Geometry of chaos in the two-center problem in general relativity

Abstract

The now-famous Majumdar-Papapetrou exact solution of the EinsteinMaxwell equations describes, in general, N static, maximally charged black holes balanced under mutual gravitational and electrostatic interaction. When N = 2, this solution denes the two-black-hole spacetime, and the relativistic two-center problem is the problem of geodesic motion on this static background. Contopoulos and a number of other workers have recently discovered through numerical experiments that in contrast with the Newtonian two-center problem, where the dynamics is completely integrable, relativistic null-geodesic motion on the two black-hole spacetime exhibits chaotic behavior. Here I identify the geometric sources of this chaotic dynamics by rst reducing the problem to that of geodesic motion on a negatively curved (Riemannian) surface.

A tale of two centres

“…As an alternative to our fractal methods, several groups [12][13][14][15] have advocated curvature as a coordinate independent tool for forecasting chaos. The idea is to extend Hadamard's [16] classic result that the geodesic flow on a compact manifold with all sectional curvatures negative at every point is chaotic.…”

Chaotic lensing around boson stars and Kerr black holes with scalar hair

“…1. Chaotic scattering in GR spacetimes has been observed and discussed in binary or multi-BH solutions (see, e.g., [45,46,[74][75][76][77][78][79][80][81][82][83][84]) and is well known in the context of manybody scattering in classical dynamics, for example, the scattering of charged particles off magnetic dipoles [85] and the three-body problem (see, e.g., [86]). KBHsSH or RBSs provide an example of chaos in geodesic motion on the background of a single compact object, which moreover solves a simple and well-defined matter model minimally coupled to GR.…”

Gravitational waves from a chaotic dynamical system

“…It is also important to study chaos in general relativity because the Einstein equations are nonlinear. Many authors have reported chaos in general relativity [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. As for a realistic astrophysical object, there has been a discussion whether or not chaos occurs in a compact binary system [21,22,23].…”