About Lorentz-M{\o}ller-Nelson transformation to rigid noninertial frame of reference

TL;DR

This paper develops and analyzes the Lorentz-Møller-Nelson transformation for noninertial frames, exploring its physical implications, applications to rotating bodies, and connections to relativistic kinematics phenomena like Thomas precession.

Contribution

It introduces a special LMN transformation for accelerated, rotating frames, derives inverse transformations, and examines their physical and kinematic consequences, including rotation matrices and relativistic effects.

Findings

Desynchronization of clocks in moving frames.

Lorentz contraction of rulers in noninertial frames.

Connection between Thomas precession and Wigner rotation.

Abstract

With a special Lorentz-M{\o}ller-Nelson (LMN) transformation found transformation of velocity from the laboratory system S to an accelerated, rotating frame of reference s. The physical sense of parameter entering into the LMN special transformation is established. For small distances, and their proper smooth motion without jerks suggested the inverse special LMN transformation. The main consequences of this transformation is considered, namely, a) the desync in moving frame of reference s of proper clocks of the pre-synchronized in the laboratory frame S and b) the Lorentz contraction of proper rulers of frame s in the frame S. The applicability of the inverse LMN transformation for real frames with maximum rigidity is established. Equations for the rotation matrix is obtained. It is shown that the intrinsic rotation of the axes s, considered with respect to S is not rigid. Found the direct and inverse transformation of affine "angular" velocity in the S to the comoving, but not rotating frame s. Also shown that for the non-inertial motion of rigidly rotating frame of reference her the kinematic deformation of coordinates system is absent in two planes. The application of this transformation to a rotating rigid body is considered. The matrice and angle of proper Wigner rotation is calculated. We find differential equations for the inverse problem of relativistic kinematics, and their decision in the case of uniformly accelerated motion. The close connection between the proper Thomas precession and the proper Wigner rotation and their mutual compensation for the case uniformly accelerated motion has shown. The difference of the uniformly accelerated motion from the hyperbolic one has been shown. Also, the basic formulas are expressed in terms of the parameter, which is solution of the equation for the inverse problem of relativistic kinematics.