FERMI DERIVATIVE AND FERMI–WALKER TRANSPORTS OVER $(\bar{L}_n, g)$-SPACES

Abstract

The notions of Fermi covariant differential operator, Fermi derivative, and Fermi–Walker transport are generalized for the case of differentiable manifolds with different (not only by sign) contravariant and covariant affine connections and metrics [[Formula: see text]-spaces]. The generalization of Fermi–Walker transport is not unique and depends on the structure of the covariant antisymmetric tensor of second rank in the construction of the Fermi–Walker transport. The existence of such type of transport over [Formula: see text]-spaces allows the determination of a proper nonrotating accelerated observer's frame of reference, if a [Formula: see text]-space is used as a model of the space–time.

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

“…2a. 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.…”

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

“…The construction of a frame of reference for an accelerated observer by means of vector fields preserving their lengths and the angles between them under a Fermi-Walker transport [1], [2], could also be related to the description of the motion of the axes of a gyroscope in a space with different (not only by sign) contravariant and covariant affine connections and metrics [(L n , g)-space]. On the other side, the problem arises how can we describe the motion of vector fields preserving the angles between them but, at the same time, changing the length of every one of them proportionally to its own length.…”

“…Remark. The most notions, abbreviations, and symbols in this paper are defined in the previous papers [1], [2]. The reader is kindly asked to refer to one of them.…”

“…If we construct the length of ξ by the use of the metric g in a (for instance) V n -space as l ξ = | g(ξ, ξ) | 1 2 and by the use of the conformal to g metric g = e 2ϕ .g, ϕ = ϕ(x k ) ∈ C r (M ), r ≥ 1, as l ξ = | g(ξ, ξ) | 1 2 = e ϕ .l ξ , then the rate of change u l ξ of the length of ξ along a contravariant vector field u leads to the relation u l ξ = (uϕ).e ϕ .l ξ + e ϕ .ul ξ = (uϕ). l ξ + e ϕ .ul ξ .…”

“…In previous papers [1], [2], we have considered Fermi-Walker transports over (L n , g) and (L n , g)-spaces leading to preservation of the lengths of two contravariant vector fields and the angle between them, when they are transported along a non-null (non-isotropic) contravariant vector field. The investigations have been based on a special form of the extended covariant differential operator e ∇ u determined over a (L n , g)-or a (L n , g)-space as e ∇ u = ∇ u − A u , where ∇ u is the covariant differential operator:…”