Novel Shadows from the Asymmetric Thin-Shell Wormhole

Physics Letters B 811 (2020) 135930

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Novel shadows from the asymmetric thin-shell wormhole ∗ Xiaobao Wang a, Peng-Cheng Li b,c, Cheng-Yong Zhang d, Minyong Guo b, a School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, PR China b Center for High Energy Physics, Peking University, No. 5 Yiheyuan Rd, Beijing 100871, PR China c Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, No. 5 Yiheyuan Rd, Beijing 100871, PR China d Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou 510632, PR China a r t i c l e i n f o a b s t r a c t

Article history: For dark compact objects such as black holes or wormholes, the shadow size has long been thought to Received 29 September 2020 be determined by the unstable photon sphere (region). However, by considering the asymmetric thin- Accepted 2 November 2020 shell wormhole (ATSW) model, we ﬁnd that the impact parameter of the null geodesics is discontinuous Available online 6 November 2020 through the wormhole in general and hence we identify novel shadows whose sizes are dependent of Editor: N. Lambert the photon sphere in the other side of the spacetime. The novel shadows appear in three cases: (A2) The observer’s spacetime contains a photon sphere and the mass parameter is smaller than that of the opposite side; (B1, B2) there’ s no photon sphere no matter which mass parameter is bigger. In particular, comparing with the black hole, the wormhole shadow size is always smaller and their difference is signiﬁcant in most cases, which provides a potential way to observe wormholes directly through Event Horizon Telescope with better detection capability in the future. © 2020 The Author(s). 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 two distinct Schwarzschild spacetimes whose difference is only the mass parameter [14]. The Generic spherically symmetric dy- The image of M87* taken by Event Horizon Telescope made its namic thin-shell traversable wormholes and their stabilities were debut and revealed the shadow of black holes in April 2019 [1], discussed in [15]. Using the backward ray-tracing method, we care- which has ignited a wave of researches into the shadows of black fully analyze the ingoing null particles that emit from the observer’ holes. As is well-known, the shadow size of a black hole is con- s position in one side and pass through the throat to the opposite sidered to be determined by the unstable photon sphere (region) side of the thin shell. Note that the static observers on both sides other than the event horizon from the earliest work [2,3]. As one of the spacetime are different, thus we first find the impact pa- of the most important predictions of general relativity (GR) be- rameter of the same photon defined in each side is different and sides black holes, wormhole spacetime [4]also contains unstable related by a simple equation. Correspondingly, the turning point photon sphere which also creates shadows which were found to in each side for the same photon becomes different. Considering a be different from the black holes in the size or the oblateness, see physical process that ingoing photons with no turning point would examples in [5–10]. Even though it has been shown a traversable fall into the thin shell and then turn into outgoing in the other wormhole with ordinary matter is not allowed in GR, if there is side, we first find some of them may hit its turning point de- some exotic matter in the universe, it becomes possible [11,12]. fined in this spacetime and then turn back to its birthplace. Thus, Furthermore, if the exotic matter distributes into a thin shell, a we can see the difference between the traversable wormhole and simple model of thin-shell wormhole can be constructed by the black hole from the null geodesic motion. It’s worth noting that for “cut and paste” technique [13,14]. And as far as we know, the a black hole, the light that enters the event horizon never returns, shadow of a asymmetric thin-shell wormhole has not been studied since the event horizon is a “one-way” membrane. before in general. Based on our new finding, we identify novel shadows in ATSW In this letter, we focus on an asymmetric thin-shell wormhole spacetime (see [16]for a similar study). In particular, these novel (ATSW) spacetime, more specially, a static thin shell connecting shadows originate from spacetime asymmetry of both sides con- nected by the throat and their sizes do not depend on the photon sphere, contrary to what we typically think about. Specifically, let’s Corresponding author. * assume we stay in the spacetime M and the radius of the thin E-mail addresses: [email protected] (X. Wang), [email protected] 1 (P.-C. Li), [email protected] (C.-Y. Zhang), [email protected] shell is R. See Fig. 1, the shadow of the wormhole seen by the (M. Guo). static observer is the same with that of the corresponding black https://doi.org/10.1016/j.physletb.2020.135930 0370-2693/© 2020 The Author(s). 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. X.Wang,P.-C.Li,C.-Y.Zhangetal. Physics Letters B 811 (2020) 135930

b2 = r2/ f (√r) has two real roots for the positive branch of r when b ≥ bc = 3 3M. For b = bc , both roots are equal to r ph = 3M is known as the radius of photon sphere. For b > bc , the roots are ad- dressed as the turning points with the larger root rl > r ph and the = s ph smaller one 2M rh < r < r , where rh is the radius of the event horizon. Also, we introduce a new parameter bl,s ≡ rl,s/ f (rl,s) that will be later used. Moreover, considering a distant static observer at (to, ro →∞, π/2, 0), the angular radius of the shadow c left by the photon sphere satisfies sin θ = b /ro and the radius of sh c the shadow is defined by r ≡ ro sin θ = b , as seen in Fig. 2. Next, let’s introduce some necessary background knowledge of thin-shell wormhole using cut-and-paste method, that is, two distinct spacetimes M1,2 with different parameters are glued by a thin shell which forms a new manifold M = M1 ∪ M2, as seen in Fig. 3. We suppose that a spherical thin shell is moving in a spherically symmetric spacetime and the metrics on both sides take in Fig. 1. The shadow of thin-shell wormhole for various parameters. There is novel this form shadow for some situations. − ds2 =−f (r )dt2 + f 1(r )dr2 + r2d2, (3) ph i i i i i i i i hole when M contains the photon sphere R ≤ r and M > M , 1 1 1 2 = that is, there is no novel shadow and the throat behaves like an where i 1, 2, and by focusing on the Schwarzschild case we have 2Mi fi (r) = 1 − , where Mi are the mass parameters. “event horizon”. However, we find novel shadows in other cases. ri ph When R ≤ r and M2 > M1, the shadow size of the wormhole The local tetrads in the neighborhood of the thin shell of each 1 √ M sh R−2M2 spacetime 1,2 becomes r = 3 3M2 − which is determined by the pho- 1 R 2M1 ton sphere of the opposite side and always smaller than that in − 1 ∂ a ea ≡ f 2 (R) , (4) Schwarzschild √spacetime since the photons with the impact pa- ti i ∂t M i rameter b1 < 3 3M1 would fall into the spacetime 2 and turn a M a ∂ back to 1 which never occurs for black hole.√ In particular, the e ≡ f (R) , (5) ri i size of the shadow can range from zero to 3 3M1, so that for ∂ri some parameters the difference can be large enough to be de- are related by the following Lorentz transformation [17] tected to distinguish the wormhole from the black hole. When ph a a there is no photon sphere in M , that is, R > r , surprisingly, e e 1 1 t2 = ( − ) t1 , (6) there also exists a novel shadow. For M > M since R is the turn- ea 1 2 ea 1 2 r2 r1 ing point for certain null geodesics with the corresponding impact R R where we have defined parameter b1 and the photons with b1 < b1 will go through the thin shell and never turn back, the shadow of the wormhole de- = cosh sinh sh = R () , (7) pends on the thin shell and its radius is r1 . While for sinh cosh f1(R) { } M M2 > M1 and max 2M2, 3M1 < R < 3M2, the side 2 contains with R the photon sphere and some null geodesics with b1 < b1 could M ˙ turn back after arriving at the spacetime 2, thus the critical im- −1 i R √ i = sinh . (8) R−2M2 pact parameter is 3 3M2 − and the novel shadow size is fi(R) √ R 2M1 sh R−2M2 R ˙ r = 3 3M2 which is smaller than b . Interestingly, this Here R is the radius of the thin shell and R denotes the velocity 1 R−2M1 1 ≤ ph of the moving thin shell. expression is the same with that for M2 > M1 and R r1 , see Fig. 1. 3. Conserved quantities and some conventions 2. Background of Schwarzschild black hole shadow and the thin Now, we consider an ingoing photon from the spacetime M shell wormhole 1 passing through the thin shell. For simplicity, we assume the in- teraction between the photon and the thin shell is only governed Let us start from some important physical quantities and by gravity which implies the 4-momentum pa is invariant during useful symbols in terms of the shadow in the background of the process passing through the thin shell. The similar treatment Schwarzschild black hole whose metric takes in this form has been applied in [18]In addition, the metric of the spacetime M M 2 2 −1 2 2 2 M is continuous by definition which means g 1 R = g 2 R . ds =−f (r)dr + f (r)dr + r d , (1) ab ( ) ab ( ) =− = Hence, the conserved quantities pti Ei and pφi Li along a where f (r) = 1 − 2M/r. Considering a null particle with momen- geodesic imply tum vector pa, the radial motion of the geodesic is described by E cosh ( − ) sinh ( − ) the equation − 2 =− 1 2 + 1 2 r1 E1 pR , f2(R) f1(R) f1(R) b2 L = L , (9) pr =±E 1 − f (r), (2) 1 2 2 r r1 at the position of the thin shell, and here we have used pR to where b = L/E is the impact parameter and ± stands for outgo- denote the value of pr1 on the thin shell. Using Eq. (2), Eq. (9)can ing and ingoing directions, respectively. The equation pr = 0gives be simplified as

2 X.Wang,P.-C.Li,C.-Y.Zhangetal. Physics Letters B 811 (2020) 135930

Fig. 2. The left picture shows behaviors of the function r3(pr )2 with three different kinds of impact parameters. There’s no turning point for b < bc , single turning point for b = bc and two turning points for b > bc . The right one presents the shadow of Schwarzschild black hole.

needs to be larger than the horizon of the black hole in the corresponding spacetime. Thus, whether the photon sphere exists in M1 is unknown, that is, both max{2k, 2} < R ≤ 3 and R > 3are possible, but if the latter is true, wormholes are even more different from black holes in terms of null geodesic’s motion. Therefore, we would like to discuss the former first. (A1). Since the relationship of size between M1 and M2 is not clear, but it matters a lot. We might as well consider 0 < k < 1 < ≤ ≥ c R/2 3/2now. In addition, for the geodesics with b1 b1, the known results in Schwarzschild spacetime containing a black hole can be applied to the thin-shell wormhole.√ Next, we carefully dis- c = cuss the ingoing photons with b1 < b1 3 3, we find √ Fig. 3. The diagram of the thin-shell wormhole spacetime. In this figure, we give √ M R − 2 an example that in 1 an ingoing photon with certain impact parameter passes √ ≡ c M M b2 < 3 3 B2. (12) through the thin shell and stops at its turning point in 2 and turns back to 1. R − 2k Furthermore, since R > 2 > 2k, there are always null geodesics b1 f2(R) R r2 = = , (10) with b2 such that pR 0. From simple analysis, we find b2 < b2 f1(R) c ≤ R ≤ B2 b2 is always true for 2 < R 3. Hence, when R > 3k, R is the larger real root making pR = 0 which implies all the out- where for simplicity we have set R˙ = 0without losing the physics 2 R nature, that is, we assume the thin-shell wormhole is static which going null geodesics with b2 < b2 starting from the thin shell M can be proven that if we tolerate the violation of null energy con- will go to the infinity in 2. In the range 0 < k < 1, the re- dition, we could always find the thin-shell wormhole to be static. gion that doesn’t satisfy the condition R > 3k is 2/3 < k < 1 and 2 < R < 3k, where R is the smaller real root. In this situation, one From this equation, we find b1 = b2 when f1(R) = f2(R), that is expects the outgoing photons bc < b < bR will stop at the turn- M1 = M2, which tells us that the spacetime M is symmetric about 2 2 2 ing point in the spacetime M and bounce back to the M , thus the thin shell. However, if b1 = b2, the mass parameters of M1 and 2 1 the radius of a novel shadow of the thin-shell wormhole would M2 are no longer equal and the spacetime M becomes asymmet- √ sh = R−2k ≡ c ric about the thin shell which we focus on in this letter. be r1 3 3k R−2 B1. However, after some algebraic calcu- Before we move to discuss the shadow of the asymmetric thin- c c lus, we find B2 < b2 always holds for 2/3 < k < 1 and 2 < R < 3k, shell wormhole, in order to calculate conveniently we would like which tells us there is no turning point for null geodesics with = = c to establish some conventions. Let’s suppose M1 1 and M2 k > b2 < B2. From the above, we conclude that when M1 > M2 and 0for the mass parameter. Thus we have R > max{2, 2k} for thin- the side M1 contains a photon sphere, the shadow of the thin- shell wormhole. Based on the previous review of the background shell wormhole observed by the static observer is the same with the corresponding black hole. of the shadow, the radius of the photon sphere of M1 and M2 ph ph is r = 3 and r = 3k, respectively. Correspondingly, the critical (A2). Next, we turn to the case b1/b2 < 1, that is, 1 < k < R/2, 1 2 √ √ ph c c ≤ = impact parameter reads b = 3 3 and b = 3 3k for each side. we have R 3 < 3k r2 which implies the photon sphere always 1 2 M R = R exists in the√ spacetime 2. Considering the ingoing null geodesics Furthermore, as mentioned before we use b1 2 to denote , f1,2(R) with b1 < 3 3in the spacetime M1, if we wish some geodesics the specific impact parameter in which case one finds pR = 0. 1,2 would turn back passing through the throat, a necessary condition Using these conventions, Eq. (10)can be rewritten as − R 2 c b2 = b1 > b , (13) b1 R − 2k 2(k − 1) − 2 = = 1 − . (11) R 2k − − b2 R 2 R 2 must be hold. Thus, we need 4. The size of the thin-shell wormhole − √ c R 2k c B = 3 3k < b1 < b , (14) 1 R − 2 1 Now, we directly face the calculation of the thin-shell wormhole shadow. Unlike horizons of black holes, the throat of thin- holds. Thus, the key is to check if this condition can be satisfied R c shell wormhole is not fixed, instead, the thin-shell throat only given 1 < k < 2 and 2 < R < 3. Note that B1 is a decreasing func-

3 X.Wang,P.-C.Li,C.-Y.Zhangetal. Physics Letters B 811 (2020) 135930

other side of the spacetime. However, different from the “one way” property of black holes, some of them above a new critical impact parameter will stop at the turning point, bounce back to the orig- inal spacetime and come into the eyes of the static observer due to the asymmetry. Thus, the novel shadow is always smaller than the analogue of black holes. The difference can be very signiﬁcant to overcome the uncertainty in the measurements of black hole mass and its distance from observers so that observing the thin- shell wormholes directly through Event horizon Telescope becomes feasible. While, the observer lives on the side with a larger mass parameter for a wormhole spacetime, the observational shadow is as same as that of a black hole, so one cannot discern them by

Fig. 4. The novel shadow for (A2) case, the spacetime M1 contains the photon imaging shadow. M M sphere and the mass parameter of 1 is smaller than that of 2, namely, 1 < Moreover, when the radius of the thin shell is larger than that ≤ k < R/2 3/2. It can be seen that the shadow of wormhole is smaller than that of the photon sphere, we also show that there are novel shadows of black hole compared to the right graph in Fig. 2. In fact, from our analysis, the even though the observer’s spacetime does not contain the photon novel shadow of wormhole is always smaller and could be arbitrarily small. sphere, which is an important complement to the related existing knowledge. tion of k in the range k ∈ 1, R , we have 0 < Bc < bc , that is to 2 1 1 The mechanism for predicting this novel shadow of the thin- say, the desired condition is always satisfied. Therefore, we find shell wormhole is very simple and elegant, we believe it could be a that for 1 < k < R ≤ 3/2, the angular radius of the shadow of the 2 universal property. Thus it would be extremely interesting to see if thin shell is always smaller than that of the corresponding black this novel shadow can be found in other wormhole models in gen- ∼ hole, see Fig. 4. In particular, for R 2k, the radius of the shadow eral relativity or modified theories of gravity. Furthermore, search- can be arbitrarily small. In terms of the novel shadow induced by ing for this novel shadow by Event Horizon Telescope is also an the asymmetric mechanism of the thin-shell wormhole, the ob- exciting task in the future to know whether the thin-shell worm- M servational size of the shadow in the spacetime 1 can be very hole exists, or not. Concerning the EHT2017 observations [1], as different between wormholes and black holes which may be de- analyzed in [19]the compact objects whose shadows exhibit qual- tected by the EHT directly, if the resolution is raised to a good itatively deviation from those of black holes can be ruled out for enough level. M87* (however, see [20]for exception). So the thin-shell wormhole (B1). Now, we move to the case R > 3, and also focus on 0 < models can be strongly constrained with the present observations. k < 1at first. Since R > 3, R must be a outer turning point for a However it may still have the chance to discover wormhole by EHT R = R class of null geodesics with the impact parameter b1 . And f1(R) according to the novel shadow feature in the future. the incident null geodesics will bounce back to the infinity when ≥ R M Declaration of competing interest b1 b1 in the spacetime 1. R R For b1 < b1 , note b1 is an increasing function of R, we have R c The authors declare that they have no known competing finan- b1 > b1. When the null geodesics pass through the wormhole cial interests or personal relationships that could have appeared to M R−2 R = throat and arrive at the spacetime 2 we have b2 < R−2k b1 influence the work reported in this paper. R R b2 . Also, we find b2 is an increasing function of R for 0 < k < 1, R c = ph Acknowledgements and obtain b2 > b2. Combining with the condition R > 3k r2 we conclude the outgoing geodesics with b < bR will go to infinity in 2 2 The work is in part supported by NSFC Grant No. 11335012, the spacetime M , thus the novel shadow radius observed in the 2 No. 11325522 and No. 11735001. MG and PCL are also sup- spacetime M is rsh = bR . 1 1 1 ported by NSFC Grant No. 11947210. And MG is also funded by (B2). Next, we pay our attention to the case k > 1. Similarly, China Postdoctoral Science Foundation Grant No. 2019M660278 we find bR > bc for R > max{2k, 3}. Thus, for R > 3k, the outgo- 2 2 and 2020T130020. PCL is also funded by China Postdoctoral Sci- ing geodesics will go to infinity in the spacetime M , that is, the 2 ence Foundation Grant No. 2020M670010. CYZ is supported by shadow radius is rsh = bR . While, for max{2k, 3} < R < 3k, the crit- 1 1 √ √ NSFC Grant No. 11947067. c = c = R−2k ical impact parameter b2 3 3k corresponds to B1 3 3k R−2 which is same with the case 2 < R ≤ 3in surprise. And after some References simple analysis, we find Bc < bR as expected. Therefore, in this 1 1 [1] K. Akiyama, et al., Event Horizon Telescope Collaboration, Astrophys. J. Lett. 875 case, the asymmetry of the spactime M results in a novel shadow (2019) L1. very interestingly, which is also worth in-depth study. [2] J.L. Synge, Mon. Not. R. Astron. Soc. 131 (3) (1966) 463. [3] J.M. Bardeen, Timelike and null geodesies in the Kerr metric, in: C. Witt, B. 5. Discussion Witt (Eds.), Proceedings, Ecole d’Ete de Physique Theorique: Les Astres Occlus, Les Houches, France, 1973, pp. 215–239. [4] M. Visser, Lorentzian Wormholes: From Einstein to Hawking, AIP, Woodbury, To summarize, by analyzing the behavior of null geodesics pass- USA, 1995, 412 p. ing through the throat of the static thin-shell wormhole model, we [5] O. Sarbach, T. Zannias, AIP Conf. Proc. 1473 (1) (2012) 223. identify novel shadows resulting from the asymmetry between the [6] C. Bambi, Phys. Rev. D 87 (2013) 107501. [7] P.G. Nedkova, V.K. Tinchev, S.S. Yazadjiev, Phys. Rev. D 88 (12) (2013) 124019, two sides of the thin shell and their sizes are not dependent on the https://doi .org /10 .1103 /PhysRevD .88 .124019, arXiv:1307.7647 [gr-qc]. photon sphere in the observer’s spacetime. When the observer’s [8] T. Ohgami, N. Sakai, Phys. Rev. D 91 (12) (2015) 124020, https://doi .org /10 . spacetime contains a phono sphere, if a static observer lives on 1103 /PhysRevD .91.124020, arXiv:1704 .07065 [gr-qc]. the side with a smaller mass parameter for a wormhole space- [9] F. Lamy, E. Gourgoulhon, T. Paumard, F.H. Vincent, Class. Quantum Gravity 35 (11) (2018) 115009, https://doi .org /10 .1088 /1361 -6382 /aabd97, arXiv:1802 . time, taking the backward ray-tracing point of view, all the ingoing 01635 [gr-qc]. geodesics√ with impact parameters smaller than the critical one [10] F.H. Vincent, M. Wielgus, M.A. Abramowicz, E. Gourgoulhon, J.P. Lasota, T. Pau- 3 3will pass through the thin shell and become outgoing in the mard, G. Perrin, arXiv:2002 .09226 [gr-qc].

4 X.Wang,P.-C.Li,C.-Y.Zhangetal. Physics Letters B 811 (2020) 135930

[11] H.G. Ellis, J. Math. Phys. 14 (1973) 104. [17] D. Langlois, K.i. Maeda, D. Wands, Phys. Rev. Lett. 88 (2002) 181301. [12] M.S. Morris, K.S. Thorne, Am. J. Phys. 56 (1988) 395. [18] K.i. Nakao, T. Uno, S. Kinoshita, Phys. Rev. D 88 (2013) 044036, https://doi .org / [13] M. Visser, Phys. Rev. D 39 (1989) 3182. 10 .1103 /PhysRevD .88 .044036, arXiv:1306 .6917 [gr-qc]. [14] M. Visser, Nucl. Phys. B 328 (1989) 203. [19] K. Akiyama, et al., Event Horizon Telescope Collaboration, Astrophys. J. 875 (1) [15] N.M. Garcia, F.S.N. Lobo, M. Visser, Phys. Rev. D 86 (2012) 044026, https://doi . (2019) L5. org /10 .1103 /PhysRevD .86 .044026, arXiv:1112 .2057 [gr-qc]. [20] C. Bambi, K. Freese, S. Vagnozzi, L. Visinelli, Phys. Rev. D 100 (4) (2019) 044057. [16] D.C. Dai, D. Stojkovic, Observing a Wormhole, Phys. Rev. D 100 (8) (2019) 083513.