Normal frames and the validity of the equivalence principle. I. Cases in a neighborhood and at a point

TL;DR

This paper investigates the conditions under which normal frames exist in a neighborhood or at a point on a manifold, providing explicit descriptions and analyzing their properties in relation to the equivalence principle.

Contribution

It establishes necessary and sufficient conditions for the existence of normal frames in the context of tensor algebra on manifolds, including symmetric and nonsymmetric connections.

Findings

Normal frames exist under specific conditions.

Explicit descriptions of normal frames are provided.

Results apply to both symmetric and nonsymmetric connections.

Abstract

A treatment in a neighborhood and at a point of the equivalence principle on the basis of derivations of the tensor algebra over a manifold is given. Necessary and sufficient conditions are given for the existence of local bases, called normal frames, in which the components of derivations vanish in a neighborhood or at a point. These frames (bases), if any, are explicitly described and the problem of their holonomicity is considered. In particular, the obtained results concern symmetric as well as nonsymmetric linear connections.