Post-Newtonian reference-ellipsoid for relativistic geodesy

We apply general relativity to construct the post-Newtonian background manifold that serves as a reference level surface in relativistic geodesy for conducting calculation of geoid's undulation. We chose the perfect homogeneous fluid uniformly rotating around a fixed axis as a source of the background manifold. We, then, reformulate and extend rotating-fluid calculations done by a number of previous researchers for astrophysical applications to the realm of relativistic geodesy to find out the algebraic equation of the post-Newtonian reference-ellipsoid. We explicitly perform all integrals characterizing gravitational potentials inside the fluid body and represent them in terms of elementary functions depending on the body's eccentricity. We fully explore the coordinate freedom of the equations describing the post-Newtonian ellipsoid and demonstrate that the fractional deviation of the post-Newtonian level surface from the Maclaurin ellipsoid can be made much smaller than the previously anticipated estimate advocated by Bardeen and Chandrasekhar. We also derive the relations of the post-Newtonian mass and angular velocity of the rotating fluid to the parameters of the post-Newtonian ellipsoid. We find out the gauge-invariant expressions for the post-Newtonian mass and angular momentum of the rotating fluid body as well as the gauge-invariant post-Newtonian formulations of the canonical Pizzetti and Clairaut theorems that are used in geodesy to connect the parameters of the reference ellipsoid to the force of gravity at its pole and on equator. Finally, we expand the post-Newtonian geodetic equations characterizing the post-Newtonian ellipsoid into the Taylor series with respect to the eccentricity of the ellipsoid and discuss them with regard to future practical applications by the International Earth Rotation Service (IERS) and the International Union of Geodesy and Geophysics (IUGG).