Fermi coordinates in Schwarzschild spacetime: closed form expressions

TL;DR

This paper derives explicit Fermi coordinate expressions in Schwarzschild spacetime, highlighting the complexity of obtaining exact local inertial frames even in symmetric gravitational fields.

Contribution

It provides closed-form expressions for Fermi coordinates around a static observer in Schwarzschild spacetime, demonstrating the challenges of explicit construction.

Findings

Exact Fermi coordinates are expressed as functions of Schwarzschild coordinates.

Construction of explicit Fermi coordinates is complex even in symmetric spacetimes.

The impact parameter is determined implicitly in the coordinate expressions.

Abstract

Fermi coordinates are constructed as exact functions of the Schwar\-zschild coordinates around the world line of a static observer in the equatorial plane of the Schwarzschild spacetime modulo a single impact parameter determined implicitly as a function of the latter coordinates. This illustrates the difficulty of constructing explicit exact Fermi coordinates even along simple world lines in highly symmetric spacetimes.