Semi-Riemannian Geometry: The Mathematical Language of General Relativity

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 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