Relativistic Euler’s three-body problem, optical geometry, and the golden ratio

Abstract

A Weyl solution describing two Schwarzschild black holes is considered. We focus on the ${\mathbb{Z}}_{2}$ invariant solution, with Arnowitt-Deser-Misner mass ${M}_{\mathrm{ADM}}=2{M}_{\mathrm{K}}$, where ${M}_{\mathrm{K}}$ is the Komar mass of each black hole. For this solution the set of fixed points of the discrete symmetry is a totally geodesic submanifold. The existence and radii of circular photon orbits in this submanifold are studied, as functions of the distance $2L$ between the two black holes. For $L\ensuremath{\rightarrow}0$ there are two such orbits, corresponding to $r=3{M}_{\mathrm{ADM}}$ and $r=2{M}_{\mathrm{ADM}}$ in Schwarzschild coordinates. As the distance increases, it is shown that the two photon orbits approach one another and merge when ${M}_{\mathrm{K}}=\ensuremath{\varphi}L$, where $\ensuremath{\varphi}$ is the golden ratio. Beyond this distance there exist no circular photon orbits. The two null orbits delimit a forbidden band for timelike circular orbits, which is interpreted in terms of optical geometry. For large $L$, timelike circular orbits are allowed everywhere, as in the analogous Newtonian problem. The analysis is generalized by considering a ${\mathbb{Z}}_{2}$ invariant Weyl solution with an array of $N$ black holes and also by charging the black holes, which connects the Weyl solution to a Majumdar-Papapetrou spacetime.