Eur. Phys. J. C (2023) 83:361 https://doi.org/10.1140/epjc/s10052-023-11486-y Regular Article - Theoretical Physics Observational appearance of a freely-falling star in an asymmetric thin-shell wormhole Yiqian Chen1,a, Peng Wang1,b, Houwen Wu1,2,c, Haitang Yang1,d 1 Center for Theoretical Physics, College of Physics, Sichuan University, Chengdu 610064, China 2 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Received: 13 February 2023 / Accepted: 6 April 2023 / Published online: 4 May 2023 © The Author(s) 2023 Abstract It has been recently reported that, at late times, the total luminosity of a star freely falling in black holes decays exponentially with time, and one or two series of flashes with decreasing intensity are seen by a specific observer, depending on the number of photon spheres. In this paper, we examine observational appearances of an infalling star in a reflection-asymmetric wormhole, which has two photon spheres, one on each side of the wormhole. We find that the late-time total luminosity measured by dis- tant observers gradually decays with time or remains roughly constant due to the absence of the event horizon. Moreover, a specific observer would detect a couple of light flashes in a bright background at late times. These observations would offer a new tool to distinguish wormholes from black holes, even those with multiple photon spheres. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . 1 2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Numerical results . . . . . . . . . . . . . . . . . . . 4 3.1 Scenario I . . . . . . . . . . . . . . . . . . . . 5 3.2 Scenario II . . . . . . . . . . . . . . . . . . . . 8 4 Conclusions . . . . . . . . . . . . . . . . . . . . . 11 References . . . . . . . . . . . . . . . . . . . . . . . . 11 a e-mail: chenyiqian@stu.scu.edu.cn b e-mail: pengw@scu.edu.cn c e-mail: hw598@damtp.cam.ac.uk (corresponding author) d e-mail: hyanga@scu.edu.cn 1 Introduction The event horizon telescope (EHT) collaboration released images of the supermassive black holes M87* [1–8] and Sgr A* [9–14], which provides a new method to test general rel- ativity in the strong field regime. The main feature displayed in these images is a central brightness depression, namely black hole shadow, surrounded by a bright ring. The edge of black hole shadow involves a critical curve in the sky of observers, which is closely related to some unstable bound photon orbits. For static spherically symmetric black holes, unstable photon orbits form photon spheres outside the event horizon. Since light rays undergo strong gravitational lens- ing near photon spheres, black hole images encode valuable information of the geometry in the vicinity of photon spheres. Therefore, black hole images have been widely studied in the context of different theories of gravity, e.g., nonlinear elec- trodynamics [15–21], the Gauss–Bonnet theory [22–25], the Chern–Simons type theory [26,27], f (R) gravity [28–30], string inspired black holes [31–34] and other theories [35– 46]. On the other hand, testing the nature of compact objects in the universe has been an important question in astrophysics for decades. Although the black hole images captured by EHT are in good agreement with the predictions of Kerr black holes, the black hole mass/distance and EHT system- atic uncertainties still leave some room within observational uncertainty bounds for black hole mimickers. An intrigu- ing type of black hole mimickers is exotic compact objects, which are more massive than neutron stars but horizonless (e.g., boson stars, gravastars and wormholes). Among exotic compact objects, those with enough compactness to possess light rings (or photon spheres in the spherically symmetric case) are called ultra compact objects (UCOs) [47]. UCOs are of particular interest since their observational signatures can be quite similar to those of black holes [48–51]. Never- 123 http://crossmark.crossref.org/dialog/?doi=10.1140/epjc/s10052-023-11486-y&domain=pdf mailto:chenyiqian@stu.scu.edu.cn mailto:pengw@scu.edu.cn mailto:hw598@damtp.cam.ac.uk mailto:hyanga@scu.edu.cn 361 Page 2 of 14 Eur. Phys. J. C (2023) 83 :361 theless, it is of great importance to seek observational signals to distinguish UCOs from black holes. For example, due to a reflective surface or an extra photon sphere, echo signals associated with the post-merger ringdown phase in the binary black hole waveforms can be found in various UCO models [52–62]. In addition, asymmetric thin-shell wormholes with two photon spheres were found to have double shadows and an additional photon ring in their images [63–67]. For black holes with one photon sphere, there is one shadow and one photon ring1 in black hole images, and no echo signal in late- time waveforms. These observational features would allow us to distinguish wormholes from black holes with one pho- ton sphere. Intriguingly, more than one photon sphere has been reported to exist outside the event horizon for a class of hairy black holes in certain parameter regions [69–73]. Multiple photon spheres can introduce distinctive features in black hole images, e.g., double shadows [73], extra photon rings [74] and tripling higher-order images [75]. Furthermore, late- time echo signals were also observed since the effective potential of a scalar perturbation possesses a multiple-peak structure [76,77]. Can we distinguish black holes with multiple photon spheres from UCOs? To answer this question, we investigate dynamic observations of a luminous object freely falling in an asymmetric thin-shell wormhole in this paper. Lately, obser- vational appearances of a star freely falling onto black holes with a single or double photon spheres have been numerically simulated [78,79]. Particularly, the total observed luminosity fades out exponentially with a declining tail, which is caused by photons orbiting around the photon sphere, in the single- photon-sphere case. In contrast, when there exist two photon spheres, the total luminosity exhibits two exponential decays and a sharp peak between them. In addition, due to photons trapped between two photon spheres, a specific observer can detect one more cascade of flashes in the double-photon- sphere case. Recently, luminous matter falling onto a black hole has been reported to occur periodically near the Cyg X-1 [80] and the Sgr A* source [81,82]. Moreover, a new way to measure the spin of Sgr A* was proposed by simulating an infalling gas cloud [83]. In practice, detecting photons cir- cling around photon spheres several times at late times could be a challenging task due to the scarcity of these photons. Interestingly, it showed that precise measurements of photon rings, which are formed of photons circling around photon spheres more than once, may be feasible with a very long 1 A photon ring consists of an infinite series of subrings, where the n-th subring is produced by photons that orbit the black hole by n half-orbits. Depending on the emitting plasma details, the subrings either cannot be visibly distinguished, which leads to a single observable photon ring, or are visibly distinct, which decomposes the photon ring to an infinite sequence of exponentially demagnified images [68]. baseline interferometry [68,84,85]. Therefore, it is timely to study observational appearances of a freely-falling star in the wormhole background, which provides a new way to detect wormholes. The rest of the paper is organized as follows. In Sect. 2, we briefly review the asymmetric thin-shell wormhole and introduce our observational settings. Numerical results are presented in Sect. 3. Finally, we conclude with a brief dis- cussion in Sect. 4. We set G = c = 1 throughout this paper. 2 Setup As introduced in [63,66,86], an asymmetric thin-shell worm- hole has two distinct spacetimes, M1 and M2, which are glued together by a thin shell at its throat. The metric of the wormhole is described as ds2 i = − fi (ri )dt 2 i + dr2 i fi (ri ) + r2 i d�2, (1) where i = 1 and 2 indicate quantities inM1 andM2, respec- tively. Focusing on the Schwarzschild spacetime, we have fi (ri ) = 1 − 2Mi ri for ri ≥ R, (2) where Mi are the mass parameters, and R is the throat radius. Since the wormhole has piecewise Schwarzschild metric, the stress-energy tensor is zero except at the throat. At the throat, discontinuity of derivatives of the metric indicates the pres- ence of a massive thin shell, whose stress-energy tensor can be obtained by the “junction condition” formalism [86]. Note that the energy density was found to be negative, showing that the thin shell is comprised of some exotic matter. With- out loss of generality, we set M1 = 1 and M2 = k in the rest of this paper. For more details of the asymmetric thin-shell wormhole, refer to [63]. In M1 and M2, the local tetrads are eti = f − 1 2 i (ri ) ∂ ∂ti , eri = f 1 2 i (ri ) ∂ ∂ri , eθi = 1 ri ∂ ∂θi , eφi = 1 ri sin(θ) ∂ ∂φi . (3) At the throat, one has et1 = et2 , er1 = −er2 , eθ1 = eθ2 and eφ1 = eφ2 , which yields the relations between the bases of the tangent space of M1 and M2, ∂ ∂t1 = Z−1 ∂ ∂t2 , ∂ ∂r1 = −Z ∂ ∂r2 , ∂ ∂θ1 = ∂ ∂θ2 , ∂ ∂φ1 = ∂ ∂φ2 , (4) where Z ≡ √ f2(R)/ f1(R). Therefore, the components of a vector at the throat in M1 and M2 are related by V t1 = ZV t2 , Vr1 = −Z−1Vr2 , V θ1 = V θ2 , V φ1 = V φ2 . (5) 123 Eur. Phys. J. C (2023) 83 :361 Page 3 of 14 361 In this paper, we study a point-like star freely falling along the radial direction at θi = π/2 and ϕi = 0, which emits pho- tons isotropically in its rest frame. With spherical symmetry, we can confine ourselves to emissions on the equatorial plane. The geodesics on the equatorial plane are described by the Lagrangian L = −1 2 [ fi (ri )ṫ 2 i + 1 fi (ri ) ṙ2 i + r2 i ϕ̇i 2 ] , (6) where dots stand for derivative with respect to an affine parameter τ . Since the Lagrangian L does not depend on coordinates ti and ϕi , the geodesics can be characterized by their conserved energy Ei and angular momentum li in Mi , Ei = −pti = fi (ri )ṫi , li = pϕi = r2 i ϕ̇i . (7) Note that, according to Eq. (5), one has E1 = E2/Z and l1 = l2. The Lagrangian of the freely-falling star obeys the con- stancy L = −1/2 when the affine parameter τ is chosen as the proper time. Since the star falls radially, its angular momentum li = 0. Due to the traversability of the worm- hole, we consider two scenarios with distinct trajectories of the star. In the scenario I, the star with energy E1 = 1/Z (E2 = 1) has a nonzero initial velocity at spatial infinity of M1. So, the star can pass through the throat and travel towards spatial infinity of M2. With the relation (7), the four-velocities of the star in M1 and M2 are given by vμ1 e (r1) = ( 1 1 − 2r−1 1 √ R − 2 R − 2k ,− √ 2k − 2 R − 2k + 2 r1 , 0, 0 ) , vμ2 e (r2) = ( 1 1 − 2kr−1 2 , √ 2k r2 , 0, 0 ) . (8) In the scenario II, the star with energy E1 = 1 is initially at rest at spatial infinity of M1. At first, the star falls freely in M1, passes through the throat and reaches a turning point in M2. Then, it moves towards the throat in M2, returns to M1 and comes to rest at spatial infinity of M1. Similarly, the four-velocities of the star in M1 and M2 are vμ1 e (r1) = ( 1 1 − 2r−1 1 ,∓ √ 2M r1 , 0, 0 ) , vμ2 e (r2) = ( 1 1 − 2kr−1 2 √ R − 2k R − 2 ,± √ −2k + 2 R − 2 + 2k r2 , 0, 0 ) , (9) where plus and minus signs represent outward and inward moving, respectively. Moreover, null geodesics on the equatorial plane are also governed by the Lagrangian (6) with L = 0, which rewrites the radial component of the null geodesic equations as ṙi 2 L2 i = 1 b2 i − Vi ,eff (ri ) , (10) where bi ≡ li/Ei is the impact parameter, and Vi ,eff (ri ) = fi (ri )r −2 i is the effective potential. Note that the impact parameters of a null geodesic in M1 and M2, namely b1 and b2, are related by b1 = Zb2. A photon sphere in Mi is constituted of unstable circular null geodesics, whose radius rph i is determined by Vi ,eff(r ph i ) = 1 (b ph i )2 , V ′ i ,eff(r ph i ) = 0, V ′′ i ,eff(r ph i ) < 0, (11) where bph i is the corresponding impact parameter. Photons with bi ≈ bph i are temporarily trapped at the photon sphere and can determine late-time observational appearances of the wormhole. If the throat radius satisfies max{2, 2k} < R < min{3, 3k}, the asymmetric thin-shell wormhole can be free of the event horizon and possess two photon spheres, which are located at rph 1 = 3 and rph 2 = 3k in M1 and M2, respec- tively. In this paper, we consider the asymmetric thin-shell wormhole with k = 1.2 and R = 2.6, whose observational appearance of an accretion disk has been discussed in [66]. We assume that the emitted photons are collected by dis- tant observers distributed on a celestial sphere located at r1 = ro in M1. To trace light rays emitting from the star to a distant observer, one needs to supply initial conditions. For a photon of four-momentum pμi , the momentum mea- sured in the rest frame of the star with four-velocity v μi e at ri = re is pt̂ = −vtie (re)pti − vrie (re)pri , pr̂ = − √[ v ti e (re) ]2 − f −1 i (re)pti ± √[ v ri e (re) ]2 + fi (re)pri , pθ̂ = 0, pϕ̂ = pϕi re , (12) where plus and minus signs correspond to negative and pos- itive v ri e , respectively. The emission angle α is defined as cos α = pr̂ pt̂ , (13) which is the angle between the propagation direction of the photon and the radial direction in the rest frame of the star. In the rest frame, the photon is emitted with proper frequency ωe = − ( v μi e pμi ) e = pt̂ . For a distant static observer with four-velocity v μ1 o = (1, 0, 0, 0), the photon is observed with frequency ωo = − ( v μ1 o pμ1 ) o = pt1 . With Eqs. (5), (12) and (13), we express the normalized frequency ωo/ωe as a func- tion of the star position re and the emission angle α for two scenarios in Table 1. Furthermore, the luminosity of photons is given by Lk = dEk/dτk , where Ek is the total energy, τk 123 361 Page 4 of 14 Eur. Phys. J. C (2023) 83 :361 Table 1 The normalized frequency ωo/ωe as a function of the star position re and the emission angle α in the scenarios I and II. Inward and outward correspond to travelling towards and away from the throat, respectively Inward Outward Scenario I M1 √ R−2 R−2k − cos(α) √ 2k−2 R−2k + 2 re / M2 / √ R−2 R−2k − cos(α) √ R−2 R−2k + √ 2k re Scenario II M1 1 − cos(α) √ 2 re 1 + cos(α) √ 2 re M2 1 − cos(α) √ R−2 R−2k √ −2k−2 R−2 + 2k re 1 + cos(α) √ R−2 R−2k √ −2k−2 R−2 + 2k re is the proper time, and k = e and o denote quantities corre- sponding to the emitter and the observer, respectively. Similar to the normalized frequency, one can define the normalized luminosity Lo Le = dEo/dτo dEe/dτe ≈ ωodno ωedne ( dto dτe )−1 , (14) where no and ne are the observed and emitted photon num- bers, respectively, and we replaced dτo by dto since they are almost the same for distant observers. 3 Numerical results In this section, we numerically study observational appear- ances of a star freely falling radially in the asymmetric thin- shell wormhole in the scenarios I and II. During the free fall of the star, photons are emitted isotropically in the rest frame of the star. Specifically, we assume that the star starts emit- ting photons at t1 = t2 = 0 and r1 = 30.65 in M1, and emits 3200 photons, which are uniformly distributed in the emission angle α, every proper time interval δτe = 0.002. It is worth emphasizing that observational appearances of the freely-falling star, especially late-time appearances, are rather insensitive to the initial position where the star starts emitting. Here, for better comparison with the Schwarzschild black hole case, we simply choose the initial position as r1 = 30.65, which is in agreement with that of [78]. Here, observational appearances of the star are studied for two kinds of observers in M1. The first kind is observers distributed on a celestial sphere at the radius ro = 100, which refers to collecting photons in the whole sky at fixed radial coordinate ro = 100 in M1. The measurement by the observers on the celestial sphere would give the frequency distribution and the total luminosity of photons that reach the celestial sphere. The second kind is a specific observer, who is located at ϕo = 0 on the equator of the celestial sphere. Among all photons collected on the celestial sphere, we select photons with cos ϕ > 0.99 to mimic photons detected by the specific observer. To calculate observed luminosities, the col- lected photons are grouped into packets of 50 (i.e., dno = 50) according to their arrival time. As shown in Fig. 1, the asymmetric thin-shell wormhole with k = 1.2 and R = 2.6 has a double-peak effective potential, corresponding to one photon sphere in M1 and one in M2. Specifically, the photon sphere in M1 is located at rph 1 = 3 with the critical impact parameter bph 1 = 3 √ 3, and that in M2 is located at rph 2 = 3.6 with the critical impact parameter bph 2 = 3.6 √ 3. To discuss how photons with dif- ferent impact parameters contribute to the observations of the star, we classify received photons into seven categories according to their impact parameter b1 in M1, • b1 < 3.579. Yellow region in Fig. 1 and yellow dots in Figs. 4, 5, 7 and 8. • 3.579 ≤ b1 < Zbph 2 . Pink region in Fig. 1 and pink dots in Figs. 4, 5, 7 and 8. In this category, photons emit- ted inward outside the photon sphere in M2 can circle around the photon sphere more than once before reaching a distant observer in M1. For example, a light ray with b1 = 3.579, which has �ϕ = 2π ,2 is displayed in the upper-left panel of Fig. 2. • Zbph 2 < b1 ≤ 3.664. Brown region in Fig. 1 and brown dots in Figs. 4, 5, 7 and 8. In this category, photons emitted inward would circle around the photon sphere in M2 roughly with �ϕ ≥ 2π before escaping to the celestial sphere in M1. For example, a light ray with b1 = 3.664, which has �ϕ = 2π , is displayed in the upper-right panel of Fig. 2. • 3.664 < b1 ≤ 4.923. Blue region in Fig. 1 and blue dots in Figs. 4, 5, 7 and 8. • 4.923 < b1 < bph 1 . Orange region in Fig. 1 and orange dots in Figs. 4, 5, 7 and 8. In this category, if photons are emitted inward outside the photon sphere in M1, they would linger for some time around the photon sphere by orbiting it approximately with �ϕ ≥ 2π . For example, a light ray with b1 = 4.923, which has �ϕ = 2π , is displayed in the lower-left panel of Fig. 2. 2 Sinceϕ1 = ϕ2 at the throat, the subscript ofϕ is omitted for simplicity. 123 Eur. Phys. J. C (2023) 83 :361 Page 5 of 14 361 Fig. 1 The effective potential of null geodesics in the asymmetric thin- shell wormhole with k = 1.2 and R = 2.6. The potential has two peaks at rph 1 = 3 (solid vertical blue line) and rph 2 = 3.6 (dashed vertical blue line), corresponding to a photon sphere with bph 1 = 3 √ 3 in M1 and another one with bph 2 = 3.6 √ 3 in M2, respectively. The vertical red line denotes the throat at r1 = r2 = R. Photons emitted in the pink, brown, orange and purple regions have impact parameters close to the impact parameters of the photon spheres, and hence can be temporar- ily trapped around the photon spheres. In particular, when photons are emitted towards the throat at r2 > rph 2 in the pink region or at r1 > rph 1 in the brown, orange and purple regions, they usually orbit the wormhole with �ϕ ≥ 2π • bph 1 < b1 ≤ 5.238. Purple region in Fig. 1 and purple dots in Figs. 4, 5, 7 and 8. In this category, photons emitted inward outside the photon sphere in M1 usually circle around the photon sphere more than once. For example, a light ray with b1 = 5.238, which has �ϕ = 2π , is displayed in the lower-right panel of Fig. 2. • b1 > 5.238. Green region in Fig. 1 and green dots in Figs. 4, 5, 7 and 8. In short, we use the orbit number of light rays emitted at r1 = 5 in M1 or r2 = 5 in M2 to determine the threshold impact parameters separating the seven categories. To sum up, light rays emitted inward at r2 = 5 in the yellow/pink cat- egory would circle around the wormhole less/more than once before being received; light rays emitted inward at r1 = 5 would circle around the wormhole less than once before being received in the blue and green categories, or more than once in the brown, orange and purple categories. Note that the orbit number of light rays with a given impact parameter depends slightly on the emitting position. So, light rays con- necting the star and the observers circle around the wormhole approximately more than once in the pink, brown, orange and purple categories, and less than once in the yellow, blue and green categories. In other words, photons in the pink, brown, orange and purple categories can be temporarily trapped near the photon spheres. 3.1 Scenario I In the scenario I, the star with energy E1 = 1/Z = √ 3 would travel through the throat and move towards spatial infinity of M2. For near-critical photons emitted with the impact parameter very close to those of the photon spheres in M1 (i.e., b1 bph 1 ) and M2 (i.e., b2 bph 2 ), their nor- malized frequencies ωo/ωe measured by observers on the celestial sphere are plotted against the emitted position re in Fig. 3. The colors of the lines in Fig. 3 match those of the corresponding emitted regions in Fig. 1. Moreover, pho- tons with b1 bph 1 and b2 bph 2 are denoted by solid and dashed lines, respectively. It is worth emphasizing that the observed frequency of a photon is determined by the gravita- tional redshift and the Doppler effect, which are controlled by the position and the velocity of the photon when it is emitted, respectively. For photons of b2 bph 2 , the normalized frequency can noticeably exceed 1 at a large re in M1 since the Doppler effect plays a more important role than the gravitational redshift. As the star falls towards the throat, the normal- ized frequency decreases due to stronger gravitational red- shift, and blueshift becomes redshift at re = 4.063 in M1, where the normalized frequency is 1. When emitted at the throat, the normalized frequency reaches the minimum. After the star enters M2, the normalized frequency increases as 123 361 Page 6 of 14 Eur. Phys. J. C (2023) 83 :361 Fig. 2 Photon trajectories in the asymmetric thin-shell wormhole with k = 1.2 and R = 2.6. The red points and circles denote the star and the throat, respectively. The blue solid and dashed circles represent the pho- ton spheres in M1 and M2, respectively. The upper-left panel shows a photon emitted at re = 5 in M2 with b1 = 3.579, and the light ray has �ϕ = 2π . Other panels show photons emitted at re = 5 in M1 with b1 = 3.664, 4.923 and 5.238, and the light rays all have �ϕ = 2π . The solid and dashed segments of the light rays correspond to the segments in M1 and M2, respectively re grows, and observed photons become bluershifted when re > 12.281. For photons of b1 bph 1 , the behavior of the normalized frequency is quite similar to those of b2 bph 2 when they are emitted outside the photon sphere in M1. When the star emits photons between the two photon spheres, inward-emitted and outward-emitted photons can both be captured by a distant observer after they circle around the photon sphere in M1, thus leading to two branches as shown in the inset. The upper and lower branches correspond to photons emitted away from and towards the observer, respec- tively. In the left panel of Fig. 4, we display the normalized fre- quency distribution of photons, which are emitted from the freely-falling star in the scenario I and collected by observers distributed on the celestial sphere at ro = 100 in M1. At early times, received photons are dominated by those emit- ted in the green region of Fig. 1, among which inward-emitted photons contribute to the high-frequency observation. When to > 160, photons emitted towards the photon sphere in M2 in the blue and brown regions start reaching the observers after orbiting around the photon sphere. Subsequently, the observers receive photons emitted towards the photon sphere in M1 in the purple and orange regions. Since time moves faster in M2 roughly by a factor of 1/Z = √ 3 relative to in M1, photons circling around the photon sphere in M2 arrive earlier. Moreover, the maximum frequency of photons emitted in the blue and brown regions is higher than that of photons emitted in the green, purple and orange regions. This is expected from Fig. 3, which shows that near-critical pho- tons with b2 bph 2 have higher normalized frequency than 123 Eur. Phys. J. C (2023) 83 :361 Page 7 of 14 361 Fig. 3 The normalized frequency ωo/ωe as a function of the emitted position re for photons in the scenario I, whose impact parameter is very close to these of the photon spheres in M1 (solid lines) and M2 (dashed lines). The observers are distributed on the celestial sphere at ro = 100 in M1. For a large re in M1, inward-emitted and near- critical photons can be blueshifted since the Doppler effect dominates over the gravitational redshift. Due to the relation (5) at the throat, near-critical photons can also be blueshifted when re is large in M2. Photons emitted inward and outward between the two photon spheres can both reach a distant observer after orbiting the photon sphere in M1, which gives two branches of the orange line in the inset. Moreover, the normalized frequency reaches the minimum at the throat, which is located at re = 2.6 these with b1 bph 1 . Afterwards, the frequency observa- tions are dominated by photons emitted in the orange region, which are trapped at the photon sphere in M1 for a longer time. At late times, the observers mostly receive photons in the yellow and pink regions, which are emitted towards the throat in M2 with a small impact parameter. The normalized total luminosity of the freely-falling star is displayed in the right panel of Fig. 4, where a dot corre- sponds to a packet of 50 photons, and the color of the dot is that having most photons in the packet. The luminosity grad- ually increases until reaching a peak around to 145, and is dominated by photons emitted in the green region roughly before to = 150, which is in agreement with the frequency observation. After to 160, photons emitted in the blue region give rise to a noticeable increase of the total luminos- ity. As the star moves towards spatial infinity of M2, emitted photons can still propagate to the observers in M1 through the throat, and a slight decrease of the total luminosity is displayed at late times. Interestingly, this late-time observa- tion is strikingly different from the black hole case, where the total luminosity has been found to decay exponentially at late times [78,79]. For a specific observer located at ϕo = 0 and θo = π/2 on the celestial sphere at ro = 100 inM1, the angular coordinate change �ϕ of light rays connecting the star with the observer is �ϕ = 2nπ, (15) where n = 0, 1, 2 . . . is the number of orbits that the light rays complete around the wormhole. To simulate observa- tional appearances of the star seen by the observer, we select photons with cos ϕ > 0.99 from all photons received on the celestial sphere. The frequency observation is presented in the left panel of Fig. 5, which shows a discrete spectrum separated by the received time. The yellow line is formed by photons with n = 0, which radially propagate to the observer. Fig. 4 The normalized frequency distribution and the total luminosity of the freely-falling star in the scenario I, measured by observers on a celestial sphere at ro = 100 in M1. Left: the observers receive photons with a wide range of frequencies. At the early stage, photons emitted in the green region of Fig. 1 give rise to the frequency observation. After- wards, photons emitted in the brown, blue, orange and purple regions are observed. In particular, photons with a near-critical impact parameter produce high frequency observations. The late-time frequency obser- vation is determined by photons in the yellow and pink regions, which are emitted at a large re in M2. Right: the luminosity is calculated by grouping received photons into packets of 50. An increase of the observed luminosity is caused by photons emitted inward in the blue region, leading to a peak at to 168. At late times, the total luminosity gradually decays with time and is mainly controlled by photons, which are emitted at a large re in M2 and travel through the throat to reach the observers 123 361 Page 8 of 14 Eur. Phys. J. C (2023) 83 :361 Fig. 5 The normalized frequency and the luminosity of the freely- falling star in the scenario I, measured by a distant observer at ro = 100, θo = π/2 and φo = 0 in M1. The colored dots denote photons emitted in the regions with the same color in Fig. 1. Left: received photons form several frequency lines indexed by the orbiting number n. The inset displays three frequency lines caused by n = 1 photons with b1 � bph 1 , b1 � bph 1 and b2 bph 2 . The time delay between the adjacent n ≥ 1 lines formed by photons orbiting around the photon sphere in M1 and M2 is roughly the period of circular null geodesics at the photon sphere, i.e., �T1 2πbph 1 33 and �T2 2π Zbph 2 23, respectively. Right: at early times, the luminosity is dominated by photons with a small impact parameter, and decreases first and then increases after the star goes through the throat. Subsequently, blueshifted n = 1 photons start to reach the observer and become the most dominant contribution, which produces a luminous flash at to 170. Later, the luminosity is mainly contributed by the n = 0 photons emitted in the yellow region of M2 and almost declines gradually at late times. In addition, a faint flash, which results from the n = 2 photons emitted in the orange region, is observed at to 200 At early times, the observed frequency of the n = 0 photons decreases with the received time as the star falls towards the throat. After the star passes through the throat, the observed frequency of the n = 0 photons increases since the gravita- tional redshift becomes weaker as the star moves further away from the throat, which results in the dip at to 150. Owing to the existence of two photon spheres, the n = 1 photons with impact parameters b1 � bph 1 , b1 � bph 1 and b2 bph 2 can form three frequency lines, which are highlighted in the inset of Fig. 5. As the star falls towards the throat, the three frequency lines decrease rapidly due to strong gravitational redshift near the throat. After the star passes through the throat, the frequency line with b2 bph 2 gradually increases. For n = 2, the frequency lines with b1 � bph 1 and b1 � bph 1 move closer and are hardly distinguishable from each other. On the other hand, the frequency line with b2 bph 2 becomes more separate from them since photons spend more time orbiting around the photon sphere in M1. Indeed, it takes �T1 2πbph 1 33 to orbit around the photon sphere in M1 one time, and �T2 2π Zbph 2 23 to orbit around that in M2 3. Therefore, for b1 � bph 1 and b1 � bph 1 (b2 bph 2 ), the time delay between the n = 1 and 2 frequency lines roughly equals to �T1 (�T2). For n = 3, because of the finite number of photons in our numerical simulation, only 3 Equation (7) leads to dt/dφ|rph = b−1V−1 eff (rph) = bph, which gives �T 2πbph. the frequency line with b2 � bph 2 can be found and is shown by orange dots around to 230. The left panel of Fig. 5 shows the observed normalized luminosity as a function of the time, which exhibits a decline before the star reaches the throat. After the star moves through the throat, the luminosity starts to increase since the fre- quency of received photons grows, which causes a dip at to 150. Around to 160, blueshifted photons with n = 1 start to play a dominant role, leading to a luminous flash around to 170. Afterwards, the luminosity is mainly dom- inated by photons emitted in the yellow region of M2, and slowly decreases except a faint flash at to 200 caused by the arrival of n = 2 photons. The flashes of photons with n ≥ 3 are much fainter and barely visible in the background of the dominant photons emitted in the yellow region. In con- trast, for a black hole with two photon spheres, a series of flashes with decreasing luminosity are observed at late times due to photons orbiting around the hairy black hole different times [79]. 3.2 Scenario II In the scenario II, the star starts falling from spatial infinity ofM1 and returns to the infinity after going through the throat twice. Similarly, the normalized frequency ωo/ωe for near- critical photons is plotted in Fig. 6. Specifically, we focus on photons with b1 bph 1 emitted in the purple and orange regions and those with b2 bph 2 emitted in the brown region, which are denoted by solid and dashed lines, respectively. For 123 Eur. Phys. J. C (2023) 83 :361 Page 9 of 14 361 Fig. 6 The normalized frequency ωo/ωe as a function of the emitted position re for photons in the scenario II, whose impact parameter b1 is very close to bph 1 (solid lines) or b2 is very close to bph 2 (dashed lines). The observers are distributed on the celestial sphere at ro = 100 in M1. The normalized frequency of near-critical photons emitted in the purple and brown regions has two branches. Specifically, the high-frequency (low-frequency) branch corresponds to photons emitted from the star falling away (towards) from the observer. Similar to the scenario I, the high-frequency branch can be blueshifted for a large re in M1. The normalized frequency reaches the global minimum ωo/ωe 0.139 at the throat for the low-frequency branch photons with b1 bph 1 emitted outside the photon sphere in M1 (i.e., the purple region) and those with b2 bph 2 , the normalized frequency has high-frequency and low-frequency branches, corresponding to the star falling away from and towards the observer, respectively. If photons are emitted inside the photon sphere in M1 with b1 bph 1 , the high- frequency (low-frequency) branch denotes ingoing and out- going (outgoing and ingoing) emissions from the star falling away from and towards the observer, respectively. For the high-frequency branches, strong gravitational lensing around the photon spheres can cause blueshifts of near-critical pho- tons emitted inward at a large re in M1. In particular, the normalized frequency with b1 bph 1 (b2 bph 2 ) reaches the maximum ωo/ωe = 4/3 (ωo/ωe = 1.392) at re = 12 (re = 8.679), becomes one at re = 5.196 (re = 3.6), and reaches the minimum ωo/ωe = 0.306 (ωo/ωe = 0.139) at the throat. In M2, the normalized frequency with b2 bph 2 reaches the maximum ωo/ωe = 1 at re = 3.6, where the star returns. The normalized frequency distribution of photons received by observers distributed on the celestial sphere is presented in the left panel of Fig. 7. When to � 200, a wide range of fre- quencies is observed for photons emitted in the green region. After near-critical photons emitted in the purple, orange, blue and brown regions start arriving at the observers around to 150, they come to dominate the high-frequency part of the frequency distribution. This early-stage frequency distri- bution bears a resemblance to the Schwarzschild black hole case, in which a star falls from spatial infinity at rest [78]. Similar to the scenario I, the maximum frequency of photons emitted inward in the blue and brown regions is greater than that of photons emitted inward in the green, purple and orange regions. After the star enters M2, the observed frequency of photons emitted in the yellow region starts to increase and reaches a maximum around to 220, which is associated with the star returning to the throat. Subsequently, photons emitted in the brown and purple regions are observed to have a wide range of frequencies after they circle around the pho- ton sphere in M1 and reach the observers. At late times, the star comes back to M1 and moves towards the observer, and Fig. 7 The normalized frequency distribution (Left) and the total lumi- nosity (Right) of the freely-falling star in the scenario II, measured by observers on the celestial sphere at ro = 100 in M1. Similar to the scenario I, photons emitted in the green region of Fig. 1 dominate the frequency and luminosity observations in the early stage. After the star enters M2, photons emitted in the yellow region, which propagates to the observers nearly in the radial direction, produce frequency and lumi- nosity peaks around to 220. Later, near-critical photons with a wide range of frequencies are observed. At late times, the emitted position re is in M1 and large, and therefore the observers would collect most of emitted photons, which leads to a nearly constant total luminosity 123 361 Page 10 of 14 Eur. Phys. J. C (2023) 83 :361 Fig. 8 The normalized frequency and the luminosity of the freely- falling star in the scenario II, measured by a distant observer at ro = 100, θ = π/2 and φ = 0 in M1. Left: the yellow line denotes radially emit- ted photons with n = 0 and has a dip (peak) near to 200 (to 220), corresponding to emission from the star at the throat. The n = 1 fre- quency lines with b1 � bph 1 , b1 � bph 1 and b2 bph 2 steadily increase to a peak followed by a sharp decrease when to � 230, and gradually increase when to � 240. For 230 � to � 240, the n = 1 frequency line with b2 bph 2 rises to another high point. Right: similar to the scenario I, the luminosity is dominated by n = 0 photons and gradually decreases before to 160. Later, blueshifted n = 1 photons start to reach the observer and then become the most dominant contribution, which results in a luminosity peak around to 180. Afterwards, due to the increasing frequency of n = 0 photons emitted in M2, the lumi- nosity rises and reaches a peak around to 220. At late times, received n = 0 photons emitted in M1 enable the luminosity to stay roughly constant thus the low-frequency distribution is dominated by photons emitted towards the throat with b1 bph 1 and b2 bph 2 . On the other hand, photons emitted towards the observers with a small impact parameter produce the high-frequency observation. The normalized total luminosity of the freely-falling star in the scenario II is displayed in the right panel of Fig. 7. Before to 200, the total luminosity behaves similarly to the Schwarzschild black hole case studied in [78], which is in consistency with the frequency observation. Afterwards, the received blueshifted photons with a small impact parameter dominate the total luminosity, resulting in a peak at to 220. At late times, the total luminosity is maintained around one since most emitted photons can be collected by the observers. In the left panel of Fig. 8, we exhibit the normalized fre- quency of photons received by an observer located at ϕ = 0 and θ = π/2 on the celestial sphere in M1 for the sce- nario II. The observed frequency of radially emitted photons with n = 0 is represented by the yellow line, which dis- plays three periods. In the first and last periods, the photons are emitted when the star moves towards and away from the throat inM1, respectively, and the n = 0 frequency line both decreases with the received time; in the intermediate period, the star emits the photons in M2, and the n = 0 frequency line increases. There appears a peak and a dip of the n = 0 frequency line, which correspond to the star going through the throat the first time and the second time, respectively. Similar to the scenario I, the n = 1 frequency lines consist of three lines with b1 � bph 1 , b1 � bph 1 and b2 bph 2 , respec- tively. The n = 1 frequency line with b2 bph 2 increases slowly until the maximum and then decreases rapidly in the first period, rises to a peak followed by a steep decline in the intermediate period, and gradually increases in the last period. For the n = 1 frequency lines with b1 � bph 1 and b1 � bph 1 , there is a sharp drop after reaching a peak when the star moves away from the observer, and a steady increase when the star moves towards the observer. For n = 2, only two frequency lines are visible, namely the b1 � bph 1 (orange dots) and b2 bph 2 (brown dots) lines. Note that the n = 2 frequency lines are quite similar to the n = 1 counterparts. In addition, only the frequency line with b1 � bph 1 is visible for n = 3. The normalized luminosity of the star in the scenario II measured by the observer is displayed in the right panel of Fig. 8. Similar to the scenario I, the luminosity decreases slowly before to 170, which is dominated by radially emit- ted photons in the yellow region. Afterwards, photons emit- ted in the blue region come to control the luminosity obser- vation and lead to a flash around to 180. Subsequently, photons emitted in the yellow region determine the luminos- ity observation again and produce a peak around to 220. At late times, the star travels towards the observer at a large re in M1, and hence radially emitted photons would make a dominant contribution to the total luminosity. In particu- lar, the late-time luminosity remains fairly constant, which is greatly different from the black hole case. 123 Eur. Phys. J. C (2023) 83 :361 Page 11 of 14 361 4 Conclusions In this paper, we investigated observational appearances of a point-like freely-falling star, which emits photons isotrop- ically in its rest frame, in an asymmetric thin-shell worm- hole connecting two spacetimes, M1 and M2. Specifically, two scenarios with different initial velocities of the star were considered. In the scenario I, the star starts with a nonzero velocity at spatial infinity of M1 and moves towards spatial infinity of M2. In the scenario II, the star falls at rest from spatial infinity of M1, reaches a turning point in M2 and returns to M1. For the two scenarios, the frequency distri- bution and luminosity of the star measured by all observers and a specific observer on a celestial sphere were obtained by numerically tracing emitted light rays. Interestingly, it was found that the absence of the event horizon and the presence of two photon spheres play a pivotal role in frequency and luminosity observations. In [78,79], observational appearances of a star freely falling in black holes with one or two photon spheres were investigated. To compare the wormhole case with the black hole one, we briefly summarize the main findings of [78,79] and this paper as follows. • Black holes with a single photon sphere: The total lumi- nosity of the star fades out with an exponentially decay- ing tail, which is determined by quasinormal modes at the photon sphere. At late times, the specific observer sees a series of flashes indexed by the orbit number, whose luminosity decreases exponentially with the orbit num- ber. Moreover, the frequency content of received photons contains a discrete spectrum of frequency lines indexed by the orbit number, which decay sharply at late limes. • Black holes with double photon spheres: At late times, the total luminosity first rises to a peak and then decreases with an exponentially decaying tail. The sub-long-lived quasinormal modes at the outer photon sphere are respon- sible for the slowly decaying exponential tail, and the leakage of photons trapped between the inner and outer photon spheres results in the luminosity peak. The spe- cific observer sees two series of flashes, which are mainly determined by photons orbiting outside the outer and inner photon spheres, respectively. Moreover, the spe- cific observer detects a discrete spectrum of frequency lines indexed by the orbit number and the photon sphere that received photons orbit around, which fall steeply at late limes. • Wormhole: At late times, the total luminosity first rises to a peak and then gradually decays with time (scenario I) or remains roughly constant (scenario II). The luminos- ity peak is caused by photons travelling between the two photon spheres (scenario I) or those emitted inM2 nearly along the radial direction (scenario II). Due to the absence of the event horizon, a considerable number of photons can still reach observers at late times, and hence an expo- nentially decaying tail would not appear. Similarly, the late-time luminosity measured by the specific observer can be sizable, and therefore he only sees a bright flash and a faint one (scenario I) or two bright flashes (scenario II) due to strong background luminance. Moreover, the specific observer detects frequency lines indexed by the orbit number and the photon sphere that received photons orbit around. The frequency lines produced by photons orbiting around the photon sphere in M1 decline sharply (scenario I) or grow steadily (scenario II) at late limes; those produced by photons orbiting around the photon sphere in M2 gradually increase at late limes. In short, we showed that the absence of the event hori- zon in wormholes gives rise to significantly different optical appearances of a luminous star accreted onto wormholes at late times. Therefore, these findings can provide us a novel tool to distinguish wormholes from black holes in future observations. Acknowledgements We are grateful to Guangzhou Guo and Qingyu Gan for useful discussions and valuable comments. This work is sup- ported in part by NSFC (Grant Nos. 11875196, 11947225, 12105191, 12275183 and 12275184). Houwen Wu is supported by the International Visiting Program for Excellent Young Scholars of Sichuan University. Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: Data sharing not applicable to our article as no datasets were generated or analyzed during the current study]. Open Access This article is licensed under a Creative Commons Attri- bution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, pro- vide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indi- cated otherwise in a credit line to the material. 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