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.