SYLLABUS
MTH 437/537

Here is a rough summary of what we have done in class. It is not intended to be comprehensive.
I will try to keep it a lecture or two ahead of where we actually are, but this may not be very accurate.

(Section numbers refer to d'Inverno.)

Week 1:

  • Introduction: Relativity is both physics and geometry.
  • Surveyor's parable (from Taylor & Wheeler).
  • Special Relativity: §2.5-§2.6.
  • Spacetime diagrams.
  • Simultaneity: §2.10-§2.11.
  • Lorentz transformations: §3.1-§3.2 (optional: §2.7-§2.9 and §2.12); my notes.
  • Length contraction: §3.3.
  • Time dilation: §3.4.

    Week 2:

  • Addition of velocities: §3.5.
  • Paradoxes
  • Twin paradox: §3.9.
    Supplemental reading: The Twin Paradox Revisited, Tevian Dray, Am. J. Phys. 58, 822-825 (1990).
    See also §1.13 of Schutz (see book list).
  • Proper Time
  • Light Cone
  • Tensor algebra: §5.1-5.8.
  • Tensors
  • Components and indices
  • Summation convention

    • Week 3:

  • Vectors as derivative operators: §5.9.
  • 1-forms as differentials
  • The metric tensor: §6.8
  • Coordinate bases and Jacobians
  • Orthonormal bases
  • Example: Polar Coordinates
  • Introduction to Tensor calculus: §6.1

    Week 4:

  • Covariant differentiation: §6.3, §6.10; my notes.
  • Curvature: §6.5, §6.12; my notes.
  • Geodesics: §6.4, §6.9.

    Week 5:

  • Energy-momentum tensor: §12.2-§12.3. (Supplemental reading: §12.4-§12.6.)
  • Geodesic deviation: §10.3. (Supplemental reading: §10.2, §10.4.)
  • Einstein's equation: § 10.5-§10.7.
  • General relativity: §13.1-§13.3.
  • Equations of motion: §13.4.

    Week 6:

  • Homogeneity & Isotropy: §22.4.
  • What is Cosmology?: §22.5-§22.6.
  • Spaces of constant curvature: §22.7-§22.8.
  • Friedmann-Robertson-Walker models: §22.9, §23.1-§23.4.
    Look especially at Figure 23.1 in §23.3.

    Week 7:

  • More Cosmological models: §23.5-§23.9.
  • Red shift: §22.10,§22.12.
  • Symmetries & Killing vectors: §7.7.

    Week 8:

  • Time-independence: Static vs. Stationary: §14.1-§14.3.
  • Spherical Symmetry: §14.4.
  • The Schwarzschild metric: §14.5-§14.6.
  • Experimental tests of general relativity: §15.

    Week 9:

  • Proper time vs. coordinate time: §16.5.
  • Acceleration: §3.7-§3.8.
  • Rindler space.
    Compare pp. 149-152 of Wald (see book list).
  • Black Holes: §16.9.
  • Maximal extension: Kruskal geometry: §17.2-§17.3.

    Week 10:

  • Penrose diagrams: §17.4-§17.5.
  • Charged black holes: §18.
  • Rotating black holes: §19.