Transformation equations for the kinetic energy of the same particle: Filling the gap in special relativity literature

TL;DR

This paper derives transformation equations for the kinetic energy of particles, including tardyons and photons, in special relativity, filling a gap in the literature by providing formulas that relate energies across different inertial frames.

Contribution

It introduces new transformation formulas for kinetic energy in special relativity, applicable to particles with mass and massless photons, extending existing theoretical frameworks.

Findings

Derived formulas for kinetic energy transformation of tardyons.

Extended results to photons, recovering relativistic Doppler formulas.

Illustrated behavior of energy transformations for electrons and photons.

Abstract

Transformation equations for the kinetic energy of a tardyon are derived in the limits of classical and of special relativity theory. Two formulas are presented. In the first one the energy of the particle in one of the involved reference frames is presented as a function of its proper energy, of the relative velocity of the two frames and of its speed in the second one. In the second one the kinetic energy in one of the involved reference frames is expressed as a function of its kinetic energy in the second one of its proper energy, of the relative velocity of the involved inertial reference frames and of its velocity relative to that frame. The obtained results are extended to the case of a photon that moves under the same geometrical conditions, recovering the formulas that account for the relativistic Doppler Effect, illustrating the behavior of a transformation equation when it should account for the properties of an electron and for those of a photon as well.