Explicit symplectic methods in black hole spacetimes

Abstract

Many Hamiltonian problems in the Solar System are separable or separate into two analytically solvable parts, and thus give a great chance to the development and application of explicit symplectic integrators based on operator splitting and composing. However, such constructions cannot in general be available for curved spacetimes in general relativity and modified theories of gravity, because these curved spacetimes correspond to nonseparable Hamiltonians without the two part splits. Recently, several black hole spacetimes such as the Schwarzschild black hole were found to allow the construction of explicit symplectic integrators, since their corresponding Hamiltonians are separable into more than two explicitly integrable pieces. Although some other curved spacetimes including the Kerr black hole do not exist such multi part splits, their corresponding appropriate time transformation Hamiltonians do. In fact, the key problem for the obtainment of symplectic analytically integrable decomposition algorithms is how to split these Hamiltonians or time transformation Hamiltonians. Considering this idea, we develop explicit sympelcetic schemes in curved spacetimes. We introduce a class of spacetimes whose Hamiltonians are directly split into several explicitly integrable terms. For example, the Hamiltonian of rotating black ring has a 13 part split. We also present two sets of spacetimes whose appropriate time transformation Hamiltonians have the desirable splits. For instance, an 8 part split exists in a time-transformed Hamiltonian of Kerr-Newman solution with disformal parameter. In this way, the proposed symplectic splitting methods will be used widely for long-term integrations of orbits in most curved spacetimes we have known.

16pages,accepted by Astrophysical Journal

References in corpus (18)

• Regular black holes with a nonlinear electrodynamics source
• Transition from Regular to Chaotic Circulation in Magnetized Coronae near Compact Objects
• Particle Motion around Black Hole in Horava-Lifshitz Gravity
• Symplectic structure of post-Newtonian Hamiltonian for spinning compact binaries
• Construction of Explicit Symplectic Integrators in General Relativity. I. Schwarzschild Black Holes
• Construction of explicit symplectic integrators in general relativity. II. Reissner-Nordstrom black holes
• On post-Newtonian orbits and the Galactic-center stars
• Construction of explicit symplectic integrators in general relativity. IV. Kerr black holes
• Inducing chaos by breaking axial symmetry in a black hole magnetosphere
• Symplectic Integration of Post-Newtonian Equations of Motion with Spin
• A Symmetric Integrator for non-integrable Hamiltonian Relativistic Systems
• 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
• Chaos in Geodesic Motion around a Black Ring
• Dynamics of charged particles moving around Kerr black hole with inductive charge and external magnetic field
• Charged particle motions near non-Schwarzschild black holes with external magnetic fields in modified theories of gravity
• Innermost stable circular orbits in Majumdar--Papapetrou dihole spacetime
• Black holes, stationary clouds and magnetic fields