On the Lorentz Transformations of Momentum and Energy

TL;DR

This paper examines all possible Lorentz transformation variants for momentum and energy, confirming that only the standard four-vector transformations are consistent with physical requirements, which supports the application of the GZK cutoff in Lorentz-invariant theories.

Contribution

It demonstrates that only the usual four-vector Lorentz transformations are physically consistent for energy and momentum, excluding alternative transformations based on natural physical constraints.

Findings

Only four-vector transformations are consistent with physical requirements.

Supports the application of the GZK cutoff in Lorentz-invariant frameworks.

Clarifies the relation between energy-momentum conservation and transformation properties.

Abstract

Motivated by ultra-high-energy cosmic ray physics, we discuss all the possible alternatives to the familiar Lorentz transformations of the momentum and the energy of a particle. Starting from natural physical requirements, we exclude all the possibilities, apart from the ones which arise from the usual four-vector transformations by means of a change of coordinates in the mass-shell. This result confirms the remark, given in a preceding paper, that, in a theory without preferred inertial frames, one can always define a linearly transforming energy parameter to which the GZK cutoff argument can be applied. We also discuss the connections between the conservation and the transformation properties of energy-momentum and the relation between energy-momentum and velocity.