Non-Linear Relativity in Position Space

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.

7 pages, revised version

References in corpus (19)

• Potential Sensitivity of Gamma-Ray Burster Observations to Wave Dispersion in Vacuo
• Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale
• Testable scenario for Relativity with minimum-length
• Generalized Lorentz invariance with an invariant energy scale
• Extension of the Cosmic-Ray Energy Spectrum Beyond the Predicted Greisen-Zatsepin-Kuz'min Cutoff
• Gravity's Rainbow
• Ultraviolet modifications of dispersion relations in effective field theory
• Small-scale anisotropy of cosmic rays above 10^19eV observed with the Akeno Giant Air Shower Array
• Planck-scale deformation of Lorentz symmetry as a solution to the UHECR and the TeV-$\gamma$ paradoxes
• Distance Measurement and Wave Dispersion in a Liouville-String Approach to Quantum Gravity
• Deformed Boost Transformations That Saturate at the Planck Scale
• Astrophysical Probes of the Constancy of the Velocity of Light
• Observer-independent quanta of mass and length
• Space-Time Foam Effects on Particle Interactions and the GZK Cutoff
• Dirac spinors for Doubly Special Relativity and $\kappa$-Minkowski noncommutative spacetime
• On the definition of velocity in doubly special relativity theories
• On the definition of velocity in theories with two observer-independent scales
• Non-Critical Liouville String Escapes Constraints on Generic Models of Quantum Gravity
• On the Lorentz Transformations of Momentum and Energy

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