A generic rule that simplifies the derivation of the transformation equations accounting for the properties of the photon

TL;DR

This paper introduces a universal rule that simplifies deriving transformation equations for photons and other waves, unifying their properties and behaviors across different reference frames.

Contribution

It presents a generic framework involving two functions that derive transformation equations for photons and tardyons, streamlining the process across various wave types.

Findings

Derived transformation equations for photon properties from a generic rule.

Unified treatment of wave transformations from acoustic to electromagnetic.

Validated the approach by recovering known photon transformation equations.

Abstract

We show that the transformation equation for the tardyon velocity involves two generic functions which in turn depend on the relative velocity of the involved reference frames, on the tardyon velocity u and on the polar angle which define the direction along which the tardyon moves. The same functions are further involved in the transformation equations for the space-time coordinates of the same event generated by a moving tardyon and for its relativistic mass, momentum and energy. Taking the limits of these functions for u approaching c we obtain exactly the transformation equations for the space-time coordinates of the same event generated by a photon and for its momentum and energy. The same procedure works also for the transition from a plane acoustic wave to an electromagnetic wave.