Class Notes Introduction to Relativity Physics 375R Under Construction

Transcription

1 Class Notes Introduction to Relativity Physics 375R Under Construction Austin M. Gleeson 1 Department of Physics University of Texas at Austin Austin, TX March 20, gleeson@physics.utexas.edu

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3 Contents 1 The Special Theory of Relativity Pre-History of concepts about light Galilean Invariance Implications of and for Maxwell s Equations Pursuit of a special frame Michelson-Morley Experiment Kinematics of special relativity Special Relativity Principles of Relativity Harry and Sally and Space Time Diagrams Introduction The Paradox of Harry and Sally The Relativity of Simultaneity Harry and Sally s Movements in a Diagram The Nature of Space-Time The Problem of Coordinates The Lorentz Transformations The Relatively Moving Clock Derivation of the Lorentz Transformation Details of the Derivation of the Lorentz Transformations Using Lorentz Transformations Time Dilation Length contraction The Doppler Effect Addition of velocities Time for Different Travelers Visual Appearence of Rapidly Moving Objects

4 4 CONTENTS 4 Events, Worldlines, Intervals Introduction Place and Path in the Two Dimensional Plane Minkowski Space-time Causality and Trajectories The Hyperbolic Hangle The same result directly using calculus Four Vectors and Invariants Event Four Vector and Four Velocity Other Four Vector and Four Tensor Elements Harry, Dorothy, and Sally Revisited Paradoxes of Relativity The Twin Paradox The Problem The Solution The Bandits and the Bullet Train The Problem The Solution The Boy in the Barn The Problem The Solution The Relativistic Manhole Cover The Problem The Solution Uniform Acceleration Events at the same proper distance from some event Uniformly accelerated motion Details of the calculation of the acceleration The Proper Time along the Trajectory Timelike Trajectories and Accelerated Motion Examples using accelerated motion Deceleration Accelerated Rocket John Bell s Problem The Accelerated Reference Frame

5 CONTENTS 5 7 Relativistic Dynamics General Principles of Dynamics Relativistic Action The Action for a Free Particle Energy and momentum of a single free particle Mass Kinetic Energy of a Single Particle Transformations of Momentum and Energy The Energy, Momentum, and Mass of Light Interactions Multi-particle Systems Rest energy of composite and elementary systems Applications of Energy Momentum Introduction to General Relativity The Problem Free Fall Observers and the Equivalence Principle The Equivalence Principle The Monkey and the Hunter Direct Effects from the Equivalence Principle Universality and Eötvös Dicke Bending of Light Rays Clocks and Accelerations in Towers Intrinsic Effects of Gravity Distortion of Elastic Bodies Gravitation and Tidal Forces Geometry and Gravitation Introduction to Geometry Gaussian Curvature Example of negative curvature: the Pringle Curvature and Geodesics The Theorema Egregium and the Line Element Proof of Theorema Egregium Examples of Two Dimensional Geometry Euclidean Two Plane in Polar Coordinates The Unit Two Sphere in Euclidean Three Space The Two Torus The Accelerated Reference Frame Revisited The Hyperbolic Space-Time Two Surface

6 6 CONTENTS 9.7 Geometry in Four or More Dimensions Some notation and nomenclature: Coordinate Labels in General Relativity Distances and Time Intervals Einstein Equations Effects of Gravitation Linearized Gravity Linearized Theory Analog to Electromagnetism Gravitational Waves Curvature around a Massive Body The Geometry and Evolution of the Universe Background Ideas Copernican Principle Olber s Paradox Hubble Expansion Implications of expansion Missing Mass The space time structure of the universe Black Body Background Problems with the Expanding Universe The Cosmological Constant The Standard Model of the Universe Large Scale and Micro-physics Structure in the Universe The Inflationary Universe String Theory A Primer on Action 209 A.1 Newtonian Dynamics A.2 Dynamics and Action A.2.1 Background on Formulation of Action A.2.2 Introduction to Action A.2.3 Definition of Action A.2.4 Trajectory of a Free Particle A.2.5 Proof that the Least Action Reproduces Newtonian Physics A.2.6 Examples of action gravitation near a flat earth

7 CONTENTS 7 A.2.7 Same Example done another way A.2.8 More Examples of Actions A.3 The Nature of Symmetry in Physics A.3.1 Discrete Transformations A.3.2 Continuous Transformations A.3.3 Identity Transformation A.3.4 Examples of symmetry in situations like physics A.3.5 Physics transformations: A.4 Examples of Symmetry in physics A.4.1 Physics transformations: A.5 Symmetry and Action A.5.1 Introduction A.5.2 Galilean invariance A.5.3 More on Symmetry and Action A.5.4 Noether s Theorem B Introduction to Field Theory 245 B.1 Action at a Distance and Field Dynamics B.1.1 Action at a Distance B.1.2 Local Field Theory B.2 The Stretched String B.3 Maxwell s Theory of Electromagnetism C Indexology 271 C.1 Vectors and tensors under rotation C.2 General Coordinates Labels C.3 Fields and Manafolds C.3.1 Tensor Fields and General Transformations C.3.2 Nomenclature D Generalized Linear Theory 283