Redshift Academy

Wolfram Alpha:         

  Search by keyword:  

Astronomy

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

Chemistry

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

Classical Mechanics

-
Blackbody Radiation .

Classical Physics

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

Climate Change

-
Keeling Curve .

Cosmology

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

Finance and Accounting

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

Game Theory

-
The Truel .

General Relativity

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

Group Theory

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

Lagrangian and Hamiltonian Mechanics

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

Macroeconomics

-
Lecture Notes on International Economics
-
Lecture Notes on Macroeconomics
-
Macroeconomic Policy

Mathematics

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

Microeconomics

-
Marginal Revenue and Cost

Nuclear Physics

-
-
Radioactive Decay

Particle Physics

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

Probability and Statistics

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

Programming and Computer Science

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

Quantum Computing

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

Quantum Field Theory

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

Quantum Mechanics

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

Semiconductor Reliability

-
The Weibull Distribution

Solid State Electronics

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

Special Relativity

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

Statistical Mechanics

-
Entropy and the Partition Function
-
The Harmonic Oscillator
-
The Ideal Gas

String Theory

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

Superconductivity

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

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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

The Standard Model

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

Topology

-

Units, Constants and Useful Formulas

-
Constants
Last modified: March 1, 2022 ✓

Hubble's Law ------------ Hubble compared recession velocities of galaxies by measuring their redshift and comparing this with their distance. The distances were obtained using Cepheid variable stars found in the galaxies. Cepheid variable stars were discovered by Henrietta Leavitt and are objects whose period of variability is related to their average luminosity (brightness). By comparing stars at assumed comparable distances in the Small Magellanic Cloud, she found that the ones with longer periods had higher luminosities. Subsequent to Leavitt's discovery, Harlow Shapley used parallax methods to measure the distance to one of these cepheid variables. This allowed him to determine its actual or absolute brightness. Knowing the relationship between distance, apparent brightness and absolute brightness, allowed for the distance to any cepheid variable to be calculated. Hubble discovered the following relationship between the recession velocity, v, and the proper distance, D, and associated this with the expansion of the universe: v = H0D Where H0 is the HUBBLE CONSTANT. This is HUBBLE'S LAW. The proper distance, D, is the separation measured at a specific time. It can be considered as the physical or actual measurement made by a giant cosmological-sized ruler between the star and the Earth. Since the Universe is expanding, the proper distance between two distant objects (which aren’t held together by gravity) will increase with time. Hubble's law tells us that velocity is dependent on distance. Gravitationally, interacting galaxies move relative to each other independent of the expansion of the Universe. These relative velocities need to be accounted for separately. Hubble's Law can be derived in terms of the expansion of space in the following manner: O a a a a a o----o----o----o----o----> x x y Point O is fixed. a is the COSMIC SCALE FACTOR and is a function of t. It represents the relative expansion of the universe. It is easy to see that as the system is stretched in the x direction, the further away the points are from O, and the faster they must be moving. D = an where n is an integer that represents the number of a's. So x->y = 2. D is the distance between the objects. . . D = v = an . = a(a/a)n . = (a/a)D = H0D where H0 is the HUBBLE CONSTANT In 3D: D = a√{(Δx)2 + (Δy)2 + (Δz)2} . . D = (a/a)D as before. Since a is function of time, the coordinate mesh expands with time. --------------- / / / / /----/----/----/ / / / / --------------- ^ ------------ | / / / / | time /---/---/---/ | / / / / ------------ a(t) --------- / / / / a(t) /--/--/--/ / / / / --------- The current estimate of H0 is about 72 (km/s)/Mpc (1 Mpc ~ 3.26 x 106 light years). So at 3.26 Mly the velocity ~ 72 km/s, at 7.52 mly it is 144 km/s and so on, i.e. <- 3.26 Mly -> o o -> v ~ 72 km/s O Note: H0 is a function of time and therefore is not actually a constant in the traditional sense. Comoving Distance ----------------- There is another distance commonly referred used by cosmologists to define distances between objects. This is called to comoving distance. The comoving distance, DCM, is defined as the proper distance divided by the scale factor. Thus, DCM = DP(t)/a(t) For objects which only get further apart as a result of the expansion of the Universe, the comoving distance between them does not change over time. The comoving distance changes if the proper distances change due to a factor other than the expansion of the Universe (i.e. gravitational attraction). Comoving distance and proper distance are defined to be equal at the present time. Relativistic Redshift --------------------- Consider a stationary source and an observer moving away from it with velocity, v. Therefore, λs + vts = cts Where ts is the period of the light in the source's frame. Rearranging gives: ts = λs/(c - v) = λs/c(1 - β) where β = v/c = 1/fs(1 - β) The observer sees clocks in the source's frame running slower relative to clocks in his own frame. The observer will measure this time to be: to = ts/γ where γ = 1/√(1 - β2) Now fo = 1/t0 = γ/ts = 1/ts√(1 - v2/c2) = fs(1 - β)/√(1 - v2/c2) fo/fs = √[(1 - β)/((1 + β)] The redshift, z, is defined as: z = (fo - fs)/fs = (fo/fs) - 1 = (λso) - 1 (f = c/λ) = (1 - β)/√(1 - v2/c2) - 1 = √[(1 - β)/(1 + β)] - 1 Using the binomial expansion, this can be written as: z = (1 - β)(1 + v2/2c2) - 1 For v << c z = -β = -v/c Cosmological Redshift --------------------- The above analysis describes Doppler shift at relativistic velocities. In Doppler shift, the wavelength of the emitted radiation depends on the motion of the object at the instant the photons are emitted. In cosmological redshift, the wavelength at which the radiation is originally emitted is stretched as it travels through expanding space. Cosmological redshift results from the expansion of space itself and NOT from the motion of an individual body. Using the v << c aproximation, z can be written in terms of the Hubble constant as: z = v/c = H0D/c (v = H0D) = H0cΔt/c (D = c.dt) = H0Δt . = (a/a)Δt = (Δa/Δt)Δt/a = Δa/a = (a(t0) - a(ts))/a(ts) = a(t0)/a(ts) - 1 ∴ z + 1 = a(t0)/a(ts) Distance Measurement -------------------- We can use the above relationships to estimate the distance to objects. D = v/Ho = βc/Ho = [((z + 1)2 - 1)/((z + 1)2 + 1)](c/Ho) If v << c we can use: v = HoD ∴ D = v/Ho z = v/c ∴ v = cz D = cz/H0 The modern discovery of TYPE 1A supernovae has enabled further refinement of these techniques by allowing measurements at much greater distances. Type Ia supernovae occur in a binary system where 2 stars orbit one another. One of the stars is a white dwarf. The other can be a giant star or even a smaller white dwarf. The white dwarf pulls material off its companion star, adding that matter to itself. Eventually, when the white dwarf reaches a certain mass, a nuclear reaction occurs causing the white dwarf to explode and produce a burst of light 5 billion times brighter than the Sun. Because the chain reaction always happens in the same way, and at the same mass, the brightness of these Type Ia supernovae are also always the same. Using this knowledge, the distance to the galaxy containing the supernova can be computed by comparing the observed and absolute intensities and again applying the inverse square law (Distance Modulus Equation). Age of Universe --------------- The reciprocal of H0 provides an estimate of the age of the universe. Note [H0] = [(L/t)/L] = [1/t] ∴ [1/H0] = [t] Using the current estimate of H0 ~ 72 (km/s)/Mpc 1 Mpc = 3.086 x 1019 km So, 1 km/1 MPc = 1/3.086 x 1019 = 3.24 x 10-20 1/H0 = 1/(72 x 3.24 x 10-20) s = 1/233 x 10-20 s = 0.0043 x 1020 s = 4.3 x 1017 s ~ 13.6 x 109 years.