Geodesics and Constant Angular Momentum in the de Sitter Manifold.

Abstract

Let M = SO(1, 4)/SO(1, 3) ' S 3 × R (a parallelizable manifold) be a submanifold in the structure (M̊ ,g̊) (hereafter called the bulk) where M̊ ' R5 and g̊ is a pseudo Euclidian metric of signature (1, 4). Let i : M → R5 be the inclusion map and let g = i∗ g̊ be the pullback metric on M . It has signature (1, 3). Let D be the Levi-Civita connection of g. We call the structure (M, g) a de Sitter manifold and M dSL = (M = R×S 3 , g, D, τg , ↑) V4 a de Sitter spacetime structure, which is of course orientable by τg ∈ sec T ∗ M and time orientable (by ↑). Under these conditions, here we want to present the results that appears in particular that if the motion of a free particle moving on M happens with constant bulk angular momentum then its motion in the structure M dSL is a timelike geodesic. Also any geodesic motion in the structure M dSL implies that the particle has constant angular momentum in the bulk. So using the Clifford and spin-Clifford formalisms and the natural hypothesis that a particle moving freely in (M, g) has constant bulk angular momentum leads naturally to the Dirac equation as found in the de Sitter structure (M, g).