Observational Signature and Additional Photon Rings of Asymmetric Thin-shell Wormhole
OObservational Signature and Additional Photon Ringsof Asymmetric Thin-shell Wormhole
Jun Peng , Minyong Guo and Xing-Hui Feng ∗ Van Swinderen Institute, University of Groningen, 9747 AG Groningen, The Netherlands Center for High Energy Physics, Peking University, Beijing 100871, China Center for Joint Quantum Studies and Department of Physics, School of Science, TianjinUniversity, Tianjin 300350, China
Abstract
Recently, a distinct shadow mechanism was proposed by Wang et al. from the asymmetricthin-shell wormhole (ATW) in [Phys. Lett. B 811 (2020) 135930]. On the other hand, Grallaet al’s work [Phys. Rev. D 100 (2019) 2, 024018] represented a nice description of photonrings in the presence of an accretion disk around a black hole. In this paper, we are inspiredto thoroughly investigate the observational appearance of accretion disk around the ATW.Although the spacetime outside an ATW with a throat could be identical to that containinga black hole with its event horizon, we show evident additional photon rings from the ATWspacetime. Moreover, a potential lensing band between two highly demagnified photon ringsis found. Our analysis provides an optically observational signature to distinguish ATWs fromblack holes. [email protected], [email protected], [email protected], ∗ Corresponding author. a r X i v : . [ g r- q c ] F e b Introduction
The first black hole image of M87* released by the Event Horizon Telescope plays a signifi-cant role in the frontier of general relativity [1]. It provides a direct and powerful observationalinformation for general relativity in the strong gravitational regime. As a result the research ofblack hole shadow becomes very popular thereafter [2–26]. Among these interesting works, Grallaet al. originality give an elegant description on black hole shadows, lensing rings and photon ringsconsidering an emission disk around black holes [21], which naturally led to lots of interestingfollow-up works [22–26]. In this context, the term ’shadow’ refers to a dark area outside the blackhole. While, the regular ‘shadow’ denotes the critical curve in the sky of observers and the criticalcurve is closely related to the spherical photon orbits which are always radially unstable [27]. Toavoid misunderstanding, for the latter we go by the name of critical curve in the rest of this article.Therefore, one can conclude that the shape of the shadow could change if different sources of lightare considered for the same black hole. On the contrary, the critical curve is invariant as long asthe spacetime geometry are given.Nevertheless, with the help of the image of M87* we are still not enough to assert that thesupermassive object in the center of M87* must be a black hole. This is because black holes arenot the only ones that have photon spheres which have the ability to shade the horizon. It hasbeen found some ultra compact objects (UCOs) also own photon spheres [27, 28]. Furthermore,some of them are found to mimic the optical appearances of black holes including shadows [29–31].Thus it is crucial to distinguish black holes from UCOs whose external spacetime geometries aresimilar or even identical to those of black holes up to the vicinity of horizon, such as gravastar,wormhole and fuzzball and so on. In fact one may distinguish black holes from some UCOs in theiracoustic properties, such as echo effect [32–35]. Roughly speaking, the essential distinction betweenblack holes and UCOs is that the event horizon of black holes is a one-way membrane while UCOshave no horizons. In this sense, black holes can be regarded as absolute black objects with zeroreflection, while UCOs have considerable reflection.Recently the shadow of an asymmetric thin-shell wormhole (ATW) has been studied in [36].The authors showed that due to reflection of photons by the wormhole there exist a novel shadowwhich is different from that of black hole in certain parameters space. Then more examples ofasymmetric shin-shell wormhole are studied in [37–39], of which the authors named their results asdouble shadows [37]. It’s worth noting that double shadows should be called double critical curvesor photon spheres in the context of our article. On the other hand, the novel shadow proposed in[36] does refer to the dark area, however, its edge is still related to the photon sphere when thethroat of the ATW is inside the photon sphere, but the photon sphere here is the one which islocated in the other side opposite the observers other than the usual one in the observers’ side foran ATW spacetime. So far there’s no discussion on the observational appearance of emission disk inATW spacetime although no matter novel shadows or double shadows are improper terminologiesin practical observation. Our motivation is to give a completed picture of observational appearanceof accretion disk around an ATW.The plan of our paper is organized as follows: we first give a brief review of the asymmetric2hin-shell wormhole in section II. Then we analysis the photon trajectories and deflection angles inthis wormhole in section III. Next we discuss the transfer functions and observational appearancesof emission disks around this wormhole in section IV. We conclude in section V.
In this section, we give a brief review of the asymmetric thin-shell wormhole model presented in[36]. We consider a thin-shell wormhole using cut-and-paste method, that is, two distinct spacetimeswith different parameters are glued by a thin shell ds i = − f i ( r i ) dt i + dr i f i ( r i ) + r i d Ω , (1)where i = 1 ,
2, and by focusing on the Schwarzschild case we have f i ( r i ) = 1 − M i r i , r ≥ R, (2)where M i are the mass parameters, and R is the position of thin-shell, i.e. the radius of throat.Thus we have R > max { M , M } , (3)Figure 1: Asymmetric shin-shell wormhole as a black hole mimicer. We assume that observers arelocated at spacetime M . So the spacetime geometry viewed by observers is identical to that ofblack hole up to the thin-shell.In each spacetime, the radial null geodesic is (cid:16) dr i dτ (cid:17) + V i,eff = 1 b i (4)with the effective potential given by V i,eff = f i ( r i ) r i (5)3
10 5 R 5 100.0.010.02 ( / b c ) ( / b c ) ( / Zb c ) M Figure 2: Plot of effective potential of radial null geodesic in asymmetric thin-shell wormhole. Wehave set M = 1 , M = 1 . , R = 2 .
6, so b c = 3 √ ≈ . Zb c = 3 . M is scaled by Z because the impact parameters intwo spacetimes are connected with each other by (6)where b i = L i E i is called the impact parameter.Without loss of generality, we suppose that observers are located in spacetime M and set M = 1 and M = k . The impact parameters in two spacetimes are connected with each other by b b = (cid:114) R − kR − ≡ Z (6)If there exists photon sphere in M , it is hardly to distinguish a wormhole from a black holebased on direct optical observation, because the horizon is shaded by the photon sphere. So we areinterested in this case, i.e. R <
3. In other words, we want to distinguish a wormhole from a blackhole through more abundant information even if they look like each other.The more abundant information indeed exists. Considering the ingoing null geodesics with b < b c = 3 √ M , certain geodesics would turn back passing through the throatwith a necessary condition b = b Z > b c = 3 √ k. (7)To summarize, the ingoing null geodesic in spacetime M whose impact parameter satisfies3 √ kZ < b < √ , (8)would drop into spacetime M and then turn back to spacetime M passing through the throat.This condition can be satisfied given 1 < k < R ≤ . (9)We show a plot of effective potential in Fig.2. It can be obviously seen from this plot that thephoton in spacetime M with impact parameter lying in the range Zb c < b < b c can turn back4 - - - b = - - - - b = - - - - b = Figure 3: Photon trajectories in the Euclidean polar coordinates ( r, φ ) with impact parameterslying in the range Zb c < b < b c . The photons are coming from far right in spacetime M . The redsolid lines stand for trajectories in spacetime M , and the blue dashed lines stand for trajectoriesin spacetime M . .when it reaches the turning point in spacetime M . We will see that this reflection mechanismby wormhole make a essentially distinction in the observational appearance between wormhole andblack hole in the following. To have a complete understanding of the observational appearance of accretion disk aroundasymmetric thin-shell wormhole through the shadow, photon rings and lensing rings, we need firstto investigate the trajectory and deflection angle of a light ray traveling in the wormhole. It isconvenient to make a coordinate transformation u i = 1 /r i . The trajectory of photon is determinedby orbit equation (cid:16) du i dφ (cid:17) = G i ( u i ) , (10)where G i ( u i ) = 1 b i + 2 M i u i − u i . (11)All trajectories can be divided into three classes. When b > b c , photon in M from infinityapproach one closest point outside the throat, and then move back to infinity in M . When Zb c < b < b c , photon in M from infinity drop into M and then turn back passing through thethroat to infinity in M . When b < Zb c , photon in M from infinity drop into M and move toinfinity in M .For b > b c , the turning point in spacetime M corresponds to the minimally positive real rootof G ( u ) = 0, which we will denote by u min . According to (10), the total change of azimuthal5ngle φ for certain trajectory with impact parameter b can be calculated by φ ( b ) = 2 (cid:90) u min du (cid:112) G ( u ) , b > b c . (12)For Zb c < b < b c , (13)we firstly focus on the trajectory in spacetime M outside the throat. The total change of azimuthalangle φ in spacetime M is obtained by φ ( b ) = (cid:90) /R du (cid:112) G ( u ) , b < b c , (14)The turning point in spacetime M corresponds to the maximally positive real root of G ( u ) = 0,which we will denote by u max . According to (10), the total change of azimuthal angle φ for certaintrajectory with impact parameter b in spacetime M can be calculated by φ ( b ) = 2 (cid:90) /Ru max du (cid:112) G ( u ) , b > b c (15)It is convenient to define three orbit numbers n ( b ) = φ ( b )2 π , n ( b ) = φ ( b ) + φ ( b /Z )2 π , n ( b ) = 2 φ ( b ) + φ ( b /Z )2 π (16)for later use. As Gralla et al’s proposal [21], we focus on the optically and geometrically thin accretion disksfor simplicity. Both static observer and accretion disk are assumed to locate at spacetime M . Thestatic observer is assumed to locate at the north pole, and the accretion disk is on the equatorialplane. The lights emitted from the accretion disk is considered isotropic in the rest frame of thestatic observer. In view of the spherical symmetry of the spacetime, we also suppose the emittedspecific intensity only depends on the radial coordinate, denoted by I em ν ( r ) with emission frequency ν in a static frame. An observer in infinity will receive the specific intensity I obs ν (cid:48) with redshiftedfrequency ν (cid:48) = √ f ν . Considering I ν /ν is conserved along a ray, i.e. I obs ν (cid:48) ν (cid:48) = I em ν ν (17)we have the observed specific intensity I obs ν (cid:48) = f / ( r ) I em ν ( r ) (18)6 b n b n , n b Figure 4: The ranges of impact parameter and plots for first three transfer functions. The black,orange and red ones correspond respectively to the first, second and third transfer functions.So the total observed intensity is an integral over all frequencies I obs = (cid:90) I obs ν (cid:48) dν (cid:48) = (cid:90) f I em ν dν = f ( r ) I em ( r ) (19)where I em = (cid:82) I em ν dν is the total emitted intensity from the accretion disk.According to ray-tracing method, if a light ray from the observer intersects with the emissiondisk, it means the intersecting point as a light source will contribute brightness to the observer.We first consider that a light ray completely travels in spacetime M . As the black hole case, alight ray whose orbit number n > / n goeslarger than 3 /
4, the light ray will bend around the wormhole, intersecting with the disk for thesecond time on the back side. Further, when n > /
4, the light ray will intersect with the diskfor the third time on the front side again, and so on. Hence, the observed intensity is a sum of theintensities from each intersection, I obs ( b ) = (cid:88) m f I em | r = r m ( b ) (20)where r m ( b ) is the so called transfer function which denotes the radial position of the m -th inter-section with the emission disk.As discussed in above sections, for asymmetric thin shell wormhole photon can turn back passingthrough the throat in proper parameters space, so we would have addition transfer functions.According to the definitions of orbit numbers, when n < / n > /
4, the reflected outgoingtrajectories in spacetime M will intersect with the disk on the back side. When n < / n > /
4, the reflected outgoing trajectories in spacetime M will intersect with the disk on thefront side. Thus the impact parameter range for new second transfer functions are determinedby n < / n > / n < / n > / dr/dφ , called the demagnificationfactor. We can see from the right plot of Fig.4 that the new third transfer function near Zb c hasa high slope like the usual third transfer function near b c . Another new third transfer function inthe left of b c has a smaller slope than the usual third transfer function near b c , but bigger slopethan the usual second transfer function in the right of b c . The new second transfer function hasa modest slope like the usual first transfer function, so we had better call the resulted image as’lensing band’.Now we take two typical emission models as examples proposed by Gralla et al. [21] to makephysical pictures more clear I em / I I em / I Figure 5: The left emission profile (model I) is sharply peaked and abruptly end at the innermoststable circular orbit (6 M ). The right emission profile (model II) decaying gradually from thephoton sphere (3 M ) to the innermost stable circular orbit (6 M )The observed intensity can be obtained according to (20). We give corresponding plots anddensityplots of the observed intensities in Fig.6 for emission model I and Fig.7 for emission modelII. We can see that in both models we have two additional photon rings resulted from correspondingtwo new third transfer functions for asymmetric shin shell wormhole. The new photon ring nearcritical curve Zb c is highly demagnified like the photon ring near critical curve b c . Another newphoton ring which is located at the inside of critical curve b c has a considerable size, but is smallerthan the lensing ring which is located at outside of critical curve b c . However in emission modelI, the new second transfer function make no contribution to observed intensity, since this transferfunction is out of the domain of emission model. While in emission model II, we have an additionalsizable lensing band between critical curves Zb c and b c resulted from the new second transferfunction in this emission model. 8 b I obs / I b I obs / I Figure 6: Observed intensity and densityplot of emission model I. The top row corresponds to blackhole and the bottom row corresponds to wormhole. The left collum are the observed intensities. Themiddle collum are the densityplots of observed intensities and the right collum are local densityplots. b I obs / I b I obs / I Figure 7: Observed intensity and densityplot of emission model II. The top row corresponds to blackhole and the bottom row corresponds to wormhole. The left collum are the observed intensities. Themiddle collum are the densityplots of observed intensities and the right collum are local densityplots.9
Conclusion
In this paper, we studied the trajectories of photons and their deflection angles in asymmetricthin-shell wormhole connecting two distinct Schwarzschild spacetimes by the throat. Typically, weplaced observers in the spacetime M of which the mass parameter is smaller than that of theother side. Our interest is mainly about the photon rings in the observers’ sky, we focused on thecase that the throat of the ATW is inside the photon sphere of M . After giving the formulasof deflection angles in each side, we constructed new orbit numbers counting the total deflectionangles and showed the completed trajectories of photons which go through the throat twice, thatis, ingoing photons of M pass through the throat and turn back after they reach the turningpoints in spacetime M , then they go through the throat again. Then, considering optically andgeometrically thin accretion disks around the wormhole, we gave the transfer functions and obtainedthe observed intensity and density plot in the sky of observers based on some emission models.From our calculations, we found the ATW and corresponding Schwarzschild spacetimes differmarkedly in the second and third transfer functions. In particular, the second transfer function isno longer a monotone function, a new segment appears before the monotone increasing part whichalso exists in corresponding Schwarzschild black hole spacetimes. As for the third transfer function,more structures were found in ATW. In addition to a new monotone increasing part, the usual onesplits into two branches. The underlying reasons for these new characteristics is that photons withcertain impact parameters could turn back after they go through the throat while these photonsin corresponding Schwarzschild black hole spacetimes would fall into the event horizon and nevercome back since the event horizon is a one-way membrane.As a result, two additional photon rings are found for ATW spacetime, one of which is highlydemagnified near Zb c (the critical curve in opposite spacetime viewed by observers’ side) like theusual photon ring near b c (the critical curve in observers’ spacetime) and the other one locatedat inside of critical curve b c is much brighter and has a considerable size, even though it’s smallerthan the lensing ring which is located at outside of critical curve b c . Besides, we also found anadditional lensing band when the emission profile overlap the domain of the new second transferfunction. Though the ATW is a constructed model, these additional photon rings or lensing rings(or bands) should be exclusive structures for ultra compact objects because of reflectivity . Ouranalysis provide an optically observational evidence to distinguish UCOs from black holes. Acknowledgments
J.P. is supported by the China Scholarship Council. M.G. is supported by China PostdoctoralScience Foundation Grant No. 2019M660278 and 2020T130020. X.H.F. is supported by NSFC(National Natural Science Foundation of China) Grant No. 11905157 and No. 11935009. In fact, the relevant study of gravastar as another UCO example has been implemented in [40]. eferences [1] K. Akiyama et al. [Event Horizon Telescope Collaboration], Astrophys. J. , no. 1, L1(2019).[2] V. Cardoso, A. S. Miranda, E. Berti, H. Witek and V. T. Zanchin, Phys. Rev. D , 064016(2009)[3] M. Zhang and M. Guo, Eur. Phys. J. C , no.8, 790 (2020)[4] X. C. Cai and Y. G. Miao, [arXiv:2101.10780 [gr-qc]].[5] C. Bambi, K. Freese, S. Vagnozzi and L. Visinelli, Phys. Rev. D , no.4, 044057 (2019)[6] H. Lu and H. D. Lyu, Phys. Rev. D , no.4, 044059 (2020)[7] X. H. Feng and H. Lu, Eur. Phys. J. C , no.6, 551 (2020)[8] M. Guo, S. Song and H. Yan, Phys. Rev. D , no.2, 024055 (2020)[9] L. Ma and H. Lu, Phys. Lett. B , 135535 (2020)[10] R. Q. Yang and H. Lu, Eur. Phys. J. C , no.10, 949 (2020)[11] M. Guo and P. C. Li, Eur. Phys. J. C , no.6, 588 (2020)[12] X. X. Zeng, H. Q. Zhang and H. Zhang, Eur. Phys. J. C , no.9, 872 (2020)[13] X. X. Zeng and H. Q. Zhang, Eur. Phys. J. C , no.11, 1058 (2020)[14] H. Yang, [arXiv:2101.11129 [gr-qc]].[15] V. Perlick, O. Y. Tsupko and G. S. Bisnovatyi-Kogan, Phys. Rev. D , no.10, 104062 (2018)[16] P. C. Li, M. Guo and B. Chen, Phys. Rev. D , no.8, 084041 (2020)[17] M. Zhang and J. Jiang, Phys. Rev. D , no.2, 025005 (2021)[18] W. L. Qian, S. Chen, C. G. Shao, B. Wang and R. H. Yue, [arXiv:2102.03820 [gr-qc]].[19] M. Wang, S. Chen and J. Jing, JCAP , 051 (2017)[20] S. W. Wei and Y. X. Liu, JCAP , 063 (2013)[21] S. E. Gralla, D. E. Holz and R. M. Wald, Phys. Rev. D , no.2, 024018 (2019)[22] M. D. Johnson, A. Lupsasca, A. Strominger, G. N. Wong, S. Hadar, D. Kapec, R. Narayan,A. Chael, C. F. Gammie and P. Galison, et al. Sci. Adv. , no.12, eaaz1310 (2020)[23] E. Himwich, M. D. Johnson, A. Lupsasca and A. Strominger, Phys. Rev. D , no.8, 084020(2020) 1124] S. E. Gralla and A. Lupsasca, Phys. Rev. D , no.12, 124003 (2020)[25] S. E. Gralla, A. Lupsasca and D. P. Marrone, Phys. Rev. D , no.12, 124004 (2020)[26] J. Peng, M. Guo and X. H. Feng, [arXiv:2008.00657 [gr-qc]].[27] M. Guo and S. Gao, [arXiv:2011.02211 [gr-qc]].[28] P. V. P. Cunha, E. Berti and C. A. R. Herdeiro, Phys. Rev. Lett. , no.25, 251102 (2017)[29] A. B. Abdikamalov, A. A. Abdujabbarov, D. Ayzenberg, D. Malafarina, C. Bambi andB. Ahmedov, Phys. Rev. D , no.2, 024014 (2019)[30] B. Narzilloev, J. Rayimbaev, S. Shaymatov, A. Abdujabbarov, B. Ahmedov and C. Bambi,Phys. Rev. D , no.4, 044013 (2020)[31] C. A. R. Herdeiro, A. M. Pombo, E. Radu, P. V. P. Cunha and N. Sanchis-Gual,[arXiv:2102.01703 [gr-qc]].[32] R. A. Konoplya, Z. Stuchl´ık and A. Zhidenko, Phys. Rev. D , no.2, 024007 (2019)[33] V. Cardoso and P. Pani, Living Rev. Rel. , no.1, 4 (2019)[34] L. Buoninfante, A. Mazumdar and J. Peng, Phys. Rev. D , no.10, 104059 (2019)[35] E. Maggio, L. Buoninfante, A. Mazumdar and P. Pani, Phys. Rev. D , no.6, 064053 (2020)[36] X. Wang, P. C. Li, C. Y. Zhang and M. Guo, Phys. Lett. B , 135930 (2020)[37] M. Wielgus, J. Horak, F. Vincent and M. Abramowicz, Phys. Rev. D102