Construction of explicit symplectic integrators in general relativity. IV. Kerr black holes

Abstract

In previous papers, explicit symplectic integrators were designed for nonrotating black holes, such as a Schwarzschild black hole. However, they fail to work in the Kerr spacetime because not all variables can be separable, or not all splitting parts have analytical solutions as explicit functions of proper time. To cope with this difficulty, we introduce a time transformation function to the Hamiltonian of Kerr geometry so as to obtain a time-transformed Hamiltonian consisting of five splitting parts, whose analytical solutions are explicit functions of the new coordinate time. The chosen time transformation function can cause time steps to be adaptive, but it is mainly used to implement the desired splitting of the time transformed Hamiltonian. In this manner, new explicit symplectic algorithms are easily available. Unlike Runge Kutta integrators, the newly proposed algorithms exhibit good long term behavior in the conservation of Hamiltonian quantities when appropriate fixed coordinate time steps are considered. They are better than same order implicit and explicit mixed symplectic algorithms and extended phase space explicit symplectic like methods in computational efficiency. The proposed idea on the construction of explicit symplectic integrators is suitable for not only the Kerr metric but also many other relativistic problems, such as a Kerr black hole immersed in a magnetic field, a Kerr Newman black hole with an external magnetic field, axially symmetric core shell systems, and five dimensional black ring metrics.