A novel gravitational lensing feature by wormholes
Rajibul Shaikh, Pritam Banerjee, Suvankar Paul, Tapobrata Sarkar
aa r X i v : . [ g r- q c ] D ec A novel gravitational lensing feature by wormholes
Rajibul Shaikh, ∗ Pritam Banerjee, † Suvankar Paul, ‡ and Tapobrata Sarkar § Department of Physics, Indian Institute of Technology, Kanpur 208016, India
Abstract
Horizonless compact objects with light rings (or photon spheres) are becoming increasingly popular inrecent years for several reasons. In this paper, we show that a horizonless object such as a wormholeof Morris-Thorne type can have two photon spheres. In particular, we show that, in addition to theone present outside a wormhole throat, the throat can itself act as an effective photon sphere. Suchwormholes 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 castsa shadow at all, then the inner set of the relativistic Einstein rings will form the outer bright edge of theshadow. Such a novel lensing feature might serve as a distinguishing feature between wormholes and othercelestial objects as far as gravitational lensing is concerned. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] . INTRODUCTION Gravitational lensing is an established observational tool for probing gravitational fields arounddifferent compact objects. It is also used to test the viability of different alternative theoriesof gravity [1]. After the theoretical prediction of light bending and its subsequent observationalverification [2, 3], interest in studying gravitational lensing has grown immensely in the last fewdecades. One of the fascinating features of gravitational lensing is that light can undergo anunboundedly large (i.e theoretically infinite) amount of bending in the presence of unstable lightrings (or photon spheres in the spherically symmetric, static case) [4–7]. As a result of such stronggravitational lensing, a large (theoretically infinite) number of relativistic images are formed.In recent years, horizonless objects have attracted much attention for several reasons, one ofthem being that these objects with light rings can act as alternatives to black holes, i.e they can be“black hole mimickers” (see [8–12] and references therein). There has also been a lot of effort onhow to distinguish such horizonless compact objects from black holes. For example, gravitationallensing and its various aspects, such as images, shadows etc by different horizonless objects suchas wormholes [13–34], naked singularities [35–42], Bosonic stars [43] etc. have been analyzed tosee whether one can distinguish between these and a black hole. Such studies show that, in somecases, lensing by these objects are qualitatively different from that by a black hole. In this work,we study gravitational lensing by wormholes and point out a novel lensing feature which somewormholes can exhibit. In particular, we show that, in addition to a photon sphere present outsidea wormhole throat, the throat can also act as an effective photon sphere. Such wormholes canexhibit interesting strong gravitational lensing features. In particular, they exhibit two sets ofrelativistic Einstein ring systems. To the best of our knowledge, such a novel lensing feature hasnot been pointed out before in the literature. This is the main result of this paper that we elaboratein sequel.The plan of the paper is as follows. In Sec. II, we work out the conditions under which thethroat of a Morris-Thorne wormhole can act as an effective photon sphere, in addition to a photonsphere present outside the throat. We consider a specific example which satisfies all the energyconditions in a modified theory of gravity and study formation of relativistic images in Sec. III.We conclude Sec. IV with a summary and discussion of the key results.2
I. A NOVEL GRAVITATIONAL LENSING FEATURE BY WORMHOLES
We consider a general spherically symmetric, static wormhole spacetime of the Morris-Thorneclass. This spacetime is generically written as [44] ds = − e r ) dt + dr − B ( r ) r + r ( dθ + sin θdφ ) , (1)where Φ( r ) and B ( r ) are the redshift function and the wormhole shape function, respectively. Thewormhole throat, where two different regions are connected, is given by (cid:16) − B ( r ) r (cid:17) (cid:12)(cid:12)(cid:12) r = 0, i.e., by B ( r ) = r , with r being the radius of the throat. B ( r ) satisfies the flare-out condition B ′ ( r ) < r ) is finite everywhere (from the throat to spatial infinity).For simplicity, we restrict ourselves to θ = π/
2. Because of the spherical symmetry, the sameresults can be applied to all θ . Therefore, the Lagrangian describing the motion of a photon in the θ = π/ L = − e r ) ˙ t + ˙ r − B ( r ) r + r ˙ φ , (2)where a dot represents a derivative with respect to the affine parameter. Since the Lagrangian isindependent of t and φ , we have two constants of motion given by p t = ∂ L ∂ ˙ t = − e r ) ˙ t = − E , p φ = ∂ L ∂ ˙ φ = r ˙ φ = L , (3)where E and L are the energy and angular momentum of the photon, respectively. Using the nullgeodesics condition g µν ˙ x µ ˙ x ν = 0, we obtain e r ) − B ( r ) r ˙ r + V eff = E , V eff = L e r ) r , (4)where V eff is the effective potential. From this, we can obtain the deflection angle given by [6] α = 2 Z ∞ r tp e Φ( r ) drr q − B ( r ) r q b − e r ) r − π , (5)where b = L/E is the impact parameter, and r tp is the turning point given by drdφ = 0. drdφ = 0 givesthe following relation between b and r tp : b = r tp e − Φ( r tp ) . (6)It is well-known that the deflection angle diverges logarithmically to infinity as the turning pointapproaches a photon sphere (which corresponds to the unstable photon orbits), and as a result,3here are an infinite number of relativistic images formed due to lensing, just outside the photonsphere. Circular photon orbits satisfy ˙ r = 0 and ¨ r = 0. Unstable photon orbits, which constitutethe so-called photon sphere, additionally satisfy ... r >
0. Generally, the conditions for the unstablephoton orbit (photon sphere) are written in terms of the conditions on the effective potential. Theseare given by V eff ( r ph ) = E , dV eff dr (cid:12)(cid:12)(cid:12) r = r ph = 0 , d V eff dr (cid:12)(cid:12)(cid:12) r = r ph < , (7)where r ph is the radius of a photon sphere. Note that a photon sphere corresponds to the maximumof the potential. For the metric of Eq.(1), the above conditions become b ph = r ph e − Φ( r ph ) , Φ ′ ( r ph ) = 1 r ph , Φ ′′ ( r ph ) < − r ph , (8)where b ph is the critical impact parameter corresponding to the photon sphere. In addition to thephoton sphere mentioned above, the effective potential can exhibit an extremum at the wormholethroat also, thereby implying circular photon orbits (either stable or unstable) at the throat. Thiscan be seen by rewriting Eq. (4) as˙ r + e − r ) (cid:18) − B r (cid:19) (cid:0) V eff − E (cid:1) = 0 . (9)Note that ˙ r = 0 is automatically satisfied at the wormhole throat r = r since B ( r ) = r there.Therefore, the throat can act as an effective photon sphere when the other two conditions, namely¨ r = 0 and ... r >
0, are satisfied. In terms of the effective potential, this becomes V eff ( r ) = E , dV eff dr (cid:12)(cid:12)(cid:12) r = r < , (10)where we have used the flare-out condition B ′ ( r ) <
1. Note that the last set of conditions forthe throat to be an effective photon sphere are different from those in (7) in the sense that theconditions in (10) require one less derivative of the effective potential than those required for thephoton sphere present outside the throat. This is due to the factor (cid:16) − B ( r ) r (cid:17) present in the radialgeodesic equation. For the metric (1), Eq. (10) becomes b = r e − Φ( r ) , Φ ′ ( r ) < r , (11)where b is the critical impact parameter corresponding to the effective photon sphere at the throat.It is to be noted that, in the Schwarzschild gauge, the effective potential does not seem to havea maxima since dV eff /dr = 0 at the throat, even though the throat acts as an effective photonsphere. This may seem confusing as traditionally the photon sphere is defined as the maximum4f the effective potential. However, things become more clear when we switch over from theSchwarschild gauge to the proper radial coordinates defined by l ( r ) = ± Z rr dr q − B r , (12)where the throat is at l ( r ) = 0, and the two signs correspond to the two different regions connectedthrough the throat. In the proper radial coordinates, the radial geodesic equation becomes e r ( l )) ˙ l + V eff = E , V eff = L e r ( l )) r ( l ) . (13)Therefore, at the throat, we have dV eff dl (cid:12)(cid:12)(cid:12) r = ± r − B ( r ) r dV eff dr (cid:12)(cid:12)(cid:12) r = 0 (14) d V eff dl (cid:12)(cid:12)(cid:12) r = (cid:18) − B ( r ) r (cid:19) d V eff dr (cid:12)(cid:12)(cid:12) r + 12 (cid:18) B r − B ′ r (cid:19) dV eff dr (cid:12)(cid:12)(cid:12) r = (1 − B ′ ( r ))2 r dV eff dr (cid:12)(cid:12)(cid:12) r (15)Now, using B ′ ( r ) < V eff ( r ) = E , dV eff dl (cid:12)(cid:12)(cid:12) r = 0 , d V eff dl (cid:12)(cid:12)(cid:12) r < , (16)which implies that, unlike in the Schwarzschild gauge (see Eq. (10)), in the proper radial coordinate,the effective potential has a maxima at the throat when the throat acts as an effective photon sphere.Therefore, in addition to the outer photon sphere, the wormhole throat can also act as aneffective photon sphere. As a result, two sets of infinite number of relativistic images may beformed due to strong gravitational lensing of light coming from a distant light source. However,for the effective photon sphere at the throat to take part in the formation of relativistic images,the potential must have higher height at the throat than at the photon sphere present outside thethroat. Therefore, the necessary and sufficient conditions that the throat also takes part in theformation of relativistic images in addition to those formed due to the outer photon sphere aregiven by Φ ′ ( r ) < r , e r ) r > e r ph ) r ph , (17)where r and r ph are given by B ( r ) = r , Φ ′ ( r ph ) = 1 r ph . (18)Note that there should be a minima between the throat and the outer photon sphere. Therefore,when the throat acts as an effective photon sphere, maxima and minima outside the throat mustcome in pairs, and the condition Φ ′′ ( r ph ) < − r ph will automatically be satisfied at the maxima.5et us now show how we can construct a wormhole metric which possesses an effective photonsphere at its throat, as well as a photon sphere outside its throat. This demands that we must have V ′ eff = 0 at two different radii outside the throat, with one of them representing the maxima atthe outer photon sphere and the remaining one representing the minima of V eff present in betweenthe throat and the outer photon sphere. Let us consider the following ansatz for V ′ eff ( r ) : V ′ eff ( r ) = − K ( r − β )( r − γ ) r n (19)where K is a positive constant, n is a positive integer and r ≤ β < γ . V ′ eff ( r ) is zero at two points( β , γ ) with γ representing maxima, i.e., V ′′ eff ( r ) (cid:12)(cid:12) γ <
0, and β representing minima of V eff ( r ), i.e. V ′′ eff ( r ) (cid:12)(cid:12) β >
0. Integrating the above form of V ′ eff ( r ), we obtain V eff ( r ) = − K Z ( r − β )( r − γ ) r n dr = Kr − n (cid:18) r n − − r ( β + γ ) n − r ( βγ ) n − (cid:19) (20)From Eqs. (4) and (20), we find e φ ( r ) = Kr − n (cid:18) r n − − r ( β + γ ) n − r ( βγ ) n − (cid:19) (21)where the constant of motion L is absorbed in K .Equation (21) shows that e φ ( r ) remains finite in the range r ∈ [ r , ∞ ) for n ≥
5. Again, theasymptotically flat condition, i.e., e φ ( r ) → r → ∞ , is satisfied only for n = 5. Puttingthis value of n in Eq. (21) and taking the r → ∞ limit, we getlim r →∞ e φ ( r ) = 1 = ⇒ K ⇒ K = 2Therefore, the final form of e φ ( r ) becomes e φ ( r ) = 2 (cid:18) − ( β + γ )3 r + ( βγ )4 r (cid:19) (22)Now we should choose β and γ in such a way that the conditions in (17) (with r ph = γ ) are satisfied.Let us illustrate by taking a specific choice β = r and γ = r . Also, we choose B ( r ) = r − r r so that B ( r ) = r , and M = r is the Arnowitt-Deser-Misner mass of the wormhole. Therefore,we have e φ ( r ) = 1 − r r + 15 r r , − B ( r ) r = 1 − r r + 5 r r . (23)The above wormhole metric was constructed by using the fact that it must have an effective photonsphere at its throat as well as a photon sphere outside its throat. In the next section, however, weconsider a wormhole example which possesses the above feature and arises as an exact solution ina modified theory of gravity. 6 II. A SPECIFIC EXAMPLE IN MODIFIED GRAVITY
We consider a wormhole solution obtained in [45]. The spacetime geometry is given by ds = − ψ ( r ) f ( r ) dt + dr f ( r ) + r (cid:0) dθ + sin θdφ (cid:1) , (24) ψ ( r ) = 1 √ κρ (25) f ( r ) = 1 + κρ − κp θ " − s − κC r κC r (cid:18) Mr − C r F (cid:20) , ,
98 ; κ C r (cid:21) + κC r F (cid:20) , ,
138 ; κ C r (cid:21)(cid:19) − κC r (cid:0) − κC r (cid:1) , (26)where ρ is the energy density given by ρ = C r − κC r , where C is an integration constant. It is to be noted that the factor (cid:0) − κC r (cid:1) present in thedenominator of the last term in Eq. (26) is absent in [45]. This was a typographic error in thatpaper. For κ <
0, the solution represents a wormhole, and the wormhole throat radius is givenby (1 + κρ ) | r = 0, which gives r = (cid:16) | κ | C (cid:17) / [45]. Additionally, we must have x = r | κ | < x >
1, the throat is covered by an event horizon, thereby giving a black holesolution. The matter supporting the above wormhole satisfies all the energy conditions. However,to make the wormhole traversable, the parameter κ , the mass M and the throat radius r of thewormhole have to satisfy the following [45]: M = r | κ | F (cid:20) , ,
98 ; 1 (cid:21) + r | κ | F (cid:20) , ,
138 ; 1 (cid:21) . (27)Figure 1 shows the effective potential for different value of x (both wormhole and black hole).In Figs. 1(a) and 1(b), only the throat acts as an effective photon sphere. In Figs. 1(c)-1(e), thereis an outer photon sphere in addition to the effective photon sphere at the throat. However, unlikethe effective photon sphere at the throat in Figs. 1(c) and 1(d), that in Fig. 1(e) will be hiddenwithin the outer photon sphere and hence will not take part in the formations of relativistic imagesin gravitational lensing since the effective potential at the throat has lesser height than that at theouter photon sphere.Figure 2 shows the numerically integrated deflection angle as a function of the impact parameterfor the metric parameter values same as those in Fig. 1 (both wormhole and black hole). Note7 a) x = 0 . x = 0 .
85 (c) x = 0 . x = 0 .
93 (e) x = 0 .
96 (f) x = 1 . FIG. 1. The effective potential V (= V eff L ) plotted for different x (= r | κ | ) and κ = − that, as expected, the deflection angle diverges as the impact parameter approaches the valuescorresponding to the effective photon sphere at the throat or the outer photon sphere. In the casewhere there is both the effective photon sphere at the throat as well as the outer photon sphere,the deflection angle diverges at the locations of both the throat and the outer photon sphere. Inthis case, as we decrease the impact parameter, the deflection angle diverges to infinity as b → b ph .As we decrease b further below b ph , the deflection angle start decreasing, but is again divergent as b approaches the critical value b corresponding to that of the effective photon sphere at the throat.To study the relativistic images formed due to strong gravitational lensing of light coming froma distant light source, we will, for simplicity, assume that the observer, the lens (the wormhole orthe black hole), and the distant point light source are all aligned. We also consider the situationwhere the observer and the light source are far away from the lens. Therefore, in the observerssky, the relativistic images will be concentric rings (known as relativistic Einstein rings) of radiigiven by the corresponding impact parameters b ( r tp ). These impact parameter values b ( r tp ) canbe obtained by solving α ≃ πn , where n is the ring number [6]. Figure 3 shows the relativisticEinstein rings in the observer’s sky. Note that there are two sets of infinite number of relativisticEinstein rings in the case when there are two photon spheres (effective photon sphere at the throatand the outer photon sphere). This is something different from the black hole case (Fig. 3(f))8 a) x = 0 . x = 0 .
85 (c) x = 0 . x = 0 .
93 (e) x = 0 .
96 (f) x = 1 . FIG. 2. The deflection angle plotted as a function of the impact parameter for different x (= r | κ | ) and κ = − where there is only one set of relativistic rings formed due to a photon sphere.Also note that, when there is only one photon sphere (effective photon sphere or the outerphoton sphere), the rings are closely clumped together. But, when there are two photon spheres,the rings at the outer photon sphere are less closely packed than those at the inner photon sphere.Here we would like to point out that photons with impact parameters less than the critical valuecorresponding to the inner photon sphere will get captured by the wormholes and travel to theother side. Therefore, if no radiation or little amount of radiation comes from the other side, thensuch a wormhole will cast a shadow [29], with the inner set of the relativistic images forming theouter bright edge of the shadow. IV. CONCLUSION
Horizonless compact objects with light rings (or photon spheres) are becoming increasinglypopular, as these objects can be potential alternatives to black holes and can be laboratories fortesting modifications to general relativity at the horizon scale. It is therefore important to studyvarious properties of these objects and see whether or how we can distinguish these objects from9 a) x = 0 . x = 0 .
85 (c) x = 0 . x = 0 .
93 (e) x = 0 .
96 (f) x = 1 . FIG. 3. The relativistic Einstein rings for different x (= r | κ | ) and κ = − black holes through observations. To this end, strong gravitational lensing, where a photon sphereplays a crucial role, is an important observational tool for probing the spacetime geometries aroundthese objects.In this work, we have shown that spherically symmetric, static wormholes of the Morris-Thorneclass can posses two photon spheres. In particular, we have shown that, in addition to a photonsphere present outside a wormhole throat, the throat can also act as an effective photon sphere. Suchwormholes exhibit two sets of relativistic Einstein ring system formed due to strong gravitationallensing of light. If such type of wormhole casts a shadow at all, then the inner set of the relativisticEinstein rings will form the outer bright edge of the shadow. We have illustrated this furtherthrough a specific example. The above-mentioned rich and novel strong lensing feature points outthe vital role that a wormhole throat can play. Such a novel lensing feature, which has not been10ointed out before, may be useful in detecting wormholes in future observations. [1] C. R. Keeton and A. O. Petters, Formalism for testing theories of gravity using lensing by compactobjects: Static, spherically symmetric case , Phys. Rev. D , 104006 (2005); S. Pal and S. Kar, Gravitational lensing in braneworld gravity: Formalism and applications , Classical Quantum Gravity , 045003 (2008).[2] F. W. Dyson, A. S. Eddington, and C. Davidson, A determination of the deflection of light by theSun’s gravitational field, from observations made at the total eclipse of May 29, 1919 , Phil. Trans. R.Soc. A , 291 (1920).[3] A. S. Eddington,
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