Existence and Stability of Circular Orbits in Time-Dependent Spherically Symmetric Spacetimes

Abstract

For a general spherically four--dimensional metric the notion of "circularity" of a family of equatorial geodesic trajectories is defined in geometrical terms. The main object turns out to be the angular--momentum function $J$ obeying a consistency condition involving the mean extrinsic curvature of the submanifold containing the geodesics. The ana\-ly\-sis of linear stability is reduced to a simple dynamical system formally describing a damped harmonic oscillator. For static metrics the existence of such geodesics is given when $J^2 > 0$, and $(J^2)' > 0$ for stability. The formalism is then applied to the Schwarzschild--de Sitter solution, both in its static and in its time--dependent cosmological version, as well to the Kerr--de Sitter solution. In addition we present an approximate solution to a cosmological metric containing a massive source and solving the Einstein field equation for a massless scalar.