This elementary introduction pays special attention to aspects of tensor calculus and relativity that students tend to find most difficult. Its use of relatively unsophisticated mathematics in the early chapters allows readers to develop their confidence within the framework of Cartesian coordinates before undertaking the theory of tensors in curved spaces and its application to general relativity theory.
Topics include the special principle of relativity and Lorentz transformations; orthogonal transformations and Cartesian tensors; special relativity mechanics and electrodynamics; general tensor calculus and Riemannian space; and the general theory of relativity, including a focus on black holes and gravitational waves. The text concludes with a chapter offering a sound background in applying the principles of general relativity to cosmology. Numerous exercises advance the theoretical developments of the main text, thus enhancing this volume’s appeal to students of applied mathematics and physics at both undergraduate and postgraduate levels. Preface. List of Constants. References. Bibliography.
Preface List of Constants Chapter 1 Special Principle of Relativity. Lorentz Transformations 1. Newton's laws of motion 2. Covariance of the laws of motion 3. Special principle of relativity 4. Lorentz transformations. Minkowski space-time 5. The special Lorentz transformation 6. Fitzgerald contraction. Time dilation 7. Spacelike and timelike intervals. Light cone Exercises 1 Chapter 2 Orthogonal Transformations. Cartesian Tensors 8. Orthogonal transformations 9. Repeated-index summation convention 10. Rectangular Cartesian tensors 11. Invariants. Gradients. Derivatives of tensors 12. Contraction. Scalar product. Divergence 13. Pseudotensors 14. Vector products. Curl Exercises 2 Chapter 3 Special Relativity Mechanics 15. The velocity vector 16. Mass and momentum 17. The force vector. Energy 18. Lorentz transformation equations for force 19. Fundamental particles. Photon and neutrino 20. Lagrange's and Hamilton's equations 21. Energy-momentum tensor 22. Energy-momentum tensor for a fluid 23. Angular momentum Exercises 3 Chapter 4 Special Relativity Electrodynamics 24. 4-Current density 25. 4-Vector potential 26. The field tensor 27. Lorentz transformations of electric and magnetic vectors 28. The Lorentz force 29. The engery-momentum tensor for an electromagnetic field Exercises 4 Chapter 5 General Tensor Calculus. Riemannian Space 30. Generalized N-dimensional spaces 31. Contravariant and covariant tensors 32. The quotient theorem. Conjugate tensors 33. Covariant derivatives. Parallel displacement. Affine connection 34. Transformation of an affinity 35. Covariant derivatives of tensors 36. The Riemann-Christoffel curvature tensor 37. Metrical connection. Raising and lowering indices 38. Scalar products. Magnitudes of vectors 39. Geodesic frame. Christoffel symbols 40. Bianchi identity 41. The covariant curvature tensor 42. Divergence. The Laplacian. Einstein's tensor 43. Geodesics Exercises 5 Chapter 6 General Theory of Relativity 44. Principle of equivalence 45. Metric in a gravitational field 46. Motion of a free particle in a gravitational field 47. Einstein's law of gravitation 48. Acceleration of a particle in a weak gravitational field 49. Newton's law of gravitation 50. Freely falling dust cloud 51. Metrics with spherical symmetry 52. Schwarzchild's solution 53. Planetary orbits 54. Gravitational deflection of a light ray 55. Gravitational displacement of spectral lines 56. Maxwell's equations in a gravitational field 57. Black holes 58. Gravitational waves Exercises 6 Chapter 7 Cosmology 59. Cosmological principle. Cosmical time 60. Spaces of constant curvature 61. The Robertson-Walker metric 62. Hubble's constant and the deceleration parameter 63. Red shifts of galaxies 64. Luminosity distance 65. Cosmic dynamics 66. Model universes of Einstein and de Sitter 67. Friedmann universes 68. Radiation model 69. Particle and event horizons Exercises 7 References Bibliography Index
