A Student's Guide to General Relativity

Description

This compact guide presents the key features of general relativity, to support and supplement the presentation in mainstream, more comprehensive undergraduate textbooks, or as a re-cap of essentials for graduate students pursuing more advanced studies. It helps students plot a careful path to understanding the core ideas and basics of differential geometry, as applied to general relativity, without overwhelming them. While the guide doesn't shy away from necessary technicalities, it emphasises the essential simplicity of the main physical arguments. Presuming a familiarity with special relativity (with a brief account in an appendix), it describes how general covariance and the equivalence principle motivate Einstein's theory of gravitation. It then introduces differential geometry and the covariant derivative as the mathematical technology which allows us to understand Einstein's equations of general relativity. The book is supported by numerous worked exampled and problems, and important applications of general relativity are described in an appendix. Contents Preface Acknowledgements 1 Introduction 1.1 Three Principles 1.2 Some Thought Experiments on Gravitation 1.3 Covariant Differentiation 1.4 A Few Further Remarks Exercises 2 Vectors, Tensors, and Functions 2.1 Linear Algebra 2.2 Tensors, Vectors, and One-Forms 2.3 Examples of Bases and Transformations 2.4 Coordinates and Spaces Exercises 3 Manifolds, Vectors, and Differentiation 3.1 The Tangent Vector 3.2 Covariant Differentiation in Flat Spaces 3.3 Covariant Differentiation in Curved Spaces 3.4 Geodesics 3.5 Curvature Exercises 4 Energy, Momentum, and Einstein’s Equations 4.1 The Energy-Momentum Tensor 4.2 The Laws of Physics in Curved Space-time 4.3 The Newtonian Limit Exercises Appendix A Special Relativity – A Brief Introduction A.1 The Basic Ideas A.2 The Postulates A.3 Spacetime and the Lorentz Transformation A.4 Vectors, Kinematics, and Dynamics Exercises Appendix B Solutions to Einstein’s Equations B.1 The Schwarzschild Solution B.2 The Perihelion of Mercury B.3 Gravitational Waves Exercises Appendix C Notation C.1 Tensors C.2 Coordinates and Components C.3 Contractions C.4 Differentiation C.5 Changing Bases C.6 Einstein’s Summation Convention C.7 Miscellaneous References Index