By Abraham A. Ungar

This booklet provides a robust technique to research Einstein's targeted idea of relativity and its underlying hyperbolic geometry within which analogies with classical effects shape the correct software. It introduces the inspiration of vectors into analytic hyperbolic geometry, the place they're known as gyrovectors.

Newtonian pace addition is the typical vector addition, that's either commutative and associative. The ensuing vector areas, in flip, shape the algebraic atmosphere for a standard version of Euclidean geometry. In complete analogy, Einsteinian pace addition is a gyrovector addition, that is either gyrocommutative and gyroassociative. The ensuing gyrovector areas, in flip, shape the algebraic environment for the Beltrami Klein ball version of the hyperbolic geometry of Bolyai and Lobachevsky. equally, MÃ¶bius addition offers upward push to gyrovector areas that shape the algebraic environment for the PoincarÃ© ball version of hyperbolic geometry.

In complete analogy with classical effects, the ebook offers a unique relativistic interpretation of stellar aberration by way of relativistic gyrotrigonometry and gyrovector addition. moreover, the e-book provides, for the 1st time, the relativistic heart of mass of an remoted process of noninteracting debris that coincided at a few preliminary time t = zero. the radical relativistic resultant mass of the approach, targeted on the relativistic heart of mass, dictates the validity of the darkish topic and the darkish power that have been brought by means of cosmologists as advert hoc postulates to give an explanation for cosmological observations approximately lacking gravitational strength and late-time cosmic speeded up growth.

the invention of the relativistic heart of mass during this booklet therefore demonstrates once more the usefulness of the examine of Einstein's exact idea of relativity by way of its underlying analytic hyperbolic geometry.

Contents: Gyrogroups; Gyrocommutative Gyrogroups; Gyrogroup Extension; Gyrovectors and Cogyrovectors; Gyrovector areas; Rudiments of Differential Geometry; Gyrotrigonometry; Bloch Gyrovector of Quantum info and Computation; unique concept of Relativity: The Analytic Hyperbolic Geometric point of view; Relativistic Gyrotrigonometry; Stellar and Particle Aberration.

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**Example text**

10). 121) of the left loop property followed by a left cancellation. 123), of the left loop property followed by a left cancellation. 112). 128) gyr[gyr[a, −b]b, a] = gyr[a, −b] for all a, b ∈ G. Proof. 131) for all a, b ∈ G. 126). 126). 127). 128) is equivalent to the first one. 128) follows from the first (third) by replacing a by −a (or, alternatively, by replacing b by −b). January 14, 2008 9:33 WSPC/Book Trim Size for 9in x 6in 42 ws-book9x6 Analytic Hyperbolic Geometry We are now in a position to find that the left gyroassociative law and the left loop property of gyrogroups have right counterparts.

In this book, accordingly, the presentation of Einstein’s special theory of relativity is solely based on Einstein velocity addition law, taking the reader to the immensity of the underlying hyperbolic geometry. Thus, more than 100 years after Einstein introduced the relativistic velocity addition law that now bears his name, this book demonstrates that placing Einstein velocity addition centrally in special relativity theory is an old idea whose time has returned [Ungar (2006a)]. Einstein’s failure to recognize and advance the gyrovector space structure that underlies his relativistic velocity addition law contributed to the eclipse of his velocity addition of relativistically admissible 3-velocities, creating a void that could be filled only with Minkowskian relativity, Minkowski’s reformulation of Einstein’s special relativity based on the Lorentz transformation of 4-velocities and on spacetime [Walter (2008)].

5 equivalence classes of pairs in our way to convert pairs of points in a gyrocommutative gyrogroup to gyrovectors and, similarly, to cogyrovectors. Following Def. 25) in a gyrogroup (G, ⊕). 18 rogroup (G, +). Proof. The gyropolygonal gyroaddition is associative in any gy- On the one hand (−a + b) + ♦ {(−b + c) + ♦ (−c + d)} = (−a + b) + ♦ (−b + d) = −a + d and on the other hand {(−a + b) + ♦ (−b + c)} + ♦ (−c + d) = (−a + c) + ♦ (−c + d) = −a + d January 14, 2008 9:33 WSPC/Book Trim Size for 9in x 6in ws-book9x6 25 Gyrogroups The gyropolygonal gyrosubtraction is just the gyropolygonal gyroaddition in the reversed direction along the gyropolygonal path.