An alternative derivation of the gravitomagnetic clock effect

TL;DR

This paper presents a new, more intuitive derivation of the gravitomagnetic clock effect, relating it to electromagnetic analogies and providing a general formula for spherical orbits with arbitrary inclination.

Contribution

It introduces an alternative derivation of the gravitomagnetic clock effect using electromagnetic analogies and extends the formula to arbitrary inclined spherical orbits.

Findings

Derived a classical analogy for the gravitomagnetic clock effect.

Provided a general expression for spherical orbits with arbitrary inclination.

Identified differences from relativistic calculations at order c^{-4}.

Abstract

The possibility of detecting the gravitomagnetic clock effect using artificial Earth satellites provides the incentive to develop a more intuitive approach to its derivation. We first consider two test electric charges moving on the same circular orbit but in opposite directions in orthogonal electric and magnetic fields and show that the particles take different times in describing a full orbit. The expression for the time difference is completely analogous to that of the general relativistic gravitomagnetic clock effect in the weak-field and slow-motion approximation. The latter is obtained by considering the gravitomagnetic force as a small classical non-central perturbation of the main central Newtonian monopole force. A general expression for the clock effect is given for a spherical orbit with an arbitrary inclination angle. This formula differs from the result of the general relativistic calculations by terms of order c^{-4}.