Einstein's Special Relativity: The Hyperbolic Geometric Viewpoint

The analytic hyperbolic geometric viewpoint of Einstein's special theory of relativity is presented. Owing to the introductio n of vectors into hyperbolic geometry, where they are called gyrovectors, the use of analytic hyperbolic geometry extends Einstein's unfinished symphony significantly, elev ating it to the status of a mathematical theory that could be emulated to the benefit of t he entire mathematical and physical community. The resulting theory involves a gyrovector space approach to hyperbolic geometry and relativistic mechanics, and could be studied with profit by anyone with a sufficient background in the common vector spac e approach to Euclidean geometry and classical mechanics. Einstein noted in his 1905 paper that founded the special theory of relativity that his velocity addition law satisfies the law of velocity parallelogram only to a first approximation. Within our hype rbolic geometric viewpoint of special relativity it becomes clear that Einstein's velo city addition law leads to a hyperbolic parallelogram addition law of Einsteinian velocities, which is supported experimentally by the cosmological effect known as stellar aberration and its relativistic interpretation. The latter, in turn, is supported experime ntally by the "GP-B" gyroscope experiment developed by NASA and Stanford University. Furthermore, the hyperbolic viewpoint of special relativity meshes extraordinarily well with the Minkowskian four- vector formalism of special relativity, revealing that the seemingly notorious relativistic mass meshes up with the four-vector formalism as well, owing to the natural emergence of dark matter. It is therefore hoped that both special relativity and its u nderlying analytic hyperbolic geometry will become part of the lore learned by all undergraduate and graduate mathematics and physics students.