Newton's Theorem of Revolving Orbits in Curved Spacetime

TL;DR

This paper extends Newton's theorem of revolving orbits to general relativity, deriving a new theorem for static, spherically symmetric spacetimes and exploring its limitations and extensions to charged particles and rotating metrics.

Contribution

It provides the first general relativistic extension of Newton's revolving orbit theorem for static, spherically symmetric metrics and examines its applicability to charged particles and rotating spacetimes.

Findings

Derived a relativistic version of Newton's revolving orbit theorem.

Confirmed the theorem's validity in the Newtonian limit.

Found no extension for rotating metrics.

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.