Einsteins Theory - A Rigorous Introduction To General Relativity For The Mathematically Untrained

Einstein’s Theory a Rigorous Introduction for the Mathematically Untrained Øyvind Grøn and Arne Næss E= 2 c m Rµ ν duµ dλ − 1 2 gµν R= + Γµαν uα uν = 0 Ad Infinitum ∞ 8π G c4 T µν Einstein’s Theory a Rigorous Introduction to General Relativity for the Mathematically Untrained by Øyvind Grøn Arne Næss ∞ Ad Infinitum Oslo Einstein’s Theory: a Rigorous Introduction to General Relativity for the Mathematically Untrained by Øyvind Grøn and Arne Næss c 2002 by Ad Infinitum AS, Oslo, Norway Copyright www.adinfinitum.no ISBN 82-92261-07-9 Contents Preface by Arne Næss xv Preface by Øyvind Grøn xx 1 . . . . . . . 24 24 25 28 33 45 51 55 . . . . 63 63 72 75 76 2 Vectors 1.1 Introduction . . . . . . . . . . . . . . 1.2 Vectors as arrows . . . . . . . . . . . . 1.3 Vector fields . . . . . . . . . . . . . . . 1.4 Calculus of vectors. Two dimensions 1.5 Three and more dimensions . . . . . 1.6 The vector product . . . . . . . . . . . 1.7 Space and metric . . . . . . . . . . . . Differential calculus 2.1 Differentiation . . . . . . . . . . . . . 2.2 Calculation of slopes of tangent lines 2.3 Geometry of second derivatives . . . 2.4 The product rule . . . . . . . . . . . . iv Contents 2.5 2.6 2.7 2.8 2.9 v The chain rule . . . . . . . . . . . . . . The derivative of a power function . . Differentiation of fractions . . . . . . . Functions of several variables . . . . . The MacLaurin and the Taylor series expansions . . . . . . . . . . . . . . . . . . 79 82 84 86 Tangent vectors 3.1 Parametric description of curves . . . . 3.2 Parametrization of a straight line . . . 3.3 Tangent vector fields . . . . . . . . . . . 3.4 Differential equations and Newton’s 2. law . . . . . . . . . . . . . . . . . . . . . 3.5 Integration . . . . . . . . . . . . . . . . 3.6 Exponential and logarithmic functions 3.7 Integrating equations of motion . . . . 103 103 108 111 4 Curvilinear coordinate systems 4.1 Trigonometric functions . . . . . . . . . 4.2 Plane polar coordinates . . . . . . . . . 135 137 155 5 The metric tensor 5.1 Basis vectors and dimension of space . 5.2 Space and spacetime . . . . . . . . . . . 5.3 Transformation of vector components . 5.4 The Galilean coordinate transformation 5.5 Transformation of basis vectors . . . . 5.6 Vector components . . . . . . . . . . . . 162 162 166 169 176 181 183 3 94 116 118 125 131 Contents 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 6 vi Tensors . . . . . . . . . . . . . . . . . . The metric tensor . . . . . . . . . . . . . Tensor components . . . . . . . . . . . The Lorentz transformation . . . . . . . The relativistic time dilation . . . . . . The line element . . . . . . . . . . . . . Minkowski diagrams and light cones . The spacetime interval . . . . . . . . . The general formula for the line element Epistemological comment . . . . . . . . Kant or Einstein . . . . . . . . . . . . . The Christoffel symbols 6.1 Geometrical calculation . . . . . . . . 6.2 Algebraic calculation . . . . . . . . . 6.3 Spherical coordinates . . . . . . . . . 6.4 Symmetry of the Christoffel symbols 186 191 200 205 219 222 232 236 246 253 256 . . . . 265 267 280 285 293 7 Covariant differentiation 7.1 Variation of vector components . . . . 7.2 The covariant derivative . . . . . . . . 7.3 Transformation of covariant derivatives 7.4 Covariant tensor components . . . . . 7.5 Connection expressed by the metric . . 295 296 301 309 313 315 8 Geodesics 8.1 Generalizing