Notes on Conservation Laws, Equations of Motion of Matter and Particle Fields in Lorentzian and Teleparallel de Sitter Spacetime Structures

Abstract: We discuss the physics of interacting tensor fields and particles living in $M=\mathrm{S0}(1,4)/\mathrm{S0} (1,3)\simeq\mathbb{R}\times S^{3}$ a submanifold of $\mathring{M}=(\mathbb{R}^{5},\boldsymbol{\mathring{g}})$, where $\boldsymbol{\mathring {g}}$ has signature $(1,4)$. Structure $(M,\boldsymbol{g})$ where $(\boldsymbol{g=i}^{\ast}\boldsymbol{\mathring{g}})$ is a Lorentzian manifold. Structure $(M,\boldsymbol{g,}\tau_{\boldsymbol{g}},\uparrow)$ is primely used to study the energy-momentum conservation law (for a system of physical fields (and particles) living in $M$ and to get the respective equations of motion. We construct two different de Sitter spacetime structures $M^{dSL}=(M,\boldsymbol{g,D},\tau_{\boldsymbol{g}},\uparrow)$ and $M^{dSTP}=(M,\boldsymbol{g,\nabla},\tau_{\boldsymbol{g} },\uparrow)$. Both (metrical compatible) connections are used only as mathematical devices. In particular $M^{dSL}$ is not supposed to be the model of any gravitational field in the(\textbf{GRT}). We clarify some misconceptions appearing in the literature. We use the Clifford and spin-Clifford bundles formalism and gives a thoughtful presentation of the concept of a Komar current $\mathcal{J}_{A}$ (in GRT}) associated to any vector field $\mathbf{A}$. A formula for the Komar current and its physical meaning are given. We show also how $F=dA$ satisfy in the Clifford bundle a Maxwell like equation encoding the contents of Einstein equation. We show that in GRT there are infinitely many conserved currents,independently of the fact that the Lorentzian spacetime possess or not Killing vector fields and that even when the appropriate Killing vector fields exist there does not exist a conserved energy-momentum covector (not a covector field) as in SRT.