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

TL;DR

The paper develops a framework to study light propagation in Kerr and higher dimensional rotating spacetimes by applying coupling-constant metamorphosis to Fermat-principle–based Hamiltonians. This yields a family of Hamiltonians quadratic in momenta, enabling explicit reduction to two decoupled nonlinear oscillators via a Mino-time reparameterization and allowing both exact and perturbative solutions (via Lindstedt–Poincaré theory). A Carter-like constant ensures Arnold–Liouville integrability of the Kerr case, and the method naturally extends to Myers–Perry metrics, including the 5D single-rotation scenario. The work provides a unified, analytically tractable approach for null geodesics in rotating black-hole spacetimes, with potential applications to gravitational lensing and black-hole optics in higher dimensions.

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--Poincaré 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.

Figures (1)

• Figure 1: The shape of $V(u)$ for $\alpha=2$, $\beta=-1.5$ and $f=0.1$.