Revisiting the gravitomagnetic clock effect

TL;DR

This paper revisits the gravitomagnetic clock effect, demonstrating through numerical integration that the actual time difference due to gravitomagnetism is four times larger than previously calculated, confirming recent analytical results.

Contribution

The study provides a numerical validation that the gravitomagnetic corrections to orbital periods are four times larger than earlier analytical estimates, refining the understanding of the effect.

Findings

Numerical integration shows larger gravitomagnetic corrections by a factor of 4.

The proportionality coefficient of the time difference is confirmed as 16π.

Results align with recent analytical calculations by the author.

Abstract

To the first post-Newtonian order, if two test particles revolve in opposite directions about a massive, spinning body along two circular and equatorial orbits with the same radius, they take different times to return to the reference direction relative to which their motion is measured: it is the so-called gravitomagnetic clock effect. The satellite moving in the same sense of the rotation of the primary is slower, and experiences a retardation with respect to the case when the latter does not spin, while the one circling in the opposite sense of the rotation of the source is faster, and its orbital period is shorter than it would be in the static case. The resulting time difference due to the stationary gravitomagnetic field of the central spinning body is proportional to the angular momentum per unit mass of the latter through a numerical factor which so far has been found to be $4\pi$. A numerical integration of the equations of motion of a fictitious test particle moving along a circular path lying in the equatorial plane of a hypothetical rotating object by including the gravitomagnetic acceleration to the first post-Newtonian order shows that, actually, the gravitomagnetic corrections to the orbital periods are larger by a factor of $4$ in both the prograde and retrograde cases. Such an outcome, which makes the proportionality coefficient of the gravitomagnetic difference in the orbital periods of the two counter-revolving orbiters equal to $16\pi$, confirms an analytical calculation recently published in the literature by the present author.