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

Waldyr Rodrigues, IMECC-UNICAMP, Brazil

In this paper we discuss the physics of interacting tensor (and spinor fields) and particles living in a de Sitter manifold $M = SO(1,4)/SO(1,3) \backsimeq \mathbb R \times S^3$ interpreted as a submanifold of $\mathring M = (\mathbb R^5,\mathring g)$, with $\mathring g$ a metric of signature $(1,4)$. The pair $(M,g)$ where $g$ is the pullback metric of $\mathring g$ ($g=i^{*}\mathring g$) is Lorentzian manifold that is oriented by $\tau_{g}$ and time oriented by $\uparrow$. It is the structure $(M,g,\tau_{g},\uparrow)$ that is primely used to study the energy-momentum conservation law for a system of physical fields living in M and the true equation of motion for free particles that can be derived using Papapetrou's method. To achieve our objectives we construct two different de Sitter spacetime structures $M^{dSL}=(M,g,D,\tau_{g},\uparrow)$ and $M^{dSTP}=(M,g,\nabla,\tau_{g},\uparrow)$, where $D$ is the Levi-Civita connection of $g$ and $\nabla$ is a metric compatible parallel connection. Both connections are introduced in our study only as mathematical devices, no special physical meaning is attributed to these objects. In particular $M^{dSL}$ is not supposed to be the model of any gravitational field in the General Relativity Theory (GRT). Our approach permit to clarify some odd misconceptions appearing in the literature. Our paper also deals with the delicate question of defining a meaningful Lie derivative for spinor fields in $(M,g)$ and we claim to have a a geometrical based definition and which agrees with the postulated Kosmann's formula for the Lie derivative of a spinor field in the direction of arbitrary vector fields. The paper makes use of the Clifford and spin-Clifford bundles formalism recalled in one of the appendices, something needed among other things for a nice presentation of the concept of a Komar current $\mathcal J_A$ (in GRT) associated to any vector field ${\mathrm A}$ generating a one parameter group of diffeomorphisms. The explicit formula for $\mathcal J_A$ in terms of the energy-momentum tensor of the fields and its physical meaning is given. Besides that we show how $F=dA$ ($A=g(\mathrm A,\ )$) satisfy a Maxwell like equation $\partial F=\mathcal J_{A}$ which encodes the contents of Einstein equation.