Mathematics 489, Section 500, Fall, 2009

Tensors and General Relativity

Announcements (reverse chronological order)

• Dec 21: The final exam key is posted below. Please study at least the multiple-choice questions. I expected better results there.
• Dec 17: Final exam and course grades are now in Vista. Curve? Yes, so violent you don't really want to know. (I think the course achieved its aims for most of you.) I will try to post at least a partial final exam solution set later this week. I was disappointed in the results of the tensor index manipulation problem. It was essentially the same as Qu. 5 of the Fall 2005 final; please read that solution. On the other hand, you did well on the open-ended question about static solutions.
• Dec 4: Announcement of next Math Club Meeting (Dec 8) (movie)
• Nov 4: Test announcement for Nov 11 and 13
• Oct 25: (a) Test key is posted below, and grades have been brought up to date in Vista. (b) Some notes on counting degrees of freedom have been added below (under "Supplementary material").
• Sep 27: No office hour on Monday, Sep 28. (I need to attend a Ph.D. oral exam.)
• Sep 16: Special procedures for week of Sep 21-25: Prof. Yasskin will handle the class in my absence.
  • Start work on the electromagnetism paper. Read the instructions all the way to the end! Dr. Yasskin will answer questions about the assignment on Friday.
  • Read Chapter 4 "superficially" (see below).
  • Read Sec. 5.1.
  • There will be no written homework on Chapters 3 and 4.
  • In Week 6 (starting Sep 28) I'll be back and we will work hard on the rest of Chapter 5!
• Sep 1: Undergraduate Research Scholars announcement
• Aug 30: Announcement of Sep 16 meeting on How to Find and Compete for Graduate Fellowships

Course handout ._._. Please see my home page for up-to-date office hours.

Track your grades on Blackboard Vista.

Electromagnetism paper ._._. Here's the TeX file, in case you want to import some of the questions into your own document. It is in Plain TeX and uses the vanilla macros. (LaTeX users will need to make some changes.)

Homework exercises (These are not to be turned in except as announced. Uncollected problems, or questions inspired by them, may show up later on exams.)

• Chapter 1: 3, 5, 13, 14, 15, 18, 19 (Turn in 18 and 19 on Sep 9, along with any thoughts on the following for extra credit.)
  Also: Make a list of the assumptions made (explicitly or implicitly) in the "proof" in Sec. 1.6. (An example of the sort of issue I have in mind here is this: In Sec. 1.5, p. 7, we read that "the event R on the t axis must be as far from the origin as event E." Does this tacitly assume an invariance under time reversal?)
• Chapter 2: 12, 13, 16, 19, 21, 22, 24, 30 (Turn in 19 and 24 on Sep 16. It may help to do 21 before 19.)
• Chapter 3: 4, 6, 9, 13, 16, 21 (No written homework.)
• Chapter 4: None (We will not "cover" this chapter, but you will want to read it at least superficially to assure continuity with the later chapters.)
• Chapter 5: 2, 7, 8, 11, 12, 13, 20, 22 (No written homework.)
• Chapter 6: 7, 9, 13, 18, 23, 25, 32, 33
  Also: Calculate the Christoffel symbols for the Robertson-Walker metric (12.13). (Work in pairs! One of you should use the geodesic Lagrangian method (see notes, pp. 41-42), and the other should check the results with eq. (5.75). (Trade jobs halfway through.) Turn in one paper per pair on Oct 23.)
• Chapter 7: 2, 7 [omit (iii)] , 10 (See next line for instructions. Hint on 10(b): There are 4 types of symmetries: space translations, time translation, rotations, Lorentz boosts.)
• Chapter 8: 4, 5, 9, 18 (From Chapters 7 and 8, turn in only Exercises 7.7, 7.10, and 8.18. These are "extra credit" -- that is, I don't expect most of you to do them, or to complete them. I'll accept papers any time before the end of classes (Monday, 7 Dec).)
• Chapter 12: 1, 4, 8, 20, 21
  Also: For the Robertson-Walker metric, calculate the Riemann tensor (20 independent components), Ricci tensor, Ricci curvature scalar, and Einstein tensor. Check that the last obeys the conservation law (contracted Bianchi identity). (Work in teams and turn in one paper per team on Nov 18.)
• to be continued

Test solutions

• Test A
• Test B (CORRECTED 10 DEC) ._. which was an unusual two-day affair
• Final

Supplementary material

• Solution to Exercise 1.3
• Solution to Exercises 2.19 and 2.21 (corrected and extended)
• Notes on the relation between matrix inverses and index shuffling (to be added to Chapter 3 notes)
• Notes on tensor calculus in hyperbolic coordinates (to be added to Chapter 5 notes)
• Notes on counting degrees of freedom of the electromagnetic field (to be added to Chapter 8 notes)
• Notes on ergospheres (to be added to Chapter 11 notes)
• Notes on div, grad, and curl in curvilinear coordinates (from Honors M. 311 notes)
• Notes on hyperbolic functions and the twin paradox (from Honors M. 311 notes)
• The latest observational cosmology, with an executive summary by our local expert.
• More books on relativity
• Special issues of journals entirely devoted to relativity

Old course home pages: Spring 2008 ._. Fall 2005

Go to home pages: Fulling ._._. Calclab ._._. Math Dept ._._. University

e-mail: fulling@math.tamu.edu