Mathematics 460, Section 500, Fall, 2013

Tensors and General Relativity

Last updated Sat 19 Jul '14

Announcements (reverse chronological order)

4 Dec:Here is an undergraduate conference you might want to attend in February.
19 Nov: Last AMUSE meeting 20 Nov (student reports on internships). See http://www.math.tamu.edu/seminars/amuse
11 Nov: Next AMUSE meeting 13 Nov (geology and ecology). See http://www.math.tamu.edu/seminars/amuse
4 Nov: Next AMUSE meeting 6 Nov (epidemiology). See http://www.math.tamu.edu/seminars/amuse
30 Oct: No office hour tomorrow (31 Oct)
29 Oct: Next AMUSE meeting 30 Oct (Nuclear engineering application, featuring a friend of our class). See http://www.math.tamu.edu/seminars/amuse
18 Oct: The midterm test key is now posted below. (Yes, of course the scores will be interpreted on a curve.) Also here are the slides from the Frontiers talks of Mark Green, and some related slides of mine that should have been here all along but weren't.
7 Oct: Next AMUSE meeting (9 Oct): Post-Quantum Cryptography. See http://www.math.tamu.edu/seminars/amuse/
30 Sep: Next AMUSE meeting (2 Oct).
16 Sep: Next AMUSE meeting (19 Sep -- unusual day and place).
13 Sep: A variety of solutions to the second homework are now visible below. Also, I changed all the Wednesday due dates to Fridays, as promised.
2 Sep: First Aggie Actuaries meeting (5 Sep)
30 Aug: First AMUSE meeting (4 Sep)
28 Aug: First Math Club meeting (2 Sep)

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

Corrections to the textbook
Lecture notes (Updated through p. 12 on 30 Aug.)
Temporary home of the extra notes on the twin "paradox"
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., 1989.

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.)
A particularly impressive format for submitting your paper :-)

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. 4.) 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. 6. ("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. 13. 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. 18.) You will need the results as input into a later assignment. Note that Omega is not a coordinate, but a shorthand for the two angular coordinates collectively.
Now find the Christoffel symbols for the static, spherically symmetric metric (see Exercise 6.35 of Schutz or p. 77 of notes; note that Omega is not a coordinate, but a shorthand for the two angular coordinates collectively). Work in pairs. Due Oct. 28.
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. 2).)
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. 15.)
Now we need all the same stuff for the static, spherically symmetric metric (Work in pairs; due Nov. 22.)

Test solutions

Track your grades on eCampus.

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)
Slides on vector calculus and topology (from Honors M. 311 notes)
Frontiers lectures of Mark Green. (Most relevant are pages 15-28.)
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 2011 ._. Fall 2009 ._. Spring 2008 ._. Fall 2005

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

e-mail: fulling@math.tamu.edu