Foundation Of The Mechanics Of Oriented Point

TL;DR

This paper develops a mechanics framework for oriented points with spin using 3D and 4D Frenet equations, linking particle trajectories with internal rotations and extending Einstein's equations through absolute parallelism geometry.

Contribution

It introduces a novel mechanics of oriented points incorporating rotational relativity and connects it with absolute parallelism geometry, generalizing Einstein's equations with torsion as a source.

Findings

Establishes equivalence between 4D oriented point equations and geodesic equations in absolute parallelism geometry.

Describes space of events with 10 degrees of freedom using Cartan structure equations.

Generalizes vacuum Einstein's equations to include geometrical torsion as a source.

Abstract

The mechanics of an oriented point (point with "spin") based on 3D and 4D Frenet equations is considered. In such mechanics there is an opportunity to describe formally any physical trajectory of a particle with own rotation. We use anholonomic rotational coordinates (Euler angles) as elements of internal space of the mechanics which generate a rotational relativity. The groups of transformations of the mechanics of an oriented point form Poincare's group with semidirect product of translations and rotations, so translational and rotational momentums appear dependent from each other. Connection of the curve torsion with Ricci rotational coefficients is shown and rotational metric is entered. Equivalence between equations of motion 4D oriented point and geodesic equations of absolute parallelism geometry is established. The space of events an arbitrary accelerated 4D frame of reference, which has 10 degrees of freedom, is described by Cartan structure equations of absolute parallelism geometry. It represent 10D coordinate space in which 4 translational coordinates describe motion of the origin $O$ 4D oriented point and 6 angular coordinates describe change of its orientation. The structural equations of absolute parallelism geometry, represent an extended set of Einstein-Yang-Mills equations with the gauge translations group defined on the base $x^i$ and with the gauge rotational group, defined in the fibre. The sources in these equations are defined through the torsion (torsion field) of geometry. The received system of the equations represents generalization vacuum Einstein's equations on a case when sources have geometrical nature.