Effects of coupling constants on chaos of charged particles in the $Einstein-\AE$ ther theory

TL;DR

This paper investigates how coupling constants in Einstein- ext{ extgreek a}ether theory influence chaotic behavior of charged particles near black holes, using symplectic integrators to analyze non-integrable Hamiltonian systems.

Contribution

It introduces a fourth-order explicit symplectic integrator to study charged particle dynamics in Einstein- ext{ extgreek a}ether black hole backgrounds with varying coupling parameters.

Findings

Chaotic and regular dynamics depend on coupling constants and initial conditions.

No universal rule for the dependence of chaos on parameters $c_{13}$ and $c_{14}$.

Distribution of chaos varies with parameter combinations and initial conditions.

Abstract

There are two free coupling parameters $c_{13}$ and $c_{14}$ in the Einstein-\AE ther metric describing a non-rotating black hole. This metric is the Reissner-Nordstr\"{o}m black hole solution when $0\leq 2c_{13}<c_{14}<2$, but it is not for $0\leq c_{14}<2c_{13}<2$. When the black hole is immersed in an external asymptotically uniform magnetic field, the Hamiltonian system describing the motion of charged particles around the black hole is not integrable. However, the Hamiltonian allows for the construction of explicit symplectic integrators. The proposed fourth-order explicit symplectic scheme is used to investigate the dynamics of charged particles because it exhibits excellent long-term performance in conserving the Hamiltonian. No universal rule can be given to the dependence of regular and chaotic dynamics on varying one or two parameters $c_{13}$ and $c_{14}$ in the two cases of $0\leq 2c_{13}<c_{14}<2$ and $0\leq c_{14}<2c_{13}<2$. The distributions of order and chaos in the binary parameter space $(c_{13},c_{14})$ rely on different combinations of the other parameters and the initial conditions.