General Relativity. The Theoretical Minimum

Description

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