Optical 2-metrics of Schwarzschild–Tangherlini spacetimes and the Bohlin–Arnold duality

Abstract

In what follows, we consider the projection of null geodesics of the Schwarzschild–Tangherlini metric in n + 1 dimensions to the space of orbits of the static Killing vector where the motion of a given light ray is seen to lie in a plane. The projected curves coincide with the unparametrized geodesics of optical 2-metrics and can be equally understood as describing the motion of a non-relativistic particle in a central force. We consider a duality between the projected null curves for pairs of values of n and interpret its mathematical meaning in terms of the optical 2-metrics. The metrics are not projectively equivalent but the correspondence can be exposed in terms of a third order differential equation. We also explore the extension of this notion of duality to the Reissner–Nordstrom case.