GENERAL RELATIVITY : the theoretical minimum

One book at a time, the Theoretical Minimum series makes the power and grandeur of physics accessible. First came classical mechanics, then quantum mechanics and special relativity. Now, physicist Leonard Susskind, assisted by a new collaborator, Andre Cabannes, returns to tackle Einstein's masterpiece: the general theory of relativity. Starting from the equivalence principle and covering the necessary mathematics of Riemannian spaces and tensor calculus, Susskind and Cabannes explain the link between gravity and geometry. They delve into black holes, establish Einstein field equations, and solve them to describe gravity waves. The authors provide vivid explanations that, to borrow a phrase from Einstein himself, are as simple as possible (but no simpler). An approachable yet rigorous introduction to one of the most important topics in physics, General Relativity is a must-read for anyone who wants a deeper knowledge of the universe's real structure. This book is the fourth volume of The Theoretical Minimum series. The first volume, The Theoretical Minimum: What You Need to Know to Start Doing Physics, covered classical mechanics, which is the core of any physics education. We will refer to it from time to time simply as volume 1. The second book, volume 2, explains quantum mechanics and its relationship to classical mechanics. Volume 3 covers special relativity and classical field theory. This fourth volume expands on that to explore general relativity. GENERAL RELATIVITY Contents Preface Lecture 1: Equivalence Principle and Tensor Analysis Introduction Equivalence Principle Accelerated Reference Frames Curvilinear Coordinate Transformations Effect of Gravity on Light Tidal Forces Non-Euclidean Geometry Riemannian Geometry page furled Metric Tensor Mathematical Interlude: Dummy Variables Mathematical Interlude: Einstein Summation Convention Y,VvUv First Tensor Rule: Contravariant Components of Vectors [ ATX(P), X2(P), ... ,XN(P)] [y^p), y2(P),... ,yN(p)] dXm Mathematical Interlude: Vectors and Tensors 9Xp = w, — Second Tensor Rule: Covariant Components of Vectors Covariant and Contravariant Components of Vectors and Tensors Lecture 2: Tensor Mathematics Introduction Flat Space Metric Tensor Scalar, Vector, and Tensor Fields s'(r) = s(X) (2) Geometric Interpretation of Contravariant and Covariant Components of a Vector v = V1^ + V2e2 + V3e3 Mathematical Interlude: Dot Product of Two Vectors Vi = V • a V = VTnem Vn = V-en Tensor Mathematics Tensor Algebra (v” H'"’' = ar^ (l'"‘ H'h, (24) More on the Metric Tensor Mathematical Interlude: The Metric is a Symmetric Tensor The Matrix Inverse of the Metric Lecture 3: Flatness and Curvature Introduction General Relativity in Modern Physics Riemannian Geometry Gaussian Normal Coordinates Covariant Derivatives - r*m vt (io) Christoffel Symbols rf = rf (id Curvature Tensor D9Drvn = ds [drvn - r‘nyf] R,^ = drr‘n - asr‘n + r?nrL. - nnr* (23) Lecture 4: Geodesics and Gravity Introduction Parallel Transport Tangent Vectors and Geodesics Example of Calculations with Christoffel Symbols More on Geodesics Space-Time t 4 Q p. At2 = At2 - AX2 ► X Special Relativity Uniform Acceleration Uniform Gravitational Field Motion of a Particle Lecture 5: Metric for a Gravitational Field Time-like, Space-like, and Light-like Intervals and Light Cones Geodesics and Euler—Lagrange Equations shortest curve A = / £(X, X) dt Schwarzschild Metric Black Holes Event Horizon of a Black Hole Motion of a Light Ray Lecture 6: Black Holes Schwarzschild Metric Schwarzschild Radius or Black Hole Event Horizon Light Ray Orbiting a Black Hole horizon orbiting particle light-ray O black hole Photon Sphere black hole Hyperbolic Coordinates Revisited Interchange of Space and Time Dimensions at the Horizon Black Hole Singularity Alice No Escaping from a Black Hole Lecture 7: Falling into a Black Hole Introduction Schwarzschild Metric, Event Horizon, and Singularity horizon r = 0 r = 1 r Fundamental Diagram of Space-Time near a Black Hole Notes on the Fundamental Diagram History of Black Holes Falling into a Black Hole Questions/Answers Session Lecture 8: Formation of a Black Hole Introduction Kruskal—Szekeres Coordinates Penrose Diagrams T=0 T=-l T=-2 T = 2 T=1 r=o T=-l T=-2 Wormholes Formation of a Black Hole and Newton’s Shell Theorem inside the shell Discussion of the Time Variable Lecture 9: Einstein Field Equations Introduction Newtonian Gravitational Field Continuity Equation Energy-Momentum Tensor Ricci Tensor and Curvature Scalar ft = ft£ = Einstein Tensor and Einstein Field Equations Questions/Answers Session Lecture 10: Gravitational Waves Introduction r = h-^ (6) Gravitational Waves x = xf + /(x', y') (14) y = y' + 5(x', y') Questions/Answers Session Einstein—Hilbert Action for General Relativity Steps to Derive the Field Equations from the Action Index The latest volume in The New York Times bestselling physics series explains Einstein's masterpiece: the general theory of relativity He taught us classical mechanics, quantum mechanics and special relativity. Now, physicist Leonard Susskind, assisted by a new collaborator, André Cabannes, returns to tackle Einstein's general theory of relativity. Starting from the equivalence principle and covering the necessary mathematics of Riemannian spaces and tensor calculus, Susskind and Cabannes explain the link between gravity and geometry. They delve into black holes, establish Einstein field equations and solve them to describe gravity waves. The authors provide vivid explanations that, to borrow a phrase from Einstein himself, are as simple as possible (but no simpler). An approachable yet rigorous introduction to one of the most important topics in physics, General Relativity is a must-read for anyone who wants a deeper knowledge of the universe's real structure. He taught us classical mechanics quantum mechanics and special relativity. Now physicist Leonard Susskind assisted by a new collaborator Andre Cabannes returns to tackle Einstein's general theory of relativity. Starting from the equivalence principle and covering the necessary mathematics of Riemannian spaces and tensor calculus Susskind and Cabannes explain the link between gravity and geometry. They delve into black holes establish Einstein field equations and solve them to describe gravity waves. The authors provide vivid explanations that to borrow a phrase from Einstein himself are as simple as possible (but no simpler). An approachable yet rigorous introduction to one of the most important topics in physics General Relativity is a must-read for anyone who wants a deeper knowledge of the universe's real structure