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

Accelerated Reference Frames ---------------------------- An uniformly accelerated observer in Special Relativity will follow a hyperbola. x2 - T2 = r2 The PROPER ACCELERATION along a particular curve, R, is: a = c2/R Dimensionally this is [L2/T2][1/L]. It is clear from this equation that as R gets smaller, a increases. Thus, x2 - T2 = c4/a2 The hyperbola is Lorentz invariant therefore a transformation will just move the observer to a different point on the curve where he/she will experience exactly the same accelertaion as before. Now, we can regard the accelerated observer in a stationary reference frame as being equivalent to a stationary observer in a unifomly accelerated reference frame. Unfortunately, Special Relativity doesn't hande this situation too well because it is based on inertial reference frames (frames moving at constant velocity). Acceleration does not enter directly into the Lorentz transform, length contraction, or time dilation, only the velocity is important. The general theory of relativity is a theoretical framework applicable to any frame of reference - inertial or accelerating. In developing this theory, Einstein wanted to produce a theory of gravitation that incorporated the theory of special relativity and the equivalence principle. Comments: - The hyperbolae are labelled r = 1, 2, 3 etc. - Note that the worldlines bunch together, reflecting the gravitational length contraction. - Any observer at rest in Rindler coordinates has constant proper acceleration, with Rindler observers closer to the Rindler horizon having greater proper acceleration. - The distances 23 = 34 = AB = BC = EF = FG remain the same no matter how long the acceleration proceeds. - Proper times intervals along two Rindler hyperbolae for the same ω are given by the following ratio: (r + Δr)/r Thus, an observer travelling along 4 to C would judge time along 2 to A passing at half the speed since (2R + 2R)/2R By analogy to the polar coordinate conversions x = rcosθ and y = rsinθ, it is possible to write the following transformations in hyperbolic space. x = rcoshω and T = rsinhω Where ω is the RINDLER TIME along the path of the observer. x and T are referred to as RINDLER COORDINATES. 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. Note: sinh2ω - cosh2ω = -1 where -∞ < ω < +∞ ∴ r2cosh2ω - r2sinh2ω = r2 Taking the differentials we get: dx = drcoshω + rsinhωdω and dT = drsinhω + rcoshωdω Therefore, dτ2 = dT2 - dx2 = -dr2 + r22 Again this is analagous to the flat spacetime metric in polar coordinates, dS2 = dr2 + r22. Now take a particular curve, R, and displace it by an infinitesimally small amount x'. Therefore, r = R + x' and we can write: dτ2 = (R2 + 2Rx' + x'2)dω2 - d(R + x')2 = (1 + 2x'/R + x'2/R2)R22 - dx'2 ... R is fixed = (1 + 2x'/R)R22 - dx'2 since R >> x' Let Rω = t ∴ R22 = dt2. Thus, dτ2 = (1 + 2x'/R)dt2 - dx'2 If we pick a value of R that avoids relativistic effects, for example a value that gives us an acceleration of g (~10 m/s2) = 1/R then we can write: dτ2 = (1 + 2x'g)dt2 - dx'2 = (1 + 2φ)dt2 - dx'2 Where φ is defined as the GRAVITATIONAL POTENTIAL = x'g. The gravitational potential is the gravitational potential energy per unit mass. This the basic version of the RINDLER METRIC in flat space time. It shows the correspondence between a UNIFORMLY accelerated reference frame and a UNIFORM gravitational field. This is the EQUIVALENCE PRINCIPLE. Note: A uniform gravitational field does not produce tidal forces. For these we need to consider the curvature of spacetime. To see the effects of curved spacetime it is necessary to use a metric that contains a mass term. Consider φ = -GM/r. The metric becomes: dτ2 = (1 - 2MG/rc2)dt2 - (1/c2){dx2 + .... } In spherical coordinates this becomes: dτ2 = (1 - 2MG/rc2)dt2 - (1/c2){dr2 + r2Ω2} The problem with this metric is that at some point the (1 - 2MG/rc2) term changes its sign and the signature of the metric will have 4 negative terms which is a violation of spacetime. It turns out from Einsteins equations that the correct metric is: dτ2 = (1 - 2MG/rc2)dt2 - (1/(1 - 2MG/rc2))dr2 - (1/c2)r2Ω2 This is equivalent to the form: dτ2 = (1 + φ/c2)dt2 - (1/(1 + φ/c2))dr2 - (1/c2)r2Ω2 Which can be written (after expanding the middle term in terms of a Taylor series) as: dτ2 = (1 + φ/c2)dt2 - (1 - φ/c2)dr2 - (1/c2)r2Ω2 Now when a sign change occurs, the time and spatial terms flip to maintain the signature of the metric. This is the 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 travelling radially. Since light follows a null geodesic (dτ = 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 slows and eventually stops at the Scwartzchild radius.