Observational appearance of a freely-falling star in an asymmetric thin-shell wormhole

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 distant 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.

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.The Event Horizon Telescope (EHT) collaboration released images of the supermassive black holes M87* [1][2][3][4][5][6][7][8] and Sgr A* [9][10][11][12][13][14], which provides a new method to test general relativity 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 lensing near photon spheres, black hole images encode valuable information of the geometry in the vicinity of photon spheres.
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 systematic uncertainties still leave some room within observational uncertainty bounds for black hole mimickers.Among all black hole mimickers, ultra compact objects (UCOs), e.g., boson stars, gravastars and wormholes, which are horizonless and possess light rings (or photon spheres in the spherically symmetric case), are of particular interest since their observational signatures can be quite similar to those of black holes [47][48][49][50].Nevertheless, 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 ECO models [51][52][53][54][55][56][57][58][59][60][61].In addition, asymmetric thin-shell wormholes with two photon spheres were found to have double shadows and an additional photon ring in their images [62][63][64][65][66].For black holes with one photon sphere, there is one shadow and one photon ring 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 photon 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 [67][68][69][70][71]. Multiple photon spheres can introduce distinctive features in black hole images, e.g., double shadows [71], extra photon rings [72] and tripling higher-order images [73].Furthermore, late-time echo signals were also observed since the effective potential of a scalar perturbation possesses a multiple-peak structure [74,75].
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, observational appearances of a star freely falling onto black holes with a single or double photon spheres have been numerically simulated [76,77].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 [78] and the Sgr A* source [79,80].Moreover, a new way to measure the spin of Sgr A* was proposed by simulating an infalling gas cloud [81].In practice, detecting photons circling 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 baseline interferometry [82][83][84].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 Section II, we briefly review the asymmetric thin-shell wormhole and introduce our observational settings.Numerical results are presented in Section III.Finally, we conclude with a brief discussion in Section IV.We set G = c = 1 throughout this paper.

II. SETUP
As introduced in [62,65,85], an asymmetric thin-shell wormhole has two distinct spacetimes, M 1 and M 2 , which are glued together by a thin shell at its throat.The metric of the wormhole is described as where i = 1 and 2 indicate quantities in M 1 and M 2 , respectively.Focusing on the Schwarzschild spacetime, we have where M i are the mass parameters, and R is throat radius.Without loss of generality, we set M 1 = 1 and M 2 = k in the rest of this paper.For more details of the asymmetric thin-shell wormhole, refer to [62].In M 1 and M 2 , the local tetrads are , At the throat, one has e t 1 = e t 2 , e r 1 = −e r 2 , e θ 1 = e θ 2 and e φ 1 = e φ 2 , which yields the relations between the bases of the tangent space of M 1 and M 2 , where Therefore, the components of a vector at the throat in M 1 and M 2 are related by In this paper, we study a point-like star freely falling along the radial direction at θ i = π/2 and ϕ i = 0, which emits photons 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 where dots stand for derivative with respect to an affine parameter τ .Since the Lagrangian L does not depend on coordinates t i and ϕ i , the geodesics can be characterized by their conserved energy E i and angular momentum l i in M i , Note that, according to eqn. ( 5), one has The Lagrangian of the freely-falling star obeys the constancy L = −1/2 when the affine parameter τ is chosen as the proper time.Since the star falls radially, its angular momentum l i = 0. Due to the traversability of the wormhole, we consider two scenarios with distinct trajectories of the star.In the scenario I, the star with energy E 1 = 1/Z (E 2 = 1) has a nonzero initial velocity at spatial infinity of M 1 .So, the star can pass through the throat and travel towards spatial infinity of M 2 .With the relation (7), the four-velocities of the star in M 1 and M 2 are given by In the scenario II, the star with energy E 1 = 1 is initially at rest at spatial infinity of M 1 .At first, the star falls freely in M 1 , passes through the throat and reaches a turning point in M 2 .Then, it moves towards the throat in M 2 , returns to M 1 and comes to rest at spatial infinity of M 1 .
Similarly, the four-velocities of the star in M 1 and M 2 are 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 where b i ≡ l i /E i is the impact parameter, and is the effective potential.Note that the impact parameters of a null geodesic in M 1 and M 2 , namely b 1 and b 2 , are related by b 1 = Zb 2 .A photon sphere in M i is constituted of unstable circular null geodesics, whose radius where b ph i is the corresponding impact parameter.Photons with b i ≈ b ph 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 r ph 1 = 3 and r ph 2 = 3k in M 1 and M 2 , respectively.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 [65].
We assume that the emitted photons are collected by distant observers distributed on a celestial sphere located at r 1 = r o in M 1 .To trace light rays emitting from the star to a distant observer, one needs to supply initial conditions.For a photon of four-momentum p µ i , the momentum measured in the rest frame of the star with four-velocity v µ i e at r i = r e is where plus and minus signs correspond to negative and positive v r i e , respectively.The emission angle α is defined as 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 ω e = − (v µ i e p µ i ) e = p t.For a distant static observer with four-velocity v µ 1 o = (1, 0, 0, 0), the photon is With eqns.( 5), ( 12) and ( 13), we express the normalized frequency ω o /ω e as a function of the star position r e and the emission angle α for two scenarios in Table .I. Furthermore, the luminosity of photons is given by L k = dE k /dτ k , where E k is the total energy, τ k is the proper time, and k = e and o denote quantities corresponding to the emitter and the observer, respectively.Similar to the normalized frequency, one can define the normalized luminosity where n o and n e are the observed and emitted photon numbers, respectively, and we replaced dτ o by dt o since they are almost the same for distant observers.

III. NUMERICAL RESULTS
In this section, we numerically study observational appearances 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 emitting photons at t 1 = t 2 = 0 and r 1 = 30.65 in M 1 , and emits 3200 photons, which are uniformly distributed in the emission angle α, every proper time interval δτ e = 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 r 1 = 30.65,which is in agreement with that of [76].
Here, observational appearances of the star are studied for two kinds of observers in M 1 .
The first kind is observers distributed on a celestial sphere at the radius r o = 100, which refers to collecting photons in the whole sky at fixed radial coordinate r o = 100 in M 1 .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 ϕ o = 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 1 Since ϕ1 = ϕ2 at the throat, the subscript of ϕ is omitted for simplicity.photons emitted in the green, purple and orange regions.This is expected from FIG. 3, which shows that near-critical photons with b 2 b ph 2 have higher normalized frequency than these with b 1 b ph 1 .Afterwards, the frequency observations are dominated by photons emitted in the orange region, which are trapped at the photon sphere in M 1 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 M 2 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 corresponds to a packet of 50 photons, and the color of the dot is that having most photons in the packet.The luminosity gradually increases until reaching a peak around t o 145, and is dominated by photons emitted in the green region roughly before t o = 150, which is in agreement with the frequency observation.After t o 160, photons emitted in the blue region give rise to a noticeable increase of the total luminosity.As the star moves towards spatial infinity of M 2 , emitted photons can still propagate to the observers in M 1 through the throat, and a slight decrease of the total luminosity is displayed at late times.Interestingly, this late-time observation is strikingly different from the black hole case, where the total luminosity has been found to decay exponentially at late times [76,77].
For a specific observer located at ϕ o = 0 and θ o = π/2 on the celestial sphere at r o = 100 in M 1 , the angular coordinate change ∆ϕ of light rays connecting the star with the observer is where n = 0, 1, 2 • • • is the number of orbits that the light rays complete around the wormhole.
To simulate observational 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.
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 M 1 (i.e., the purple region) and those with b 2 b ph 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 M 1 with b 1 b ph 1 , the highfrequency (low-frequency) branch denotes ingoing and outgoing (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 nearcritical photons emitted inward at a large r e in M 1 .In particular, the normalized frequency with spatial infinity at rest [76].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 M 2 , the observed frequency of photons emitted in the yellow region starts to increase and reaches a maximum around t o 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 photon sphere in M 1 and reach the observers.At late times, the star comes back to M 1 and moves towards the observer, and thus the low-frequency distribution is dominated by photons emitted towards the throat with b 1 b ph 1 and b 2 b ph 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 t o 200, the total luminosity behaves similarly to the Schwarzschild black hole case studied in [76], 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 t o 220.At late times, the total luminosity is maintained around one since most emitted photons can be collected by the observers.Subsequently, photons emitted in the yellow region determine the luminosity observation again and produce a peak around t o 220.At late times, the star travels towards the observer at a large r e in M 1 , and hence radially emitted photons would make a dominant contribution to the total luminosity.In particular, the late-time luminosity remains fairly constant, which is greatly different from the black hole case.

IV. CONCLUSIONS
In this paper, we investigated observational appearances of a point-like freely-falling star, which emits photons isotropically in its rest frame, in an asymmetric thin-shell wormhole connecting two spacetimes, M 1 and M 2 .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 M 1 and moves towards spatial infinity of M 2 .In the scenario II, the star falls at rest from spatial infinity of M 1 , reaches a turning point in M 2 and returns to M 1 .For the two scenarios, the frequency distribution 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 [76] and [77], 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 [76,77] and this paper as follows.
• Black holes with a single photon sphere: The total luminosity of the star fades out with an exponentially decaying 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 number.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 responsible 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 specific observer sees two series of flashes, which are mainly determined by photons orbiting outside the outer and inner photon spheres, respectively.
Moreover, the specific 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 luminosity peak is caused by photons travelling between the two photon spheres (scenario I) or those emitted in M 2 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 exponentially 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 M 1 decline sharply (scenario I) or grow steadily (scenario II) at late limes; those produced by photons orbiting around the photon sphere in M 2 gradually increase at late limes.
In short, we showed that the absence of the event horizon 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.

3 √ 3 6 √ 3 .
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 r ph 1 = 3 (solid vertical blue line) and r ph 2 = 3.6 (dashed vertical blue line), corresponding to a photon sphere with b ph 1 = 3 √ 3 in M 1 and another one with b ph 2 = 3.6 √ 3 in M 2 , respectively.The vertical red line denotes the throat at r 1 = r 2 = 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 temporarily trapped around the photon spheres.In particular, when photons are emitted towards the throat at r 2 > r ph 2 in the pink region or at r 1 > r ph 1 in the brown, orange and purple regions, they usually orbit the wormhole with ∆ϕ ≥ 2π.

FIG. 2 .
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 photon spheres in M 1 and M 2 , respectively.The upper-left panel shows a photon emitted at r e = 5 in M 2 with b 1 = 3.579, and the light ray has ∆ϕ = 2π.Other panels show photons emitted at r e = 5 in M 1 with b 1 = 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 M 1 and M 2 , respectively.

• 4 .
FIG.3.The normalized frequency ω o /ω e as a function of the emitted position r e for photons in the scenario I, whose impact parameter is very close to these of the photon spheres in M 1 (solid lines) and M 2 (dashed lines).The observers are distributed on the celestial sphere at r o = 100 in M 1 .For a large r e in M 1 , 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 r e is large in M 2 .Photons emitted inward and outward between the two photon spheres can both reach a distant observer after orbiting the photon sphere in M 1 , 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 r e = 2.6.

FIG. 5 .2πb ph 1 33 and ∆T 2 2πZb ph 2 23,
FIG. 5.The normalized frequency and the luminosity of the freely-falling star in the scenario I, measured by a distant observer at r o = 100, θ o = π/2 and φ o = 0 in M 1 .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 b 1 b ph 1 , b 1 b ph 1 and b 2 b ph 2 .The time delay between the adjacent n ≥ 1 lines formed by photons orbiting around the photon sphere in M 1 and M 2 is roughly the period of circular null geodesics at the photon sphere, i.e., ∆T 1 2πb ph 1

b ph 1 , b 1 b ph 1 and b 2 b ph 2 b ph 1 and b 1 b ph 1
photon spheres, the n = 1 photons with impact parameters b 1 b ph 1 , b 1 b ph 1 and b 2 b ph 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 b 2 b ph 2 gradually increases.For n = 2, the frequency lines with b 1 b ph 1 and b 1 b ph 1 move closer and are hardly distinguishable from each other.On the other hand, the frequency line with b 2 b ph 2 becomes more separate from them since photons spend more time orbiting around the photon sphere in M 1 .Indeed, it takes ∆T 1 2πb ph 1

b 1 b ph 1 (b 2 b ph 2 )
FIG. 7. The normalized frequency distribution (Left) and the total luminosity (Right) of the freely-falling star in the scenario II, measured by observers on the celestial sphere at r o = 100 in M 1 .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 M 2 , photons emitted in the yellow region, which propagates to the observers nearly in the radial direction, produce frequency and luminosity peaks around t o 220.Later, near-critical photons with a wide range of frequencies are observed.At late times, the emitted position r e is in M 1 and large, and therefore the observers would collect most of emitted photons, which leads to a nearly constant total luminosity.

FIG. 8 .
FIG. 8.The normalized frequency and the luminosity of the freely-falling star in the scenario II, measured by a distant observer at r o = 100, θ = π/2 and φ = 0 in M 1 .Left: The yellow line denotes radially emitted photons with n = 0 and has a dip (peak) near t o 200 (t o 220), corresponding to emission from the star at the throat.The n = 1 frequency lines with b 1 b ph 1 , b 1 b ph 1 and b 2 b ph 2 steadily increase to a peak followed by a sharp decrease when t o 230, and gradually increase when t o 240.For 230 t o 240, the n = 1 frequency line with b 2 b ph 2 rises to another high point.Right: Similar to the scenario I, the luminosity is dominated by n = 0 photons and gradually decreases before t o 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 t o 180.Afterwards, due to the increasing frequency of n = 0 photons emitted in M 2 , the luminosity rises and reaches a peak around t o 220.At late times, received n = 0 photons emitted in M 1 enable the luminosity to stay roughly constant.

TABLE I .
The normalized frequency ω o /ω e as a function of the star position r e and the emission angle α