Analytic Hyperbolic Geometry And Albert Einstein's Special Theory Of Relativity (Second Edition)This 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. The premise of analogy as a study strategy is to make the unfamiliar familiar. Accordingly, this book introduces the notion of vectors into analytic hyperbolic geometry, where they are called gyrovectors. Gyrovectors turn out to be equivalence classes that add according to the gyroparallelogram law just as vectors are equivalence classes that add according to the parallelogram law. In the gyrolanguage of this book, accordingly, one prefixes a gyro to a classical term to mean the analogous term in hyperbolic geometry. As an example, the relativistic gyrotrigonometry of Einstein's special relativity is developed and employed to the study of the stellar aberration phenomenon in astronomy.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. It turns out that the invariant mass of the relativistic center of mass of an expanding system (like galaxies) exceeds the sum of the masses of its constituent particles. This excess of mass suggests a viable mechanism for the formation of dark matter in the universe, which has not been detected but is needed to gravitationally 'glue' each galaxy in the universe. 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 hyperbolic geometry. |
Contents
| 1 | |
| 15 | |
3 Gyrocommutative Gyrogroups | 55 |
4 Gyrogroup Extension | 105 |
5 Gyrovectors and Cogyrovectors | 135 |
6 Gyrovector Spaces | 157 |
7 Rudiments of Differential Geometry | 249 |
8 Gyrotrigonometry | 273 |
The Analytic Hyperbolic Geometric Viewpoint | 417 |
The Analytic Hyperbolic Geometric Viewpoint | 469 |
12 Relativistic Gyrotrigonometry | 569 |
13 Stellar and Particle Aberration | 657 |
Special Relativity of Signature mn | 681 |
Notation and Special Symbols | 719 |
Bibliography | 723 |
| 743 | |
9 Bloch Gyrovector of Quantum Information and Computation | 393 |
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Common terms and phrases
addition law algebra automorphism b]gyr[b ball barycentric coordinates chain of equations classical CMM frame coaddition cogyroline Definition density matrix disc Einstein addition Einstein gyrovector space Einstein velocity addition Einsteinian equivalent Euclidean geometry following theorem Galilei gamma factor given gyr[a gyr[b gyr[u gyr[x gyration gyro gyroangle gyroarea gyrobarycentric coordinates gyrocentroid gyrocommutative gyrogroup gyrodiagonal gyrogroup gyrogroup G gyroline gyromidpoint gyroparallelepiped gyrorays gyrosegment gyrosquare gyrotriangle gyrotriangle ABC gyrovec gyrovector translation Hence hyperbolic geometry identity inner product invariant mass left cancellation left gyrotranslations Lemma Lorentz boost Lorentz group Lorentz transformation Möbius gyrovector plane Möbius gyrovector space model of hyperbolic Newtonian notation numbered parallel transport Phys Poincaré points Proof quantum qubit relativistic CMM relativistic mass right cancellation Rnxm rooted gyrovector scalar Sect shown in Fig side gyrolengths signature space G spacetime special relativity Thomas precession triangle Ungar vector space vertex Σο