Mathematics 460, Section 500, Fall, 2011

Tensors and General Relativity

Announcements (reverse chronological order)

• Jan 1: The final exam key is now available at the proper link below. I am sorry for the delay.
• Dec 14: Grades are posted. I will put up a solution set for the final eventually; please come back and look at it.
• Dec 5: I will not have an office hour today. I expect to be here Wednesday and Thursday afternoons.
• Nov 28: Announcement of next Applied Math Undergraduate SEminar (Nov 30, room changed to BLOC 166) (Math 442, student projects) ._. AMUSE Web page
• Nov 22: Schwarzschild curvature calculations are now due NEXT Wednesday (Nov 30).
• Nov 11: Next Monday (Nov 14) office hour will start at 4:00 (or slightly later), not 3:00.
• Nov 8: Announcement of next Math Club Meeting (Nov 14) (W. Bangerth, scientific computing) ._. Math Club Web page
• Oct 31: Reminder: Colloquium on The Accelerating Universe (the 2011 Nobel Prize in Physics, with a TAMU connection), Nov 3
• Oct 28: The final version of the E&M paper is now due Monday, Nov 7.
• Oct 16: This week's Wednesday office hour will be at 3, not 4 (Oct 19). Thursday office hour is cancelled (Oct 20).
• Oct 11: Due date for final version of E&M paper extended to Nov 2. Note that on the intervening Wednesdays you have some heavy-duty Christoffel calculations due, on which I strongly encourage you to work in teams.
• Special procedures for Week 4 (19-23 Sept): I will be away all week. Prof. Philip Yasskin will handle the course, concentrating on helping you with the electromagnetism assignment.
  • Start work on the electromagnetism paper. (At least read the instructions and think about what you need to do!) A TeX file is available for those who know what to do with it).
  • Read Chapter 4.
  • Read Section 5.1 and the associated pages of the notes.
  • Consult the solutions to the second week's homework problems. The 2009 class came up with a variety of approaches, which I synthesized into one document.

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

Lecture notes
Chapter on covariant derivatives and non-Abelian gauge theories with bibliography from Aspects of Quantum Field Theory in Curved Space-Time, S. A, Fulling, Cambridge U. P., 2008.

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 Sept. 7.) Also: Answer the 3 questions on pp. 5-6 of the notes (2 "canards" and one "topic for class discussion"). If you want to use concrete numbers in the Lorentz contraction-dilation discussion, I suggest taking speed 3/5. Turn in these essays on Sept. 9. ("Essay" does not mean a major, multipage production, but it should be a paragraph in intelligible English.)
• Chapter 2: 12, 13, 16, 19, 21, 22, 24, 30 (Turn in 19 and 24 on Sept. 14. 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 (on Christoffel symbols) per pair on Oct. 19.) You will need the results as input into a later assignment.
• Now find the Christoffel symbols for the static, spherically symmetric metric (see Exercise 6.35 of Schutz or p. of notes). This time you can work in teams of 4 if you like. Due Oct. 26.
• 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 tough, so 2/3 of the points will be "extra credit" -- that is, the actual maximum point value of all homework will be 120, not 100. I'll accept papers any time before the end of classes (Monday, Dec. 5).)
• 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 pairs and turn in one paper per pair on Nov. 16.)
• Now we need all the same stuff for the static, spherically symmetric metric (Work in teams or pairs; due Nov. 23.)

Test solutions

• Midterm
• Final

Track your grades on Blackboard Vista.

Supplementary material

• Solution to Exercise 1.3
• MathCad session on composition of Lorentz transformations
• Solution to Exercises 2.19 and 2.21
• Notes on hyperbolic functions and the twin paradox (from Honors M. 311 notes)
• Notes on div, grad, and curl in curvilinear coordinates (from Honors M. 311 notes)
• Recent observational cosmology, with an executive summary by our local expert.
• Nobel Prize scientific background document on the accelerating universe
• Special issues of journals entirely devoted to relativity

Old course home pages: Fall 2009 ._. Spring 2008 ._. Fall 2005

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

e-mail: fulling@math.tamu.edu