Overview

This chapter commemorates 2005 as the 100th anniversary of Albert Einstein's (1879 1955) miraculous year, and the 50th anniversary of his death in April 18, 1955. In 1905 he published the paper [Einstein (1905); Einstein (1998)] that founded the special theory of relativity, a term that he coined about ten years later. As we will see throughout this chapter, in the framework of analytic hyperbolic geometry (gyrogeometry) special theory of relativity reveals more of its intricate beauty, where Einstein velocity addition becomes a gyrovector addition. Analytic hyperbolic geometry, thus, significantly extends Einstein's unfinished symphony. As such, this chapter prepares non-Euclidean geometers and physicists to practice twenty-first century special relativity.

The majestic scientific achievement of the 20th century in mathematical beauty and experimental verifications has been the special theory of relativity that Einstein introduced a century ago in 1905 [Einstein (1905)] [Einstein (1998), p. 141]. Einstein's special relativity theory is one of the foundation blocks of modern theoretical physics. Yet, in the year of 2005 Einstein's special relativity appears the first classical theory that defies common sense. Dedicated to the centenary of the birth of the special theory of relativity, 1905 2005, this book presents the analytic theory of hyperbolic geometry in terms of analogies that it shares with the analytic theory of Euclidean geometry. The resulting unification of analytic Euclidean and hyperbolic geometry allows the extension of our intuitive understanding from Euclidean to hyperbolic geometry, and similarly, from classical mechanics to relativistic mechanics as well.

Coincidentally, it is the mathematical abstraction...