Relativistic effects in local inertial frames

Abstract

The concept of a generalized Fermi frame is introduced with the aim of describing the relativistic effects due to a third, distant body (such as the Sun) upon the motion of an Earth satellite. This extends Fermi’s construction of a local inertial frame to the case in which there are local gravitating masses. This is done in the slow-motion, weak-field approximation by splitting the metric into an external part and a local part; Fermi’s construction of local inertial coordinates defined with respect to the external metric is then used to transform the complete metric. The results show that the main relativistic effects on an Earth satellite are due to the nonlinear correction in the Earth’s own Schwarzschild field. There are much smaller relativistic corrections in the tidal field of the Sun, and an Earth-Sun interaction term. The spatial axes of the local frame also undergo geodetic precession. Particular care must be taken with respect to the definition of the time coordinate in the generalized Fermi frame in order that the unit of time be consistent with readings of reasonable physical clocks on Earth’s surface. Also discussed more rigorously is the generalized Fermi frame for a system of two bodies revolving in circular orbits around a common barycenter.