, numbers: PACS techn and r encountere experiments dedicated where be purpose. new geometry might this that that JMN suggest difficulties with and practical images case caust the the radial out not the point is that which contras shown sphere, in was are photon the it here of where obtained existence singularities results the naked the unravel Also obse to sphere. the us photon the allow that inside principle, suggests formed in study be might, wid This can a which hole. of for black some case Schwarzschild another, ima the relativistic one is many from finitely which apart get sphere, we that photon show we the spacetime, relativ of inner absence the and Eins the axis and in optic images from angle relativistic critical many spaccertain infinitely this with z of hole with signature T black fluid lensing a radius. gravitational of finite sphere collapse a gravitational photon of at state exterior end the th possible to at w singularity metric Schwarzschild naked which soluti a the spacetime contains symmetric and The spherically pressure a anisotropic represents holes. sing which black naked metric from JMN not distinguished or be whether investigate can to censorship, cosmic nti ae esuygaiainllnigi h togfie strong the in lensing gravitational study we paper this In .INTRODUCTION I. aybaaSahu Satyabrata oiBah od ubi400,India 400005, Mumbai Road, Bhabha Homi aaIsiueo udmna Research Fundamental of Institute Tata ∗ adrPatil Mandar , † .Narasimha D. , 1,1] h tde h eairo ulgeodesics and null hole black of Schwarzschild behavior of regime the field studied strong who in 12], question. [11, same the address of to position recently [8– done the spacetimes been Tamimatsu-Sato and has and 10] shape Kerr [7] in the spacetime shadow of JNW the investigation of the generalization rotating and well as the [6] strong for in geometry lensing as the Kerr 5], that for [4, framework here spacetime post-Newtonian note lensing JNW We in gravitational lensing the perspective. singular- gravitational explore naked a we such the paper of from this existence In either the to out ities. observations rule the or the- with confirm computed compared singularities are naked oretically the of of consequences occurrence a the the take where could approach, one Thus an phenomenological in scenario. equations it Einstein realistic as the astrophysically solve investigations is to theoretical difficult otherwise purely extremely or is from occurrence infer Their to singularities hard nature. naked Thus in 3]. occur [2, starting might data field initial matter regular reasonable a col- of from cloud gravitational naked matter continual as a well a recently of as lapse in studies holes formed black many are the were that singularities after shown There decades was it several forward. where even put the proved was censorship in yet cosmic it singularities not the naked However is the [1]. conjecture of us rid around get world real to order in rose conjecture. from censorship important cosmic also the is of This perspective the singularity. naked and hole al ok nsrn edlnigwr yDarwin by were lensing field strong on works Early Pen- by proposed was conjecture censorship Cosmic si mgsalcupdtgte.However, together. clumped all images istic tm sietclt hto Schwarzschild of that to identical is etime r ailpesr.I h rsneo the of presence the In pressure. radial ero etr M emtyi ace with matched is geometry JMN center. e e n isenrnssae reasonably spaced rings Einstein and ges enrns l fte oae eoda beyond located them of all rings, tein ihteeririvsiaino JNW on investigation earlier the with t ae iglrt nteasneo the of absence the in singularity naked vto frltvsi mgsadrings and images relativistic of rvation nt h isenfil qain with equations field Einstein the to on lrte,i talte xs nnature, in exist they all at if ularities, nteosraino h relativistic the of observation the in the d absence the in present always is ic i ercwsrcnl hw ob a be to shown recently was metric his qe aeb eeoe nftr for future in developed be have iques ag fprmtrvle nthis in values parameter of range e rtclagefrtecorresponding the for angle critical xlr rmti esetv is perspective this from explore e ‡ dlmtfo h esetv of perspective the from limit ld da asi sasn.W also We absent. is caustic adial akjS Joshi S. Pankaj , rte rmbakholes? black from arities § 2 pointed out the divergence of Einstein deflection angle exactly identical to Schwarzschild black hole case while as the distance of closest approach of the geodesics ap- in the absence of photon sphere it is greatly different. proaches photon sphere. Strong field lensing with a lens In this work, the galactic supermassive compact object equation was studied by Virbhadra and Ellis [13], who ex- is analyzed as a strong gravity lens to illustrate these amined strong field lensing in Schwarzschild black holes characteristics. and showed that there could in principle be infinite rel- This paper is organized as follows. In section II we ativistic images on each side of the black hole when a introduce the basic formalism in brief. In section III we light ray with small enough impact parameter ( distance discuss the lens model with galactic supermassive dark of closest approach close enough to photon sphere) goes object as the lens and in section IV we discuss the lensing around one or several times around the black hole be- signatures when it is modeled as a Schwarzschild black fore reaching the observer. Earlier, lens equation for hole. We discuss the naked singularity spacetime we in- spherically symmetric static spacetimes that goes beyond tend to study and lensing in this background in V and the weak field small-angle approximation was studied by compare this with Schwarzschild back hole and JNW so- Virbhadra ,Narasimha and Chitre in [4]. The Virbhadra- lution in VI & VII respectively. In section VIII, we dis- Ellis type lens equation has also been applied to boson cuss the implications of going beyond point source ap- star by D¸abrowski and Schunck [14], to a fermion star proximation for our study and in IX we briefly discuss by Bili´c, Nikoli´cand Viollier [15]. As one of the first how binary systems could be useful for probing question steps towards using strong field lensing to probe the cos- of cosmic censorship via gravitational lensing. Finally, mic censorship question, Virbhadra and Ellis have used we discuss the main results and conclude with a general this lens equation to study and compare gravitational discussion in section X. lensing by normal black holes and by naked singularities modeled by the Janis, Newman, Winicour metric (JNW solution)[5]. II. BASIC FORMALISM It is worthwhile to extend this line of work to other novel, more interesting and if possible more realistic In this section we review the standard gravitational naked singularity models. With this in mind we con- lensing formalism [4, 13] used in this paper to compute sider here the class of solutions recently obtained by the location and properties of the images. Joshi, Malafarina and Narayan [16] as end state of cer- We assume that the spacetime under consideration tain dynamical collapse scenarios in a toy example. JMN that is to be thought of as a gravitational lens is spher- metric is a solution of Einstein field equations with an ically symmetric, static and asymptotically flat. We as- anisotropic pressure fluid and has a naked singularity at sume that the source and the observer are located suffi- the center. It is matched to the Schwarzschild metric ciently far away from the lens so that they can be taken to at a certain radius. We refer to it here as JMN naked be at infinity for all practical purposes. We also assume singularity from now on. It is worthwhile to mention that the source is a point-like object, although towards that, not only the presence of the central naked singular- the end of the paper we describe how the results based on ity but also the value of the radius at which the interior the point source assumption would change if the source solution is matched to exterior Schwarzschild geometry has a finite extent instead of it being point-like. We as- plays a crucial role in determining gravitational lensing sume that the geometrical optics approximation holds observables. good. However we note that if we go arbitrarily close We should also mention that exact lens equations were to the singularity, the Riemann curvature might become proposed by [17] for arbitrary spacetime and also by [18] comparable to the wavelength of the light leading to the for spherically symmetric case. Bozza et al. have defined breakdown of the geometrical optics approximation. and analytically calculated strong field limit observables The gravitational lensing calculations has two impor- in spherically symmetric spacetimes endowed with a pho- tant parts. First one is the lens equation which relates ton sphere [19, 20]. In such a situation strong lensing the location of the source to the location of the image from various alternatives/modifications of Schwarzschild given the amount of deflection suffered by the light from geometry in modeling the galactic center has been stud- source to the observer as it passes by the the gravitational ied. For example lensing from regular black holes was lens. The second important component is the deflection studied in [21] and lensing from stringy black holes was of the light encoded in the Einstein deflection angleα ˆ(θ) studied in [22]. However the basic qualitative features which we define later. We note that the deflection angle in a lensing scenario in the presence of a photon sphere is the only input from the General theory of Relativity, is very similar to Schwarzschild case and is ineffective in and it can be computed by integrating the null geodesics. probing the geometry beyond the photon sphere. Strong field lensing would be much easily able to probe differ- ences from Schwarzschild spacetime if geometry being A. Lens equation studied will be without a photon sphere. As we will see for the family of solutions studied in this paper, when the The lens equation essentially relates the position of the geometry has a photon sphere the lensing signatures are source to that of image. Fig1 is the lens diagram. It is
3
diagram we get J sin θ = . (1) Dd The location of the source β and the image θ can be related to each other by the following relation from the D lens diagram. ds αˆ tan β = tan θ α, (2)
Ds −
where
D α ds [tan θ +tan(ˆα θ)] . (3) Dd ≡ Ds − β From the diagram above it it clear that what enters into θ the lens equation is the deflection angleα ˆ modulo 2π. We note that the many versions of the lens equation have been used in the literature depending on the need
and convenience. The lens equation 2 used in this paper
was derived by Virbhadra and Ellis [13]. It allows for a arbitrarily large deflection of the light. We note that one FIG. 1: The lens diagram: Positions of the source, lens, ob- of the coauthors of this paper (DN) along with Virbhadra server and the image are given by S, L, O and I. The dis- and Chitre [4] had worked on a different lens equation in a tances between lens-source, lens-observer and source-observer first investigation of the strong lensing phenomenon with are given by Dds, Dd and Ds. The angular location of the large deflection; but none of the features we describe here source and the image with respect to optic axis are given by change if we use that equation. β and θ. The impact parameter is given by J.
B. Deflection angle same as lens diagram given in [13]. The spherically sym- metric spacetime under consideration is to be thought of One requires the knowledge of the metric of the space- as a lens denoted by L in the lens diagram. The source time to derive the expression for the Einstein deflec- S and observer O are located faraway as compared to tion angle. Consider general spherically symmetric static the Schwarzschild radius, from the center of the space- spacetime. The metric in the Schwarzschild-like coordi- time in the asymptotic flat region. The line joining lens nates (t,r,ν,φ) can be written as and the observer is known as the optic axis of the lensing 1 geometry. In the absence of the lens light would have ds2 = g(r)dt2 + dr2 + r2dΩ2, (4) traveled along the line SO and would have made an an- − f(r) gle β with respect to the optic axis. Thus β is the source location. In the presence of the lens light gets bent. Let where g(r) and f(r) are arbitrary functions. Asymptotic flatness demands that g(r )= f(r ) = 1. SC and OC be the tangents drawn to the trajectories of → ∞ → ∞ the light at the source and the observer. The angle OC In a gravitational lensing scenario under consideration, makes with the optic axis namely θ depicts the location source, observer and the lens define a plane. In a spher- of the image I. The angle SCI is the Einstein deflection ically symmetric spacetime, the trajectory of the photon angleα ˆ which we calculate later in this section. The dis- is confined to a plane passing through the center which tances from source to lens, observer to lens and source to by the appropriate gauge choice can be taken to be the observer are given by D ,D and D respectively. equatorial plane (ϑ = π/2). Thus only those light rays ds d s emitted by the source, which travel in this plane can pos- The light mostly travels on the lines SC and CO except sibly reach the observer, ultimately leading to the forma- for the region close to lens where curvature is large and tion of images and Einstein rings, and this plane can be it suffers from a deflection. When the deflection is large, taken to be the equatorial plane without loss of general- light can go around the lens multiple times. ity. The equation of motion for the light ray can be written Here we assume that β is very small i.e. the observer, as lens and the source are aligned to a very good approxi- mation. Let LN be the perpendicular drawn to OC from 1 1 U t = ,U ϑ =0,U φ = (5) the lens. J here is the impact parameter. From the lens Jg(r) r2 4 and the radial motion is described by the equation C. Lensing observables
2 g(r) dr 1 We now describe the important lensing observables. + V (r)= (6) f(r) dλ eff J 2 For a fixed position of the source, we compute the posi- tion of images and their magnifications. where All those values of θ that satisfy the lens equation (2) for fixed values of the source position β yield us the lo- g(r) cation of the images. In order to do that we must write Veff = (7) r2 down the deflection angle as a function of the source po- sitionα ˆ(θ). This can be achieved using (1),(8). can be thought of as an effective potential for the radial t r ϑ φ The cross-section of the bundle of rays gets modified motion. Here U = U ,U ,U ,U stands for the veloc- due to the lensing. Liouville’s theorem implies that the ity of the photon, λ is the affine parameter and as stated surface brightness is preserved. Thus the magnification earlier, J is the impact parameter. i.e. ratio of the flux of the image to the flux of the source We can relate the impact parameter J and the distance dr is the ratio of the solid angle subtended by the image to of closest approach r0 using (5),(6) and by setting dφ =0 that of the source at the location of the observer. in the following way The total magnification is defined as
− 1 sin β dβ 1 J(r )= r . (8) µ . (11) 0 o g(r ) ≡ sin θ dθ s o The total deflection suffered by the light ray as it travels which can be broken down into the tangential and radial from the source to the observer (i.e. the deflection angle) magnification in the following way. as a function of a distance of the closest approach of the −1 −1 ∞ r light ray to the lens, isα ˆ(r )=2 U dφ π So it is sin β dβ 0 r0 U φ − µt , µr (12) given by ≡ sin θ ≡ dθ R −1/2 The sign of the magnification of an image gives the par- ∞ 1/2 2 1 r g(r0) dr ity of the image. The singularities of the tangential and αˆ (r )=2 1 π, 0 f(r) r g(r) − r − radial magnification yield the tangential critical curves Zr0 " 0 # (9) (TCCs) and radial critical curves (RCCs), respectively One important question for lensing in strong field in the lens plane and tangential caustic (TC) and radial regime is the presence/absence of photon sphere which caustics (RCs) respectively in the source plane. is a r = const null geodesics. As the distance of closest It is obvious from the expression for the tangential approach asymptotically approaches the photon sphere, magnification that β = 0 gives the TC and the corre- the photon revolves around the lens more and more num- sponding values of θ are the TCCs, also known as Ein- ber of times and the bending angleα ˆ diverges as the stein rings (ER). Thus Einstein rings can be obtained by solving for lens equation for β = 0 i.e, in aligned config- distance of closest approach tends to photon sphere rph. uration of source, lens and observer. The maxima/minima of Veff give unstable/stable pho- ton spheres. Thus equation for photon sphere is given Using the lens equation (2) the radial magnification by (12) can be written in the following way: dβ D sec2 θ D sec2 (ˆα θ) dαˆ dg(r) 2g(r) = 1 ds ds − 1 (13) = (10) dθ − D sec2 β − D sec2 β dθ − dr r s s dαˆ The effective potential for photons in this geometry It is clear from the expression above that if dθ < 0 i.e. gives an idea of the radial behavior of photon trajectories, whenα ˆ is a monotonically decreasing function, we have dβ in particular the turning points for photons. Hence it dθ > 0 and the radial magnification will never diverge. gives an idea as to when photons coming from infinity get Thus the radial critical curves would be absent, which captured and when they can escape back to asymptotia. will be the case for Schwarzschild as well as JMN naked If J >J(rph) then the photon turns back from radius singularity geometry dealt in this paper later . r > rph before it reaches the photon sphere. On the other hand if J The central supermassive dark object in our galaxy is gravitational lensing is identical to that in Schwarzschild modeled initially as Schwarzschild black hole and in the spacetime. In the next section we discuss change in the later section as a naked singularity. The mass of this gravitational lensing properties due to the presence of the 6 object is taken to be M =2.8 10 M⊙ which is the mass naked singularity and make a critical comparison with of the supermassive black hole× in our galaxy. Distance of Schwarzschild results. The gravitational lensing by the the source from the center of the galaxy is taken to be the Schwarzschild black hole was explored in detail in [13]. distance of the sun from the galactic center Dd =8.5kpc. We discuss the relevant details and results here. Thus, in our example, in the near-aligned configuration The Schwarzschild metric is given by the lens is situated midway between the source and the observer i.e. D /D =1/2. Ratio of mass of the lens to ds s 2M 2M −1 the distance to the observer which would later appear in ds2 = 1 dt2 + 1 dr2 + r2dΩ2 −11 r r the calculations is M/Dd 1.57 10 . − − − ≈ × (14) For convenience we work in the dimensionless variable r r0 IV. GRAVITATIONAL LENSING BY x = 2M . The distance of closest approach is x0 = 2M . SCHWARZSCHILD BLACK HOLE In this section we provide a brief overview of the A. Deflection angle and photon sphere results related to the gravitational lensing of light in Schwarzschild black hole geometry. As we discuss later, the naked singularity geometry under investigation in We now compute the deflection angle as a function this paper matches with the Schwarzschild geometry at of the distance of minimum approach. From (9), the the finite radial coordinate. Therefore for the light rays Einstein deflection angle in the Schwarzschild spacetime that stay in the Schwarzschild regime all the time, the in terms of the dimensionless variables is given by −1/2 ∞ 1 1/2 x 2 1 1 dx − x0 αˆ (x0)=2 1 1 1 π (15) 1 x0 1 − x − Zx0 − x " − x ! # As stated earlier the Einstein deflection angle diverges B. Images when the distance of minimum approach is very close to the radius of the photon sphere as the light circles around the center multiple times. It turns out that there We now qualitatively describe the images formed due is a photon sphere in a Schwarzschild spacetime that can to the deflection of the light by Schwarzschild black hole be obtained by solving (10) which is located at the ra- in the galactic central supermassive object scenario. The dius r = rph =3M or in terms of dimensionless variable images’ locations can be obtained by solving the lens at x = xph = 1.5. Using (1) and (8) the distance of equation for the chosen source location. The image is minimum approach x0 can be translated into the image said to be relativistic if the deflection of the light ray is location θ as π larger than 3 2 as per the convention used in [13]. 2M x sin θ = 0 (16) In the weak field limit when the impact parameter is Dd 1 1 large and the deflection angle is small a pair of nonrela- − x0 tivistic images are formed. They have opposite parities. q For small enough impact parameter, we get relativistic which allows us to write deflection angleα ˆ as a function images with large deflection. Theoretically there are in- of image location θ). We have plotted θ(x0) in Fig2. It finitely many images formed on both sides of the optic is a monotonically increasing function of x0. axis i.e. with both positive and negative values of θ. The We plot the Einstein deflection angleα ˆ(θ) in Fig3 relativistic images are bunched together around θ 16.8 We have made use here of the values of the different microarcseconds . This is an extremely important≈ point quantities we have chosen in a galactic central supermas- that no images are formed between the the optic axis sive black hole scenario we discussed earlier. It is a mono- and θ 16.8 microarcseconds . As we discuss later the tonically decreasing function of θ. It diverges as we ap- situation≈ can be significantly different in the case of the proach the photon sphere which corresponds to θ 16.8 naked singularity. More details on the location of the microarcseconds. ≈ images and the magnification can be found in [13]. 6 ` C. Einstein rings Α As the Einstein deflection angle is a monotonically de- 8 creasing function as seen from Fig3, there is no radial critical curve present in the geometry. Location of the 6 Einstein rings can be obtained by solving the lens equa- tion with β = 0. The Einstein rings are said to be rel- 4 ativistic if the deflection angle larger that 2π as per the convention used in [13]. 2 With our choice of parameters for galactic super- massive object scenario we get an equation tan θ = Θ tan(ˆα θ), which admits a solutionα ˆ =2nπ+2θ. There 17 18 19 20 21 22 23 is a nonrelativistic− Einstein ring which can be obtained by solving this equation for θ with n = 0. It is located at FIG. 3:α ˆ vs θ: Einstein deflection angleα ˆ (in radian) is plot- θ = 1.15 arcsecond. There are infinitely many Einstein ted as a function of the image location θ ( in microarcseconds ) rings that can be obtained by solving the above equation in Schwarzschild spacetime for a Galactic supermassive black with different values of n. All the relativistic Einstein hole scenario.α ˆ(θ) is monotonically decreasing. It diverges ≈ rings are located close to θ 16.8 microarcseconds . As around θ 16.8 microarcseconds which corresponds to the ≈ photon sphere. The horizontal lines correspond to 3π/2 and in the case of images there are no Einstien rings located 2π for dashed(red) and thick(purple) lines respectively which between the optic axis and θ = 16.8 microarcseconds . In mark the onset of the relativistic deflection of light. Relativis- case of the naked singularity geometry that we are about tic images and Einstein rings can be seen between the photon to discuss, the situation is significantly different. sphere and the intersection points of horizontal lines with αˆ curve. V. GRAVITATIONAL LENSING BY JMN NAKED SINGULARITY state of dynamical collapse from regular initial conditions for a fluid with zero radial pressure but non-vanishing In this section we study the gravitational lensing by a tangential pressure. The solution has a naked singularity spacetime containing naked singularity. We imagine a hy- at the center and matches to a Schwarzschild spacetime pothetical situation where the galactic supermassive ob- across the boundary r = Rb. Basic features of accretion ject is modeled by a naked singularity solution described disks in such a model was studied by the same authors by a specific metric. We study the images and the Ein- and differences with black hole case were pointed out. In stein rings in the same situation and make a comparison. the same spirit, gravitational lensing in this background The spacetime geometry (to be referred to as JMN solu- would also be an interesting observational probe of the tion henceforth in this paper) we will be dealing with is toy model. a naked singularity solution obtained in [16] as the end A. JMN naked singularity geometry Θsch 40 The spacetime is divided into two parts. (a) The inte- rior region which is described by the following metric: 30 M0 r 1−M0 dr2 ds2 = (1 M ) dt2 + + r2dΩ2 . e − − 0 R 1 M b − 0 20 (17) It can be easily shown that the curvature blows up at the center and thus it corresponds to a strong curvature time–like singularity. (b) The exterior region is described 10 by a Schwarzschild solution 2 2 M0Rb 2 dr 2 2 0 x0 ds = 1 dt + + r dΩ . 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 − − R (1 M R /r) 0 b − (18) FIG. 2: θ vs x0 in Schwarzschild spacetime. The image loca- There is no event horizon in this geometry and thus the tion θ (in microarcseconds ) is plotted as a function of closest singularity at the center is exposed to the asymptotic approach x0 (dimensionless) in Schwarzschild spacetime for a observer at infinity. Therefore it is a naked singularity. galactic supermassive black hole scenario. It is a monotoni- It can be easily verified that the two metrics are con- cally increasing function. 2 nected across the boundary R = Rb via C matching. 7 Veff Veff sphere or otherwise and its location we look for the max- g(r) 0.14 0.6 imum of the effective potential Veff = r2 or in other 0.12 0.5 words we solve the equation (10). 0.10 0.4 2 0.08 0.3 When M the boundary between the interior and 0.06 0 3 ≥ 1 3 0.04 0.2 exterior Schwarzschild region is at xb = M0 2 . So 0.02 0.1 ≤ x xthe boundary is below the photon sphere in the exterior 1 2 3 4 5 1 2 3 4 5 6 Schwarzschild region x xph. The effective potential for the radial motion of the≤ light ray is plotted in Fig4(left FIG. 4: Veff vs x: Plotted here is the effective potential part). The effective potential goes on decreasing below Veff (dimensionless) for the radial motion for the light rays the photon sphere and asymptotically goes to zero. It as a function of x (dimensionless). The plot on the left corre- is clear from the behavior of the effective potential that sponds to the parameter values M0 ≥ 2/3. The thick(green) the light ray which enters the photon sphere never turns vertical line corresponds to the photon sphere, whereas the back and it eventually hits the naked singularity. Thus dashed(red) line corresponds to the boundary between the it is captured. Photons can turn back from the region interior naked singularity region and exterior Schwarzschild exterior to the photon sphere and deflection angle goes geometry. The effective potential monotonically goes to zero in the interior region. The plot on right corresponds to the on increasing indefinitely as we approach approach it. parameter values M0 < 2/3, where the photon sphere is ab- This implies that the lensing will be exactly identical to sent. The vertical red line is the boundary. The effective Schwarzschild case which we discussed in the previous potential blows up as we approach the singularity. section as it is not possible for photons coming from and going back to a large distance probe the metric interior to photon sphere. Thus the gravitational lensing cannot There are two parameters in the solution, M0 which is a unravel the possible existence of the naked singularity at 2 dimensionless parameter and, R which is the boundary the center for M0 . b ≥ 3 radius at which the interior naked singularity metric is 2 1 When M0 < the boundary is located at xb = > matched to the exterior Schwarzschild metric. We must 3 M0 3 . Thus there is no Schwarzschild photon sphere in the have 0 < M0 < 1. The Schwarzschild mass is related 2 0 exterior region, since the boundary is outside the loca- to these two parameters by a relation M = M Rb . We 2 tion of the Schwarzschild photon sphere x > x . The fix the Schwarzschild mass to be same as we had cho- b ph effective potential for the radial motion for the light rays sen in the previous section for the sake of comparison. is plotted in Fig4(right part). The effective potential is Thus for fixed M the only free parameter happens be monotonically decreasing function in the interior region. M . The boundary between the two region is related to 0 Since it does not admit any extremum, no photon sphere M as R = 2M . 0 b M0 is present in the interior region as well. The effective po- As in the Schwarzschild black hole case we introduce tential in fact blows up at the singularity which implies the dimensionless variable x = r , where all the dis- 2M that no light ray can reach it. tances are expressed in units of the Schwarzschild radius. r0 2 From now on wards we focus on the case M0 < . Thus the impact parameter is x0 = 2M and the bound- 3 ary radius becomes simply the inverse of the parameter If the distance of minimum approach is larger than R 1 1 M i.e. x = b = . Thus larger the parameter M , x0 xb = , then the light ray travels in the exte- 0 b 2M M0 0 ≥ M0 smaller is the radius of the boundary for a given mass M rior Schwarzschild geometry and the images and Einstein and we have a more compact object. rings formed due to the lensing are identical to those dis- cussed in the previous section. Thus we focus on the case where the distance of minimum approach is less that the 1 B. Deflection angle boundary radius x0 < xb = M0 . So that the light rays travel partly in the external Schwarzschild metric and We now compute the deflection angle. The first ques- partly in the interior metric and it can actually probe tion one would like to ask for lensing in strong field regime the interior, containing naked singularity. We would like is whether photon sphere is present in the spacetime, to understand the formation of images and the Einstein since the bending angleα ˆ would diverge as the distance rings due to the light rays passing through the interior region. of closest approach tends to xph where x = xph is the location of the photon sphere. The Einstein deflection as a function of distance of 1 In order to investigate the existence of the photon closest approach when x0 < M0 is given by 8 1 1/2 2 −1/2 1/2 2 −1/2 M γ ∞ γ 0 1 x x0 dx 1 x (1 M0)(x0M0) dx αˆ (x0)=2 1 +2 − 1 π 1 1 1 x0 1 M0 " x0 x − # x 1 " x0 1 − # x − Z − Z M0 − x − x (19) ` M0 Α where γ = 1−M0 . The first term corresponds to the con- tribution to the deflection angle from the interior region 70 0 .6 5 and the second term corresponds to the contribution from 60 0 .6 3 the exterior Schwarzschild region. 0 .6 1 Using the relationship between θ and x , i.e. 50 0 .55 0 0 .475 40 2M x0 sin θ = (20) 30 γ Dd (1 M0)(x0M0) − 20 p we can express deflection angle as a function of image 10 locationα ˆ(θ). We have plotted θ(x0) for the geometry in 0 Θ Fig5 andα ˆ(θ) in Fig6. 5 10 15 20 25 Presence of the photon sphere guarantees the relativis- tic deflection of light, though, as we pointed out earlier, FIG. 6:α ˆ vs θ for different M0: Einstein deflection angle it prevents probes to the interior density structure. But αˆ (in radian) in JMN spacetime is plotted against the im- 2 age location θ (in microarcseconds ). The value of M0 for we are investigating the parameter regime M0 < 3 where the photon sphere is absent in the geometry. So the rel- the curves are shown in the legends.α ˆ is a monotonically de- ativistic deflection may or may not occur. Firstly we creasing function. As we increase M0 the maximum deflection angle goes on increasing. The deflection angle will be larger would like to find out the parameter range M where we 0 than 2π and thus relativistic images can form for M0 > 0.475. expect relativistic deflection of light to happen. The curves for different values of M0 cross because ultimately Looking at the Fig3 and Fig 6, Einstein deflection angle all the curves have to match to the deflection angle curve is monotonically increasing with decreasing x0 for both for Schwarzschild spacetime beyond the boundary and since 1 0 Schwarzschild and JMN. In Schwarzschild geometry the boundary xb = M0 is at the smaller radius for larger M . deflection angle reaches 3π at x 1.605 i.e. if M > 2 0 ∼ 0 0.62. So any xb less than that (and hence M0 > 0.62) will definitely give relativistic defection. This is sufficient and Einstein rings can form beyond this parameter value. but not necessary condition. More detailed calculation Note that this is independent any value of Schwarzschild shows that the maximum value of the deflection angle is mass. larger than 2π for M0 > 0.475. Thus relativistic images Θ jmn C. Images 20 We now describe the properties of the images formed due to the gravitational lensing of the light passing 15 through the interior region. The relativistic images are possible only beyond the parameter value M0 > 0.475. These images probe the interior geometry and can un- 10 ravel existence of the naked singularity. We calculate the location of the images and their mag- nification for galactic supermassive object scenario with 5 a given source location that is in the near-aligned config- uration. We solve the lens equation for fixed β = 0.075 and for given M0 > 0.475. For different values of M0 we 0 x0 0.0 0.5 1.0 1.5 get different number of images on the same side as well as on the opposite side of the optic axis. The number of FIG. 5: θ vs x0 in JMN: The image location θ (in microarcsec- image goes on increasing as we increase M0. onds ) is plotted as a function of closest approach x0 (dimen- In Table I,II we make a list of image locations and sionless) in JMN spacetime for a galactic central supermassive magnifications for a specific value of M0 = 0.63. There object scenario in an interior region. It is a monotonically in- are four images on the same same side as well as on op- creasing function. posite side of the optic axis. Images are well separated 9 would be absent. The caustic happens to be a point TABLE I: Images and magnification (same side): In this table β = 0, since we are dealing with spherically symmetric we list the location of the relativistic images and magnifica- tions on the same side as source for the galactic central super- massive object scenario for M0 = 0.63 and β = 0.075 radian. θ is in microarcseconds . The images are well separated and have comparable magnifications Image θ µt µr µ TABLE III: Einstein rings: In this table we list the location of −9 −12 −21 I 16.74 1.9 × 10 1.8 × 10 1.9 × 10 the relativistic Einstein rings for M0 = 0.63 for the galactic 9 12 21 II 14.56 0.9 × 10− 4.7 × 10− 4.4 × 10− central supermassive scenario and the corresponding values 9 12 21 III 10.75 0.7 × 10− 6.8 × 10− 5.0 × 10− of the deflection angle, θ is in microarcseconds andα ˆ is in 9 12 21 IV 5.82 0.4 × 10− 8.0 × 10− 3.2 × 10− radian. The Einstein rings are well separated. No. θE αˆ I 16.715 2π + 0.00068 TABLE II: Images and magnification (opposite side): In this II 14.485 4π + 0.00042 table we list the location of the relativistic images and magni- III 10.469 6π + 0.00076 fications on the opposite side as source for the galactic central IV 5.700 8π + 0.00173 supermassive object scenario for M0 = 0.63 and β = 0.075 ra- dian. θ is in microarcseconds . The images are well separated spacetimes. In order to compute the critical curves i.e. and have comparable magnifications. the location of the Eisntein rings we solve the lens equa- Image θ µt µr µ tion with β = 0. As discussed in the Schwarzschild case I 16.68 −1.1 × 10−9 1.9 × 10−12 −2.1 × 10−21 for the galactic supermassive object scenario, we have to −9 −12 −21 solve the equation tan θ = tan(ˆα θ), which holds good II 14.41 −0.9 × 10 4.8 × 10 −4.4 × 10 − III 10.54 −0.7 × 10−9 6.9 × 10−12 −4.7 × 10−21 whenα ˆ =2nπ +2θ. Knowingα ˆ we can solve the previ- IV 5.58 −0.4 × 10−9 8.1 × 10−12 −3.0 × 10−21 ous equation to get the angular locations of the Einstein rings. For a given value of M0, the number of solutions to this equation would be either αˆmax or αˆmax 1 which 2π 2π − from one another with angular separation of around 2-5 will be the number of the relativistic Einstein rings. The microacrsecond. The magnification of all the images is number of the relativistic Einstein rings goes on increas- of the same order of magnitude. The radial parity of the ing as we increase M0. As in the Schwarzschild case there image is always positive. The tangential parity and thus will be a nonrelativistic Einstein ring located at 1.15 arc- the total parity is positive for the images on the same side second which corresponds to the solution of the equation of the optic axis while it is negative for the images on the with n = 0. opposite side. Also it is worthwhile to mention here that the position of relativistic images does not change much For M0 =0.63 we have four relativistic Einstein rings. with changes in source position which can also be inferred The location of the relativistic Einstein rings and the from the fact that the radial magnification of the images corresponding Einstein deflection angles is given in Table dθ −12 (which is dβ ) is of the order of 10 as shown in tables III. The rings are well separated from one another with I and II. separation of the order of 2-5 microarcseconds. In Fig 7, 8, 9, we show the variation of the tangential, D. Einstein rings radial and total magnification with θ near the relativistic Einstein rings. As expected tangential and consequently Since the deflection angle is a monotonically decreasing total magnification diverges at the Einstein rings and falls function of the image location the radial critical curves rapidly as we move away from it. VI. COMPARISON WITH SCHWARZSCHILD and Einstein ring, infinite number of relativistic images BLACK HOLE and Einstein rings are formed. But they are clumped together around θ 16.8 microarcseconds . No images or rings are formed≈ between the optic axis and θ 16.8 In this section we make a critical comparison of the ≈ results obtained for gravitational lensing in Schwarzschild microarcseconds which happens to be the forbidden re- black hole and naked singularity geometry described by gion. The separation between the first and rest of the JMN solution with the same mass in a galactic central images clumped together is of the order of 0.1 microarc- supermassive object scenario. seconds and the ratio of magnifications is of the order of 500 [13, 19, 20]. In the Schwarzschild black hole case the photon sphere is present. Apart from the outer nonrelativistic images In JMN geometry when M 2 Schwarzschild photon 0 ≥ 3 10 Μt Μt 2000 3000 2000 1000 1000 0 Θ 0 Θ 16.715 16.716 16.717 16.718 14.483 14.484 14.485 14.486 -1000 -1000 -2000 -2000 -3000 Μ t Μt 600 150 400 100 200 50 0 Θ 0 Θ 10.645 10.650 10.655 10.660 5.68 5.69 5.70 5.71 5.72 5.73 5.74 -200 -50 -400 -100 -150 -600 FIG. 7: µt vs θ for JMN spacetime: Plotted here is the tangential magnification near the Relativistic Einstein Rings for 9 M0 = 0.63. θ is in microarcseconds and y-axis scale has been multiplied by 10 for clarity. Tangential magnification blows up at the location of the Einstein ring and decreases as we go away from it. Μr Μr 1.895 4.812 1.890 4.811 1.885 4.810 1.880 4.809 1.875 1.870 Θ 4.808 Θ 16.714 16.715 16.716 16.717 16.718 14.482 14.483 14.484 14.485 14.486 Μ Μ r r 8.4 6.908 6.906 8.3 6.904 6.902 8.2 6.900 8.1 6.898 6.896 Θ 10.645 10.650 10.655 10.660 8.0 Θ 5.67 5.68 5.69 5.70 5.71 5.72 5.73 5.74 FIG. 8: µr vs θ for JMN spacetime: Plotted here is the radial magnification near the Relativistic Einstein Rings for M0 = 0.63. θ is in microarcseconds and y-axis scale has been multiplied by 1012 for clarity 11 Μ Μ 4000 15 000 10 000 2000 5000 0 Θ 0 Θ 16.715 16.716 16.717 16.718 14.483 14.484 14.485 14.486 -5000 -2000 -10 000 -4000 -15 000 Μ Μ 1500 3000 1000 2000 500 1000 0 Θ 0 Θ 5.68 5.69 5.70 5.71 5.72 5.73 5.74 -1000 10.645 10.650 10.655 10.660 -500 -2000 -1000 -3000 -1500 FIG. 9: µ vs θ for JMN spacetime: Plotted here is the total magnification near the Relativistic Einstein Rings for M0 = 0.63. θ is in microarcseconds and y-axis scale has been multiplied by 1021 for clarity. Total magnification blows us at the location of the Einstein ring and decreases as we go away from it. sphere is present and the gravitational lensing is identical tative features of lensing are very similar to Schwarzschild to that of Schwarzschild black hole. If M0 0.475 no case. ≤ q √ relativistic images are formed. However, for large scalar charge M > 3 photon The interesting regime in parameter space is 0.475 < sphere is absent in JNW spacetime. It was stated in 2 M0 < 3 . A number of relativistic images and Einstein [5] that the relativistic deflection and images are absent rings are formed depending on how large is M0. Unlike completely in the absence of the photon sphere. How- Schwarzschild black hole case the images are not clumped ever, a careful investigation in this range of parameters together. They are well separated from one another. Im- shows that in an extremely small range of parameter √ q ages and Einstein rings can appear in the region between 3 < M 1.746, relativistic lensing and images are optic axis and θ 16.8 microarcseconds which is a for- formed even≤ in the absence of the photon sphere. Rela- ≈ q bidden region for Schwarzschild black hole. The images tivistic images are absent when M > 1.746. But inter- have comparable magnifications. estingly as mentioned in [5] the radial caustic is always Thus there are qualitative differences in the images present in the absence of the photon sphere. This is a formed in Schwarzschild black hole geometry and JMN consequence of the fact that as we decrease the impact 2 spacetime when M0 < 3 . These features can in princi- parameter the deflection angle initially increases, it at- ple lead to the observational distinction between the two tains a maximum and goes on decreasing. Eventually it spacetimes. settles down to a constant negative value π in the limit where impact parameter approaches zero.− In the JMN spacetime, in the absence of the photon VII. COMPARISON WITH JNW NAKED sphere we get relativistic images in a rather wide range of SINGULARITY 2 parameter values 0.475 2 multiple times. Time difference between the disappear- 3 . The photon sphere is absent. But the relativistic im- ance and reappearance of the consecutive Einstein rings ages and Einstein rings can form and their number in- will be influenced by the presence of the naked singular- creases with increasing value of the parameter M0. The ity in the binary system. Thus it could be used to infer images and rings are well separated from one another and the presence of naked singularity, as against a black hole. happen to lie in the forbidden region for Schwarzschild Such a situation is however extremely difficult to model. black hole, within a distance from the optic axis of It is beyond the scope of this paper and might be dealt θ = 16.8 microarcseconds . Their magnification is also with later. comparable. Thus the strong gravitational lensing sig- nature is qualitatively different from Schwarzschild black hole. X. CONCLUSION AND DISCUSSION The gravitational lensing in the absence of the pho- ton sphere is qualitatively different in JMN and JNW In this paper we studied the strong gravitational lens- spacetimes. In both the geometries relativistic images ing from the perspective of cosmic censorship and ex- are present in an appropriate parameter range. However, plored the possibility of distinguishing black holes from there are no radial caustics in the JMN geometry, while naked singularities. We modeled the galactic central su- radial caustic is always present in the JNW spacetime in permassive dark object initially by a black hole and then the absence of the photon sphere. by naked singularity. We studied the gravitational lens- However, there are practical difficulties as far as obser- ing of the source in a near aligned configuration at a dis- vation of relativistic images and rings are concerned with tance from a galactic center approximately comparable the telescopes and techniques currently being used. We to distance of the sun from the center. require microarcseconds resolution which can be achieved The Schwarzschild black hole has a photon sphere. with VLBI. However magnification of the images which Thus apart from a pair of nonrelativistic images and a is of the order of µ = 10−22 is too small. Relativistic nonrelativistic Einstein ring, infinitely many relativistic Einstein rings formed due to the lensing of the star with images and Einstein rings clumped together. No images the size comparable to sun, will be 1017 times weaker as and Einstein rings lie in the region between the optic axis compared to the nonrelativistic Schwarzschild Einstein and θ = 16.8 microarcseconds . Also all the images that ring and thus will not be seen since the current instru- are clumped together are highly demagnified as compared ments do not have dynamical range over seventeen orders to the first relativistic image with a small separation be- of magnitude of brightness. tween them of the order of 0.1 microarcseconds . Keeping this in mind, new techniques and instruments We then model the galactic center object as JMN solu- must be developed in the future which will be able to tion which was recently shown to occur as an end state of observe the Einstein rings and can unravel the nature of the gravitational collapse of a fluid with zero radial pres- the galactic central supermassive object. sure but non-vanishing tangential pressure. This solution We also suggest that the appearance and disappear- has two parameters, namely mass and another parameter ance of the outer Einstein ring as the source crosses di- M0. The spacetime is divided into two parts. Exterior amond shaped caustic more than once can possibly shed metric is Schwarzschild spacetime with same mass as that light on the possible existence of the naked singularity in of the Schwarzschild black hole considered earlier. Inte- the binary system of a naked singularity and a massive rior metric contains a central naked singularity with the star. We wish to explore this situation in the future. 2M In this paper we studied a naked singularity geometry boundary located at the radius Rb = M0 .The two metrics are connected across the boundary by C2 matching. arising out of a toy calculation of dynamical collapse of 2 a matter with only the tangential pressure. It would be In the parameter range M0 3 the Schwarzschild pho- ton sphere is present in the≥ geometry and the gravita- interesting to study more realistic cases e.g. with the tioanl lensing signature of JMN spacetime is identical to inclusion of the radial pressure. that of the Schwarzschild black hole. When M0 0.475, the photon sphere is absent. But no relativitic≤ bending of light and thus no relativistic Acknowledgments images possible. This behavior is different from the Schwarzschild black hole. 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