Analytic Hyperbolic Geometry in N Dimensions: An by Abraham Albert Ungar

By Abraham Albert Ungar

The idea of the Euclidean simplex is critical within the examine of n-dimensional Euclidean geometry. This booklet introduces for the 1st time the concept that of hyperbolic simplex as a tremendous idea in n-dimensional hyperbolic geometry.

Following the emergence of his gyroalgebra in 1988, the writer crafted gyrolanguage, the algebraic language that sheds traditional mild on hyperbolic geometry and specified relativity. a number of authors have effectively hired the author’s gyroalgebra of their exploration for novel effects. Françoise Chatelin famous in her ebook, and in other places, that the computation language of Einstein defined during this booklet performs a common computational position, which extends a long way past the area of designated relativity.

This ebook will motivate researchers to take advantage of the author’s novel options to formulate their very own effects. The publication presents new mathematical tools, such as hyperbolic simplexes, for the examine of hyperbolic geometry in n dimensions. It also presents a brand new examine Einstein’s distinct relativity idea.

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65). Moreover, we have the following result. 7 (Gyration–Thomas Precession Angle). Let u, v, x ∈ Rns be relativistically admissible velocities such that u  −v (so that u⊕v  0). 66) Proof. 22), pp. 53). 31), p. 29, coincide. Special attention to three dimensional gyrations, which are of interest in physical applications, is paid in Chapter 13 in the study of Thomas precession. 8 From Einstein Velocity Addition to Gyrogroups Guided by analogies with groups, the key features of Einstein groupoids (Rns, ⊕), n = 1, 2, 3, .

Interestingly, it is Einstein coaddition that captures the hyperbolic parallelogram (gyroparallelogram) law, u ⊞ v, for the composition of Einsteinian velocity gyrovectors. Experimental evidence that supports the physical significance of Einstein gyroparallelogram law of velocity addition is provided by the relativistic interpretation of the cosmological stellar aberration phenomenon, as explained in Sect. 3 and, in detail, in [129, Chapter 13]. In Euclidean geometry, the extension of the parallelogram law of addition of two vectors to a parallelotope law of addition of more than two vectors is obvious.

The book is divided into six parts: 1. Part I: Einstein Gyrogroups and Gyrovector Spaces. The first part of the book reveals the emergence of mathematical beauty and regularity that results from decoding the algebraic structures that the Einstein relativistic velocity addition law encodes. Part I of the book, Chapters 2–4, presents the Einstein velocity addition law of special relativity theory, revealing the novel algebra, called gyroalgebra, that it encodes. The resulting gyroalgebra stems from the notions of a) the gyrogroup, which is a natural generalization of the group concept in algebra; and b) the gyrovector space, which is a natural generalization of the vector space concept in algebra.

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