Conformal derivative and conformal transports over spaces with contravariant and covariant affine connections and metrics

TL;DR

This paper introduces and investigates conformal transports over spaces with affine connections and metrics, generalizing Fermi-Walker transports, and explores conditions for harmonic oscillations of vector lengths, with potential applications in gravitational theories.

Contribution

It defines conformal transports and derivatives in spaces with affine connections, extending existing transport concepts and analyzing their properties and special cases.

Findings

Conformal transports preserve angles but alter lengths proportionally.

Conditions for vector length oscillations as harmonic oscillators are established.

Different types of conformal transports include Fermi-Walker transports as special cases.

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 spaces with contravariant and covariant affine connections (whose components differ not only by sign) and metrics. 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 the above mentioned 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. PACS numbers: 04.90.+e; 04.50.+h; 12.10.Gq; 02.40.Vh