On the gravitomagnetic clock effect

TL;DR

This paper derives a general formula for the gravitomagnetic clock effect on elliptical orbits around a slowly rotating mass, extending previous results limited to circular or spherical orbits.

Contribution

It introduces a new general formula for the gravitomagnetic clock effect applicable to arbitrary elliptical orbits in slow rotation regimes.

Findings

Derived a formula for elliptical orbits

Extended previous results to more general orbits

Discussed implications of the new formula

Abstract

General relativity predicts that two freely counter-revolving test particles in the exterior field of a central rotating mass take different periods of time to complete the same full orbit; this time difference leads to the gravitomagnetic clock effect. The effect has been derived for circular equatorial orbits; moreover, it has been extended via azimuthal closure to spherical orbits around a slowly rotating mass. In this work, a general formula is derived for the main gravitomagnetic clock effect in the case of slow motion along an arbitrary {\it elliptical} orbit in the exterior field of a slowly rotating mass. Some of the implications of this result are briefly discussed.