Phase-plane analysis of orbital motion in Schwarzschild-(anti-)de Sitter spacetimes

Abstract

We use dynamical systems methods to understand qualitative aspects of geodesic motion of a particle in the Schwarzschild-(anti-)de Sitter (S-(a)dS) spacetimes. We express the equations of motion in dimensionless, first-order form, and find that the types of motion are fully characterized by the number and nature of fixed points in the (σ,n) plane, where σ and n are dimensionless parameters that involving the orbital angular momentum (L) and the cosmological constant Λ, respectively. We study the phase portraits and their dependence on these parameters, and with these provide a complete classification of the possible motions in the S-(a)dS spacetimes.