 
Abstract/Syllabus:

Tegmark, Max, 8.033 Relativity, Fall 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 09 Jul, 2010). License: Creative Commons BYNCSA
Relativity
Fall 2006
Albert Einstein first published his theory of special relativity in 1905. (Image courtesy of Wikipedia.)
Course Highlights
This course features a complete set of lecture notes, assignments, and exams.
Course Description
This course, which concentrates on special relativity, is normally taken by physics majors in their sophomore year. Topics include Einstein's postulates, the Lorentz transformation, relativistic effects and paradoxes, and applications involving electromagnetism and particle physics. This course also provides a brief introduction to some concepts of general relativity, including the principle of equivalence, the Schwartzschild metric and black holes, and the FRW metric and cosmology.
Syllabus
Course Description
This course is normally taken by physics majors in their sophomore year.
Major Topics Include

Einstein's Postulates, and their Consequences for

Simultaneity

Time Dilation

Length Contraction

Clock Synchronization

Lorentz Transformation

Relativistic Effects and Paradoxes

Minkowski Diagrams

Relativistic Invariants and FourVectors

Relativistic Momentum, Energy, and Mass

Relativistic Particle Collisions

Relativity and Electricity

Coulomb's Law

Magnetic Fields

Introduction to Key Concepts of General Relativity
Prerequisites
Physics I (8.01), Calculus II (18.02)
Texts
Resnick, Robert. Introduction to Special Relativity. New York, NY: Wiley, 1968.
French, A. P. Special Relativity. Massachusetts Institute of Technology Education Research Center: MIT Introductory Physics Series. New York, NY: Norton, 1968.
Taylor, Edwin F., and John A. Wheeler. Exploring Black Holes: Introduction to General Relativity. San Francisco, CA: Addison Wesley Longman, 2000. ISBN: 9780201384239.
Grading
Grading criteria.
ACTIVITIES 
PERCENTAGES 
Weekly Problem Sets (9 total) 
20% 
Quiz 1 (1 hour) 
20% 
Quiz 2 (1 hour) 
20% 
Final Exam 
40% 
Calendar
Course calendar.
SES # 
TOPICS 
KEY DATES 
1 
Course Overview
 Overview of Course Contents
 Practical Issues and Advice
 Related Subjects; Brief History of Physics


2 
Symmetry and Invariance

Background and History

Galilean Transformation, Inertial Reference Frames

Classical Wave Equations; Transformation to Other Frames

MichelsonMorley Experiment; Aether


3 
Symmetry and Invariance (cont.)
 Postulates of Special Relativity
 First Discussion of Minkowski Diagrams, World Lines

Problem set 1 due 
4 
Relativistic Kinematics
 Derivation of LorentzEinstein Transformations
 Introduction of FourVectors


5 
Relativistic Kinematics (cont.)
 Time Dilation and Length Contraction
 Decay of Atmospheric Muons
 Pole Vaulter Problem
 Alternative Looks at Time Dilation and Length Contraction
 Spacetime Intervals
 First Discussion of Accelerated Clocks

Problem set 2 due 
6 
Relativistic Kinematics (cont.)
 Addition of Velocities
 Angle Transformation for Trajectories
 Doppler Effect
 Classical Doppler Effect for Sound
 Relativistic Doppler Effect
 Astrophysical Examples; Relativistic and Superluminal Jets


7 
Relativistic Kinematics (cont.)
 Stellar Aberration
 Doppler Effect and Angle Transformation via Transformation of Phase of Plane Waves
 Fully Calibrated Minkowski Diagrams
 PoleVaulter Problem
 Twin Paradox with Constant Velocity Plus a Reversal
 Twin Paradox with Arbitrary Acceleration

Problem set 3 due 
8 
Variational Calculus
 Short Discourse on the Calculus of Variations
 Extremization of Path Integrals
 The EulerLagrange Equations and Constants of the Motion
 Brachistochrone Problem
 Extremal Aging for Inertially Moving Clocks
 Optional Problems in the Use of the Calculus of Variations as Applied to Lagragian Mechanics and Other Problems in the Extremization of Path Integrals


9 
Relativistic Dynamics and Particle Physics
 Relativistic Momentum Inferred from Gedanken Experiment with Inelastic Collisions
 Relativistic Relations between Force and Acceleration
 Relativistic version of WorkEnergy Theorem
 Kinetic Energy, Rest Energy, Equivalence of MassEnergy
 E^{2}  p^{2} Invariant
 Nuclear Binding Energies
 Atomic Mass Excesses, SemiEmpirical Binding Energy Equation
 Nuclear Reactions
 Solar pp Chain

Problem set 4 due 
10 
Relativistic Dynamics and Particle Physics (cont.)
 Relativistic Motion in a B Field, Lorentz Force
 Further Gedanken Experiments Relating to MassEnergy Equivalence, Relativistic Momentum
 Quantum Nature of Light
 Photoelectric Effect, Photons
 betaDecay and the Inference of Neutrino


11 
Quiz 1 

12 
Relativistic Dynamics and Particle Physics (cont.)
 Absorption and Emission of Light Quanta
 Atomic and Nuclear Recoil
 Mössbauer Effect
 PoundRebka Experiment
 Collisions
 Between Photons and Moving Atoms
 Elastic
 Compton
 Inverse Compton
 Between Photon and Relativistic Particle

Problem set 5 due 
13 
Relativistic Dynamics and Particle Physics (cont.)
 Particle Production
 Threshold Energy
 Colliding Particle Beams
 Two Photons Producing an Electron/Positron Pair


14 
Relativistic Dynamics and Particle Physics (cont.)
 Formal Transformation of E and P as a FourVector
 Revisit the Relativistic Doppler Effect
 Relativistic Invariant E^{2}  p^{2} for a Collection of Particles

Problem set 6 due 
15 
Relativity and Electromagnetism
 Coulomb's Law
 Transformation of Coulomb's Law
 Force on a Moving Test Charge
 Magnetic Field and Relativity
 Derivation of Lorentz Force


16 
Relativity and Electromagnetism (cont.)
 General Transformation Laws for E and B
 Magnetic Force due to CurrentBearing Wire
 Force between CurrentBearing Wires

Problem set 7 due 
17 
The Equivalence Principle and General Relativity
 Strong and Weak Principles of Equivalence
 Local Equivalence of Gravity and Acceleration
 Elevator Thought Experiments
 Gravitational Redshift
 Light Bending
 Relative Acceleration of Test Particles in Falling Elevator of Finite Size
 Definition of the Metric Tensor
 Analogy between the Metric Tensor and the Ordinary Potential, and between Einstein's Field Equations and Poisson's Equation


18 
General Relativity and Cosmology
 Cosmological Redshifts and the Hubble Law


19 
General Relativity and Cosmology (cont.)
 Cosmology
 Dynamical Equations for the Scale Factor a  Including Ordinary Matter, Dark Matter, and Dark Energy
 Critical Closure Density; Open, Closed, Flat Universes
 Solutions for Various Combinations of Omega_{m}, Omega_{Lambda} and Omega_{k}


20 
General Relativity and Cosmology (cont.)
 Cosmology (cont.)
 Age of the Universe, Brief History
 Relation between Scale Factor and Z from the Doppler Shift
 Lookback Age as a Function of Z for Various Values of Omega_{m}, Omega_{Lambda} and Omega_{k}
 Acceleration Parameter as a Function of Scale Factor
 Current S status of Cosmology, Unsolved Puzzles



Quiz 2 

22 
General Relativity and Cosmology (cont.)
 Handout Defining Einstein Field Equations, Einstein Tensor, StressEnergy Tensor, Curvature Scalar, Ricci Tensor, Christoffel Symbols, Riemann Curvature Tensor
 Symmetry Arguments by Which 6 Schwarzschild Metric Tensor Components Vanish
 Symmetry Arguments for Why the Nonzero Components are Functions of Radius Only
 The Differential Equations for G00 and G11
 Shell Radius vs. Bookkeepers Radial Coordinate


23 
General Relativity and Black Holes
 Gravitational Redshift
 Application to the GPS System
 Particle Orbits
 Use Euler Equations (for External Aging) in Connection with the Schwarzschild Metric to find Constants of the Motion E and L
 Derive the Full Expression for the Effective Potential


24 
General Relativity and Black Holes (cont.)
 Derive Analytic Results for Radial Motion
 Compare Speeds and Energies for Bookkeeper and Shell Observers
 Equations of Motion for a General Orbit
 Explain How these can be Numerically Integrated
 Expand the Effective Potential in the WeakField Limit


25 
General Relativity and Black Holes (cont.)
 Keplers Third Law in the Schwarzschild Metric
 Relativistic Precession in the WeakField Limit
 TaylorHulse Binary Neutron Star System
 Derivation of the Last Stable Circular Orbit at 6M
 Analytic E and L for Circular Orbits

Problem set 9 due 
26 
General Relativity and Black Holes (cont.)
 Photon Trajectories
 Derive Differential Equation for the Trajectories
 Critical Impact Parameter
 Derive Expression for Light Bending in the WeakField Limit
 Shapiro Time Delay





Further Reading:

Readings
Course readings.
SES # 
TOPICS 
READINGS 
1 
Course Overview
 Overview of Course Contents
 Practical Issues and Advice
 Related Subjects; Brief History of Physics

Resnick: Chapter 1 and beginning of Chapter 2.
French: Chapters 1 and 2.
Course Study Guide Handout

2 
Symmetry and Invariance

Background and History

Galilean Transformation, Inertial Reference Frames

Classical Wave Equations; Transformation to Other Frames

MichelsonMorley Experiment; Aether

Resnick: Chapter 1 and beginning of Chapter 2.
French: Chapters 1 and 2.
Symmetry Handout

3 
Symmetry and Invariance (cont.)
 Postulates of Special Relativity
 First Discussion of Minkowski Diagrams, World Lines

Resnick: Chapter 1 and beginning of Chapter 2.
French: Chapters 1 and 2.
Symmetry Handout

4 
Relativistic Kinematics
 Derivation of LorentzEinstein Transformations
 Introduction of FourVectors

Resnick: Chapter 2.
French: Chapters 3 and 4.
Matrix Primer Handout
Kinematics Handout

5 
Relativistic Kinematics (cont.)
 Time Dilation and Length Contraction
 Decay of Atmospheric Muons
 Pole Vaulter Problem
 Alternative Looks at Time Dilation and Length Contraction
 Spacetime Intervals
 First Discussion of Accelerated Clocks

Resnick: Chapter 2.
French: Chapters 3 and 4.
Matrix Primer handout
Kinematics Handout

6 
Relativistic Kinematics (cont.)
 Addition of Velocities
 Angle Transformation for Trajectories
 Doppler Effect
 Classical Doppler Effect for Sound
 Relativistic Doppler Effect
 Astrophysical Examples; Relativistic and Superluminal Jets

French: Chapter 5.
Kinematics Handout

7 
Relativistic Kinematics (cont.)
 Stellar Aberration
 Doppler Effect and Angle Transformation via Transformation of Phase of Plane Waves
 Fully Calibrated Minkowski Diagrams
 PoleVaulter Problem
 Twin Paradox with Constant Velocity Plus a Reversal
 Twin Paradox with Arbitrary Acceleration

French: Chapter 5.
Kinematics Handout

8 
Variational Calculus
 Short Discourse on the Calculus of Variations
 Extremization of Path Integrals
 The EulerLagrange Equations and Constants of the Motion
 Brachistochrone Problem
 Extremal Aging for Inertially Moving Clocks
 Optional Problems in the Use of the Calculus of Variations as Applied to Lagragian Mechanics and Other Problems in the Extremization of Path Integrals

Resnick: Supplementary Topics A and B in pages 188209. 
9 
Relativistic Dynamics and Particle Physics
 Relativistic Momentum Inferred from Gedanken Experiment with Inelastic Collisions
 Relativistic Relations between Force and Acceleration
 Relativistic Version of WorkEnergy Theorem
 Kinetic Energy, Rest Energy, Equivalence of MassEnergy
 E^{2}  p^{2} Invariant
 Nuclear Binding Energies
 Atomic Mass Excesses, SemiEmpirical Binding Energy Equation
 Nuclear Reactions
 Solar pp Chain

Resnick: Supplementary Topics A and B in pages 188209.
Relativistic Dynamics Handout
Particle Physics Handout

10 
Relativistic Dynamics and Particle Physics (cont.)
 Relativistic Motion in a B Field, Lorentz Force
 Further Gedanken Experiments Relating to MassEnergy Equivalence, Relativistic Momentum
 Quantum Nature of Light
 Photoelectric Effect, Photons
 betaDecay and the Inference of Neutrino

Relativistic Dynamics Chapters in Resnick and French.
Relativistic Dynamics Handout
Particle Physics Handout

11 
Quiz 1 

12 
Relativistic Dynamics and Particle Physics (cont.)
 Absorption and Emission of Light Quanta
 Atomic and Nuclear Recoil
 Mössbauer Effect
 PoundRebka Experiment
 Collisions
 Between Photons and Moving Atoms
 Elastic
 Compton
 Inverse Compton
 Between Photon and Relativistic Particle

Relativistic Dynamics Chapters in Resnick and French.
Relativistic Dynamics Handout
Particle Physics Handout

13 
Relativistic Dynamics and Particle Physics (cont.)
 Particle Production
 Threshold Energy
 Colliding Particle Beams
 Two Photons Producing an Electron/Positron Pair

Resnick: Chapter 3.
French: Chapters 6 and 7.
Relativistic Dynamics Handout
Particle Physics Handout

14 
Relativistic Dynamics and Particle Physics (cont.)
 Formal Transformation of E and P as a FourVector
 Revisit the Relativistic Doppler Effect
 Relativistic Invariant E^{2}  p^{2} for a Collection of Particles

Resnick: Chapter 3.
French: Chapters 6 and 7.
Relativistic Dynamics Handout
Particle Physics Handout

15 
Relativity and Electromagnetism
 Coulomb's Law
 Transformation of Coulomb's Law
 Force on a Moving Test Charge
 Magnetic Field and Relativity
 Derivation of Lorentz Force

Resnick: Chapter 4.
French: Chapter 8.
Note: As stressed in lecture, please don't get bogged down with their excessive E&M algebra.
Electromagnetism Handout

16 
Relativity and Electromagnetism (cont.)
 General Transformation Laws for E and B
 Magnetic Force due to CurrentBearing Wire
 Force between CurrentBearing Wires

Resnick: Chapter 4.
French: Chapter 8.
Electromagnetism Handout

17 
The Equivalence Principle and General Relativity
 Strong and Weak Principles of Equivalence
 Local Equivalence of Gravity and Acceleration
 Elevator Thought Experiments
 Gravitational Redshift
 Light Bending
 Relative Acceleration of Test Particles in Falling Elevator of Finite Size
 Definition of the Metric Tensor
 Analogy between the Metric Tensor and the Ordinary Potential, and between Einstein's Field Equations and Poisson's Equation

Taylor and Wheeler: Until pp. 218, in addition to Project G.
Please also read the following:
Cosmology: Popular Overview
Lemonick, Michael D. "The End." Time, June 25, 2001, 4856.
Cosmology: Spacetime Overview
Tegmark, Max. "Spacetime." Science 296 (2002): 14271433.
Cosmology: Ned Wright's Tutorial.

18 
General Relativity and Cosmology
 Cosmological Redshifts and the Hubble Law

Taylor and Wheeler: Until page 218, also project G.
Please also read the following:
Cosmology: Popular Overview
Lemonick, Michael D. "The End." Time, June 25, 2001, 4856.
Cosmology: Spacetime Overview
Tegmark, Max. "Spacetime." Science 296 (2002): 14271433.
Cosmology: Ned Wright's Tutorial.

19 
General Relativity and Cosmology (cont.)
 Cosmology
 Dynamical Equations for the Scale Factor a  Including Ordinary Matter, Dark Matter, and Dark Energy
 Critical Closure Density; Open, Closed, Flat Universes
 Solutions for Various combinations of Omega_{m}, Omega_{Lambda} and Omega_{k}

Taylor and Wheeler: Until pp. 218, also Project G.
Please also read the following:
Cosmology: Popular Overview
Lemonick, Michael D. "The End." Time, June 25, 2001, 4856.
Cosmology: Spacetime Overview
Tegmark, Max. "Spacetime." Science 296 (2002): 14271433.
Cosmology: Ned Wright's Tutorial.

20 
General Relativity and Cosmology (cont.)
 Cosmology (cont.)
 Age of the Universe, Brief History
 Relation between Scale Factor and Z from the Doppler Shift
 Lookback Age as a Function of Z for Various Values of Omega_{m}, Omega_{Lambda} and Omega_{k}
 Acceleration Parameter as a Function of Scale Factor
 Current S Status of Cosmology, Unsolved Puzzles

Taylor and Wheeler: Until pp. 218, also Project G.
Please also read the following:
Cosmology: Popular Overview
Lemonick, Michael D. "The End." Time, June 25, 2001, 4856.
Cosmology: Spacetime Overview
Tegmark, Max. "Spacetime." Science 296 (2002): 14271433.
Cosmology: Ned Wright's Tutorial.


Quiz 2 

22 
General Relativity and Cosmology (cont.)
 Handout Defining Einstein Field Equations, Einstein Tensor, StressEnergy Tensor, Curvature Scalar, Ricci Tensor, Christoffel Symbols, Riemann Curvature Tensor
 Symmetry Arguments by Which 6 Schwarzschild Metric Tensor Components Vanish
 Symmetry Arguments for Why the Nonzero Components are Functions of Radius Only
 The Differential Equations for G00 and G11
 Shell Radius vs. Bookkeepers Radial Coordinate

Taylor and Wheeler: Chapters 2, 3, 4, 5, and Project D.
Note: You will be responsible only for the corresponding material that was actually covered in the lectures. Project E should also be understandable, but this topic will be mentioned only very briefly in lecture.
General Relativity Handout

23 
General Relativity and Black Holes
 Gravitational Redshift
 Application to the GPS System
 Particle Orbits
 Use Euler Equations (for External Aging) in Connection with the Schwarzschild Metric to find Constants of the Motion E and L
 Derive the Full Expression for the Effective Potential

Taylor and Wheeler: Chapters 2, 3, 4, 5, and Project D.
General Relativity Handout
Einstein's Field Equations Handout

24 
General Relativity and Black Holes (cont.)
 Derive Analytic Results for Radial Motion
 Compare Speeds and Energies for Bookkeeper and Shell Observers
 Equations of Motion for a General Orbit
 Explain How these can be Numerically Integrated
 Expand the Effective Potential in the WeakField Limit

Taylor and Wheeler: Chapters 2, 3, 4, 5, and Project D.
General Relativity Handout
Einstein's Field Equations Handout

25 
General Relativity and Black Holes (cont.)
 Keplers Third Law in the Schwarzschild Metric
 Relativistic Precession in the WeakField Limit
 TaylorHulse Binary Neutron Star System
 Derivation of the Last Stable Circular Orbit at 6M
 Analytic E and L for Circular Orbits

Taylor and Wheeler: Chapters 2, 3, 4, 5, and Project D.
General Relativity Handout
Einstein's Field Equations Handout

26 
General Relativity and Black Holes (cont.)
 Photon Trajectories
 Derive Differential Equation for the Trajectories
 Critical Impact Parameter
 Derive Expression for Light Bending in the WeakField Limit
 Shapiro Time Delay

Taylor and Wheeler: Chapters 2, 3, 4, 5, and Project D.
General Relativity Handout
Einstein's Field Equations Handout




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