8.033 | Fall 2006 | Undergraduate

Relativity

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 (PDF)

2

Symmetry and Invariance

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

Resnick: Chapter 1 and beginning of Chapter 2.

French: Chapters 1 and 2.

Symmetry Handout (PDF)

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 (PDF)

4

Relativistic Kinematics

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

Resnick: Chapter 2.

French: Chapters 3 and 4.

Matrix Primer Handout (PDF)

Kinematics Handout (PDF)

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 (PDF)

Kinematics Handout (PDF)

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 (PDF)

7

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

French: Chapter 5.

Kinematics Handout (PDF)

8

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

Resnick: Supplementary Topics A and B in pages 188-209.
9

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

Resnick: Supplementary Topics A and B in pages 188-209.

Relativistic Dynamics Handout (PDF)

Particle Physics Handout (PDF)

10

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

Relativistic Dynamics Chapters in Resnick and French.

Relativistic Dynamics Handout (PDF)

Particle Physics Handout (PDF)

11 Quiz 1  
12

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

Relativistic Dynamics Chapters in Resnick and French.

Relativistic Dynamics Handout (PDF)

Particle Physics Handout (PDF)

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 (PDF)

Particle Physics Handout (PDF)

14

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

Resnick: Chapter 3.

French: Chapters 6 and 7.

Relativistic Dynamics Handout (PDF)

Particle Physics Handout (PDF)

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 (PDF)

16

Relativity and Electromagnetism (cont.)

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

Resnick: Chapter 4.

French: Chapter 8.

Electromagnetism Handout (PDF)

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. 2-18, in addition to Project G.

Please also read the following:

Cosmology: Spacetime Overview 
Tegmark, Max. “Spacetime.” Science 296 (2002): 1427-1433.

Cosmology: Ned Wright’s Tutorial.

18

General Relativity and Cosmology

  • Cosmological Redshifts and the Hubble Law

Taylor and Wheeler: Until page 2-18, also project G.

Please also read the following:

Cosmology: Popular Overview 
Lemonick, Michael D. “The End.” Time, June 25, 2001, 48-56.

Cosmology: Spacetime Overview 
Tegmark, Max. “Spacetime.” Science 296 (2002): 1427-1433.

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 Omegam, OmegaLambda and Omegak

Taylor and Wheeler: Until pp. 2-18, also Project G.

Please also read the following:

Cosmology: Popular Overview 
Lemonick, Michael D. “The End.” Time, June 25, 2001, 48-56.

Cosmology: Spacetime Overview 
Tegmark, Max. “Spacetime.” Science 296 (2002): 1427-1433.

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 Omegam, OmegaLambda and Omegak
    • Acceleration Parameter as a Function of Scale Factor
    • Current S Status of Cosmology, Unsolved Puzzles

Taylor and Wheeler: Until pp. 2-18, also Project G.

Please also read the following:

Cosmology: Popular Overview 
Lemonick, Michael D. “The End.” Time, June 25, 2001, 48-56.

Cosmology: Spacetime Overview 
Tegmark, Max. “Spacetime.” Science 296 (2002): 1427-1433.

Cosmology: Ned Wright’s Tutorial.

  • 21
Quiz 2  
22

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

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 (PDF)

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 (PDF)

Einstein’s Field Equations Handout (PDF)

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 Weak-Field Limit

Taylor and Wheeler: Chapters 2, 3, 4, 5, and Project D.

General Relativity Handout (PDF)

Einstein’s Field Equations Handout (PDF)

25

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

Taylor and Wheeler: Chapters 2, 3, 4, 5, and Project D.

General Relativity Handout (PDF)

Einstein’s Field Equations Handout (PDF)

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 Weak-Field Limit
  • Shapiro Time Delay

Taylor and Wheeler: Chapters 2, 3, 4, 5, and Project D.

General Relativity Handout (PDF)

Einstein’s Field Equations Handout (PDF)

Course Info

Instructor
Departments
As Taught In
Fall 2006
Learning Resource Types
Problem Sets
Exams with Solutions
Lecture Notes