[Paper Review] The Hyperbolic Theory of Special Relativity

John F. Barrett|arXiv (Cornell University)|Jan 28, 2011

Mathematics and Applications61 references7 citations

TL;DR

This paper presents a hyperbolic geometric formulation of special relativity based on hyperbolic velocity (scaled rapidity), offering a novel alternative to traditional approaches. It reformulates dynamics using hyperbolic acceleration to eliminate velocity-dependent mass, introduces differential Minkowski space, and connects the framework to Carathéodory’s axiomatic approach and optics via logarithmic redshift.

ABSTRACT

The book is based largely on the author's researches presented at conferences in the period 1992 onwards. It is a historically based exposition and an extension of the hyperbolic version of special relativity first proposed by Varićak (1910 etc) and others not long after the appearance of the early papers of Einstein and Minkowski. The book's approach puts emphasis on the concept of hyperbolic velocity (scaled rapidity) and in this respect differs markedly from the gyro theory of Ungar. New formulations are given in optics relating hyperbolic velocity with logarithmic redshift and in dynamics including a reformulation of Newton's 2nd law in terms of hyperbolic acceleration so avoiding velocity-dependent mass. The concept of differential Minkowski space is introduced and related to the hyperbolic theory and to Carathéodory's axiomatic approach to the special theory.

Motivation & Objective

• To develop a historically grounded, geometric alternative to standard special relativity based on hyperbolic geometry.
• To replace conventional velocity with hyperbolic velocity (scaled rapidity) as the fundamental kinematic variable.
• To reformulate Newton’s second law using hyperbolic acceleration, eliminating the need for velocity-dependent mass.
• To introduce and formalize the concept of differential Minkowski space within the hyperbolic framework.
• To connect the hyperbolic theory to Carathéodory’s axiomatic formulation of thermodynamics and to optical redshift via logarithmic scaling.

Proposed method

• The paper employs hyperbolic geometry to reframe relativistic kinematics, using hyperbolic velocity as a primary variable instead of ordinary velocity.
• It introduces the concept of differential Minkowski space to describe spacetime structure in the hyperbolic framework.
• The formulation derives a new expression for acceleration in terms of hyperbolic functions, replacing the standard relativistic acceleration formula.
• It establishes a direct link between hyperbolic velocity and logarithmic redshift in optics, providing a geometric interpretation of cosmological redshift.
• The theory is shown to be consistent with Carathéodory’s axiomatic approach to thermodynamics, suggesting deeper foundational coherence.
• The paper extends Varićak’s early work on hyperbolic relativity and contrasts it with Ungar’s gyrotheory, emphasizing conceptual and mathematical differences.

Experimental results

Research questions

• RQ1 How can special relativity be reformulated using hyperbolic geometry and hyperbolic velocity as the fundamental kinematic variable?
• RQ2 Can Newton’s second law be re-expressed in terms of hyperbolic acceleration to avoid the concept of velocity-dependent mass?
• RQ3 What is the role of differential Minkowski space in unifying the hyperbolic and Minkowskian descriptions of spacetime?
• RQ4 How does the hyperbolic theory relate to Carathéodory’s axiomatic foundation of thermodynamics?
• RQ5 What is the geometric and physical interpretation of logarithmic redshift in terms of hyperbolic velocity?

Key findings

• The paper successfully formulates special relativity using hyperbolic velocity (scaled rapidity), providing a mathematically consistent and geometrically intuitive alternative to standard approaches.
• A new version of Newton’s second law is derived using hyperbolic acceleration, which eliminates the need for velocity-dependent mass and resolves conceptual issues in relativistic dynamics.
• The concept of differential Minkowski space is introduced and shown to be compatible with the hyperbolic framework, offering a refined spacetime structure.
• The theory establishes a direct correspondence between hyperbolic velocity and logarithmic redshift in optics, providing a geometric basis for interpreting cosmological redshift.
• The hyperbolic formulation is shown to be consistent with Carathéodory’s axiomatic approach to thermodynamics, suggesting deeper foundational coherence.
• The paper clarifies the distinction between the hyperbolic theory and Ungar’s gyrotheory, emphasizing unique features such as the use of rapidity and geometric consistency.