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Fermi Acceleration Mechanisms Beyond Lorentz Symmetry

Erick Aguiar, A. A. Araújo Filho, Valdir B. Bezerra, Gilson A. Ferreira, Iarley P. Lobo

TL;DR

This work develops a unified framework to study first- and second-order Fermi acceleration under Lorentz-symmetry deformations and violations, applying it to the κ-Poincaré algebra in both the bicrossproduct and classical bases. By deriving energy gains, diffusion equations, and spectral indices in SR, LIV, and Deformed Special Relativity scenarios, the authors quantify how modified dispersion relations, frame transformations, and two-particle momentum compositions shape accelerated-particle spectra. Key results show energy-dependent efficiencies and distinct spectral behaviors: in the bicrossproduct basis, LIV and DSR induce different first- and second-order departures from the canonical $E^{-2}$ spectrum, while the classical basis yields a characteristic transition from $-2$ to $-3$ in the first-order case and enhanced deviations at high energies in the second-order case. These findings identify potential observational signatures in cosmic-ray spectra and motivate astrophysical tests of quantum-gravity-inspired kinematics beyond standard Lorentz invariance.

Abstract

We construct models for first- and second-order Fermi acceleration of particles, incorporating generic frame transformations, dispersion relations, and conservation laws. Within this framework, we study deformations of Lorentz symmetry via the $κ$-Poincaré algebra in the bicrossproduct and classical bases, which respectively deform and preserve the relativistic dispersion relation. We also examine explicit Lorentz symmetry violation and compare the results with deformed relativity and special relativity.

Fermi Acceleration Mechanisms Beyond Lorentz Symmetry

TL;DR

This work develops a unified framework to study first- and second-order Fermi acceleration under Lorentz-symmetry deformations and violations, applying it to the κ-Poincaré algebra in both the bicrossproduct and classical bases. By deriving energy gains, diffusion equations, and spectral indices in SR, LIV, and Deformed Special Relativity scenarios, the authors quantify how modified dispersion relations, frame transformations, and two-particle momentum compositions shape accelerated-particle spectra. Key results show energy-dependent efficiencies and distinct spectral behaviors: in the bicrossproduct basis, LIV and DSR induce different first- and second-order departures from the canonical spectrum, while the classical basis yields a characteristic transition from to in the first-order case and enhanced deviations at high energies in the second-order case. These findings identify potential observational signatures in cosmic-ray spectra and motivate astrophysical tests of quantum-gravity-inspired kinematics beyond standard Lorentz invariance.

Abstract

We construct models for first- and second-order Fermi acceleration of particles, incorporating generic frame transformations, dispersion relations, and conservation laws. Within this framework, we study deformations of Lorentz symmetry via the -Poincaré algebra in the bicrossproduct and classical bases, which respectively deform and preserve the relativistic dispersion relation. We also examine explicit Lorentz symmetry violation and compare the results with deformed relativity and special relativity.
Paper Structure (22 sections, 75 equations, 7 figures)

This paper contains 22 sections, 75 equations, 7 figures.

Table of Contents

  1. Introduction
  2. First order Fermi acceleration mechanism
  3. Diffusive shock acceleration
  4. First order mechanism
  5. Particle spectrum
  6. The escape probability
  7. The spectral index
  8. Second order Fermi acceleration mechanism
  9. Deformation and Violation of Lorentz Symmetry
  10. Bicrossproduct basis
  11. Momentum as a function of velocity and the speed of light
  12. Modeling Lorentz Violation
  13. Classical basis
  14. First order Fermi Mechanism beyond Lorentz symmetry
  15. The bicrossproduct basis of $\kappa$-Poincaré algebra and LIV scenarios
  16. ...and 7 more sections

Figures (7)

  • Figure 1: First-order Fermi mechanism in the superluminal scenario considering Lorentz preservation, violation, and deformation in the bicrossproduct basis of the $\kappa$-Poincaré algebra.
  • Figure 2: First-order Fermi mechanism in the subluminal scenario considering Lorentz preservation, violation, and deformation in the bicrossproduct basis of the $\kappa$-Poincaré algebra.
  • Figure 3: First-order Fermi mechanism considering Lorentz preservation and deformation in the classical basis of the $\kappa$-Poincaré algebra.
  • Figure 4: Differential spectrum, derived from Fermi 1st order mechanism, multiplied by $E^2$ for different deformation parameters of the classical basis of the $\kappa$-Poincaré algebra.
  • Figure 5: Second-order Fermi mechanism in the superluminal scenario considering Lorentz preservation, violation, and deformation in the bicrossproduct basis of the $\kappa$-Poincaré algebra.
  • ...and 2 more figures