Lorentz invariance with an invariant energy scale

TL;DR

This paper introduces a modified special relativity theory where an invariant energy scale, such as the Planck energy, coexists with the speed of light, achieved through a non-linear Lorentz transformation on momentum space.

Contribution

It proposes a novel non-linear Lorentz group action that preserves an invariant energy scale while maintaining relativity of inertial frames and compatibility with Einstein's theory at low energies.

Findings

Invariant energy scale can be incorporated into special relativity.

Modified Lorentz transformations involve a dilatation added to boosts.

Implications for field theory and gravity are discussed.

Abstract

We propose a modification of special relativity in which a physical energy, which may be the Planck energy, joins the speed of light as an invariant, in spite of a complete relativity of inertial frames and agreement with Einstein's theory at low energies. This is accomplished by a non-linear modification of the action of the Lorentz group on momentum space, generated by adding a dilatation to each boost in such a way that the Planck energy remains invariant. The associated algebra has unmodified structure constants, and we highlight the similarities between the group action found and a transformation previously proposed by Fock. We also discuss the resulting modifications of field theory and suggest a modification of the equivalence principle which determines how the new theory is embedded in general relativity.