Plenary Talk

Binary Black Hole Shadows: Chaos in General Relativity

November 23, 9:00-11:00 Am, Central Time 

Miguel A. F. Sanjuán

Professor, corresponding member of the Spanish Royal Academy of Sciences, foreign member of the Lithuanian Academy of Sciences, regular member of the Academia Europaea-The Academy of Europe,
Departamento de Física,
Universidad Rey Juan Carlos, Spain

Abstract: General relativity — itself a nonlinear field theory — naturally leads to deterministic chaos. For example, the fate of a photon approaching a pair of black holes can be essentially indeterminate, even though it is governed by a deterministic set of equations. Here we explore a topic of interest to astronomers, relativists, and nonlinear dynamicists alike: the intricate structure of the shadow cast by the event horizons of a pair of black holes (BHs). An exciting era for gravitational astronomy is underway. In 2015, the first direct observation of gravitational waves (GWs), by the LIGO/Virgo collaboration, confirmed that binary black holes exist in Nature. The Event Horizon Telescope (EHT) has begun observing nearby galactic centres, such as Sagittarius A* and M87. On April 10, 2019, the first picture of a BH shadow at the center of the M87 galaxy was shown. A BH shadow is associated with the set of all photons which, when traced backwards in time from the observer, asymptote towards the event horizon of the BH. In the language of nonlinear dynamics, a BH shadow is an exit basin in an open Hamiltonian dynamical system. Motivated by the GW detections from merging binary BHs, and the future prospects of the EHT, a strand of recent work has focused on what the shadow of a pair of BHs would look like. The null geodesic equations, which describe the propagation of photons, are non-integrable, and chaotic scattering of photons emerges naturally [1]. One of the hallmarks of chaos is the presence of fractal structures in phase space. In a binary BH system, a photon meets one of three possible fates: it falls into the first BH, the second BH, or it escapes to infinity. Thus, it is natural to define three exit basins. As we shall show, across the phase space the shadow may exhibit both a regular and a fractal structure. Furthermore, in certain parts of the phase space, the three basins have the more restrictive property of Wada, with all three basins sharing a common fractal boundary. For the binary BH system, this means that a photon which starts close to a Wada boundary in phase space is uncertain and could end up in one of three final states: the photon could fall into either of the black holes, or escape to spatial infinity. Here we apply a recently-developed numerical method, the merging method [2] to test for the Wada property to study the fractal structures that arise in a binary BH model in general relativity. We have verified the Wada property in both (i) the exit basins in phase space; and (ii) exit basins on an image plane which define the shadow cast by the BHs. We have demonstrated that the BH shadow can exhibit either the partial Wada or the full Wada property, depending on the value of the BH separation. To our knowledge, this work [3] represents the first demonstration of the Wada property for a general-relativistic system. As well as demonstrating that tools from the field of chaos theory can be used to understand the rich dynamics of scattering processes in general relativity, this work highlights that there exist novel dynamical systems in gravitational physics which can be fruitfully explored by nonlinear dynamicists. This is a joint work with A. Daza (Spain), J. Shipley, and S. Dolan (UK).

[1] Jake O. Shipley and Sam R. Dolan. Binary black hole shadows, chaotic scattering and the Cantor set. Classical Quantum Gravity 33, 175001 (2016).
[2] Alvar Daza, Alexandre Wagemakers, Miguel A.F. Sanjuán. Ascertaining when a basin is Wada: the merging method. Scientific Reports 8, 9954 (2018)
[3] Alvar Daza, Jake O. Shipley, Sam R. Dolan and Miguel A. F. Sanjuan. Wada structures in a binary black hole system. Phys. Rev. D 98, 084050 (2018)