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# Einstein's General Theory of Relativity

Einstein's General Theory of Relativity yvind Gr n and Sigbj rn Hervik Contents Preface xv Notation xvii I I NTRODUCTION : N EWTONIAN P HYSICS AND S PECIAL R ELATIVITY 1. 1 Relativity Principles and Gravitation 3. Newtonian mechanics .. 3. Galilei Newton's principle of Relativity .. 4. The principle of Relativity .. 5. Newton's law of Gravitation .. 6. Local form of Newton's Gravitational law .. 8. Tidal forces .. 10. The principle of equivalence .. 14. The covariance principle .. 15. Mach's principle .. 16. Problems .. 17. 2 The Special Theory of Relativity 21. Coordinate systems and Minkowski-diagrams .. 21. Synchronization of clocks .. 23. The Doppler effect .. 23. Relativistic time-dilatation .. 25. The Relativity of simultaneity .. 26. The Lorentz-contraction .. 28. The Lorentz transformation .. 30. Lorentz-invariant interval .. 32. The twin-paradox .. 34. Hyperbolic motion .. 35.

List of Problems Chapter 1 17 1.1 The strength of gravity compared to the Coulomb force . . . . 17 1.2 Falling objects in the gravitational eld of the Earth ...

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### Transcription of Einstein's General Theory of Relativity

1 Einstein's General Theory of Relativity yvind Gr n and Sigbj rn Hervik Contents Preface xv Notation xvii I I NTRODUCTION : N EWTONIAN P HYSICS AND S PECIAL R ELATIVITY 1. 1 Relativity Principles and Gravitation 3. Newtonian mechanics .. 3. Galilei Newton's principle of Relativity .. 4. The principle of Relativity .. 5. Newton's law of Gravitation .. 6. Local form of Newton's Gravitational law .. 8. Tidal forces .. 10. The principle of equivalence .. 14. The covariance principle .. 15. Mach's principle .. 16. Problems .. 17. 2 The Special Theory of Relativity 21. Coordinate systems and Minkowski-diagrams .. 21. Synchronization of clocks .. 23. The Doppler effect .. 23. Relativistic time-dilatation .. 25. The Relativity of simultaneity .. 26. The Lorentz-contraction .. 28. The Lorentz transformation .. 30. Lorentz-invariant interval .. 32. The twin-paradox .. 34. Hyperbolic motion .. 35.

2 Energy and mass .. 37. Relativistic increase of mass .. 38. Tachyons .. 39. Magnetism as a relativistic second-order effect .. 40. Problems .. 42. II T HE M ATHEMATICS OF THE. G ENERAL T HEORY OF R ELATIVITY 49. 3 Vectors, Tensors, and Forms 51. Vectors .. 51. iv Contents Four-vectors .. 52. One-forms .. 54. Tensors .. 55. Forms .. 57. Problems .. 60. 4 Basis Vector Fields and the Metric Tensor 63. Manifolds and their coordinate-systems .. 63. Tangent vector fields and the coordinate basis vector fields .. 65. Structure coefficients .. 71. General basis transformations .. 71. The metric tensor .. 73. Orthonormal basis .. 75. Spatial geometry .. 78. The tetrad field of a comoving coordinate system .. 80. The volume form .. 81. Dual forms .. 82. Problems .. 85. 5 Non-inertial Reference Frames 89. Spatial geometry in rotating reference frames .. 89. Ehrenfest's paradox .. 90. The Sagnac effect.

3 93. Gravitational time dilatation .. 94. Uniformly accelerated reference frame .. 95. Covariant Lagrangian dynamics .. 98. A General equation for the Doppler effect .. 103. Problems .. 107. 6 Differentiation, Connections and Integration 109. Exterior Differentiation of forms .. 109. Electromagnetism .. 113. Integration of forms .. 115. Covariant differentiation of vectors .. 120. Covariant differentiation of forms and tensors .. 128. Exterior differentiation of vectors .. 129. Covariant exterior derivative .. 133. Geodesic normal coordinates .. 136. One-parameter groups of diffeomorphisms .. 137. The Lie derivative .. 139. Killing vectors and Symmetries .. 143. Problems .. 146. 7 Curvature 149. Curves .. 149. Surfaces .. 151. The Riemann Curvature Tensor .. 153. Extrinsic and Intrinsic Curvature .. 159. The equation of geodesic deviation .. 162. Spaces of constant curvature .. 163. Problems.

4 170. Contents v III E INSTEIN ' S F IELD E QUATIONS 175. 8 Einstein's Field Equations 177. Deduction of Einstein's vacuum field equations from Hilbert's variational principle .. 177. The field equations in the presence of matter and energy .. 180. Energy-momentum conservation .. 181. Energy-momentum tensors .. 182. Some particular fluids .. 184. The paths of free point particles .. 188. Problems .. 188. 9 The Linear Field Approximation 191. The linearised field equations .. 191. The Newtonian limit of General Relativity .. 194. Solutions to the linearised field equations .. 195. Gravitoelectromagnetism .. 197. Gravitational waves .. 199. Gravitational radiation from sources .. 202. Problems .. 206. 10 The Schwarzschild Solution and Black Holes 211. The Schwarzschild solution for empty space .. 211. Radial free fall in Schwarzschild spacetime .. 216. The light-cone in a Schwarzschild spacetime.

5 217. Particle trajectories in Schwarzschild spacetime .. 221. Analytical extension of the Schwarzschild spacetime .. 226. Charged and rotating black holes .. 229. Black Hole thermodynamics .. 241. The Tolman-Oppenheimer-Volkoff equation .. 247. The interior Schwarzschild solution .. 249. Problems .. 251. IV C OSMOLOGY 259. 11 Homogeneous and Isotropic Universe Models 261. The cosmological principles .. 261. Friedmann-Robertson-Walker models .. 262. Dynamics of Homogeneous and Isotropic cosmologies .. 265. Cosmological redshift and the Hubble law .. 267. Radiation dominated universe models .. 272. Matter dominated universe models .. 275. The gravitational lens effect .. 277. Redshift-luminosity relation .. 283. Cosmological horizons .. 287. Bang in an infinite Universe .. 288. Problems .. 290. vi Contents 12 Universe Models with Vacuum Energy 297. Einstein's static universe .. 297. de Sitter's solution.

6 298. The de Sitter hyperboloid .. 301. The horizon problem and the flatness problem .. 302. Inflation .. 304. The Friedmann-Lema tre model .. 311. Universe models with quintessence energy .. 317. Dark energy and the statefinder diagnostic .. 320. Cosmic density perturbations .. 326. fluctuations in the CMB .. 331. History of our Universe .. 338. Problems .. 349. 13 An Anisotropic Universe 357. The Bianchi type I universe model .. 357. The Kasner solutions .. 360. The energy-momentum conservation law in an anisotropic uni- verse .. 361. Models with a perfect fluid .. 363. Inflation through bulk viscosity .. 366. A universe with a dissipative fluid .. 367. Problems .. 369. V A DVANCED T OPICS 373. 14 Covariant decomposition, Singularities, and Canonical Cosmology 375. Covariant decomposition .. 375. Equations of motion .. 378. Singularities .. 380. Lagrangian formulation of General Relativity .

7 385. Hamiltonian formulation .. 388. Canonical formulation with matter and energy .. 390. The space of three-metrics: Superspace .. 392. Problems .. 395. 15 Homogeneous Spaces 399. Lie groups and Lie algebras .. 399. Homogeneous spaces .. 402. The Bianchi models .. 405. The orthonormal frame approach to the Bianchi models .. 409. The 8 model geometries .. 414. Constructing compact quotients .. 416. Problems .. 419. 16 Israel's Formalism: The metric junction method 425. The relativistic Theory of surface layers .. 425. Einstein's field equations .. 427. Surface layers and boundary surfaces .. 429. Spherical shell of dust in vacuum .. 431. Problems .. 436. Contents vii 17 Brane-worlds 439. Field equations on the brane .. 439. Five-dimensional brane cosmology .. 442. Problem with perfect fluid brane world in an empty bulk .. 445. Solutions in the bulk .. 445. Towards a realistic brane cosmology.

8 447. Inflation in the brane .. 450. Dynamics of two branes .. 453. The hierarchy problem and the weakness of gravity .. 455. The Randall-Sundrum models .. 457. Problems .. 460. 18 Kaluza-Klein Theory 463. A fifth extra dimension .. 463. The Kaluza-Klein action .. 465. Implications of a fifth extra dimension .. 469. Conformal transformations .. 472. Conformal transformation of the Kaluza-Klein action .. 476. Kaluza-Klein cosmology .. 477. Problems .. 480. VI A PPENDICES 483. A Constants of Nature 485. B Penrose diagrams 487. Conformal transformations and causal structure .. 487. Schwarzschild spacetime .. 489. de Sitter spacetime .. 489. C Anti-de Sitter spacetime 493. The anti-de Sitter hyperboloid .. 493. Foliations of AdSn .. 494. Geodesics in AdSn .. 495. The BTZ black hole .. 496. AdS3 as the group SL(2, R) .. 497. D Suggested further reading 499. Bibliography 503. Index 511. List of Problems Chapter 1 17.

9 The strength of gravity compared to the Coulomb force .. 17. Falling objects in the gravitational field of the Earth .. 17. Newtonian potentials for spherically symmetric bodies .. 17. The Earth-Moon system .. 18. The Roche-limit .. 18. A Newtonian Black Hole .. 18. Non-relativistic Kepler orbits .. 19. Chapter 2 42. Two successive boosts in different directions .. 42. Length-contraction and time-dilatation .. 43. Faster than the speed of light? .. 44. Reflection angles off moving mirrors .. 44. Minkowski-diagram .. 44. Robb's Lorentz invariant spacetime interval formula .. 45. The Doppler effect .. 45. Abberation and Doppler effect .. 45. A traffic problem .. 46. The twin-paradox .. 46. Work and rotation .. 47. Muon experiment .. 47. Cerenkov radiation .. 47. Chapter 3 60. The tensor product .. 60. Contractions of tensors .. 60. Four-vectors .. 61. The Lorentz-Abraham-Dirac equation.

10 62. Chapter 4 85. Coordinate-transformations in a two-dimensional Euclidean plane .. 85. Covariant and contravariant components .. 86. The Levi-Civit symbol .. 86. Dual forms .. 87. Chapter 5 107. Geodetic curves in space .. 107. Free particle in a hyperbolic reference frame .. 107. Spatial geodesics in a rotating RF .. 108. x List of Problems Chapter 6 146. Loop integral of a closed form .. 146. The covariant derivative .. 147. The Poincar half-plane .. 147. The Christoffel symbols in a rotating reference frame with plane polar coordinates .. 147. Chapter 7 170. Rotation matrices .. 170. Inverse metric on S n .. 170. The curvature of a curve .. 170. The Gauss-Codazzi equations .. 171. The Poincar half-space .. 171. The pseudo-sphere .. 172. A non-Cartesian coordinate system in two dimensions .. 172. The curvature tensor of a sphere .. 172. The curvature scalar of a surface of simultaneity.