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.
Table of Contents
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 Rm 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 Rm 195
10 Analysis in Rm 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
III Smooth Manifolds and Semi-Riemannian Manifolds 333
14 Smooth Manifolds 337
14.1 Smooth Manifolds 337
14.2 Functions and Maps 340
14.3 Tangent Spaces 344
14.4 Differential of Maps 351
14.5 Differential of Functions 353
14.6 Immersions and Diffeomorphisms 357
14.7 Curves 358
14.8 Submanifolds 360
14.9 Parametrized Surfaces 364
15 Fields on Smooth Manifolds 367
15.1 Vector Fields 367
15.2 Representation of Vector Fields 372
15.3 Lie Bracket 374
15.4 Covector Fields 376
15.5 Representation of Covector Fields 379
15.6 Tensor Fields 382
15.7 Representation of Tensor Fields 385
15.8 Differential Forms 387
15.9 Pushforward and Pullback of Functions 389
15.10 Pushforward and Pullback of Vector Fields 391
15.11 Pullback of Covector Fields 393
15.12 Pullback of Covariant Tensor Fields 398
15.13 Pullback of Differential Forms 401
15.14 Contraction of Tensor Fields 405
16 Differentiation and Integration on Smooth Manifolds 407
16.1 Exterior Derivatives 407
16.2 Tensor Derivations 413
16.3 Form Derivations 417
16.4 Lie Derivative 419
16.5 Interior Multiplication 423
16.6 Orientation 425
16.7 Integration of Differential Forms 432
16.8 Line Integrals 435
16.9 Closed and Exact Covector Fields 437
16.10 Flows 443
17 Smooth Manifolds with Boundary 449
17.1 Smooth Manifolds with Boundary 449
17.2 Inward-Pointing and Outward-Pointing Vectors 452
17.3 Orientation of Boundaries 456
17.4 Stokes's Theorem 459
18 Smooth Manifolds with a Connection 463
18.1 Covariant Derivatives 463
18.2 Christoffel Symbols 466
18.3 Covariant Derivative on Curves 472
18.4 Total Covariant Derivatives 476
18.5 Parallel Translation 479
18.6 Torsion Tensors 485
18.7 Curvature Tensors 488
18.8 Geodesics 497
18.9 Radial Geodesics and Exponential Maps 502
18.10 Normal Coordinates 507
18.11 Jacobi Fields 509
19 Semi-Riemannian Manifolds 515
19.1 Semi-Riemannian Manifolds 515
19.2 Curves 519
19.3 Fundamental Theorem of Semi-Riemannian Manifolds 519
19.4 Flat Maps and Sharp Maps 526
19.5 Representation of Tensor Fields 529
19.6 Contraction of Tensor Fields 532
19.7 Isometries 535
19.8 Riemann Curvature Tensor 539
19.9 Geodesics 546
19.10 Volume Forms 550
19.11 Orientation of Hypersurfaces 551
19.12 Induced Connections 558
20 Differential Operators on Semi-Riemannian Manifolds 561
20.1 Hodge Star 561
20.2 Codifferential 562
20.3 Gradient 566
20.4 Divergence of Vector Fields 568
20.5 Curl 572
20.6 Hesse Operator 573
20.7 Laplace Operator 575
20.8 Laplace-de Rham Operator 576
20.9 Divergence of Symmetric 2-Covariant Tensor Fields 577
21 Riemannian Manifolds 579
21.1 Geodesics and Curvature on Riemannian Manifolds 579
21.2 Classical Vector Calculus Theorems 582
22 Applications to Physics 587
22.1 Linear Isometries on Lorentz Vector Spaces 587
22.2 Maxwell's Equations 598
22.3 Einstein Tensor 603
IV Appendices 609
A Notation and Set Theory 611
B Abstract Algebra 617
B.1 Groups 617
B.2 Permutation Groups 618
B.3 Rings 623
B.4 Fields 623
B.5 Modules 624
B.6 Vector Spaces 625
B.7 Lie Algebras 626
Further Reading 627
Index 629
