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.