Frames of reference in spaces with affine connections and metrics

TL;DR

This paper proposes a generalized geometric framework for defining frames of reference in spaces with affine connections and metrics, applying it to models of space-time and Einstein's gravitation.

Contribution

It introduces a new definition of frames of reference based on differential-geometric objects and explores their applications in space-time models and gravity theories.

Findings

Defines frames of reference using vector fields, subspaces, and affine connections.

Derives auto-parallel equations as Euler-Lagrange equations.

Shows Einstein's gravitation as a special frame of reference determined by metrics and matter distribution.

Abstract

A generalized definition of a frame of reference in spaces with affine connections and metrics is proposed based on the set of the following differential-geometric objects: (a) a non-null (non-isotropic) vector field, (b) the orthogonal to the vector field sub space, (c) an affine connection and the related to it covariant differential operator determining a transport along the given non-null vector filed. On the grounds of this definition other definitions related to the notions of accelerated, inertial, proper accelerated and proper inertial frames of reference are introduced and applied to some mathematical models for the space-time. The auto-parallel equation is obtained as an Euler-Lagrange's equation. Einstein's theory of gravitation appears as a theory for determination of a special frame of reference (with the gravitational force as inertial force) by means of the metrics and the characteristics of a material distribution. PACS numbers: 0490, 0450, 1210G, 0240V