Chronometry: On the redshift and relativistic gravity potential determination in GR

We derive exact expressions for the relativistic redshift and timing between observers in various configurations in stationary spacetimes for the purpose of chronometry, i.e., relativistic gravimetry based on clocks. These observers are assumed to be equipped with standard clocks and move along arbitrary worldlines. It is shown how redshift observations can be used to infer the (mass) multipole moments of the underlying spacetime, i.e., a decomposition of the gravito-electric potential. In particular, an Earth-bound observer is considered that is meant to model a standard clock on the Earth's surface (or on the geoid).  Its clock is continuously compared with a clock on a satellite to determine from redshift measurements a relativistic gravity potential in the vicinity of the Earth. Results shown here are in agreement with the Newtonian potential determination from the so-called energy approach. The framework is intended for applications within relativistic geodesy and is exemplified in different exact vacuum spacetimes for illustration.