8.033 | Fall 2006 | Undergraduate


Lecture Notes


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


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


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

(PDF - 1.1 MB)

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


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


Relativistic Dynamics and Particle Physics (cont.)

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

(PDF - 1.2 MB)

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

(PDF - 2.0 MB)

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


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

(PDF - 1.4 MB)
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

(PDF - 4.1 MB)

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

(PDF - 1.1 MB)

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


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

(PDF - 1.8 MB)

Course Info

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