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Astronomy

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Celestial Coordinates
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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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Classical Physics

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Archimedes Principle
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Bernoulli Principle
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Blackbody (Cavity) Radiation and Planck's Hypothesis
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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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Rotational Dynamics
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Specific Heat, Latent Heat and Calorimetry
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Wien's Displacement Law
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Young's Modulus

Climate Change

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Cosmology

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Penrose Diagrams
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Geometries of the Universe
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Inflation Theory
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Finance and Accounting

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Annuities
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Brownian Model of Financial Markets
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Lecture Notes on International Financial Management
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis

General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes
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Dirac Equation in Curved Spacetime
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Einstein's Field Equations
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Tensors
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The Metric Tensor
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Vierbein (Frame) Fields
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World Lines Refresher

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics
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Macroeconomics

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Lecture Notes on International Economics
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Macroeconomic Policy

Mathematics

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Amplitude, Period and Phase
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Building Groups From Other Groups
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Complex Numbers
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Contravariant and Covariant Components of a Vector
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Grassmann and Clifford Algebras
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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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Symmetric Groups
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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 Integers Modulo n Under + and x
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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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Vector Calculus
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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity and Chirality
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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
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The Vacuum Catastrophe

Probability and Statistics

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Box and Whisker Plots
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Categorical Data - Crosstabs
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Chebyshev's Theorem
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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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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Set Theory - Venn Diagrams
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Stacked and Unstacked Data
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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 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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Basic Relationships
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Bell's Theorem
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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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Entangled States
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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 Exclusion Principle
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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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Schrodinger Wave Equation (continued)
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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 Observer Effect
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The Qubit
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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 4 - Potential
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Energy and Momentum, E = mc2
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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 Lorentz Group
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Velocity Addition

Statistical Mechanics

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Black Body Radiation
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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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BCS Theory
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Introduction to Superconductors
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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

test

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test

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
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Formulas
Last modified: January 4, 2020

The Essential Mathematics of General Relativity ----------------------------------------------- We will begin our discussion with a review of vectors and dual vectors in flat spacetime and then transition the discussion to curved spacetime. Coordinate Transformations in Flat Spacetime -------------------------------------------- Vectors exist independently of coordinate systems. For example, a velocity is a velocity regardless of the coordinate system we choose. If we are moving in a car at 50 MPH in a westerly direction the speedometer reads 50 MPH and a compass will read W without any notion of an coordinate system. Therefore, it is a real physical quantity. Note: A position vector is different. Its direction will depend on the coordinate system because it is defined with respect to one coordinate system. However, a displacement vector (the difference between 2 points) is coordinate independent. If a coordinate basis is chosen, then the components of vector with respect to that basis can be written. We can see this easily. V = v1e1 + v2e2 + v3e3 Where, v1, v2, v3 are the components (numbers). e1, e2, e3 is the basis (i.e. i, j, k). In Einstein notation this can be written as V = Vμeμ It should be obvious that as the coordinate basis changes, the components of the vector must change accordingly to maintain the invariance. We will now look at this in more detail. Vectors ------- Vectors, V, live in vector space, V. x2 | e2 | ^ | | | | | -----> e1 ---------------- x1 Consider a vector, V: V = Vμeμ -> V' = Vμ'eμ'    = Λμ'μVμeμ'    = (Λμ'μVμ)(Λσμ'eσ)    = (component)(basis) Now, Λμ'μΛσμ' = δσμ so the implication is that Λσμ' has to be the inverse of Λμ'μ. So if the components change in one way, the basis has to change in the opposite way for ds to be invariant. Thus, Vμ -> Vμ' = Λμ'νdxν and, eμ -> eμ' = Λσμ'eσ Mathematically, components are said to transform contravariantly and bases are said to transform covariantly. Dual Vectors ------------ Dual vectors, ω, live in dual vector space, V*. Like vectors They are invariant and can be expressed in terms of components and dual basis vectors. They linearly 'eat' vectors to produce scalars (more on this later). x2 | θ2 | ^ | | | | | -----> θ1 ---------------- x1 We will define 'eating' in terms of basis vectors and dual basis vectors as follows: θμeν = δμν     This is interpreted as θμ acting on eν to produce a scalar. The bases transform as: θμeν = δμν -> θμ'eν' = δμ'ν' Now eν' = Λνν'eν and θμ' = Λμ'μθμ Therefore, δμ'ν' = (Λμ'μθμ)(Λνν'eν) = Λμ'μΛνν'θμeν So, in order for this to hold true: Λμ'μ ≡ Λ-1 and, Λνν' ≡ Λ In spacetime the simplest example of a dual vector is the gradient of a scalar function, φ, which is given by: ω = (∂φ/∂xμμ Proof: We can rewrite ∂φ/∂xμ using the Chain Rule as: ∂φ/∂xμ -> ∂φ/∂xμ' = (∂xμ/∂xμ')(∂φ/∂xμ) = λμμ'(∂φ/∂xμ) λμμ' is the transformation law for dual vector components so we can conclude that ∂φ/∂xμ is a dual vector. The Dot Product --------------- Consider a dual vector, ω, eating a vector, V. ωV = ωμθμvνeν = ωμvνδμν = ω0v0 + w1v1 + w2v2 + w3v3 This looks like the dot product. Therefore, the dot product is 'secretely' the act of taking one of the vectors and turning it into a dual vector by the application of the metric tensor i.e., A.B = ημνAμeμBνeν = AνθμBνeν = AνBνδμν The Lorentz Transformation -------------------------- It is conventional to write the inverse Lorentz transformation as: (Λαβ)-1 ≡ Λαβ Consider a boost in the x direction. The partial boost matrix is: - - B = | γ -βγ | = Bμν | -βγ γ | - - With inverse, - - B-1 = | γ βγ | = Bμσ    | βγ γ | - - ----------------------------------------------------- Digression: Consider a boost in the x direction: - - - - - - | γ -βγ || ct | = | γt - βγx | | -βγ γ || x | | -βγt + γx | - - - - - - And the inverse transformation: - - - - - - | γ βγ || ct' | = | γt' + βγx' | | βγ γ || x' | | βγt' + γx' | - - - - - - From these we get: x' = γx - βγct x = γx' + βγct' t' = γct - γβx t = γct' + βγx' ----------------------------------------------------- The components transform as: v0' = Bμνvν = B00v0 + B01v1 = γv0 - βγv1 v1' = Bμνvν = B10v0 + B11v1 = -βγv0 + γv1 The basis transforms as: e0' = Bμσeσ = B00e0 + B01e1 = γe0 + βγe1 e1 = Bμσeσ = B10e0 + B11e1 = βγe0 + γe1 The transformed vector is identical to the vector before the transformation. Proof: v0'e0' + v1'e1' = (γv0 - βγv1)(γe0 + βγe1) + (-βγv0 + γv1)(βγe0 + γe1) = (γ2v0e0 + βγ2v0e1 - βγ2v1e0 - β2γ2v1e1) + (-β2γ2v0e0 - βγ2v0e1 + βγ2v1e0 + γ2v1e1) = (γ2v0e0 - β2γ2v1e1) + (-β2γ2v0e0 + γ2v1e1) = γ2(1 - β2)v0e0 + γ2(1 - β2)v1e1 = v0e0 + v1e1 Now consider a rotation about the z axis. The partial rotation matrix is: - - R(90°) = | 0 1 | = Rμν   | -1 0 | - - With inverse, - - R(-90°) = | 0 -1 | = Rμσ   | 1 0 | - - The components transform as: v1' = Rμνvν = R11v1 + R12v2 = v2 v2' = Rμνvν = R21v1 + R22v2 = -v1 The basis transforms as: e1' = Rμσeσ = R11e1 + R12e2 = e2 e2' = Rμσeσ = R21e1 + R22e2 = -e1 e2 e1' (= e2) | | | | | -> | | | --------- e1 e2' --------- (= -e1) The transformed vector is identical to the vector before the transformation. Proof: v1'e1' + v2'e2' = v2e2 + (-v1)(-e1) = v1e1 + v2e2 Lorentz Transform as a Coordinate Transformation ------------------------------------------------ The Lorentz transformation is just a special case of a coordinate transformation. Again, consider a boost in the x direction: Λμ'μ | v - - - - - - | coshζ -sinhζ 0 0 || t | = | coshζt - sinhζx | | -sinhζ coshζ 0 0 || x | | -sinhζt + coshζx | | 0 0 1 0 || y | | y | | 0 0 0 1 || z | | z | - - - - - - - - | t' | = | x' | | y' | | z' | - - We can write this as a coordinate transformation as a JACOBEAN MATRIX:     - -     | ∂t'/∂t ∂t'/∂x ∂t'/∂y ∂t'/∂z | Λμ'μ = | ∂x'/∂t ∂x'/∂x ∂x'/∂y ∂x'/∂z |     | ∂y'/∂t ∂y'/∂x ∂y'/∂y ∂y'/∂z |     | ∂z'/∂t ∂z'/∂x ∂z'/∂y ∂z'/∂z |     - -     - -     | coshζ -sinhζ 0 0 |     = | -sinhζ coshζ 0 0 |     | 0 0 1 0 |     | 0 0 0 1 |     - - Alternatively, we can see this in another way: t' = γct - γβx x' = γx - βγct y' = y z' = z Therefore, ∂t'/∂t = γc ∂t'/∂x = -γβ ∂t'/∂y = ∂t'/∂z = 0 And, ∂x'/∂t = -γβc ∂x'/∂x = γ In general we can write: Λμ'μ = ∂xμ'/∂xμ This produces the matrix: - -     | γ -γβ 0 0 | Λμ'μ = | -γβ γ 0 0 |     | 0 0 1 0 |     | 0 0 0 1 | - - Now consider the inverse transform given by: t = γct' + βγx' x = γx' + βγct' y = y' z = z' In this case we get: ∂t/∂t' = γc ∂t/∂x' = βγ ∂t/∂y' = ∂t/∂z' = 0 And, ∂x/∂t' = βcγ ∂x/∂x' = γ In general we can write: Λμ'μ = ∂xμ/∂xμ' This produces the inverse transformation matrix we had before: - -     | γ γβ 0 0 | Λμ'μ = | γβ γ 0 0 |     | 0 0 1 0 |     | 0 0 0 1 | - - Summary of Transformation Laws ------------------------------ Vector Components: Vμ -> Vμ' = Λμ'μVμ Basis vectors: eμ -> eμ' = Λσμ'eσ Dual Vector Components: ωμ -> ωμ' = Λμμ'ωμ Dual Basis vectors: θμ -> θμ' = Λμ'σθσ Vectors: V -> V' = Λμ'μVμΛσμ'eσ = V Dual Vectors: ω -> ω' = Λμμ'ωμΛμ'σθσ = ω Λσμ'Λμ'μ = δσμ The direction of the transformation is always from the bottom index to the top index. Coordinate Transformations in Curved Spacetime ---------------------------------------------- Manifolds --------- A CP manifold, M, is a collection of points in spacetime with a maximal atlas. We can break up M into PATCHES. All of M needs to be covered. We have only shown a few patches to illustrate the concept. A COORDINATE CHART (aka chart) is a subset of M with a 1-to-1 map, φ: φ: Uα -> ℝn where ℝn is n-dimensional Euclidean space. (Note: R ≡ ℝ in diagrams). Such that the image of φ is open. Open means the interior of an n - 1 dimensonal closed surface (i.e. the edges of the image are not included). Note that charts must have the same dimensionality as the manifold. The fact that manifolds look locally like ℝn, which is manifested by the construction of coordinate charts, introduces the possibility of analysis on manifolds, including operations such as differentiation and integration. In other words φ(p) converts a point, p, in U on M to a point with coordinates xμ in ℝn. Thus, φ(p) = (x1(p), x2(p) ... xn(p)) or, in general: φμ(p) = xμ(p) Where xμ(p) are COORDINATE FUNCTIONS. They are scalar functions that return the μth coordinate of P as a real number. In a fixed coordinate system, each of the 4 coordinates, xμ, can be thought of as a function on spacetime. Therefore, xμ = xμ(p) For example: . . --------- . uα . | ℝ2| . p--.-- φ(p) --|--->xμ | . . | | --------- This takes a point, p, in Uα on a 2D manifold, M2 and sends it to (x,y) ∈ ℝ2. An important thing to recognize is that if a manifold can be covered by a single chart, we can have a global coordinate system on M. If it cannot be covered by a single chart then it is not possible to have a system of global coordinates. Atlases ------- An ATLAS on M is a collection of charts {(Uα,φ)} such that: 1. The union of all charts Uα is equal to M 2. The charts are sewn together with CP TRANSITION FUNCTIONS. CP Transition Functions ----------------------- Suppose that 2 charts overlap. Now, P3 = φ3(p) = φ3 o φ4-1(P4): ℝn -> ℝn Now, the interesting thing is that P3 and P4 are both in ℝn. Therefore, we are taking the point P4 in one set of coordinates and expressing it as the point P3 in another set of coordinates. In other words, this is just a ccoordinate transformation. P4 with coordinates (x4, y4, z4 ...) -> P3 with coordinates (x3, y3, z3 ...) The transition function, φ43 is defined as: φ43 = φ3 o φ4-1 Where φ3 and φ4-1 are coordinate and inverse coordinate functions respectively. Now that we are dealing with functions from ℝn to ℝn we can talk about using calculus, in particular we can consider the derivatives of φαβ. If the pth derivative of all φαβ exists and is continuous then φαβ is Cp. If P = ∞ (i.e. C) then we call φαβ smooth. Maximal Atlas ------------- It is possible that a manifold may be covered by different choices of atlases. In this case each atlas is an equally good representation of the manifold so they belong to the maximal atlas. Therefore, a Cp n-dimensional manifold is a set, M, with a maximal atlas. In summary, we can construct an atlas of charts and use the maps φα: uα -> ℝn to coordinatize M on each chart. We can then sew them together with transition functions which are equivalent to coordinate transformations. Manifolds in SR and GR ---------------------- With curved manifolds it is important to remove the idea that a vector stretches from one point to another, but instead is just an object associated with a single point (the tangent space - see next) that facilitates the use of multivariable calculus. Manifolds can then be equipped with a metric at each point that allows the existence of the inner product. Consider the infinitesimally small displacement vector, ds: ds2 = gαβdxαdxβ = g00dx0dx0 + g01dx0dx1 + g10dx1dx0 + g11dx1dx1 If ds2 is always positive the manifold is RIEMANNIAN If ds2 can take any value it is PSEUDO-RIEMANNIAN. If ds2 takes the form: ds2 = -dt2 + dx2 + dy2 + dz2 it is LORENTZIAN. Because ds2 can have different signs in GR, the manifolds are pseudo-Riemannian. Armed with this knowledge we can now proceed with a discussion of tangent and cotangent spaces on manifolds. Tangent Space ------------- Consider an n-dimensional mamifold, M. Consider a SCALAR field, ψ(p), that lives in the manifold. γ(λ) maps λp in ℝ1 to the 1 dimensional curve, γ(λp), that passes through a particular point, p, on the manifold. So as λp changes, P will move along the curve γ. f is a scalar function that maps any point, P, on the manifold to αp in ℝ1. Therefore, f takes any point on the manifold and assigns to it a real number (i.e. a scalar). Note, λ and α are not charts since they do not have the same dimensions as the manifold. We want to find how the arbitrary function, f, changes as we change λp. Since f, gives us αp this is equivalent to seeing how αp changes with λp. Thus, df/dλ = d(f o γ)/dλ This is called the DIRECTIONAL DERIVATIVE OF f ALONG THE CURVE γ Note: F o G ≡ F(G(x)) so the order of execution is from right to left. This is equivalent to the tangent at P along the curve γ. If we repeat this for other curves through P we wlll construct set of directional derivatives that forms a vector space of tangents at P called the TANGENT SPACE, Tp, at p. This tangent space is an example of a vector spaces. The set of all the tangent spaces of a manifold is called the TANGENT BUNDLE. Now that we have a tangent space we can now make use of charts to setup a basis and, therefore, coordinate representations of vectors in that tangent space. λp -> P -> xμ -> P -> αp f o γ = f o φ-1 o φ o γ: λp -> αp df/dλ = d[(f o φ-1) o (φ o γ)]/dλ Now φ o γ: ℝ1 -> ℝn so we can call it (φ o γ)μ Using the chain rule: df/dλ = [∂(f o φ-1)/∂(φ o γ)μ][d(φ o γ)μ)/dλ] Since (φ o γ)μ: ℝ -> ℝn we can call it xμ(λ) Since (f o φ-1): ℝn -> ℝ we can call it f(xμ) Therefore; df/dλ = (∂f/∂xμ)(dxμ/dλ) Or, d/dλ = (∂/∂xμ)(dxμ/dλ) Where, (dxμ/dλ) are the components of the TV. and, (∂/∂xμ) = ∂μ are the bases of the tangent space ≡ eμ d/dλ is a tangent vector in tangent space. Therefore, we should be able to generalize this for any vector, V, in that space. Therefore, we can write: V = Vμμ The basis vectors in the primed frame are given by the chain rule. Therefore, ∂μ -> ∂μ' = (∂xμ/∂xμ')∂μ Since V = Vμμ must be invariant under a transformation this implies: Vμμ = Vμ'μ' = Vμ'(∂xμ/∂xμ')∂μ Summarizing: Directional Derivative: d/dλ = (∂/∂xμ)(dxμ/dλ) Transformation law for basis vectors: ∂μ -> ∂μ' = (∂xμ/∂xμ')∂μ Transformation law for vector components: Vμ -> Vμ' = (∂xμ'/∂xμ)Vμ So, as in the case of flat spacetime, the basis transforms in the opposite direction to the components to preserve the invariance of the vector. Basis vectors are usually not written explicitly with the understanding that a change of coordinates induces a change of basis. ----------------------------------------------------- A slightly different view: Consider a curve on a manifold. We can define a displacement (separation) vector Δp along the curve. Δp has to short so that we can assume that the surface local to P is flat and we can use basic calculus to figure out the tangent vector, A, to the curve at p. The tangent vector to the curve, A, is given by: A = dp/dε ≡ lim [P(ε + Δε) - P(ε)]/Δε Δε->0 Any point on the curve can be written in terms of coordinates, xα. Coordinate basis: eα ≡ ∂p/∂xα Components of vector: Aα = dp/dε ∂xα/∂ε We can write A = dp/dε as: A = (dxα/dε)(∂p/∂xα) .. 1. ≡ Aαeα Therefore, we can identify the components of the vector as: Aα = dxα/dε Directional Derivative ---------------------- Consider a scalar field, φ(p), that lives in the manifold at position, p, on the curve. The derivative of φ along A, ∂Aφ, is given by: ∂Aφ(P(ε)) = Aα∂[φ(P(ε))]/∂xα Or, ∂A = Aα∂/∂xα = (dxα/dε)(∂/∂xα) ... 2. = (d/dε) along the curve. If we compare 1. with 2. we see that there is a direct 1-to-1 correspondence between the tangent vector and the directional derivative. Thus, A ≡ ∂A = ∂p/∂ε = ∂/∂ε Cotangent (Dual) Space ---------------------- In addition to tangent space it is also possible to construct COTANGENT SPACES, Tp* at each point on the manifold. Just as vectors are defined as elements of the tangent space, dual vectors are defined as elements of the cotangent space. The set of all cotangent spaces over a manifold is called the COTANGENT BUNDLE. A dual space, V*, is a vector space of linear maps from a vector space, V, to its field of scalars. The maps are also referred to as linear functionals, linear forms, one-forms, covectors or dual vectors. In layman's terms a linear functionals at P 'eat' the tangent vector at P producing a scalar. Therefore, ω: V -> ℝ Where ω ∈ V* (aka Tp*) and V ∈ V (aka Tp) Note that dim V = dim V*. In tangent space we used charts to set up coordinates that enabled us to define the basis for the tangent vector along λ. With cotangent space we use the gradients of the scalar coordinate functions, φμ(p) (≡ xμ(p)), denoted by (df), to determine the basis. Note that dxμ is not the same as dxμ which is an infinitesimal displacement vector. Therefore, we can write: (df): TpV -> ℝ Where (df) = (∂f/∂xμ) Dual spaces have to satisfy the operation of addition and scalar multiplication. (df)(V + V') = (df)(V) + (df)(V') (df)(αV) = α(df)(V) We now let (df) 'act' on the tangent vector d/dλ. Note 'act' in this case does NOT mean differentiate. Therefore, (∂f/∂xμ)(d/dλ) Now by the definition of linearity: (df)V = V o f V ∈ V, f ∈ V* Therefore, we get: (df)(d/dλ) = (d/dλ) o f = df/dλ We now write the tangent vector in terms of its components and bases. df(dxμ/dλ)(∂μ) = df/dλ We are free to write this as: (dxμ/dλ)(df)(∂μ) = df/dλ Now the RHS can be rewritten using the chain rule. (dxμ/dλ)(df)(∂μ) = (dxμ/dλ)(∂f/∂xμ) By comparison: (df)(∂μ) ≡ ∂f/∂xμ But f = xν(P), the coordinate functions, so: (dxν)(∂μ) = ∂xν/∂xμ = δμν In conclusion, θμ = (dxμ) Now, we had said previously that dual vectors 'eat' vectors to produce scalars, i.e. θμeν = δμν Or, dxμν = δμν Therefore, for invariance, dxμν = dxμ'ν' So, dxμ' = (∂xμ'/∂xμ)dxμ If this is the transformation law for dual basis vectors then the transformation law for dual vector components has to be: ωμ -> ωμ' = (∂xμ/∂xμ'μ Comparison with Quantum Mechanics --------------------------------- Dual spaces play an important role in Quantum Mechanics. QM uses complex functions to represent vectors and dual vectors. Since vectors live in complex Hilbert space, the dual vectors have to live in a conjugate space to ensure that the resulting wavefunctions are scalars. The state |α> does not represent a wave function until it is projected onto a basis in position or momentum space. For example, ψ(p) = <p|α> or ψ(x) = <x|α> ψ is a scalar and <p|α> just specifies the 'components' of |α> in that basis. We can also project <α| onto a basis: ψ*(p) = <α|p>* or ψ*(x) = <α|x> = <p|α>* and ψ*(x) = <x|α>* Likewise, the overlap of 2 states <φ|ψ> is a scalar since probabilities have to be postive numbers. Dual Space Vector Space ---------- ------------ Dual vector Vector One-form Tangent vector Covariant vector Contravariant vector Row vector Column vector Complex conjugate function Complex function (QM) Summary of Transformation Laws ------------------------------ SR: Vμ -> Vμ' = Λμ'μVμ and eμ -> eμ' = Λμμ'eμ ωμ -> ωμ' = Λμμ'ωμ and θμ -> θμ' = Λμ'μθμ Where Λμ'μ and Λμμ' allows global transformations of (t,x,y,z). GR: Vμ -> Vμ' = (∂xμ'/∂xμ)Vμ and ∂μ -> ∂μ' = (∂xμ/∂xμ')∂μ ωμ -> ωμ' = (∂xμ/∂xμ'μ and dxμ -> dxμ' = (∂xμ'/∂xμ)dxμ Where ∂xμ'/∂xμ and ∂xμ/∂xμ' allows local or coordinate dependent transformations of (t,x,y,z). Vectors: V -> V' = Λμ'μVμΛσμ'eσ = V Dual Vectors: ω -> ω' = Λμμ'ωμΛμ'σθσ = ω Λσμ'Λμ'μ = δσμ Pullbacks and Pushforwards -------------------------- Consider 2 manifolds M and N. The map: φ: M -> N φ is called a DIFFEOMORPHISM if it is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth. The PULLBACK of the function, f, from N to M is: f o φ (≡ f(φ)) Which is written as: φ*f While we can pull functions back there is no construction that allows them to be pushed forward. However, a tangent vector is defined as Aμμ. This allows us to define the PUSHFORWARD of a tangent vector, V, from M to N, as: φ o f (≡ φ(f)) Which is written as: φ*V The action of the pushed forward vector on f is defined to be: (φ*V)f = V(φ*f) What this means is that pushing V forward to N and acting on f is equivalent to pulling f back to M and acting on it with V. To see how this works in practice, apply the pushed forward vector Vαα to an arbitrary function, f. Therefore, (φ*V)ααf = Vμμ*f) = Vμμ(f o φ) = Vμ(∂φ/∂xμ)(∂f/∂yα) = Vμ(∂yα/∂xμ)(∂f/∂yα) = Vμ(∂yα/∂xμ)∂αf We can write this as: (φ*V)αα = (φ*)αμVμα Where (φ*)αμ = ∂yα/∂xμ is a matrix that represents the vector transformation law under a change of coordinates. When M and N are the same manifold the constructions are identical and ∂yα/∂xμ = 1. Example: Consider the map: - - φ(x,y) = | xy | or (xy,x + y) | x + y | - - And an arbitrary curve, γ(t), on the manifold, M, defined as: γ(t) = (1 + 3t,2 - 2t) Here we have replaced the parameter, λ, with the parameter, t. If we plot γ(t) for different values of t we get a straight line with slope = -2/3. Taking the differential of γ(t) w.r.t. t gives the vector (3,-2) which we can define at the point (1,2), i.e. The image curve of γ(t), β(t), is: β(t) = φ o γ = ((1 + 3t)(2 - 2t),(1 + 3t) + (2 - 2t)) = ((2 + 4t - 6t2),(3 + t)) β(0) = (2,3) dβ(t)/dt = ((4 - 12t),1) The tangent vector to the image curve at (2,3) when t = 0 is (4,1). Again, we define dβ(t)/dt at (2,3). i.e. Let us now compute this in a different way. We let: x = a(t), y = b(t) where a(t) and b(t) are coordinate functions. Therefore, - - β(t) = φ o γ = | a(t)b(t) | | a(t) + b(t) | - - and, - - dβ(t)/dt = | a(t)db(t)/dt + b(t)da(t)/dt | | da(t)/dt + db(t)/dt | - - - - - - = | a(t) b(t) || db(t)/dt | | 1 1 || da(t)/dt | - - - - The first matrix is just the Jacobean: - - J = | ∂fx/dx ∂fx/dy | | ∂fy/dx ∂fy/dy | - - - - = | x y | | 1 1 | - - The vertical sum of each column gives the coordinates of the pushed forward vector. x: a(t) + 1 = 1 + 1 = 2 y: b(t) + 1 = 2 + 1 = 3 To get the vector at (2,3) we apply the Jacobean to the vector at (1,2): - - - - - - | 1 2 || -2 | = | 4 | | 1 1 || 3 | | 1 | - - - - - - This can also be found as: (3∂x - 2∂y)(xy) = 3y - 2x = 6 - 2 = 4 (3∂x - 2∂y)(x + y) = 3 - 2 = 1 Vectors can be pushed forward from M to N but there is no construction that allows them to be pulled back. On the other hand, cotangent vectors, ω, behave in just the opposite manner. They can be pulled back from N to M but there is no construction to allow them to push forward from N to M. The action of the pulled back dual vector on V is defined as: (φ*ω)V = ω(φ*V) What this means is that pulling ω back to M and acting on V to produce a scalar is equivalent to pushing forward V to N and acting on it with ω. We can repeat the above process and write: (φ*ω)μ = (φ*)αμωα Where (φ*)αμ = ∂yα/∂xμ is the same matrix as before except that we now contract on a different index. We can extend the idea of pulling back dual vectors to tensors with an arbitrary number of lower indices. One common occurrence of a map between two manifolds is when M is actually a smooth embedded submanifold of N. Under these circumstances there is a map from M to N which just takes an element of M to the same element of N. Therefore, every embedded smooth submanifold inherits a metric from being embedded in a Riemannian manifold. For example, in String Theory there are 2 manifolds that are involved in string propagation. - The spacetime in which the string propagates. - The worldsheet of the string itself. The 2 dimensional world sheet is considered to be embedded in the spacetime manifold such that points (σ11) on the worldsheet correspond to points in the manifold xμ11). The worldheet then inherits its metric from the metric on the ambient spacetime manifold. Orthonormal and Coordinate Bases -------------------------------- Up until now we have defined the bases of tangent and cotangent space as: eμ = ∂μ and θν = dxν Where we set up the basis vectors to point along the coordinate axes. In 2D this corresponds to the following diagram. We can define the commutator [∂μeμ,∂νeν] = 0 There is nothing to stop us, however, from setting up any bases we like. We will choose these basis vectors to be orthonormal (orthogonal and of unit length) and denote them as ea and eb. In 2D this corresponds to the following diagram. The left is a flat space and the commutator equals 0. However, when we move to a curved surface the commutator no longer equals 0 but represents a difference vector (A). In general we can write: [eσ,eρ] = cσρλeλ Where cσρλ is the commutation coefficient. There is a theorem that says a basis is a coordinate basis if and only if cσρλ = 0. Calculations are easier in the coordinate basis. On the other hand, the orthonormal basis is the most familiar to people and is better for physical interpretation. For example, Quantum Mechanics, SR and classical physics generally are developed in terms of orthonormal bases. From the above diagram it should be clear that we cannot just impose an orthonormal basis on curved spacetime. What we need, therefore, is a mechanism that enables us to move from one coordinate system to the other. The approach in GR that replaces the coordinate basis with an orthonormal basis is referred to as the TETRAD FORMALISM. This was proposed by by Albert Einstein in 1928. Tetrads ------- In 4 dimensions, the set of vector fields in an orthonomal basis is called a TETRAD (aka VIERBEIN or FRAME FIELD). We can express the coordinate basis in terms of the orthonormal (tetrad) basis as: Vectors: eμ = eaμea or ea = eμaeμ One-forms: θμ = eμaθa or θa = eaμθμ Components of a vector written in a coordinate basis can be written in an orthonormal basis as: Va = eaμVμ alternatively, Vμ = eμaVa The components of eaμ etc. form n x n invertible matrices where the Latin indeces represent the orthonormal basis and the Greek indeces represent the coordinate basis. If the distinction between a vector and its components is dropped, eaμ etc. are regarded as the vierbeins. Naturally, they are functions of position on the manifold. They have the following properties: eμa = (eaμ)-1 eμaeaν = δμν eaμeμb = δab g(ea,eb) = ea.eb = ηab g(eμ,eν) = eμ.eν = gμν Now, ea.eb = eμaeνbeμ.eν = eμaeνbgμν Therefore, ηab = eμaeνbgμν or equivalently, gμν = eaμebνηab This last result expresses the fact that eaμ and ebν are very much like √gμν. Tetrads can be looked at as linear maps from tangent space to Minkowski space that preserves the inner product. The fact that we can write the Minkowski metric in terms of gμν gives us the freedom to perform LOCAL LORENTZ TRANSFORMATIONS at every point in space. Therefore, Λa'aΛbbηab = ηa'b' ea -> ea' = Λa'aea θa -> θa' = Λa'aθa Where (Λ-1)ν'μ = Λν'μ We also retain the freedom to make general coordinates transformations at the same time. Therefore, we can define a mixed tensor transformation law as follows: Ta'μ'b'ν' = Λa'a(∂xμ'/∂xμb'b(∂xν/∂xν')T Spin Connection --------------- The introduction of tetrads requires changing the covariant derivative. ∇μeaν must transform like a (1,1) tensor and a 4-vector. In order to account for this we need to modify the idea of a connection. Now ∇μηab ≡ ∂μηab = 0 so we can impose the same condition on the tetrad. Therefore, ∇μeaν = 0 ∇μeaν = ∂μeaν - Γρμνeaρ + ωaebν = 0 Multiply through by eνb to get: ωa = eνbΓρμνeaρ - eνbμeaν ωais called the SPIN CONNECTION. We can define a curvature associated with this connection: Rμνab = ∂μωνab - ∂νωμab + ωeμaωνeb - ωeνaωμeb Note that: eaρebαRμνab = Rμνρα Which is the Riemann tensor. Why Tetrads? ------------ Tetrads provide the ability to describe spinor fields on spacetime and take their covariant derivatives. They are the most natural way to represent a relativistic quantum field theory in curved space. It is also gauge field theory for gravity with the tetrad playing the role of a gauge field but not exactly like the vector potential field does in Yang-Mills theory. Covariant Derivative -------------------- We have established the transformation laws for vectors and dual vectors as well as scalars and partial derivatives. In summary, Scalars: φ -> φ Gradients of scalars: ∂φ/∂xμ -> ∂φ/∂xμ' = (∂φ/∂xμ)(∂xμ/∂xμ') Partial derivatives: ∂μ -> ∂μ' = (∂/∂xμ')∂μ Vectors: Vμ -> Vμ' = (∂xμ'/∂xμ)Vμ Dual vectors: Vμ -> Vμ' = (∂xμ/∂xμ')Vμ Tensors: Tμν -> Tμ'ν' = (∂xμ/∂xμ')(∂xν/∂xν')Tμν THe question is how do the derivatives of vectors and tensors transform?. Consider, a vector: ∂μVν = lim [Vν(xμ + εμ) - Vν(xμ)]/εμ ε->0 The problem is that Vν(xμ + εμ) and Vν(xμ) live in different tangent spaces so the coordinate system is not constant. This problem manifests itself as follows: ∂μVν -> ∂μ'Vν' = (∂xμ/∂xμ')∂μ(∂xν'/∂xν)Vν = (∂xμ/∂xμ')(∂xν'/∂xν)∂μVν + (∂xμ/∂xμ')Vνμ(∂xν'/∂xν) The first term is a tensor but the second is not. Therefore, since ∂μ'Vν' does not ransform like a tensor it is not a tensor. We can write the above equation as follows: ∇μVν = ∂μVν + ΓνμλVλ Where ∇μ is the COVARIANT DERIVATIVE and Γνμλ are the CHRISTOFFEL SYMBOLS. The Christoffel symbols solves the problem by taking Vν(xμ + εμ) and PARALLEL TRANSPORTING it back to xμ before performing the subtraction. Therefore, ∇μVν = lim [V//ν(xμ + εμ) - Vν(xμ)]/εμ ε->0 Note. ∇μ of a scalar is the same as ∂μ. Thus, ∇μφ = ∂μφ How do the symbols transform? ∇μVν -> ∇μ'Vν' = (∂xμ/∂xμ')(∂xν'/∂xν)∇μVν Γν'μ'λ' = (∂xμ/∂xμ')(∂xν'/∂xν)(∂xλ/∂xλ'νμλ + (∂xμ/∂xμ')(∂xλ/∂xλ')(∂2xν'/∂xμxλ) The first term is a tensor but the second term is not. It is this second term that counters the second term in the covariant derivative to produce a result that is a tensor. Computing the Christoffel Symbols --------------------------------- Before we do this we need to consider 2 things - torsion and metric compatibility. Torsion Tensor -------------- The torsion tensor is defined as: Tμνλ = Γλμν - Γλνμ Torsion is a difficult concept to explain. Loosely speaking it characterizes how tangent spaces twist about a curve when they are parallel transported whereas curvature describes how the tangent spaces roll along the curve. Suffice to say that in GR we assume the torsion to be 0 meaning that the Christoffel symbols are symmetric under the interchange of their lower indeces. Likewise the metric is also symmetric under the interchange its indeces. Metric Compatibility -------------------- A connection is metric compatible if the covariant derivative of the metric with respect to that connection is everywhere zero, i.e., ∇ρgμν = 0 Now we're ready to compute the Christoffel symbols. Lets expand out the equation of metric compatibility for three different permutations of the indices: 1. ∇ρgμν = ∂ρgμν - Γλρμgλν - Γλρνgμλ = 0 2. ∇μgνρ = ∂μgνρ - Γλμνgλρ - Γλμρgνλ = 0 ^ | ≡ Γλρμgλν 3. ∇νgρμ = ∂νgρμ - Γλνρgλμ - Γλνμgρλ = 0 ^ | ≡ Γλρνgμλ 1. - 2. - 3. gives: ∂ρgμν - ∂μgνρ - ∂νgρμ + 2Γλμνgλρ Γλμν = (1/2)(-∂ρgμν + ∂μgνρ + ∂νgρμ)/gλρ Noting that 1/gλρ = gσρ (i.e. gλρgσρ = δσλ) Γσμν = (1/2)gσρ(∂μgνρ + ∂νgρμ - ∂ρgμν) Note that Γσμν -> 0 as we get closer and closer to the local coordinate frame where the derivative of the metric = 0 and the space can be considered to be flat (tangent space). Γσμν = 0 does not by itself imply that the overall space is flat. This determination is made by examining the 2 second derivative of the metric as discussed above. Covariant Directional Derivative -------------------------------- Recall, for a scalar the directional derivative is: d/dλ = (dxμ/dλ)∂μ For a vector/tensor this becomes we can define the DIRECTIONAL COVARIANT DERIVATIVE as: D/dλ = (dxμ/dλ)∇μ Therefore, for a vector, Vν, DVν/dλ = (dxμ/dλ)∇μVν = lim [V//ν(xμ(λ + δλ)) - Vν(xμ(λ))]/Δxμ Δxμ->0μVν picks a coordinate, μ, and shifts Vν along xμ. DVν/dλ shifts Vν along the curve xμ(λ). Parallel Transport ------------------ From the above we have the CDD: DVν/dλ = (dxμ/dλ)∇μVν To transport Vν(xμ + εμ) along xμ to P requires keeping the vector as PARALLEL TO ITSELF AS POSSIBLE as it is transported. This condition implies that: DVν/dλ = (dxμ/dλ)∇μVν = 0 = (dxμ/dλ)∂μVν + Γνμρ(dxμ/dλ)Vρ = 0 In other words, the components of the tangent vector dxμ/dλ don't change as it is transported. Now, (dxμ/dλ)∂μ is just the ordinary DD so we can write: dVν/dλ + ΓνμρVρ = 0 This is a differential equation that can be solved for any point on xμ(λ). Geodesics --------- Geodesics are paths taken by objects under the influence of gravity. They are curves xμ(λ) that parallel transport their tangent vectors. In other words, if xμ(λ) is a geodesic then dxμ/∂λ (the components of the tangent vector) should be constant along the path (i.e. should be parallel to each other. The 4-velocity, uμ, along the curve is given by the directional derivative: uμ = dxμ/dλ The covariant directional derivative of uμ along the curve is given by: Duμ/dλ = D(dxμ/dλ)/dλ The fact that the components of the tangent vector should be constant implies D(dxμ/dλ)/dλ = 0. Therefore, D(dxμ/dλ)/dλ ≡ (dxν/dλ)∇ν(dxμ/dλ) = (dxν/dλ)[∂ν(dxμ/dλ) + Γμνα(dxα/dλ)] = (dxν/dλ)∂ν(dxμ/dλ) + Γμνα(dxν/dλ)(dxα/dλ) = (dxν/dλ)(∂(dxμ/dλ)/∂xν) + Γμνα(dxν/dλ)(dxα/dλ) = d2xμ/dλ2 + Γμνα(dxν/dλ)(dxα/dλ) = 0 Therefore, d2xμ/dλ2 = -Γμνα(dxν/dλ)(dxα/dλ) If we now substitute the proper time, τ, for λ to create a world line we get: d2xμ/dτ2 = -Γμνα(dxν/dτ)(dxα/dτ) Flatness and Curvature ---------------------- Consider the 2 sphere which represents the surface of a sphere (not the interior). The 2 sphere is not flat and has the metric: - - gμν = | R2 0   |    | 0 R2sin2θ | - - Where R is the radius. Which can be written as a line element as: ds2 = R22 + R2sinθ22 This is derived using spherical coordinates as follows: x = Rsinθcosφ y = Rsinθsinφ z = Rcosθ ds2 = dx2 + dy2 + dz2 Which leads to: ds2 = dR2 + R22 + R2sin2θdφ2 Consider a point, P, at the North pole where θ = 0. If we do this we get a horrible metric. - - gμν = | R2 0 |    | 0  0 | - - Which is degenerate. We need to find a 'good' set of coordinates that gets rid of this degeneracy. Such coordinates are referred to as RIEMANN NORMAL COORDINATES. The basic idea behind Riemann normal coordinates is to use the geodesics through a given point to define the coordinates at a nearby point that lies on the same geodesic. We choose a point, P, to be at xμ(0) in the tangent space. The components of the tangent vector at P are given by: Vμ = dxμ/dλ|λ= 0 We now push Vμ along the geodesic in n tiny increments to the nearby point, Q. We can write this as: λQ = lim (1 + λ/n)n n->∞ ~ 1 So we have increased λ by 1 unit of distance. The change from xμ(0) to xμ(1) gives a map from the set of tangent vectors at P to points in the manifold, and locally this map is one-to-one. In this way we can use Vμ to define a local coordinate system on the manifold near P. Now consider the geodesic equation: d2xμ/dλ2 = -Γμνα(dxν/dλ)(dxα/dλ) If the space at P is flat Γμνα = 0 and d2xμ/dλ2 = 0 Geodesics that satisfy this differential equation have the form: xμ(λ) = λVμ Therefore, any geodesic that passes through P allows us to define a local coordinate system on the manifold. Now lim (1 + λ/n)n is also the definition of the n->∞ exponential function so the map from vectors at P to points in the manifold is called the EXPONENTIAL MAP. exp: V in Tp -> M In other words, expP(Vμ) = xμ(1) A 'better' set of coordinates can be found as follows. Consider a point very close to the North pole where sinθ ~ θ and the z coordinate can be regarded as constant. By doing this we are effectively emulating the exponential map discussed above. Therefore, we can approximate x and y as: x = Rθcosφ y = Rθsinφ So that: x2 + y2 = R2θ2(sin2φ + cos2φ) Therefore, θ = √(x2 + y2)/R and, dθ = (1/2)(x2 + y2)-1/2[2xdx + 2ydy]/R = (xdx + ydy)/R√(x2 + y2) y/x = tanφ Therefore, φ = arctan(y/x) d(arctan(y/x))= -ydx/(x2 + y2) and d(arctan(y/x)) = xdy/(x2 + y2) Therefore, dφ = (xdy - ydx)/(x2 + y2) The line element for the 2 Sphere is: ds2 = R2 + R22 + R2sin2θdφ2 For constant R we get: ds2 = R22 + R2sin2θdφ2 Taking this term by term and substituting gives: R22 = R2(xdx + ydy)2/R2(x2 + y2) = (x2dx2 + y2dy2 + 2xydydx)/(x2 + y2) and, R2sin2θdφ2 = R2sin2(√(x2 + y2)/R)(x2dy2 + y2dx2 - 2xdydx)/(x2 + y2)2 Therefore, R22 + R2sin2θdφ2 equals: [x2/(x2 + y2) + R2sin2(√(x2 + y2)/R)(y2/(x2 + y2)2]dx2 + [y2/(x2 + y2) + R2sin2(√(x2 + y2)/R)(x2/(x2 + y2)2]dy2 - 2[xy/(x2 + y2) - R2sin2(√(x2 + y2)/R)(xy/(x2 + y2)2]dydx Now sin2a = a2 - a4/3 + .... (Taylor series) Therefore, sin2(√(x2 + y2)/R) = (√(x2 + y2))2/R2 - (√(x2 + y2))4/3R4 ... For the dx2 term we get: [y2/(x2 + y2) + R2{(√(x2 + y2))2/R2 - (√(x2 + y2))4/3R4)}(y2/(x2 + y2)2]dx2 = (1 - y2/3R2)dx2 Similarly, for the dy2 we get: (1 - x2/3R2)dy2 and for the dxdy term we get: (2xy/3R2)dxdy Therefore, the line elements becomes: ds2 = (1 - y2/3R2)dx2 + (1 - x2/3R2)dy2 + (2xy/3R2)dxdy and the metric looks like: - - gμν = | 1 - y2/3R2 2xy/3R2 |    | 2xy/3R2 1 - x2/3R2 | - - At P, x = y = 0 the metric becomes: - - gμν = | 1 0 |    | 0 1 | - - and the first derivative vanishes. What this tells us is that just looking at the metric and its first derivative to see it vanishes, doesn't tell us if the overall space we are working in is flat. It is possible to be working in a local coordinate system without knowing it ahead of time, and that the space is in fact curved. However, in general the second derivative of the metric will not vanish if the space is curved even if you happen to be in a local coordinate system. For example, ∂2(1 - x2/3R2)/∂x2 = 2/R3 ≠ 0 for any x. If the second derivative does vanish then it means the the space is truly flat. This is the reason why the second derivative of the metric plays a key role in the structure of the Riemann curvature tensor that we will discuss next. Riemann normal coordinates, with the associated basis vectors constitute a local Lorentz frame. They are the best approximation to flat space that is available. The fact that the metric at P looks like that of flat space to first order support the idea that small enough regions of spacetime look like Minkowski space. This implies that locally all physics looks the same, which is the famous equivalence principle. The Riemann Curvature Tensor ---------------------------- Parallel transport can detect curvature. Previously we have talked about curvature how this might be measured using the second derivative of the metric. We now look at this in more detail. Consider the following schematic representation of the parallel transport of a tangent vector, VAλ, around a square in flat space. VAλ, VA'λ -Δxν A <-------------   | ^   | | Δxμ | | -Δxμ   | |   v | -------------> B VBλ, VB'λ Δxν Let Δxμ, Δxν, -Δxμ and -Δxν act as infinitessimal operators that do the shifting. Moving counter- clockwise we get: (-Δxν)(-Δxμ)(Δxν)(Δxμ)VAλ = VA'λ Therefore, VA'λ - VAλ = δVλ Alternatively, we can split this into 2 steps Counter-clockwise from A to B: (Δxν)(Δxμ)VAλ = VBλ Clockwise from A to B: (Δxμ)(Δxν)VAλ = VB'λ Therefore, VB'λ - VBλ = [(Δxμ),(Δxν)]Vλ = δVλ We can do the same thing with the 2-Sphere which is a curved surface. So again the idea is to transport the tangent vector in suvh a way that it remains as parallel as possible from point to point (the vector has to lie in the tangent plane and cannot protrude above it.). If this operation is perform on a manifold that is not flat, the original vector and the transported vector will not coincide if brought back to the same point along a closed trajectory. Now, we have previously seen that the covariant derivative parallel transports a vector along a curve. Therefore, we can replace the Δs with ∇s. [∇μ,∇ν]Vλ = δVλ If we expand this out we get: [∇μ,∇ν]Vλ = (∂μΓλνρ - ∂νΓλμρ + ΓλμαΓανρ - ΓλναΓαμρ)Vρ = RλρμνVρ Rλρμν is the RIEMANN CURVATURE TENSOR. It is comprised of Christoffel symbols and their partial derivatives. In turn, the Christoffel symbols are comprised of the metric tensor and its derivatives. Therefore, the Riemann tensor consists of 2nd derivativeS of metric which, from the previous discussion, is a measure of the curvature. The Riemann tensor can be shown to transform in the following way proving indeed that it is a tensor. Rλρμν -> Rλ'ρ'μ'ν' = (∂xλ'/∂xλ)(∂xρ/∂xρ')(∂xμ/∂xμ')(∂xν/∂xν')Rλρμν Rλρμν ρ corresponds to the transported vector μ and μ correspond to the paths taken. λ corresponds to the components of the difference vector. If one is interested only in certain aspects of curvature, it is possible to form other tensors from the Riemann tensor. The simplest of those is the Ricci tensor, which is formed by contracting the upper index with the lower middle index. This the trace of the Riemann tensor. Symmetries ---------- The Riemann tensor has the following symmetries: Rαρμν = gαλRλρμν Rαρμν = -Rαρνμ Rαρμν = -Rραμν Rαρμν = Rμνρα Rαρμν + Rανρμ + Rαμνρ = 0 The original Riemann tensor has 256 components but after these symmetries have been applied, the number of independent components reduces to 20.. Geodesic Deviation Equation --------------------------- We start by looking at the geodesic deviation under Newtonian gravity. Consider the following diagram. At the P the geodesics are parallel and the separation vector, ξ is, orthogonal to both geodesics. Therefore, dξ/dt = 0 and ξ.t = 0 The acceleration can be written as: d2ξ/dt2 = -𝕋(_,ξ) 𝕋(_,ξ) is referred to as the TIDAL TENSOR. Alternatively, we can write this in component (index) form as: d2ξj/dt2 = -Tjkξk The equation of motions of particles moving along A and B are given by: (d2xj/dt2)A = -(∂φ/∂xj)A ... 1. and, (d2xj/dt2)B = -(∂φ/∂xj)B Where φ is the gravitational potential. Note that φ = mgh ∴ F = ma = φ/d so the RHS is equivalent to a force as one would expect. We can expand (φ)A in terms of ξ using a Taylor series as follows: (φ)A = (φ)B + ξ(∂φ/∂xj)B If we substitute this into 1 we get:. (d2xj/dt2)A = -(∂φ/∂xj + ξ∂2φ/∂xj∂xj)B Now, d2ξ/dt2 = (d2xj/dt2)A - (d2xj/dt2)B so Therefore, d2ξ/dt2 = -(∂φ/∂xj + ξ∂2φ/∂xj∂xj)B + (∂φ/∂xj)B = -ξ∂2φ/∂xj∂xj Restoring the indeces leads to: d2ξj/dt2 = -(∂2φ/∂xj∂xkk This is the geodesic deviation in 3D space. Lets now see how this looks in General Relativity. We replace d with D which is the covariant directional derivative of ξ along τ to get: D2ξ/dτ2 = -R(_,u,ξ,u) Where u is the 4-velocity = dx/dτ and the geodesic is treated as a world line parametized by τ instead of λ. Note that R(_,ξ,u,u) doesn't work because of the asymmetry in these indeces. In component form this is: D2ξα/dτ2 = -Rαβγδ(dxβ/dτ)ξγ(dxδ/dτ) The Riemann tensor produces the rate of change of separation between neighbouring geodesics over time. This is a vector since it has both magnitude and direction. It is the link between curvature and a force (≡ accelerration) that causes the geodesics to converge. These are the tidal forces of GRAVITY.