Spinning Equations for Objects of Some Classes in Finslerian Geometry
- PDF / 530,756 Bytes
- 8 Pages / 612 x 792 pts (letter) Page_size
- 101 Downloads / 174 Views
Spinning Equations for Objects of Some Classes in Finslerian Geometry Magd E. Kahil1, 2* 1
Faculty of Engineering, Modern Sciences and Arts University, Giza, Egypt 2 Egyptian Relativity Group, Cairo, Egypt Received January 10, 2020; revised May 23, 2020; accepted May 27, 2020
Abstract—Spinning equations of Finslerian gravity, the counterpart of the Mathisson-Papapetrou spinning equations of motion are obtained. Two approaches of Finslerian geometries are formulated and discussed, the Cartan-Rund and Finsler-Cartan ones, as well as their corresponding spinning equations. The significance of the nonlinear connection and its relevance on spinning equations is noticed, and their deviations are examined. DOI: 10.1134/S0202289320030093
1. INTRODUCTION The problem of motion of a test particle has been discussed by many authors in Finslerian geometry as an introductory point to express the behavior of particles defined within the framework of this geometry [1]. This type of achievement is always compared with its counterpart in Riemannian geometry. It is worth mentioning that the most important significance of Finslerian geometry is a description, at each point in its manifold, by coordinates and a direction, it leads to replacing manifolds with tangent bundles [2]. Accordingly, a new appearing quantities may be expected, able to interpret it from a physical point of view. This type of technicality is due to implementing the concept of geometrization of physics. In our study, we are going to obtain the spinning equations for the Cartan-Rund approach for the orthodox Finsler space [3] and the Finsler-Cartan approach which is based on introducing a nonlinear connection [4], which plays a vital role in revisiting the falsifiability of geometrizing fundamental theories in physics and cosmology such as general relativity, brane, stings and gauge theories (see [5] and references therein). The aim of this work is searching for a suitable Lagrangian function, inspired from the famous Bazanski Lagrangian [6] as expressed in the previous works [7–9]. Thus it is vital to derive their corresponding geodesic and geodesic deviation equations to obtain the version of the spinning object for short, using a specific type of transformation explained in our present work [9]. Consequently, this step will enable us to make sure the reliability of the obtained equation. This will *
guide us to determine its corresponding Lagrangian function, a step toward generalization, to describe the spinning equation of different objects . Also, we present a technique of commutation to obtain their deviation equation rather than the traditional method of Bazanski [6], which is based on variation with respect to the four-velocity, as well as using some identities to preserve the appearance of covariance in the system of equations [7–9] . The advantage of this method may give rise to examining the case of rate of change of the spinning object and its associated deviation equation. 2. FINSLERIAN GEOMETRY: A BRIEF INTRODUCTION Finslerian spaces are n-dimensional ma
Data Loading...
