Discussions on the integrable dynamics of charged particles around Kerr-Newman black holes in combined gravitational and electromagnetic fields

There are three types of electromagnetic potentials in Kerr-Newman backgrounds. The first type potential is a self-consistent potential, which was considered by Carter in 1968. This potential is a self-consistent solution of the Einstein and Maxwell equations. The second type potential is the Wald potential as an exact solution of the source-free Maxwell equations when the rotating black hole is uncharged. In fact, it describes an external electromagnetic test field based on the Kerr geometry. The third type potential is the generalized potential of Azreg-Aïnou, which is still associated with an external test field in the Kerr-Newman metric. In the later two potentials, the self-gravitating fields are ignored. Such magnetic fields are too small to make contributions to the geometries of spacetimes, but they would strongly affect the dynamics of charged particles near the black holes. The charged particle dynamics are integrable and regular in the self-consistent potential due to the existence of the Carter constant. Nevertheless, they are nonintegtrable and can allow the onset of chaos in both the Wald potential and the generalized potential. Here, the Wald potential in the Kerr metric is assumed to be the potential of an external test field in the Kerr-Newman spacetime and the inductive charge is supposed as the black hole charge. In this way, the dynamics of charged particles can be compared in the Wald potential and the generalized one. Under some circumstances, chaos is much easily induced in the generalized potential than in the Wald potential. As the spin and charge of the black hole increase, the degree of chaos is weakened in the Wald potential, but seems to have no explicit change in the generalized potential.