Reference-Ellipsoid and Normal Gravity Field in Post-Newtonian Geodesy

Modern geodesy is undergoing a crucial transformation from the Newtonian paradigm to the Einstein theory of general relativity. This is motivated by advances in developing quantum geodetic sensors including gravimeters and gradientometers, atomic clocks and fiber optics for making ultra-precise measurements of geoid and multipolar structure of Earth’s gravitational field. At the same time, Very Long Baseline Interferometry, Satellite Laser Ranging and Global Navigation Satellite System have achieved an unprecedented level of accuracy in measuring spatial coordinates of reference points of the International Terrestrial Reference Frame and the world height system. The main geodetic reference standard to which gravimetric measurements of Earth’s gravitational field are referred, is called normal gravity field which is represented in the Newtonian gravity by the field of a uniformly rotating, homogeneous Maclaurin ellipsoid having mass and quadrupole momentum equal to the total mass and (tide-free) quadrupole moment of the gravitational field of Earth. The present chapter extends the concept of the normal gravity field from the Newtonian theory to the realm of general relativity. We focus on the calculation of the post-Newtonian approximation of the normal field that would be sufficiently precise for near-future practical applications. We show that in general relativity the level surface of homogeneous and uniformly rotating fluid is no longer described by the Maclaurin ellipsoid in the most general case but represents an axisymmetric spheroid of the fourth order (PN spheroid) with respect to the geodetic Cartesian coordinates. At the same time, admitting post-Newtonian inhomogeneity of mass density in the form of concentric elliptical shells allows us to preserve the level surface of the fluid as an exact ellipsoid of rotation. We parametrize the mass density distribution and the level equipotential surface with two parameters which are intrinsically connected to the existence of the residual gauge freedom, and derive the post-Newtonian normal gravity field of the rotating spheroid both inside and outside of the rotating fluid body. The normal gravity field is given, similarly to the Newtonian gravity, in a closed form by a finite number of the ellipsoidal harmonics. We employ transformation from the ellipsoidal to spherical coordinates to deduce a more conventional post-Newtonian multipolar expansion of scalar and vector gravitational potentials of the rotating spheroid. We compare these expansions with that of the normal gravity field generated by the Kerr metric and demonstrate that the Kerr metric has a fairly limited application in relativistic geodesy as it does not match the normal gravity field of the Maclaurin ellipsoid already in the Newtonian limit. We derive the post-Newtonian generalization of the Somigliana formula for the normal gravity field measured on the surface of the rotating PN spheroid and employed in practical work for measuring the Earth gravitational field anomalies. Finally, we discuss the possible choice of the gauge-dependent parameters of the normal gravity field model for practical applications and compare it with the existing EGM2008 model of gravitational field.