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

Lorentz Transformation of the Electromagnetic Field --------------------------------------------------- An observer measures a charge at rest in a frame F. He will see a static electric field but no current because the charge is not moving. Since there is no cuurent there is no magnetic field. Another observer in a frame F' moving at a velocity, v, w.r.t. F, however, will see an electric current and hence a magnetic field. Consider the electromagnetic tensor:    - -    | 0  Ex/c Ey/c Ez/c | Fμν = | -Ex/c 0  Bz -By |    | -Ey/c -Bz 0  Bx |    | -Ez/c By -Bx 0  |    - - Which is contructed as follows: Fμν = ∂νAμ - ∂μAν Where, Aμ = (φ,-Ax,-Ay,-Az)   = (A0,-A1,-A2,-A3) Therefore, F01 = ∂1A0 - ∂0A1    = -∂Ax/∂t - ∂φ/∂x    = -Ex F23 = ∂2A3 - ∂3A2    = -∂Az/∂y + ∂Ay/∂z    = (∇ x A)x    = Bx Now add a boost:     - -     | γ -βγ 0 0 | Λμν = Λ = ΛT = | -βγ γ 0 0 |     | 0 0 1 0 |     | 0 0 0 1 |     - Therefore, we get: Fμν' = ΣΣΛμσΛμρFσρ (see note on Lorentz Group) σρ Where σ amd ρ run from 0 to 3. For F01 we get: F01' = ΣΣΛ0σΛ1ρFσρ = Ex'/c    = Λ00Λ11F01 + Λ01Λ10F10 (all other terms are 0)    = γ2Ex/c - β2γ2Ex/c    = (1 - β22Ex/c    = Ex/c We get the same answer using the matrix form: F' = ΛTFΛ - - - - - - | p -q 0 0 || 0 a b c || p -q 0 0 | = | -q p 0 0 || -a 0 d e || -q p 0 0 | | 0 0 1 0 || -b -d 0 f || 0 0 1 0 | | 0 0 0 1 || -c e -f 0 || 0 0 0 1 | - - - - - - - - | 0 ap2 - aq2 bp - dq cp + eq | = | aq2 - ap2 0 dp - bq -cq - ep | | dq - bp bq - dp 0 f   | | -cp - eq cq + ep -f 0   | - - Therefore, F01' = ap2 - aq2 = γ2Ex/c - γ2β2Ex = Ex/c If we repeat the process for all of the components of F we get: Bx' = Bx By' = γ(By + (v/c2)Ez) Bz' = γ(Bz - (v/c2)Ey) Ex' = Ex Ey' = γ(Ey - vBz) Ez' = γ(Ez + vBy) What appears to be a magnetic field to one observer looks like an electric field to another, and vice versa. Maxwell's Equations ------------------- Maxwell's equation can be written as: μ0jμ = ∂μFμν and, ∂μFμν = 0 μ0j0 = ∂F00/∂t + ∂F10/∂x + ∂F20/∂y + ∂F30/∂z    = (1/c)(0 - ∂Ex/∂x - ∂Ey/∂y - ∂Ey/∂z)    = -∇.E ... Gauss's Law μ0j1 = ∂F01/∂t + ∂F11/∂x + ∂F21/∂y + ∂F31/∂z    = (1/c)∂Ex/∂t + 0 - ∂Bz/∂y + ∂By/∂z μ0jx = [(1/c)∂E/∂t - ∇ x B]x ... Ampere's Law If we make the substitution E/c -> B and B -> -E/c we get the DUAL ELECTROMAGNETIC TENSOR, Gμν. Note this operation leaves the 6 boost equations shown above unchanged. When we do this and repeat the above procedure for ∂μFμν = 0 we get: 0 = ∂G00/∂t + ∂G10/∂x + ∂G20/∂y + ∂G30/∂z = 0 + ∂Bx/∂x + ∂By/∂y + ∂By/∂z = ∇.B And, 0 = ∂G01/∂t + ∂G11/∂x + ∂G21/∂y + ∂G31/∂z = ∂Bx/∂t + 0 + (1/c)∂Ez/∂y - (1/c)∂Ey/∂z = [∇ x E + ∂B/∂t]x ... Faraday's Law Alternative Derivation ---------------------- Aμ = (-φ/c,A) ... the vector potential jμ = (cρ,j) ... the 4 current. μ0jμ = ∂μFμν = ∂μ(∂μAν - ∂νAμ) = ∂μμAν - ∂μνAμ j0 = (∂00A0 - ∂00A0) + (∂11A0 - ∂10A1) + (∂22A0 - ∂20A2) + (∂33A0 - ∂30A3)   = 0 + ∂11A0 + ∂22A0 + ∂33A0 - ∂10A1 - ∂20A2 - ∂30A3   = -∂2φ/∂x2 - ∂2φ/∂y2 - ∂2φ/∂z2 - ∂/∂t(∂A1/∂x) - ∂/∂t(∂A2/∂y) - ∂/∂t(∂A3/∂z)   = -∂2φ/∂x2 - ∂2φ/∂y2 - ∂2φ/∂z2 - ∂/∂t(∂A1/∂x + ∂A2/∂y + ∂A3/∂z) ρ/ε0 = -∇2φ - ∂/∂t(∇.A) This is equivalent to: ∇ x E = -∂B/∂t Proof: The fields E and B are expressed in terms of the potentials as: B = ∇ x A and E = -∇φ - ∂A/∂t Therefore, ∇ x E = -∂B/∂t = -∂(∇ x A)/∂t = -∂(∇ x A)/∂t ∇ x E + ∂(∇ x A)/∂t = 0 ∇ x (E + ∂A/∂t) = 0 Now, ∇ x ∇φ = 0 (curl of the gradient of a scalar = 0). Therefore, by comparison: ∇φ = (E + ∂A/∂t) ∇.(∇φ) = ∇.E - ∇.(∂A/∂t) ∇2φ = -ρ/ε0 - ∂(∇.A)/∂t Which is what we had before. Lorentz Invariance ------------------ ∂μFμν = μ0jν and ∂μFμν = 0 arn not Lorentz invariant because they have a dangling index, ν, (i.e. they are not scalars like FμνFμν or xμxμ). However, we will show that the equations derived from these equations are, in fact, Lorentz invariant. - - - - - - | γ -γβ 0 0 || -cρ | | -γcρ - γβjx | | -γβ γ 0 0 || jx | = | γjx + γβcρ | | 0 0 1 0 || jy | | jy | | 0 0 0 1 || jx | | jz | - - - - - - - - - - - - | γ -γβ 0 0 || -φ/c | | -γφ/c - γβAx | | -γβ γ 0 0 || Ax | = | γAx + γβφ/c | | 0 0 1 0 || Ay | | Ay | | 0 0 0 1 || Az | | Az | - - - - - - Length contraction has an effect on charge density and current density, and time dilation has an effect on the rate of flow of charge. Therefore, the charge and current distributions must transform in a related way under a boost. Consider a boost in the x directions. The y component is: ∂Ex'/∂z' - ∂Ez'/∂x' = -∂By'/∂t Substituting Ez' = γ(Ez + vBy) and By' = γ(By + (v/c2)Ez) The equation becomes: ∂Ex'/∂z' - γ∂(Ez + vBy)/∂x' = -γ∂(By + (v/c2)Ez)/∂t' ∂Ex'/∂z' - γ∂Ez/∂x' + γv∂By/∂x' = -γ∂By/∂t' - (v/c2)γ∂Ez/∂t' Rearranging we get: ∂Ex'/∂z' + (v/c2)γ∂Ez/∂t' - γ∂Ez/∂x' = vγ∂By/∂x' - γ∂By/∂t' ... Equation 1. In order to go any further it is necessary to see how the derivatives of the E and B components transform in the direction of the boost (x). To do this we make use of the chain rule. ∂/∂x = (∂/∂x')(∂x'/∂x) + (∂/∂t')(∂t'/∂x) and ∂/∂t = (∂/∂x')(∂x'/∂t) + (∂/∂t')(∂t'/∂t) Also; ∂/∂t' = (∂/∂x)(∂x/∂t') + (∂/∂t)(∂t/∂t') The transformations in the y and z directions are simply: ∂/∂y = ∂/∂y' ∂/∂z = ∂/∂z' For a boost in the x direction the cordinates transform as: x' = γ(x - vt) x = γ(x' - v't') ≡ γ(x' + vt') y' = y z' = z t' = γ(t - (v/c2)x) t = γ(t' - (v'/c2)x') ≡ γ(t' + (v/c2)x') Therefore, ∂x'/∂x = γ ∂t'/∂t = γ ∂t/∂t' = γ ∂x'/∂t = -γv ∂x/∂t' = γv ∂t'/∂x = -γv/c2 Therefore, ∂Ez/∂x = (∂Ez/∂x')(∂x'/∂x) + (∂Ez/∂t')(∂t'/∂x)   = (∂Ez/∂x')(γ) + (∂Ez/∂t')(-γv/c2)   = γ∂Ez/∂x' - (γv/c2)∂Ez/∂t' ∂By/∂t = (∂By/∂x')(∂x'/∂t) + (∂By/∂t')(∂t'/∂t)   = (∂By/∂x')(-γv) + (∂By/∂t')(γ)   = γ∂By/∂t' - γv∂By/∂x' Armed with this information we can rewrite equation 1 as: ∂Ex/∂z - ∂Ez/∂x = -∂By/∂t Or, (∇ x E)y = -∂By/∂t Q.E.D The x component is a little simpler to calculate: ∂Ez'/∂y' - ∂Ey'/∂z' = -∂Bx'/∂t' Again, Ey' = γ(Ey - vBz) and Ez' = γ(Ez + vBy). Therefore, the equation becomes: (γ∂Ez/∂y' + γv∂By/∂y') - (γ∂Ey/∂z' - γv∂Bz/∂z') = -∂Bx'/∂t' Using the chain rule from above, and noting that Bx' = Bx, the RHS becomes: ∂Bx'/∂t' = (γ∂Bx/∂t + γv∂Bx/∂x) Therefore, (γ∂Ez/∂y + γv∂By/∂y) - (γ∂Ey/∂z - γv∂Bz/∂z) = -(γ∂Bx/∂t + γv∂Bx/∂x) Note: The derivatives for the y and z directions do not transform in accordance with the chain rule as described for the x direction. Now ∇.B = 0 meaning that ∂Bx/∂x = ∂By/∂y = ∂Bz/∂z = 0. Therefore, γ∂Ez/∂y - γ∂Ey/∂z = -γ∂Bx/∂t Or, (∇ x E)x = -∂Bx/∂t Q.E.D Using similar techniques we can demonstrate that the other Maxwell equations are also Lorentz invariant.