Riemann normal coordinates, Fermi reference system and the geodesic deviation equation

TL;DR

This paper develops integral formulas for calculating tetrads and metrics in Riemann normal and Fermi coordinates, extending their validity, and applies these to analyze geodesic deviation and plane-wave metrics in gravitational fields.

Contribution

It introduces new integral formulas for Riemann and Fermi coordinates that broaden their applicability and demonstrates their use in gravitational wave analysis.

Findings

Extended the validity range of Riemann and Fermi coordinates.

Derived integral formulas for metric and tetrad computations.

Applied formulas to gravitational wave and plane-wave metrics.

Abstract

We obtain the integral formulae for computing the tetrads and metric components in Riemann normal coordinates and Fermi coordinate system of an observer in arbitrary motion. Our approach admits essential enlarging the range of validity of these coordinates. The results obtained are applied to the geodesic deviation in the field of a weak plane gravitational wave and the computation of plane-wave metric in Fermi normal coordinates.