[Paper Review] Unphysical Predictions of Some Doubly Special Relativity Theories

Jakub Rembieliński, Kordian A. Smoliński|ArXiv.org|Jul 3, 2002

Statistical Mechanics and Entropy14 citations

TL;DR

This paper critically analyzes a doubly special relativity (DSR) theory proposed by Magueijo and Smolin, which introduces a fundamental length scale (Planck length) into special relativity via nonlinear Lorentz transformations in momentum space. It demonstrates that the theory leads to unphysical kinematical features—such as velocity dependence on mass and non-physical energy bounds—and renders statistical mechanics and thermodynamics incoherent due to divergent partition functions and non-additive energy, undermining its physical viability.

ABSTRACT

A kind of doubly special relativity theory proposed by J. Magueijo and L. Smolin [Phys. Rev. Lett. 88, 190403 (2002)] is analysed. It is shown that this theory leads to serious physical difficulties in interpretation of kinematical quantities. Moreover, it is argued that statistical mechanics and thermodynamics cannot be resonably formulate within the model proposed in the mentioned paper.

Motivation & Objective

• To assess the physical consistency of a doubly special relativity (DSR) theory that introduces the Planck length as a fundamental scale in special relativity.
• To investigate the interpretational problems in kinematical quantities such as energy, momentum, and velocity within the proposed DSR framework.
• To determine whether statistical mechanics and thermodynamics can be consistently formulated in this DSR model.
• To evaluate the mathematical structure of the momentum space and the implications of nonlinear Lorentz transformations on physical observables.

Proposed method

• Analyzes the nonlinear Lorentz transformations in momentum space, parameterized by a dimensionless parameter λ, with the Planck mass mP and speed of light c.
• Derives the invariant dispersion relation m²c² = (p₀² − p²)/(1 − λp₀/(mPc))², showing singularities at p₀ = mPc/λ.
• Identifies momentum space orbits as algebraic conics (hyperbola, parabola, ellipse, or half-lines), with only lower branches being physically acceptable due to global Lorentz group realization constraints.
• Introduces a coordinate transformation kμ = pμ / (1 − λp₀/(mPc)) to linearize the momentum space, mapping to standard Minkowski momentum space with linear Lorentz transformations.
• Evaluates the one-particle partition function in 3D momentum space using the invariant measure dΓ, showing divergence near the momentum boundary |p| → mPc/λ.
• Considers both relativistic and non-relativistic momentum measures, demonstrating that the partition function diverges or leads to temperature-independent internal energy in the thermodynamic limit.

Experimental results

Research questions

• RQ1 Does the DSR model proposed by Magueijo and Smolin allow for a consistent interpretation of kinematical quantities such as energy, momentum, and velocity?
• RQ2 Can the nonlinear Lorentz transformations in momentum space be consistently mapped to a linear realization without introducing unphysical features?
• RQ3 Is statistical mechanics, including the partition function and thermodynamic quantities, well-defined within this DSR framework?
• RQ4 What are the implications of the momentum space boundary at |p| = mPc/λ for the existence of physical states and thermodynamic limits?
• RQ5 Does the non-additivity of energy in the N-particle system imply a breakdown of standard thermodynamic behavior?

Key findings

• The momentum space is bounded, with p₀ < mPc/λ, and the invariant measure dΓ becomes singular at |p| → mPc/λ, causing the one-particle partition function to diverge.
• The partition function Z₁ diverges due to the singularity in dΓ, rendering the internal energy and entropy ill-defined in the standard statistical mechanics framework.
• Even with a non-relativistic measure d³p, the N-particle internal energy approaches mPc²/λ in the thermodynamic limit, independent of temperature, violating standard thermodynamic behavior.
• The velocity defined via p and p₀ does not transform consistently under Lorentz boosts; it depends on mass and is not equivalent to the canonical velocity vL = k/k₀.
• The nonlinear transformation law is not an essential nonlinearity but an artifact of a nonlinear coordinate system in momentum space, as it can be linearized via kμ = pμ / (1 − λp₀/(mPc)).
• The model fails to support a consistent formulation of statistical mechanics and thermodynamics due to divergent partition functions and non-additive energy, undermining its physical viability.