Redshift Academy

Wolfram Alpha:         

  Search by keyword:  

Astronomy

-
Celestial Coordinates
-
Celestial Navigation
-
Distance Units
-
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

-
Penrose Diagrams
-
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
-
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

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 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 Qubit
-
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

test

-
test

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

Einstein's Field Equations -------------------------- We will first state the Einstein field equations and then walk through the general rationale that Einstein used to derive them (without getting into the enormously complicated mathematics). The equations are: Gμν = 8πGTμν Where, Gμν = Rμν - (1/2)gμνR Where, Rμν is the RICCI tensor. gμν is the METRIC tensor. R is the SCALAR CURVATURE. R = gμνRμν. This corresponds to the trace of the Ricci curvature tensor. Tμν is the STRESS-ENERGY tensor. The Einstein equation actually consists of 16 separate equations. In a nutshell, they show how energy and momentum (aka mass) determine the curvature of spacetime. Energy and mass are represented on the right hand side of the equations while curvature is represented on the left hand side. The Riemann tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives. In turn, the Christoffel symbols are comprised of the partial derivatives of the metric tensor. Therefore, the Riemann tensor is comprised of the first and second derivatives of the metric tensor. Thus, for the Rindler metric we would have: - - gμν = | (1 + 2φ) 0 0 0 |    | 0 -1 0 0 |    | 0 0 -1 0 |    | 0 0 0 -1 | - - Rμνσλ = ∂νΓλμσ - ∂μΓλνσ + ΓρμσΓλρν - ΓρνσΓλρμ Where, Γabc = (1/2)gad{∂gdc/∂xb + ∂gdb/∂xc - ∂gbc/∂xd} = (1/2)gad{∂bgdc + ∂cgdb - ∂dgbc} In Newtonian mechanics, a freely falling object accelerating towards the earth implies a force acting between the object and the earth corresponding to gravity. In general relativity, however, the earth and the object are considered to be both moving along geodesics (along straight lines with zero proper acceleration) and it is the acceleration produced by the convergence of these geodesics as a result of the curvature of the spacetime that we observed as the object falls. For example, imagine 2 ships starting a certain distance apart at the equator of the earth and sailing due north along different lines of longitude (geodesics). As they proceed, their east west separation diminishes until they finally meet at the north pole. We can imagine this 'coming together' as being the result of a force (acceleration) with both transverse and longitudinal components acting on both objects. Thus, Mathematially, if the separation 4-vector between the earth and the object at the same proper time, τ, as both advance along their respective geodesics is ξμ, the coordinate acceleration between them is given by the GEODESIC DEVIATION EQUATION (not proven here but fairly straightforward to work out): D2ξμ/Dτ2 = -Rμνρσ(dxν/dτ)(dxρ/dτ)ξσ Where D is the covariant derivative and Rμνρσ is the RIEMANN tensor. The geodesic deviation equation assigns TIDAL forces to the Riemann tensor. The equation yields a component of acceleration along the geodesic (the Γnmr(dxm/dτ)(dxr/dτ) term in the original geodesic equation) and components transverse to the direction of motion. Together these accelerations act together to produce tidal forces on the objects. The components along the geodesic result in elongation while the tranvserse components result in compression. Another way to see this is to imagine an object in flat spacetime moving towards a region of curvature. When the object encounters the region it must 'adjust' dimensionally in the transverse and longitudinal (radial) directions in order to follow the metric of the curved spacetime at each point. In the process of adjusting, the object experiences non-uniform forces and becomes stretched in the direction of motion and squeezed in transverse direction. Uniform gravitational fields that are 'simulated' by accelerated reference frames do not have curvature. Such fields do not exist in nature (real gravitational fields involve mass and curvature). However, the gravitational field near the surface of a large object over a relatively small solid angle can be considered to be approximately uniform. Since they do not have curvature, a free falling object in a uniform gravitational field will not experience any tidal forces. The Weak Field Approximation ---------------------------- The weak field approximation is analagous to perturbation theory and allows simplifying the study of many problems in general relativity while still producing useful approximate results. Equations can be obtained by assuming the spacetime metric is only slightly different from the Minkowski metric, η, where η is constant. Thus, gμν = ημν + hμν and gμν = ημν - hμν The Geodesic Equation ----------------------- Look at acceleration of a particle along a geodesic. In the limit this should reduce to the familiar Newtonian force law. d2xn/dτ2 = -Γnmr(dxm/dτ)(dxr/dτ) x0 = t = τ, m = 1, and dτ2 = gμνdxμdxν, v << c If the velocity is small compared to c, the spatial components can be ignored. Therefore, d2x/dt2 + Γμ00(dx0/dτ)(dx0/dτ) = 0 d2x/dt2 = -Γμ00 since dx0/dτ = dτ/dτ = 1 From before we can write Γ in terms of the metric, Γabc = (1/2)gad{∂bgdc + ∂cgdb - ∂dgbc} Γa00 = (1/2)(ηax - hax){∂0hx0 + ∂0hx0 - ∂xh00} Since they very small we can neglect all terms that involve products of h and its derivatives. Thus: Γa00 = (1/2)ηax{∂0hx0 + ∂0h0x - ∂xh00} = (1/2)ηax{2∂0hx0 - ∂xh00} = -(1/2){2∂0hx0 - ∂xh00} If the velocity is assumed to be small, the derivatives with respect to space dominate over the derivatives with respect to time. Therefore, Γa00 = (1/2)∂xh00 Therefore, we can write: d2x/dt2 = -(1/2)∂xh00 However, in Newtonian mechanics force F = -∂Φ/∂x (m = 1) where φ is the GRAVITATIONAL POTENTIAL ENERGY. Thus, we can write: F = -∂Φ/∂x = d2x/dt2 Substitution yields, 2∂Φ/∂x = ∂h00/∂x So, h00 = 2φ + constant In flat space (φ = 0) h00 = 1 therefore the constant equals 1. We end up with: h00 = g00 = 1 + 2φ Which is the same form found in the Rindler metric associated with accelerated reference frames. dτ2 = (1 + 2φ)dt2 - dx2 The Field Equations ------------------ Rμν - (1/2)gμνR = 8πGTμν gμνRμν - (1/2)gμνgμνR = gμν8πGTμν R - (1/2)δμμR = 8πGT - -      | 1 0 0 0 | Now δμμ = δνν = | 0 1 0 0 | = 4      | 0 0 1 0 |      | 0 0 0 1 | - - Thus, R is just the trace of Rμν R - 2R = 8πGT -R = 8πGT Substituting this back into the EFEs gives: Rμν + (1/2)gμν8πGT = 8πGTμν Rμν = 8πG(Tμν - (1/2)gμνT) T = T00 + T11 + T22 + T33 ... the trace of Tμν Consider 00 component with T00 = ρ. Thus, R00 = 8πG(ρ - (1/2)g00ρ) R00 = 8πG(ρ - (1/2)ρ) R00 = 4πGρ ... 1. From before, the Riemann tensor is: Rμνσλ = ∂νΓλμσ - ∂μΓλνσ + ΓρμσΓλρν - ΓρνσΓλρμ The ΓΓ terms can be neglected since they are the products of the derivatives of h. Therefore, Rμνσλ = ∂μΓλνσ - ∂νΓλμσ Γμνσ = (1/2)ημλ(∂νhλσ + ∂σhλν - ∂λhνσ) Γλνσ = (1/2)ηλρ(∂νhρσ + ∂σhρν - ∂ρhνσ) Γμμσ = (1/2)ηλρ(∂μhρσ + ∂σhρμ - ∂ρhμσ) Rμνσλ = (1/2)ηλρ(∂μνhρσ + ∂μσhρν - ∂μρhνσ - ∂νμhρσ - ∂νσhρμ + ∂νρhμσ) Rμνσλ = (1/2)ηλρ(∂μσhρν - ∂μρhνσ - ∂νσhρμ + ∂νρhμσ) Multiply by ηνρ to get the Ricci tensor: Rμσ = (1/2)ηνρ(∂μσhρν - ∂μρhνσ - ∂νσhρμ + ∂νρhμσ) Multiply by ησρ to get the Ricci tensor: Rμν = (1/2)ησρ(∂μσhρν - ∂μρhνσ - ∂νσhρμ + ∂νρhμσ) Pick the 00 component: R00 = (1/2)ηνρ(∂00hρν - ∂0ρhν0 - ∂ν0hρ0 + ∂νρh00) As before, if the velocity is assumed to be small, the derivatives with respect to space dominate over the derivatives with respect to time. Therefore, R00 = (1/2)ηνρνρh00 R00 = -(1/2)∂ijh00 ... 2. From 1. and 2. we can write: -(1/2)∂ijh00 = 4πGρ or ∇2h00 = -8πGρ But h00 = 1 + 2φ. Therefore, ∇2φ = 4πGρ Which is the Newtonian equation for the gravitational field. Einstein's Process for Developing the Field Equations ----------------------------------------------------- From Newton's Law of Gravity: ∇2φ = 4πGρ The solution to this equation is the familiar: φ = -GM/r and F = ma = -m∇φ = GmM/r2 Einstein wanted an equation of a similar form. Geometry and curvature on the LHS, Energy density/momentum (the source of the gravitational field) on the RHS. From the weak field theory g00 = 1 + 2φ ∴ φ = (1/2)g00 - 1/2 ∴ (1/2)∇2φ = 4πGρ ∴ ∇2g00 = 8πGρ ∴ ∇2g00 = 8πGT00 This was Einstein's first clue that somehow mass effects the curvature of space. Now, the energy-momentum tensor has 2 indeces, is symmetric and its divergence equals 0. Thus, Tμν = Tνμ and DμTμν = 0 So the object representing the curvature on the LHS has to be a tensor with the same properties. The obvious choice was the Ricci tensor but while it is symmetric, it does not have a divergence equal to 0. Einstein started with the Riemann tensor and used the second BIANCHI IDENTITIES in conjunction of the asymmetric nature of the Riemann tensor, to find something that did. The second BIANCHI identities are written as: DλRαβμν + DνRαβλμ + DμRαβνλ = 0 The Riemann tensor exhibits the following symmetries: Rαβμν = -Rβαμν = -Rαβνμ = Rμναβ Note: Rαβμν = gδνRαβδμ Also, we can make use of the fact that Dg = 0 so we can move g inside or outside of the covariant derivatives at will. If we multiply the above Bianchi identity by gαμ we get: gαμDλRαβμν + gαμDνRαβλμ + gαμDμRαβνλ = 0 Now gαμRαβλμ = -gαμRαβμλ= -Rβλ So we get: DλRβν - DνRβλ + DμRμβνλ = 0 If we multiply the above by gβν we get: gβνDλRβν - gβνDνRβλ + gβνDμRμβνλ = 0 ∴ DλR - DβRβλ + DμβRμνβνλ = 0 ∴ DλR - DβRβλ + DμβRμνβνλ = 0 Interchange μ and ν and from the asymmetry properties we get: DλR - DβRβλ - DνβRνμβμλ = 0 ∴ DλR - DβRβλ - DβRβλ = 0 ∴ DλR - 2DβRβλ = 0 Alternatively, 2DβRβλ - DλR = 0 ∴ DβRβλ - (1/2)DλR = 0 ∴ DβRβλ - (1/2)gβλDβR = 0 ∴ Dβ[Rβλ - (1/2)gβλR] = 0 Gβλ = Rβλ - (1/2)gβλR Gβλ is both symmetric and has a divergence of 0. Einstein could now write: Gβλ = 8πGρTβλ Einstein - Hilbert Derivation ----------------------------- The Einstein field equations can also be obtained very elegantly through the principle of least action. Consider an small area of the gravitational potential energy field, φ, in curved spacetime. √(gxx)dx ---- √(gyy)dy | φ |    | | ---- Area = √(gxxgyy)dxdy This is a special case of a metric with only diagonal components. In reality the metric will also have gxy and gyx components so we can generalize by writing: dA = √(|g|)dxdy where |g| is the determinant of g (i.e. gxxgyy - gxygyx) We can further generalize this to a small volume element in curved spacetime by writing: dV = √(-|gμν|)d4x The minus sign arises because the time component is negative. Therefore, V = ∫√(-|gμν|)d4x This integrand has the form of a Lagrangian density which is an invariant quantity. Now, it is possible to multiply the integrand by a scalar with no loss of the invariance. The scalar takes the form of the curvature scalar, R, plus terms that represent the energy/matter fields, LM. Thus we can write: S = ∫√(-|gμν|)(k'R + LM)d4x where k' is defined to be c4/16πG After a lot of work that involves starting with the Riemann tensor (constructed out of Christoffel symbols and derivatives of Christoffel symbols, which are turn constructed out of derivatives of the metric), contracting indeces to get the Ricci tensor, contracting again to get the curvature scalar, multiplying by the square root of the determinant of the metric tensor, factoring in the mass/energy terms and then minimizing the action using the Euler-Lagrange equations, one finally obtains the EFEs. The Cosmological Constant ------------------------- Einstein recognized there was an ambiguity in his equations. In order to achieve a stationary universe he had to include another term as: Gμν + Λgμν = kTμν The CC is only a factor on galactic scales since the term is otherwise very small and can be ignored.