  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.
