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Tegmark, Max, 8.033 Relativity, Fall 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), (Accessed 09 Jul, 2010). License: Creative Commons BY-NC-SA


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.


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 Four-Vectors
  • Relativistic Momentum, Energy, and Mass
  • Relativistic Particle Collisions
  • Relativity and Electricity
    • Coulomb's Law
    • Magnetic Fields
  • Introduction to Key Concepts of General Relativity
    • Principle of Equivalence
    • Cosmology
    • Friedman-Robertson-Walker Metric
    • Black Holes
    • Schwarzchild Metric
    • Gravitational Red Shift
    • Particle and Light Trajectories
    • Shapiro Delay
    • Gravitational Lensing


Physics I (8.01), Calculus II (18.02)


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.


Weekly Problem Sets (9 total) 20%
Quiz 1 (1 hour) 20%
Quiz 2 (1 hour) 20%
Final Exam 40%



Course Overview

  • Overview of Course Contents
  • Practical Issues and Advice
  • Related Subjects; Brief History of Physics

Symmetry and Invariance

  • Background and History
  • Galilean Transformation, Inertial Reference Frames
  • Classical Wave Equations; Transformation to Other Frames
  • Michelson-Morley Experiment; Aether

Symmetry and Invariance (cont.)

  • Postulates of Special Relativity
  • First Discussion of Minkowski Diagrams, World Lines
Problem set 1 due

Relativistic Kinematics

  • Derivation of Lorentz-Einstein Transformations
    • Matrix Representation
  • Introduction of Four-Vectors

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

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

Relativistic Kinematics (cont.)

  • Stellar Aberration
  • Doppler Effect and Angle Transformation via Transformation of Phase of Plane Waves
  • Fully Calibrated Minkowski Diagrams
    • Pole-Vaulter Problem
    • Twin Paradox with Constant Velocity Plus a Reversal
    • Twin Paradox with Arbitrary Acceleration
Problem set 3 due

Variational Calculus

  • Short Discourse on the Calculus of Variations
    • Extremization of Path Integrals
      • The Euler-Lagrange 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

Relativistic Dynamics and Particle Physics

  • Relativistic Momentum Inferred from Gedanken Experiment with Inelastic Collisions
  • Relativistic Relations between Force and Acceleration
  • Relativistic version of Work-Energy Theorem
    • Kinetic Energy, Rest Energy, Equivalence of Mass-Energy
    • E2 - p2 Invariant
  • Nuclear Binding Energies
    • Atomic Mass Excesses, Semi-Empirical Binding Energy Equation
    • Nuclear Reactions
    • Solar p-p Chain
Problem set 4 due

Relativistic Dynamics and Particle Physics (cont.)

  • Relativistic Motion in a B Field, Lorentz Force
    • Cyclotrons, Synchrotrons
  • Further Gedanken Experiments Relating to Mass-Energy Equivalence, Relativistic Momentum
  • Quantum Nature of Light
    • Photoelectric Effect, Photons
  • beta-Decay and the Inference of Neutrino
11 Quiz 1  

Relativistic Dynamics and Particle Physics (cont.)

  • Absorption and Emission of Light Quanta
    • Atomic and Nuclear Recoil
    • Mössbauer Effect
    • Pound-Rebka Experiment
  • Collisions
    • Between Photons and Moving Atoms
    • Elastic
    • Compton
    • Inverse Compton
    • Between Photon and Relativistic Particle
Problem set 5 due

Relativistic Dynamics and Particle Physics (cont.)

  • Particle Production
    • Threshold Energy
      • Colliding Particle Beams
      • Two Photons Producing an Electron/Positron Pair

Relativistic Dynamics and Particle Physics (cont.)

  • Formal Transformation of E and P as a Four-Vector
    • Revisit the Relativistic Doppler Effect
  • Relativistic Invariant E2 - p2 for a Collection of Particles
Problem set 6 due

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

Relativity and Electromagnetism (cont.)

  • General Transformation Laws for E and B
  • Magnetic Force due to Current-Bearing Wire
  • Force between Current-Bearing Wires
Problem set 7 due

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

General Relativity and Cosmology

  • Cosmological Redshifts and the Hubble Law

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 Omegam, OmegaLambda and Omegak

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 Omegam, OmegaLambda and Omegak
    • Acceleration Parameter as a Function of Scale Factor
    • Current S status of Cosmology, Unsolved Puzzles
  • Problem set 8 due
  • 21
Quiz 2  

General Relativity and Cosmology (cont.)

  • Handout Defining Einstein Field Equations, Einstein Tensor, Stress-Energy 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 Non-zero Components are Functions of Radius Only
  • The Differential Equations for G00 and G11
  • Shell Radius vs. Bookkeepers Radial Coordinate

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

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 Weak-Field Limit

General Relativity and Black Holes (cont.)

  • Keplers Third Law in the Schwarzschild Metric
  • Relativistic Precession in the Weak-Field Limit
    • Taylor-Hulse Binary Neutron Star System
  • Derivation of the Last Stable Circular Orbit at 6M
    • Analytic E and L for Circular Orbits
Problem set 9 due

General Relativity and Black Holes (cont.)

  • Photon Trajectories
    • Derive Differential Equation for the Trajectories
    • Critical Impact Parameter
    • Derive Expression for Light Bending in the Weak-Field Limit
  • Shapiro Time Delay   Tell A Friend