[Paper Review] Interpretation of Lorentz boosts in conformally deformed special relativity theory

Alexei A. Deriglazov|ArXiv.org|Oct 3, 2005

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper proposes that Lorentz boosts in conformally deformed special relativity (CDSR) describe transformations between observers undergoing specific accelerated motion, with the acceleration governed by a new invariant scale R. The model exhibits nontrivial kinematics due to deformed momentum-space geometry and noncommutative phase space, and it is shown that the accelerated trajectories of these observers satisfy a geodesic equation, suggesting a relativistic foundation for the MOND program to address the dark matter problem.

ABSTRACT

Conformally deformed special relativity is mathematically consistent example of a theory with two observer independent scales. As compare with recent DSR proposals, it is formulated starting from the position space. In this work we propose interpretation of Lorentz boosts of the model as transformations among accelerated observers. We point further that the model can be considered as relativistic version of MOND program and thus may be interesting in context of dark matter problem.

Motivation & Objective

• To provide a physical interpretation of Lorentz boosts in conformally deformed special relativity (CDSR), a theory with two observer-independent scales: c and R.
• To demonstrate that the deformed Lorentz transformations correspond to changes between observers moving along accelerated, geodesic trajectories in the model’s spacetime.
• To explore the connection between CDSR and the MOND program by showing that the model's kinematics reproduce MOND-like dynamics in the nonrelativistic limit.
• To establish that the theory remains mathematically consistent, with a well-defined Hamiltonian formulation and conserved momentum under the deformed Lorentz group.
• To clarify the role of the deformation parameter λ, showing it relates to the Hubble constant and governs the rate of acceleration in the model.

Proposed method

• The paper uses a conformal transformation $ U_eta $ to deform the standard Lorentz group action, leading to a new realization $ \Lambda_\lambda = (U_\lambda)^{-1} \Lambda U_\lambda $, which acts non-linearly on spacetime coordinates.
• The deformed boost transformations are derived explicitly, showing they map inertial observers to accelerated ones, with the acceleration parameterized by $ \lambda $.
• The dynamics of a relativistic particle are derived from an action principle invariant under the deformed Lorentz group, yielding equations of motion that include a position-dependent effective mass and force term.
• The geodesic equation for the model is derived from the equations of motion, and it is shown that the specific accelerated trajectory $ x^i(t) = Vt + H_0 V (1 - V^2/c^2) t^2 $ satisfies this equation.
• The Hamiltonian formulation is used to show that the conjugate momentum $ p^\mu $ transforms linearly under the deformed Lorentz group, while the physical momentum $ P^\mu $ is non-linearly related to $ p^\mu $, leading to noncommutative phase space geometry.
• The noncommutative structure of the phase space is confirmed by computing $[x^\mu, P^\nu]$ and $[P^\mu, P^\nu]$, which are found to be non-vanishing and deformed, indicating a Planck-scale-dependent structure.

Experimental results

Research questions

• RQ1 How can Lorentz boosts in conformally deformed special relativity be physically interpreted beyond their mathematical transformation role?
• RQ2 What kind of physical motion do the transformed observers undergo, and is it consistent with the dynamics of the model?
• RQ3 Does the deformed theory reproduce MOND-like non-relativistic dynamics, and if so, under what conditions?
• RQ4 Can the accelerated trajectories in the model be shown to satisfy a geodesic equation, thereby replacing inertial motion in standard relativity?
• RQ5 What is the physical meaning of the deformation parameter $ \lambda $, and how does it relate to cosmological scales such as the Hubble constant?

Key findings

• The Lorentz boosts in CDSR are physically interpreted as transformations between observers moving along accelerated, geodesic trajectories, with the acceleration parameterized by $ \lambda $.
• The specific trajectory $ x^i(t) = Vt + H_0 V (1 - V^2/c^2) t^2 $, where $ H_0 \sim \lambda c $, satisfies the geodesic equation of the model, confirming its dynamical consistency.
• The model’s equations of motion in the nonrelativistic limit reduce to a modified Newtonian dynamics with a constant acceleration scale $ a_0 \sim \lambda c^2 $, characteristic of the MOND program.
• The phase space of the model exhibits noncommutative geometry: $[x^\mu, P^\nu] \neq 0$ and $[P^\mu, P^\nu] \neq 0$, indicating a deformation of canonical commutation relations dependent on position.
• The conserved four-momentum $ P^\mu $ transforms linearly under the deformed Lorentz group, resolving the total momentum problem present in other DSR models.
• The action and Hamiltonian formulation are consistent with the deformed symmetries, and the theory reduces to standard special relativity in the limit $ \lambda \to 0 $.