Nonstandard Lorentz-Einstein transformations

TL;DR

This paper introduces nonstandard Lorentz transformations that relate clock readings in inertial frames more physically accurately, considering various cases like radar detection and dielectric media, extending standard Lorentz transformations.

Contribution

It develops nonstandard Lorentz transformations that connect synchronized and non-synchronized clock readings, addressing physical scenarios not covered by standard transformations.

Findings

Derived nonstandard transformations for radar detection.

Extended transformations to dielectric media.

Analyzed relativity of particle positions.

Abstract

The standard Lorentz transformations establish a relationship between the space-time coordinates of the same event when detected from two inertial reference frames I and I' in the standard arrangement. This event is characterized by the space-time coordinates E(x,tE) and E'(x',t'E), tE and t'E representing the readings of the standard synchronized clocks C(x) and C'(x') located in the two frames where the event takes place. We obtain the nonstandard Lorentz transformations establishing a "physically" correct relationship between the readings of the standard synchronized clocks and the readings of other clocks (ta,t'a) of the same inertial reference frames. This relationship of the type tE=f(x,ta), expresses the standard Lorentz transformations as a function of ta and t'a respectively. We present several cases of nonstandard Lorentz transformation (the case of radar detection, the case when one reference frame is filled with an ideal transparent dielectric and the case of relativity of the apparent, actual and synchronized positions of the same moving particle).