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

2016; Birkhäuser; Volume: 27; Issue: 2 Linguagem: Inglês

10.1007/s00006-016-0729-z

1661-4909

Waldyr A. Rodrigues, Samuel A. Wainer,

Advanced Differential Geometry Research

Let $${M = SO(1, 4)/SO(1,3) \simeq S^{3}\times \mathbb{R}}$$ (a parallelizable manifold) be a submanifold in the structure $${(\mathring{M}, {\mathring{g}})}$$ (hereafter called the bulk) where $${\mathring {M} \simeq \mathbb{R}^{5}}$$ and $${{\mathring{g}}}$$ is a pseudo Euclidian metric of signature (1,4). Let $${{i}:M\rightarrow\mathbb{R}^{5}}$$ be the inclusion map and let $${{g}={i}^{\ast}{\mathring{g}}}$$ be the pullback metric on M. It has signature (1,3) Let $${{D}}$$ be the Levi-Civita connection of $${{g} }$$ . We call the structure $${(M,{g})}$$ a de Sitter manifold and $${M^{dSL} = (M = {\mathbb{R}} \times S^{3},g,D,\tau_{g},\uparrow)}$$ a de Sitter spacetime structure, which is of course orientable by $${\tau_{g} \in {\rm sec} \bigwedge^{4} T^{\ast}M}$$ and time orientable (by $${\uparrow}$$ ). Under these conditions (and without using any General Relativity theory concept) we prove that if the motion of a particle restricted to move on M (without the action of any force as detected by observers living in M) happens with constant 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.