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 Physics

-
Archimedes Principle
-
Bernoulli Principle
-
Blackbody (Cavity) Radiation and Planck's Hypothesis
-
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
-
Laplace's and Poisson's Equations
-
Lorentz Force Law
-
Maxwell's Equations
-
Moments and Torque
-
Nuclear Spin
-
One Dimensional Wave Equation .
-
Pascal's Principle
-
Phase and Group Velocity
-
Planck Radiation Law .
-
Poiseuille's Law
-
Radioactive Decay
-
Refractive Index
-
Rotational Dynamics
-
Simple Harmonic Motion
-
Specific Heat, Latent Heat and Calorimetry
-
Stefan-Boltzmann Law
-
The Gas Laws
-
The Laws of Thermodynamics
-
The Zeeman Effect .
-
Wien's Displacement Law
-
Young's Modulus

Climate Change

-
Keeling Curve .

Cosmology

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

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

-
Accelerated Reference Frames - Rindler Coordinates
-
Catalog of Spacetimes .
-
Curvature and Parallel Transport
-
Dirac Equation in Curved Spacetime
-
Einstein's Field Equations
-
Geodesics
-
Gravitational Time Dilation
-
Gravitational Waves
-
One-forms
-
Quantum Gravity
-
Relativistic, Cosmological and Gravitational Redshift
-
Ricci Decomposition
-
Ricci Flow
-
Stress-Energy Tensor
-
Stress-Energy-Momentum Tensor
-
Tensors
-
The Area Metric
-
The Equivalence Principal
-
The Essential Mathematics of General Relativity
-
The Induced Metric
-
The Metric Tensor
-
Vierbein (Frame) Fields
-
World Lines Refresher

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
-
Basic Group Theory
-
Basic Representation Theory
-
Binomial Theorem (Pascal's Triangle)
-
Building Groups From Other Groups
-
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
-
Sets, Groups, Modules, Rings and Vector Spaces
-
Similar Matrices and Diagonalization .
-
Spherical Trigonometry
-
Stirling's Approximation
-
Sum and Differences of Squares and Cubes
-
Symbolic Logic
-
Symmetric Groups
-
Tangent and Normal Line
-
Taylor and Maclaurin Series .
-
The Essential Mathematics of Lie Groups
-
The Integers Modulo n Under + and x
-
The Limit Definition of the Exponential Function
-
Tic-Tac-Toe Factoring
-
Trapezoidal Rule
-
Unit Vectors
-
Vector Calculus
-
Volume Integrals

Microeconomics

-
Marginal Revenue and Cost

Particle Physics

-
Feynman Diagrams and Loops
-
Field Dimensions
-
Helicity and Chirality
-
Klein-Gordon and Dirac Equations
-
Regularization and Renormalization
-
Scattering - Mandelstam Variables
-
Spin 1 Eigenvectors .
-
The Vacuum Catastrophe

Probability and Statistics

-
Box and Whisker Plots
-
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

-
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

-
Basic Relationships
-
Bell's Theorem
-
Bohr Atom
-
Clebsch-Gordan Coefficients .
-
Commutators
-
Dyson Series
-
Electron Orbital Angular Momentum and Spin
-
Entangled States
-
Heisenberg Uncertainty Principle
-
Ladder Operators .
-
Multi Electron Wavefunctions
-
Pauli Exclusion Principle
-
Pauli Spin Matrices
-
Photoelectric Effect
-
Position and Momentum States
-
Probability Current
-
Schrodinger Equation for Hydrogen Atom
-
Schrodinger Wave Equation
-
Schrodinger Wave Equation (continued)
-
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 4 - Potential
-
Energy and Momentum, E = mc2
-
Lorentz Invariance
-
Lorentz Transform
-
Lorentz Transformation of the EM Field
-
Newton versus Einstein
-
Spinors - Part 1 .
-
Spinors - Part 2 .
-
The Lorentz Group
-
Velocity Addition

Statistical Mechanics

-
Black Body Radiation
-
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

-
BCS Theory
-
Introduction to Superconductors
-
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
-
Formulas
Last modified: January 26, 2018

Gravitational Waves ------------------- The Einstein Field Equations are: Rμν - (1/2)gμνR = 8πGTμν Rμν represents the part of curvature that derives from the presence of matter (i.e. that part due to the stress energy tensor Tμν). The remaining components of the Riemann tensor (the WEYL tensor) represents the part of the gravitational field which can propagate as a gravitational wave through a region of spacetime containing no matter or nongravitational fields. In other words, these are the parts of the curvature derived from the dynamics of the gravitational field itself independent of what matter the spacetime contains. An empty spacetime containing only gravitational radiation will satisfy Rμν = 0 but will also have Rμνσλ ≠ 0. Regions of spacetime in which Rμν = 0 are referred to as being RICCI FLAT. Einstein predicted the existence of gravitational waves. We start with the field equations in the absence of any mass. This is a good approximation if there is a large distance between the source and the observer. Therefore, Rμν - (1/2)gμνR = 0 Therefore, gμνRμν - (1/2)gμνgμνR = 0 Or, R - (1/2)δμμR = 0 Now δμμ = 4. Therefore, R - 2R = 0 This implies that R = 0. Thus, we can write: Rμν = 0 Note: R is also the trace of the Ricci tensor. Now consider empty flat space so that gμν = ημν. Consider a small perturbation is spacetime due to some kind of cosmic 'event'. The metric tensor can be written as: gμν = ημν - hμν(x,t) Where hμν(x,t) represents the perturbation. The Ricci tensor is comprised of the first derivatives and products of the Christoffel symbols. Thus, schematicaly and ignoring all indeces in the interest of brevity: R ~ ∂Γ + ΓΓ In turn, the Christoffel symbols consist of the metric tensor and its derivatives. Again, schematically: Γ ~ (1/2)g-1∂g Where g-1 is the inverse metric tensor (gμν). Therefore, Γ ~ (η - h)∂h If h is very small then the product h∂h can be ignored. ~ η∂h ~ ∂h Therefore, the Ricci tensor can be written: R = ∂2h + ∂h∂h Again, if is small then ∂h∂h -> 0. Therefore, R = ∂2h This is not just one equation. There is a separate equation for each μ and ν. Thus, Rμν = [∂2h]μν = 0 [∂2h]μν = 0 has the form of the wave equation: [∂2h/∂t2 - ∂2h/∂x2]μν = 0 ... 1. NOTE: In the case where the source is close by, the above equation becomes: [∂2h]μν = kT'μν where k is a constant. Where T'μν represents the stress–energy tensor plus quadratic terms involving hμν. h is involved because the wave field has an effect on Tμν. While the equations are valid everywhere they are difficulty to solve analytically because of the complicated source term. Under certain circumstances (flat space assumption) the quadratic terms involving h can be ignored making the solutions easier. However, for most purposes, the zero source solutions are perfectly adequate. Solutions to 1.0 are of the form: hμν(t,z) = h0μνsin(k(t - z)) The waves are transverse meaning that waves can only have components in the plane perpendicular to the propagation of the wave. y | |\ | | \ | | \ ---- -----\-------- z \ | | \ \ | x \ | \| Therefore, the time and z components of h are 0. - - | 0 0   0   0 | | 0 hxx hxy 0 | | 0 hyx hyy 0 | | 0 0   0   0 | - - Now, h is a symmetric tensor so hxy = hyx. Furthermore, there is another constraint that hxx = -hyy (The trace of hμν = 0). Therefore, we arrive at 3 equations. hxx(t,z) = h0xxsin(k(t - z)) so that, gxx(t,z) = 1 + hxx This corresponds to coordinates that are stretched and contracted in the x direction. The contraction occurs with the change in sign of sin(k(t - z). Similarly, there is stretching and contracting in the y direction: hyy(t,z) = h0yysin(k(t - z)) so that, gyy(t,z) = 1 + hyy So there is stretching and contracting in the y direction: Thus, in the x-y plane, this looks like. y ^ : ^ | <....<---- |----->...> x | v : v Finally, there is a third solution: hxy(t,z) = h0xysin(k(t - z)) so that, gxy(t,z) = 1 + hxy This corresponds to simultaneous stretching and contracting along the diagonal direction. In summary, if a binary-pulsar is being observed, the gravitational waves produced will be observed in the plane perpendicular to the line of sight. Wave Origins ------------ Gravitational waves are radiated by objects whose motion involves asymmetric accelerations. Two objects spinning about a central axis will not produce gravitational waves but if they tumble end over end, as in the case of two objects orbiting each other, they will radiate gravitational waves. The heavier the masses , and the faster they tumbles, the greater the gravitational radiation emitted. In summary, the the following will/will not emit gravitational waves: - Two objects orbiting each other. - A supernova if the explosion asymmetric. - An isolated non-spinning solid object moving at a constant velocity will not radiate (conservation of linear momentum) but one that is accelerating will. - A spinning disk will not radiate (conservation of angular momentum). - A spherically pulsating spherical star will not radiate. Orbiting Bodies --------------- Gravitational waves carry energy away from their sources and, in the case of orbiting bodies, this is associated with a decrease in the orbit. For a simple system of two masses, the radiated power is given by: P = dE/dt = -32G4(m1m2)2(m1 + m2)/5c5r5 The amount of radiation emitted corresponds to the loss of angular momentum of the orbiting body. In the case of the Earth orbiting the Sun the power emitted is about 200 watts! The emission of radiation first circularizes their orbits and then gradually shrinks their radius increasing the orbital frequency. The rate of decrease in the orbital radius is given by: dr/dt = -64G3(m1m2)(m1 + m2)/5c5r3 In the case of the Earth orbiting the Sun, the Earth's orbit shrinks by 1.1 x 10-20 meter per second. At this rate it will take about 1013 times more than the current age of the Universe to spiral onto the Sun! Verification ------------ On 2/11/2016 the first experimental evidence of gravitational waves was announced to the world, thus proving Einstein's original assertion. The waves were detected by the Laser Interferometer Gravitational Wave Observatory (LIGO) that consists of widely separated mirrors that are suspended like pendulums in a clock. The apparatus is designed such that the tidal forces associated with gravitational waves causes the distance between the mirrors to change. The change in distance between the mirrors is detected using the technique of laser interferometry. The discovery of gravitational waves promises to have a profound impact on our understanding of how the Universe works.