## Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of RelativityThis book presents a powerful way to study Einstein's special theory of relativity and its underlying hyperbolic geometry in which analogies with classical results form the right tool. It introduces the notion of vectors into analytic hyperbolic geometry, where they are called gyrovectors . Newtonian velocity addition is the common vector addition, which is both commutative and associative. The resulting vector spaces, in turn, form the algebraic setting for the standard model of Euclidean geometry. In full analogy, Einsteinian velocity addition is a gyrovector addition, which is both gyrocommutative and gyroassociative . The resulting gyrovector spaces, in turn, form the algebraic setting for the Beltrami-Klein ball model of the hyperbolic geometry of Bolyai and Lobachevsky. Similarly, MAbius addition gives rise to gyrovector spaces that form the algebraic setting for the Poincar(r) ball model of hyperbolic geometry. In full analogy with classical results, the book presents a novel relativistic interpretation of stellar aberration in terms of relativistic gyrotrigonometry and gyrovector addition. Furthermore, the book presents, for the first time, the relativistic center of mass of an isolated system of noninteracting particles that coincided at some initial time t = 0. The novel relativistic resultant mass of the system, concentrated at the relativistic center of mass, dictates the validity of the dark matter and the dark energy that were introduced by cosmologists as ad hoc postulates to explain cosmological observations about missing gravitational force and late-time cosmic accelerated expansion. The discovery of the relativistic center of mass in this book thus demonstrates once again the usefulness of the study of Einstein's special theory of relativity in terms of its underlying analytic hyperbolic geometry. Sample Chapter(s). Chapter 1: Introduction (145 KB). Contents: Gyrogroups; Gyrocommutative Gyrogroups; Gyrogroup Extension; Gyrovectors and Cogyrovectors; Gyrovector Spaces; Rudiments of Differential Geometry; Gyrotrigonometry; Bloch Gyrovector of Quantum Information and Computation; Special Theory of Relativity: The Analytic Hyperbolic Geometric Viewpoint; Relativistic Gyrotrigonometry; Stellar and Particle Aberration. Readership: Undergraduates, graduate students, researchers and academics in geometry, algebra, mathematical physics, theoretical physics and astronomy." |

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### 內容

1 Introduction | 1 |

2 Gyrogroups | 15 |

3 Gyrocommutative Gyrogroups | 51 |

4 Gyrogroup Extension | 101 |

5 Gyrovectors and Cogyrovectors | 131 |

6 Gyrovector Spaces | 153 |

7 Rudiments of Differential Geometry | 239 |

8 Gyrotrigonometry | 261 |

The Analytic Hyperbolic Geometric Viewpoint Part I Einstein Velocity Addition and its Consequences | 397 |

The Analytic Hyperbolic Geometric Viewpoint Part II Lorentz Transformation and its Consequences | 445 |

12 Relativistic Gyrotrigonometry | 537 |

13 Stellar and Particle Aberration | 577 |

Notation And Special Symbols | 601 |

605 | |

621 | |

9 Bloch Gyrovector of Quantum Information and Computation | 375 |

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### 常見字詞

addition law algebra automorphism barycentric coordinates classical coaddition cogyroangle cogyroline Definition density matrix disc eaeb Einstein addition Einstein gyrovector space Einstein velocity addition Einsteinian equivalent Euclidean geometry following theorem gamma factor given gyr[b gyra gyration gyro gyroangle gyroassociative law gyroautomorphic inverse gyrocentroid gyrocommutative gyrogroup gyrocosine gyrodiagonal gyrogroup G gyrogroup operation gyroline gyrometric gyromidpoint gyroparallelepiped gyroquadrilateral gyrorays gyrosegment gyrosquare gyrotriangle gyrotriangle ABC gyrotrigonometry gyrovec gyrovector PQ gyrovector translation Hence hyperbolic geometry identity inner product invariant mass left cancellation left gyrotranslations Lemma Let G Lorentz boost Lorentz transformation Möbius addition Möbius gyrovector plane Möbius gyrovector space model of hyperbolic Newtonian notation parallel transport particle aberration Poincaré points Proof proper velocity qubit qubit density relativistic mass right cancellation rooted cogyrovector PQ rooted gyrovector scalar shown in Fig side gyrolengths space G special relativity Thomas precession triangle Ungar vector space velocity addition