Constants of Motion for Constrained Hamiltonian Systems: A Particle around a Charged Rotating Black Hole

TL;DR

This paper develops a hierarchical framework for identifying constants of motion in constrained Hamiltonian systems, specifically applied to particles around charged rotating black holes, including electromagnetic and gravitational fields.

Contribution

It generalizes the Killing tensor equation to include constraints and conformal Killing tensors, providing new methods to find conserved quantities in complex spacetime backgrounds.

Findings

Derived hierarchical equations for constants of motion.

Found conserved quantities for charged particles in black hole spacetimes.

Identified conditions under which conformal Killing tensors yield constants of motion.

Abstract

We discuss constants of motion of a particle under an external field in a curved spacetime, taking into account the Hamiltonian constraint which arises from reparametrization invariance of the particle orbit. As the necessary and sufficient condition for the existence of a constant of motion, we obtain a set of equations with a hierarchical structure, which is understood as a generalization of the Killing tensor equation. It is also a generalization of the conventional argument in that it includes the case when the conservation condition holds only on the constraint surface in the phase space. In that case, it is shown that the constant of motion is associated with a conformal Killing tensor. We apply the hierarchical equations and find constants of motion in the case of a charged particle in an electro-magnetic field in black hole spacetimes. We also demonstrate that gravitational and electro-magnetic fields exist in which a charged particle has a constant of motion associated with a conformal Killing tensor.