Non-Linear Relativity in Position Space

TL;DR

This paper explores two methods to derive the dual of non-linear relativity in position space, leading to different Lorentz transformations and implications for gravity theories with invariant scales.

Contribution

It introduces two novel approaches for dualizing non-linear relativity in position space, expanding the theoretical framework and physical implications.

Findings

Two distinct dual formulations of non-linear relativity in position space.

Different Lorentz transformation structures for each approach.

Implications for gravity theories with invariant energy or length scales.

Abstract

We propose two methods for obtaining the dual of non-linear relativity as previously formulated in momentum space. In the first we allow for the (dual) position space to acquire a non-linear representation of the Lorentz group independently of the chosen representation in momentum space. This requires a non-linear definition for the invariant contraction between momentum and position spaces. The second approach, instead, respects the linearity of the invariant contraction. This fully fixes the dual of momentum space and dictates a set of energy-dependent space-time Lorentz transformations. We discuss a variety of physical implications that would distinguish these two strategies. We also show how they point to two rather distinct formulations of theories of gravity with an invariant energy and/or length scale.