关于de Sitter流形中自由质点的运动
On the Motion of a Free Particle in the de Sitter Manifold
摘要：
令$M=SO(1,4)/SO(1,3)\simeq S^{3}\times\mathbb{R}$（可并行流形）是结构$(\mathring{M}% ,\boldsymbol{\mathring{g}})$（以下称为体）中的子流形，其中$\mathring {M}\simeq\mathbb{R}^{5}$和$\boldsymbol{\mathring{g}}$是签名$(1,4)$的伪欧几里德度量。令$\boldsymbol{i}:M\rightarrow\mathbb{R}^{5}$为包含图，并令\ $\boldsymbol{g}=\boldsymbol{i}^{\ast }\boldsymbol{\mathring{g}}$为$M$上的回调度量。它具有签名$(1,3)$令$\boldsymbol{D}$是$\boldsymbol{g}% $的Levi-Civita连接。我们称结构$(M,\boldsymbol{g})$为德西特流形，$M^{dSL}=(M=\mathbb{R\times}S^{3},\boldsymbol{g},\boldsymbol{D},τ_{\boldsymbol{g}},\uparrow)$为德西特时空结构，它当然可以由$τ_{\boldsymbol{g}}\in\sec% %TCIMACRO{\tbigwedge \nolimits^{4}}% %BeginExpansion {\textstyle\bigwedge\nolimits^{4}} %EndExpansion T^{\ast}M$定向，也可以由$\uparrow$定向。在这些条件下，我们证明了，如果运动的自由粒子的$M$发生与恒定的\emph{bulk}角动量，那么它的运动在结构$M^{dSL}$是类时测地线。此外，结构$M^{dSL}$中的任何测地线运动都意味着粒子在体积中具有恒定的角动量。
英文摘要：
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 $M^{dSL}=(M=\mathbb{R\times}S^{3},\boldsymbol{g},\boldsymbol{D},τ_{\boldsymbol{g}},\uparrow)$ a de Sitter spacetime structure, which is \ of course orientable by $τ_{\boldsymbol{g}}\in\sec% %TCIMACRO{\tbigwedge \nolimits^{4}}% %BeginExpansion {\textstyle\bigwedge\nolimits^{4}} %EndExpansion T^{\ast}M$ 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.
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