Semi-Riemannian Geometry - The Mathematical Language of General Relativity

An introduction to semi-Riemannian geometry as a foundation for general relativity
 
Semi-Riemannian Geometry: The Mathematical Language of General Relativity is an accessible exposition of the mathematics underlying general relativity. The book begins with background on linear and multilinear algebra, general topology, and real analysis. This is followed by material on the classical theory of curves and surfaces, expanded to include both the Lorentz and Euclidean signatures. The remainder of the book is devoted to a discussion of smooth manifolds, smooth manifolds with boundary, smooth manifolds with a connection, semi-Riemannian manifolds, and differential operators, culminating in applications to Maxwell's equations and the Einstein tensor. Many worked examples and detailed diagrams are provided to aid understanding. This book will appeal especially to physics students wishing to learn more differential geometry than is usually provided in texts on general relativity.

Sommario

I Preliminaries 1
 
1 Vector Spaces 5
 
1.1 Vector Spaces 5
 
1.2 Dual Spaces 17
 
1.3 Pullback of Covectors 19
 
1.4 Annihilators 20
 
2 Matrices and Determinants 23
 
2.1 Matrices 23
 
2.2 Matrix Representations 27
 
2.3 Rank of Matrices 32
 
2.4 Determinant of Matrices 33
 
2.5 Trace and Determinant of Linear Maps 43
 
3 Bilinear Functions 45
 
3.1 Bilinear Functions 45
 
3.2 Symmetric Bilinear Functions 49
 
3.3 Flat Maps and Sharp Maps 51
 
4 Scalar Product Spaces 57
 
4.1 Scalar Product Spaces 57
 
4.2 Orthonormal Bases 62
 
4.3 Adjoints 65
 
4.4 Linear Isometries 68
 
4.5 Dual Scalar Product Spaces 72
 
4.6 Inner Product Spaces 75
 
4.7 Eigenvalues and Eigenvectors 81
 
4.8 Lorentz Vector Spaces 84
 
4.9 Time Cones 91
 
5 Tensors on Vector Spaces 97
 
5.1 Tensors 97
 
5.2 Pullback of Covariant Tensors 103
 
5.3 Representation of Tensors 104
 
5.4 Contraction of Tensors 106
 
6 Tensors on Scalar Product Spaces 113
 
6.1 Contraction of Tensors 113
 
6.2 Flat Maps 114
 
6.3 Sharp Maps 119
 
6.4 Representation of Tensors 123
 
6.5 Metric Contraction of Tensors 127
 
6.6 Symmetries of (0, 4)-Tensors 129
 
7 Multicovectors 133
 
7.1 Multicovectors 133
 
7.2 Wedge Products 137
 
7.3 Pullback of Multicovectors 144
 
7.4 Interior Multiplication 148
 
7.5 Multicovector Scalar Product Spaces 150
 
8 Orientation 155
 
8.1 Orientation of R¯m 155
 
8.2 Orientation of Vector Spaces 158
 
8.3 Orientation of Scalar Product Spaces 163
 
8.4 Vector Products 166
 
8.5 Hodge Star 178
 
9 Topology 183
 
9.1 Topology 183
 
9.2 Metric Spaces 193
 
9.3 Normed Vector Spaces 195
 
9.4 Euclidean Topology on R¯m 195
 
10 Analysis in R¯m 199
 
10.1 Derivatives 199
 
10.2 Immersions and Diffeomorphisms 207
 
10.3 Euclidean Derivative and Vector Fields 209
 
10.4 Lie Bracket 213
 
10.5 Integrals 218
 
10.6 Vector Calculus 221
 
II Curves and Regular Surfaces 223
 
11 Curves and Regular Surfaces in R³ 225
 
11.1 Curves in R³ 225
 
11.2 Regular Surfaces in R³ 226
 
11.3 Tangent Planes in R³ 237
 
11.4 Types of Regular Surfaces in R³ 240
 
11.5 Functions on Regular Surfaces in R³ 246
 
11.6 Maps on Regular Surfaces in R³ 248
 
11.7 Vector Fields along Regular Surfaces in R³ 252
 
12 Curves and Regular Surfaces in R³v 255
 
12.1 Curves in R³v 256
 
12.2 Regular Surfaces in R³v 257
 
12.3 Induced Euclidean Derivative in R³v 266
 
12.4 Covariant Derivative on Regular Surfaces in R³v 274
 
12.5 Covariant Derivative on Curves in R³v 282
 
12.6 Lie Bracket in R³v 285
 
12.7 Orientation in R³v 288
 
12.8 Gauss Curvature in R³v 292
 
12.9 Riemann Curvature Tensor in R³v 299
 
12.10 Computations for Regular Surfaces in R³v 310
 
13 Examples of Regular Surfaces 321
 
13.1 Plane in R³0 321
 
13.2 Cylinder in R³0 322
 
13.3 Cone in R³0 323
 
13.4 Sphere in R³0 324
 
13.5 Tractoid in R³0 325
 
13.6 Hyperboloid of One Sheet in R³0 326
 
13.7 Hyperboloid of Two Sheets in R³0 327
 
13.8 Torus in R³0 329
 
13.9 Pseudosphere in R³1 330
 
13.10 Hyperbolic Space in R³1 331
 

Info autore

STEPHEN C. NEWMAN is Professor Emeritus at the University of Alberta, Edmonton, Alberta, Canada. He is the author of Biostatistical Methods in Epidemiology and A Classical Introduction to Galois Theory, both published by Wiley.

Riassunto

An introduction to semi-Riemannian geometry as a foundation for general relativity

Semi-Riemannian Geometry: The Mathematical Language of General Relativity is an accessible exposition of the mathematics underlying general relativity. The book begins with background on linear and multilinear algebra, general topology, and real analysis. This is followed by material on the classical theory of curves and surfaces, expanded to include both the Lorentz and Euclidean signatures. The remainder of the book is devoted to a discussion of smooth manifolds, smooth manifolds with boundary, smooth manifolds with a connection, semi-Riemannian manifolds, and differential operators, culminating in applications to Maxwell's equations and the Einstein tensor. Many worked examples and detailed diagrams are provided to aid understanding. This book will appeal especially to physics students wishing to learn more differential geometry than is usually provided in texts on general relativity.

Dettagli sul prodotto

• Autori: Sc Newman, Stephen C Newman, Stephen C. Newman, Stephen C. (University of Alberta Newman, Newman Stephen C.
• Editore: Wiley, John and Sons Ltd
• Lingue: Inglese
• Formato: Copertina rigida
• Pubblicazione: 31.07.2019
• EAN: 9781119517535
• ISBN: 978-1-119-51753-5
• Pagine: 656
• Categorie: Scienze naturali, medicina, informatica, tecnica > Matematica > Geometria; Physik, Mathematik, Geometrie, Mathematische Physik, Riemannsche Geometrie, Mathematics, Differentialgleichungen, Physics, Differential equations, Geometrie u. Topologie, Geometry & Topology, Mathematical & Computational Physics