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Recently, the Event Horizon Telescope (EHT) Collaboration et al. revealed the first images of the supermassive black hole at the centre of the giant elliptical galaxy M87 [1–7] and Milky Way Galaxy [8–13]. This is highly useful for black hole physics and astrophysics and has drawn an increasing number of researchers into the study of black hole images. An important element in a black hole image is the black hole shadow [14–17], which is a dark silhouette. A dark shadow appears because light rays close to the event horizon are captured by the black hole, thereby leaving a black shadow in the observer's sky. Because the fingerprints of the geometry around the black hole can be reflected in the shape and size of the shadow, research on black hole shadows plays a vital role in the study of black holes (constraining black hole parameters) [18–24], probing fundamental physics issues, including dark matter [25–29], and the verification of various gravity theories [30–36]. These shadows are a perfect black disk for Schwarzschild black holes, and they gradually become a D-shaped silhouette with an increase in the spin parameter for Kerr black holes [15]. In the space-time of a Kerr black hole with Proca hair and a Konoplya-Zhidenko rotating non-Kerr black hole, a cusp silhouette of black hole shadows emerges [37, 38]. Self-similar fractal structures appear in black hole shadows originating from chaotic lensing [39–48]. Many other black hole shadows with other parameters in various theories of gravity have been recently investigated in Refs. [49–91]. It is hoped that the information imprinted in black hole shadows can be captured in future astronomical observations, including the upgraded Event Horizon Telescope and BlackHoleCam [92], to study black holes and verify various gravity theories.
It is widely believed that a massive halo, ring, or other shell-like distributions of matter may be concentrated around the black hole in a galactic center. The first image of a Schwarzschild black hole with a thin accretion disk was announced by Luminet [16], revealing primary and secondary images of the thin accretion disk around a shadow. The superposition of a black hole and exterior matter will reorganize the space-time structure and induce a significant change to the black hole shadow. We previously researched the shadows of a Schwarzschild black hole surrounded by a Bach-Weyl ring [49] and found that the black hole shadow becomes an "8" shaped silhouette and possesses self-similar fractal structures. Cunha et al. [50] researched the shadows of a black hole surrounded by a heavy Lemos-Letelier accretion disk and found that the black hole shadow becomes more prolate with increasing accretion disk mass. In this paper, we consider a solution to the Einstein equations representing the superposition of a Schwarzschild black hole with a halo containing quadrupolar and octopolar contributions [93]. For this space-time, Vieira et al. detected the timelike geodesic orbits of test particles and found that the octopolar term could bring about chaotic motions [93]. Through further numerical analysis, they found that the quadrupolar term also gives rise to a chaotic behavior in addendum [94]. Dubeibe further investigated the effects of the quadrupole and octupole moments on the motion of a test particle using modern color-coded basin diagrams [95] and found that the final state of the test particle was highly affected. Therefore, the nonlinear superposition of a Schwarzschild black hole with a halo may also influence the null geodesic motions of photons, thus affecting the black hole shadow. In this paper, we investigate the effect of quadrupolar and octopolar terms on the Schwarzschild black hole shadow.
The paper is organized as follows: In Sec. II, we briefly review the space-time of a Schwarzschild black hole with a halo and reveal the influences of quadrupolar and octopolar terms on the surface geometry of the black hole. In Sec. III, we numerically present the shadows of a Schwarzschild black hole with a halo and analyze the new features of the black hole shadow arising from the quadrupolar and octopolar terms. Finally, we conclude the paper with a summary.
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The space-time of a Schwarzschild black hole with a halo is vacuum static and axially symmetric; hence, it can be described by the Weyl metric
ds2=−e2νdt2+e2λ−2ν(dρ2+dz2)+ρ2e−2νdϕ2,
(1) where only ν and λ are the functions of ρ and z. The Einstein equations reduce to
ν,ρρ+ν,ρρ+ν,zz=0,
(2) λ,ρ=ρ(ν,ρ)2−ρ(ν,z)2,λ,z=2ρν,ρν,z.
(3) The function
ν(ρ,z) satisfies the Laplace equation and behaves as the gravitational potential in Newtonian theory; thus, it can be linearly superposed. However, the functionλ(ρ,z) does not exhibit such linear superposition. In the solution of a Schwarzschild black hole with a halo, the functions ν and λ for the entire system can be written asν=νSchw+νhalo andλ=λSchw+λhalo+λint , respectively [96]. The functionsν,λ are the solution of Schwarzschild space-time, expressed asνSchw=12lnd1+d2−2Md1+d2+2M,
(4) λSchw=12ln(d1+d2)2−4M24d1d2,
(5) where M is the mass of a Schwarzschild black hole, and
d1,2=√ρ2+(z∓M)2 . The functionsνhalo,λhalo are the solution of the halo structure, andλint represents the interaction between the black hole and halo.In Schwarzschild coordinates, the space-time of a Schwarzschild black hole with a halo can be described by the metric [93]
ds2=−(1−2Mr)e(νQ+νO)dt2+e(λQ+λO+λQO−νQ−νO)×[(1−2Mr)−1dr2+r2dθ2]+e−(νQ+νO)r2sin2θdϕ2,
(6) where
νQ=(Q/3)(3u2−1)(3v2−1),νO=(O/5)uv(5u2−3)(5v2−3),λQ=−4Qu(1−v2)+(Q2/2)[9u4v4−10u4v2−10u2v4+12u2v2+u4+v4−2u2−2v2+1],λO=−2O[3u2v−3u2v3+v3−95v+45]+2O2[758u6v6−1178u6v4−1178u4v6+458u6v2+458u2v6+1898u4v4−38740u4v2−38740u2v4+891200u2v2−38u6−38v6+2740u4+2740v4−81200u2−81200v2+21200],
λQO=2QO[9u5v5−12u5v3+3u5v−12u3v5+845u3v3−245u3v+3uv5−245uv3+95uv],
(7) and
u=12M[√ρ2+(z+M)2+√ρ2+(z−M)2],v=12M[√ρ2+(z+M)2−√ρ2+(z−M)2].
(8) The transformation between the Schwarzschild coordinates (
r,θ ) and Weyl coordinates (ρ,z ) isρ=√r(r−2M)sinθ,z=(r−M)cosθ.
(9) The exterior halo is a multipolar structure containing quadrupolar and octopolar terms, and
Q andO are the quadrupole and octopole strengths, respectively. The metric will reduce to a Schwarzschild solution whenQ=O=0 .The existence of a halo does not change the radial coordinate of the event horizon,
rh=2M ; however, the surface geometry of a Schwarzschild black hole with a halo would be affected by the quadrupole strengthQ and octopole strengthO . The event horizon of a Schwarzschild black hole with a halo can be described by the two-dimensional line elementds2h=r2he(−νQ−νO)|r=rh(dθ2+sin2θdϕ2).
(10) Embedding diagrams [97, 98] for the event horizon of a Schwarzschild black hole with a halo in Euclidean 3-space with different
Q andO are exhibited in Fig. 1. Because the event horizon in the (x,y ) plane is centrosymmetric, we only show the event horizon in the (x,z ) plane, wherez=0 represents the equatorial plane. As shown in Fig. 1(a) withO=0 , the event horizon is prolate at the quadrupole strengthQ<0 and may become more prolate via stretching along the vertical direction and squeezing along the horizontal direction asQ decreases. However, atQ>0 , the event horizon becomes more oblate in the opposite manner asQ increases. As shown in Fig. 1(b) withQ=0 , the octopolar term (O≠0 ) breaks the reflection symmetry of the event horizon with respect to the equatorial plane, whereas the event horizon in the equatorial plane does not change withO . The event horizon stretches upward atO<0 and downward atO>0 . The two event horizons with oppositeO are symmetrical to each other about the equatorial plane. Figure 1(c) shows the joint effort of the quadrupole strengthQ and octopole strengthO on the event horizon atO=0.1 andQ=−0.1 , 0, 0.1. That is, the quadrupole term causes the Schwarzschild black hole to be prolate atQ<0 and oblate atQ>0 , and the octopole term causes the shadow to stretch upward atO<0 and downward atO>0 .Figure 1. (color online) Embedding diagrams for the surface geometry of a Schwarzschild black hole with a halo in Euclidean 3-space with different quadrupole strengths
Q and octopole strengthsO .To quantitatively illustrate the influences of the quadrupolar and octopolar terms on a Schwarzschild black hole with a halo, we compute the equatorial circumference
Ce and polar circumferenceCp of the event horizon, which are given byCe=∫2π0√gϕϕdϕ|(r=rh,θ=π2)=∫2π0rhe−12(νQ+νO)|(r=rh,θ=π2)dϕ,
(11) Cp=2∫π0√gθθdθ|r=rh=2∫π0rhe−12(νQ+νO)|r=rhdθ.
(12) Figures 2(a) and (b) show the plots of the equatorial circumference
Ce and polar circumferenceCp of the event horizon as a function ofQ . Here,2πrh is the circumference of the Schwarzschild black hole event horizon (Q=O=0 ). With increasingQ , we find thatCe increases andCp decreases. The equatorial circumferenceCe is independent ofO , which is consistent with the results in Fig. 1(b). The two polar circumferencesCp with oppositeO share the same value and increase as|O| increases for fixedQ . In addition, the change inCp withO is very small, which signifies that the octopole strengthO has little effect on the polar circumferenceCp . The distortion parameter δ, defined as(Ce−Cp)/Ce , can determine the departure of the event horizon from spherical symmetry. Figure 2(c) shows the behavior of the distortion parameter δ withQ andO . The distortion parameter δ almost depends only onQ and is nearly less than0 atQ<0 , indicating a prolate event horizon, whereasδ>0 indicates an oblate event horizon. The surface area A of the event horizon is given byFigure 2. (color online) Change in the equatorial circumference
Ce , polar circumferenceCp , distortion parameter δ, and surface area A of the event horizon with the quadrupole strengthQ for different octopole strengthsO . Here, we set the Schwarzschild black hole massM=1 .A=∫π0∫2π0√gϕϕgθθdθdϕ|r=rh=∫π0∫2π0r2hsinθe−(νQ+νO)|r=rhdθdϕ.
(13) Figure 2(d) shows the change in the surface area A of the event horizon with
Q andO , where4πr2h is the surface area of the Schwarzschild black hole. The surface area A first decreases and then increases with increasingQ , which indicates that prolate and oblate event horizons both have large surface areas. The increase in the surface area A with|O| means that the octopole term can also enlarge the surface area of the Schwarzschild black hole.The Hamiltonian
H of photon propagation along null geodesics in the space-time of a Schwarzschild black hole with a halo (6) can be described asH=grrp2r+gθθp2θ+Veff=0,
(14) where the effective potential
Veff is defined asVeff=gttE2+gϕϕL2z=E2(gtt+gϕϕη2).
(15) E and
Lz are two constants of motion for null geodesics motion, that is, the energy and z-component of the angular momentum; therefore, the impact parameterη=Lz/E is also a constant of photon motion. The boundary of the shadow is determined using the photon sphere caused by unstable photon circular orbits. Light rays that enter the photon sphere are captured by the black hole; light rays that do not enter the photon sphere move away to infinity; and light rays that spiral asymptotically toward the photon sphere compose the boundary of the black hole shadow. Now, let us study the spherical photon orbits in the equatorial plane, which are also known as light rings. Light rings must satisfyθ=π/2,Veff=0,∂Veff∂r=0.
(16) Moreover, a light ring with
∂2Veff/∂r2<0 is unstable, whereas a light ring with∂2Veff/∂r2>0 is stable. By solving Eq. (16), we find that the radius of light ringsrLR only depends on the quadrupole strengthQ , as shown in Fig. 3. There is a critical value of the quadrupole strengthQc≈−0.0209445 for light rings in Fig. 3. Light rings do not exist in the space-time of a Schwarzschild black hole with a halo atQ<Qc ; both unstable (red dashed line) and stable (black line) light rings exist atQc<Q<0 ; and only one unstable (red dashed line) light ring exists atQ>0 . Our previous study [51] manifested that the existence of a stable light ring ensures that photons in stable orbits are always moving around the black hole and cannot enter the black hole or escape to infinity. The inexistence of light rings causes a panoramic (equatorial) shadow to appear [51, 52]. In this study, we only research the case in which light rings exist, and the halo structures could be considered a perturbation of a black hole. After this research, we decide to set the quadrupole strengthQ to the order of10−4 and the quadrupole strengthO to the order of10−6 . In this case, the halo structures are more in line with the actual astronomical situation. Moreover, the radius of stable light rings is larger than the radiusrobs of the observer we set (forQ=−1×10−4 ,rLR=70.7M>robs=50M ), in which the observer directly observes the black hole shadow. -
Now, we study the shadow of the Schwarzschild black hole with a halo using the backward ray-tracing technique [39–45]. In this method, we assume that the static observer is located at (
robs,θobs ) in the zero-angular-moment-observers (ZAMOs) reference frame [15], and evolved light rays from the observer move backward in time. The shadow of a black hole is caused by light rays falling into the event horizon of the black hole. The coordinates of a photon image in the observer's sky can be expressed as [39–45]x=−robs√1−2MrLzrsinθe(12λQ+12λO+12λQO−νQ−νO)˙r|(robs,θobs),y=robs√r(r−2M)˙θ˙r|(robs,θobs).
(17) In Fig. 4, we show the influence of the quadrupole term on the Schwarzschild black hole shadow with different quadrupole strengths
Q . Here, we setM=1 and the static observer atrobs=50 with the inclination angleθobs=π/2 . In this paper, we set a light-emitting celestial sphere as the background light source marked by four different colored quadrants and brown grids as the longitude and latitude, which is the same as the celestial sphere in Refs. [39–41]. The shadow of a Schwarzschild black hole with only the quadrupole structure is symmetric about the equatorial plane. The shadow of a Schwarzschild black hole with a halo becomes more oblate by only stretching along the horizontal direction asQ decreases atQ<0 . However, atQ>0 , the black hole shadow becomes more prolate by only squeezing along the horizontal direction asQ increases. With increasing|Q| , the Einstein ring, the white ring around the black hole shadow, is torn into an Einstein cross. In Fig. 5, we show the influence of the octopolar term on the Schwarzschild black hole shadow with different octopolar strengthsO . The black hole shadow shifts upward asO decreases atO<0 and downward asO increases atO>0 . The octopolar term also breaks the reflection symmetry of the Schwarzschild black hole shadow with respect to the equatorial plane; however, the two black hole shadows with oppositeO are symmetrical to each other about the equatorial plane. In Fig. 6, we show the joint effort of the quadrupole and octopole terms on the shadow of the Schwarzschild black hole. That is, the black hole shadow stretches and squeezes along the horizontal direction atQ<0 andQ>0 , respectively, and the black hole shadow shifts upward atO<0 and downward atO>0 . In Fig. 7, we present the light rays that form the shadow boundary to explain the emergence of the extraordinary patterns of the black hole shadow with quadrupole and octopole terms. Figures 7(a) and (b) show the light rays withQ=−2×10−4 andQ=2×10−4 , respectively, in they=0 plane (the equatorial plane), where y is the celestial coordinate (17) in the observer's sky. As shown, the two light rays (red lines) propagate from the observer backward in time and spiral asymptotically toward the photon sphere. However, for the observer, the light rays propagate along the red dash lines, which determine the left and right margins of the black hole shadow, respectively. The black region on the x axis represent the black hole shadow on the celestial sphere, where x is the celestial coordinate (17). The angle between the two red dash lines, namely, the angular radius of the black hole shadow, is significantly larger forQ=−2×10−4 and significantly smaller forQ=2×10−4 , which results in the stretching and squeezing of the black hole shadow in the horizontal direction atQ<0 andQ>0 , respectively. Figures 7(c) and (d) show the light rays withO=−2×10−6 andO=2×10−6 , respectively, in thex=0 plane. We also present the light rays (red lines) spiraling asymptotically toward the photon sphere and their tangents (red dash lines) at the observer, which determine the upper and lower margins of the shadow. As shown, the tangents slope upward forO=−2×10−6 and downward forO=2×10−6 , which causes the shadow to shift upward and downward along the y axis of the celestial sphere atO<0 andO>0 , respectively.Figure 4. (color online) Shadows of a Schwarzschild black hole with a halo for the quadrupole strength
Q=−2×10−4,−1×10−4, 0, 1×10−4, 2×10−4 , and octopolar strengthO=0 . Here, we setM=1 and the static observer atrobs=50 with the inclination angleθobs=π/2 .Figure 5. (color online) Shadows of a Schwarzschild black hole with a halo for the quadrupole strength
Q=0 , and octopolar strengthO=−2×10−6, −1×10−6, 0,1×10−6, 2×10−6 . Here, we setM=1 and the static observer atrobs=50 with the inclination angleθobs=π/2 .Figure 6. (color online) Shadows of a Schwarzschild black hole with a halo for the quadrupole strength
Q=−2×10−4, −1×10−4, 0,1×10−4, 2×10−4 , and octopolar strengthO=−1×10−6 . Here, we setM=1 and the static observer atrobs=50 with the inclination angleθobs=π/2 .Figure 7. (color online) Light rays (red lines) spiraling asymptotically toward the photon sphere and their tangents (red dash lines) at the observer, which determine the boundary of the black hole shadow. The black hole shadow stretches and squeezes on the celestial sphere at
Q<0 andQ>0 , respectively, and shifts upward atO<0 and downward atO>0 .Black holes in the universe can be perturbed by a halo with quadrupole and octopolar terms; hence, we estimate several observables of the black hole shadow to determine the quadrupole strength
Q and octopolar strengthO in astronomical observations. To characterize the shadow of a Schwarzschild black hole with a halo, we should first introduce four important points for the shadow: the leftmost point (xl ,yl ), rightmost point (xr ,yr ), topmost point (xt ,yt ), and bottommost point (xb ,yb ), as shown in Fig. 8. We can then define the observable values of the black hole shadow: the widthW=(xr−xl)/Rs , heightH=(yt−yb)/Rs , and oblatenessK=W/H , whereRs is the radius of the Schwarzschild black hole shadow (Q=O=0 ). Figure 9 shows the various widths W, heights H, and oblateness K of the black hole shadow with the quadrupole strengthQ for different octopolar strengthsO . As shown in Fig. 9(a), the width W of the black hole shadow decreases asQ increases; almost all width values are larger than2Rs atQ<0 and less than2Rs atQ>0 . Moreover, the octopolar strengthO has little effect on the width W of the black hole shadow. This indicates that only the quadrupole term stretches and squeezes the black hole shadow along the horizontal direction atQ<0 andQ>0 , respectively. As shown in Fig. 9(b), the height H of the black hole shadow increases asQ increases, but the change in height is considerably smaller than the change in width. Thus, the change in oblateness K of the black hole shadow withQ is almost the same as the change in width W. The black hole shadow is oblate (K>1 ) atQ<0 and prolate (K<1 ) atQ>0 . In addition, the height H of the black hole shadow increases as|O| increases for fixedQ . The main effect of the octopolar strengthO is moving the black hole shadow along the vertical direction. Therefore, we define the center of the black hole shadow as(xc,yc)=(xl+xr2,yt+yb2) and make use ofYc (yc/Rs ) to describe the deviation of the shadow away from the equatorial plane. In Fig. 10(a), we show variousYc with the octopolar strengthO for differentQ . The deviation|Yc| increases as|O| increases, andYc>0 atO<0 , whereasYc<0 atO>0 . Furthermore, the deviation|Yc| is larger at largerQ . This indicates that a prolate shadow amplifies the deviation of the shadow, whereas an oblate shadow diminishes the deviation. Unfortunately, in actual observations of a black hole shadow, we cannot determine the coordinates of the shadow center(xc,yc) to estimate the value ofO . The octopolar term, meanwhile, can cause a slight distortion along the vertical direction in the black hole shadow that isyc is not equal toyl oryr , as shown in Fig. 8. Hence, we can define a distortion parameterδc=(yc−yl)/Rs to describe the distortion caused by the octopolar term. Figure 10(b) shows the various values of the distortion parameterδc with the octopolar strengthO for differentQ . The distortion parameter|δc| increases as|O| increases, andδc<0 atO<0 , whereasδc>0 atO>0 . In addition, the distortion parameter|δc| is smaller at largerQ . This indicates that a prolate shadow diminishes the distortion, whereas an oblate shadow amplifies the distortion. We hope that the Event Horizon Telescope and BlackHoleCam will observe the shadow of a black hole perturbed by a halo with quadrupole and octopolar terms in future astronomical observations.Figure 8. (color online) Leftmost point (
xl ,yl ), rightmost point (xr ,yr ), topmost point (xt ,yt ), bottommost point (xb ,yb ), and center(xc,yc)=(xl+xr2,yt+yb2) of the black hole shadow. The distortion parameterδc is defined as(yc−yl)/Rs , whereRs is the radius of the Schwarzschild black hole shadow (Q=O=0 ).Figure 9. (color online) Various values of the width W, height H, and oblateness K of the black hole shadow with the quadrupole strength
Q for differentO .Figure 10. (color online) Various values of the deviation
Yc and distortion parameterδc of the black hole shadow with the octopolar strengthO for differentQ .In Figs. 11 and 12, we present the shadows of a Schwarzschild black hole with a halo for the observer inclination angle
θobs=0 andπ/4 , respectively. As shown in Fig. 11, the black hole shadows are always circular with differentQ andO atθobs=0 . However, interestingly, the black hole shadow becomes larger with increasingQ orO . Via a comparison with the Schwarzschild black hole shadow (Fig. 11(e) withQ=O=0 ), we find that a negativeQ orO causes the black hole shadow to shrink, whereas a positiveQ orO causes the black hole shadow to expand. For the observer inclination angleθobs=π/4 (Fig. 12), the black hole shadow not only becomes more prolate but also shifts upward with increasingQ orO . -
We study the surface geometry and shadows of a Schwarzschild black hole with a halo. The exterior halo is a multipolar structure containing quadrupolar and octopolar terms. We find that the quadrupole term causes the Schwarzschild black hole to be prolate at quadrupole strength
Q<0 and oblate atQ>0 , and the octopole term causes the shadow to stretch upward at octopolar strengthO<0 and downward atO>0 . The radius of the light ringsrLR only depends onQ in the space-time of a Schwarzschild black hole with a halo. Light rings do not exist whenQ is larger than a criticalQc ; both unstable and stable light rings exist atQc<Q<0 ; and only one unstable light ring exists atQ>0 . The shadow of a Schwarzschild black hole with a halo stretches and squeezes along the horizontal direction atQ<0 andQ>0 , respectively. Meanwhile, the black hole shadow shifts upward atO<0 and downward atO>0 . We show the light rays spiraling asymptotically toward the photon sphere and their tangents at the observer, which determine the margins of the shadow. The angle between the two tangents, namely, the angular radius of the black hole shadow, is larger atQ<0 and smaller atQ>0 , which results in the stretching and squeezing of the black hole shadow in the horizontal direction. The tangents slope upward atO<0 , and downward atO> , which causes the shadow to shift upward and downward in the vertical direction. From the observable width W, height H, oblateness K, and distortion parameterδc of the black hole shadow, one can determine the value ofQ andO of a Schwarzschild black hole with a halo. Black hole shadows are always circular and become larger asQ orO increases for the observer inclination angleθobs=0 . Black hole shadows not only become more prolate but also shift upward with increasingQ orO forθobs=π/4 . Our results show that the quadrupolar and octopolar terms yield a series of interesting patterns for the shadow of a Schwarzschild black hole with a halo.
