The Fermi coordinate system in the post-Newtonian framework

The Fermi coordinate system (or the non-rotating proper reference frame) is studied within the framework of the post-Newtonian approximation of general relativity. First its general functional form is found to consist of three parts; (1) the motion of the space origin of the Fermi coordinate system relative to the background one, (2) the Fermi-Walker tetrad transported along the world line of the space origin, and (3) the spacelike geodesic starting from the space origin. Next, the post-Newtonian expressions of the latter two are obtained under the condition that the first is given. Then the full coordinate transformation formula connecting the Fermi coordinate system to the background one is derived explicitly. The effectiveness of the Fermi coordinate system is discussed and the effective region is found to be a cylinder with the radius of about 0.5 kpc for the Fermi coordinate system comoving with the Earth. The mathematical way to derive the generalized Fermi coordinate system which Ashby and Bertotti defined is also shown.