Lectures on General Relativity

This book is an elaboration of lecture notes for the graduate course on General Rela­ tivity given by the author at Boston University in the spring semester of 1972. It is an introduction to the subject only, as the time available for the course was limited. The author of an introduction to General Relativity is faced from the beginning with the difficult task of choosing which material to include. A general criterion as­ sisting in this choice is provided by the didactic character of the book: Those chapters have to be included in priority, which will be most useful to the reader in enabling him to understand the methods used in General Relativity, the results obtained so far and possibly the problems still to be solved. This criterion is not sufficient to ensure a unique choice. General Relativity has developed to such a degree, that it is impossible to include in an introductory textbook of a reasonable length even a very condensed treatment of all important problems which have been discussed until now and the author is obliged to decide, in a more or less subjective manner, which of the more recent developments to omit. The following lines indicate by means of some examples the kind of choice made in this book.

I/Tensor Calculus. - 1. Scalars and Vectors. Tensors. - 2. Algebraic Operations. Symmetry Properties. - 3. Tensor Densities. - Exercises. - II/Covariant Differentiation. - 4. Differentiation. - 5. The Connection. - 6. Rules for Covariant Differentiation. - 7. Parallel Transport. - 8. Geodesies. - 9. The Curvature Tensor. - Exercises. - III/Riemannian Geometry. - 10. Riemannian Space. - 11. The Determinant of g? ? . Metrical Densities. - 12. The Connection of a Riemannian Space: Christoffel Symbols. - 13. Geodesics in a Riemannian Space. - 14. The Curvature of a Riemannian Space: The Riemann Tensor. - 15. Algebraic Classification of the Weyl Tensor. - 16. Lie Derivatives. Isometries, Killing Vectors. - Exercises. - IV/The Gravitational Field. - 17. Introductory Remarks. - 18. The Principle of Equivalence. - 19. The Field Equations of General Relativity. - V/The Schwarzschild Solution. - 20. Metrics with Spherical Symmetry. - 21. The Schwarzschild Solution. Theorem of Birkhoff. - 22. Geodesies in the Schwarzschild Space. - 23. Advance of the Perihelion of a Planet. - 24. The Deflection of Light Rays. - 25. Red Shift of Spectral Lines. - 26. The Schwarzschild Sphere. The (Event) Horizon. Kruskal Coordinates. - 27. Gravitational Collapse. Black Holes. - Exercises. - VI /Some Other Exact Solutions. - 28. Fluid Without Pressure. Comoving Coordinates. - 29. The Tolman Solution. - 30. The Kerr Solution. - Exercises. - VII/Weak Gravitational Fields. - 31. The Linear Approximation. - 32. Applications. Mass and Angular Momentum. - Exercise. - VIII/Variational Principle. Identities, Conservation Laws. - 33. Variational Principle. - 34. Identities Corresponding to the Lagrangian L? . - 35. Conservation Laws. - Exercises. - IX/The Einstein-Maxwell Equations. - 36. Gravitational and Electromagnetic Field. - 37. The Reissner-NordströmSolution. - 38. The Singularity and the Horizon of the Reissner-Nordström Solution. - 39. Some Remarks on `Unified Field Theories . - Exercise. - X/Equations of Motion in General Relativity. - 40. The Problem of the Equations of Motion. - 41. The `Astronomical Problem. - 42. The Motion of Test Particles. - Exercise. - XI/Gravitational Radiation. - 43. Introductory Remarks. - 44. The Linear Approximation. - 45. The Gravitational Poynting Vector. - 46. Gravitational Multipoles. The 4-Pole Radiation. - 47. Gravitational Shock Waves. General Discussion. - 48. Local Structure and Propagation of Discontinuities. - 49. Radiation Coordinates (Bondi Coordinates). - 50. Null Tetrads. - 51. The Method of Newman and Penrose. - Exercises. - XII/The Cosmological Problem. - 52. Historical Survey. - 53. The Cosmological Principle. Form of the Metric. - 54. Some Cosmological Solutions. - Exercises.