On the Motion of a Free Particle in the de Sitter Manifold

Abstract

Let $M=SO(1,4)/SO(1,3)\simeq S^3\times \mathbb{R}$ (a parallelizable manifold) be a submanifold in the structure $(\mathring{M}$ (hereafter called the bulk) where $\mathring{M}\simeq {\mathbb{R}}^5$ and $\mathring{\mathbf{g}}$ is a pseudo Euclidian metric of signature $(1,4)$. Let $\mathbf{i}:M\to {\mathbb{R}}^5$ be the inclusion map and let \ $\mathbf{g}={\mathbf{i}}^{\ast }\mathring{\mathbf{g}}$ be the pullback metric on $M$. It has signature $(1,3)$ Let $\mathbf{D}$ be the Levi-Civita connection of $\mathbf{g}$. We call the structure $(M,\mathbf{g})$ a de Sitter manifold and $M^{dSL}=(M=\mathbb{R}\times S^3,\mathbf{g},\mathbf{D},{\tau }_{\mathbf{g}},\uparrow )$ a de Sitter spacetime structure, which is \ of course orientable by ${\tau }_{\mathbf{g}}\in \sec$ and time orientable (by $\uparrow$).\ Under these conditions we prove that if the motion of a free particle moving on $M$ happens with constant \emph{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.

References in corpus (1)

• A Clifford Bundle Approach to the Differential Geometry of Branes