The Spinning Equations of Motion for Objects in AP-Geometry

Abstract

Equations of spinning objects are obtained in Absolute Parallelism Geometry [AP], a special class of non-Riemannian geometry admitting specific types having non-vanishing curvature and torsion simultaneously. This new set of equations is the counterpart of the Papapetrou equations in the Riemannian geometry. Applying, the concept of geometerization of physics, it may give rise to describe the spin tensor as parameterized commutation relation between path and path deviation equations in both Riemannian and non-Riemannian geometries.

Spinning and Spinning Deviation Equations for Special Types of Gauge Theories of Gravity

“…From the above equations, we can figure out that, the spinning and spinning deviation equations as similar as the famous spinning and spinning equations [21] of general relativity. Such a similarity is due to replacing the metric tensor g µν (x) by g µν (ψ).…”

“…where | is the associated covariant derivative flat space. Comparing (20) and (21) with and (25) and (26), we find out that the effect of different covariant derivatives appear effective, if the object is regarded its intrinsic properties on the spinning deviation equations. such a metric in the following way [28].…”

“…Equations of motion for a spinning objects may be derived in twofold , one of them from geodesic equations as being a deviated path from geodesic satisfying the following relation [21]:…”

Motion in Clifford Space

“…the geodesic equation. The spinning object has been studied by many authors long time ago, Mathisson [1] started the idea; Papapetrou amended its content [2] and then it was developed to include charged objects by Dixon [3], which led many of their followers to obtain the corresponding equations of motion of moving objects in different types of geometries [4][5][6][7][8][9]. Not only these path equations but also their deviation equations play a fundamental role in regulating the stability of objects [10].…”

“…Equations of geodesic and geodesic deviation equations in Riemannian geometry are required to examine many problems of motion for different test particles in gravitational fields. This led many authors to derive them by various methods, one of the most applicable ones is the Bazanski approach [27] in which from one single Lagrangian one can obtain simultaneously equation of geodesic and geodesic deviations which has been applied in different theories of gravity [4][5][6][7][8][9],and [28][29][30]. Thus, by analogy this technique in case of Poly-vectors to become [31], to obtain the geodesic deviation equation,…”