Relativistic Euler’s 3-body problem, optical geometry and the golden ratio

A Weyl solution describing two Schwarzschild black holes is considered. We focus on the Z2 invariant solution, with ADM mass MADM = 2MK , where MK is the Komar mass of each black hole. For this solution the set of fixed points of the discrete symmetry is a totally geodesic sub-manifold. The existence and radii of circular photon orbits in this sub-manifold are studied, as functions of the distance 2L between the two black holes. For L → 0 there are two such orbits, corresponding to r = 3MADM and r = 2MADM in Schwarzschild coordinates. As the distance increases, it is shown that the two photon orbits approach one another and merge when MK = φL, where φ is the golden ratio. Beyond this distance there exist no circular photon orbits. The two null orbits delimit a forbidden band for time-like circular orbits, which is interpreted in terms of optical geometry. For large L, time-like circular orbits are allowed everywhere, as in the analogous Newtonian problem. The analysis is generalised by considering a Z2 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.