Boundary Orbits: 1 Static Spacetimes

The study of circular orbits in spacetime is of astrophysical importance. The identification and classification of circular orbits in both static and stationary spacetimes remains an active area of interest. Even in the simplest static spherically symmetric case, it is well known that the introduction of a cosmological constant in vacuum leads to the study of quartic polynomials in order to locate \textit{boundary orbits}, those that straddle between stable and unstable orbits. These orbits are often referred to as `marginally stable orbits' or `indifferently stable orbits'. A comprehensive study of texts offers little clarification as to the stability or instability of these boundary orbits. Here we argue that the direct use of second order perturbation theory immediately shows that these boundary orbits are unstable in the perturbative sense. Our study here includes the two-particle Curzon-Chazy solution.