Flows and particles with shear-free and expansion-free velocities in (bar-Lng)- and Weyl spaces

Abstract

Conditions for the existence of flows with non-null shear-free and expansionfree velocities in spaces with affine connections and metrics are found. On their basis, generalized Weyl's spaces with shear-free and expansion-free conformal Killing vectors as velocities vectors of spinless test particles moving in a Weyl's space are considered. The necessary and sufficient conditions are found under which a free spinless test particle could move in spaces with affine connections and metrics on a curve described by means of an auto-parallel equation. In Weyl's spaces with Weyl's covector, constructed by the use of a dilaton field, the dilaton field appears as a scaling factor for the rest mass density of the test particle.

Elliptic Curves and Algebraic Geometry Approach in Gravity Theory I.The General Approach

“…And this is indeed the case, because the relation between the two connections Γ k ij and Γ k ij is given by formulae (2.4) Γ s kl := g is g im Γ m kl . In other words, since these two connections are not considered to be "separately introduced" and so they do not depend on one another by means of the equality (2.13), this particular investigated case does not fall within the classification of spaces with covariant and contravariant metrics and connections (Table I in a previous paper [47]). This is an important "terminological" clarification, since it turns out that it is possible to have a theory with (separate) covariant and contravariant metrics, but not (with separate) connections as well.…”

“…Evidently, with respect to the metric g ij (and its inverse contravariant one g jk ),we have the usual gravitational theory with the contravariant Γ so that with respect to these connections the theory can be considered a GTCCMC. This also means that Table I in [47] correctly does not account for theories with different covariant and contravariant metrics only, because the different GTCCMC are in principle with different covariant and contravariant metrics and with different connections.…”