Theoretical Aspects of Relativistic Geodesy

Abstract

In this thesis, I show how fundamental geodetic notions can be defined within a general relativistic framework. Among the concepts that are analyzed there are the relativistic gravity potential, the geoid, the normal gravity field and its potential, as well as the genuinely relativistic definition of chronometric height. Moreover, a simple procedure for the operational preservation of a chosen level surface of the relativistic gravity potential is investigated. For all these concepts, the respective Newtonian notions are recovered in the weak-field limit. In the first-order (parametrized) post-Newtonian expansion the results previously published in the literature are obtained and it is shown how they are embedded into the present framework. Magnitudes of leading-order relativistic corrections to the geoid as well as redshift and acceleration measurements are calculated in a simple gravity field model. After the most important geodetic notions are introduced, the theory of General Relativity and the mathematical formalism are briefly discussed. Emphasis lies on some exact solutions to Einstein's vacuum field equation. These spacetimes are used in the following to either estimate relativistic effects or generalize geodetic concepts. Proceeding to a relativistic theory of gravity changes the underlying stage on which all physics takes place. The involved mathematical structure, related to the description of a curved spacetime, causes conventional geodetic notions to become ill-defined in the framework of General Relativity. Here, it is shown how relativistic generalizations of these notions can be constructed, working without any kind of weak-field approximation. The approach is mainly based on a so-called redshift potential of which the level sets foliate a stationary spacetime into isochronometric surfaces. It gives rise to the definition of a relativistic gravity potential which is used intensively. In particular, using a parametrized post-Newtonian spacetime for the Earth, the magnitude of relativistic corrections to the geoid is investigated in a simple Earth model. In the last part, the relation between proper time on the geoid and the defining constant L g in the IAU resolutions is discussed and a consistent relativistic definition for chronometric heights is proposed. Finally, relativistic orbital effects are compared to non-gravitational perturbations of satellite orbits and relativistic gravity gradiometry is investigated to link geodesic deviation to the curvature of spacetime, which is determinable by geodetic measurements.