The Mathematics of Minkowski Space-Time
With an Introduction to Cummunitative Hypercomplex Numbers
(Sprache: Englisch)
This book arose out of original research on the extension of well-established applications of complex numbers related to Euclidean geometry and to the space-time symmetry of two-dimensional Special Relativity. The system of hyperbolic numbers is...
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Klappentext zu „The Mathematics of Minkowski Space-Time “
This book arose out of original research on the extension of well-established applications of complex numbers related to Euclidean geometry and to the space-time symmetry of two-dimensional Special Relativity. The system of hyperbolic numbers is extensively studied, and a plain exposition of space-time geometry and trigonometry is given. Commutative hypercomplex systems with four unities are studied and attention is drawn to their interesting properties.
Inhaltsverzeichnis zu „The Mathematics of Minkowski Space-Time “
- The Mathematics of Minkowski Space-Time: 1 N-Dimensional Hypercomplex Numbers and the associated Geometries
- Commutative Hypercomplex Number Systems
- The General Two-Dimensional System
- Linear Transformations and Geometries
- The Geometries Associated with Hypercomplex Numbers
- Conclusions
2 Trigonometry in the Minkowski Plane
- Geometrical Representation of Hyperbolic Numbers
- Basics of Hyperbolic Trigonometry
- Geometry in Pseudo-Euclidean Cartesian Plane
- Trigonometry in the Pseudo-Euclidean Plane
- Theorems on Equilateral Hyperbolas in the Pseudo-Euclidean Plane
- Some Examples of Triangle Solutions in the Minkowski Plane
- Conclusions
3 Uniform and Accelerated Motions in the Minkowski Space-Time (Twin Paradox)
- Inertial Motions
- Inertial and Uniformly Accelerated Motions
- Non-uniformly Accelerated Motions
- Conclusions
4 General Two-Dimensional Hypercomplex Numbers.-Geometrical Representation
- Geometry and Trigonometry in Two-Dimensional Algebras
- Some Properties of Fundamental Conic Section
- Numerical Examples
5 Functions of a Hyperbolic Variable
- Some Remarks on Functions of a Complex Variable
- Functions of Hypercomplex Variables
- The Functions of a Hyperbolic Variable
- The Elementary Functions of a Canonical Hyperbolic Variable
- H-Conformal Mappings
- Commutative Hypercomplex Systems with Three Unities
6 Hyperbolic Variables on Lorentz Surfaces
- Introduction
- Gauss: Conformal Mapping of Surfaces
- Extension of Gauss Theorem: Conformal Mapping of Lorentz Surfaces
- Beltrami: Complex Variables on a Surface
- Beltrami's Integration of Geodesic Equations
- Extension of Beltrami's Equation to Non-Definite Differential Forms
7 Constant Curvature Lorentz Surfaces
- Introduction
- Constant Curvature Riemann Surfaces
- Constant Curvature Lorentz Surfaces
- Geodesics and Geodesic Distances on Riemann and Lorentz Surfaces
- Conclusions
8 Generalization of Two-Dimensional Special Relativity (Hyperbolic
... mehr
Transformations and the Equivalence Principle)
- Physical Meaning of Transformations by Hyperbolic Functions
- Physical Interpretation of Geodesics on Riemann and Lorentz Surfaces with Positive Constant Curvature
- Einstein's Way to General Relativity
- Conclusions
- II An Introduction to Commutative Hypercomplex Numbers
9 Commutative Segre's Quaternions
- Introduction
- Hypercomplex Systems with Four Units
- Historical Introduction of Segre's Quaternion
- Algebraic Properties of Commutative Quaternions
- Functions of a Quaternion Variable
- Mapping by Means of Quaternion Functions
- Elementary Functions of the Quaternions
- Elliptic-Hyperbolic Quaternions
- Elliptic-Parabolic Generalized Segre's Quaternions
10 Constant Curvature Segre's Quaternion Spaces
- Introduction
- Quaternion differential geometry and geodesic equations
- Orthogonality in Segre's Quaternion Space
- Constant Curvature Quaternion Spaces
- Geodesic Equations in Quaternion Space
- Beltrami's Integration Method for Quaternion Spaces
- Beltrami's Integration Method for Quaternion Spaces
- Conclusions
11 A Matrix Formalization for Commutative Hypercomplex Systems
- Mathematical Operations
- Properties of the Characteristic Matrix M
- Functions of Hypercomplex Variable
- Functions of a Two-Dimensional Hypercomplex Variable
- Derivatives of a Hypercomplex Function
- Characteristic Differential Equation
- A Equivalence Between the Formalizations of Hypercomplex Numbers.
- Physical Meaning of Transformations by Hyperbolic Functions
- Physical Interpretation of Geodesics on Riemann and Lorentz Surfaces with Positive Constant Curvature
- Einstein's Way to General Relativity
- Conclusions
- II An Introduction to Commutative Hypercomplex Numbers
9 Commutative Segre's Quaternions
- Introduction
- Hypercomplex Systems with Four Units
- Historical Introduction of Segre's Quaternion
- Algebraic Properties of Commutative Quaternions
- Functions of a Quaternion Variable
- Mapping by Means of Quaternion Functions
- Elementary Functions of the Quaternions
- Elliptic-Hyperbolic Quaternions
- Elliptic-Parabolic Generalized Segre's Quaternions
10 Constant Curvature Segre's Quaternion Spaces
- Introduction
- Quaternion differential geometry and geodesic equations
- Orthogonality in Segre's Quaternion Space
- Constant Curvature Quaternion Spaces
- Geodesic Equations in Quaternion Space
- Beltrami's Integration Method for Quaternion Spaces
- Beltrami's Integration Method for Quaternion Spaces
- Conclusions
11 A Matrix Formalization for Commutative Hypercomplex Systems
- Mathematical Operations
- Properties of the Characteristic Matrix M
- Functions of Hypercomplex Variable
- Functions of a Two-Dimensional Hypercomplex Variable
- Derivatives of a Hypercomplex Function
- Characteristic Differential Equation
- A Equivalence Between the Formalizations of Hypercomplex Numbers.
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Bibliographische Angaben
- Autoren: Francesco Catoni , Dino Boccaletti , Paolo Zampetti , Vincenzo Catoni , Enrico Nichelatti , Roberto Cannata
- 2008, 2008, 256 Seiten, Maße: 17 x 24,4 cm, Kartoniert (TB), Englisch
- Mitarbeit:Boccaletti, Dino; Catoni, Francesco; Cannata, Roberto
- Verlag: Springer
- ISBN-10: 3764386134
- ISBN-13: 9783764386139
- Erscheinungsdatum: 17.04.2008
Sprache:
Englisch
Rezension zu „The Mathematics of Minkowski Space-Time “
From the reviews:"It is worth pointing out that the book is mainly a text about commutative hypercomplex numbers and some of their applications to a 2-dimensional Minkowski spacetime. ... This book should be interesting to anybody who is interested in applications of hypercomplex numbers ... . In conclusion, I recommend this book to anyone who wants to learn about hypercomplex numbers." (Emanuel Gallo, Mathematical Reviews, Issue 2010 d)
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