Abstract
We discuss the physics of interacting fields and particles living in a de Sitter Lorentzian manifold (dSLM), a submanifold of a 5-dimensional pseudo-Euclidean (5dPE) equipped with a metric tensor inherited from the metric of the 5dPE space. The dSLM is naturally oriented and time oriented and is the arena used to study the energy-momentum conservation law and equations of motion for physical systems living there. Two distinct de Sitter space-time structuresMdSLandMdSTPare introduced given dSLM, the first equipped with the Levi-Civita connection of its metric field and the second with a metric compatible parallel connection. Both connections are used only as mathematical devices. Thus, for example,MdSLis not supposed to be the model of any gravitational field in the General Relativity Theory (GRT). Misconceptions appearing in the literature concerning the motion of free particles in dSLM are clarified. Komar currents are introduced within Clifford bundle formalism permitting the presentation of Einstein equation as a Maxwell like equation and proving that in GRT there are infinitely many conserved currents. We prove that in GRT even when the appropriate Killing vector fields exist it is not possible to define a conserved energy-momentum covector as in special relativistic theories.
Highlights
In this paper we study some aspects of Physics of fields living and interacting in a manifold M = SO(1, 4)/SO(1, 3) ≃R × S3
Our main objective is the following: taking (M, g) as the arena where physical fields live and interact, how do we formulate conservation laws of energy-momentum and angular momentum for the system of physical fields? In order to give a meaningful meaning to this question we recall the fact that in Lorentzian space-time structures that are models of gravitational fields in the General Relativity Theory (GRT) there are no genuine conservation laws of energymomentum for a closed system of fields and there are no genuine energymomentum and angular momentum conservation laws for the system consisting of nongravitational plus the gravitational field
We show that even if we can get four conserved currents given one timelike and three spacelike vector fields and get four scalar invariants, these objects cannot be associated with the components of a momentum covector4 for the system of fields producing the energymomentum tensor T
Summary
In GRT there are an infinite number of conserved currents In. Appendix C.1 we derive from the Lagrangian formalism conserved currents for fields living in a general Lorentzian spacetime structure and the corresponding generalized covariant energy-momentum “conservation” law. We prove that from the equation describing the motion of a single-pole the geodesic equation follows automatically
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