Explicit symplectic methods in black hole spacetimes

TL;DR

This paper develops explicit symplectic integrators for curved spacetimes in general relativity by identifying Hamiltonian splits, enabling long-term stable simulations of orbits around black holes and other exotic objects.

Contribution

It introduces a class of spacetimes with directly split Hamiltonians and time transformation Hamiltonians, expanding the applicability of symplectic methods in curved spacetimes.

Findings

Hamiltonians of certain black hole spacetimes can be split into multiple integrable parts.

Explicit symplectic schemes are constructed for spacetimes like rotating black rings and Kerr-Newman.

These methods facilitate long-term orbit integrations in complex curved 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.