Explicit Symplectic Methods in Black Hole Spacetimes

Abstract

Many Hamiltonian problems in the solar system are separable into two analytically solvable parts, and thus serve as a great chance to develop and apply explicit symplectic integrators based on operator splitting and composing. However, such constructions are not in general 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 for 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 have such multipart splits, their corresponding appropriate time-transformation Hamiltonians do. In fact, the key problem in obtaining symplectic analytically integrable decomposition algorithms is how to split these Hamiltonians or time-transformation Hamiltonians. Considering this idea, we develop explicit symplectic 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 a 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 eight-part split exists in a time-transformed Hamiltonian of a Kerr–Newman solution with a disformal parameter. In this way, the proposed symplectic splitting methods can be used widely for long-term integrations of orbits in most curved spacetimes we know of.

A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes

“…That is to say, the numerical results should be correct for the original coefficients used in the published article and in Yang et al (2022) and Zhang et al (2022). In spite of this, the new coefficients in the double-precision environment should be recommended in the published article, in Wu et al (2022), Yang et al (2022), andZhang et al (2022), and in future works.…”

Explicit K-symplectic-like algorithms for guiding center system

“…The multi-part splitting method was also used to construct explicit symplectic integrators in references [7][8][9][10][11][12][13].…”

“…When the considered Hamiltonian is separable to the variables or can be split into two explicitly integrable terms, explicit symplectic integrators can be easily obtained by exactly solving the two subsystems with explicitly integrable sub-Hamiltonians and then composing the two exact solutions. Although most of Hamiltonians associated with curved spacetimes in general relativity lack such a property, they or their corresponding time-transformed Hamiltonians have more than two explicitly integrable splitting parts, and thus allow for the construction of explicit symplectic integrators [7][8][9][10][11][12][13]. In these general cases that the Hamiltonian can be separated into multi-parts, explicit symplectic methods can be constructed by the above splitting and composing approach.…”

“…In these general cases that the Hamiltonian can be separated into multi-parts, explicit symplectic methods can be constructed by the above splitting and composing approach. Explicit symplectic methods were constructed for charged particles around the magnetized Schwarzschild Black Holes [7], the Reissner-Nordström Black Holes [8], the Reissner-Nordström-(anti)-de Black Holes [9], the Kerr Black Holes [10], the Black Hole Spacetimes [11] and the Schwarzschild Spacetimes [12]. The explicit symplectic methods have been used to detect chaos of charged particles around magnetized Kerr black hole [14], magnetized Reissner-Nordström spacetimes [15], magnetized modified gravity Schwarzschild spacetime [16], and Einstein-aether theory [17].…”

Contrasting the Implicit Method in Incoherent Lagrangian and the Correction Map Method in Hamiltonian

“…However, Einstein's equations of general relativity describe the motion of strong gravitational systems for which exact solutions are very difficult to obtain, but there have been some efforts to address this issue. For example, Xin Wu [X. Wu(2022)] developed the explicit symplectic integrators for Hamiltonian systems in curved spacetimes, particularly for black hole spacetimes. The papers [Zhou(2022), Zhou(2023)] discuss the motion of charged particles around the Schwarzschild black hole with an external magnetic field.…”