An Introduction to General Relativity and Cosmology

Transcription

1 An Introduction to General Relativity and Cosmology Jerzy Plebariski Centro de Investigacion y de Estudios Avanzados Instituto Politecnico Nacional Apartado Postal , Mexico D.F., Mexico Andrzej Krasiriski Centrum Astronomiczne im. M. Kopernika, Polska Akademia Nauk, Bartycka 18, Warszawa, Poland CAMBRIDGE UNIVERSITY PRESS

2 Contents List of figures page xiii The scope of this text xvii Acknowledgements xix 1 How 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 4 Part I Elements of differential geometry 7 2 A short sketch of 2-dimensional differential geometry Constructing parallel straight lines in a flat space Generalisation of the notion of parallelism to curved surfaces 10 3 Tensors, tensor densities ; What are tensors good for? Differentiable manifolds Scalars Contravariant vectors Co variant vectors Tensors of second rank Tensor densities Tensor densities of arbitrary rank Algebraic properties of tensor densities Mappings between manifolds The Levi-Civita symbol Multidimensional Kronecker deltas Examples of applications of the Levi-Civita symbol and of the multidimensional Kronecker delta Exercises 25

3 vi Contents 4 Covariant derivatives Differentiation of tensors Axioms of the covariant derivative A field of bases on a manifold and scalar components of tensors The affine connection The explicit formula for the covariant derivative of tensor density fields Exercises 32 5 Parallel transport and geodesic lines Parallel transport Geodesic lines Exercises 35 6 The curvature of a manifold; flat manifolds The commutator of second covariant derivatives The commutator of directional covariant derivatives The relation between curvature and parallel transport Covariantly constant fields of vector bases A torsion-free flat manifold Parallel transport in a flat manifold Geodesic deviation Algebraic and differential identities obeyed by the curvature tensor Exercises 47 7 Riemannian geometry The metric tensor Riemann spaces, The signature of a metric, degenerate metrics Christoffel symbols The curvature of a Riemann space Flat Riemann spaces Subspaces of a Riemann space Flat Riemann spaces that are globally non-euclidean The Riemann curvature versus the normal curvature of a surface The geodesic line as the line of extremal distance Mappings between Riemann spaces Conformally related Riemann spaces Conformal curvature Timelike, null and spacelike intervals in a 4-dimensional spacetime Embeddings of Riemann spaces in Riemann spaces of higher dimension The Petrov classification Exercises 72

4 Contents vii 8 Symmetries of Riemann spaces, invariance of tensors Symmetry transformations The Killing equations The connection between generators and the invariance transformations Finding the Killing vector fields Invariance of other tensor fields The Lie derivative The algebra of Killing vector fields Surface-forming vector fields Spherically symmetric 4-dimensional Riemann spaces * Conformal Killing fields and their finite basis * The maximal dimension of an invariance group Exercises 91 9 Methods to calculate the curvature quickly - Cartan forms and algebraic computer programs The basis of differential forms 94 9^2 The connection forms The Riemann tensor Using computers to calculate the curvature Exercises The spatially homogeneous Bianchi type spacetimes The Bianchi classification of 3-dimensional Lie algebras The dimension of the group versus the dimension of the orbit Action of a group on a manifold Groups acting transitively, homogeneous spaces Invariant vector fields The metrics of the Bianchi-type spacetimes The isotropic Bianchi-type (Robertson-Walker) spacetimes Exercises * The Petrov classification by the spinor method What is a spinor? Translating spinors to tensors and vice versa The spinor image of the Weyl tensor The Petrov classification in the spinor representation The Weyl spinor represented as a 3 x 3 complex matrix The equivalence of the Penrose classes to the Petrov classes ' The Petrov classification by the Debever method Exercises 122

5 viii Contents Part II The theory of gravitation The Einstein equations and the sources of a gravitational field Why Riemannian geometry? Local inertial frames Trajectories of free motion in Einstein's theory Special relativity versus gravitation theory The Newtonian limit of relativity Sources of the gravitational field The Einstein equations Hilbert's derivation of the Einstein equations The Palatini variational principle The asymptotically Cartesian coordinates and the asymptotically flat spacetime The Newtonian limit of Einstein's equations Examples of sources in the Einstein equations: perfect fluid and dust Equations of motion of a perfect fluid The cosmological constant An example of an exact solution of Einstein's equations: a Bianchi type I spacetime with dust source * Other gravitation theories The Brans-Dicke theory The Bergmann-Wagoner theory The conformally invariant Canuto theory The Einstein-Cartan theory The bi-metric Rosen theory Matching solutions of Einstein's equations The weak-field approximation to general relativity Exercises The Maxwell and Einstein-Maxwell equations and the Kaluza-KIein theory The Lorentz-covariant description of electromagnetic field The covariant form of the Maxwell equations The energy-momentum tensor of an electromagnetic field The Einstein-Maxwell equations * The variational principle for the Einstein-Maxwell equations * The Kaluza-KIein theory Exercises Spherically symmetric gravitational fields of isolated objects The curvature coordinates Symmetry inheritance 172

6 Contents ix 14.3 Spherically symmetric electromagnetic field in vacuum The Schwarzschild and Reissner-Nordstrom 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-Nordstrom solution * Motion of particles in the Reissner-Nordstrom spacetime with e 2 < m Exercises Relativistic hydrodynamics and thermodynamics Motion of a continuous medium in Newtonian mechanics Motion of a continuous medium in relativistic mechanics The equations of evolution of 6, a a p, a) a p and ii a ; the Raychaudhuri equation Singularities and singularity theorems Relativistic thermodynamics Exercises \ Relativistic cosmology I: general geometry A continuous medium as a model of the Universe Optical observations in the Universe - part I The geometric optics approximation Theredshift The optical tensors The apparent horizon * The double-null tetrad * The Goldberg-Sachs theorem * Optical observations in the Universe - part II The area distance The reciprocity theorem Other observable quantities Exercises 260

7 x Contents 17 Relativistic cosmology II: the Robertson-Walker geometry The Robertson-Walker metrics as models of the Universe Optical observations in an R-W Universe Theredshift The redshift-distance relation Number counts The Friedmann equations and the critical density The Friedmann solutions with A = The redshift-distance relation in the A = 0 Friedmann models The Newtonian cosmology The Friedmann solutions with the cosmological constant Horizons in the Robertson-Walker models The inflationary models and the 'problems' they solved The value of the cosmological constant The'history of the Universe' Invariant definitions of the Robertson-Walker models Different representations,of the R-W metrics Exercises Relativistic cosmology III: the Lemattre-Tolman geometry The comoving-synchronous coordinates The spherically symmetric inhomogeneous models The Lemaitre-Tolman model Conditions of regularity at the centre Formation of voids in the Universe Formation of other structures in the Universe \ Density to density evolution Velocity to density evolution Velocity to velocity evolution The influence of cosmic expansion on planetary orbits * Apparent horizons in the L-T model * Black holes in the evolving Universe * Shell crossings and necks/wormholes < E = E> Theredshift The influence of inhomogeneities in matter distribution on the cosmic microwave background radiation Matching the L-T model to the Schwarzschild and Friedmann solutions 332

8 Contents xi * General properties of the Big Bang/Big Crunch singularities in the L-T model * Extending the L-T spacetime through a shell crossing singularity * Singularities and cosmic censorship Solving the 'horizon problem' without inflation * The evolution of R(t, M) versus the evolution of p(t, M) * Increasing and decreasing density perturbations * L&T curio shop Lagging cores of the Big Bang Strange or non-intuitive properties of the L-T model Chances to fit the L-T model to observations An 'in one ear and out the other' Universe A'string of beads' Universe Uncertainties in inferring the spatial distribution of matter Is the matter distribution in our Universe fractal? General results related to the L-T models Exercises Relativistic oosmology IV: generalisations of L-T and related geometries The plane-and hyperbolically symmetric spacetimes G 3 /S 2 -symmetric dust solutions with R, r ^ G 3 /S 2 -symmetric dust in electromagnetic field, the case R, r ^ Integrals of the field equations Matching the charged dust metric to the Reissner-Nordstrom metric Prevention of the Big Crunch singularity by electric charge * Charged dust in curvature and mass-curvature coordinates Regularity conditions at the centre * Shell crossings in charged dust The Datt-Ruban solution The Szekeres-Szafron family of solutions The /3, z = 0 subfamily The/3,,^0 subfamily Interpretation of the Szekeres-Szafron coordinates Common properties of the two subfamilies * The invariant definitions of the Szekeres-Szafron metrics The Szekeres solutions and their properties The /3, z =0 subfamily The )3, z ^ 0 subfamily * The )3, z = 0 family as a limit of the /3, z ^0 family Properties of the quasi-spherical Szekeres solutions with /3,, ^= 0 = A Basic physical restrictions The significance of 404

9 xii Contents Conditions of regularity at the origin Shell crossings Regular maxima and minima The apparent horizons Szekeres wormholes and their properties The mass-dipole * The Goode-Wainwright representation of the Szekeres solutions Selected interesting subcases of the Szekeres-Szafron family The Szafron-Wainwright model The toroidal Universe of Senin * The discarded case in (19.103)-(19.112) Exercises The Kerr solution The Kerr-Schild metrics The derivation of the Kerr solution by the original method Basic properties " * Derivation of the Kerr metric by Carter's method - from the separability of the Klein-^Gordon equation The event horizons and the stationary limit hypersurfaces General geodesies Geodesies in the equatorial plane * The maximal analytic extension of the Kerr spacetime * The Penrose process Stationary-axisymmetric spacetimes and locally nonrotating observers * Ellipsoidal spacetimes A Newtonian analogue of the Kerr solution A source of the Kerr field? Exercises Subjects omitted from this book 498 References 501 Index 518