The relativistic geoid

Based on the formalism of General Relativity, we analyze generalizations of concepts used in conventional geodesy. One such concept is the Earth's geoid. We present our definition of the relativistic geoid in terms of the level sets of a time-independent redshift potential. Such a potential exists for any congruence of Killing observers, i.e. for any rigidly moving object associated with a stationary spacetime in the outer region. The level surfaces of the redshift potential foliate the three dimensional space into isochronometric surfaces, which can be determined with the help of standard clocks. Two such clocks on the same surface will show zero redshift when their frequencies are compared. One of these level surfaces, singled out by a suitable convention, defines the relativistic geoid in our framework. At the same time, the redshift potential is also an acceleration potential for the congruence of observers. Hence, the isochrono-metric surfaces are orthogonal to the acceleration of freely falling objects, i.e. they are orthogonal to the local plumb line. Therefore, two independent kinds of measurements can contribute to the determination of the relativistic geoid. It can be shown that clocks, which are connected by optical fiber links, can be used to determine the redshift potential; this gives the operational foundation of our framework. Moreover, we show that our definition reduces to the well-known Newtonian and post-Newtonian notions in the respective limits. To illustrate our framework, we consider analytic examples of spacetimes, for which we calculate the level surfaces of the redshift potential and illustrate their intrinsic geometry by an embedding into flat Euclidean space. We emphasize that our definition of the geoid in terms of relativistic concepts is valid for arbitrarily strong gravitational fields. We do not use any approximation in the sense of weak fields or post-Newtonian expansion schemes. Hence, the definition can also be applied to very compact objects such as neutron stars.