Lectures on General Relativity

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.