Coupling constant metamorphosis, Fermat principle and light propagation in Kerr metric

TL;DR

This paper simplifies the complex Hamiltonian describing light geodesics in Kerr and Myers-Perry metrics using coupling constant metamorphosis, enabling easier analysis of light propagation in rotating black hole spacetimes.

Contribution

It introduces a novel application of coupling constant metamorphosis to reduce the Hamiltonian for Kerr and Myers-Perry metrics, simplifying the analysis of geodesics.

Findings

Hamiltonian reduced to quadratic form in momenta

Dynamics described as two nonlinear oscillators

Applicable to perturbation theory with Lindstedt--Poincaré algorithm

Abstract

The geodesics of Kerr's metric are described by the four-dimensional Hamiltonian dynamics integrable in the Arnold--Liouville sense. It can be reduced to two-dimensional one by the use of Fermat's principle. The resulting Hamiltonian is, however, rather complicated. We show how one can apply the coupling constant metamorphosis to simplify the Hamiltonian to the one quadratic in momenta and depending on the initial "energy" as parameter. It describes a simple dynamics of two non-linear oscillators and can be integrated directly or evaluated in the framework of perturbation theory by adopting the elegant Lindstedt--Poincar\'e algorithm. The idea of coupling constant metamorphosis is also applied to the Myers--Perry metric -- a five dimensional generalization of Kerr's metric. The case of single rotation parameter is considered in some detail.