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

TL;DR

This paper demonstrates that free particles with constant bulk angular momentum follow timelike geodesics in de Sitter spacetime, establishing a link between bulk angular momentum and particle trajectories.

Contribution

It proves the equivalence between constant bulk angular momentum and geodesic motion in de Sitter spacetime, providing new insights into particle dynamics in this manifold.

Findings

Constant bulk angular momentum implies timelike geodesic motion.

Geodesic motion corresponds to constant angular momentum in the bulk.

The results connect bulk angular momentum with particle trajectories in de Sitter space.

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.