Expansions of the affinity, metric and geodesic equations in Fermi normal coordinates about a geodesic

Abstract

A coordinatefree derivation of a generalized geodesic deviation equationFermi normal coordinates about a geodesic form a natural coordinate system for the nonrotating geodesic (freely falling) observer. Expansions of the affinity, metric, and geodesic equations in these coordinates in powers of proper distance normal to the geodesic are calculated here to third order, fourth order, and third order, respectively. An iteration scheme for calculation to higher orders is also given. For generality, we compute the affinity and the geodesic equations in an arbitrary affine manifold, and compute the metric in a Riemannian manifold with arbitrary signature.

Physics in precision-dependent normal neighborhoods

“…Following this, we show sample calculations for the patch size in Sec. IV and also establish the connection between our result and the familiar patch size estimate presented, for example, in [5] or [6]. In Sec.…”

“…The size of normal neighborhoods has until now only been estimated based on curvature arguments, see, for example, [4][5][6][7]. Such an estimate is sufficient for calculations aiming for a proof of concept, i.e.…”

General transformation formulas for Fermi–Walker coordinates

“…Some general results are known for expansions of the metric in these coordinates. In [13] Li and Ni derived the third order expansion of the metric and second order expansion of the equations of motion in Fermi-Walker coordinates for general spacetimes, and in [14] they found expansions of the connection coefficients, metric, and geodesic equations in Fermi coordinates to third order, fourth order, and third order respectively, and gave an iteration scheme for calculation to higher order. Marzlin investigated weak gravitational fields in [7] and found the expansion of the Minkowski metric with small perturbations to infinite order in Fermi-Walker coordinates.…”

Measuring the Gravitational Field in General Relativity: From Deviation Equations and the Gravitational Compass to Relativistic Clock Gradiometry

“…From an experimentalists perspective so-called (generalized) Fermi coordinates appear to be realizable operationally. There have been several suggestions for such coordinates in the literature in different contexts [21,22,23,24,25,26,2,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]. In the following we are going to derive the line element in the vicinity of a world line, representing an observer in an arbitrary state of motion, in generalized Fermi coordinates.…”