Newton's Theorem of Revolving Orbits in Curved Spacetime

Abstract

Newton's theorem of revolving orbits states that one can multiply the angular speed of a Keplerian orbit by a factor $k$ by applying a radial inverse cubed force proportional to $(1-k^2)$. In this paper we derive an extension of this theorem in general relativity, valid for the motion of massive particles in any static, spherically symmetric metrics. We verify the Newtonian limit of this extension and demonstrate that there is no such generalization for rotating metrics. Further we also extend the theory to the case of charged particles in the Einstein-Maxwell and Kaluza-Klein theories.