Mathematics 460, Section 500, Fall, 2019
Tensors and General Relativity
Last updated Mon 16 Dec 2019
Announcements (reverse chronological order)
- 16 Dec: Final grades are posted. Some explanations:
- eCampus does not display meaningful statistics on letter grades.
I'll just say that the median grade was a high B.
- The scaled homework grade is 0.52632 times the raw score, to bring it
down to the scale stated in the handout and assignments.
- In addition, there were a few points of class participation (mostly
from the in-class exercise on metrics of the (...) du dv type)
that I added as bonus points.
I decided not to count absences; you are grownups.
- 9 Dec: Final exam answers are posted below. Sorry for the
bad wording of the hint on the first question (inherited from an old
answer key): "2 x 2 matrix" should have been "square matrix". (The
matrix is, of course, 6 x 6.)
- 5 Dec: Your final graded homework is ready to be picked up at
my office (BLOC 620H). On Friday (12/6) I'll be in the vicinity of the
office (probably in BLOC 620F) in the morning from 9:30 to 11:00, and in
the office in the afternoon approximately from 2:30 to 5:00. As usual,
don't hesitate to knock or interrupt.
- 2 Dec: Tomorrow (12/3) I'll
be in my office from 1 to 4. During the first half of that time you'll
have competition from my research students.
- 2 Dec:
Next
Math Club meeting Tuesday Dec. 3 (Ramanujan movie)
- 8 Nov:
Next
Math Club meeting Tuesday Nov. 12
- 20 Oct: The link to the vanilla TeX macros below has been repaired.
- 16 Oct: Due date for the EM paper has been extended to
Monday, Oct. 28. (Christoffel symbols for static, spherical metric
are still due on Oct. 30.)
- 11 Oct:
Next
Math Club meeting Tuesday Oct. 15
- 10 Oct: Due date for the first round of Christoffel symbols has been
put off to Monday, 21 Oct.
On Monday 14 Oct we will make sure that everybody has a partner for
these exercises.
- 18 Sep:
The grader has provided a
solution for Exercise 2.19.
- 17 Sep (corrected 18 Sep): IMPORTANT ._. The midterm test will be on
Monday, October 7, one class day earlier than previously announced.
(Oct. 9 is a major religious holiday for some people. Sorry.)
._._.
Also, recall that a rough draft of your electromagnetism paper (see
instructions below) is due on Oct. 2.
Don't worry if you haven't reached the end of the assignment;
turn in a copy of whatever you have finished for constructive comments.
- 12 Aug: I will be out of town the week of September 2-6.
Course handout (updated with new test day)
._._.
Please see
my home page for up-to-date office hours.
Corrections to the textbook
Lecture notes
(subject to further revisions)
Projector version of the notes
(Printing out is discouraged.)
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 (subject to change)
(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. 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.
11. 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 in depth,
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. 21.) 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 (see (10.2)).
- 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. 30.
- 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
and class participation, after rescaling, will
be something more than 100. I'll accept papers
any time before the end of classes (Wednesday, Dec. 4).)
- 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. 13.)
- Now we need all the same stuff for the static, spherically symmetric metric
(Work in pairs; due Nov. 25.)
Test solutions
Track your grades on eCampus.
Supplementary material
Old course home pages:
Fall 2017 ._.
Fall
2015 ._.
Fall 2013 ._.
Fall 2011 ._.
Fall 2009 ._.
Spring 2008 ._.
Fall 2005
Go to home pages:
Fulling ._._.
Calclab ._._.
Math Dept ._._.
University
e-mail: fulling@math.tamu.edu