Geometry of chaos in the two-center problem in General Relativity

TL;DR

This paper investigates the chaotic behavior of null-geodesic motion in the two-black-hole spacetime described by the Majumdar-Papapetrou solution in General Relativity, linking it to the geometry of negatively curved surfaces.

Contribution

It identifies the geometric origins of chaos in relativistic two-center problems by reducing the dynamics to geodesic motion on a negatively curved surface.

Findings

Relativistic null-geodesics exhibit chaos unlike the integrable Newtonian case.

Chaos is linked to the negative curvature of the underlying geometric surface.

The geometric analysis explains the source of chaos in the two-black-hole spacetime.

Abstract

The now-famous Majumdar-Papapetrou exact solution of the Einstein-Maxwell equations describes, in general, $N$ static, maximally charged black holes balanced under mutual gravitational and electrostatic interaction. When $N=2$, this solution defines 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 first reducing the problem to that of geodesic motion on a negatively curved (Riemannian) surface.