نظریهٔ انیشتین: مقدمه‌ای دقیق بر نسبیت عام برای غیرمتخصصان ریاضی

This introduction to The General Theory of Relativity and its mathematics is written for all those, young and old, who lack or have forgotten the necessary mathematical knowledge to cope with already published introductions. Some of these introductions seem, at the start to require only moderately much mathematics. Very soon, however, there are frightful "jumps" in the exposition, or suddenly new concepts or notations appear as if nearly self evident. The present text starts at a lower level than any other, and leads the reader slowly and faithfully all the way to the heart of relativity: Einstein's field equations.One day, early in the Autumn 1985, the seventy three year old philosopher Arne Næss appeared at Professor Gron's graduate course on General Relativity. He immediatly decided that a new type of introduction to the general theory of relativity is needed; an introduction designed to meet the requirements of non-science educated people wanting to get a thorough understanding of this, most remarkable, theory. This book is the result of the combined effort of a philosopher wanting to understand every logical step in the derivation of Einstein's field equations, and an experienced physicist having a thorough knowledge of these steps. Starting from a freshman level in mathematics the reader is guided along the long and winding road to Einstein's field equations, black holes and relativistic cosmology. Cover......Page 1 Contents......Page 4 List of figures......Page 10 Preface by Arne Næss......Page 15 Preface by Øyvind Grøn......Page 20 1.1 Introduction......Page 24 1.2 Vectors as arrows......Page 25 1.3 Vector fields......Page 28 1.4 Calculus of vectors. Two dimensions......Page 33 1.5 Three and more dimensions......Page 45 1.6 The vector product......Page 51 1.7 Space and metric......Page 55 2.1 Differentiation......Page 63 2.2 Calculation of slopes of tangent lines......Page 72 2.3 Geometry of second derivatives......Page 75 2.4 The product rule......Page 76 2.5 The chain rule......Page 79 2.6 The derivative of a power function......Page 82 2.7 Differentiation of fractions......Page 84 2.8 Functions of several variables......Page 86 2.9 The MacLaurin and the Taylor series expansions......Page 94 3.1 Parametric description of curves......Page 103 3.2 Parametrization of a straight line......Page 108 3.3 Tangent vector fields......Page 111 3.4 Differential equations and Newton's 2. law......Page 116 3.5 Integration......Page 118 3.6 Exponential and logarithmic functions......Page 125 3.7 Integrating equations of motion......Page 131 4 Curvilinear coordinate systems......Page 135 4.1 Trigonometric functions......Page 137 4.2 Plane polar coordinates......Page 155 5.1 Basis vectors and dimension of space......Page 162 5.2 Space and spacetime......Page 166 5.3 Transformation of vector components......Page 169 5.4 The Galilean coordinate transformation......Page 176 5.5 Transformation of basis vectors......Page 181 5.6 Vector components......Page 183 5.7 Tensors......Page 186 5.8 The metric tensor......Page 191 5.9 Tensor components......Page 200 5.10 The Lorentz transformation......Page 205 5.11 The relativistic time dilation......Page 219 5.12 The line element......Page 222 5.13 Minkowski diagrams and light cones......Page 232 5.14 The spacetime interval......Page 236 5.15 The general formula for the line element......Page 246 5.16 Epistemological comment......Page 253 5.17 Kant or Einstein......Page 256 6 The Christoffel symbols......Page 265 6.1 Geometrical calculation......Page 267 6.2 Algebraic calculation......Page 280 6.3 Spherical coordinates......Page 285 6.4 Symmetry of the Christoffel symbols......Page 293 7 Covariant differentiation......Page 295 7.1 Variation of vector components......Page 296 7.2 The covariant derivative......Page 301 7.3 Transformation of covariant derivatives......Page 309 7.4 Covariant tensor components......Page 313 7.5 Connection expressed by the metric......Page 315 8 Geodesics......Page 320 8.1 Generalizing `flat space concepts'......Page 321 8.2 Parallel transport: unexpected difficulties......Page 323 8.3 Definition of parallel transport......Page 327 8.4 The general geodesic equation......Page 329 8.5 Local Cartesian coordinates......Page 334 9.1 The curvature of plane curves......Page 337 9.2 The curvature of surfaces......Page 342 9.3 Curl......Page 356 9.4 The Riemann curvature tensor......Page 361 10.1 Introduction......Page 372 10.2 Divergence......Page 375 10.3 The equation of continuity......Page 380 10.4 The stress tensor......Page 384 10.5 The net surface force acting on a fluid element......Page 388 10.6 The material derivative......Page 393 10.7 The equation of motion......Page 396 10.8 Four-velocity......Page 398 10.9 Newtonian energy-momentum of a perfect fluid......Page 403 10.10 The basic conservation laws......Page 407 10.11 Relativistic energy-momentum of a perfect fluid......Page 413 11.1 A new conception of gravitation......Page 415 11.2 The Ricci curvature tensor......Page 418 11.3 The Bianchi identity and Einstein's tensor......Page 426 12.1 Newtonian kinematics......Page 442 12.2 Forces......Page 444 12.3 Newton's theory of gravitation......Page 451 12.4 Special relativity and gravity......Page 453 12.5 The general theory of relativity......Page 463 12.6 The Newtonian limit of general relativity......Page 479 12.7 Repulsive gravitation......Page 489 12.8 `Geodesic postulate' and the field equations......Page 491 12.9 Constants of motion......Page 495 12.10 Conceptual structure of general relativity......Page 498 12.11 General relativity versus Newton's theory......Page 499 12.12 Epistemological comment......Page 502 13 Some applications......Page 505 13.1 Rotating reference frame......Page 507 13.2 The gravitational time dilation......Page 510 13.3 The Schwarzschild solution......Page 517 13.4 The Pound--Rebka experiment......Page 526 13.5 The Hafele--Keating experiment......Page 530 13.6 Mercury's perihelion precession......Page 535 13.7 Gravitational deflection of light......Page 543 13.8 Black holes......Page 548 14 Relativistic universe models......Page 561 14.1 Observations......Page 562 14.2 Homogeneous and isotropic models......Page 566 14.3 Einstein's cosmological field equations......Page 572 14.4 The Friedmann models......Page 577 14.5 The matter and radiation dominated periods......Page 587 14.6 Problems with the standard model......Page 591 14.7 Inflationary cosmology......Page 594 A The Laplacian in a spherical coordinate system......Page 605 B Ricci tensor of a static, spherically symmetric metric......Page 614 C Ricci tensor of the Robertson--Walker metric......Page 624 Bibliography......Page 631 Index......Page 633