Normal frames and the validity of the equivalence principle

TL;DR

This paper explores the conditions under which the equivalence principle holds in gravitational theories, demonstrating that adherence along all paths implies the use of linear connections, and introduces normal bases where derivation components vanish.

Contribution

It proves the existence of normal bases in tensor algebra where derivation components vanish along paths and links the validity of the equivalence principle to the use of linear connections.

Findings

Normal bases exist where derivation components vanish along paths

The equivalence principle along all paths implies the use of linear connections

Normal bases can be explicitly constructed and analyzed

Abstract

We investigate the validity of the equivalence principle along paths in gravitational theories based on derivations of the tensor algebra over a differentiable manifold. We prove the existence of local bases, called normal, in which the components of the derivations vanish along arbitrary paths. All such bases are explicitly described. The holonomicity of the normal bases is considered. The results obtained are applied to the important case of linear connections and their relationship with the equivalence principle is described. In particular, any gravitational theory based on tensor derivations which obeys the equivalence principle along all paths, must be based on a linear connection.