Jerzy Plebanski, Andrzej Krasinski - An Introduction to General Relativity and Cosmology-Cambridge U

This preview shows page 1 out of 556 pages.

Unformatted text preview: This page intentionally left blank An Introduction to General Relativity and Cosmology General relativity is a cornerstone of modern physics, and is of major importance in its applications to cosmology. Experts in the field Pleba´nski and Krasi´nski provide a thorough introduction to general relativity to guide the reader through complete derivations of the most important results. An Introduction to General Relativity and Cosmology is a unique text that presents a detailed coverage of cosmology as described by exact methods of relativity and inhomogeneous cosmological models. Geometric, physical and astrophysical properties of inhomogeneous cosmological models and advanced aspects of the Kerr metric are all systematically derived and clearly presented so that the reader can follow and verify all details. The book contains a detailed presentation of many topics that are not found in other textbooks. This textbook for advanced undergraduates and graduates of physics and astronomy will enable students to develop expertise in the mathematical techniques necessary to study general relativity. An Introduction to General Relativity and Cosmology Jerzy Pleba´nski Centro de Investigación y de Estudios Avanzados Instituto Politécnico Nacional Apartado Postal 14-740, 07000 México D.F., Mexico Andrzej Krasi´nski Centrum Astronomiczne im. M. Kopernika, Polska Akademia Nauk, Bartycka 18, 00 716 Warszawa, Poland Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , UK Published in the United States of America by Cambridge University Press, New York Information on this title: © J. Plebanski and A. Krasi nski 2006 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2006 - - ---- eBook (EBL) --- eBook (EBL) - - ---- hardback --- hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents List of figures The scope of this text Acknowledgements 1 How 1.1 1.2 1.3 1.4 Part I page xiii xvii xix the theory of relativity came into being (a brief historical sketch) Special versus general relativity Space and inertia in Newtonian physics Newton’s theory and the orbits of planets The basic assumptions of general relativity 1 1 1 2 4 Elements of differential geometry 7 2 A short sketch of 2-dimensional differential geometry 2.1 Constructing parallel straight lines in a flat space 2.2 Generalisation of the notion of parallelism to curved surfaces 9 9 10 3 Tensors, tensor densities 3.1 What are tensors good for? 3.2 Differentiable manifolds 3.3 Scalars 3.4 Contravariant vectors 3.5 Covariant vectors 3.6 Tensors of second rank 3.7 Tensor densities 3.8 Tensor densities of arbitrary rank 3.9 Algebraic properties of tensor densities 3.10 Mappings between manifolds 3.11 The Levi-Civita symbol 3.12 Multidimensional Kronecker deltas 3.13 Examples of applications of the Levi-Civita symbol and of the multidimensional Kronecker delta 3.14 Exercises 13 13 13 15 15 16 16 17 18 18 19 22 23 v 24 25 vi Contents 4 Covariant derivatives 4.1 Differentiation of tensors 4.2 Axioms of the covariant derivative 4.3 A field of bases on a manifold and scalar components of tensors 4.4 The affine connection 4.5 The explicit formula for the covariant derivative of tensor density fields 4.6 Exercises 26 26 28 29 30 31 32 5 Parallel transport and geodesic lines 5.1 Parallel transport 5.2 Geodesic lines 5.3 Exercises 33 33 34 35 6 The curvature of a manifold; flat manifolds 6.1 The commutator of second covariant derivatives 6.2 The commutator of directional covariant derivatives 6.3 The relation between curvature and parallel transport 6.4 Covariantly constant fields of vector bases 6.5 A torsion-free flat manifold 6.6 Parallel transport in a flat manifold 6.7 Geodesic deviation 6.8 Algebraic and differential identities obeyed by the curvature tensor 6.9 Exercises 36 36 38 39 43 44 44 45 46 47 7 Riemannian geometry 7.1 The metric tensor 7.2 Riemann spaces 7.3 The signature of a metric, degenerate metrics 7.4 Christoffel symbols 7.5 The curvature of a Riemann space 7.6 Flat Riemann spaces 7.7 Subspaces of a Riemann space 7.8 Flat Riemann spaces that are globally non-Euclidean 7.9 The Riemann curvature versus the normal curvature of a surface 7.10 The geodesic line as the line of extremal distance 7.11 Mappings between Riemann spaces 7.12 Conformally related Riemann spaces 7.13 Conformal curvature 7.14 Timelike, null and spacelike intervals in a 4-dimensional spacetime 7.15 Embeddings of Riemann spaces in Riemann spaces of higher dimension 7.16 The Petrov classification 7.17 Exercises 48 48 49 49 51 51 52 53 53 54 55 56 56 58 61 63 70 72 Contents vii 8 Symmetries of Riemann spaces, invariance of tensors 8.1 Symmetry transformations 8.2 The Killing equations 8.3 The connection between generators and the invariance transformations 8.4 Finding the Killing vector fields 8.5 Invariance of other tensor fields 8.6 The Lie derivative 8.7 The algebra of Killing vector fields 8.8 Surface-forming vector fields 8.9 Spherically symmetric 4-dimensional Riemann spaces 8.10 * Conformal Killing fields and their finite basis 8.11 * The maximal dimension of an invariance group 8.12 Exercises 74 74 75 77 78 79 80 81 81 82 86 89 91 9 Methods to calculate the curvature quickly – Cartan forms and algebraic computer programs 9.1 The basis of differential forms 9.2 The connection forms 9.3 The Riemann tensor 9.4 Using computers to calculate the curvature 9.5 Exercises 94 94 95 96 98 98 10 The spatially homogeneous Bianchi type spacetimes 10.1 The Bianchi classification of 3-dimensional Lie algebras 10.2 The dimension of the group versus the dimension of the orbit 10.3 Action of a group on a manifold 10.4 Groups acting transitively, homogeneous spaces 10.5 Invariant vector fields 10.6 The metrics of the Bianchi-type spacetimes 10.7 The isotropic Bianchi-type (Robertson–Walker) spacetimes 10.8 Exercises 99 99 104 105 105 106 108 109 112 11 * The Petrov classification by the spinor method 11.1 What is a spinor? 11.2 Translating spinors to tensors and vice versa 11.3 The spinor image of the Weyl tensor 11.4 The Petrov classification in the spinor representation 11.5 The Weyl spinor represented as a 3 × 3 complex matrix 11.6 The equivalence of the Penrose classes to the Petrov classes 11.7 The Petrov classification by the Debever method 11.8 Exercises 113 113 114 116 116 117 119 120 122 viii Part II Contents The theory of gravitation 12 The Einstein equations and the sources of a gravitational field 12.1 Why Riemannian geometry? 12.2 Local inertial frames 12.3 Trajectories of free motion in Einstein’s theory 12.4 Special relativity versus gravitation theory 12.5 The Newtonian limit of relativity 12.6 Sources of the gravitational field 12.7 The Einstein equations 12.8 Hilbert’s derivation of the Einstein equations 12.9 The Palatini variational principle 12.10 The asymptotically Cartesian coordinates and the asymptotically flat spacetime 12.11 The Newtonian limit of Einstein’s equations 12.12 Examples of sources in the Einstein equations: perfect fluid and dust 12.13 Equations of motion of a perfect fluid 12.14 The cosmological constant 12.15 An example of an exact solution of Einstein’s equations: a Bianchi type I spacetime with dust source 12.16 * Other gravitation theories 12.16.1 The Brans–Dicke theory 12.16.2 The Bergmann–Wagoner theory 12.16.3 The conformally invariant Canuto theory 12.16.4 The Einstein–Cartan theory 12.16.5 The bi-metric Rosen theory 12.17 Matching solutions of Einstein’s equations 12.18 The weak-field approximation to general relativity 12.19 Exercises 123 125 125 125 126 129 130 130 131 132 136 136 136 140 143 144 145 149 149 150 150 150 151 151 154 160 13 The Maxwell and Einstein–Maxwell equations and the Kaluza–Klein theory 13.1 The Lorentz-covariant description of electromagnetic field 13.2 The covariant form of the Maxwell equations 13.3 The energy-momentum tensor of an electromagnetic field 13.4 The Einstein–Maxwell equations 13.5 * The variational principle for the Einstein–Maxwell equations 13.6 * The Kaluza–Klein theory 13.7 Exercises 161 161 161 162 163 164 164 167 14 Spherically symmetric gravitational fields of isolated objects 14.1 The curvature coordinates 14.2 Symmetry inheritance 168 168 172 Contents 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13 14.14 14.15 14.16 14.17 Spherically symmetric electromagnetic field in vacuum The Schwarzschild and Reissner–Nordström solutions Orbits of planets in the gravitational field of the Sun Deflection of light rays in the Schwarzschild field Measuring the deflection of light rays Gravitational lenses The spurious singularity of the Schwarzschild solution at r = 2m * Embedding the Schwarzschild spacetime in a flat Riemannian space Interpretation of the spurious singularity at r = 2m; black holes The Schwarzschild solution in other coordinate systems The equation of hydrostatic equilibrium The ‘interior Schwarzschild solution’ * The maximal analytic extension of the Reissner–Nordström solution * Motion of particles in the Reissner–Nordström spacetime with e2 < m2 Exercises ix 172 173 176 183 186 189 191 196 200 202 203 206 207 217 219 15 Relativistic hydrodynamics and thermodynamics 15.1 Motion of a continuous medium in Newtonian mechanics 15.2 Motion of a continuous medium in relativistic mechanics 15.3 The equations of evolution of     and u˙  ; the Raychaudhuri equation 15.4 Singularities and singularity theorems 15.5 Relativistic thermodynamics 15.6 Exercises 222 222 224 16 Relativistic cosmology I: general geometry 16.1 A continuous medium as a model of the Universe 16.2 Optical observations in the Universe – part I 16.2.1 The geometric optics approximation 16.2.2 The redshift 16.3 The optical tensors 16.4 The apparent horizon 16.5 * The double-null tetrad 16.6 * The Goldberg–Sachs theorem 16.7 * Optical observations in the Universe – part II 16.7.1 The area distance 16.7.2 The reciprocity theorem 16.7.3 Other observable quantities 16.8 Exercises 235 235 237 237 239 240 242 243 245 253 253 256 259 260 228 230 231 234 x Contents 17 Relativistic cosmology II: the Robertson–Walker geometry 17.1 The Robertson–Walker metrics as models of the Universe 17.2 Optical observations in an R–W Universe 17.2.1 The redshift 17.2.2 The redshift–distance relation 17.2.3 Number counts 17.3 The Friedmann equations and the critical density 17.4 The Friedmann solutions with  = 0 17.4.1 The redshift–distance relation in the  = 0 Friedmann models 17.5 The Newtonian cosmology 17.6 The Friedmann solutions with the cosmological constant 17.7 Horizons in the Robertson–Walker models 17.8 The inflationary models and the ‘problems’ they solved 17.9 The value of the cosmological constant 17.10 The ‘history of the Universe’ 17.11 Invariant definitions of the Robertson–Walker models 17.12 Different representations of the R–W metrics 17.13 Exercises 18 Relativistic cosmology III: the Lemaître–Tolman geometry 18.1 The comoving–synchronous coordinates 18.2 The spherically symmetric inhomogeneous models 18.3 The Lemaître–Tolman model 18.4 Conditions of regularity at the centre 18.5 Formation of voids in the Universe 18.6 Formation of other structures in the Universe 18.6.1 Density to density evolution 18.6.2 Velocity to density evolution 18.6.3 Velocity to velocity evolution 18.7 The influence of cosmic expansion on planetary orbits 18.8 * Apparent horizons in the L–T model 18.9 * Black holes in the evolving Universe 18.10 * Shell crossings and necks/wormholes 18.10.1 E < 0 18.10.2 E = 0 18.10.3 E > 0 18.11 The redshift 18.12 The influence of inhomogeneities in matter distribution on the cosmic microwave background radiation 18.13 Matching the L–T model to the Schwarzschild and Friedmann solutions 261 261 263 263 265 265 266 269 270 271 273 277 282 286 287 290 291 293 294 294 294 296 300 301 303 304 306 308 309 311 316 321 325 327 327 328 330 332 Contents 18.14 * General properties of the Big Bang/Big Crunch singularities in the L–T model 18.15 * Extending the L–T spacetime through a shell crossing singularity 18.16 * Singularities and cosmic censorship 18.17 Solving the ‘horizon problem’ without inflation 18.18 * The evolution of R t M versus the evolution of t M 18.19 * Increasing and decreasing density perturbations 18.20 * L&T curio shop 18.20.1 Lagging cores of the Big Bang 18.20.2 Strange or non-intuitive properties of the L–T model 18.20.3 Chances to fit the L–T model to observations 18.20.4 An ‘in one ear and out the other’ Universe 18.20.5 A ‘string of beads’ Universe 18.20.6 Uncertainties in inferring the spatial distribution of matter 18.20.7 Is the matter distribution in our Universe fractal? 18.20.8 General results related to the L–T models 18.21 Exercises 19 Relativistic cosmology IV: generalisations of L–T and related geometries 19.1 The plane- and hyperbolically symmetric spacetimes 19.2 G3 /S2 -symmetric dust solutions with Rr = 0 19.3 G3 /S2 -symmetric dust in electromagnetic field, the case Rr = 0 19.3.1 Integrals of the field equations 19.3.2 Matching the charged dust metric to the Reissner–Nordström metric 19.3.3 Prevention of the Big Crunch singularity by electric charge 19.3.4 * Charged dust in curvature and mass-curvature coordinates 19.3.5 Regularity conditions at the centre 19.3.6 * Shell crossings in charged dust 19.4 The Datt–Ruban solution 19.5 The Szekeres–Szafron family of solutions 19.5.1 The z = 0 subfamily 19.5.2 The z = 0 subfamily 19.5.3 Interpretation of the Szekeres–Szafron coordinates 19.5.4 Common properties of the two subfamilies 19.5.5 * The invariant definitions of the Szekeres–Szafron metrics 19.6 The Szekeres solutions and their properties 19.6.1 The z = 0 subfamily 19.6.2 The z = 0 subfamily 19.6.3 * The z = 0 family as a limit of the z = 0 family 19.7 Properties of the quasi-spherical Szekeres solutions with z = 0 =  19.7.1 Basic physical restrictions 19.7.2 The significance of  xi 332 337 339 347 348 349 353 353 353 357 357 359 359 362 362 363 367 367 369 369 369 375 377 379 382 383 384 387 388 392 394 396 397 399 399 400 401 403 403 404 xii Contents 19.7.3 Conditions of regularity at the origin 19.7.4 Shell crossings 19.7.5 Regular maxima and minima 19.7.6 The apparent horizons 19.7.7 Szekeres wormholes and their properties 19.7.8 The mass-dipole 19.8 * The Goode–Wainwright representation of the Szekeres solutions 19.9 Selected interesting subcases of the Szekeres–Szafron family 19.9.1 The Szafron–Wainwright model 19.9.2 The toroidal Universe of Senin 19.10 * The discarded case in (19.103)–(19.112) 19.11 Exercises 407 410 413 414 418 419 421 426 426 428 431 435 20 The Kerr solution 20.1 The Kerr–Schild metrics 20.2 The derivation of the Kerr solution by the original method 20.3 Basic properties 20.4 * Derivation of the Kerr metric by Carter’s method – from the separability of the Klein–Gordon equation 20.5 The event horizons and the stationary limit hypersurfaces 20.6 General geodesics 20.7 Geodesics in the equatorial plane 20.8 * The maximal analytic extension of the Kerr spacetime 20.9 * The Penrose process 20.10 Stationary–axisymmetric spacetimes and locally nonrotating observers 20.11 * Ellipsoidal spacetimes 20.12 A Newtonian analogue of the Kerr solution 20.13 A source of the Kerr field? 20.14 Exercises 438 438 441 447 21 Subjects omitted from this book 498 References Index 501 518 452 459 464 466 475 486 487 490 493 494 495 Figures 1.1 1.2 2.1 2.2 2.3 6.1 7.1 7.2 7.3 8.1 8.2 11.1 12.1 12.2 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13 14.14 Real planetary orbits. page 3 A vehicle flying across a light ray. 5 Parallel straight lines. 9 Parallel transport on a curved surface. 11 Parallel transport on a sphere. 11 One-parameter family of loops. 41 A light cone. 61 A non geodesic null line. 62 The Petrov classification. 71 A mapping of a manifold. 74 Surface-forming vector fields. 82 The Penrose–Petrov classification. 117 Fermi coordinates. 127 Gravitational field of a finite body. 157 Deflection of light rays. 185 Measuring the deflection of light, Eddington’s method. 187 Measuring the deflection of microwaves. 188 A gravitational lens. 189  r − 1. 193 Graph of r = r + 2m ln  2m The Kruskal diagram. 195 The surface t = const  = /2 in the Schwarzschild spacetime. 197 Embedding of the Schwarzschild spacetime in six dimensions projected onto Z1  Z2  Z3 . 198 Embedding of the Schwarzschild spacetime in six dimensions projected onto Z3  Z4  Z5 . 199 211 The maximally extended Reissner–Nordström spacetime, e2 < m2 . The ‘throat’ in the Schwarzschild and in the R–N spacetime. 213 Embeddings of the v = 0 surface. 214 Surfaces of Fig. 14.12 placed in correct positions. 214 Maximal extension of the extreme R–N metric. 216 xiii xiv 14.15 15.1 16.1 16.2 17.1 17.2 17.3 17.4 17.5 17.6 17.7 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9 18.10 18.11 18.12 18.13 18.14 18.15 19.1 19.2 19.3 19.4 19.5 19.6 19.7 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 Figures Embeddings of the t = const  = /2 surface of the extreme R–N metric. An everywhere concave function. Refocussing of light in the Universe. Reciprocity theorem. R t in Friedmann models. ˙ = 0 in the R  plane. Curves R Recollapsing Friedmann models.  = E Friedmann models. Remaining Friedmann models. Illustration to (17.62). The ‘horizon problem’ in R–W. Black hole in the E < 0 L–T model. 3-d graph of black hole formation. Contours of constant R-value. The compactified diagram of Fig. 18.1. The event horizon in the frame of Fig. 18.1. A neck. Radial rays in around central singularity. A shell crossing in comoving coordinates. A shell crossing in Gautreau coordinates. A naked shell crossing. Solutions of s = S − sS  . Solution of the ‘horizon problem’ in L–T. Evolution of the t r subspace in (18.198). The model of (18.202)–(18.205). A ‘string of beads’ Universe. Stereographic projection to Szekeres–Szafron coordinates. Circles C1 and C2 projected as disjoint. Circles C1 and C2 projected one inside the other. A Szekeres wormhole as a handle. Szafron–Wainwright model. A 2-torus. The 3-torus with the metric (19.311). Ellipsoids and hyperboloids. A surface of constant . Space t = const in the Kerr metric, case a2 < m2 . Space t = const in the Kerr metric, case a2 = m2 . Space t = const in the Kerr metric, case a2 > m2 . Light cones in the Kerr spacetime. Emin r /0 − 1 for different values of Lz . Analogue of Fig. 20.7 for null geodesics. 217 231 255 256 270 274 275 276 277 280 283 318 319 320 322 323 326 335 339 340 343 346 348 356 358 360 396 417 417 419 428 428 429 449 450 460 461 462 463 468 470 Figures 20.9 20.10 20.11 20.12 20.13 20.14 20.15 20.16 20.17 Allowed ranges of  and for null geodesics, case a2 < m2 . Allowed ranges of  and for null geodesics, case a2 = m2 . Allowed ranges of  and for null geodesics, case a2 > m2 . Allowed ranges of  and for timelike geodesics. Extending r > r+ along -field and k-field. Maximally extended Kerr spacetime. Axial cross-section through (20.154). Maximally extended extreme Kerr spacetime. A discontinuous time coordinate. xv 472 473 473 474 478 482 485 486 489 The scope of this text General relativity is the currently accepted theory of gravitation. Under this heading one could include a huge amount of material. For the needs of this theory an elaborate mathematical apparatus was created. It has partly become a self-standing sub-discipline of mathematics and physics, and it keeps developing, providing input or inspiration to phy...
View Full Document

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture