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.

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.

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 R³_v 255

12.1 Curves in R³_v 256

12.2 Regular Surfaces in R³_v 257

12.3 Induced Euclidean Derivative in R³_v 266

12.4 Covariant Derivative on Regular Surfaces in R³_v 274

12.5 Covariant Derivative on Curves in R³_v 282

12.6 Lie Bracket in R³_v 285

12.7 Orientation in R³_v 288

12.8 Gauss Curvature in R³_v 292

12.9 Riemann Curvature Tensor in R³_v 299

12.10 Computations for Regular Surfaces in R³_v 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

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.