Spacetime Symmetries and Kepler's Third Law

TL;DR

This paper demonstrates how Kepler's third law and its post-Newtonian generalization can be derived from spacetime symmetries, specifically helical Killing vectors, providing educational insights into key general relativity concepts.

Contribution

It introduces a covariant method to derive Kepler's law from spacetime symmetries, linking orbital dynamics with Killing vectors in curved spacetime.

Findings

Kepler's third law can be derived from the norm of a helical Killing vector.

The approach generalizes to post-Newtonian regimes.

Educational utility in illustrating core GR concepts.

Abstract

The curved spacetime geometry of a system of two point masses moving on a circular orbit has a helical symmetry. We show how Kepler's third law for circular motion, and its generalization in post-Newtonian theory, can be recovered from a simple, covariant condition on the norm of the associated helical Killing vector field. This unusual derivation can be used to illustrate some concepts of prime importance in a general relativity course, including those of Killing field, covariance, coordinate dependence, and gravitational redshift.