  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
$2\pi$. 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.
