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Generalized gravitomagnetic clock effect

In General Relativity, the rotation of a gravitating body like the Earth influences the motion of orbiting test particles or satellites in a non-Newtonian way. This causes, e.g., a precession of the orbital plane known as the Lense-Thirring effect and a precession of the spin of a gyroscope known as the Schiff effect. Here, we discuss a third effect first introduced by Cohen and Mashhoon called the gravitomagnetic clock effect. It describes the difference in proper time of counterrevolving clocks after a revolution of $2π$. For two clocks on counterrotating equatorial circular orbits around the Earth, the effect is about $10^{-7}$ seconds per revolution, which is quite large. We introduce a general relativistic definition of the gravitomagnetic clock effect which is valid for arbitrary pairs of orbits. This includes rotations in the same direction and different initial conditions, which are crucial if the effect can be detected with existing satellites or with payloads on nondedicated missions. We also derive the post-Newtonian expansion of the general relativistic expression and calculate the effect for the example of a satellite of a Global Navigation Satellite System compared to a geostationary satellite.