Fermi-normal coordinates for the Newtonian approximation of gravity

TL;DR

This paper derives the metric in Fermi-normal coordinates for a weak gravitational field around a spherical mass, showing local measurements match between general relativity and Newtonian approximation at quadratic order.

Contribution

It introduces a method to compute the metric in Fermi-normal coordinates for weak fields, simplifying calculations and comparing local measurements in GR and Newtonian gravity.

Findings

Local measurements in Schwarzschild field match Newtonian approximation at quadratic order.

Simplified expression for radial geodesic in Fermi-normal coordinates.

Method using Cartan formalism enhances calculation efficiency.

Abstract

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.