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 $(\mathring{M}% ,\boldsymbol{\mathring{g}})$ (hereafter called the bulk) where $\mathring {M}\simeq\mathbb{R}^{5}$ and $\boldsymbol{\mathring{g}}$ is a pseudo Euclidian metric of signature $(1,4)$. Let $\boldsymbol{i}:M\rightarrow\mathbb{R}^{5}$ be the inclusion map and let \ $\boldsymbol{g}=\boldsymbol{i}^{\ast }\boldsymbol{\mathring{g}}$ be the pullback metric on $M$. It has signature $(1,3)$ Let $\boldsymbol{D}$ be the Levi-Civita connection of $\boldsymbol{g}% $. We call the structure $(M,\boldsymbol{g})$ a de Sitter manifold and $M^{dSL}=(M=\mathbb{R\times}S^{3},\boldsymbol{g},\boldsymbol{D},\tau _{\boldsymbol{g}},\uparrow)$ a de Sitter spacetime structure, which is \ of course orientable by $\tau_{\boldsymbol{g}}\in\sec% %TCIMACRO{\tbigwedge \nolimits^{4}}% %BeginExpansion {\textstyle\bigwedge\nolimits^{4}} %EndExpansion T^{\ast}M$ 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.