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Einstein's General Theory of Relativity Einstein’s General Theory of Relativity Einstein’s General Theory of Relativity by Asghar Qadir Einstein’s General Theory of Relativity By Asghar Qadir This book first published 2020 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A cat- alogue record for this book is available from the British Library Copyright c 2020 by Asghar Qadir All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmit- ted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-4428-1 ISBN (13): 978-1-5275-4428-4 Dedicated To My Mentors: Manzur Qadir Roger Penrose Remo Ruffini John Archibald Wheeler and My Wife: Rabiya Asghar Qadir Contents List of Figures xi Preface xv 1 Introduction 1 1.1 Equivalence Principle . 5 1.2 Field Theory . 9 1.3 The Lagrange Equations . 10 1.4 Lagrange Equations extension to Fields . 11 1.5 Relativistic Fields . 13 1.6 Generalize Special Relativity . 15 1.7 The Principle of General Relativity . 17 1.8 Principles Underlying Relativity . 18 1.9 Exercises . 21 2 Analytic Geometry of Three Dimensions 23 2.1 Review of Three Dimensional Vector Notation . 24 2.2 Space Curves . 29 2.3 Surfaces . 33 2.4 Coordinate Transformations on Surfaces . 37 2.5 The Second Fundamental Form . 38 2.6 Examples . 41 2.7 Gauss’ Formulation of the Geometry of Surfaces . 45 2.8 The Gauss-Codazzi Equations and Gauss’ Theorem . 49 2.9 Exercises . 53 3 Tensors and Differential Geometry 55 3.1 Space Curves in Flat n-Dimensional Space . 55 vii viii CONTENTS 3.2 Manifolds . 58 3.3 Vectors in Curved Spaces . 63 3.4 Penrose’s Abstract Index Notation . 66 3.5 The Metric Tensor and Covariant Differentiation . 69 3.6 The Curvature Tensors and Scalars . 78 3.7 Curves in Manifolds . 85 3.8 Isometries and Killing’s Equations . 97 3.9 Miscellaneous Topics in Geometry . 103 3.10 Exercises . 107 4 Unrestricted Theory of Relativity 109 4.1 Stress-Energy Tensor . 109 4.2 The Stress-Energy Tensor for Fields . 112 4.3 The Einstein Field Equations . 114 4.4 Newtonian Limit . 115 4.5 The Schwarzschild Solution . 119 4.6 The Relativistic Equation of Motion . 123 4.7 The First Three Tests . 126 4.8 The Gravitational Red-Shift . 129 4.9 The Gravitational Deflection of Light . 131 4.10 The Perihelion Shift of Mercury . 133 4.11 Exercises . 137 5 Field Theory of Gravity 139 5.1 Re-derivation of Einstein’s Equations . 139 5.2 The Schwarzschild Interior Solution . 143 5.3 The Reissner-Nordström Metric . 145 5.4 The Kerr and Charged Kerr Metrics . 146 5.5 Gravitational Waves and Linearised Gravity . 148 5.6 Exact Gravitational Wave Solutions . 151 5.7 Interpretation of Gravitational Waves . 154 5.8 The (3 + 1) Split of Spacetime . 156 5.9 General Relativity in Terms of Forces . 161 5.10 Exercises . 166 6 Black Holes 167 CONTENTS ix 6.1 The Classical Black Hole . 167 6.2 Escape Velocity for Schwarzschild Metric . 168 6.3 The Black Hole Horizon . 170 6.4 Convenient Coordinates for Black Holes . 172 6.5 Physics Near and Inside a Black Hole . 180 6.6 The Charged Black Hole . 183 6.7 Convenient Coordinates for Charged Black Holes . 184 6.8 The Kerr Black Hole . 189 6.9 Naked Singularities and Cosmic Censorship . 192 6.10 Foliating the Schwarzschild Spacetime . 195 6.11 Black Hole Thermodynamics . 199 6.12 Exercises . 208 7 Relativistic Cosmology 209 7.1 The Cosmological Principle . 210 7.2 Strong Cosmological Principle . 212 7.3 Enter the Cosmological Constant . 214 7.4 Measuring Cosmological Distances . 218 7.5 The Hubble Expansion of the Universe . 221 7.6 The Einstein-Friedmann Models . 222 7.7 Saving the Strong Cosmological Principle . 229 7.8 The Hot Big Bang Model Of Gamow . 229 7.9 The Microwave Background Radiation . 232 7.10 The Geometry of the Universe . 234 7.11 Digression Into High Energy Physics . 237 7.12 Attempts at Further Unification . 241 7.13 The Chronology of the Universe . 244 7.14 The Composition of the Universe . 245 7.15 Accelerated Expansion of the Universe . 251 7.16 Non-Baryonic Dark Matter . 252 7.17 Problems of the Standard Cosmological Model . 255 7.18 The Inflationary Models . 261 7.19 Exercises . 268 8 Some Special Topics 269 8.1 Two-component Spinors . 269 x CONTENTS 8.2 Spacetime Symmetries . 274 8.3 More on Gravitational Waves . 280 8.4 Collapsed Stars and Black Holes . 283 8.5 Attempts to Unify Quantum Theory and GR . 289 References 297 Subject Index 307 List of Figures 1.1 The E¨otvos experiment. 8 1.2 The “wotld-tube”. 13 2.1 Cartesian coordinates (x, y, z). 25 2.2 Spherical coordinates (r, θ, φ). 25 2.3 Cylindrical coordinates (r, θ, z). 25 2.4 Cartesian basis vectors e1, e2, e3................... 26 2.5 Spherical basis vectors. 26 2.6 Cylindrical basis vectors. 27 2.7 Translation of the origin from O to O0 by a vector a. 28 2.8 The change of basis due to rotation exactly balances the change of components of the vector. 28 2.9 A curve, x(u), through three points Q, P , R. 30 2.10 A car going up a mountain road. 31 2.11 The helix like a coil of spring. 33 2.12 A generic surface x(u; v) with a point, P , on it. 34 2.13 Two intersecting curves '1(u; v) = 0 and '2(u; v) = 0, on the surface x(u; v). ............................ 36 2.14 The height of Q above the tangent plane, P, to the surface x(u; v) at the point P ............................. 38 2.15 A sphere parameterized by spherical coordinates u = θ, v = '. 41 2.16 A cylinder parameterized by cylindrical polar coordinates u = θ, v = z. ................................. 42 2.17 (a) The hyperboloid of one sheet, x2 +y2 −z2 = a2 parameterized by orthogonal (hyperbolic) coordinates u; v. (b) The hyperboloid z2 − x2 − y2 = a2 parameterized by orthogonal (hyperbolic) co- ordinates u; v.............................. 43 3.1 Problems with assigning one coordinate for the entire circle. 59 3.2 The unit circle S1 covered by two open sets A = S1 n fNg and B = (−a; a)............................... 59 3.3 An example of a non-Hausdorff space. 61 3.4 An infinitely differentiable curve that is not analytic. 62 3.5 A manifold Mn covered (here) by four open sets. 63 3.6 The effect of curvature of a space on the vectors defined on it. 64 3.7 Meaning of polar components of a vector. 64 3.8 The use of polar coordinates to discuss the motion of a particle under a central force. 73 3.9 A curve γ in a manifold Mn .................... 86 3.10 Lie and parallel transport of a derivation p. 90 xi xii LIST OF FIGURES 3.11 Geodesic deviation between two geodesics of a family with unit tangent t................................ 91 4.1 The force dF acts on an area element dS with centre at P . 110 4.2 A force dF acts on an area element dS with centre at P . 117 4.3 Perihelion shift of a planet. 127 4.4 GR gravitational deflection of light . 133 5.1 Time slicing the Minkowski spacetime by t = const: lines. 157 5.2 Time slicing Minkowski spacetime from by observer moving at constant speed relative to previous observer. 157 5.3 diagram with (t0; r0) coordinates drawn in place of (t; r). 158 5.4 Minkowski spacetime for accelerated observer. 158 5.5 Time slicing by a sequence of parabolic hyper-cylinders. 158 5.6 The Newtonian tidal force. 161 5.7 An accelerometer measures GR tidal forces. 162 6.1 The ( ; χ) coordinates in terms of (θ; φ) or Cartesian coordinates. 169 6.2 The Schwarzschild spacetime in Schwarzschild coordinates with two dimensions suppressed. 173 6.3 The Schwarzschild spacetime in Schwarzschild coordinates with only one dimension suppressed. 173 6.4 The interior of the black hole in “Schwarzschild-like” coordinates. 174 6.5 A top view of the sequence of light cones entering a black hole. 175 6.6 The Schwarzschild spacetime in Eddington-Finkelstein retarded advanced coordinates. 176 6.7 The Kruskal picture of the Schwarzschild spacetime. 177 6.8 The Carter-Penrose diagram in compactified null and compacti- fied Kruskal-Szekres coordinates. 179 6.9 An intrepid observer in his rocket ship goings into a black hole. 180 6.10 Signals going astray and even crossing over. 181 6.11 Another intrepid observer exploring the black hole interior fol- lowed in by us. 181 6.12 A Carter-Penrose representation of me falling freely into the black hole before you do. 182.
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