Relative velocity and relative acceleration induced by the torsion in (pseudo) Riemannian spaces with torsion and in spaces with an affine connection and metrics

Abstract

The influence of the torsion on the relative velocity and on the relative acceleration between particles (points) in spaces with an affine connection and a metric [$(L_n,g)$-spaces] and in (pseudo) Riemannian spaces with torsion ($U_n$-spaces) is considered. Necessary and sufficient conditions as well as only necessary and only sufficient conditions for vanishing deformation, shear, rotation and expansion are found. The notion of relative acceleration and the related to it notions of shear, rotation and expansion accelerations induced by the torsion are determined. It is shown that the kinematic characteristics induced by the torsion (shear acceleration, rotation acceleration and expansion acceleration) could play the same role as the kinematic characteristics induced by the curvature and can (under given conditions) compensate their action as well as the action of external forces. The change of the rate of change of the length of a deviation vector field is given in explicit form for $(L_n,g)$- and $U_n$-spaces. PACS numbers: 04.90+e, 04.50+h, 12.10.Gq, 03.40.-t