In general relativity, the generators of symmetries are called Killing vector elds. The spacetime geometry of a system of two point masses moving on a circular orbit has a helical Killing vector (HKV). We show how Kepler’s third law for circular orbits, and its generalization in post-Newtonian theory, can be recovered from a simple, covariant condition on the norm of that HKV. This unusual derivation can be used to illustrate some concepts of prime importance in a general relativity course, including those of Killing vector, covariance, coordinate dependence, and gravitational redshift.

Kepler's third law for circular orbits can be recovered from a simple, covariant condition on the norm of a helical Killing vector in general relativity.

Paper

Spacetime Symmetries and Kepler's Third Law

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