  Clock synchronization is the backbone of applications such as high-accuracy
satellite navigation, geolocation, space-based interferometry, and
cryptographic communication systems. The high accuracy of synchronization
needed over satellite-to-ground and satellite-to-satellite distances requires
the use of general relativistic concepts. The role of geometrical optics and
antenna phase center approximations are discussed in high accuracy work. The
clock synchronization problem is explored from a general relativistic point of
view, with emphasis on the local measurement process and the use of the tetrad
formalism as the correct model of relativistic measurements. The treatment
makes use of J. L. Synge's world function of space-time as a basic coordinate
independent geometric concept. A metric is used for space-time in the vicinity
of the Earth, where coordinate time is proper time on the geoid. The problem of
satellite clock syntonization is analyzed by numerically integrating the
geodesic equations of motion for low-Earth orbit (LEO), geosynchronous orbit
(GEO), and highly elliptical orbit (HEO) satellites. Proper time minus
coordinate time is computed for satellites in these orbital regimes. The
frequency shift as a function of time is computed for a signal observed on the
Earth's geoid from a LEO, GEO, and HEO satellite. Finally, the problem of
geolocation in curved space-time is briefly explored using the world function
formalism.
