We investigate the qualitative features of binary black hole shadows using the model of two extremally charged black holes in static equilibrium (a Majumdar--Papapetrou solution). Our perspective is that binary spacetimes are natural exemplars of chaotic scattering, because they admit more than one fundamental null orbit, and thus an uncountably-infinite set of perpetual null orbits which generate scattering singularities in initial data. Inspired by the three-disc model, we develop an appropriate symbolic dynamics to describe planar null geodesics on the double black hole spacetime. We show that a one-dimensional (1D) black hole shadow may constructed through an iterative procedure akin to the construction of the Cantor set; thus the 1D shadow is self-similar. Next, we study non-planar rays, to understand how angular momentum affects the existence and properties of the fundamental null orbits. Taking slices through 2D shadows, we observe three types of 1D shadow: regular, Cantor-like, and highly chaotic. The switch from Cantor-like to regular occurs where outer fundamental orbits are forbidden by angular momentum. The highly chaotic part is associated with an unexpected feature: stable and bounded null orbits, which exist around two black holes of equal mass $M$ separated by $a_1 < a < \sqrt{2} a_1$, where $a_1 = 4M/\sqrt{27}$. To show how this possibility arises, we define a certain potential function and classify its stationary points. We conjecture that the highly chaotic parts of the 2D shadow possess the Wada property. Finally, we consider the possibility of following null geodesics through event horizons, and chaos in the maximally-extended spacetime.
We give a brief review on the formation and the calculation of black hole shadow. Firstly, we introduce the conception of black hole shadow and the current works on a variety of black hole shadows. Secondly, we present main methods of calculating photon sphere radius and shadow radius, and then explain how the photon sphere affect the boundary of black hole shadow. We review the analytical calculation for black hole shadows which have analytic expressions for shadow boundary due to the integrable photon motion system. And we introduce the fundamental photon orbits which can explain the pattern…
We have studied numerically the shadows of a non-Kerr rotating compact object with quadrupole mass moment, which belongs to Manko-Novikov family. The non-integrable photon motion caused by quadrupole mass moment affects sharply the shadow of the compact object. As the deviation parameter related to quadrupole mass moment is negative, the shadow of compact object is prolate and there are two disconnected main shadows with eyebrows located symmetrically on both sides of the equatorial plane. As the deviation parameter is positive, the shadow becomes oblate and the main shadow is joined together …
We have studied numerically the shadows of Bonnor black dihole through the technique of backward ray-tracing. The presence of magnetic dipole yields non-integrable photon motion, which affects sharply the shadow of the compact object. Our results show that there exists a critical value for the shadow. As the magnetic dipole parameter is less than the critical value, the shadow is a black disk, but as the magnetic dipole parameter is larger than the critical one, the shadow becomes a concave disk with eyebrows possessing a self-similar fractal structure. These behavior are very similar to those…
We discuss the emergence of black hole shadow and photon-sphere in the context of $f(R)$ gravity. It is shown that the shadow is exponentially sensitive to linear instabilities of metric coming from some $f(R)$ solutions. Thus, the instabilities of photon circular trajectories, delimiting the black hole photon-sphere, are double exponentialized. Specifically we individuate two Lyapunov exponents, rather than only one, related to two different sources of chaos in geodesic orbits as a sort of butterfly effect. Such a result violates the black hole chaos bound proposed by Maldacena, Shenker and S…
The data from the event horizon telescope have provided a novel view of the vicinity of the horizon of a black hole (BH), by imaging the region around the light-ring. They have also raised hopes for measuring in the near future, features of the image (or the shadow) related to higher order effects of photons traveling in these regions, such as the appearance of higher order bright rings. While the prospect of measuring these fine features of Kerr BHs is very interesting in itself, there are some even more intriguing prospects for observing novel features of possible non-Kerr objects, in the ca…
We study the physical properties of four-dimensional, string-theoretical, horizonless "fuzzball" geometries by imaging their shadows. Their microstructure traps light rays straying near the would-be horizon on long-lived, highly redshifted chaotic orbits. In fuzzballs sufficiently near the scaling limit this creates a shadow much like that of a black hole, while avoiding the paradoxes associated with an event horizon. Observations of the shadow size and residual glow can potentially discriminate between fuzzballs away from the scaling limit and alternative models of black compact objects.
We have studied the shadows of a Schwarzschild black hole surrounded by a Bach-Weyl ring through the backward ray-tracing method. The presence of Bach-Weyl ring leads to that the photon dynamical system is non-integrable and then chaos would appear in the photon motion, which affects sharply the black hole shadow. The size and shape the black hole shadow depend on the black hole parameter, the Bach-Weyl ring mass and the Weyl radius between black hole and ring. Some self-similar fractal structures also appear in the black hole shadow, which originates from the chaotic lensing. We also study th…
We investigate the null geodesic flow and in particular the light rings (LRs), fundamental photon orbits (FPOs) and shadows of a black hole (BH) immersed in a strong, uniform magnetic field, described by the Schwarzschilld-Melvin electrovacuum solution. The empty Melvin magnetic Universe contains a tube of planar LRs. Including a BH, for weak magnetic fields, the shadow becomes oblate, whereas the intrinsic horizon geometry becomes prolate. For strong magnetic fields (over-critical solutions), there are no LRs outside the BH horizon, a result explained using topological arguments. This feature…
In this thesis we study aspects of strong-field gravitational lensing by black holes in general relativity, with a particular focus on the role of integrability and chaos in geodesic motion. We first investigate binary black hole shadows using the Majumdar-Papapetrou static binary black hole (or di-hole) solution. It is shown that the propagation of null geodesics on this spacetime background is a natural example of chaotic scattering. We demonstrate that the binary black hole shadows exhibit a self-similar fractal structure akin to the Cantor set. Next, we use techniques from the field of non…
The recently obtained hairy Kerr black holes, due to additional sources or surrounding fluid, like dark matter, with conserved energy-momentum tensor, have a deviation $α$ and primary hair $l_0$, apart from rotation parameter $a$ and mass $M$. In the wake of the \textit{Event Horizon Telescope} (\textit{EHT}) observations of the supermassive black hole M87*, a recent surge in interest in black hole shadows suggests comparing the black holes in general relativity (GR) and modified theories of gravity (MoG) to assess these models' differences. Motivated by this, we take on an extensive study of …
The spacetime singularities in classical general relativity predicted by the celebrated singularity theorems are formed at the end of gravitational collapse. Quantum gravity is the expected theory to resolve the singularity problem, but we are now far from it. Therefore attention has shifted to models of regular black holes free from the singularities. A spherically symmetric regular toy model was obtained by Dymnikova (1992) which we demonstrate as an exact solution of Einstein's field equations coupled to nonlinear electrodynamics for a Lagrangian with parameter $b$ related to magnetic charg…
We have studied the spacetime of a Kerr black hole immersed in Melvin magnetic field, and found not only unstable light rings could exist, but also stable light rings could exist. Both the prograde and retrograde unstable light rings radiuses increase with the magnetic field parameter $B$, but it is the opposite for stable light rings. The existence of unstable, stable light rings depend on both the rotation parameter $a$ and the magnetic field parameter $B$. For a certain $a$, there are both the prograde and retroprade unstable (stable) light rings when $B$ is less than a critical value $B_{c…
Probing Quantum Gravity through Chaotic Orbits and Strong-Field Effects in Kerr Black Holes Embedded in Perfect Fluid Dark Matter
The Institute of Mathematical Sciences, C.I.T. Campus（泰米鲁纳德邦金奈塔拉马尼 CIT 校区的数学科学研究所）；Khazar University（阿塞拜疆巴库的哈萨尔大学）；University of Mazandaran（伊朗巴巴萨尔的马赞达兰大学）；Virtual University of Pakistan（巴基斯坦拉合尔的虚拟大学）；Hemwati Nandan Bahuguna Garhwal University（印度北阿坎德邦斯里纳加的赫姆瓦蒂·南丹·巴胡加纳加尔瓦尔大学）；Harbin Institute of Technology（中国哈尔滨的哈尔滨工业大学）；Zhejiang Normal University（中国金华的浙江师范大学）
We consider a static, axially symmetric spacetime describing the superposition of a Schwarzschild black hole (BH) with a thin and heavy accretion disk. The BH-disk configuration is a solution of the Einstein field equations within the Weyl class. The disk is sourced by a distributional energy-momentum tensor and it is located at the equatorial plane. It can be interpreted as two streams of counter-rotating particles, yielding a total vanishing angular momentum. The phenomenology of the composed system depends on two parameters: the fraction of the total mass in the disk, $m$, and the location …
Distorting Kerr Images with Parity-Odd Scalar Hair
South China University of Technology（华南理工大学）；Shanghai Jiao-Tong University（上海交通大学）；University of Jinan（济南大学）；Beijing Normal University（北京师范大学）；School of Physical Science and Technology, Ningbo University（宁波大学物理科学与技术学院）；Peking University（北京大学）
