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

10 pages,2 figures

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Cited by in corpus (9)

• Construction of explicit symplectic integrators in general relativity. II. Reissner-Nordstrom black holes
• Construction of explicit symplectic integrators in general relativity. IV. Kerr black holes
• Applying explicit symplectic integrator to study chaos of charged particles around magnetized Kerr black hole
• A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes
• Dynamics of charged particles moving around Kerr black hole with inductive charge and external magnetic field
• Extended phase-space symplectic-like integrators for coherent post-Newtonian Euler-Lagrange equations
• Charged particle motions near non-Schwarzschild black holes with external magnetic fields in modified theories of gravity
• Energy-conserving integrator for conservative Hamiltonian systems with ten-dimensional phase space
• Adjustment of force-gradient operator in symplectic methods