The task of the relativistic oscillator in a non-inertial frame of reference

TL;DR

This paper derives Lorentz-like transformations for non-inertial frames, linking classical mechanics and relativity through a vibrating system analogy involving sine-Gordon equations.

Contribution

It introduces generalized transformations for non-inertial frames that preserve space-time interval invariance, extending Lorentz transformations beyond inertial reference frames.

Findings

Transformations reduce to Lorentz transformations for inertial frames.

Relates classical mechanics to relativity via sine-Gordon type equations.

Shows time dependence on vibration amplitude in relativistic context.

Abstract

In this paper the analogues of the Lorentz transformations for non-inertial reference frames have been obtained. A common case when the movement speed of one coordinate frame in relation to another one can have time derivatives of higher orders. The obtained transformations conserve invariance of the space-time interval, and in the particular case of inertial frames become the well-known Lorentz transformations. It is shown that the transition from classical mechanics to the theory of relativity is analogous to the consideration of the vibrating system described by the equation of the sine-Gordon type. In this case, if the amplitude of the elliptic functions is $k\to 0$ the fluctuations can be considered small, and that leads to classical mechanics. With $k\to 1$ time depends on the vibration amplitude, which leads to the theory of relativity. In the case of inertial frames the amplitude is $k=\beta ={v}/{c}\;$.