Introduction to Einstein’s Theory of Relativity-Springer (2020)

Undergraduate Texts in Physics Øyvind Grøn Introduction to Einstein’s Theory of Relativity From Newton’s Attractive Gravity to the Repulsive Gravity of Vacuum Energy Second Edition Undergraduate Texts in Physics Series Editors Kurt H Becker, NYU Polytechnic School of Engineering, Brooklyn, NY, USA Jean-Marc Di Meglio, Matière et Systèmes Complexes, Université Paris Diderot, Bâtiment Condorcet, Paris, France Sadri D Hassani, Department of Physics, Loomis Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL, USA Morten Hjorth-Jensen, Department of Physics, Blindern, University of Oslo, Oslo, Norway Michael Inglis, Patchogue, NY, USA Bill Munro, NTT Basic Research Laboratories, Optical Science Laboratories, Atsugi, Kanagawa, Japan Susan Scott, Department of Quantum Science, Australian National University, Acton, ACT, Australia Martin Stutzmann, Walter Schottky Institute, Technical University of Munich, Garching, Bayern, Germany Undergraduate Texts in Physics (UTP) publishes authoritative texts covering topics encountered in a physics undergraduate syllabus Each title in the series is suitable as an adopted text for undergraduate courses, typically containing practice problems, worked examples, chapter summaries, and suggestions for further reading UTP titles should provide an exceptionally clear and concise treatment of a subject at undergraduate level, usually based on a successful lecture course Core and elective subjects are considered for inclusion in UTP UTP books will be ideal candidates for course adoption, providing lecturers with a firm basis for development of lecture series, and students with an essential reference for their studies and beyond More information about this series at http://www.springer.com/series/15593 Øyvind Grøn Introduction to Einstein’s Theory of Relativity From Newton’s Attractive Gravity to the Repulsive Gravity of Vacuum Energy Second Edition 123 Øyvind Grøn OsloMet—Oslo Metropolitan University Oslo, Norway ISSN 2510-411X ISSN 2510-4128 (electronic) Undergraduate Texts in Physics ISBN 978-3-030-43861-6 ISBN 978-3-030-43862-3 (eBook) https://doi.org/10.1007/978-3-030-43862-3 1st edition: © Springer Science+Business Media, LLC 2009 2nd edition: © Springer Nature Switzerland AG 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface to the Second Edition These notes are a transcript of lectures delivered by Øyvind Grøn during the spring of 1997 at the University of Oslo The manuscript has been revised in 2019 The present version of this document is an extended and corrected version of a set of Lecture Notes which were written down by S Bard, Andreas O Jaunsen, Frode Hansen and Ragnvald J Irgens using LATEX2 Sven E Hjelmeland has made many useful suggestions which have improved the text The manuscript has been revised in 2019 In this version, solutions to the exercises have been included Most of these have been provided by Håkon Enger I thank all my good helpers for enthusiastic work which was decisive for the realization of the book I hope that these notes are useful to students of general relativity and look forward to their comments accepting all feedback with thanks The comments may be sent to the author by e-mail to oyvind.gron.no@gmail.com Oslo, Norway Øyvind Grøn v Preface to the First Edition These notes are a transcript of lectures delivered by Øyvind Grøn during the spring of 1997 at the University of Oslo The present version of this document is an extended and corrected version of a set of Lecture Notes which were typesetted by S Bard, Andreas O Jaunsen, Frode Hansen and Ragnvald J Irgens using LATEX2 Svend E Hjelmeland has made many useful suggestions which have improved the text I would also like to thank Jon Magne Leinaas and Sigbjørn Hervik for contributing with problems and Gorm Krogh Johnsen for help with finishing the manuscript I also want to thank Prof Finn Ravndal for inspiring lectures on general relativity While we hope that these typeset notes are of benefit particularly to students of general relativity and look forward to their comments, we welcome all interested readers and accept all feedback with thanks All comments may be sent to the author by e-mail E-mail: Oyvind.Gron@iu.hio.no Øyvind Grøn vii Contents Newton’s Theory of Gravitation 1.1 The Force Law of Gravitation 1.2 Newton’s Law of Gravitation in Local Form 1.3 Newtonian Incompressible Star 1.4 Tidal Forces 1.5 The Principle of Equivalence 1.6 The General Principle of Relativity 1.7 The Covariance Principle 1.8 Mach’s Principle 1.9 Exercises References 10 14 17 17 18 19 22 The Special Theory of Relativity 2.1 Coordinate Systems and Minkowski Diagrams 2.2 Synchronization of Clocks 2.3 The Doppler Effect 2.4 Relativistic Time Dilation 2.5 The Relativity of Simultaneity 2.6 The Lorentz Contraction 2.7 The Lorentz Transformation 2.8 Lorentz Invariant Interval 2.9 The Twin Paradox 2.10 Hyperbolic Motion 2.11 Energy and Mass 2.12 Relativistic Increase of Mass 2.13 Lorentz Transformation of Velocity, Momentum, Energy and Force 2.14 Tachyons 2.15 Magnetism as a Relativistic Second-Order Effect 23 23 25 26 28 30 33 34 37 40 41 44 45 47 50 51 ix x Contents Exercises Reference 54 58 Vectors, Tensors and Forms 3.1 Vectors 3.1.1 Four-Vectors 3.1.2 Tangent Vector Fields and Coordinate Vectors 3.1.3 Coordinate Transformations 3.1.4 Structure Coefficients 3.2 Tensors 3.2.1 Transformation of Tensor Components 3.2.2 Transformation of Basis One-Forms 3.2.3 The Metric Tensor 3.3 The Causal Structure of Spacetime 3.4 Forms 3.4.1 The Volume Form 3.4.2 Dual Forms Exercises 59 59 60 62 65 68 69 71 71 72 76 78 80 82 85 Accelerated Reference Frames 4.1 The Spatial Metric Tensor 4.2 Einstein Synchronization of Clocks in a Rotating Reference Frame 4.3 Angular Acceleration in the Rotating Frame 4.4 Gravitational Time Dilation 4.5 Path of Photons Emitted from the Axis in a Rotating Reference Frame 4.6 The Sagnac Effect 4.7 Non-integrability of a Simultaneity Curve in a Rotating Frame 4.8 Orthonormal Basis Field in a Rotating Frame 4.9 Uniformly Accelerated Reference Frame 4.10 The Projection Tensor Exercises 89 89 92 95 98 99 99 101 102 105 113 115 Covariant Differentiation 5.1 Differentiation of Forms 5.1.1 Exterior Differentiation 5.1.2 Covariant Derivative 5.2 The Christoffel Symbols 5.3 Geodesic Curves 5.4 The Covariant Euler–Lagrange Equations 5.5 Application of the Lagrange Formalism to Free Particles 5.5.1 Equation of Motion from Lagrange’s Equations 5.5.2 Geodesic World Lines in Spacetime 119 119 119 122 122 125 127 129 129 133 Contents xi 5.5.3 Acceleration of Gravity 5.5.4 Gravitational Shift of Wavelength 5.6 Connection Coefficients 5.6.1 Structure Coefficients 5.7 Covariant Differentiation of Vectors, Forms and Tensors 5.7.1 Covariant Differentiation of Vectors 5.7.2 Covariant Differentiation of Forms 5.7.3 Covariant Differentiation of Tensors of Arbitrary Rank 5.8 The Cartan Connection 5.9 Covariant Decomposition of a Velocity Field 5.9.1 Newtonian 3-Velocity 5.9.2 Relativistic 4-Velocity 5.10 Killing Vectors and Symmetries 5.11 Covariant Expressions for Gradient, Divergence, Curl, Laplacian and D’Alembert’s Wave Operator 5.12 Electromagnetism in Form Language Exercises Curvature 6.1 The Riemann Curvature Tensor 6.2 Differential Geometry of Surfaces 6.2.1 Surface Curvature Using the Cartan Formalism 6.3 The Ricci Identity 6.4 Bianchi’s Identity 6.5 Bianchi’s Identity 6.6 Torsion 6.7 The Equation of Geodesic Deviation 6.8 Tidal Acceleration and Spacetime Curvature 6.9 The Newtonian Tidal Tensor 6.10 The Tidal and Non-tidal Components of a Gravitational Field Exercises 135 138 140 143 144 144 145 146 147 151 151 153 155 157 163 169 173 173 179 183 184 185 186 187 188 190 191 192 195 Einstein’s Field Equations 7.1 Newtonian Fluid 7.2 Perfect Fluids 7.2.1 Lorentz Invariant Vacuum Energy—LIVE 7.2.2 Energy–Momentum Tensor of an Electromagnetic Field 7.3 Einstein’s Curvature Tensor 7.4 Einstein’s Field Equations 7.5 The “Geodesic Postulate” as a Consequence of the Field Equations 197 197 199 200 201 201 202 204 ... Newton’s theory of gravity In Newton’s theory an inertial frame is defined as a reference frame moving along a straight line with constant velocity The fundamental laws of Newton’s theory of. .. Switzerland AG 2020 Ø Grøn, Introduction to Einstein’s Theory of Relativity, Undergraduate Texts in Physics, https://doi.org/10.1007/978-3-030-43862-3_1 Newton’s Theory of Gravitation In a non-inertial... Theory of Relativity 11.5 The Levi-Civita—Bertotti—Robinson Solution of Einstein’s Field Equations 11.6 The Source of the Levi-Civita—Bertotti—Robinson