Gyrations: The Missing Link Between Classical Mechanics with Its Underlying Euclidean Geometry and Relativistic Mechanics with Its Underlying Hyperbolic Geometry

The present article on the hyperbolic geometric interpretation of the relativistic mechanical effect known as Thomas precession is dedicated to the 80th Anniversary of Steve Smale for his leadership and commitment to excellence in the field of geometric mechanics. A study of Thomas precession in terms of its underlying hyperbolic geometry and elegant algebra is presented here in order to clarify the concept of Thomas precession. We review the studies of both Thomas precession and its abstract version, gyration. Based on the review we derive the correct Thomas precession angular velocity. We demonstrate here convincingly that the Thomas precession angle ε and its generating angle θ have opposite signs. We present the path from Einstein velocity addition to the gyroalgebra of gyrogroups and gyrations, and to the gyrogeometry that coincides with the hyperbolic geometry of Bolyai and Lobachevsky. We, then, demonstrate that the concept of Thomas precession in Einstein’s special theory of relativity is a concrete realization of the abstract concept of gyration in gyroalgebra.