Director of UVITOR Co., Ltd, 38/2 Ladprao, Soi 15, Bangkok 10900, Thailand*
Deparment of Physics, Faculty of Science Chulalongkorn University, Bangkok 10330, Thailand
The mechanics of an orientable 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 orientable 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 A4(6). It represent 10D coordinate space in which 4 translational coordinates x0 = ct, x1 = x, x2 = y, x3 = z describe motion of the origin O 4D orientable point and 6 angular coordinates φ1= φ, φ2 = ψ, φ3 = θ, φ4 = ϑx, φ5 = ϑy , φ6 = ϑz describe change of its orientation.
The structural equations of absolute parallelism geometry A4(6), represent an extended set of Einstein-Yang-Mills equations with the gauge translations group T4 deﬁned on the base xi and with the gauge rotational group O(1.3), deﬁned in the ﬁbre e ia. The sources in these equations are deﬁned through the torsion (torsion ﬁeld) of A4(6) geometry. The received system of the equations represents generalization vacuum Einstein’s equations on a case when sources have geometrical nature. On the basis of the Vaidya-like solution of the Einstein-Yang-Mills equations correspondence with the Einstein’s equations is established.