General Relativity, Astrophysics, and Cosmology, Softcover reprint of the original 1st ed. 1992 Astronomy and Astrophysics Library Series

For about half a century the general theory of relativity attracted little attention from physicists. However, the discovery of compact objects such as quasars and pulsars, as well as candidates for black holes on the one hand, and the microwave background radiation on the other hand completely changed the picture. In addition, developments in elementary particle physics, such as predictions of the behavior of matter at the ultrahigh energies that might have prevailed in the early stages of the big bang, have greatly en­ hanced the interest in general relativity. These developments created a large body of readers interested in general relativity, and its applications in astrophysics and cosmology. Having neither the time nor the inclination to delve deeply into the technical literature, such readers need a general introduction to the subject before exploring applica­ tions. It is for these readers that the present volume is intended. Keeping in mind the broad range of interests and wanting to avoid mathematical compli­ cations as much as possible, we have ventured to combine all three topics­ relativity, astrophysics, and cosmology-in a single volume. Naturally, we had to make a careful selection of topics to be discussed in order to keep the book to a manageable length.

I. The General Theory of Relativity.- 1. Introduction.- 1.1. The Case for Nonflat Space—Time.- 1.2. The Principle of Equivalence.- 1.3. Conflict Between the Equivalence Principle and the Pseudo-Euclidean Metric: Gravitational Redshift.- 1.4. A Fifth Force.- 2. Tensor Calculus and Riemannian Geometry.- 2.1. Riemannian Geometry and the Metric Tensor.- 2.2. Vectors and Tensors.- 2.3. Invariant Volume and Volume Integral.- 2.4. Affine Connection—Parallel Transport.- 2.5. Covariant Differentiation.- 2.6. The Differential Equation of a Geodesic.- 2.7. The Integrability of Parallel Displacement.- 2.8. The Riemann—Christoffel Tensor.- 2.9. The Bianchi Identity.- 2.10. The Ricci Tensor and the Einstein Tensor.- 2.11. The Weyl Tensor.- 2.12. Geodesic Deviation.- 3. Einstein’s Field Equations.- 3.1. Einstein’s Formulation of the Field Equations.- 3.2. Weak Field Approximation (Static Case).- 3.3. Gravitational Waves in Weak Field Approximation.- 3.4. Detection of Gravitational Waves.- 3.5. Integration of the Linearized Equations for a Stationary Axially Symmetric Distribution.- 3.6. The Action Principle and the Energy—Momentum Tensors.- 3.7. The Energy—Stress Tensor.- 3.8. The Einstein Equations from the Variational Principle.- 4. The Schwarzschild Metric and Crucial Tests.- 4.1. The Schwarzschild Solution.- 4.2. Birkhoff’s Theorem.- 4.3. Three Crucial Tests.- 4.4. The PPN Formalism.- 4.5. The Schwarzschild or the Spherically Symmetric Black Hole.- 4.6. Frequency Shift of Spectral Lines of Light Emitted by a Collapsing/Exploding Spherical Body.- 4.7. Fall in Apparent Luminosity of a Collapsing Body.- 4.8. Kruskal—Szekeres Coordinates.- 4.9. Historical Note on the Schwarzschild Black Hole.- 5. Electromagnetism in General Relativity.- 5.1. Introduction.- 5.2. The Field of a Charged Particle.- 5.3. Static Electrovac.- 5.4. The Already Unified Field Theory.- 6. Axially Symmetric Fields.- 6.1. The Lie Derivative and the Killing Equation.- 6.2. Static and Stationary Metrics.- 6.3. The Axially Symmetric Static Metric.- 6.4. Weyl’s Canonical Form.- 6.5. The Case of Two Mass Particles.- 6.6. The Schwarzschild Metric in the Form (6.21).- 6.7. Stationary Axisymmetric Vacuum Solutions.- 7. The Kerr Metric or the Rotating Black Hole.- 7.1. The Kerr Metric in Boyer—Lindquist Coordinates.- 7.2. The Black Hole Property.- 7.3. Locally Nonrotating Observers.- 7.4. The Horizon as a Null Surface.- 7.5. The Kerr—Newmann Metric.- 7.6. The Penrose Process.- 8. The Energy—Momentum Pseudotensor of the Gravitational Field and Loss of Energy by Gravitational Radiation.- 8.1. The Pseudo-Energy—Momentum Tensor.- 8.2. Historical Note.- 8.3. Loss of Energy by Gravitational Radiation.- 8.4. The Case of a Binary Star.- 9. Analysis of the Observational Data of the Hulse—Taylor Pulsar. Confirmation of the Einstein Quadrupole Radiation Formula.- II. Relativistic Astrophysics.- 10. White Dwarf Stars.- 10.1. Introduction.- 10.2. The Contraction of a Radiating Star in the Absence of Energy Generation.- 10.3. Degeneracy and the Equation of State.- 10.4. Limiting Mass for White Dwarfs.- 10.5. A Simple Argument for the Mass Limit.- 10.6. Critique of Chandrasekhar’s Result and Later Works.- 10.7. Historical Note.- 10.8. Observational Data on White Dwarfs.- 10.9. The Cooling and Age of White Dwarfs.- 11. Stellar Evolution, Supernovae, and Compact Objects.- 11.1. Introduction.- 11.2. The Evolution of Stars.- 11.3. The Dynamical Collapse.- 11.4. Some Numerical Results.- 11.5. Explosive Processes.- 11.6. Supernova 1987 A.- 12. Pulsars.- 12.1. Introduction.- 12.2. Distance from Dispersion Measure.- 12.3. Identification of Pulsars as Neutron Stars.- 12.4. The Energetics of Pulsar Emission.- 12.5. The Magnetic Field at the Pulsar Surface.- 12.6. The Age of Pulsars.- 12.7. Calculation of the Braking Index.- 12.8. The Nonvacuum Model.- 12.9. Observational Determination of Pulsar Masses.- 12.10. Cooling of Neutron Stars—Theory and Observation.- 12.11. The Influence of Superfluidity.- 12.12. The Influence of Pion Condensation.- 12.13. The Influence of Quarks.- 13. Spherically Symmetric Star Models.- 13.1. Introduction.- 13.2. The Tolman, Oppenheimer—Volkoff Equation.- 13.3. The Equation of State for Cold Catalyzed Matter.- 13.4. A Model of a Neutron Star and the Mass Limits.- 13.5. The Problems of the Upper Mass Limit of Neutron Stars.- 13.6. The Influence of Rotation, etc., on the Mass Limit.- 13.7. Note on the Stability of Compact Objects.- 14. Black Holes.- 14.1. Introduction.- 14.2. The No-Hair Theorem.- 14.3. The Laws of Black Hole Physics.- 14.4. Black Hole Thermodynamics.- 14.5. The Identification of a Black Hole—Cygnus X-1.- 14.6. The Possible Locale of the Occurrence of Black Holes.- 14.7. The Quasi-Steller Objects (Quasars).- 14.8. Gravitational Lens.- 15. Accretion onto Compact Objects.- 15.1. Introduction—Spherically Symmetric Accretion.- 15.2. Disk Accretion.- 15.3. Compact X-Ray Sources.- III. Cosmology.- 16. The Standard Cosmological Model.- 16.1. Introduction to the Friedmann Metric.- 16.2. Elementary Discussion of Standard Cosmology.- 16.3. The Observational Background of Cosmology.- 16.4. Summary.- 17. The Singularity Problem.- 17.1. Introduction.- 17.2. The Raychaudhuri Equation.- 17.3. The Meaning of Shear, Vorticity, and Expansion.- 17.4. An Elementary Singularity Theorem.- 17.5. The Gödel Universe.- 17.6. General Singularity Theorems.- 18. Thermal History of the Universe—Cosmological Nucleosynthesis.- 18.1. The Thermal History.- 18.2. Cosmological Nucleosynthesis.- 19. Structure Formation in the Universe.- 19.1. The Problem.- 19.2. The Linear Growth Formula.- 19.3. Finite Perturbation.- 19:4. Structure Formation with Dark Matter.- 20. Grand Unified Theory and Spontaneous Symmetry Breaking.- 20.1. Introduction.- 20.2. Gauge Fields.- 20.3. Weak Interaction.- 20.4. Strong Interaction and Grand Unification.- 20.5. Baryon Asymmetry and the Baryon/Photon Ratio.- 21. The Inflationary Scenario.- 21.1. Introduction.- 21.2. The Problems in Terms of Entropy.- 21.3. The Vacuum Energy—Stress Tensor and the de Sitter Phase.- 21.4. The Different Models of Inflation.- 21.5. A Critique of the Inflationary Models.- 21.6. Fluctuations in the Inflationary Models.- 22. Concluding Remarks.- Appendix. Differential Forms.- A.1. Introductory Ideas and Definitions.- A.2. Connection 1-Forms and Ricci Rotation Coefficients.- A.3. Cartan’s Equations of Structure.- A.4. Bianchi Identities and Symmetry Properties of the Riemann—Christoffel Tensor.- A.5. An Example of the Calculation of the Riemann—Christoffel Tensor.- References.