| Description |
1 online resource (xix, 628 pages) : illustrations |
| Contents |
1. Introduction. 1.1. A vector space approach to Euclidean geometry and a gyrovector space approach to hyperbolic geometry. 1.2. Gyrolanguage. 1.3. Analytic hyperbolic geometry. 1.4. The three methods. 1.5. Applications in quantum and special relativity theory -- 2. Gyrogroups. 2.1. Definitions. 2.2. First gyrogroup theorems. 2.3. The associative gyropolygonal gyroaddition. 2.4. Two basic gyrogroup equations and cancellation laws. 2.5. Commuting automorphisms with gyroautomorphisms. 2.6. The gyrosemidirect product group. 2.7. Basic gyration properties -- 3. Gyrocommutative gyrogroups. 3.1. Gyrocommutative gyrogroups. 3.2. Nested gyroautomorphism identities. 3.3. Two-divisible two-torsion free gyrocommutative gyrogroups. 3.4. From Möbius to gyrogroups. 3.5. Higher dimensional Möbius gyrogroups. 3.6. Möbius gyrations. 3.7. Three-dimensional Möbius gyrations. 3.8. Einstein gyrogroups. 3.9. Einstein coaddition. 3.10. PV gyrogroups. 3.11. Points and vectors in a real inner product space. 3.12. Exercises -- 4. Gyrogroup extension. 4.1. Gyrogroup extension. 4.2. The gyroinner product, the gyronorm, and the gyroboost. 4.3. The extended automorphisms. 4.4. Gyrotransformation groups. 4.5. Einstein gyrotransformation groups. 4.6. PV (proper velocity) gyrotransformation groups. 4.7. Galilei transformation groups. 4.8. From gyroboosts to boosts. 4.9. The Lorentz boost. 4.10. The (p:q)-gyromidpoint. 4.11. The (p1:p2 ... :pn)-gyromidpoint -- 5. Gyrovectors and cogyrovectors. 5.1. Equivalence classes. 5.2. Gyrovectors. 5.3. Gyrovector translation. 5.4. Gyrovector translation composition. 5.5. Points and gyrovectors. 5.6. The gyroparallelogram addition law. 5.7. Cogyrovectors. 5.8. Cogyrovector translation. 5.9. Cogyrovector translation composition. 5.10. Points and cogyrovectors. 5.11. Exercises -- 6. Gyrovector spaces. 6.1. Definition and first gyrovector space theorems. 6.2. Solving a system of two equations in a gyrovector space. 6.3. Gyrolines and cogyrolines. 6.4. Gyrolines. 6.5. Gyromidpoints. 6.6. Gyrocovariance. 6.7. Gyroparallelograms. 6.8. Gyrogeodesics. 6.9. Cogyrolines. 6.10. Carrier cogyrolines of cogyrovectors. 6.11. Cogyromidpoints. 6.12. Cogyrogeodesics. 6.13. Various gyrolines and cancellation laws. 6.14. Möbius gyrovector spaces. 6.15. Möbius cogyroline parallelism. 6.16. Illustrating the gyroline gyration transitive law. 6.17. Turning the Möbius gyrometric into the Poincaré metric. 6.18. Einstein gyrovector spaces. 6.19. Turning Einstein gyrometric into a metric. 6.20. PV (proper velocity) gyrovector spaces. 6.21. Gyrovector space isomorphisms. 6.22. Gyrotriangle gyromedians and gyrocentroids. 6.23. Exercises -- 7. Rudiments of Differential geometry. 7.1. The Riemannian line element of Euclidean metric. 7.2. The gyroline and the cogyroline element. 7.3. The gyroline element of Möbius gyrovector spaces. 7.4. The cogyroline element of Möbius gyrovector spaces. 7.5. The gyroline element of Einstein gyrovector spaces. 7.6. The cogyroline element of Einstein gyrovector spaces. 7.7. The gyroline element of PV gyrovector spaces. 7.8. The cogyroline element of PV gyrovector spaces. 7.9. Table of Riemannian line elements -- 8. Gyrotrigonometry. 8.1. Vectors and gyrovectors are equivalence classes. 8.2. Gyroangles. 8.3. Gyrovector translation of gyrorays. 8.4. Gyrorays parallelism and perpendicularity. 8.5. Gyrotrigonometry in Möbius gyrovector spaces. 8.6. Gyrotriangle gyroangles and side gyrolengths. 8.7. The Gyroangular defect of right gyroangle gyrotriangles. 8.8. Gyroangular defect of the gyrotriangle. 8.9. Gyroangular defect of the gyrotriangle -- a synthetic proof. 8.10. The gyrotriangle side gyrolengths in terms of its gyroangles. 8.11. The semi-gyrocircle gyrotriangle. 8.12. Gyrotriangular gyration and defect. 8.13. The equilateral gyrotriangle. 8.14. The Möbius gyroparallelogram. 8.15. Gyrotriangle defect in the Möbius gyroparallelogram. 8.16. Gyroparallelograms inscribed in a gyroparallelogram. 8.17. Möbius gyroparallelogram addition law. 8.18. The gyrosquare. 8.19. Equidefect gyrotriangles. 8.20. Parallel transport. 8.21. Parallel transport vs. gyrovector translation. 8.22. Gyrocircle gyrotrigonometry. 8.23. Cogyroangles. 8.24. The cogyroangle in the three models. 8.25. Parallelism in gyrovector spaces. 8.26. Reflection, gyroreflection, and cogyroreflection. 8.27. Tessellation of the Poincaré disc. 8.28. Bifurcation approach to non-Euclidean geometry. 8.29. Exercises -- 9. Bloch gyrovector of quantum information and computation. 9.1. The density matrix for mixed state qubits. 9.2. Bloch gyrovector. 9.3. Trace distance and Bures fidelity. 9.4. The real density matrix for mixed state qubits. 9.5. Extending the real density matrix. 9.6. Exercises -- 10. Special theory of relativity: the analytic hyperbolic geometric viewpoint pt. I: Einstein velocity addition and its consequences. 10.1. Introduction. 10.2. Einstein velocity addition. 10.3. From Thomas gyration to Thomas precession. 10.4. The relativistic gyrovector space. 10.5. Gyrogeodesics, gyromidpoints and gyrocentroids. 10.6. The midpoint and the gyromidpoint -- Newtonian and Einsteinian mechanical interpretation. 10.7. Einstein gyroparallelograms. 10.8. The relativistic gyroparallelogram law. 10.9. The parallelepiped. 10.10. The pre-gyroparallelepiped. 10.11. The gyroparallelepiped. 10.12. The relativistic gyroparallelepiped addition law. 10.13. Exercises -- 11. Special Theory of Relativity: The analytic hyperbolic geometric viewpoint pt. II: Lorentz transformation and its consequences. 11.1. The Lorentz transformation and its gyro-algebra. 11.2. Galilei and Lorentz transformation links. 11.3. (t1:t2)-gyromidpoints as CMM velocities. 11.4. The hyperbolic theorems of Ceva and Menelaus. 11.5. Relativistic two-particle systems. 11.6. The covariant relativistic CMM frame velocity. 11.7. The relativistic invariant mass of an isolated particle system. 11.8. Relativistic CMM and the kinetic energy theorem. 11.9. Additivity of relativistic energy and momentum. 11.10. Bright (baryonic) and dark matter. 11.11. Newtonian and relativistic kinetic energy. 11.12. Barycentric coordinates. 11.13. Einsteinian gyrobarycentric coordinates. 11.14. The proper velocity Lorentz group. 11.15. Demystifying the proper velocity Lorentz group. 11.16. The standard Lorentz transformation revisited. 11.17. The inhomogeneous Lorentz transformation. 11.18. The relativistic center of momentum and mass. 11.19. Relativistic center of mass: example 1. 11.20. Relativistic center of mass: example 2. 11.21. Dark matter and dark energy. 11.22. Exercises -- 12. Relativistic gyrotrigonometry. 12.1. The relativistic gyrotriangle. 12.2. Law of gyrocosines, SSS to AAA conversion law. 12.3. The AAA to SSS conversion law. 12.4. The law of gyrosines. 12.5. The relativistic equilateral gyrotriangle. 12.6. The relativistic gyrosquare. 12.7. The Einstein gyrosquare with [symbol] = [symbol]/3. 12.8. The ASA to SAS conversion law. 12.9. The relativistic gyrotriangle defect. 12.10. The right-gyroangled gyrotriangle. 12.11. The Einsteinian gyrotrigonometry. 12.12. The relativistic gyrotriangle gyroarea. 12.13. The gyrosquare gyroarea. 12.14. The gyrotriangle constant principle. 12.15. Ceva and Menelaus, revisited. 12.16. Saccheri gyroquadrilaterals. 12.17. Lambert gyroquadrilaterals. 12.18. Exercises -- 13. Stellar and particle aberration. 13.1. Particle aberration: the classical interpretation. 13.2. Particle aberration: the relativistic interpretation. 13.3. Particle aberration: the geometric interpretation. 13.4. Relativistic stellar aberration. 13.5. Exercises |
| Summary |
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. 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, Möbius addition gives rise to gyrovector spaces that form the algebraic setting for the Poincaré 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 |
| Bibliography |
Includes bibliographical references (pages 605-620) and index |
| Notes |
Print version record |
| Subject |
Special relativity (Physics)
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Geometry, Hyperbolic.
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SCIENCE -- Physics -- Relativity.
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Geometry, Hyperbolic
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Special relativity (Physics)
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| Form |
Electronic book
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| ISBN |
9789812772305 |
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9812772308 |
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9812772294 |
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9789812772299 |
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1281911992 |
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9781281911995 |
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