Conformal derivative and conformal transports over spaces with an affine connection and metrics

TL;DR

This paper introduces and studies conformal transports over spaces with affine connections and metrics, generalizing Fermi-Walker transports, and explores their properties and potential applications in gravitational theories.

Contribution

It defines conformal transports and derivatives, generalizes Fermi-Walker transports, and establishes conditions for harmonic oscillations of vector lengths in such spaces.

Findings

Conformal transports preserve angles but alter lengths proportionally.

A conformal covariant differential operator is constructed.

Conditions for harmonic oscillations of vector lengths are derived.

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 one affine connection and metric. 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 spaces with one affine connection and metric. 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