Introduction to general relativity, black holes, and cosmology / / Yvonne Choquet-Bruhat ; with a foreword by Thibault Damour.

• Superior document: Oxford scholarship online
• VerfasserIn: Choquet-Bruhat, Yvonne,
• Place / Publishing House: Oxford : : Oxford University Press,, 2023.
• Year of Publication: 2023
• Edition: First edition.
• Language: English
• Series: Oxford scholarship online.
• Physical Description: 1 online resource (301 pages)
• Notes: This edition also issued in print: 2023.

General relativity is a beautiful geometric theory, simple in its mathematical formulation but leading to numerous consequences with striking physical interpretations: gravitational waves, black holes, cosmological models, and so on. This introductory textbook is written for mathematics students int...

Table of Contents

• Cover; Foreword; Preface; Notation; Contents; Part A Fundamentals; I Riemannian and Lorentzian geometry; I.1 Introduction; I.2 Differentiable manifolds and mappings; I.2.1 Differentiable manifolds; I.2.2 Differentiable mappings; I.2.3 Submanifolds; I.2.4 Tangent and cotangent spaces; I.2.5 Vector fields and 1-forms; I.2.6 Moving frames; I.3 Tensors and tensor fields; I.3.1 Tensors, products and contraction; I.3.2 Tensor fields. Pullback and Lie derivative; I.3.3 Exterior forms; I.4 Structure coefficients of moving frames; I.5 Pseudo-Riemannian metrics; I.5.1 General properties
• I.5.2 Riemannian metricsI.5.3 Lorentzian metrics; I.6 Causality; I.6.1 Causal and null cones; I.6.2 Future and past; I.6.3 Spacelike submanifolds; I.6.4 Length and geodesics; I.7 Connections; I.7.1 Linear connection; I.7.2 Riemannian connection; I.8 Geodesics, another definition; I.8.1 Pseudo-Riemannian manifolds; I.8.2 Riemannian manifolds; I.8.3 Lorentzian manifolds; I.9 Curvature; I.9.1 Definitions; I.9.2 Symmetries and antisymmetries; I.9.3 Differential Bianchi identity and contractions; I.10 Geodesic deviation; I.11 Linearized Ricci tensor; I.11.1 Linearized Bianchi identities
• I.12 Physical commentI.13 Solutions of selected exercises; I.14 Problems; I.14.1 Liouville theorem; I.14.2 Codifferential δ and Laplacian of an exterior form; I.14.3 Geodesic normal coordinates; I.14.4 Cases d=1, 2, and 3; I.14.5 Wave equation satisfied by the Riemann tensor; I.14.6 The Bel-Robinson tensor; I.14.7 The Weyl tensor; I.14.8 The Cotton-York tensor; I.14.9 Linearization of the Riemann tensor; I.14.10 Second derivative of the Ricci tensor; II Special relativity; II.1 Introduction; II.2 Newtonian mechanics; II.2.1 The Galileo-Newton Spacetime
• II.2.2 Newtonian dynamics. Galileo groupII.2.3 Physical comment; II.2.4 The Maxwell equations in Galileo-Newton spacetime; II.3 The Lorentz and Poincaré groups; II.4 Lorentz contraction and dilation; II.5 Electromagnetic field and Maxwell equations in Minkowski spacetime M4; II.6 Maxwell equations in arbitrary dimensions; II.7 Special Relativity; II.7.1 Proper time; II.7.2 Proper frame and relative velocities; II.8 Some physical comments; II.9 Dynamics of a pointlike mass; II.9.1 Newtonian law; II.9.2 Relativistic law; II.9.3 Newtonian approximation of the relativistic equation
• II.9.4 Equivalence of mass and energyII.9.5 Particles with zero rest mass; II.10 Continuous matter; II.10.1 Case of dust (incoherent matter), massive particles; II.10.2 Perfect fluids; II.10.3 Yang-Mills fields; II.11 Problems; II.11.1 Lorentz transformation of the Maxwell equations; II.11.2 The relativistic Doppler-Fizeau effect; III General Relativity; III.1 Introduction; III.2 Principle of general covariance; III.3 The Galileo-Newton equivalence principle; III.4 General Relativity; III.4.1 Einstein equivalence principles; III.4.2 Conclusion; III.5 Constants and units of measurement
• III.6 Classical fields in General Relativity