[Paper Review] Doubly Special Relativity

Giovanni Amelino-Camelia|arXiv (Cornell University)|Jul 12, 2002

Noncommutative and Quantum Gravity Theories8 references200 citations

TL;DR

This paper introduces Doubly Special Relativity (DSR), a relativistic theory postulating two observer-independent scales: the speed of light $c$ and the Planck momentum/length. It proposes that spacetime symmetries are deformed at the Planck scale to resolve the cosmic-ray paradox, where ultra-high-energy cosmic rays appear to exceed the GZK cutoff, though the model's predicted effects are too weak to fully explain the data.

ABSTRACT

I give a short non-technical review of the results obtained in recent work on "Doubly Special Relativity", the relativistic theories in which the rotation/boost transformations between inertial observers are characterized by two observer-independent scales (the familiar velocity scale, $c$, and a new observer-independent length/momentum scale, naturally identified with the Planck length/momentum). I emphasize the aspects relevant for the search of a solution to the cosmic-ray paradox.

Motivation & Objective

• To address the cosmic-ray paradox, where ultra-high-energy cosmic rays appear to exceed the GZK energy threshold predicted by Special Relativity.
• To explore whether introducing a second observer-independent scale—Planck momentum/length—can resolve inconsistencies between high-energy cosmic ray observations and relativistic kinematics.
• To investigate whether a modification of Special Relativity with two invariants (c and Planck scale) can provide a consistent theoretical framework for quantum gravity effects.
• To examine the implications of such a theory for macroscopic vs. microscopic systems, particularly regarding momentum composition and the behavior of composite particles.
• To assess the role of $κ$-Poincaré Hopf algebras in formulating the transformation laws of DSR theories and their viability in multi-particle sectors.

Proposed method

• Proposes a new relativistic framework where both the speed of light $c$ and the Planck momentum/length are observer-independent, extending Einstein's Special Relativity.
• Uses deformed dispersion relations and modified energy-momentum conservation laws to describe particle kinematics at Planck-scale energies.
• Applies the structure of $κ$-Poincaré Hopf algebras to define infinitesimal boost and rotation transformations between inertial observers in the one-particle sector.
• Analyzes the threshold energy $E_{GZK}$ for pion production in cosmic-ray interactions using deformed kinematics to test consistency with observations.
• Considers the challenge of defining total momentum for multi-particle and bound-state systems in DSR, where macroscopic bodies may exhibit different relativistic properties than free particles.
• Evaluates the possibility of constructing DSR theories without relying on $κ$-Poincaré algebras, exploring alternative mathematical frameworks for the two-scale symmetry.

Experimental results

Research questions

• RQ1 Can a relativistic theory with two observer-independent scales—$c$ and the Planck scale—resolve the apparent violation of the GZK cutoff in ultra-high-energy cosmic ray observations?
• RQ2 How can the laws of energy-momentum composition in DSR be consistently extended to multi-particle and bound-state systems, given that macroscopic bodies may not follow the same rules as free particles?
• RQ3 What is the role of $κ$-Poincaré Hopf algebras in formulating the transformation laws of DSR, and can DSR be constructed without them?
• RQ4 Can DSR naturally accommodate a separation between macroscopic and microscopic physical behavior, particularly in the context of momentum addition and relativistic invariance?
• RQ5 What experimental signatures, such as energy-dependent photon speed, could confirm or rule out DSR models in the near future?

Key findings

• The proposed DSR framework with two observer-independent scales—$c$ and the Planck scale—does not face any fundamental obstruction and is mathematically consistent.
• The model predicts new kinematic effects that could explain the cosmic-ray paradox, but their magnitude is too weak to account for the observed ultra-high-energy cosmic rays.
• Deformed dispersion relations in DSR are consistent with observations of fundamental particles but conflict with macroscopic body behavior, necessitating a distinction between micro and macro systems.
• The total momentum of a composite system in DSR cannot be simply additive from constituent particles, suggesting a breakdown of standard momentum composition rules at high energies.
• The role of $κ$-Poincaré Hopf algebras remains unclear in multi-particle sectors, and their use may not be essential, leaving open the possibility of alternative mathematical formulations.
• Future experiments, such as those on the GLAST space telescope, may test the prediction of wavelength-dependent photon speeds in some DSR models, offering a potential empirical test.