  In this work, we compute the metric corresponding to a static and spherically
symmetric mass distribution in the general relativistic weak field
approximation to quadratic order in Fermi-normal coordinates surrounding a
radial geodesic. To construct a geodesic and a convenient tetrad transported
along it, we first introduce a general metric, use the Cartan formalism of
differential forms, and then specialize the space-time by considering the
nearly Newtonian metric. This procedure simplifies the calculations
significantly, and the expression for the radial geodesic admits a simple form.
We conclude that in quadratic order, the effects of a Schwarzschild
gravitational field measured locally by a freely falling observer equals the
measured by an observer in similar conditions in the presence of a Newtonian
approximation of gravitation.
