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

  • Wolfgang Graf
Research Article


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 \((J^2)^{\prime } > 0\) 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.


Relativistic astrophysics General relativity Cosmology Symmetries 



I thank Peter Aichelburg for helpful discussions.


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Authors and Affiliations

  • Wolfgang Graf
    • 1
  1. 1.ViennaAustria

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