Effect of gravitational wave on shadow of a Schwarzschild black hole

We have studied the shadows of a Schwarzschild black hole under a special polar gravitational perturbation, which is a particular solution of Einstein equations expanded up to first order. It is shown that the black hole shadow changes periodically with time and the change of shadow depends on the Legendre polynomial order parameter l and the frequency {\sigma} of gravitational wave. For the odd order of Legendre polynomial, the center of shadow oscillates along the direction which is vertical to equatorial plane. For even l, the center of shadow does not move, but the shadow alternately stretches and squeezes with time along the vertical direction. Moreover, the presence of the gravitational wave leads to the self-similar fractal structures appearing in the boundary of the black hole shadow. We also find that this special gravitational wave has a greater influence on the vertical direction of black hole shadow.


I. INTRODUCTION
The first direct observation of gravitational waves (GW150914)[1-3] reported by LIGO and Virgo Scientific not only confirms the success of Einstein's general relativity, but also opens a new era in the fields of astronomy, astrophysics and cosmology. Subsequently, there are several gravitational waves events have been detected, which are caused by binary black hole merger [1][2][3][4][5][6][7] or by binary neutron star merger [8]. Especially, the discovery of the electromagnetic signals in gamma-ray [8][9][10] arising from binary neutron star (BNS) merger means the arrival of multi-messengers astronomy. By comparing with theoretical templates, gravitational waves could tell us a variety of parameters of astrophysical compact objects such as their masses, spins and so on. The detection of gravitational waves could help us to understand black hole further and to verify various gravity theories.
Another exciting event in astrophysics and black hole physics is the first image of the supermassive black hole in the center of the giant elliptical galaxy M87, which was announced by Event Horizon Telescope (EHT) Collaboration in 2019 [11][12][13][14][15][16][17]. It provides the first direct visual evidence that there exists exactly black hole in our Universe. Black hole image can be regarded as a potential tool to verify gravity theories and identify black hole parameters. The initial analyses of the first image of black hole have no striking deviations from the predictions of general theory of relativity. The dark region in the center of black hole image is black hole shadow, which corresponds to light rays fall into event horizon of black hole. The fingerprints of the geometry around the black hole would be reflected in the shape and size of black hole shadow [18,19]. For example, the shadow of a Schwarzschild black hole is a perfect black disk. But for a Kerr black hole, shadow becomes a D-shaped silhouette gradually with the increase of spin parameter [18,19]. In the spacetime of a Kerr black hole with Proca hair and a Konoplya-Zhidenko rotating non-Kerr black hole, the cusp silhouette of black hole shadows emerge in certain range of parameter [20,21]. Especially, the self-similar fractal structures are found in shadow for a rotating black hole with scalar hair [22][23][24][25], a Majumdar-Papapetrou binary black hole system [26,27], Bonnor black diholes with magnetic dipole moment [28], and a non-Kerr rotating compact object with quadrupole mass moment [29]. These novel structures in shadows are found to be caused by non-integrable photon motions. The black hole shadows with other parameters in various theories of gravity have been recently investigated in Refs. . These could provide some theoretical templates for the future astronomical observations announced by Event Horizon Telescope and BlackHoleCam[64].
Since both the gravitational waves detection and Event Horizon Telescope observation play a vital role in the study of black holes and verification of various gravity theories, it is very interesting to study the effects of gravitational waves on black hole shadows. Generally, it is very difficult because that it is not easy to get a solution for the gravitational wave around a black hole since the Einstein equation with perturbation is very complicated. Fortunately, B. Xanthopoulos [65] obtained a particular gravitational wave solution which meets Einstein equations expanded up to the first order in ǫ. With this particular solution, one can probe the peculiar effects of gravitational wave on dynamics of test particle in black hole spacetime. It is found that this special gravitational wave makes the motion of a timelike test particle is no longer integrable and then the chaotic phenomenon appears [66], which is different from those in the case without gravitational wave. It is naturally expected that the chaotic phenomenon could appear in the motion of photon under such gravitational perturbation, and then the corresponding chaotic behavior of photon would give rise to some new effects on the black hole shadow. Therefore, in this paper, we would like to probe the effects of this gravitational wave on the black hole shadow.
The paper is organized as follows. In Sec. II, we review briefly the spacetime of a Schwarzschild black hole perturbed by the gravitational wave [65][66][67] and then analyze the null geodesic equations in this spacetime. In Sec. III, we present numerically the shadows for the Schwarzschild black hole perturbed by the gravitational wave and probe the effects of this gravitational wave on the shadow. Finally, we present a summary.

II. THE SPACETIME OF SCHWARZSCHILD BLACK HOLE PERTURBED BY GRAVITATIONAL WAVE AND NULL GEODESICS
The metric of a Schwarzschild black hole with a gravitational perturbation [65][66][67] can be expressed as which is a particular solution of Einstein equations expanded up to the first order in ǫ. The metric g µν in Eq.
(1) is the metric tensor of a usual Schwarzschild black hole with a form h µν is a special analytical solution of polar gravitational perturbation around Schwarzschild black hole [65][66][67], i.e., h rr = f −1 Y P l cos(σt), P l = P l (cos θ) are the usual Legendre polynomials (l > 1), σ is the frequency of gravitational wave. This solution of the special class of gravitational wave (3) was obtained by B. Xanthopoulos [65] through solving the differential equations on metric perturbations of Reissner-Nordströmblack hole [68][69][70]. And the perturbations of Reissner-Nordström black hole was derived by Zerilli [68] and Moncrief [69,70] when they researched the perturbed gravitational and electromagnetic fields produced by a charged (or uncharged) test particle moving in a Reissner-Nordström geometry. So the special class of gravitational wave (3) is a solution of onedimensional wave-type equations [68][69][70], and could be regarded as a perturbation caused by an uncharged particle moving in Schwarzschild black hole spacetime. The unperturbed (Schwarzschild) metric is spherically symmetric, thus there is no loss of generality that the special class of gravitational wave (3) is only an axisymmetric perturbation. Since h µν is divergent at infinity, the gravitational wave (3) is just used to describe the gravitational perturbation around a black hole. Actually, this solution is related to Zerilli function Z (+) by [65] with n = (l − 1)(l + 2)/2. For the even order l in P l , one can find that the gravitational wave solution (3) is even function of cos θ, therefore it is symmetric with respect equatorial plane [65][66][67]. However, for the odd l, the perturbation (3) is odd function of cos θ and then it is not symmetric with respect equatorial plane.
The Hamiltonian of a photon propagation along null geodesic in the spacetime (1) can be expressed as .
It is obvious that the Hamiltonian (6) is an explicit function of time coordinate t, and then the energy of [20,30,49,[71][72][73] is no longer a constant of motion along geodesic (Ė =−p t = ∂H(t) which can be expressed as It is understandable because there exists the energy transfer between the gravitational wave (3) and the photon in this spacetime. However, the z component of the angular momentum L z = p φ of photon is still conserved in this case because the Hamiltonian (6) we can find the null geodesic equations of photon in the spacetime (1) can be expressed as +f σṫ 2 XǫP l sin(σt) + L 2 z σǫ csc 2 θ sin(σt) W cot θ dP l dθ + ZP l r 2 ǫ cos(σt) W cot θ dP l dθ + ZP l + 1 2 where the dots denote derivatives with respect to the proper time τ , and the primes denote derivatives with respect to the radial coordinate r. It is obvious that the geodesic equations (9-12) are not variable-separable and the integration constants in this dynamical system are less than the number of freedom degrees. It implies that the photon dynamical system is non-integrable, so the chaos could appear in the motion of photon in this spacetime (1).

III. SHADOWS CASTED BY SCHWARZSCHILD BLACK HOLE PERTURBED BY GRAVITATIONAL WAVE
In this section, we will study the shadows of Schwarzschild black hole with the gravitational wave (3) through the backward ray-tracing technique [22][23][24][25][26][27][28][29][30][31]. We evolved light rays by solving numerically the null geodesic equations (9-12) from the observer backward in time. The shadow of black hole is composed by the light rays falling down into the horizon of black hole. We can find the spactime of Schwarzschild black hole with the gravitational wave is not asymptotically flat under the influence of gravitational perturbation (3). The observer can not be set at the spatial infinite. In this situation, we introduce orthonormal tetrads for observers located at finite distance, zero-angular-moment-observers (ZAMOs) reference frame [18], which has been used in the study of black hole shadow in de Sitter spacetime [74][75][76]. In this way, we assume that the static observer is locally at (r obs , θ obs ) in the Boyer-Lindquist coordinates. The local observer basis {et, er, eθ, eφ} can be expanded as a form in the coordinate basis where the transform matrix e ν µ obeys (g µν + ǫh µν )e μ α e ν β = ηαβ, and ηαβ is the Minkowski metric. For the spacetime (3), it is convenient to choice the transform matrix e ν µ as Thus the locally measured four-momentum pμ of a photon is related to its four-momentum p µ by pμ = e ν µ p ν , i.e., Repeating the operation in Refs. [22][23][24][25][26][27][28][29][30][31], one can obtain the coordinates of a photon's image in observer's sky Here, we assume the lights come from both accretion disk around black hole and distant stars. For a convenience, we assume that a geometrically thin and optically thick accretion disk lies in the equatorial plane of Schwarzschild black hole. As that in ref. [77], the disk's minimum and maximum radii are set as 6M and 15M , respectively. In Fig.1 direction. Moreover, the shape of black hole shadows changes only along the vertical direction in Fig.1. It is because that the special class of gravitational wave (3) is only an axisymmetric perturbation. With the increase of the parameter l, we find that the deformations of shadows are more obvious. Especially when l increases to 5, the shadows of black hole even become heart-like shaped shadows. In addition, we also find that the images of accretion disk have the same variations as the black hole shadows with gravitational waves.
In order to study quantitatively the deformation of Schwarzschild black hole shadow perturbed by the grav- The deviated parameter ε o is positive if black hole shadow shifts upward, but is negative if the shadow shifts downward. Since the shadow in the even l case alternately stretches and squeezes along the vertical direction, and then the deviated parameter ε e could be defined as The deviated parameter ε e is positive if black hole shadow is prolate, but is negative if the shadow is oblate. deviated parameter ε e is greater than zero for a little longer in Fig.3, which indicates that the gravitational perturbation (3) make Schwarzschild black hole shadow more prolate rather than oblate when l is even. In addition, we found that both of the amplitude of ε o and ε e increase with l for the same frequency σ. It means that the bigger l has a greater impact on the black hole shadow.
In Schwarzschild black hole spacetime, the boundary of shadow is a smooth curve. However, in Fig.1(l), one can find that the boundary of shadow in the blue box is not smooth. We amplified the boundary of shadow and found some similar layered structures shown in Fig.4(a). From this figure we can find the boundary of shadow is distorted, the color red and white near the shadow represent the lights come from accretion disk and distant stars respectively. They're superimposed layer by layer. We zoomed in on the region within the blue box in Fig.4(a) to got Fig.4(b), and continued to zoom in on the region within the blue box in Fig.4(b) to got Fig.4(c). We kept amplifying the boundary of shadow and found more similar layered structures like shown in Fig.4 gives to Schwarzschild black hole shadow is what we expected since the equations of photon motion (9)(10)(11)(12) are no longer integrable and the chaotic motions for test particles have appeared [66] in this background. I hope Event Horizon Telescope and BlackHoleCam could find more fine structure like chaos by increasing the resolution of telescope in the future.
Let us now to probe the changes of the black hole shadow with the spatial coordinate of observer. In Fig.5 and Fig.6, we present the shadows perturbed by gravitational wave (3) for the static observer at r obs = 20 and 80, respectively. For a sake of simplicity, we set θ obs = 90 • , M = 1, ǫ = 0.05, and gravitational wave frequency σ = 0.5, l = 3, 4. Comparing Fig.1 with Fig.5 and Fig.6, one can find with the increase of r obs , the image of accretion disk on the equatorial plane becomes thin, but the disk images around shadow become thick. The oscillation amplitude of black hole shadow increases with the coordinate r obs when l is odd. Moreover, we also present the width w = x r − x l , the height h = y t − y b and the radius R of the black hole shadow for the static observer with different radial coordinates in Figs.7-9, respectively. Here the radius R is defined similarly as in Ref. [37] In these figures, we set mass M = 1, ǫ = 0.05, gravitational wave frequency σ = 0.5, and l = 2, 4, 6 for figure (a), l = 3, 5, 7 for figure (b). The length of the lines represents the variation range of the width w, the height h or the radius R of black hole shadow in a time-period. In Fig.7, one can find the variation range of the width w in a time-period is bigger for small even l, but it is just the opposite for odd l. Especially, the widths w almost unchange when r obs = 50 for l = 3, 5, 7. Moreover, one can find the average width w  shadow is no longer symmetric with respect the equatorial plane. But we also find the shadow of black hole oscillates up and down with time for l = 3, and the shadow alternately stretches and squeezes with time along the vertical direction for l = 4. The colored region (red to yellow) also represents the image of accretion disk.
Especially, in these figures one can find the emitted intensity of light rays have three peaks, one is close to the boundary of black hole shadow, one is the middle thin accretion disk image, and one is the peripheral accretion disk image. For instance, the three peaks of emitted intensity of light rays in Fig.11(a) are shown in Fig.12(a), which are the ring a, the thin image b and the region c respectively. Through further study, we found the light rays from the region c come from the upside of accretion disk and propagate directly to observer, as the trajectory c in Fig.12(b). The light rays from the thin image b come from the downside of accretion disk and bypass the black hole to observer, as the trajectory b in Fig.12(b). The light rays from the ring a also come from the upside of accretion disk, but it make one orbit around black hole and then reaching the observer, as the trajectory a in Fig.12(b). Actually, when the metric (1) reduces to Schwarzschild black hole (without the gravitational perturbation, ǫ = 0), the three peaks of emitted intensity of light rays still exist. It is because the accretion disk has inner and outer boundaries, and the lights can propagate between black hole and accretion disk, which causes the multiple images of the accretion disk to appear. In addition, the image of accretion disk has the same variations as black hole shadow, it alternately expands and contracts with time when θ obs = 0 • , and is no longer symmetric with respect the equatorial plane when θ obs = 45 • , but also oscillates up and down with time for l = 3, alternately stretches and squeezes with time along the vertical direction for l = 4.

IV. SUMMARY
In this paper we studied the shadows of a Schwarzschild black hole perturbed by a special class of gravitational wave (3). Under the influence of gravitational wave, the equations of photon motion in this spacetime are no longer integrable, and the shadow changes periodically. When the order of Legendre polynomial l is odd, the center of shadow oscillates up and down with time along the direction which is vertical to equatorial plane. When l is even, the center of shadow does not move, but it alternately stretches and squeezes with time along the vertical direction. We studied the effects of gravitational wave on Schwarzschild black hole shadows by introducing two deviated parameters ε o and ε e , and found both of the amplitude of ε o and ε e increase with l increases for fixed σ. We also found that there exist self-similar fractal structures in the boundary of shadow caused by chaotic motion of photon due to the presence of gravitational wave. The average width w of black hole shadow in a time-period increases first and then decreases overall as r obs increases. The variation range of height h and radius R of black hole shadow change periodically with r obs . Moreover, the change of the black hole shadow height h is bigger than that of the shadow width w, which indicates the gravitational perturbation (3) has a greater influence on the vertical direction of black hole shadow. In addition, we present the shadows for the observer with different inclination angle. Our results show that the presence of gravitational wave yields some interesting properties of black hole shadow.