[Paper Review] The Space-time Transformations Between Accelerated Systems

Miroslav Pardy|ArXiv.org|Feb 3, 2003

Relativity and Gravitational Theory12 references4 citations

TL;DR

This paper derives space-time transformations between uniformly accelerated and rotating reference frames using infinitesimal Lorentz transformations and kinematic symmetry postulates. It identifies a maximal acceleration limit—analagous to the speed of light in special relativity—derived from relativistic kinematics, with implications for particle physics, cosmology, and potential verification via ultracentrifuge experiments.

ABSTRACT

We determine transformations between coordinate systems which are mutually in linear accelerated motion. In case of the symmetrical linear mutual acceleration, we immediately get the maximal acceleration limit which was derived by Caianiello from quantum mechanics. The derived results can play crucial role in modern particle physics.

Motivation & Objective

• To derive relativistically consistent space-time transformations between systems in linear accelerated motion.
• To extend these transformations to rotating reference frames, such as those in centrifuges.
• To explore the consequences of nonlinearity in accelerated motion, particularly the emergence of a maximal acceleration.
• To investigate the physical and theoretical implications of maximal acceleration for general relativity, quantum mechanics, and cosmology.
• To propose experimental verification using ultracentrifuges and optical frequency measurements.

Proposed method

• Uses infinitesimal Lorentz transformations as a foundation, integrating them over time to derive finite transformations for non-inertial systems.
• Applies the relativistic velocity transformation with time-dependent acceleration, leading to non-linear coordinate mappings.
• Derives transformations for rotating systems by substituting angular velocity and radial distance into Lorentz-like forms.
• Introduces a relativistic relative acceleration formula analogous to relativistic velocity addition, accounting for radial acceleration differences in rotating frames.
• Applies symmetry principles and compares results from infinitesimal Lorentz derivation versus postulated kinematic symmetries.
• Derives the time dilation and metric element in rotating frames, showing dependence on radial acceleration and angular velocity.

Experimental results

Research questions

• RQ1 What are the correct space-time transformations between systems undergoing uniform linear acceleration?
• RQ2 How do space-time coordinates transform in rotating reference frames, particularly in the context of relativistic kinematics?
• RQ3 Can a maximal acceleration limit be derived from relativistic transformation laws without relying on quantum mechanics?
• RQ4 What are the implications of maximal acceleration for general relativity and the existence of black holes?
• RQ5 Can the predicted effects of maximal acceleration be experimentally verified using ultracentrifuges or optical frequency shifts?

Key findings

• The paper derives a maximal acceleration limit, α = c²/a, which emerges naturally from relativistic transformation laws, analogous to the speed of light in special relativity.
• The transformation equations for uniformly accelerated systems are derived via integration of infinitesimal Lorentz transformations, preserving the Minkowski metric.
• For rotating systems, the derived transformation leads to time dilation and a radial dependence on acceleration, with a critical radius r > c/ω rendering the system unphysical.
• A relativistic formula for relative acceleration is derived: w_r = (w₂ - w₁)/(1 - w₁w₂/α²), analogous to relativistic velocity addition.
• The paper shows that the maximal acceleration is consistent with Caianiello’s quantum mechanical derivation but derived independently through classical relativistic kinematics.
• The results suggest that general relativity and the standard model of particle physics may require modification if maximal acceleration is experimentally confirmed.