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

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.

A MAXWELL-LIKE FORMULATION OF GRAVITATIONAL THEORY IN MINKOWSKI SPACE–TIME

Using the Clifford bundle formalism, a Lagrangian theory of the Yang–Mills type (with a gauge fixing term and an auto interacting term) for the gravitational field in Minkowski space–time is presented. It is shown how two simple hypotheses permit the interpretation of the formalism in terms of effective Lorentzian or teleparallel geometries. In the case of a Lorentzian geometry interpretation of the theory, the field equations are shown to be equivalent to Einstein's equations.

THE DIRAC–HESTENES EQUATION FOR SPHERICAL SYMMETRIC POTENTIALS IN THE SPHERICAL AND CARTESIAN GAUGES

In this paper, using the apparatus of the Clifford bundle formalism, we show how straightforwardly solve in Minkowski space–time the Dirac–Hestenes equation — which is an appropriate representative in the Clifford bundle of differential forms of the usual Dirac equation — by separation of variables for the case of a potential having spherical symmetry in the Cartesian and spherical gauges. We show that, contrary to what is expected at a first sight, the solution of the Dirac–Hestenes equation in both gauges has exactly the same mathematical difficulty.

THE EINSTEIN–HILBERT LAGRANGIAN DENSITY IN A TWO-DIMENSIONAL SPACETIME IS AN EXACT DIFFERENTIAL

Recently Kiriushcheva and Kuzmin1 claimed to have shown that the Einstein–Hilbert Lagrangian density cannot be written in any coordinate gauge as an exact differential in a two-dimensional spacetime. Since this is contrary to other statements on the subject found in the literature, as e.g., by Deser,2 Deser and Jackiw,3 Jackiw4 and Grumiller, Kummer and Vassilevich5 it is necessary to decide who has reason. This is done in this paper in a very simply way using the Clifford bundle formalism.