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An Introduction to General Relativity and Cosmology Author(s): Jerzy Plebanski, Andrzej Krasinski Publisher: Cambridge University Press Number of pages: 556 A completely bookmarked/paginated pdf file. A great resource for physicits and curious minds interested in physics/universe/cosmology and science in general :) Enjoy! General relativity is a cornerstone of modern physics, and is of major importance in its applications to cosmology. Plebanski and Krasinski are experts in the field and provide a thorough introduction to general relativity, guiding the reader through complete derivations of the most important results. Providing coverage from a unique viewpoint, geometrical, physical and astrophysical properties of inhomogeneous cosmological models are all systematically and clearly presented, allowing the reader to follow and verify all derivations. Many topics are included that are not found in other textbooks. ------------------------------------------------------------------------------------------------------------------ I share scienctific and mathematical ebooks/textbooks to help students and curious minds. If you like my uploads/torrents then support me by seeding, sharing, commenting and rating them :) https://kickass.so/user/Charm-Quark/uploads/ P.S. If you like this eBook and can afford it, then support the author by buying it :) ------------------------------------------------------------------------------------------------------------------ Contents : 1.2 Space and inertia in Newtonian physics......Page 23 1.3 Newton’s theory and the orbits of planets......Page 24 1.4 The basic assumptions of general relativity......Page 26 Part I Elements of differential geometry......Page 29 2.1 Constructing parallel straight lines in a flat space......Page 31 2.2 Generalisation of the notion of parallelism to curved surfaces......Page 32 3.2 Differentiable manifolds......Page 35 3.4 Contravariant vectors......Page 37 3.6 Tensors of second rank......Page 38 3.7 Tensor densities......Page 39 3.9 Algebraic properties of tensor densities......Page 40 3.10 Mappings between manifolds......Page 41 3.11 The Levi-Civita symbol......Page 44 3.12 Multidimensional Kronecker deltas......Page 45 3.13 Examples of applications of the Levi-Civita symbol and of the multidimensional Kronecker delta......Page 46 3.14 Exercises......Page 47 4.1 Differentiation of tensors......Page 48 4.2 Axioms of the covariant derivative......Page 50 4.3 A field of bases on a manifold and scalar components of tensors......Page 51 4.4 The affine connection......Page 52 4.5 The explicit formula for the covariant derivative of tensor density fields......Page 53 4.6 Exercises......Page 54 5.1 Parallel transport......Page 55 5.2 Geodesic lines......Page 56 5.3 Exercises......Page 57 6.1 The commutator of second covariant derivatives......Page 58 6.2 The commutator of directional covariant derivatives......Page 60 6.3 The relation between curvature and parallel transport......Page 61 6.4 Covariantly constant fields of vector bases......Page 65 6.6 Parallel transport in a flat manifold......Page 66 6.7 Geodesic deviation......Page 67 6.8 Algebraic and differential identities obeyed by the curvature tensor......Page 68 6.9 Exercises......Page 69 7.1 The metric tensor......Page 70 7.3 The signature of a metric, degenerate metrics......Page 71 7.5 The curvature of a Riemann space......Page 73 7.6 Flat Riemann spaces......Page 74 7.8 Flat Riemann spaces that are globally non-Euclidean......Page 75 7.9 The Riemann curvature versus the normal curvature of a surface......Page 76 7.10 The geodesic line as the line of extremal distance......Page 77 7.12 Conformally related Riemann spaces......Page 78 7.13 Conformal curvature......Page 80 7.14 Timelike, null and spacelike intervals in a 4-dimensional spacetime......Page 83 7.15 Embeddings of Riemann spaces in Riemann spaces of higher dimension......Page 85 7.16 The Petrov classification......Page 92 7.17 Exercises......Page 94 8.1 Symmetry transformations......Page 96 8.2 The Killing equations......Page 97 8.3 The connection between generators and the invariance transformations......Page 99 8.4 Finding the Killing vector fields......Page 100 8.5 Invariance of other tensor fields......Page 101 8.6 The Lie derivative......Page 102 8.8 Surface-forming vector fields......Page 103 8.9 Spherically symmetric 4-dimensional Riemann spaces......Page 104 8.10 * Conformal Killing fields and their finite basis......Page 108 8.11 * The maximal dimension of an invariance group......Page 111 8.12 Exercises......Page 113 9.1 The basis of differential forms......Page 116 9.2 The connection forms......Page 117 9.3 The Riemann tensor......Page 118 9.5 Exercises......Page 120 10.1 The Bianchi classification of 3-dimensional Lie algebras......Page 121 10.2 The dimension of the group versus the dimension of the orbit......Page 126 10.4 Groups acting transitively, homogeneous spaces......Page 127 10.5 Invariant vector fields......Page 128 10.6 The metrics of the Bianchi-type spacetimes......Page 130 10.7 The isotropic Bianchi-type (Robertson–Walker) spacetimes......Page 131 10.8 Exercises......Page 134 11.1 What is a spinor?......Page 135 11.2 Translating spinors to tensors and vice versa......Page 136 11.4 The Petrov classification in the spinor representation......Page 138 11.5 The Weyl spinor represented as a 3×3 complex matrix......Page 139 11.6 The equivalence of the Penrose classes to the Petrov classes......Page 141 11.7 The Petrov classification by the Debever method......Page 142 11.8 Exercises......Page 144 Part II The theory of gravitation......Page 145 12.2 Local inertial frames......Page 147 12.3 Trajectories of free motion in Einstein’s theory......Page 148 12.4 Special relativity versus gravitation theory......Page 151 12.6 Sources of the gravitational field......Page 152 12.7 The Einstein equations......Page 153 12.8 Hilbert’s derivation of the Einstein equations......Page 154 12.11 The Newtonian limit of Einstein’s equations......Page 158 12.12 Examples of sources in the Einstein equations: perfect fluid and dust......Page 162 12.13 Equations of motion of a perfect fluid......Page 165 12.14 The cosmological constant......Page 166 12.15 An example of an exact solution of Einstein’s equations: a Bianchi type I spacetime with dust source......Page 167 12.16.1 The Brans–Dicke theory......Page 171 12.16.4 The Einstein–Cartan theory......Page 172 12.17 Matching solutions of Einstein’s equations......Page 173 12.18 The weak-field approximation to general relativity......Page 176 12.19 Exercises......Page 182 13.2 The covariant form of the Maxwell equations......Page 183 13.3 The energy-momentum tensor of an electromagnetic field......Page 184 13.4 The Einstein–Maxwell equations......Page 185 13.6 * The Kaluza–Klein theory......Page 186 13.7 Exercises......Page 189 14.1 The curvature coordinates......Page 190 14.3 Spherically symmetric electromagnetic field in vacuum......Page 194 14.4 The Schwarzschild and Reissner–Nordström solutions......Page 195 14.5 Orbits of planets in the gravitational field of the Sun......Page 198 14.6 Deflection of light rays in the Schwarzschild field......Page 205 14.7 Measuring the deflection of light rays......Page 208 14.8 Gravitational lenses......Page 211 14.9 The spurious singularity of the Schwarzschild solution at r = 2m......Page 213 14.10 * Embedding the Schwarzschild spacetime in a flat Riemannian space......Page 218 14.11 Interpretation of the spurious singularity at r = 2m; black holes......Page 222 14.12 The Schwarzschild solution in other coordinate systems......Page 224 14.13 The equation of hydrostatic equilibrium......Page 225 14.14 The ‘interior Schwarzschild solution’......Page 228 14.15 * The maximal analytic extension of the Reissner–Nordström solution......Page 229 14.16 * Motion of particles in the Reissner–Nordström spacetime with e < m......Page 239 14.17 Exercises......Page 241 15.1 Motion of a continuous medium in Newtonian mechanics......Page 244 15.2 Motion of a continuous medium in relativistic mechanics......Page 246 15.3 The equations of evolution of and ˙u; the Raychaudhuri equation......Page 250 15.4 Singularities and singularity theorems......Page 252 15.5 Relativistic thermodynamics......Page 253 15.6 Exercises......Page 256 16.1 A continuous medium as a model of the Universe......Page 257 16.2.1 The geometric optics approximation......Page 259 16.2.2 The redshift......Page 261 16.3 The optical tensors......Page 262 16.4 The apparent horizon......Page 264 16.5 * The double-null tetrad......Page 265 16.6 * The Goldberg–Sachs theorem......Page 267 16.7.1 The area distance......Page 275 16.7.2 The reciprocity theorem......Page 278 16.7.3 Other observable quantities......Page 281 16.8 Exercises......Page 282 17.1 The Robertson–Walker metrics as models of the Universe......Page 283 17.2.1 The redshift......Page 285 17.2.3 Number counts......Page 287 17.3 The Friedmann equations and the critical density......Page 288 17.4 The Friedmann solutions with Lambda = 0......Page 291 17.4.1 The redshift–distance relation in the Lambda = 0 Friedmann models......Page 292 17.5 The Newtonian cosmology......Page 293 17.6 The Friedmann solutions with the cosmological constant......Page 295 17.7 Horizons in the Robertson–Walker models......Page 299 17.8 The inflationary models and the ‘problems’ they solved......Page 304 17.9 The value of the cosmological constant......Page 308 17.10 The ‘history of the Universe’......Page 309 17.11 Invariant definitions of the Robertson–Walker models......Page 312 17.12 Different representations of the R–W metrics......Page 313 17.13 Exercises......Page 315 18.2 The spherically symmetric inhomogeneous models......Page 316 18.3 The Lemaître–Tolman model......Page 318 18.4 Conditions of regularity at the centre......Page 322 18.5 Formation of voids in the Universe......Page 323 18.6 Formation of other structures in the Universe......Page 325 18.6.1 Density to density evolution......Page 326 18.6.2 Velocity to density evolution......Page 328 18.6.3 Velocity to velocity evolution......Page 330 18.7 The influence of cosmic expansion on planetary orbits......Page 331 18.8 * Apparent horizons in the L–T model......Page 333 18.9 * Black holes in the evolving Universe......Page 338 18.10 * Shell crossings and necks/wormholes......Page 343 18.11 The redshift......Page 350 18.12 The influence of inhomogeneities in matter distribution on the cosmic microwave background radiation......Page 352 18.14 * General properties of the Big Bang/Big Crunch singularities in the L–T model......Page 354 18.15 * Extending the L–T spacetime through a shell crossing singularity......Page 359 18.16 * Singularities and cosmic censorship......Page 361 18.17 Solving the ‘horizon problem’ without inflation......Page 369 18.18 * The evolution of R(t,M) versus the evolution of p(t M)......Page 370 18.19 * Increasing and decreasing density perturbations......Page 371 18.20.2 Strange or non-intuitive properties of the L–T model......Page 375 18.20.4 An ‘in one ear and out the other’ Universe......Page 379 18.20.5 A ‘string of beads’ Universe......Page 381 18.20.8 General results related to the L–T models......Page 384 18.21 Exercises......Page 385 19.1 The plane- and hyperbolically symmetric spacetimes......Page 389 19.3.1 Integrals of the field equations......Page 391 19.3.2 Matching the charged dust metric to the Reissner–Nordström metric......Page 397 19.3.3 Prevention of the Big Crunch singularity by electric charge......Page 399 19.3.4 * Charged dust in curvature and mass-curvature coordinates......Page 401 19.3.5 Regularity conditions at the centre......Page 404 19.3.6 * Shell crossings in charged dust......Page 405 19.4 The Datt–Ruban solution......Page 406 19.5 The Szekeres–Szafron family of solutions......Page 409 19.5.1 The Beta = 0 subfamily......Page 410 19.5.2 The Beta…......Page 414 19.5.3 Interpretation of the Szekeres–Szafron coordinates......Page 416 19.5.4 Common properties of the two subfamilies......Page 418 19.5.5 * The invariant definitions of the Szekeres–Szafron metrics......Page 419 19.6.1 The Beta = 0 subfamily......Page 421 19.6.2 The Beta = 0 subfamily......Page 422 19.6.3 * The Beta = 0 family as a limit of the Beta…......Page 423 19.7.1 Basic physical restrictions......Page 425 19.7.2 The significance of Epsilon......Page 426 19.7.3 Conditions of regularity at the origin......Page 429 19.7.4 Shell crossings......Page 432 19.7.5 Regular maxima and minima......Page 435 19.7.6 The apparent horizons......Page 436 19.7.7 Szekeres wormholes and their properties......Page 440 19.7.8 The mass-dipole......Page 441 19.8 * The Goode–Wainwright representation of the Szekeres solutions......Page 443 19.9.1 The Szafron–Wainwright model......Page 448 19.9.2 The toroidal Universe of Senin......Page 450 19.10 * The discarded case in (19.103)–(19.112)......Page 453 19.11 Exercises......Page 457 20.1 The Kerr–Schild metrics......Page 460 20.2 The derivation of the Kerr solution by the original method......Page 463 20.3 Basic properties......Page 469 20.4 * Derivation of the Kerr metric by Carter’s method – from the separability of the Klein–Gordon equation......Page 474 20.5 The event horizons and the stationary limit hypersurfaces......Page 481 20.6 General geodesics......Page 486 20.7 Geodesics in the equatorial plane......Page 488 20.8 * The maximal analytic extension of the Kerr spacetime......Page 497 20.9 * The Penrose process......Page 508 20.10 Stationary–axisymmetric spacetimes and locally nonrotating observers......Page 509 20.11 * Ellipsoidal spacetimes......Page 512 20.12 A Newtonian analogue of the Kerr solution......Page 515 20.13 A source of the Kerr field?......Page 516 20.14 Exercises......Page 517 21 Subjects omitted from this book......Page 520 References......Page 523 Index......Page 540 Sharing Widget |