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arXiv · 2211.12780

Fermi-normal coordinates for the Newtonian approximation of gravity

Antonio C. Gutiérrez-Piñeres
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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.

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Antonio C. Gutiérrez-Piñeres. 2023-11-25. Fermi-normal coordinates for the Newtonian approximation of gravity. https://doi.org/10.1088/1402-4896%2Fad0fc7

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