General Relativity and Cosmology: A First Encounter

Description

Gravitational physics has now become a mainstream topic in physics and physics teaching. In particular cosmology and gravitational wave physics are at the focus of a great deal of current research. Thus it is important to introduce students to General Relativity as soon as reasonable. This textbook offers a brief but comprehensive treatment accessible to advanced undergraduate students, graduate students, and any physicist or mathematician interested in understanding the material in a short time. The author, an experienced teacher of the subject, has included numerous examples and exercises to help students consolidate the ideas they have learned. Preface Contents Part ISpecial Relativity in Review 1 A Brief Stroll in Special Relativity 1.1 The Trouble with Absolute Time 1.2 The Simplest Lorentz Transformation 1.3 Some Elementary Properties and Applications 2 Lorentz Transformations 2.1 The Lorentz Group 2.2 Four-Vectors and Tensors 3 The Motion of Particles 3.1 Energy and Momentum 3.2 Acceleration 3.3 Accelerated Motion 3.4 Curves and Arc Lengths Part IIVectors and Tensors 4 Riemann Spaces and Tensors 4.1 Riemann Spaces 4.2 Vectors, Component View 4.3 Vectors and 1-Forms, Abstract View 4.4 Tensors, Component View 4.5 Tensors, Abstract View 4.6 Tetrads and n-Trads 4.7 Volume Elements Appendix 1: Differential Manifolds Appendix 2: The Signature Theorem in Two Dimensions 5 Affine Connections and Geodesics 5.1 Affine Connections, Component View 5.2 Transformation of the Affine Connections 5.3 Parallel Displacement 5.4 Geodesics as Self-parallel Curves 5.5 Geodesics as Extremum Curves 5.6 Affine Connections, Abstract View Appendix 1: A Special Coordinate System Appendix 2: The Extremum Problem and the Euler-Lagrange Equations Appendix 3: Christoffel Connections as Fictitious Forces 6 Tensor Analysis 6.1 Covariant Derivatives, Component View 6.2 Covariant Derivatives, Abstract View 6.3 The Divergence and Laplacian Appendix 1: Curve Derivatives as Vectors Appendix 2: p-Forms and Exterior Derivatives Part IIIGeneral Relativity 7 Classical Gravity and Geometry 7.1 Newtonian Gravity 7.2 The Equivalence Principle 7.3 Gravity as a Geometric Phenomenon 8 Curved Space and Gravity 8.1 Curved Space and the Riemann Tensor 8.2 Symmetries of the Riemann Tensor 8.3 The Einstein Equations for the Gravitational Field in Vacuum 8.4 The Non-vacuum Field Equations 8.5 The Intrinsic Signature of Gravity Appendix 1: Tangent Spaces Appendix 2: The Riemann Tensor as a 6 by 6 Matrix 9 Spherically Symmetric Gravitational Fields 9.1 The Schwarzschild Solution 9.2 Orbit of a Planet 9.3 Deflection of Light 9.4 Observational Tests of General Relativity Appendix 1: Isotropic Form of the Metric, Eddington Parameters 10 Black Holes and Gravitational Collapse 10.1 Schwarzschild Black Hole 10.2 Null Surfaces 10.3 Stellar Evolution, Very Briefly 10.4 Collapse of a Dust Star 10.5 Spinning Black Holes and the Kerr Metric 10.6 Black Holes in the Real Universe 10.7 Hawking Radiation from a Black Hole 11 Linearized General Relativity and Gravitational Waves 11.1 The Field Equations of the Linearized Theory 11.2 The Classical Limit 11.3 Gravitational Plane Waves 11.4 Motion of Test Bodies in Gravitational Waves 11.5 Gravitational Wave Sources 11.6 Detection of Gravitational Waves Appendix 1: Solutions for Retarded Potentials Appendix 2: Electromagnetic Plane Waves Appendix 3: Electromagnetic Wave Sources Part IVCosmology 12 The Einstein Field Equations for Cosmology 12.1 The Field Equations and Energy-Momentum Conservation 12.2 Field Equations and the Cosmic Fluid Source 12.3 The Cosmological Constant as Vacuum or Dark Energy 12.4 Summary 13 Cosmological Preliminaries 13.1 Basic Observations and Assumptions 13.2 The Cosmological FLRW Metric 13.3 Consequences of the Metric 13.4 De Sitter Space Appendix 1: Measured Values for the Hubble Constant 14 The Dynamical Equations of Cosmology 14.1 The Einstein Field Equations for Cosmology 14.2 Critical Density and the Shape of the Universe 14.3 Observed Dark Matter and Dark Energy Densities 14.4 Evolution of Cosmic Fluid Constituents 14.5 The Friedmann Master Equation Appendix 1: The Einstein Tensor for the FLRW Metric 15 Solutions for the Present Universe 15.1 The Positive Cosmological Constant 15.2 Complete Solution of the Friedmann Master Equation 15.3 Cosmological Constant Dominance 15.4 Matter Dominance 15.5 The LCDM Universe Appendix 1: A Mechanical Analogy Appendix 2: Newtonian View of Dark Energy Appendix 3: Some Discarded Cosmological Models 16 Some Properties of the LCDM Universe 16.1 Diverse Cosmological Observations 16.2 Cosmological Parameter Values 16.3 The Hubble Function and the Age of the Universe 16.4 Transition Time for Matter to Dark Energy Dominance 16.5 Density Ratios and the Shape of the Universe 16.6 Horizons and the Size of the Observable Universe 16.7 Conformal Time 17 Earlier Times and Radiation 17.1 Radiation and Temperature in Earlier Times 17.2 The Scale Factor and Basic Properties of the Radiation Era 17.3 The Isotropic CMB and the Horizon Puzzle 17.4 The Anisotropies of the CMB 18 A Brief Historical Overview of the Universe 18.1 Overview 18.2 Condensation of Stars and Galaxies 18.3 Condensation of Atoms 18.4 Condensation of Nuclei 18.5 Condensation of Nucleons 18.6 Inflation 18.7 Planck Era 19 Inflation and Some Questions 19.1 Basic Ideas of Inflation 19.2 Inflation Via Scalar Fields 19.3 Origin of Structure 19.4 The Physical Nature of Dark Energy 19.5 The Physical Nature of Dark Matter 19.6 The Planck Era and Quantum Physics Appendix 1: Power Law Inflation Appendix 2: Scalar Field Theory Appendix 3: Black Hole Remnants as Dark Matter References Index