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Astronomy

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Astronomical Distance Units .
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Celestial Coordinates .
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Chemistry

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

Classical Mechanics

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Classical Physics

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Archimedes Principle
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Comparison Between Gravitation and Electrostatics
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Compton 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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Inflation Theory
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Introduction to Black Holes .
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Planck Units .
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Finance and Accounting

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Game Theory

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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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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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Tensors
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The Area Metric
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The Equivalence Principal
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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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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics .
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Symmetry and Conservation Laws - Noether's Theorem .

Macroeconomics

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Macroeconomic Policy

Mathematics

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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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Matrix Exponential
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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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Polynomial Division
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Quaternions 1 .
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Quaternions 2 .
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Related Rates
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Spherical Trigonometry
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Trapezoidal Rule
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Unit Vectors
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Volume Integrals

Microeconomics

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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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Factor Analysis
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Set Theory - Venn Diagrams
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Programming and Computer Science

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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: April 26, 2022 ✓

Hyperbolic Motion and Rindler Coordinates ----------------------------------------- Accelerated Coordinate Systems ------------------------------ In flat space Newtonian mechanics, the motion of a uniformly accelerated object is described by a parabola and follows the familiar SUVAT equation, s = (1/2)at2. Things are more complicated in Special Relativity. Since v = at, as t increases, v would eventually be greater than the velocity of light, which is not allowed. Instead, we must think of the motion of an object in terms of hyperbolas through flat space. The hyperbola is asymptotic to the light cone and, therefore, the velocity of light cannot be exceeded. (A light cone is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through spacetime). The equation of the hyperbola is: X2 - c2T2 = R2 Where X, T are inertial coordinates in Minkowski space and R is the distance along the horizontal axis from the event horizon when T = 0. The PROPER ACCELERATION along a particular curve, R, is: α = c2/R ∴ R = c2/α Dimensional check: [α] = [(L2/T2).(1/L)] = [L/T2] ✓ [R] = [(L2/T2)/(L/T2)] = [L] ✓ The proper acceleration along each curve is constant. It is clear from this equation that as R gets smaller, α increases. Thus, X2 - c2T2 = c42 When T = 0, X = R = c2/α. The hyperbola is drawn in Minkowski spacetime and represents a worldline. It is Lorentz invariant because X2 - c2T2 is an invariant quantity. Therefore, such a transformation will just move the observer to a different point on the worldline where they will experience exactly the same acceleration as before. By analogy with the polar coordinate conversions for the circle, x = rcosθ and y = rsinθ, it is possible to write the following transformations in hyperbolic space. X = Rcosh((α/c)ω) and cT = Rsinh((α/c)ω) Where (α/c)ω is the rapidity, ξ. Dimension check: [(α/c)] = ((L/T2)/(L/T)) = 1/T [ω] = T [(α/c)ω] = ((L/T2)/(L/T).T) = dimensionless. ✓ The corresponding inverse transformations are: R = √(X2 - c2T2) ω = (c/α)arctanh(cT/X) R and ω are the RINDLER COORDINATES of the accelerating object. They are the coordinates for an observer moving with proper acceleration, α. It is important to note that ω are lines of constant time and are NOT the proper time measured along the curve of constant R. Unlike the Newtonian case, a uniformly accelerated motion in a Minkowski spacetime cannot cover the entire spacetime. It is restricted to the wedge of spacetime shown, bounded by the light cone. This is referred to as the RINDLER WEDGE or RINDLER SPACE. The Rindler Metric ------------------ Since sinh2(αω/c) - cosh2(αω/c) = -1 where -∞ < ω < +∞ R2cosh2(αω/c) - R2sinh2(αω/c) = R2 (X2 - cT2 = R2) Taking the differentials we get: dT = dRsinh(αω/c) + (Rαdω/c)cosh(αω/c) ∴ dT2 = dR2sinh2(αω/c) + (Rαdω/c)2cosh2(αω/c) + 2(Rαdω/c)sinh(αω/c)cosh(αω/c) and, dX = dRcosh(αω/c) + (Rαdω/c)sinh(αω/c) ∴ dX2 = dR2cosh2(αω/c) + (Rαdω/c)2sinh2(αω/c) + 2(Rαdω/c)sinh(αω/c)cosh(αω/c) The metric is: dτ2 = c2dT2 - dX2 = (Rα/c)22 - dR2 - - - - dτ2 = | (Rα/c)2 0 || dω2 | | 0  -1 || dR2 | - - - - Dimension check: [(Rα/c).dω] = L.(L/T2)/(L/T).T = L ✓ This the basic version of the RINDLER METRIC in flat spacetime that is analogous to the metric for the circle in polar coordinates, ds2 = dr2 + r22. The invariant spacetime interval is: dτ2 = d(ct)2 - dx2 = R2sinh2(αω/c) - R2cosh2(αω/c) = -R2 Time Dilation ------------- The proper time along a worldline in inertial coordinates can be found as follows: The first step is to parameterize the hyperbola using λ. cT(λ) = Rsinhλ X(λ) = Rcoshλ The tangent vector to the worldline (hyperbola) is: ds/dλ = (d(cT)/dλ)(∂s/∂(cT)) + (dX/dλ)(∂s/∂X) = (Rcoshλ)eT + (Rsinhλ)eX ∴ (ds/dλ)2 = [(Rcoshλ)eT + (Rsinhλ)eX][(Rcoshλ)eT + (Rsinhλ)eX] From the Minkowski metric: - - | 1 0 | | 0 -1 | - - eT.eT = 1; eX.eT = 0; eX.eX = -1 (ds/dλ)2 = R2 cτ = ∫|ds/dλ|dλ = ∫√[(ds/dλ)(ds/dλ)]dλ = ∫Rdλ = R[λfinal - λinitial] The proper time along a worldline in Rindler coordinates: R = √(X2 - c2T2) ω = (c/α)arctanh(cT/X) = (c/α)arctanh(Rsinhλ/Rcoshλ) = (c/α)arctanh(tanhλ) = (c/α)λ ds/dλ = (dω/dλ)(∂s/∂ω) + (dR/dλ)(∂s/∂R) = (c/α)eω + 0eR (ds/dλ)2 = (c/α)2eω.eω From the Minkowski metric: - - | (Rα/c)2 0 | | 0  -1 | - - eω.eω = (Rα/c)2; eR.eω = 0; eR.eR = -1 (ds/dλ)2 = (c/α)2(Rα/c)2 = R2 cτ = ∫√[(ds/dλ)(ds/dλ)]dλ = R∫dλ = R[λfinal - λinitial] Therefore, the conclusion is that the proper time, τ, for a given Rindler observer is the same in both coordinate systems. Let λ = ω cτ = ∫|ds/dω|dω = ∫√[(ds/dω)(ds/dω)]dω = ∫√[eω.eω]dω = ∫√[(Rα/c)2]dω = (Rα/c)∫dω = (Rα/c)[ωF - ωI) If ωI = 0 then: cτ = (α/c)Rω Or, τ = (α/c2)Rω If all Rindler observers set their clocks to 0 at T = 0 (i.e. T = ω = τ = 0) then there is a choice of which Rindler observer's proper time, τ, will be equal to the Rindler time, ω. It is conventional to define the Rindler coordinate system such that the Rindler observer whose proper time, τ, matches the Rindler time, ω. This corresponds to the curve where (α/c2)R = 1. For other Rindler observers at different distances from the Rindler horizon, the proper time will equal some constant multiple of the proper time at the Rindler coordinate (1,1). Ex: Let c = 1. R = 1/α ∴ α = 1 => R = 1 α ω (R,ω) τ = αRω --- - ----- ------- 1.0 0 (1,0) 0.0 1 (1,1) 1.0 2 (1,2) 2.0 Therefore, from the perspective of an observer at B, A's clock would appear to be running at 1/2 the rate of his own. Likewise, from A's perspective, B's clock would appear to be running at twice the rate of his own. Now consider, The distance from the origin to where the 1st hyperbola intersects the X axis (T = 0) is D. The proper acceleration along this hyperbola is by convention set to c2/α = D = 1. D2 = c2T2 - X2 = [(c2/α)2sinh2((α/c)τ)]2 - [(c2/α/c)cosh2((α/c)τ)]2 = c42 ∴ D = c2/α 1st hyperbola: D = c2/α 2nd hyperbola: 2D = 2c2/α (≡ α/2) 3rd hyperbola: 3D = 3c2/α (≡ α/3) We can replace the initial distance, D, with general distance, R, which is a constant multiple of D and τ with the general coordinate, ω. Therefore, cT = (Nc2/α)sinh((α/c)τ) => Rsinh((α/c)ω) X = (Nc2/α)cosh((α/)τ) => Rcosh((α/)ω) From the point of a stationary observer in the inertial reference frame at X = 0, we get: cT = (c2/α)sinh(ατ/c) and, cτ = (c2/α)arcsinh(αT/c) ∴ τ= (c/α)arcsinh(αT/c) Note: arcsinh(x) = ln(x + √(x2 + 1)) For hyperbola 1, let α = 0.1c => c/α = 10 and α/c = 0.1. τ = (10)arcsinh(0.1T) For hyperbola 2, α = 0.05c => c/α = 20 and α/c = 0.05. τ = (20)arcsinh(0.05T) This gives: α = 0.1c α = 0.05c T τ1 τ2 - ----------- ----------- 1 0.998340789 0.999583801 2 1.986901103 1.996681578 Notes: - τ2 at 2s. is exactly twice τ1 at 1s. - The stationary observer at the event horizon would see a clock in the accelerated frame moving progressively faster and faster as the object accelerated towards the horizon and progressively slower and slower as the object moves away from the event horizon. Likewise, a stationary observer well away from event horizon will see the opposite effect. - It should be emphasized that all observers see time running normally in their own frames. There is nothing wrong with their own clocks. Time itself is slowing down or speeding up because of the because of the effects of acceleration. Length Contraction ------------------ Consider 2 stationary spaceships one at A and the other at B. The ships are attached to each other by a tight string of length, D. If both ships start accelerating at the same time with the same constant rate they will have identical instantaneous velocities. According to SR, a stationary observer would see the length of the string connecting the ships themselves contracting in length causing the string to break. Now consider the motion of the ships in the uniformly accelerated reference frame. If the 2 ships follow their respective hyperbolae then the distance as measured from point of view of each spaceship will always remain the same. This can be seen as follows: A = (X1,cT1) B = (X2,cT2) Distance AB = √[(X2 - X1)2 + (cT2 - cT1)2] (Pythagoras) T = 0 => Distance AB = √[(X2 - X1)2] = Distance 23 In Rindler coordinates: X1 = R1coshξ1 cT = R1sinhξ1 X2 = R2coshξ2 cT = R2sinhξ2 ξ1 = ξ2 = 0 X1 = R1 cT = 0 X2 = R2 cT = 0 (X2 - X1)2 = (R2 - R1)2 Or, (X2 - X1) = (R2 - R1) Therefore, the distances 23 = 34 = AB = BC = EF = FG remain the same no matter how long the acceleration proceeds. Effectively, this means that the trailing ship needs to accelerate just a little bit harder to maintain the correct distance between the ships. Now from the point of view of a stationary observer traveling along the T axis (X = 0) it appears that the the back ship is catching up with the front ship and the distance between the two is diminishing. Again we can see this from the following: cT1 = cT2 = cT. X12 - cT2 = R12 X22 - cT2 = R22 ∴ X22 - R22 = X12 - R12 ∴ X22 - X12 = R22 - R12 ∴ (X2 - X1)(X2 + X1) = R22 - R12 But R22 - R12 is fixed = K. ∴ (X2 - X1) = K/(X2 + X1) At very large T, (X2 + X1) will be very large and (X2 - X1) will be very small. So the conclusion is that the string will break if both spaceships accelerate with the same proper acceleration but if the ship in the back accelerates just a little harder than the one in front so that distance, D, remains constant, the string will not break. This is BELL'S SPACESHIP PARADOX. The Equivalence Principle ------------------------- Take a particular curve, R, and displace it by a tiny amount ΔR. We can write: dτ2 = (α/c)2(R + ΔR2)22 - d(R + ΔR)2 = (α/c)2(R2 + 2RΔR + ΔR2)dω2 - d(R + ΔR)2 = (α/c)2(1 + 2ΔR/R + ΔR2/R2)R22 - dΔR2 (dR = 0) = (Rα/c)2(1 + 2ΔR/R)dω2 - dΔR2 since R >> ΔR = (1 + 2ΔR/R)c2dT2 - dΔX2 Now, α = c2/R. Therefore, dτ2 = (1 + 2ΔRg)dT2 - dΔX2 = (1 + 2φ)dT2 - dΔX2 Where φ = ΔRg is defined as the GRAVITATIONAL POTENTIAL ENERGY per unit mass (c.f. mgh with m = 1 and h ≡ ΔR). Equations of motion: d2xμ/dτ2 = -Γρσμ(dxρ/dτ)(dxσ/dτ) For a small spatial velocity this becomes: d2x1/dτ2 = -Γ001(dx0/dτ)(dx0/dτ) = -Γ001(1)(1) = (1/2)∂g00/∂x1 = -g [-g00 = (1 + 2ΔRg)] Rindler coordinates show the correspondence between uniformly accelerated reference frames and uniform gravitational fields. In other words they show the equivalence of an accelerated observer with no gravity, to an observer in a local gravitational field. This is the original statement of the Equivalence Principle. They also demonstrate that an event horizon is not unique to General Relativity, and gravity is not a requirement. An observer that is accelerating at a constant rate will see an apparent horizon. The Schwarzchild Metric ----------------------- The analysis to date is for flat spacetime in the absence of matter. Uniformally accelerated reference frames correspond to homogeneous gravitational fields and do not produce tidal forces. Tidal forces are a consequence of curved spacetime produced by the presence of gravitating masses. Therefore, to see the effects of curved spacetime it is necessary to use a metric that contains a gravitating mass term. Consider φ = -GM/r where M is the mass and r is the distance from the center of M. The above metric becomes: dτ2 = (1 - 2MG/rc2)dt2 - (1/c2){dx2 + .... } In spherical coordinates this becomes: dτ2 = (1 - 2MG/rc2)dt2 - (1/c2){dr2 + r2Ω2} [i.e. dx2 + dy2 + dz2 = dr2 + r2(dθ2 + sin2θdφ2)] The spatial term is split into a radial component, dr2, and an angular term, r2(dθ2 + sin2θdφ2). The problem with this metric is that at some point the (1 - 2MG/rc2) term changes its sign and the time coordinate becomes spacelike. Under these circumstances the metric will have 4 negative terms which violates spacetime. It turns out from Einstein's equations that the correct form of the metric is: dτ2 = (1 - 2MG/rc2)dt2 - (1/(1 - 2MG/rc2))dr2 - (1/c2)r2Ω22 Sometimes this is written as: dτ2 = (1 - rS/r)dt2 - (1/(1 - rS/r))dr2 - (1/c2)r2Ω22 Where rS = 2MG/c2 is called the SCHWARZCHILD RADIUS. Dimension check: [2GM/rc2] = [kg/(m2/s2).(m3/kg.s2).(1/m)] = [1/1] Now when a sign change occurs the time and spatial terms flip to maintain the signature of the metric. This is the famous Schwarzchild metric that describes the spacetime around a spherically symmetric gravitating object such as a black hole (or earth for that matter). Consider a light ray traveling radially towards the mass. Light is a massless particle that travels along the edges of the light cone referred to as the NULL GEODESIC. It corresponds to the light like case, dτ2 = 0. The above equation becomes: (1 - 2MG/rc2)dt2 = (1/(1 - 2MG/rc2))dr2 Or, dr/dt = ±(1 - 2MG/rc2) = ±(1 - rs/r) Thus, as the light ray approaches the event horizon it appears to a distant observer to become progressively slower and slower compared to their coordinate time, t, eventually becoming 'frozen' at the Scwartzchild radius. However, a person running along with the light wave would not see this slowing and would measure the velocity of the ray to be that of the speed of light using their proper time, τ. The apparent slowing of the light ray is a consequence of gravitational time dilation. Mathematically. dτ2 = (1 - 2MG/rc2)dt2 - (1/(1 - 2MG/rc2))dr2 - (1/c2)r2Ω22 Focus on 1st term on RHS. dτ2 = (1 - 2MG/rc2)dt2 ∴ dτ = √(1 - 2MG/rc2)dt ∴ dt = dτ/√(1 - 2MG/rc2) This has a similar form to the Lorentz Transform from SR: dt = dτ/√(1 - v2/c2) By comparison v = √(2MG/r). Time dilation causes light to oscillate at a lower frequency so the light becomes more and more shifted towards a longer wavelength as it approaches the event horizon. λ↑ = c/f↓ = ct(↑). At the horizon, the frequency becomes so low, its wavelength so large, that it appears to get stuck and never pass through the surface. Interestingly enough, an observer inside the event horizon can easily see outside. The event horizon prevents light from traveling to someone outside, but it does not prevent light from traveling from outside to the observer. It also does not prevent two infalling observers from exchanging light signals on their way to doom. Gravitational Redshift ---------------------- Consider a photon in an elevator undergoing a uniform constant acceleration: ------- | | | .x=Δx | K' = hf', U' = mgΔx = gΔxhf/c2 | | | | | | | .x=0 | K = hf, U = 0 | | ------- | v α Conservation of energy requires: K + U = K' + U' ∴ hf + 0 = hf' + gΔxhf/c2 ∴ f + 0 = f' + gΔxf/c2 ∴ f' = f(1 - gΔx/c2) Let f' = fO (O = observer) and f = fS (S = source). The Doppler equation is: fO = [(c - vO)/(c + vS)]fS = [(c - vO)/c]fS (vS = 0) = [(1 - vO/c)]fS By comparison: fO = (1 - gΔx/c2)fS (m = 1) Dimension check: [gΔx/c2)] = (L/T2).L/(L2/T2) [vO/c] = (L/T)/(L/T) fO = (1 - mgΔx/c2)fS = (1 - φ/c2)fS Using the Schwarzchild metric gives: dt = dτ/√(1 - v2/c2) dt/dτ = tO/tS = λOS = fS/fO = 1/√(1 - 2MG/rc2) dτ/dt = tS/tO = λSO = fO/fS = √(1 - 2MG/rc2) ~ (1 - MG/rc2) (Binomial expansion) = (1 - φ/c2) By convention: λOS - 1 = z Therefore, the gravitation redshift is written as: 1 + z = λOS = 1/√(1 - 2MG/rc2) In the weak field case, this becomes z ~ -αΔx/c2 Therefore, 1 + z = (1 - αΔx/c2) = (1 - φ/c2) Length Contraction in GR? ------------------------- A meter stick falling towards a black hole would seem to be length contracted from the point of view of a distant observer because the front end of the stick would appear to be moving slower and slower such that the back end effectively 'catches up'. What is really happening is that light from the front end of the stick is becoming increasingly redshifted with respect to the back end. Ultimately, the stick would appear to have no length and would be 'smeared' on the surface of the event horizon in a sedimentary way. However, because there is now a gravitating mass present, a meter stick will experience tidal forces that will stretch it in the radial direction. These tidal forces exist because the gravitational field is not homogeneous (uniform). The stretching is an effect that can be measured by an observer thatis comoving with the object. If it is stretched by a factor, s, where s > 1 then its length will measure to be sL. However, determining the length contraction from afar depends on the ability of the observer to measure length in curved spacetime. If the observer is close to the stick and stationary, it is possible to measure length if the velocity measurement is made in a time that is very short compared to the reciprocal of the acceleration. Under these conditions curvature can be ignored and the length can be calculated as sL/γ. The two effects are, therefore, in a competition but are independent. This means that in some cases the tidal forces will prevail, while in other cases the length contraction will prevail. Because of the ambiguity associated with measuring length at distances in curved spacetime, referring to length contraction in GR without specifying how it is measured is not that meaningful. Consequently, length contraction is not a central concept in GR. In other words, the the length contraction of a distant object only really makes sense in the context of flat spacetime and Special Relativity. Spaghettification ----------------- The point at which tidal forces destroy an object or kill a person will depend on the black hole's size. For a supermassive black hole, the spaghettification point lies inside the event horizon, so a person may cross the event horizon without noticing any real squashing and pulling. However, it will be only a matter of time before the person is 'spaghettified'. In contrast, for small black holes, the tidal forces would kill even before the person reaches the event horizon. The Rindler Metric versus the Schwarzchild Metric ------------------------------------------------- Return to the Rindler metric and introduce a new variable, ξ, defined as follows. R2 = 4ξ ∴ 2RdR = 4dξ ∴ dR = 4dξ/2R ∴ dR2 = dξ2/ξ ω = t/2 ∴ dω = dt/2 ∴ dω2 = dt2/4 dτ2 = (Rα/c)22 - dR2 = (α/c)2R22 - dξ2/ξ = (α/c)24ξdω2 - - dξ2/ξ = (α/c)24ξdt2/4 - dξ2/ξ = (α/c)2ξdt2 - dξ2/ξ Let (α/c)2 = 1. This corresponds to α = c. Therefore, R = c2/α = c2/c = c = 3 x 108m = 186,000 miles The metric becomes: dτ2 = ξdt2 - dξ2/ξ Dimension check: [dξ2/ξ] = L4/L2 = L2 [(α/c)2ξdt2] = (L2/T4)/(L2/T2).L2.T2 = L2 Now return to the Schwarzchild metric: dτ2 = (1 - 2MG/rc2)dt2 - (1/(1 - 2MG/rc2))dr2 - (1/c2)r2Ω22 Rewrite (1 - 2MG/c2r) as (1 - rS/r) and set rS = 1. This changes r in the metric so it now measures the distance from the event horizon rather that the distance from the center of the gravitating mass. |<-- R --> | .<--- | -- r -->. COM | | rS = 2MG/c2 COM = center of gravitating mass. The Schwarzchild metric becomes: dτ2 = (r - 1)dt2 - dr2/(r - 1) Let ξ = (r - 1) ∴ dξ = dr dτ2 = ξdt2 - dr2/ξ This metric has exactly the same form as the Rindler metric in flat spacetime. However, to determine the true flatness of the spacetime requires computation of the curvature tensor, which involves taking second derivatives of the metric. Appendix -------- 4-velocity and 4-acceleration Along the Hyperbola ------------------------------------------------- The general form of the 4-velocity and 4-acceleration are: U = dX/dτ = (γc,γv) A = dU/dτ = (cdγ/dτ,d(γv)/dτ) = γ(cdγ/dt,d(γv)/dt) = (γ4(v.a)/c,γ2a + γ4v(v.a)/c2) (not derived) Where v, a are the 3-velocity and 3-acceleration respectively. For the hyperbola these become: X = (c2/α)cosh((α/c)τ) and cT = (c2/α)sinh((α/c)τ) UT = dcT/dτ = ccosh((α/c)τ) UX = dX/dτ = csinh((α/c)τ) U2 = (UT)2 - (UX)2 = c2 AT = d2cT/dτ2 = αsinh((α/c)τ) AX = d2X/dτ2 = αcosh((α/c)τ) A2 = (AT)2 - (AX)2 = -α2 d(U.U)/dτ = (dU/dτ)U + UdU/dτ = 2U.A But U.U = c2 ∴ d(U.U)/dτ = 0 so U.A = 0. Rapidity -------- Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with distance and time coordinates. The rapidity of an object passing constantly from one inertial frame to another in such a way that its change of speed in a fixed time interval is always the same, must be increasing at a constant rate with respect to the object's proper time. This rate of change of rapidity, ξ, with respect to proper time, τ, is the proper acceleration, α. Therefore, α = cdξ/dτ = cd(ατ/c)/dτ = α The rapidity is related to the coordinate velocity, v, by the equation: v = ctanhξ = ctanh(ατ/c) Proof: X = √(R2 + c2T2) v = dX/dT = c2T/√(R2 + c2T2) = cRsinh(ατ/c)/√(R2 + R2sinh2(ατ/c)) = csinh(ατ/c)/√(cosh2(ατ/c)) = ctanh(ατ/c) = ctanhξ The 'proper' velocity, w, is given by: w = dX/dτ = (dX/dT)(dt/dτ) = γctanhξ = γv Therefore, w is made up of the spacelike components of the object's 4-vector velocity (i.e. it is a 3-vector). Now γ = dt/dτ = coshξ (see note on Lorentz Transform). Therefore, w = coshξctanhξ = csinhξ The name proper velocity is a misnomer since it is not Lorentz invariant. Its name derives from being a derivative with respect to proper time, τ. To avoid this confusion, it is often referred to as the CELERITY. Note: U.U = c2 is invariant where U is the 4-velocity. The coordinate acceleration, a, is given by: a = dv/dT = c2T/√(R2 + c2T2) = c2R2/(R2 + c2T2)3/2 = c2R2/(R2 + R2sinh2(ατ/c))3/2 = c2R2/(R2)3/2(1 + sinh2(ατ/c))3/2 = c2/Rcosh3(ατ/c) = α/cosh3(ατ/c) = α/cosh3ξ = α/γ3 or, α = γ3a The proper acceleration, α, is given by: U = (γc,γv) = (γc,w) A = dU/dτ = (cdγ/dτ,dw/dτ) = (γ4(v.a)/c,γ2a + γ4v(v.a)/c2) Therefore, dw/dτ is made up of the spacelike components of the object's 4-vector acceleration (i.e. a 3-vector). This can be better understood by introducing the idea of a MOMENTARY COMOVING INERTIAL REFERENCE FRAME (MCRF). Lorentz transformations only work with inertial frames and cartesian coordinate systems where the worldlines are straight. Therefore, they don't work in accelerated reference frames because the world lines are not straight and the coordinate systems are curvilinear. MCRFs are inertial frames traveling at the same instantaneous velocity as the object at any moment. In other words, for a specific point on a non-inertial worldline, they have the same 4-velocity and 4-position for an infinitesimal period of time. A curved worldline can then be thought of as a series of these instantaneous inertial reference frames. This means that a Lorentz transformation from a stationary inertial frame to an MCRF at each point on the curve can be made. Consider an accelerating rocket, R. S S' | | | -|- -> u' (wrt S') | | R | -> u (wrt S) | -|- -> α --|-----------|-------------- x | | | | -> v (wrt S) | -> a S' is the instantaneous inertial MCRF that always moves along at the same speed and direction as the accelerated frame. Therefore, u' = 0 and u = v. Now, - - - - - - A = d| γc |/dτ = γd| γc |/dt = | cγ(dγ/dt) | | γv | | γv | | vγ(dγ/dt) + γ(dv/dt) | - - - - - - In the MCRF, v = 0 ∴ γ = 1 ∴ dγ/dt = 0. Therefore, - - A = | 0 | = α | dv/dt | - - So, A.A = α2 Also ds2 = c22 - dx2 in the accelerating frame. = c2dt'2 - dx'2 in the MCRF. Then at the moment the particle is at rest dx2 = dx'2 = 0. Then, c22 = c2dt'2 Or, dτ = dt' In other words, the accelerated clock's rate is identical to the clock rate in a MCRF. Vector Formulation ------------------ The tangent to the worldline at any point, p, is a straight line. These vectors can be identified in terms of basis vectors in Minkowski space as: (x0,x1,x2,x3) <=> x0e0|p + x1e1|p + x2e2|p + x3e3|p Where eμ = ∂/∂xμ = ∂μ - -   | 1 | e0 = | 0 | etc.     | 0 |   | 0 | - - A distance between 2 events in Minkowski spacetime as defined by the invariant interval (ct)2 - x2, is the same in any orthonormal spacetime basis. Proof: t | (t=5,x=3) ≡ (t'=4,x'=0) |  / |  /S |  / et ^  ^et' |/ ex --->-------------- x \ ex' v (t,x) basis: S2 = (ct)2 - x2 = 52 - 32 = 16 (t',x') basis: S = 4et' + 0ex' S2 = (ct)2 - x2 = 42 - 02 = 16 In the MCRF: U = ceω + 0eR (since U.U = c2) U is always along the worldline, parallel to eω and eR = 0. Therefore, the observer in each of the MCRFs thinks they are stationary in space (i.e. their spatial velocity is 0). Now, A.U = 0 implies that: A = 0eω + aReR The observer in each of the series of instantaneous f rames has a purely spatial acceleration of aR. If this is the constant proper acceleration, α, then, A = αeR. For the observer in the stationary (not MCRF) reference frame the time and space component will both change. Therefore, A = aTeT + aXeX Because the dot product of 4-vectors is invariant regardless of the reference frame, the observer in the stationary reference frame will see the same acceleration as the observer in the instantaneous frame. Therefore, A = aTeT + αeX and, A.A = (aT)2et2 + α2ex2 = -α2 Acceleration = Rate of Change of Rapidity ----------------------------------------- Proof that the proper acceleration is the rate of change of the rapidity with respect to the proper time can be obtained by using the chain rule. dw/dt = (dw/dξ)(dξ/dτ)(dτ/dt) ∴ α = (dw/dξ)(dξ/dτ)(1/γ) Now, w = csinhξ ∴ dw/dξ = ccoshξ = cγ ∴ α = cγ(dξ/dτ)(1/γ) = cdξ/dτ Now, ξ = ατ/c ∴ dξ/dτ = α/c So, α = cdξ/dτ = α Q.E.D