1 Tensor Analysis on Differentiable Manifolds .................. 1
1.1 Differentiable Manifolds ................................ 1
1.2 Tensor Fields Over Differentiable Manifolds ............ 16
1.3 Riemannian and Pseudo-Riemannian Manifolds ............. 40
1.4 Extrinsic Curvature .................................... 88
2 The Pseudo-Riemannian Space-Time Manifold M4 ............... 105
2.1 Review of the Special Theory of Relativity ............ 105
2.2 Curved Space-Time and Gravitation ..................... 136
2.3 General Properties of Tij ............................. 174
2.4 Solution Strategies, Classification, and
Initial-Value Problems ................................ 195
2.5 Fluids, Deformable Solids, and Electromagnetic
Fields ................................................ 210
3 Spherically Symmetric Space-Time Domains ................... 229
3.1 Schwarzschild Solution ................................ 229
3.2 Spherically Symmetric Static Interior Solutions ....... 246
3.3 Nonstatic, Spherically Symmetric Solutions ............ 258
4 Static and Stationary Space-Time Domains ................... 277
4.1 Static Axially Symmetric Space-Time Domains ........... 277
4.2 The General Static Field Equations .................... 290
4.3 Axially Symmetric Stationary Space-Time Domains ....... 317
4.4 The General Stationary Field Equations ................ 331
5 Black Holes ................................................ 351
5.1 Spherically Symmetric Black Holes ..................... 351
5.2 Kerr Black Holes ...................................... 384
5.3 Exotic Black Holes .................................... 403
6 Cosmology .................................................. 419
6.1 Big Bang Models ....................................... 419
6.2 Scalar Fields in Cosmology ............................ 440
6.3 Five-Dimensional Cosmological Models .................. 456
7 Algebraic Classification of Field Equations ................ 465
7.1 The Petrov Classification of the Curvature Tensor ..... 465
7.2 Newman-Penrose Equations .............................. 503
8 The Coupled Einstein-Maxwell-Klein-Gordon Equations ........ 537
8.1 The General E-M-K-G Field Equations ................... 537
8.2 Static Space-Time Domains and the E-M-K-G Equations ... 542
8.3 Spherical Symmetry and a Nonlinear Eigenvalue
Problem for a Theoretical Fine-Structure Constant ..... 551
Appendix 1 Variational Derivation of Differential
Equations ......................................... 569
Appendix 2 Partial Differential Equations .................... 585
Appendix 3 Canonical Forms of Matrices ....................... 605
Appendix 4 Conformally Flat Space-Times and "the Fifth
Force" ............................................ 617
Appendix 5 Linearized Theory and Gravitational Waves ......... 625
Appendix 6 Exotic Solutions: Wormholes, Warp-Drives,
and Time Machines ................................. 633
Appendix 7 Gravitational Instantons .......................... 647
Appendix 8 Computational Symbolic Algebra Calculations ....... 653
References .................................................... 661
Index ......................................................... 669
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