Redshift Academy

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Astronomy

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Astronomical Distance Units .
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Celestial Coordinates .
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Celestial Navigation .
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Location of North and South Celestial Poles .

Chemistry

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Avogadro's Number .
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Balancing Chemical Equations
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Stochiometry
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The Periodic Table .

Classical Mechanics

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Blackbody Radiation .

Classical Physics

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Archimedes Principle
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Bernoulli Principle
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Center of Mass Frame
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Comparison Between Gravitation and Electrostatics
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Compton Effect .
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Coriolis Effect
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Cyclotron Resonance
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Dispersion
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Doppler Effect
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Double Slit Experiment
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Elastic and Inelastic Collisions .
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Electric Fields
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Error Analysis
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Fick's Law
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Fluid Pressure
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Gauss's Law of Universal Gravity .
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Gravity - Force and Acceleration
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Hooke's law
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Ideal and Non-Ideal Gas Laws (van der Waal)
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Impulse Force
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Inclined Plane
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Inertia
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Kepler's Laws
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Kinematics
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Kinetic Theory of Gases .
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Kirchoff's Laws
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Maxwell's Equations .
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Moments and Torque
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Nuclear Spin
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One Dimensional Wave Equation .
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Pascal's Principle
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Phase and Group Velocity
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Poiseuille's Law
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Refractive Index
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Rotational Dynamics
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Simple Harmonic Motion
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Specific Heat, Latent Heat and Calorimetry
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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect .
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Young's Modulus

Climate Change

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Keeling Curve .

Cosmology

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Baryogenesis
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Cosmic Background Radiation and Decoupling .
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CPT Symmetries
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Dark Matter .
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Friedmann-Robertson-Walker Equations .
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Hubble's Law
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Inflation Theory
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Introduction to Black Holes .
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Olbers' Paradox .
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Planck Units .
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Stephen Hawking's Last Paper .
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Stephen Hawking's PhD Thesis .
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The Big Bang Model
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Vacuum Energy .

Finance and Accounting

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Amortization
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Annuities
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Brownian Model of Financial Markets .
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Capital Structure
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Dividend Discount Formula
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Lecture Notes on International Financial Management
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NPV and IRR
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis

Game Theory

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The Truel .

General Relativity

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Basis One-forms .
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Catalog of Spacetimes .
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Curvature and Parallel Transport
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Einstein's Field Equations
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Geodesics
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Gravitational Waves
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Hyperbolic Motion and Rindler Coordinates .
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Quantum Gravity
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Ricci Decomposition .
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Ricci Flow .
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Stress-Energy-Momentum Tensor .
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Tensors
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The Area Metric
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The Dirac Equation in Curved Spacetime .
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The Equivalence Principal
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The Essential Mathematics of General Relativity
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The Induced Metric
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The Light Cone .
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The Metric Tensor .
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The Principle of Least Action in Relativity .
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Vierbein (Frame) Fields

Group Theory

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Basic Group Theory .
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Basic Representation Theory .
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Building Groups From Other Groups .
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Sets, Groups, Modules, Rings and Vector Spaces
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Symmetric Groups .
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The Integers Modulo n Under + and x .

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory .
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Euler-Lagrange Equation .
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics .
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Hamiltonian Formulation .
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Liouville's Theorem
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Symmetry and Conservation Laws - Noether's Theorem .

Macroeconomics

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Lecture Notes on International Economics
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Lecture Notes on Macroeconomics
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Macroeconomic Policy

Mathematics

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Amplitude, Period and Phase
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Arithmetic and Geometric Sequences and Series .
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Asymptotes
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Augmented Matrices and Cramer's Rule
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Binomial Theorem (Pascal's Triangle)
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Completing the Square
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Complex Numbers
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Composite Functions
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Conformal Transformations .
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Conjugate Pair Theorem
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Contravariant and Covariant Components of a Vector
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Derivatives of Inverse Functions
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Double Angle Formulas
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Eigenvectors and Eigenvalues
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Euler Formula for Polyhedrons
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Factoring of a3 +/- b3
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Fourier Series and Transforms .
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Fractals
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Gauss's Divergence Theorem
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Grassmann and Clifford Algebras .
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Heron's Formula
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Index Notation (Tensors and Matrices) .
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Inequalities
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Integration By Parts
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Introduction to Conformal Field Theory .
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Inverse of a Function
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Law of Sines and Cosines
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Line Integrals, ∮
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Logarithms and Logarithmic Equations
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Matrices and Determinants
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Matrix Exponential
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Mean Value and Rolle's Theorem
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Modulus Equations
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Orthogonal Curvilinear Coordinates .
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Parabolas, Ellipses and Hyperbolas
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Piecewise Functions
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Polar Coordinates
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Polynomial Division
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Quaternions 1 .
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Quaternions 2 .
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Regular Polygons
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Related Rates
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Similar Matrices and Diagonalization .
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Spherical Trigonometry
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Stirling's Approximation
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Sum and Differences of Squares and Cubes
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Symbolic Logic
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Tangent and Normal Line
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Taylor and Maclaurin Series .
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The Essential Mathematics of Lie Groups
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The Limit Definition of the Exponential Function
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Tic-Tac-Toe Factoring
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Trapezoidal Rule
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Unit Vectors
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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Nuclear Physics

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Radioactive Decay

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity, Chirality and Weyl Spinors .
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Klein-Gordon and Dirac Equations .
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Regularization and Renormalization
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Scattering - Mandelstam Variables
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Spin 1 Eigenvectors .

Probability and Statistics

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Box and Whisker Plots
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Buffon's Needle .
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Categorical Data - Crosstabs
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Chebyshev's Theorem
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Chi Squared Goodness of Fit
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Conditional Probability
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Confidence Intervals
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Data Types
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Expected Value
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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Monte Carlo Methods
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Non Parametric Tests
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One-Way ANOVA
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Pearson Correlation
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Permutations and Combinations
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Pooled Variance and Standard Error
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Probability Distributions
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Probability Rules
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Sample Size Determination
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Sampling Distributions
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Set Theory - Venn Diagrams
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Stacked and Unstacked Data
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Stem Plots, Histograms and Ogives
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Survey Data - Likert Item and Scale
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Tukey's Test
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Two-Way ANOVA

Programming and Computer Science

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Hashing
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How this site works ...
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More Programming Topics
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MVC Architecture
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Open Systems Interconnection (OSI) Standard - TCP/IP Protocol
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Public Key Encryption

Quantum Computing

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Density Operators and Mixed States .
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Entangled States .
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The Qubit .

Quantum Field Theory

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Creation and Annihilation Operators
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Field Operators for Bosons and Fermions
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Lagrangians in Quantum Field Theory
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Path Integral Formulation
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Relativistic Quantum Field Theory

Quantum Mechanics

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Bohr Atom
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Clebsch-Gordan Coefficients .
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Commutators
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Dyson Series
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Electron Orbital Angular Momentum and Spin
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Heisenberg Uncertainty Principle
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Ladder Operators .
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Multi Electron Wavefunctions .
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Pauli Spin Matrices
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Photoelectric Effect .
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Position and Momentum States .
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Probability Current
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Schrodinger Equation for Hydrogen Atom .
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Schrodinger Wave Equation
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Spin 1/2 Eigenvectors
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The Differential Operator
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The Essential Mathematics of Quantum Mechanics
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The Quantum Harmonic Oscillator .
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The Schrodinger, Heisenberg and Dirac Pictures
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The WKB Approximation
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Time Dependent Perturbation Theory
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Time Evolution and Symmetry Operations
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Time Independent Perturbation Theory
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Wavepackets

Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids .
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Fermi-Dirac Statistics .
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Intrinsic and Extrinsic Semiconductors .
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The MOSFET
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The P-N Junction

Special Relativity

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4-vectors .
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Electromagnetic (Faraday) Tensor .
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Energy and Momentum in Special Relativity, E = mc2 .
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Invariance of the Velocity of Light .
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Lorentz Invariance .
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Lorentz Transform .
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Lorentz Transformation of the EM Field .
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Newton versus Einstein
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Spinors - Part 1 .
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Spinors - Part 2 .
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The Continuity Equation .
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The Lorentz Group .

Statistical Mechanics

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Entropy and the Partition Function
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The Harmonic Oscillator
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The Ideal Gas

String Theory

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Bosonic Strings
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Extra Dimensions
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Introduction to String Theory
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Kaluza-Klein Compactification of Closed Strings
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Strings in Curved Spacetime
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Toroidal Compactification

Superconductivity

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Bardeen–Cooper–Schrieffer Theory
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BCS Theory
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Cooper Pairs
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Introduction to Superconductivity .
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Superconductivity (Lectures 1 - 10)
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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields
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Generators of a Supergroup
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Grassmann Numbers
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Introduction to Supersymmetry
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The Gauge Hierarchy Problem

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)
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Gauge Theories (Yang-Mills)
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Gravitational Force and the Planck Scale
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Introduction to the Standard Model
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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
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Quantum Flavordynamics and Quantum Chromodynamics
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Special Unitary Groups and the Standard Model - Part 1 .
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Special Unitary Groups and the Standard Model - Part 2
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Special Unitary Groups and the Standard Model - Part 3 .
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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Constants
Last modified: March 21, 2022 ✓

Expansion of the Universe -------------------------- Mass Dominated Period ------------------------- . . . . . M . . . . + .m (x = 1) . 0 . . . . . . . <-- D --> Consider the case of a mass, m sitting on the surface of a sphere with mass, M, that is being ejected from M at velocity, v, we can use the conservation of energy and Newton's law that states: 1. A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its center. 2. If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object's location within the shell. We can write: mv2/2 - GMm/D = E Where mv2/2 is the KE of m and GMm/D is the gravitational PE due to M. This equation can be written as: v2 - 2GM/D = 2E/m Now D = ax where a is the scale factor and x is a distance. For x = 1: . . v = D = xa So, . x2a2 - 2GM/xa = 2E/m Now rewrite M in terms of mass density, M = (4/3)πx3a3ρ: . ∴ x2a2 - (2G/xa)(4/3)πx3a3ρ = 2E/m . ∴ x2a2 - (8/3)πGx2a2ρ = 2E/m . ∴ x2(a/a)2 - (8/3)πGx2ρ = 2E/a2m Dimensionally, [m2.(m2/s2)/m2] - [(m3/kg.s2).m2.kg/m3] = [(kg.m2/s2)/m2.kg] [(m2/s2)] - [(m2/s2)] = [(1/s2)] In order to make this equation correct, the RHS needs to multiplied by a factor of x2. Thus, . x2(a/a)2 - (8/3)πGx2ρ = 2Ex2/a2m . (a/a)2 - (8/3)πρG = 2E/a2m = κ/a2 (where κ = 2E/m) Dimension check: [(m/s)/m]2 - [(kg/m3).(m3/kg.s2)] = [(kg.m2/s2)/(kg.m2)] ✓ [s-1]2 - [s-1]2 = [s-1]2 Note that κ is a dimensionless. This is the FRIEDMANN-ROBERTSON-WALKER equation for the MASS DOMINATED PERIOD. ρ is a function of the scale factor (hence time). As the volume expands, the density will decrease if the amount of mass in the universe is assumed to be constant (conserved). To account for this, ρ can be written as: ρ = ν/a3 where ν is now fixed. k = 0 Case ---------- Note when the RHS is equal to 0 we get the familiar formula for the escape velocity. The energy is just enough to prevent the sphere from re-collapsing but not enough to make it expand further. Therefore, . (a/a)2 = (8/3)πρG Reinstate ρ = ν/a3 and let (8/3)πGν = α. Therefore, . (a/a)2 = α/a3 . a = √α/√a da/dt = √α/√a ∴ dt/da = √a/√α With solution: t = (2/3)a3/2/√α Rearranging: a = (3√α/2)2/3t2/3 Therefore, during the mass dominated period, the universe grew at a rate according to: a ∝ t2/3 Radiation Dominated Period --------------------------- This occurred before the mass dominated period. The Universe has 109 photons for every proton. Photons are massless but have energy by virtue of their motion. Ep = hν = c/λ (de Broglie) As the universe expands, λ, gets stretched and so the energy (and temperature) diminish by a factor of 1/a. There will now be an extra a in the denominator of the FRW equation. Thus, . (a/a)2 = (8/3)πMG/a4 Solve this equation in the same way as before to get: p = 1/2. So, a = (6πMG)1/3t1/2 . ∴ H = (a/a) = 1/(2t) Therefore, during the radiation dominated period, the universe grew according to a power law of 1/2. Vacuum Dominated Period ----------------------- First, we need to talk about pressure and energy density for radiation dominated period. Consider many photons sloshing backwards and forwards inside a 1D box: |--------------------<>|<- wall <------ L ------> F = dp/dt The change in photon momentum is 2p as it bounces off the wall. The bouncing creates a pressure on the wall. Time between collisions = 2L/c F = 2p/(2L/c) = pc/L = E/L = ρ in 1D. Now consider a cube with side, a: -->x ------ / /| -----/ |-> P (pressure) | |A| | | / | |/ ----- <- L -> F = E/L Therefore P = E/LA => P = ρ But this is only for one direction. To get the correct answer for 3D we need to divide by 3, Thus, P = ρR/3 Now consider the above box again. Let's move the face, A, of the cube by an amount dL. The work done is PAdL or PdV. The decrease in the internal energy of the box is -PdV. Now E inside the box is: E = ρV Therefore, dE = ρdV + Vdρ = -PdV or, Vdρ = -(P + ρ)dV Now introduce the parameter, w, as P = ρw Vdρ = -(w + 1)ρdV dρ = -(w + 1)ρdV/V dρ/ρ = -(w + 1)dV/V lnρ = -(w + 1)lnV after integration = ln{1/Vw+1} + c' Where, c' is the constant of integration. ρ = c'/Vw+1 = c'/a3(w+1) since V = a3 Rewrite the FRW equation (a/a)2 + k/a2 = (8/3)πρG, as follows: . (a/a)2 + k/a2 = (8/3)πGc'/a3(1 + w) We can now use w to describe any combination of periods. Radiation Dominated: w = 1/3 => (8/3)πGc'/a4 => (8/3)πGρR Mass Dominated: w = 0 => (8/3)πGc'/a3 => (8/3)πGρM Vacuum Dominated (Dark Energy): w = -1 => c' => (8/3)πGρ0 The quantity 8πGρo is the COSMOLOGICAL CONSTANT, Λ. de SITTER SPACE --------------- A universe where the only source of energy comes from the vacuum (ρ0 ≠ 0, ρM = ρR = 0) is referred to as de SITTER SPACE. Thus, . (a/a)2 = H02 = Λ/3 - k/a2 Where Λ/3 = (8/3)πGρo. For the k = 0 case we get: . (a/a) = H0 ∴ da/dt = aH0 This has the solution: a = Aexp(H0t) Therefore, in the vacuum dominated phase the Universe is growing exponentially. The metric becomes: dτ2 = dt - Aexp(2H0t)(dx2 + dy2 + dz2) Recall that: v = H0D = D√(8πGρ0/3) For v = c this gives: DH = c/√(8πGρ0/3) Since ρ0 is a constant, DH is fixed. This distance corresponds to the distance at which we cannot see beyond because the universe is expanding at the speed of light. DH is referred to as the COSMIC HORIZON. The implication of this is that as the universe continues to expand, the objects that we see in the night sky today will eventually slip through the cosmic horizon and become invisible. Another way of saying this is that the light emitted at the horizon is effectively redshifted to infinite wavelengths. Observers in the distant future will look up at the heavens and see a different universe than we see today. Ultimately, the night sky will become completely black! The radius of the cosmic horizon is approximately 46.6 billion light years. The radius of the surface of last scattering is about 45.7 billion light years. Eventually the SOLS will pass through the cosmic horizon and the CMB will disappear. Note: The age of the universe is estimated to be 13.8 billion years. However, the radius of the observable universe is not this number. In the real universe, spacetime is expanding and the distances obtained by multiplying the speed of light multiplied by this amount of time has no direct physical meaning. It is worth noting that while the universe as a whole is expanding, it can be contracting locally due to gravity. Summary ------- Period | Density | Expansion | P | w ----------+----------+-------------+------+-------- Radiation | ∝ 1/a4 | ∝ t1/2 | ρR/3 | w = 1/3 Mass | ∝ 1/a3 | ∝ t2/3 | 0  | w = 0 Vacuum |constant | ∝ exp(H0t)  | -ρ0 | w = -1 Where H0 = √(Λ/3) = √(8πGρ0/3) The Universe grew fairly rapidly during the radiation period, slowed down during the mass period and grows exponentially during the vacuum period. This is illustrated below. Geometry of the Universe ------------------------ To begin this discussion it is necessary to consider the geometry and topology of n-spheres. n-spheres can be viewed in terms of embedding spaces. However, n-spheres can also be described without an embedding if we only consider their boundaries or surfaces. For example, the description of a circle on a plane requires 2D space, (x,y) or (r,φ) but the boundary does not need to be embedded in the same 2D space to have a meaning. A person living on a circle can travel around the entire circle by going backwards or forwards. To them, the world is a just 1-dimensional space. With this in mind it is possible to describe n-spheres in terms of spatial metrics as follows: Metric for the 0-Sphere ----------------------- The 0-sphere is the pair of points at the ends of a line segment, r. The spatial metric is: ds2 = dr2 Metric for the 1-Sphere ----------------------- The 1-sphere is a circle defined in terms of polar coordinates (r,φ) where φ ranges from 0 to 2π. The spatial metric is: ds2 = dr2 + r22 = dr2 + r212 For the unit 1-sphere, r = 1. Therefore: ds2 = dφ2 = dΩ12 Metric for the 2-Sphere ----------------------- The 2-sphere represents the 2 dimensional surface of conventional sphere defined in 3D space. The 2-sphere can be looked at as the 'stacking' of 1-spheres on top of each other where the circle diameters vary continuously from 0 to 1 and then back to 0 in accordance with sinθ. All you need to reach every spot on the surface of a sphere is to be able to go left or right, up or down. Thus, a person living on the surface of a 2-sphere would see the world in 2 dimensions. The spatial metric is: ds2 = dθ2 + sin2θdΩ12 = dΩ22 Metric for the 3-Sphere ----------------------- A 3-sphere is the 4D analog of a 2-sphere. Although a 2-sphere exists in 3D space, its surface is 2 dimensional. Similarly, a 3-sphere has a 3 dimensional surface which is embedded into 4D space. Since there is no such space in our physical world it is impossible to imagine it in the same way as we imagine 3d space. Our minds just don't have any experience of being in 4d space. In this sense, the 4th dimension of the 3-sphere is unobservable, therefore unknown. However, it is possible to do math in 4D, and, in fact, in any-dimensional space. Note: 4D in this context refers to space only and should not be confused with 4D spacetime. The 3-sphere is also referred to as a hypersphere. It can be viewed as the nesting of concentric 2-spheres where the sphere diameters vary continuously from 0 to 1 and then back to 0 in accordance with sinχ. The spatial metric is: ds2 = dχ2 + sin2χdΩ22 = dΩ32 For the 2-Sphere embedded in 3D flat space: ds2 = dx2 + dy2 + dz2 This can be written in terms of spherical coordinates x = χcosθ, y = χsinθcosφ and z = χsinθsinφ as: - - - -   | 1 0   0   || dr2 | ds2 = | 0 χ2 0   || dθ2 |   | 0 0 χ2sin2θ || dφ2 | - - - - = dχ2 + χ2(dχ2 + sin2θdφ2) = dχ2 + χ2(dχ2 + sin2θdΩ12) = dχ2 + χ222 = dΩ32 Positive Spacial Curvature, k = +1 ---------------------------------- ds2 = dχ2 + sin2χdΩ22   = dΩ32 Flat Spacial Curvature, k = 0 ----------------------------- ds2 = dx2 + dy2 + dz2 ≡ dχ2 + χ222 Negative Spacial Curvature, k = -1 ---------------------------------- For the negative curvature case we have to use the ideas of hyperbolic geometry. It turns out that in this regime the metric is given by: ds2 = dχ2 + sinh2χdΩ22   = dℋ32 dℋ32 is called the hyperbolic plane. Note: sinhχ = (eχ - e)/2 where χ is not an angle but a distance. Unlike sinχ, which is periodic with π, sinhχ is not and proceeds exponentially to ∞ as χ increases. The FRW Metric -------------- For an n-sphere, we can incorporate the expansion of space by modifying the metric as follows: ds2 = a(t)2n-12 The spacetime metric is: dτ2 = dt2 - a(t)2n-12 We can now summarize all of the above cases as follows: dτ2 = dt2 - a(t)2[dχ2 + ξ(χ)222] Where, ξ2(χ) = χ2 k = 0 ξ2(χ) = sin2χ k = +1 ξ2(χ) = sinh2χ k = -1 Now, let r ≡ ξ(χ). r = χ case: dr2 = dχ2 ∴ dχ2 = dr2/(1 - 0.r2) r = sinχ case: dr2 = cos2χdχ2 ∴ dχ2 = dr2/(1 - sin2χ) = dr2/(1 - 1.r2) r = sinhχ case: dr2 = cosh2χdχ2 ∴ dχ2 = dr2/(1 + sinh2χ) = dr2/(1 + 1.r2) In each case the denominator can be written as: (1 + kr2) Where, k = 0 (flat), +1 (spherical) or -1 (hyperbolic). Using the above. ds2 = dt2 - a2(t)(dχ2 + ξ2(χ)[dθ2 + sin2dθdφ2] = dt2 - a2(t)(dr2/(1 - kr2) + r2[dθ2 + sin2dθdφ2] In matrix form the FRW metric is: - - - -    | 1 0    0 0     || dt2 | gμν = | 0 -a2(t)/(1 - kr2) 0 0     || dr2 |    | 0 0 -a2(t)r2 0     || dθ2 |    | 0 0    0 -a2(t)r2sin2θ || dφ2 | - - - - The FRW metric fullfills the COSMOLOGICAL PRINCIPLE that the distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale. General Relativity and the FRW Equation --------------------------------------- The FRW equations can be derived from Einstein's field equations using the above metric. The Christoffel symbols are found from: Γμνρ = (1/2)gρσ(∂μgνσ + ∂νgμσ - ∂σgμν) Example: Γ110 = (1/2)g00(∂1g10 + ∂1g10 - ∂0g11) = -(1/2)g000g11 = -(1/2)[-∂ta2/(1 - kr2) . = aa/(1 - kr2) The Ricci tensor is found from: Rμν = ∂ρΓμνρ - ∂νΓμρρ + ΓμνρΓρσσ - ΓμσρΓνρσ Therefore, .. R00 = -3(a/a) .. . R11 = (a.a + 2aa + 2k)/(1 - kr2) .. . R22 = (a.a + 2aa + 2k)r2 .. . R33 = (a.a + 2aa + 2k)r2sin2θ The Scalar Curvature is found from: R = gμνRμν Therefore, .. . R = -6(a/a) + (a2/a2) + (k/a2) The Einsten Field Equations are given by: Gμν = Rμν - (1/2)gμνR G00 = R00 - (1/2)g00R .. .. . = -3(a/a) + (1/2)6((a/a) + a2/a2 + k/a2) . = 3(a2 + k)/a2 The Stress-Energy Tesnsor is: - -    | ρ 0 0 0 | Tμν = | 0 -p 0 0 |    | 0 0 -p 0 |    | 0 0 0 -p | - - Therefore, . 3(a2 + k)/a2 = 8πGρ + Λ Or, . (a/a)2 + k/a2 = 8πGρ/3 + Λ/3 This is the FRIEDMANN EQUATION. Next, find the trace inverted form of the EFE. This form is more appropriate for some types of calculations. Rμν - (1/2)gμνR + gμνΛ = 8πGTμν ... 1. gμνRμν - (1/2)gμνgμνR + gμνgμνΛ = gμν8πGTμν Now gμνgμν = D where D is the spacetime dimension. Therefore, R - (1/2)4R + 4Λ = 8πGT -R + 4Λ = 8πGT Multiply this by (1/2)gμν to get: (-1/2)gμνR + 2gμνΛ = (1/2)gμν8πGT ∴ (-1/2)gμνR = -2gμνΛ + (1/2)gμν8πGT Substitute this into 1 to get: Rμν - gμνΛ = 8πGTμν - (1/2)gμν8πGT ∴ Rμν = 8πG(Tμν - (1/2)gμνT) + gμνΛ R00 = 8πG(T00 - (1/2)g00T) + g00Λ .. -3(a/a) = 8πG(ρ - (1/2)(ρ - 3p) - Λ or, .. (a/a) = -4πG(ρ + 3p)/3 - Λ/3 This is often called the FRIEDMANN ACCELERATION EQUATION. This equation states that both the volumetric mass density, ρ, and the pressure, p, cause the rate of expansion of the universe to decrease. On the other hand, Λ, acts in the opposite way to increase the rate of expansion. The connection between the EFE and Newton is shown below: . . . . . M . . . . + .m (x = 1) . 0 . . . . . . . <-- D --> <-- R --> D = aR . v = aR .. A = a.R (acceleration) .. F = -GmM/R2 = a.R .. a/a = -MG/a3R3 . = -(4/3)πGρ(a) [V = (4/3)πD3 = (4/3)πa3R3] = -(4/3)πGν/a3 D(t) = a(t)D0 D0 = D(t0) => a(t0) = 1 . [v] = [aR] = [(1/t)L] Measuring Curvature of the Universe ----------------------------------- In flat space angles subtended by objects at a distance, r, with a diameter, d, vary as: A ^ / x \ / x x \ r / x x d = true size / x x \ /dθ x v o------------ B r d2 = dr2 + r22 dr2 = 0 (r same at A and B) ∴ dθ = d/r In positively curved space, angles subtended by objects vary as: d2 = sin2rdθ2 ∴ dθ = d/sinr In negatively curved space angles subtended by objects vary as: d2 = sinh2rdθ2 ∴ dθ = d/sinhr These are just really statements that the sum of the angles of a triangle is > 180° for k = +1, < 180° for k = -1 and = 180° for k = 0. These results can be summarized as follows (d = 1): r 1/r 1/sinr 1/sinhr -- ----- ------ ------- 0 ∞ ∞ ∞ π/8 2.55 2.61 2.48 2π/8 1.27 1.41 1.15 3π/8 0.85 1.08 0.68 4π/8 0.64 1.00 0.43 5π/8 0.51 1.08 0.29 6π/8 0.42 1.41 0.19 7π/8 0.36 2.62 0.13 π 0.32 ∞ 0.09 If we know the distance, r, and the true size of a very distant object (i.e. we pick a object that has similar characteristics to an object, like our own galaxy, whose size is known) we can determine if the subtended angle is =, >, or < than the value for flat space at the same distance. Another, indication of curvature is the number of objects observed as a function of distance. If the universe is flat it would be expected that the number of objects increases proportionally to r2. If the universe has positive curvature it would be expected that the number of galaxies increases more slowly and then turns over and starts to decrease again in accordance with sin2r. If the universe has negative curvature it would be expected that the number of galaxies increases faster than in flat space since now N ∝ sinh2r. This can be seen in the following table: r r2 sin2r sinh2r -- ---- ----- ------ 0 0 0 0 π/8 0.15 0.15 0.16 2π/8 0.62 0.50 0.75 3π/8 1.39 0.85 2.16 4π/8 2.47 1.00 5.30 5π/8 3.86 0.85 12.20 6π/8 5.55 0.50 27.35 7π/8 7.56 0.15 60.58 π 9.87 0 133.48 These scenarios can be illustated schematically as follows. Hyperbolic space is difficult to visualize, but the drawing on the right by MC Escher for dℋ22 is a good proxy. The Critical Density -------------------- The general case of the FRW equation can be written as: . (a/a)2 = 8πGρR/3 + 8πGρM/3 + Λ/3 - k/a2 H02 = 8πGρR/3 + 8πGρM/3 + Λ/3 - k/a2 1 = 8πGρR/3H02 + 8πGρM/3H02 + Λ/3H02 - k/a2H02 1 = ΩR + ΩM + Ω0 - k/a2H02 The ratio 3H02/8πG is called the CRITICAL DENSITY, ρC. It corresponds to the case where k = 0 (at the escape velocity). This is calculated today to be 9.47 x 10-27kg/m3. Ω = ρ/ρC is the DENSITY PARAMETER. If ΩR + ΩM + Ω0 = 1 then k = 0 If ΩR + ΩM + Ω0 > 1 then k = -1 If ΩR + ΩM + Ω0 < 1 then k = +1 Determination of the Ω's is an ongoing process in cosmology. Direct observations indicate that the radiation component can be ignored. To get values for the remaining terms, complex mathematical models have been developed that mimic experimental results. One such model is the Lambda-CDM (Cold Dark Matter) model. Basically, the model describes the the existence and structure of the CMB, the accelerating expansion of the universe observed in the redshifted light from distant galaxies and supernovae, the abundances of the light elements hydrogen, helium and lithium and the large scale distribution of galaxies. It is based on the FRW metric (see below) and the FRW equations and their solutions. The inputs consists of a number of different parameters that include the baryonic density, the dark matter density and the dark energy density. An initial 'guess' is made for the values of each of these parameters, and the model output is compared against physical observations. The process is iterative as parameters are adjusted to improve the fit to the observed data. Values of the Ω's at the present time are indicated to be: ΩR = 0 ΩM ~ 0.3 ΩΛ ~ 0.7 This, along with astronomical measurements described before, leads to the conclusion that k ~ 0. In other words, the Universe is remarkably flat.