Imaging black holes through AdS/CFT

Clarifying conditions for the existence of a gravitational picture for a given quantum field theory (QFT) is one of the fundamental problems in the AdS/CFT correspondence. We propose a direct way to demonstrate the existence of the dual black holes: Imaging an Einstein ring. We consider a response function of the thermal QFT on a two-dimensional sphere under a time-periodic localized source. The dual gravity picture, if exists, is a black hole in an asymptotic global AdS4 and a bulk probe field with a localized source on the AdS boundary. The response function corresponds to the asymptotic data of the bulk field propagating in the black hole spacetime. We find a formula which converts the response function to the image of the dual black hole: The view of the sky of the AdS bulk from a point on the boundary. Using the formula, we demonstrate that, for a thermal state dual to the Schwarzschild-AdS4 spacetime, the Einstein ring is constructed from the response function. The evaluated Einstein radius is found to be determined by the total energy of the dual QFT.


Introduction
One of the definite goals of the research of the holographic principle, or the AdS/CFT correspondence [1][2][3], is to find what class of quantum field theories (QFTs) or quantum materials possesses their gravity dual. Is there any direct test for the existence of a gravity dual for a given material?
Among various gravitational physics, one of the most peculiar astrophysical objects is the black hole. Gravitational lensing is one of fundamental phenomena by strong gravity. Let us consider that there is a light source behind a gravitational body. When the light source, the gravitational body, and observers are in alignment, the observers will see a ring-like image of the light source, i.e., the so-called Einstein ring. If the gravitational body is a black hole, some light rays are so strongly bended that they can go around the black hole many times, and especially infinite times on the photon sphere. As a result, multiple Einstein rings corresponding to winding numbers of the light ray orbits emerge and infinitely concentrate on the photon sphere. The dark area inside the photon sphere, in which light rays have been (a) Image of the AdS black hole reconstructed from the response function. (b) Gravitationally lensed galaxy SDP.81 taken by ALMA [6]. Image credit: ALMA (NRAO/ESO/NAOJ); B. Saxton NRAO/AUI/NSF. absorbed by the black hole, is called black hole shadow [4]. (For example, Event Horizon Telescope [5] is an observational project of the shadow of the supermassive black hole in our galaxy.) In this paper, we propose a direct method to check the existence of a gravity dual from measurements in a given thermal QFTimaging the dual black hole as an "Einstein ring".
We demonstrate explicitly construction of holographic "images" of the dual black hole from the response function of the boundary QFT with external sources, as follows. As the simplest example, we consider a (2+1)-dimensional boundary conformal field theory on a 2-sphere S 2 at a finite temperature, and study a one-point function of a scalar operator O with its conformal dimension ∆ O = 3, under a time-dependent localized source J O . The gravity dual is a black hole in the global AdS 4 and a probe massless bulk scalar field in the spacetime. The schematic picture of our setup is shown in Fig. 1. The source J O , for which we employ a time-periodic localized Gaussian source with the frequency ω, amounts to an AdS boundary condition for the scalar field. Due to the time periodic boundary condition, a bulk scalar wave is injected into the bulk from the AdS boundary. The scalar wave propagates inside the black hole spacetime and reaches other points on the S 2 of the AdS boundary. We measure the local response function e −iωt O( x) which has the information about the bulk geometry of the black hole spacetime.
Using a wave-optical method, we find a formula which converts the response function O( x) to the image of the dual black hole |Ψ S ( x S )| 2 on a virtual screen: where x = (x, y) and x S = (x S , y S ) are Cartesian-like coordinates on boundary S 2 and the virtual screen, respectively, and we have set the origin of the coordinates to an observation point. This operation is mathematically a Fourier transformation of the response function on a small patch with the radius d around the observation point. Note that f describes magnification of the image on the screen. In terms of the wave optics, we have virtually used a lens with the focal length f and the radius d to form the image. Figure 2(a) shows a typical image of the AdS black hole computed from the response function through our method. For a comparison, we also show an observational image of Einstein ring caused by a galaxy in Fig. 2 Several criteria for QFTs to have a gravity dual have been proposed in some previous works. For example, in a conformal field theory, the existence of a planer expansion and a large gap in the spectrum of anomalous dimensions has been conjectured to be the criterion for the existence of a gravity dual [7]. There is also recent progress on emergent gravity based on quantum entanglement [8], although the entanglement entropy itself is not a physical observable. The strong redshift of the black hole has been also used as the condition for the existence of gravity dual [9][10][11]. Our approach, checking the dual black hole by its image, gives an alternative. The method is simple and can be applied to any QFT on a sphere, thus probing efficiently a black hole of its possible gravity dual. Once we have a strongly correlated material on S 2 , we can apply a localized external source such as electromagnetic waves and measure its response in principle. Then, from Eq. (1.1), we would be able to construct the image of the dual black hole if it exists. The holographic image of black holes in a material, if observed by a tabletop experiment, may serve as a novel entrance to the world of quantum gravity.
The organization of this paper is as follows. In Section 2, we review how to obtain the response function under a source in AdS/CFT, from a scalar field dynamics in Schwarzschild-AdS geometry, and describe our time-oscillatory source in the boundary QFT. In Section 3, we review image formation in wave optics. This leads us to our main formula for the dual quantity of the Einstein ring (1.1). In Section 4, null geodesics in the black hole geometry is described, to gain insight on our images. Then in Section 5, we provide our concrete images for the AdS black holes seen from the QFT with the formula (1.1). They show clear holographic images of black holes as well as the Einstein rings. In Section 6, we evaluate the Einstein radius (the size of the rings) in the images, and provide its consistent understanding by geodesics. Then in Section 7, we provide analytic examples of the pure AdS 4 and the BTZ black hole, for a comparison. Finally, Section 8 is for our conclusion and discussions. Appendix A describes detailed numerical evaluation of solutions of the scalar field in the bulk. Appendix B is for the validity of the approximation of the geometric optics used in Section 4.

Scalar field in Schwarzschild-AdS 4 spacetime
We consider Schwarzschild-AdS 4 (Sch-AdS 4 ) with the spherical horizon: where R is the AdS curvature radius (Λ = −3/R 2 ) and G is the Newton constant. This is the black hole solution in the global AdS 4 : The dual CFT is on R t × S 2 . The event horizon is located at r = r h defined by F (r h ) = 0 1 . Using the horizon radius, the mass M is written as This corresponds to the total energy of the dual CFT. In what follows, we take the unit of R = 1. We focus on dynamics of a massless scalar field in the Sch-AdS 4 . The massless scalar field equation, Φ(t, r, θ, ϕ) = 0, is written as where D 2 is the scalar Laplacian on unit S 2 .
Near the AdS boundary (r = ∞), the asymptotic solution of the scalar field becomes In the asymptotic expansion, we have two independent functions, J O and O , which depend on boundary coordinates (t, θ, ϕ). In the AdS/CFT, the leading term J O corresponds to the scalar source in the boundary CFT. On the other hand, the subleading term O corresponds to its response function [13]. We consider that an axisymmetric and monochromatically oscillating Gaussian source is localized at the south pole (θ = π) of the boundary S 2 : Note that we will ignore a tiny value of the Gaussian tail at the north pole because we suppose σ ≪ π. In the bulk point of view, this source J O is the boundary condition of the scalar field at the AdS boundary. We also impose the ingoing boundary condition on the horizon of the Sch-AdS 4 . A schematic picture of our setup is shown in Fig. 1. The scalar wave generated at the south pole of the AdS boundary propagates inside the bulk black hole spacetime and reaches the other points at the AdS boundary. Imposing these boundary conditions, we solve Eq. (2.3) numerically and determine the solution of the scalar field in the bulk. We summarized the detailed method to solve the Klein-Gordon equation in Appendix A. We read off the response from the coefficient of 1/r 3 in Eq. (2.4). Due to the symmetries of the source (2.5), the response function does not depend on ϕ and its time dependence is just given by e −iωt . Hence, we can write the response function as As an example, Fig. 3 shows the absolute square of the response function for r h = 0.3, ω = 20 and σ = 0.2. We can observe the interference pattern resulting from the diffraction of the scalar wave by the black hole. However, yet we cannot directly find the "black hole like image" from the response function. To obtain the image of the black hole, we need to look at the response function through a "convex lens" virtually as we will see in section 3.

Image formation in wave optics
We introduce the convex lens in the wave optics for the purpose of reconstructing the image of the black hole from the response function [14]. (See also Refs. [15][16][17].) What role the convex lens plays in the wave optics is as follows. The lens is regarded as a "converter" between plane and spherical waves as in Fig. 4(a). The lens with focal length f is located at z = 0 and its focus is at z = ±f . We assume that the size of the lens is much smaller than the focal length f and the lens is infinitely thin. Imagine that a plane wave is irradiated to the lens from the left hand side as shown in the figure. Such a plane wave is converted into (a part of) spherical wave and it converges at the focus located at z = f . Inversely, if we consider a spherical wave emitted from the focus, it should be converted into the plane wave in z < 0. Let Ψ and Ψ T denote the incident wave and the transmitted wave, respectively. Then, mathematically, the role of the convex lens for the wave functions with frequency ω on the lens can be simply expressed as where x = (x, y) are coordinates on the lens located at z = 0. For example, an incident plane wave propagating along z-axis is written as Ψ( x) = 1 on the lens, whose phase does not depend on x. Then, from Eq. (3.1), the transmitted wave becomes This phase dependence describes that of the spherical wave converging on the focus at z = f in the Fresnel approximation (| x| ≪ f ). Let us consider a spherical screen located at (x, y, z) The transmitted wave converted by the lens is focusing and imaging on this screen. The wave function on the screen is given by where d is the radius of the lens and L is the distance between (x, y, 0) on the lens and (x S , y S , z S ) on the screen: This implies that the image on the screen can be obtained by the Fourier transformation of the incident wave within a finite domain of the lens. Equation (3.4) motivates us to regard the observable defined in (1.1) as the dual quantity of the Einstein ring.

Null geodesics: geometrical optics
To help intuitive understanding of the image of the AdS black hole which will be shown in the following sections, we consider null geodesics in the Sch-AdS 4 (that is, geometrical optics) in this section. In Appendix B, we derive the geodesic equation from the Klein-Gordon equation and examine the validity of the Eikonal approximation in asymptotically AdS spacetimes. It is well known that in the spherically symmetric spacetime an orbit of geodesics lies in a plane passing through the center of the black hole. Therefore, we can always rotate an orbital plane of the null geodesic to coincide with the equatorial plane, without loss of generality. In this section, for simplicity, we will study null geodesics on the equatorial plane (θ = π/2). Then, the conserved energy and angular momentum are written as where · = d/dλ and λ is an affine parameter. From the null condition, we havė We note that, since affine parameters of null geodesics can be rescaled by a constant factor, the null orbits depend only on the ratio of the conserved quantities, ℓ/ω. The effective potential v(r) has a maximum value: at r = r max ≡ 3r h (r 2 h + 1)/2. The maximum of the effective potential corresponds to the photon sphere, i.e., the unstable circular orbit of null geodesics. The schematic functional profile of the effective potential v(r) is shown in Fig. 5. We are now interested in null orbits between two points on the AdS boundary. Then, we have to tune the parameters so that 1 < ω 2 /ℓ 2 < v max is satisfied. When ω 2 /ℓ 2 is close to (but less than) v max , the geodesic goes through the vicinity of the photon sphere and can wind around the black hole. Consequently, there exist infinite geodesics labeled by the winding number, N w , which connect fixed two points on the AdS boundary, and in the vicinity of the photon sphere infinitely many orbits accumulate. Figure 6 shows some geodesics stretched between antipodal points on the AdS boundary for r h = 0.3. In particular, for the geodesic with N w = ∞ (i.e. geodesic from the photon sphere), the angular momentum per unit energy becomes Figure 6. Null geodesics between ϕ = 0 and π for winding number N w = 0, 1, 2. The horizon radius of the Sch-AdS 4 is fixed as r h = 0.3. ϑ i denotes the angle of incidence to the AdS boundary.
We can naturally define the angle of incidence of the null geodesic to the AdS boundary by cos ϑ i ≡ g ij u i n j /(|u||n|)| r=∞ , where u i is the spatial component of the 4-velocity of the geodesic, n i is the normal vector to the AdS boundary and g ij is the induced metric on the t = const. surface. (|u| and |n| are the norms of u i and n i with respect to g ij .) Using Eqs. (4.1) and (4.2), we can explicitly calculate the angle of incidence as Combining Eqs. (4.4) and (4.5), we can determine the angle of incidence of the null geodesic from the photon sphere as a function of the horizon radius r h . In the geometrical optics, this angle ϑ i gives the angular distance of the image of the incident ray from the zenith if an observer on the AdS boundary looks up into the AdS bulk. If two end points of the geodesic and the center of the black hole are in alignment, the observer see a ring image with a radius corresponding to the incident angle ϑ i because of axisymmetry. To make an optical system, we virtually consider the flat 3-dimensional space (x, y, z) as shown in the right side of Fig. 7. We set the convex lens on the (x, y)- plane. The focal length and radius of the lens will be denoted by f and d. We also prepare the spherical screen at (x, y, z) = (x S , y S , z S ) with x 2 S + y 2 S + z 2 S = f 2 . We copy the response function around the observation point as the wave function on the lens and observe its image.

Response on AdS boudary
We map the response function defined on S 2 onto the lens as follows. We introduce new polar coordinates (θ ′ , ϕ ′ ) as sin θ ′ cos ϕ ′ + i cos θ ′ = e iθ obs (sin θ cos ϕ + i cos θ) , (5.1) such that the direction of the north pole is rotated to align with the observation point: θ ′ = 0 ⇐⇒ (θ, ϕ) = (θ obs , 0). Then, we define Cartesian coordinates as (x, y) = (θ ′ cos ϕ ′ , θ ′ sin ϕ ′ ) on the AdS boundary S 2 . In this coordinate system, we regard the response function around the observation point as the wave function on the lens: Ψ( x) = O(θ) . The image on the screen can be obtained by Eq. (1.1): We perform the Fourier transformation within a finite domain on the lens, that is, applying an appropriate window function. We now summarize our results on image formations of Sch-AdS black holes. Figure 8 shows images of the black hole observed at various observation points: The horizon radius is varied as r h = 0.6, 0.3, 0.1. We fix other parameters as ω = 80, σ = 0.01 and d = 0.5. For θ obs = 0 • , a clear ring is observed. As we will see for details in the next section, this ring corresponds to the light rays from the vicinity of the photon sphere of the Sch-AdS 4 . Not only the brightest ring, we can also see some concentric striped patterns. They are caused by a diffraction effect with imaging, which is not directly related to properties of the black hole, because we find these patterns change depending on the lens radius d and frequency ω. As we change the angle of the observer, the ring is deformed. We observed similar images as those for asymptotically flat black hole [15][16][17]. In particular, at θ obs ∼ 90 • , we can observe double image of the point source. They correspond to light rays which are clockwisely and anticlockwisely winding around the black hole on the plane of ϕ = 0, π. The size of the ring becomes bigger as the horizon radius grows. Figure 9 shows the the image of the Sch-AdS black hole for relatively small frequency, ω = 10, 20, 30. Other parameters are fixed as r h = 0.3, θ obs = 0, σ = 0.01 and d = 0.5. For ω = 20, 30, we can only see the blurred ring because the wave effect is not negligible. For ω = 10, the ring disappears. In the geometric optics, we can only see the outside of the photon sphere. In the wave optics, however, we have a chance to probe the region between photon sphere and event horizon due to some wave effects. Do these images change if we modify the metric inside the photon sphere? This is one of the interesting direction as a future work.

Einstein radius
To study the property of the brightest ring, we set the observation point at θ obs = 0 and search x S = x ring at which |Ψ S ( x S )| 2 has maximum value. (Since the image has the rotational symmetry in (x S , y S )-plane for θ obs = 0, we only focus on the x S -axis in this section.) We will refer to the angle of the Einstein ring θ ring = sin −1 (x ring /f ) as the Einstein radius. Figure 10 shows the Einstein radius θ ring by the purple points varying the horizon radius r h . Note that the horizon radius relates to the total energy of the system by Eq. (2.2). Although the Einstein radius fluctuates as the function of r h , it has an increasing trend as r h is enlarged. As we have seen in the geodesic analysis, there is an infinite number of geodesics connecting antipodal points on the AdS boundary, which are labeled by the winding number N w . Which geodesic in the geometrical optics corresponds to the ring found in the image in the wave optics? From the analysis of null geodesics, the (specific) angular momentum of the null geodesic provides the angle of incidence ϑ i to the AdS boundary as shown in Eq. (4.5). In the wave optics, the null geodesic with angular momentum ℓ should be a wave packet realized by the superposition of the spherical harmonics Y ℓ ′ 0 (θ) with ℓ − ∆ℓ ≤ ℓ ′ ≤ ℓ + ∆ℓ (∆ℓ ≪ ℓ). For a sufficiently large ℓ, the spherical harmonics behaves as Y ℓ0 ∼ e iℓθ . Applying the Fourier transformation (3.4) onto the spherical harmonics, we have a peak in the image at x S /f ≃ ℓ/ω. Thus, the angular distance of the image on the screen, sin −1 (x S /f ), coincides with the angle of incidence of the null geodesic to the AdS boundary, ϑ i = sin −1 (ℓ/ω).
Since we found that the angular momentum of the null geodesics from the photon sphere is given by Eq. (4.4), they are expected to form the ring at θ ring = sin −1 (ℓ/ω) = sin −1 (1/ √ v max ). The Einstein radius calculated from geodesic analysis is shown by the green curve in Fig. 10. The curve seems consistent with the Einstein radius from the response function. This indicates that the major contribution to the brightest ring in the image is originated by the "light rays" from the vicinity of the photon sphere, which are infinitely accumulated. Although there are expected to be multiple Einstein rings corresponding to light lays with winding numbers N w = 0, 1, 2, · · · , the contribution for the image from small N w may be so small that we cannot resolve them within our numerical accuracy. The deviation of the Einstein radius θ ring (r h ) from the geodesic prediction can be considered as some wave effects. In the AdS cases, whether the geometrical optics can adapt to imaging of black holes is not so trivial even for a large value of ω. As studied in Appendix B, the Eikonal approximation, which supports the geometrical optics, will inevitably break down near the AdS boundary, while we have given the source J O and read the response O on the AdS boundary. Our results based on the wave optics imply that the geometrical optics is qualitatively valid but gives a non-negligible deviation even for a large ω.

Analytic examples of the images
To illustrate our methods, in this section we provide examples in which images are obtained analytically. The first example is a (2 + 1)-dimensional CFT on a sphere at zero temperature, which is dual to the pure AdS 4 geometry. The second example is a (1 + 1)-dimensional CFT on a circle at a finite temperature, which is dual to the BTZ black hole.

Imaging AdS 4
In the absence of the black hole horizon, there should not be holographic images of the black hole. To demonstrate it, here we study the case of pure AdS geometry. It is a gravity dual of a CFT at zero temperature, and we have an analytic expression for the response function.
For the pure AdS 4 (r h = 0), we can solve the scalar field equation analytically as Its asymptotic behavior is Therefore, the response function is given by where c ℓ = dθdϕ sin θ g(θ)Y l 0 (θ) whose explicit expression is in Eq. (A.12). The response diverges for ω = ℓ + 2n+ 3 (n = 0, 1, 2, · · · ). This corresponds to the normal modes of the pure AdS 4 . Applying the Fourier transformation (1.1) onto Eq. (7.3), we obtain the image of the pure AdS 4 as in Fig. 11. For θ obs = 0 • , we can observe the bright spot at the center. This corresponds to the "straight" null geodesic from the south pole θ = π to the north pole θ = 0. This indicates that there is no black hole shadow in pure AdS 4 as expected.
Note that there exists a ring in addition to the bright center. The angle seems θ ring = π/2. What is the origin of the ring in the pure AdS? For r h = 0, the effective potential for null geodesics is simply given by v(r) = 1+1/r 2 . This effective potential has the "maximum" at r = ∞. Therefore, if we tune the angular momentum per unit energy as ℓ/ω = 1, we can realize the null geodesic propagating along the AdS boundary (r = ∞). As shown in (4.5) in Section 6, the incident angle ϑ i = π/2 given by ℓ/ω = 1 corresponds to θ ring = π/2. The ring found in Fig. 11 would be originate from the null lay along the AdS boundary. Even for the Sch-AdS 4 spacetime, there should be null geodesics along the AdS boundary. However, the ring formed by such geodesics is much weaker than the ring by the photon sphere.

Imaging BTZ black holes
Another analytic example of the image is the (1 + 1)-dimensional CFT at a finite temperature. As its gravity dual, we consider the BTZ black hole: This spacetime is locally AdS and the Klein-Gordon equation Φ = 0 can be analytically solved as [18] φ m = Γ(a + 1)Γ(b + 1) where we decompose the scalar field as Φ = e −iωt+mϕ φ m (r) and define This solution satisfies ingoing boundary condition at the horizon. The asymptotic form of the solution near the AdS boundary is given by φ m = 1 + iω 2r h + ab ψ(a + 1) + ψ(b + 1) ǫ + ab ǫ ln(e 2γ−1 ǫ) + O(ǫ 2 ln ǫ) , (7.7) where ǫ = 1 − x = r 2 h /r 2 , γ is the Euler's constant and ψ(x) = Γ ′ (x)/Γ(x) is the polygamma function. We consider the time-periodic Gaussian source around ϕ = π as Φ| r=∞ = e −iωt g(ϕ) where g is the Gaussian function introduced in Eq. (2.5). Then, the response function becomes where c m = (2π) −1/2 dϕg(ϕ)e imϕ ≃ (−1) m e −σ 2 m 2 /2 . Setting observation point at ϕ = 0, we can obtain the image of the BTZ black hole as potential for null geodesics in the BTZ spacetime is given by v(r) = 1 − r 2 h /r 2 . This implies that all null geodesics with ω 2 /ℓ 2 > 1 fall into the black hole. However, like as the pure AdS case, if we tune the angular momentum per unit energy as ℓ/ω = 1, we can realize the null geodesic propagating along the AdS boundary. The ring found in the image of the BTZ would originate from the null geodesic on the boundary. Therefore, the "ring" found in the image of BTZ does not have much information about the black hole spacetime. However, the dual gravitational theory does not need to be pure Einstein in general. For example, it has been conjectured that the dual gravitational theory for a high T c superconductor is given by Einstein-Maxwellcharged scalar system [19][20][21]. Also, there can be a higher derivative corrections if we take into account the quantum gravity effect. Then, the dual black hole space time will deviate from BTZ and we would be able to observe images of black holes even for (1 + 1)-dimensional materials.

Conclusion and Discussion
We have studied how we can construct the holographic image of the AdS black hole using observables in its dual field theory. We have considered a thermal CFT with a scalar operator, which corresponds in the AdS/CFT to the massless scalar field in Sch-AdS 4 spacetime with a spherical horizon. We have put a time periodic localized source for the operator and have computed its response function by the AdS/CFT dictionary. Applying the Fourier transformation (1.1) onto the response function, we have observed the Einstein ring as the image of the AdS black hole (Fig. 8). The Einstein radius has an increasing trend as a function of the horizon radius r h . It is also consistent with the angle of the photon sphere calculated from the geodesic analysis on the basis of the geometrical optics.
We have shown that, if the dual black hole exists, we can construct the image of the AdS black hole from the observable in the thermal QFT. In other words, being able to observe the image the AdS black hole in the thermal QFT can be regarded as a necessary condition for the existence of the dual black hole. Finding conditions for the existence of the dual gravity picture for a given quantum field theory is one of the most important problems in the AdS/CFT. We would be able to use the imaging of the AdS black hole as a test for it. One of the possible applications is superconductors. It is known that some properties of high T c superconductors can be captured by the black hole physics in AdS [19][20][21]. One of the other interesting applications is the Bose-Hubbard model. It has been conjectured that the Bose-Hubbard model at the quantum critical regime has a gravity dual [22][23][24]. If we can realize these materials on S 2 , they will be appropriate targets for observing AdS black holes by experiments. Applying localized sources on such materials and measuring their responses, we would be able to observe Einstein rings by tabletop experiments.
Can we distinguish the AdS black hole from thermal AdS by the observation of the Einstein ring? Results of section.7.1 indicate that a "ring" will be also observed even for the thermal AdS. The angle of the ring is, however, always fixed at θ ring = π/2 irrespective of the temperature of the thermal radiation in the global AdS. This implies that we would be able to distinguish the AdS black hole from thermal AdS by the temperature dependence of the Einstein radius.
where Y ℓ 0 (θ) is the scalar spherical harmonics with zero magnetic quantum number (m = 0). Since the boundary condition (2.5) is axisymmetric, the scalar field can also be axisymmetric and decomposed only by Y ℓ 0 (θ). Also the time dependence of the scalar field can be factorized by e −iωt because the boundary condition is monochromatic. For actual computations, it is convenient to introduce the tortoise coordinate as In terms of the tortoise coordinate, r * = −∞ and 0 correspond to the horizon (r = r h ) and AdS boundary (r = ∞), respectively. Near the AdS boundary, the relation between r * and r becomes Substituting Eq. (A.1) into Eq. (2.3), we obtain the equation for φ ℓ as The asymptotic solution at horizon becomes where we took the ingoing mode. The asymptotic solution at the AdS boundary is given by φ ℓ = p 0 (1 + p 2 r 2 * + p 4 r 4 * + p 5 r 5 * + · · · ) + q 3 (r 3 * + q 5 r 5 * + · · · ) . (A.6) where Here, p 0 and q 3 are not determined by the asymptotic expansion, which correspond to the source and response. We numerically integrate Eq. (A.4) from horizon to AdS boundary. From the ingoing condition (A.5), we impose the initial condition at r * = r min * ≃ −3 as Using the obtained complex asymptotic value p 0 , we normalize the numerical solution asφ This solution satisfiesφ ℓ → 1 (r → ∞). For notational simplicity, hereafter, we will omit the "bar" of the scalar field.
To obtain the response q 3 , we differentiate Eq. (A.4) by r * . Then, d n φ ℓ /r n * (n = 0, 1, 2, 3) appear in the resultant equation. The second derivative d 2 φ ℓ /r 2 * can be eliminated by Eq. (A.4). As the result, we can compute d 3 φ ℓ /r 3 * from our numerical data of dφ ℓ /r * and φ ℓ . From the third order derivative, we obtain q 3 as For σ ≪ 1, we can decompose the Gaussian source by the spherical harmonics as Therefore, the response function in θ-space is given by Substituting this into Eq. (1.1) and performing the 2-dimensional Fourier transformation, we obtain the images in Fig. 8.

B Validity of geometrical optics approximation in AdS
It is well known that, if we adopt the Eikonal approximation, the massless Klein-Gordon equation yields the Hamilton-Jacobi equation for null geodesic in general curved spacetime. We assume that the scalar field is Φ = a(x µ )e iS(x µ ) and the gradient of the phase function, ∂ µ S, has typically the same scale as frequency ω. Substituting this ansatz into the field equation and taking ω ≫ 1, we have the Eikonal equation from the leading in ω as g µν ∂ µ S∂ ν S = 0. (B.1) This is nothing but the Hamilton-Jacobi equation for massless particle, where p µ ≡ ∂ µ S is the 4-momentum. In fact, differentiating Eq. (B.1), we can easily derive null geodesic equation as 0 = ∇ α (g µν ∂ µ S∂ ν S) = 2p µ ∇ µ p α (B.2) together with the null condition g µν p µ p ν = 0.
For the null geodesic equation, we consider the solution with the conserved energy and angular momentum given by (4.1), so that Hamilton's function is S(t, r, θ, ϕ) = −ωt + ℓϕ + dr F (r) Thus, it turns out that the wave front characterized by S(x µ ) will propagate along trajectories of null geodesics under the Eikonal approximation, that is, the geometrical optics. Now, let us deal with the field equation more continously. We focus on an eigenstate with frequency ω and angular momentum ℓ. By defining ψ ≡ rφ ℓ , the field equation (A.4) can be rewritten as is satisfied, both the Eikonal approximation and the WKB approximation lead to the same effective potential and phase function for ω, ℓ ≫ 1. However, since F (r)F ′ (r)/r ≃ 2r 2 near the AdS boundary, even for any large frequency ω the above condition will violate as goes to the AdS boundary. Roughly speaking, this is because the gradient of the metric function, ∂ r g µν , will be larger than the frequency due to a so-called AdS potential. As a result, in order to predict the wave propagation of the scalar field, inside the AdS bulk including the black hole we can apply the geometric optics for a sufficiently large frequency ω. Near the AdS boundary, we should deal with it in the WKB method together with appropriate matching procedures.