The mathematics of Minkowski space-time : with an introduction to commutative hypercomplex numbers

Hyperbolic numbers are proposed for a rigorous geometric formalization of the space-time symmetry of two-dimensional Special Relativity. The system of hyperbolic numbers as a simple extension of the field of complex numbers is extensively studied in the book. In particular, an exhaustive solution of the "twin paradox" is given, followed by a detailed exposition of space-time geometry and trigonometry. Finally, an appendix on general properties of commutative hypercomplex systems with four unities is presented.

Preface	p. vii
1 Introduction	p. 1
2 N-Dimensional Commutative Hypercomplex Numbers	p. 5
2.1 N-Dimensional Hypercomplex Numbers	p. 5
2.1.1 Equality and Sum	p. 5
2.1.2 The Product Operation	p. 6
2.1.3 Characteristic Matrix and Characteristic Determinant	p. 7
2.1.4 Invariant Quantities for Hypercomplex Numbers	p. 9
2.1.5 The Division Operation	p. 10
2.1.6 Characteristic Equation and Principal Conjugations	p. 10
2.1.7 Decomposable Systems	p. 12
2.2 The General Two-Dimensional System	p. 12
2.2.1 Canonical Two-Dimensional Systems	p. 16
2.2.2 The Two-Dimensional Hyperbolic System	p. 16
3 The Geometries Generated by Hypercomplex Numbers	p. 19
3.1 Linear Transformations and Geometries	p. 19
3.1.1 The Continuous Lie Groups	p. 19
3.1.2 Klein's Erlanger Programm	p. 19
3.2 Groups Associated with Hypercomplex Numbers	p. 20
3.2.1 Geometries Generated by Complex and Hyperbolic Numbers	p. 23
3.3 Conclusions	p. 24
4 Trigonometry in the Minkowski Plane	p. 27
4.1 Geometrical Representation of Hyperbolic Numbers	p. 28
4.1.1 Hyperbolic Exponential Function and Hyperbolic Polar Transformation	p. 28
4.1.2 Hyperbolic Rotations as Lorentz Transformations of Special Relativity	p. 30
4.2 Basics of Hyperbolic Trigonometry	p. 31
4.2.1 Complex Numbers and Euclidean Trigonometry	p. 31
4.2.2 Hyperbolic Rotation Invariants in Pseudo-Euclidean Plane Geometry	p. 32
4.2.3 Fjelstad's Extension of Hyperbolic Trigonometric Functions	p. 35
4.3 Geometry in the Pseudo-Euclidean Cartesian Plane	p. 37
4.4 Goniometry and Trigonometry in the Pseudo-Euclidean Plane	p. 40
4.4.1 Analytical Definitions of Hyperbolic Trigonometric Functions	p. 41
4.4.2 Trigonometric Laws in the Pseudo-Euclidean Plane	p. 42
4.4.3 The Triangle's Angles Sum	p. 43
4.5 Theorems on Equilateral Hyperbolas in the Pseudo-Euclidean Plane	p. 44
4.6 Examples of Triangle Solutions in the Minkowski Plane	p. 52
5 Uniform and Accelerated Motions in the Minkowski Space-Time (Twin Paradox)	p. 57
5.1 Inertial Motions	p. 58
5.2 Inertial and Uniformly Accelerated Motions	p. 61
5.3 Non-uniformly Accelerated Motions	p. 69
5.3.1 Frenet's Formulas in the Minkowski Space-Time	p. 70
5.3.2 Proper Time in Non-Uniformly Accelerated Motions	p. 70
6 General Two-Dimensional Hypercomplex Numbers	p. 73
6.1 Geometrical Representation	p. 73
6.2 Geometry and Trigonometry in Two-Dimensional Algebras	p. 76
6.2.1 The "Circle" for Three Points	p. 76
6.2.2 Hero's Formula and Pythagoras' Theorem	p. 77
6.2.3 Properties of "Orthogonal" Lines in General Algebras	p. 79
6.3 Some Properties of Fundamental Conic Sections	p. 79
6.3.1 "Incircles" and "Excircles" of a Triangle	p. 79
6.3.2 The Tangent Lines to the Fundamental Conic Section	p. 82
6.4 Numerical Examples	p. 83
7 Functions of a Hyperbolic Variable	p. 87
7.1 Some Remarks on Functions of a Complex Variable	p. 87
7.2 Functions of Hypercomplex Variables	p. 89
7.2.1 Generalized Cauchy-Riemann Conditions	p. 89
7.2.2 The Principal Transformation	p. 91
7.2.3 Functions of a Hypercomplex Variable as Infinite-Dimensional Lie Groups	p. 92
7.3 The Functions of a Hyperbolic Variable	p. 93
7.3.1 Cauchy-Riemann Conditions for General Two-Dimensional Systems	p. 93
7.3.2 The Derivative of Functions of a Canonical Hyperbolic Variable	p. 94
7.3.3 The Properties of H-Analytic Functions	p. 95
7.3.4 The Analytic Functions of Decomposable Systems	p. 95
7.4 The Elementary Functions of a Canonical Hyperbolic Variable	p. 96
7.5 H-Conformal Mappings	p. 97
7.5.1 H-Conformal Mappings by Means of Elementary Functions	p. 99
7.5.2 Hyperbolic Linear-Fractional Mapping	p. 109
7.6 Commutative Hypercomplex Systems with Three Unities	p. 114
7.6.1 Some Properties of the Three-Units Separable Systems	p. 115
8 Hyperbolic Variables on Lorentz Surfaces	p. 119
8.1 Introduction	p. 119
8.2 Gauss: Conformal Mapping of Surfaces	p. 121
8.2.1 Mapping of a Spherical Surface on a Plane	p. 123
8.2.2 Conclusions	p. 124
8.3 Extension of Gauss Theorem: Conformal Mapping of Lorentz Surfaces	p. 125
8.4 Beltrami: Complex Variables on a Surface	p. 126
8.4.1 Beltrami's Equation	p. 127
8.5 Beltrami's Integration of Geodesic Equations	p. 130
8.5.1 Differential Parameter and Geodesic Equations	p. 130
8.6 Extension of Beltrami's Equation to Non-Definite Differential Forms	p. 133
9 Constant Curvature Lorentz Surfaces	p. 137
9.1 Introduction	p. 137
9.2 Constant Curvature Riemann Surfaces	p. 140
9.2.1 Rotation Surfaces	p. 140
9.2.2 Positive Constant Curvature Surface	p. 143
9.2.3 Negative Constant Curvature Surface	p. 148
9.2.4 Motions	p. 149
9.2.5 Two-Sheets Hyperboloid in a Semi-Riemannian Space	p. 151
9.3 Constant Curvature Lorentz Surfaces	p. 153
9.3.1 Line Element	p. 153
9.3.2 Isometric Forms of the Line Elements	p. 153
9.3.3 Equations of the Geodesics	p. 154
9.3.4 Motions	p. 156
9.4 Geodesics and Geodesic Distances on Riemann and Lorentz Surfaces	p. 157
9.4.1 The Equation of the Geodesic	p. 157
9.4.2 Geodesic Distance	p. 159
10 Generalization of Two-Dimensional Special Relativity (Hyperbolic Transformations and the Equivalence Principle)	p. 161
10.1 The Physical Meaning of Transformations by Hyperbolic Functions	p. 161
10.2 Physical Interpretation of Geodesics on Riemann and Lorentz Surfaces with Positive Constant Curvature	p. 164
10.2.1 The Sphere	p. 165
10.2.2 The Lorentz Surfaces	p. 165
10.3 Einstein's Way to General Relativity	p. 166
10.4 Conclusions	p. 167
Appendices
A Commutative Segre's Quaternions	p. 169
A.1 Hypercomplex Systems with Four Units	p. 170
A.1.1 Historical Introduction of Segre's Quaternions	p. 171
A.1.2 Generalized Segre's Quaternions	p. 171
A.2 Algebraic Properties	p. 172
A.2.1 Quaternions as a Composed System	p. 176
A.3 Functions of a Quaternion Variable	p. 177
A.3.1 Holomorphic Functions	p. 178
A.3.2 Algebraic Reconstruction of Quaternion Functions Given a Component	p. 182
A.4 Mapping by Means of Quaternion Functions	p. 183
A.4.1 The "Polar" Representation of Elliptic and Hyperbolic Quaternions	p. 183
A.4.2 Conformal Mapping	p. 185
A.4.3 Some Considerations About Scalar and Vector Potentials	p. 186
A.5 Elementary Functions of Quaternions	p. 187
A.6 Elliptic-Hyperbolic Quaternions	p. 191
A.6.1 Generalized Cauchy-Riemann Conditions	p. 193
A.6.2 Elementary Functions	p. 193
A.7 Elliptic-Parabolic Generalized Segre's Quaternions	p. 194
A.7.1 Generalized Cauchy-Riemann conditions	p. 195
A.7.2 Elementary Functions	p. 196
B Constant Curvature Segre's Quaternion Spaces	p. 199
B.1 Quaternion Differential Geometry	p. 200
B.2 Euler's Equations for Geodesics	p. 201
B.3 Constant Curvature Quaternion Spaces	p. 203
B.3.1 Line Element for Positive Constant Curvature	p. 204
B.4 Geodesic Equations in Quaternion Space	p. 206
B.4.1 Positive Constant Curvature Quaternion Space	p. 210
C Matrix Formalization for Commutative Numbers	p. 213
C.1 Mathematical Operations	p. 213
C.1.1 Equality, Sum, and Scalar Multiplication	p. 214
C.1.2 Product and Related Operations	p. 215
C.1.3 Division Between Hypercomplex Numbers	p. 218
C.2 Two-dimensional Hypercomplex Numbers	p. 221
C.3 Properties of the Characteristic Matrix M	p. 222
C.3.1 Algebraic Properties	p. 223
C.3.2 Spectral Properties	p. 223
C.3.3 More About Divisors of Zero	p. 227
C.3.4 Modulus of a Hypercomplex Number	p. 227
C.3.5 Conjugations of a Hypercomplex Number	p. 227
C.4 Functions of a Hypercomplex Variable	p. 228
C.4.1 Analytic Continuation	p. 228
C.4.2 Properties of Hypercomplex Functions	p. 229
C.5 Functions of a Two-dimensional Hypercomplex Variable	p. 230
C.5.1 Function of 2 x 2 Matrices	p. 231
C.5.2 The Derivative of the Functions of a Real Variable	p. 233
C.6 Derivatives of a Hypercomplex Function	p. 236
C.6.1 Derivative with Respect to a Hypercomplex Variable	p. 236
C.6.2 Partial Derivatives	p. 237
C.6.3 Components of the Derivative Operator	p. 238
C.6.4 Derivative with Respect to the Conjugated Variables	p. 239
C.7 Characteristic Differential Equation	p. 239
C.7.1 Characteristic Equation for Two-dimensional Numbers	p. 241
C.8 Equivalence Between the Formalizations of Hypercomplex Numbers	p. 242
Bibliography	p. 245
Index	p. 251

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