Normal frames and the validity of the equivalence principle. III. The case along smooth maps with separable points of self-intersection

TL;DR

This paper rigorously examines the equivalence principle on differentiable manifolds, establishing conditions for normal frames and demonstrating its universal validity at points and along paths in gravitational theories based on linear connections.

Contribution

It provides a mathematically rigorous analysis of the equivalence principle, identifying conditions for the existence of normal frames along smooth maps with self-intersections.

Findings

The equivalence principle holds at points and along paths in all linear connection-based gravitational theories.

Normal frames exist under certain conditions on subsets of the manifold.

Higher-dimensional submanifolds may only satisfy the principle in exceptional cases.

Abstract

The equivalence principle is treated on a mathematically rigorous base on sufficiently general subsets of a differentiable manifold. This is carried out using the basis of derivations of the tensor algebra over that manifold. Necessary and/or sufficient conditions of existence, uniqueness, and holonomicity of these bases in which the components of the derivations of the tensor algebra over it vanish on these subsets, are studied. The linear connections are considered in this context. It is shown that the equivalence principle is identically valid at any point, and along any path, in every gravitational theory based on linear connections. On higher dimensional submanifolds it may be valid only in certain exceptional cases.