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

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.

22 pages, one figure. The abstract changed; new section concerning higher dimensional example added, accepted for publication in Nuclear Physics B

References in corpus (6)

• Higher-Dimensional Black Holes: Hidden Symmetries and Separation of Variables
• Charged particle in higher dimensional weakly charged rotating black hole spacetime
• Dynamics of test particles in the general five-dimensional Myers-Perry spacetime
• Kerr Geodesics in Terms of Weierstrass Elliptic Functions
• Five-Dimensional Black Hole Capture Cross-Sections
• Fermat's Principle in General Relativity via Herglotz Variational Formalism