Construction of Explicit Symplectic Integrators in General Relativity. I. Schwarzschild Black Holes

TL;DR

This paper develops explicit symplectic integrators for Schwarzschild black hole spacetimes, enabling efficient and accurate long-term simulations of relativistic systems by splitting the Hamiltonian into integrable parts.

Contribution

The authors construct second- and fourth-order explicit symplectic integrators for Schwarzschild spacetime by splitting the Hamiltonian into four analytically solvable parts, improving efficiency and accuracy.

Findings

The new algorithms maintain bounded Hamiltonian errors over long simulations.

They are computationally more efficient than implicit methods.

Effective for modeling chaotic particle motion near black holes.

Abstract

Symplectic integrators that preserve the geometric structure of Hamiltonian flows and do not exhibit secular growth in energy errors are suitable for the long-term integration of N-body Hamiltonian systems in the solar system. However, the construction of explicit symplectic integrators is frequently difficult in general relativity because all variables are inseparable. Moreover, even if two analytically integrable splitting parts exist in a relativistic Hamiltonian, all analytical solutions are not explicit functions of proper time. Naturally, implicit symplectic integrators, such as the midpoint rule, are applicable to this case. In general, these integrators are numerically more expensive to solve than same-order explicit symplectic algorithms. To address this issue, we split the Hamiltonian of Schwarzschild spacetime geometry into four integrable parts with analytical solutions as explicit functions of proper time. In this manner, second- and fourth-order explicit symplectic integrators can be easily made available. The new algorithms are also useful for modeling the chaotic motion of charged particles around a black hole with an external magnetic field. They demonstrate excellent long-term performance in maintaining bounded Hamiltonian errors and saving computational cost when appropriate proper time steps are adopted.