CONFORMAL DERIVATIVE AND CONFORMAL TRANSPORTS OVER $\bm{({\bar L}_n,g)}$-SPACES

Abstract

Transports preserving the angle between two contravariant vector fields but changing their lengths proportional to their own lengths are introduced as "conformal" transports and investigated over (Ln, g)-spaces. They are more general than the Fermi-Walker transports. In an analogous way as in the case of Fermi-Walker transports a conformal covariant differential operator and its conformal derivative are defined and considered over (Ln, g)-spaces. Different special types of conformal transports are determined inducing also Fermi-Walker transports for orthogonal vector fields as special cases. Conditions under which the length of a non-null contravariant vector field could swing as a homogeneous harmonic oscillator are established. The results obtained regardless of any concrete field (gravitational) theory could have direct applications in such types of theories.

Deviation operator and deviation equations over spaces with affine connections and metrics

“…• Fermi-Walker transports and conformal transports exist in (L n , g)-spaces as generalizations of these types of transports in V n -spaces [14], [15].…”

On the Existence of a Gyroscope in Spaces with Affine Connections and Metrics

“…under which conditions the lengths of the vector fields ξ ⊥ and η ⊥ , as well as the angle between them, do not change under a transport along the vector field u. Transports which preserve lengths and angles between vector fields are called Fermi-Walker transports [14], [17]. We can now apply the method, developed for finding out Fermi-Walker transports in spaces with affine connections and metrics with given metrics g and g, to the same type of spaces with determined projective metrics h u and h u .…”

“…In spaces with affine connections and metrics special types of transports (called Fermi-Walker transports) [14] ÷ [16] exist which do not deform a Lorentz basis, 3c. There also exist other type of transports (called conformal transports) [17], [18] under which a light cone does not deform. 4a.…”

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

“…• there also exist other types of transports (called conformal transports) [20], [21] under which a light cone does not deform,…”