We add physical appeal to Einstein velocity addition law of relativistically admissible velocities, thereby gaining new analogies with classical mechanics and invoking new insights into the special theory of relativity. We place Einstein velocity addition in the foundations of both special relativity and its underlying hyperbolic geometry, enabling us to present special relativity in full three space dimensions rather than the usual one-dimensional space, using three-geometry instead of four-geometry. Doing so we uncover unexpected analogies with classical results, enabling readers to understand the modern and unfamiliar in terms of the classical and familiar. In particular, we show that while the relativistic mass does not mesh up with the four-geometry, it meshes extraordinarily well with the three-geometry, providing unexpected insights that are not easy to come by, by other means.

Question #57. Thomas precession

Einstein's special relativity: Unleashing the power of its hyperbolic geometry

We review why the Thomas rotation is a crucial facet of special relativity, that is just as fundamental, and just as “unintuitive” and “paradoxical,” as such traditional effects as length contraction, time dilation, and the ambiguity of simultaneity. We show how this phenomenon can be quite naturally introduced and investigated in the context of a typical introductory course on special relativity, in a way that is appropriate for, and completely accessible to, undergraduate students. We also demonstrate, in a more advanced section aimed at the graduate student studying the Dirac equation and relativistic quantum field theory, that careful consideration of the Thomas rotation will become vital as modern experiments in particle physics continue to move from unpolarized to polarized cross sections.

The Thomas rotation

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 e 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.

background

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

Question #57. Thomas precession

Citations

Einstein's special relativity: Unleashing the power of its hyperbolic geometry

The Thomas rotation

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

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