A Novel Gravitational Lensing Feature by Wormholes ∗ Rajibul Shaikh , Pritam Banerjee, Suvankar Paul, Tapobrata Sarkar

Physics Letters B 789 (2019) 270–275

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Physics Letters B

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A novel gravitational lensing feature by wormholes ∗ Rajibul Shaikh , Pritam Banerjee, Suvankar Paul, Tapobrata Sarkar

Department of Physics, Indian Institute of Technology, Kanpur 208016, India a r t i c l e i n f o a b s t r a c t

Article history: Horizonless compact objects with light rings (or photon spheres) are becoming increasingly popular in Received 25 November 2018 recent years for several reasons. In this paper, we show that a horizonless object such as a wormhole Accepted 13 December 2018 of Morris–Thorne type can have two photon spheres. In particular, we show that, in addition to the Available online 18 December 2018 one present outside a wormhole throat, the throat can itself act as an effective photon sphere. Such Editor: M. Cveticˇ wormholes exhibit two sets of relativistic Einstein ring systems formed due to strong gravitational lensing. We consider a previously obtained wormhole solution as a speciﬁc example. If such type of wormhole casts a shadow at all, then the inner set of the relativistic Einstein rings will form the outer bright edge of the shadow. Such a novel lensing feature might serve as a distinguishing feature between wormholes and other celestial objects as far as gravitational lensing is concerned. © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction qualitatively different from that by a black hole. In this work, we study gravitational lensing by wormholes and point out a novel Gravitational lensing is an established observational tool for lensing feature which some wormholes can exhibit. In particular, probing gravitational fields around different compact objects. It is we show that, in addition to a photon sphere present outside a also used to test the viability of different alternative theories of wormhole throat, the throat can also act as an effective photon gravity [1]. After the theoretical prediction of light bending and sphere. Such wormholes can exhibit interesting strong gravitational its subsequent observational verification [2,3], interest in study- lensing features. In particular, they exhibit two sets of relativistic ing gravitational lensing has grown immensely in the last few Einstein ring systems. To the best of our knowledge, such a novel decades. One of the fascinating features of gravitational lensing is lensing feature has not been pointed out before in the literature. that light can undergo an unboundedly large (i.e. theoretically infi- This is the main result of this paper that we elaborate in sequel. nite) amount of bending in the presence of unstable light rings (or The plan of the paper is as follows. In Sec. 2, we work out photon spheres in the spherically symmetric, static case) [4–7]. As the conditions under which the throat of a Morris–Thorne worm- a result of such strong gravitational lensing, a large (theoretically hole can act as an effective photon sphere, in addition to a photon infinite) number of relativistic images are formed. sphere present outside the throat. We consider a specific exam- In recent years, horizonless objects have attracted much atten- ple which satisfies all the energy conditions in a modified theory tion for several reasons, one of them being that these objects with of gravity and study formation of relativistic images in Sec. 3. We light rings can act as alternatives to black holes, i.e. they can be conclude Sec. 4 with a summary and discussion of the key results. “black hole mimickers” (see [8–12] and references therein). There has also been a lot of effort on how to distinguish such horizon- 2. A novel gravitational lensing feature by wormholes less compact objects from black holes. For example, gravitational lensing and its various aspects, such as images, shadows etc by We consider a general spherically symmetric, static wormhole different horizonless objects such as wormholes [13–34], naked spacetime of the Morris–Thorne class. This spacetime is generically singularities [35–42], Bosonic stars [43]etc. have been analyzed to written as [44] see whether one can distinguish between these and a black hole. 2 2 2(r) 2 dr 2 2 2 2 Such studies show that, in some cases, lensing by these objects are ds =−e dt + + r (dθ + sin θdφ ), (1) − B(r) 1 r where (r) and B(r) are the redshift function and the wormhole * Corresponding author. shape function, respectively. The wormhole throat, where two dif- E-mail addresses: [email protected] (R. Shaikh), [email protected] (P. Banerjee), − B(r) = [email protected] (S. Paul), [email protected] (T. Sarkar). ferent regions are connected, is given by 1 r 0, i.e., by r0 https://doi.org/10.1016/j.physletb.2018.12.030 0370-2693/© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. R. Shaikh et al. / Physics Letters B 789 (2019) 270–275 271

B(r0) = r0, with r0 being the radius of the throat. B(r) satisfies Note that r˙ = 0is automatically satisfied at the wormhole throat B the flare-out condition (r0) < 1[44]. The redshift function (r) r = r0 since B(r0) = r0 there. Therefore, the throat can act as an is finite everywhere (from the throat to spatial infinity). effective photon sphere when the other two conditions, namely ... For simplicity, we restrict ourselves to θ = π/2. Because of r¨ = 0 and r > 0, are satisfied. In terms of the effective potential, the spherical symmetry, the same results can be applied to all θ. this becomes Therefore, the Lagrangian describing the motion of a photon in the 2 dVef f θ = π/2 plane of the spacetime geometry (1)is given by V ef f (r0) = E , < 0 , (10) dr r=r0 2 r˙ where we have used the flare-out condition B (r ) < 1. Note that 2L =−e2(r)t˙2 + + r2φ˙2, (2) 0 − B(r) the last set of conditions for the throat to be an effective pho- 1 r ton sphere are different from those in (7)in the sense that the where a dot represents a derivative with respect to the affine pa- conditions in (10)require one less derivative of the effective po- rameter. Since the Lagrangian is independent of t and φ, we have tential than those required for the photon sphere present outside two constants of motion given by − B(r) the throat. This is due to the factor 1 r present in the radial ∂L ∂L = =− 2(r)˙ =− = = 2 ˙ = geodesic equation. For the metric (1), Eq. (10) becomes pt ˙ e t E , pφ ˙ r φ L , (3) ∂t ∂φ 1 −(r0) b0 = r0e , (r0)< , (11) where E and L are the energy and angular momentum of the pho- r0 μ ν ton, respectively. Using the null geodesics condition gμν x˙ x˙ = 0, where b0 is the critical impact parameter corresponding to the ef- we obtain fective photon sphere at the throat. e2(r) e2(r) It is to be noted that, in the Schwarzschild gauge, the effective ˙2 + = 2 = 2 B r V ef f E , V ef f L , (4) potential does not seem to have a maximum since dVef f /dr = 0 1 − (r) r2 r at the throat, even though the throat acts as an effective pho- where V ef f is the effective potential. From this, we can obtain the ton sphere. This may seem confusing as traditionally the photon deflection angle given by [6] sphere is defined as the maximum of the effective potential. How- ever, things become more clear when we switch over from the ∞ e(r)dr Schwarzschild gauge to the proper radial coordinates defined by α = 2 − π , (5) B 2(r) r r2 1 − (r) 1 − e rtp r 2 2 dr b r l(r) =± , (12) B where b = L/E is the impact parameter, and r is the turning 1 − tp r0 r dr = dr = point given by dφ 0. dφ 0gives the following relation between where the throat is at l(r0) = 0, and the two signs correspond b and rtp: to the two different regions connected through the throat. In the − (rtp) proper radial coordinates, the radial geodesic equation becomes b = rtpe . (6) 2(r(l)) It is well-known that the deflection angle diverges logarithmically 2(r(l))˙2 2 2 e e l + V ef f = E , V ef f = L . (13) to infinity as the turning point approaches a photon sphere (which r2(l) corresponds to the unstable photon orbits), and as a result, there Therefore, at the throat, we have are an infinite number of relativistic images formed due to lensing, ˙ just outside the photon sphere. Circular photon orbits satisfy r = 0 dVef f B(r) dVef f =± 1 − = 0 (14) and r¨ = 0. Unstable photon orbits, which constitute the so-called ... dl r0 r dr r0 photon sphere, additionally satisfy r > 0. Generally, the conditions 2 2 d V ef f B(r) d V ef f 1 B B dVef f for the unstable photon orbit (photon sphere) are written in terms = 1 − + − dl2 r r dr2 r 2 r2 r dr r of the conditions on the effective potential. These are given by 0 0 0 (1 − B (r0)) dVef f 2 = (15) dVef f d V ef f V (r ) = E2, = 0, < 0 , (7) 2r0 dr r0 ef f ph 2 dr r=rph dr r=rph Now, using B (r0) < 1 and Eq. (10), we obtain where r is the radius of a photon sphere. Note that a photon ph 2 dVef f d V ef f sphere corresponds to the maximum of the potential. For the met- V (r ) = E2, = 0, < 0, (16) ef f 0 2 ric of Eq. (1), the above conditions become dl r0 dl r0 1 1 which implies that, unlike in the Schwarzschild gauge (see −(rph) bph = rphe , (rph) = , (rph)<− , (8) Eq. (10)), in the proper radial coordinate, the effective potential r 2 ph rph has a maximum at the throat when the throat acts as an effective photon sphere. where b is the critical impact parameter corresponding to the ph Therefore, in addition to the outer photon sphere, the worm- photon sphere. In addition to the photon sphere mentioned above, hole throat can also act as an effective photon sphere. As a result, the effective potential can exhibit an extremum at the wormhole two sets of infinite number of relativistic images may be formed throat also, thereby implying circular photon orbits (either stable due to strong gravitational lensing of light coming from a distant or unstable) at the throat. This can be seen by rewriting Eq. (4)as light source. However, for the effective photon sphere at the throat B to take part in the formation of relativistic images, the potential ˙2 −2(r) 2 r + e 1 − V ef f − E = 0 . (9) r must have higher height at the throat than at the photon sphere 272 R. Shaikh et al. / Physics Letters B 789 (2019) 270–275 present outside the throat. Therefore, the necessary and sufficient The above wormhole metric was constructed by using the fact conditions that the throat also takes part in the formation of rela- that it must have an effective photon sphere at its throat as well tivistic images in addition to those formed due to the outer photon as a photon sphere outside its throat. In the next section, how- sphere are given by ever, we consider a wormhole example which possesses the above feature and arises as an exact solution in a modified theory of 2(r ) 2(r ) 1 e 0 e ph gravity. (r0)< , > , (17) r 2 2 0 r0 rph 3. A specific example in modified gravity where r0 and rph are given by We consider a wormhole solution obtained in [45]. The space- 1 B(r0) = r0, (rph) = . (18) time geometry is given by r ph dr2 Note that there should be a minimum between the throat and the ds2 =−ψ 2(r) f (r)dt2 + + r2 dθ 2 + sin2 θdφ2 , (24) outer photon sphere. Therefore, when the throat acts as an effec- f (r) tive photon sphere, maxima and minima outside the throat must 1 1 ψ(r) = √ (25) come in pairs, and the condition (r ) < − will automatically ph r2 1 + κρ ph ⎡ be satisfied at the maxima. − κC0 1 + κρ 1 4 Let us now show how we can construct a wormhole metric f (r) = ⎣1 − 2r κC which possesses an effective photon sphere at its throat, as well 1 − κ pθ 1 + 0 2r4 as a photon sphere outside its throat. This demands that we must 2 2 have V = 0at two different radii outside the throat, with one 2M C0 1 1 9 κ C0 ef f × − 2 F1 , , ; of them representing the maxima at the outer photon sphere and r r2 2 8 8 4r8 ⎤ the remaining one representing the minima of V ef f present in be- 2 2 2 2 tween the throat and the outer photon sphere. Let us consider the κC0 1 5 13 κ C0 κC0 ⎦ + F , , ; − ,(26) following ansatz for V (r): 6 2 1 8 ef f 30r 2 8 8 4r 6r6 1 − κC0 2r4 (r − β)(r − γ ) V (r) =−K (19) where ρ is the energy density given by ef f rn ≤ C0 1 where K is a positive constant, n is a positive integer and r0 ρ = , 4 κC β<γ . V (r) is zero at two points (β, γ ) with γ representing r 1 − 0 ef f 2r4 maxima, i.e., V (r) < 0, and β representing minima of V ef f (r), ef f γ where C0 is an integration constant. It is to be noted that the κC0 i.e. V ef f (r) β > 0. Integrating the above form of V ef f (r), we obtain factor 1 − present in the denominator of the last term in 2r4 − − Eq. (26)is absent in [45]. This was a typographic error in that =− (r β)(r γ ) V ef f (r) K dr paper. For κ < 0, the solution represents a wormhole, and the rn wormhole throat radius is given by (1 + κρ)| = 0, which gives 3 2 + r0 −n r r (β γ ) r(βγ ) | | 1/4 r2 = Kr − + (20) r = κ C0 [45]. Additionally, we must have x = 0 < 1for n − 3 n − 2 n − 1 0 2 |κ| the wormhole. For x > 1, the throat is covered by an event hori- From Eqs. (4) and (20), we find zon, thereby giving a black hole solution. The matter supporting 3 2 + the above wormhole satisfies all the energy conditions. However, 2φ(r) 2−n r r (β γ ) r(βγ ) e = Kr − + (21) to make the wormhole traversable, the parameter κ, the mass M n − 3 n − 2 n − 1 and the throat radius r0 of the wormhole have to satisfy the fol- where the constant of motion L is absorbed in K . lowing [45]: Equation (21)shows that e2φ(r) remains finite in the range 3 3 r ∈ [r0, ∞) for n ≥ 5. Again, the asymptotically flat condition, i.e., r0 1 1 9 r0 1 5 13 M = 2 F1 , , ; 1 + 2 F1 , , ; 1 . (27) e2φ(r) → 1in the limit r →∞, is satisfied only for n = 5. Putting |κ| 2 8 8 15|κ| 2 8 8 this value of n in Eq. (21) and taking the r →∞ limit, we get Fig. 1 shows the effective potential for different value of x (both K wormhole and black hole). In Figs. 1(a) and 1(b), only the throat lim e2φ(r) = 1 =⇒ = 1 =⇒ K = 2 r→∞ 2 acts as an effective photon sphere. In Figs. 1(c)–1(e), there is an outer photon sphere in addition to the effective photon sphere Therefore, the final form of e2φ(r) becomes at the throat. However, unlike the effective photon sphere at the 1 (β + γ ) (βγ ) throat in Figs. 1(c) and 1(d), that in Fig. 1(e) will be hidden within e2φ(r) = 2 − + (22) 2 3r 4r2 the outer photon sphere and hence will not take part in the forma- tions of relativistic images in gravitational lensing since the effec- Now we should choose β and γ in such a way that the conditions tive potential at the throat has lesser height than that at the outer = in (17)(with rph γ ) are satisfied. Let us illustrate by taking a photon sphere. = 3 = 5 B = 8r0 − Fig. 2 shows the numerically integrated deflection angle as a specific choice β 2 r0 and γ 2 r0. Also, we choose (r) 3 5r2 function of the impact parameter for the metric parameter values 0 so that B(r ) = r , and M = 4 r is the Arnowitt–Deser–Misner 3r 0 0 3 0 same as those in Fig. 1 (both wormhole and black hole). Note that, mass of the wormhole. Therefore, we have as expected, the deflection angle diverges as the impact param- 2 B 2 eter approaches the values corresponding to the effective photon 2φ(r) 8r0 15r0 (r) 8r0 5r0 e = 1 − + , 1 − = 1 − + . (23) sphere at the throat or the outer photon sphere. In the case where 3r 8r2 r 3r 3r2 R. Shaikh et al. / Physics Letters B 789 (2019) 270–275 273

V r2 Fig. 1. The effective potential V (= ef f ) plotted for different x (= 0 )andκ =−1. L2 |κ|

r2 = 0 =− Fig. 2. The deflection angle plotted as a function of the impact parameter for different x ( |κ| )andκ 1. there is both the effective photon sphere at the throat as well as To study the relativistic images formed due to strong gravita- the outer photon sphere, the deflection angle diverges at the lo- tional lensing of light coming from a distant light source, we will, cations of both the throat and the outer photon sphere. In this for simplicity, assume that the observer, the lens (the wormhole or case, as we decrease the impact parameter, the deflection angle the black hole), and the distant point light source are all aligned. diverges to infinity as b → bph. As we decrease b further below We also consider the situation where the observer and the light bph, the deflection angle start decreasing, but is again divergent source are far away from the lens. Therefore, in the observer’s as b approaches the critical value b0 corresponding to that of the sky, the relativistic images will be concentric rings (known as rel- effective photon sphere at the throat. ativistic Einstein rings) of radii given by the corresponding impact 274 R. Shaikh et al. / Physics Letters B 789 (2019) 270–275

r2 = 0 =− Fig. 3. The relativistic Einstein rings for different x ( |κ| )andκ 1. parameters b(rtp). These impact parameter values b(rtp) can be ob- holes through observations. To this end, strong gravitational lens- tained by solving α 2πn, where n is the ring number [6]. Fig. 3 ing, where a photon sphere plays a crucial role, is an important shows the relativistic Einstein rings in the observer’s sky. Note that observational tool for probing the spacetime geometries around there are two sets of inﬁnite number of relativistic Einstein rings these objects. in the case when there are two photon spheres (effective photon In this work, we have shown that spherically symmetric, static sphere at the throat and the outer photon sphere). This is some- wormholes of the Morris–Thorne class can posses two photon thing different from the black hole case (Fig. 3(f)) where there is spheres. In particular, we have shown that, in addition to a photon only one set of relativistic rings formed due to a photon sphere. sphere present outside a wormhole throat, the throat can also act Also note that, when there is only one photon sphere (effective as an effective photon sphere. Such wormholes exhibit two sets of photon sphere or the outer photon sphere), the rings are closely relativistic Einstein ring system formed due to strong gravitational clumped together. But, when there are two photon spheres, the lensing of light. If such type of wormhole casts a shadow at all, rings at the outer photon sphere are less closely packed than those then the inner set of the relativistic Einstein rings will form the at the inner photon sphere. Here we would like to point out that outer bright edge of the shadow. We have illustrated this further photons with impact parameters less than the critical value cor- through a speciﬁc example. The above-mentioned rich and novel responding to the inner photon sphere will get captured by the strong lensing feature points out the vital role that a wormhole wormholes and travel to the other side. Therefore, if no radiation throat can play. Such a novel lensing feature, which has not been or little amount of radiation comes from the other side, then such pointed out before, may be useful in detecting wormholes in fu- a wormhole will cast a shadow [29], with the inner set of the rel- ture observations. ativistic images forming the outer bright edge of the shadow. References 4. Conclusion

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