Existence and stability of circular orbits in time-dependent spherically symmetric spacetimes

• Published in: General relativity and gravitation Vol. 51; no. 3; pp. 1 - 22
• Main Author: Graf, Wolfgang
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2019; Springer Nature B.V
• Subjects: Angular momentum; Astronomy; Astrophysics and Cosmology; Circular orbits; Classical and Quantum Gravitation; Curvature; Differential Geometry; Dynamic stability; Einstein equations; Geodesy; Gravity; Manifolds (mathematics); Mathematical and Computational Physics; Orbital stability; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Research Article; Stability analysis; Theoretical; Time dependence; Cosmology; Symmetries; Relativistic astrophysics; General relativity
• ISSN: 0001-7701; 1572-9532
• DOI: 10.1007/s10714-018-2427-8

For a general spherically-symmetric 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 analysis of linear stability is reduced to a simple dynamical system. 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.