A Circular Geometric Framework for Teaching Relativistic Kinematics and Dynamics
Abstract
Special Relativity is traditionally taught through hyperbolic spacetime geometry, where coordinate speed is asymptotically bounded by c and Lorentz boosts correspond to hyperbolic rotations parameterized by rapidity φ. While mathematically standard, the non-compact nature of hyperbolic functions introduces well-documented pedagogical obstacles for introductory and intermediate physics students. In this work, we develop a comprehensive pedagogical framework for single-frame relativistic kinematics based on the exact velocity identity:
v² + 𝕋² = c²
where 𝕋 ≡ c(dτ/dt) = c/γ represents the chronokinetic velocity—the coordinate rate of proper-time progression within an inertial frame. Building upon the intuitive spacetime heuristics of Epstein and the trigonometric parameterizations of Bartlett, we formalize the Gudermannian bridge (θ = gd(φ)) into a unified instructional curriculum.
The kinematic pitch angle θ = arcsin(v/c), measured as a zenith angle from the vertical proper-time axis, maps the velocity domain onto a compact interval θ ∈ [0, π/2), unifying: • The Lorentz factor: γ = sec θ • Relativistic spatial momentum: p = mc tan θ • Total relativistic energy: E = mc² sec θ • The metric dispersion invariant: (mc²)² + (pc)² = E² ⟺ 1 + tan²θ = sec²θ
This formulation eliminates the problematic historical construct of "relativistic mass" by attributing the nonlinear growth of momentum to the geometric depletion of the orthogonal proper-time rate 𝕋. Furthermore, we demonstrate that the longitudinal relativistic optical Doppler factor simplifies from a radical expression into an elementary stereographic half-angle projection: D = sec θ + tan θ = tan(π/4 + θ/2). Inelastic collision thresholds are similarly solved via straightforward right-triangle bookkeeping without frame transformations. We explicitly delineate the operational boundaries of this representation: while multi-frame spacetime transformations remain fundamentally governed by the pseudo-Riemannian Minkowski metric (+, -, -, -), the circular ledger provides an intuitive, mathematically exact tool for single-frame state analysis, energy-momentum bookkeeping, and optical Doppler phenomena.
Related
- Entanglement‑Strain Angle: Geometric Unification of Relativistic Kinematics, Gravitation, and Quantum Phase
- Emergence of 1PN kinematics and relativistic optics via phase space thermodynamics in flat spacetime
- Third-order relativistic kinematics and Siacci shock decompositions via localised Darboux frames
- The propagation-induced circular polarization of fast radio bursts in relativistic plasma
- FBT08A: A Geometric Bridge from Berry–Locked Symplectic Structure to Lorentzian Kinematics
BibTeX
@misc{bray2026circular,
title = {{A Circular Geometric Framework for Teaching Relativistic Kinematics and Dynamics}},
author = {Brett A. Bray},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23671047},
url = {https://zenodo.org/records/23671047},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Circular Geometric Framework for Teaching Relativistic Kinematics and Dynamics
AU - Brett A. Bray
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23671047
UR - https://zenodo.org/records/23671047
AB - Special Relativity is traditionally taught through hyperbolic spacetime geometry, where coordinate speed is asymptotically bounded by c and Lorentz boosts correspond to hyperbolic rotations parameterized by rapidity φ. While mathematically standard, the non-compact nature of hyperbolic functions introduces well-documented pedagogical obstacles for introductory and intermediate physics students. In this work, we develop a comprehensive pedagogical framework for single-frame relativistic kinematics based on the exact velocity identity: v² + 𝕋² = c² where 𝕋 ≡ c(dτ/dt) = c/γ represents the chronokinetic velocity—the coordinate rate of proper-time progression within an inertial frame. Building upon the intuitive spacetime heuristics of Epstein and the trigonometric parameterizations of Bartlett, we formalize the Gudermannian bridge (θ = gd(φ)) into a unified instructional curriculum. The kinematic pitch angle θ = arcsin(v/c), measured as a zenith angle from the vertical proper-time axis, maps the velocity domain onto a compact interval θ ∈ [0, π/2), unifying: • The Lorentz factor: γ = sec θ • Relativistic spatial momentum: p = mc tan θ • Total relativistic energy: E = mc² sec θ • The metric dispersion invariant: (mc²)² + (pc)² = E² ⟺ 1 + tan²θ = sec²θ This formulation eliminates the problematic historical construct of "relativistic mass" by attributing the nonlinear growth of momentum to the geometric depletion of the orthogonal proper-time rate 𝕋. Furthermore, we demonstrate that the longitudinal relativistic optical Doppler factor simplifies from a radical expression into an elementary stereographic half-angle projection: D = sec θ + tan θ = tan(π/4 + θ/2). Inelastic collision thresholds are similarly solved via straightforward right-triangle bookkeeping without frame transformations. We explicitly delineate the operational boundaries of this representation: while multi-frame spacetime transformations remain fundamentally governed by the pseudo-Riemannian Minkowski metric (+, -, -, -), the circular ledger provides an intuitive, mathematically exact tool for single-frame state analysis, energy-momentum bookkeeping, and optical Doppler phenomena.
ER -