[Paper Review] Lorentz Symmetry and Ultra-High-Energy Cosmic Rays

M. Toller|ArXiv.org|Nov 7, 2002

Noncommutative and Quantum Gravity Theories37 references3 citations

TL;DR

This paper investigates whether ultra-high-energy cosmic rays (UHECRs) exceeding the GZK cutoff can be explained while preserving the relativity principle and deformed Lorentz symmetry. It argues that deforming the mass-shell or dispersion relations fails due to Lie group rigidity, but proposes that energy-momentum non-conservation in atmospheric proton-nucleus collisions—rather than modified kinematics—could explain the GZK violation, shifting focus from intergalactic gamma interactions to high-altitude nuclear collisions.

ABSTRACT

We discuss the possibility of explaining the observation of ultra-high-energy cosmic rays with energy above the GZK cutoff, saving the relativity principle and the (possibly deformed) Lorentz symmetry, as proposed recently by several authors. Since it is known that the Lie group structure of the Lorentz group cannot be deformed, we study the deformations (up to isomorphisms) of the mass-shell, considered as an abstract three-dimensional homogeneous space. We find that in the massive case the mass-shell cannot be deformed and in the massless case there are deformations, but their physical interpretation is problematic. The components of the four-momentum are considered as (redundant) coordinates on the abstract mass-shell. Reinterpreting an old result, we note that if the four-momentum is conserved its components must be the usual ones, with linear Lorentz transformation properties. Even if four-momentum is not conserved at high center-of-mass energies, the linearly transforming coordinates can always be used to describe in a convenient way the kinematics of collision processes and they satisfy the GKZ cutoff. We suggest that, if one wants to save the relativity principle, one should look for new physics in the collisions between the ultra-high-energy cosmic rays and the nuclei of the high atmosphere.

Motivation & Objective

• To assess whether UHECRs above the GZK cutoff can be explained without violating the relativity principle.
• To examine the feasibility of deforming Lorentz symmetry via modified dispersion relations or mass-shell geometry.
• To challenge the assumption that GZK violation implies Lorentz symmetry breaking, proposing instead a breakdown in four-momentum conservation during atmospheric collisions.
• To shift the focus of new physics from intergalactic proton-gamma interactions to high-energy proton-nucleus collisions in Earth's atmosphere.
• To explore the theoretical implications of non-conserved four-momentum for quantum gravity and spacetime symmetry.

Proposed method

• Analyzes deformations of the mass-shell as a three-dimensional homogeneous space under the Lorentz group, using group quotient construction (L/H).
• Applies Lie group rigidity theorems to show that the Lorentz group cannot be deformed into a non-isomorphic Lie group, implying no viable deformation of the Lorentz algebra.
• Considers four-momentum components as redundant coordinates on the mass-shell and shows that linear Lorentz transformation properties are required if four-momentum is conserved.
• Proposes that the observed high-energy cosmic rays result from non-conservation of four-momentum during collisions with atmospheric nuclei, even if the relativity principle holds.
• Introduces a general functional form (mπ⁰ = (mEp)¹ᐟ²F((mEp)⁻¹ᐟ²p⁰)) relating measured energy to linear energy, allowing GZK cutoff compliance.
• Argues that non-conservation of four-momentum is a more radical departure from standard physics than introducing privileged frames, requiring new mechanisms in quantum gravity.

Experimental results

Research questions

• RQ1 Can UHECRs above the GZK cutoff be explained while preserving the relativity principle and deformed Lorentz symmetry?
• RQ2 Is it possible to deform the mass-shell or dispersion relation in a way that avoids the GZK cutoff while maintaining Lorentz group structure?
• RQ3 What are the implications of four-momentum non-conservation in high-energy proton-nucleus collisions for cosmic ray observations?
• RQ4 Why is the failure of four-momentum conservation a more profound challenge to theoretical physics than the introduction of privileged inertial frames?
• RQ5 Can a functional relation between measured and linear energy (e.g., mπ⁰ = (mEp)¹ᐟ²F((mEp)⁻¹ᐟ²p⁰)) reconcile UHECR observations with the GZK cutoff?

Key findings

• The Lorentz group's Lie algebra is rigid, meaning any small deformation is isomorphic to the original group, ruling out non-trivial deformations of the Lorentz symmetry algebra.
• In the massive case, the mass-shell cannot be deformed; in the massless case, deformations exist but lack a clear physical interpretation.
• If four-momentum is conserved, its components must transform linearly under Lorentz transformations, implying standard kinematics.
• The observed GZK violation may stem from non-conservation of four-momentum during proton-nucleus collisions in Earth's atmosphere, not from modified dispersion relations.
• A general functional form mπ⁰ = (mEp)¹ᐟ²F((mEp)⁻¹ᐟ²p⁰) can reconcile measured energies above the GZK cutoff with the cutoff if F is suitably chosen.
• Non-conservation of four-momentum poses a deeper challenge to theoretical physics than Lorentz symmetry breaking via privileged frames, requiring new mechanisms in quantum gravity.