Optical 2-metrics of Schwarzschild-Tangherlini Spacetimes and the Bohlin-Arnold Duality

TL;DR

This paper investigates the geometric properties of null geodesics in higher-dimensional Schwarzschild-Tangherlini spacetimes, revealing a duality in optical metrics and extending it to Reissner-Nordstrom black holes.

Contribution

It introduces a novel duality between projected null geodesics for different dimensions and interprets it through a third order differential equation, extending the concept to charged black holes.

Findings

Identifies a duality between optical 2-metrics in different dimensions.

Shows the duality is described by a third order differential equation.

Extends the duality concept to Reissner-Nordstrom spacetime.

Abstract

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 unparametrised 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.