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Course announcements
05.18.09
Solution set #11 posted.
05.18.09
Grades: Grades are now done; you can pick up graded
psets at my office. You can determine your grade from your
cumulative score:
score > 98%: A+
91% < score < 97%: A
89% < score < 90%: A-
85% < score < 88%: B+
80% < score < 84%: B
76% < score < 79%: B-
73% < score < 75%: C+
65% < score < 72%: C
60% < score < 64%: C-
50% < score < 59%: D
score < 50%: F
Contact me if you have any questions.
05.17.09
Kerr black hole orbits: A great paper detailing
many of the key properties of Kerr black holes is
Rotating
Black Holes: Locally Nonrotating Frames, Energy Extraction and
Scalar Synchrotron Radiation, J. M. Bardeen, W. H. Press,
and S. A. Teukolsky, Astrophys. J. 178, 347 (1972).
05.17.09
No hair theorem: I mentioned in lecture that the "no
hair theorem" is less a simple theorem than it is a kind of
aphorism summarizing the accumulated wisdom of multiple
analyses. Here are some of the key results:
Event
Horizons in Static Vacuum Space-Times, W. Israel,
Phys. Rev. 164, 1776 (1967). This paper proved that a
spacetime which is static and contained an event horizon must
also be spherical; hence, it must be the Schwarzschild
solution.
Axisymmetric
Black Hole Has Only Two Degrees of Freedom, B. Carter,
Phys. Rev. Lett. 26, 331 (1971).
Uniqueness
of the Kerr Black Hole, D. C. Robinson, Phys. Rev. Lett.
34, 905 (1975). Taken together, these papers
generalize Werner Israel's result to the case of rotation,
establishing that the only stationary spacetimes which
contain even horizons are Kerr black holes (with spin
parameter a <= M).
05.13.09
Solution set #10 posted,
with accompanying mathematica
notebook.
05.06.09
Problem set #11 posted.
05.06.09
Solution set #9 posted.
05.04.09
Numerical troubles: Some people are finding the
numerical integration for the neutron star model to be a bit
tricky. Here are some notes that may help.
04.30.09
Problem Set #10 posted,
along with an example
mathematica notebook illustrating how to numerically solve
a system of ordinary differential equations.
04.29.09
Comment on Pset 9, Problem 4: In recitation today, I
had a few questions regarding how to go about solving for the
evolution of a deviation from spatial flatness. It turns out
that, as formulated, the problem is a little trickier than I
realized --- accounting for the influence of the term kappa in
the Friedman equations makes the solution less simple than I
had realized. The solution is to either reformulate the
problem, or to systematically account for the impact of a
small but nonzero kappa. I've written some notes explaining the issue.
04.29.09
End of semester, extensions, and scheduling: As the end
of the semester approaches, I anticipate quite a few extension
requests --- lots of thesis deadlines are approaching, final
projects, general overload, etc etc. We will do what we can
to accommodate any reasonable extension request. Be aware,
however, that the MIT registrar sets a rather hard deadline
(usually the Monday after the last day of classes) for me to
submit the final grades. Accordingly, I cannot give
much of an extension beyond the end of class.
04.23.09
Problem Set #9 posted.
04.18.09
Announcements re course schedule:
1. There are no MIT classes on Tuesday, April 21st.
2. Pset #9 will be assigned on Thursday, April 23rd.
3. Since there's no pset due this coming week, and since I
will be on travel for the first half of the week, there will
be no recitation on Wednesday, April 22nd. Please contact me
via email if there is anything in particular you would like to
discuss.
04.18.09
Solution set #8 posted.
04.15.09
Solution set #7 posted.
Again, apologies for the delay --- there were quite a few
extensions this week. People seem rather busy ...
04.12.09
Here are links to several papers that formed the background of
material that was presented in lecture on April 10. The
original presentation of the averaging procedure that was used
was developed in
Method
of the Self-Consistent Field in General Relativity and its
Application to the Gravitational Geon, by Dieter R. Brill
and James B. Hartle, Phys. Rev. 135, B271 (1964).
The application of these ideas to gravitational radiation was
developed by Richard Isaacson in his Ph.D. thesis work:
Gravitational
Radiation in the Limit of High Frequency. I. The Linear
Approximation and Geometrical Optics, Richard A. Isaacson,
Phys. Rev. 166, 1263 (1968),
Gravitational
Radiation in the Limit of High Frequency. II. Nonlinear Terms
and the Effective Stress Tensor, Richard A. Isaacson,
Phys. Rev. 166, 1272 (1968).
The first paper lays out, in slightly different language than
we used, the derivation of the wave equation linearizing about
a curved background. The second one lays out the calculation
of the effective stress energy tensor. It also provides in an
appendix an explicit example of the averaging in action.
04.09.09
Problem Set #8 posted.
04.07.09
Slight adjustment to syllabus: I am going to be on an
airplane returning from the Denver APS Meeting during the time
of our May 5th lecture. I am currently working out a guest
lecture, and will hopefully find someone to give an
interesting lecture on something topical and relevant in that
slot. I've adjusted the course schedule slightly to account
for the "missing" day in our schedule. (Note, there's no
content missing: I had erroneously scheduled 3 lectures for
topics in cosmology, when I should have scheduled 2. I was
planning to use the extra time for a "special topics" lecture
anyway.)
Update: Prof. Edward Farhi has kindly agreed to recap
his ever-popular "Time travel in general relativity" lecture
for us that day.
04.07.09
Solution set #6 posted.
Sorry for the delay on this; several people had extensions, plus
I've been swamped trying to meet an upcoming deadline.
04.01.09
Problem Set #7 posted.
04.01.09
Comment on Pset 6, Problem 1: In recitation today, I
got a question from Nickolas Vanmeter regarding whether one
needed to be careful about the placement of indices in using
the Bianchi identity to show that Einstein splits into
"constraint" and "evolution" equations. After a bit of back
and forth, I've determined that he is absolutely right; the
problem as written will only work if we look at the
upstairs-upstairs components of the Einstein tensor. Many
thanks to Nickolas for the question! I've written some notes explaining the
issue.
03.24.09
Solution set #5 posted,
along with output from Mathematica
to organize part of the solution.
03.19.09
Problem Set #6 posted.
03.15.09
Solution set #4 posted,
along with output from Mathematica
to organize part of the solution.
03.12.09
Problem Set #5 posted.
03.12.09
Late extension requests: Please try to get extension
requests to me with reasonable prior notice. I'm generally
fairly lenient with requests, but you've got to meet me
halfway. I realize that people are busy; but, such busy-ness
is a constant of the motion at MIT. You should have a good
sense with more than 24 hours notice whether you'll be able to
get the pset done on time or not. For future psets, I would
like 24 hours notice on requests or a penalty will be
assessed; 48 hours would be preferred, but 24 will do in a
pinch. Please plan ahead!!
03.12.09
More tools: The discussion of Mathematica notebooks for
computing Christoffels and curvature tensors has sparked some
suggestions for other tools people might like. If you are a
Maple user, you might want to give GRTensor a whirl. I'm
not a Maple user, so I can't vouch for it, but I've definitely
heard people swear by it. Mathematica also has a set of
supported packages called MathTensor that I've likewise heard
good things about, though I've never used myself.
03.11.09
More efficient Mathematica notebooks: Connor McEntee pointed out to me
that people are sometimes using GRTool.nb for their research
projects. It was never intended for this! --- GRTool is
something that former 8.962 TA Will Farr hacked together fairly
quickly a few years ago to illustrate the calculation of
curvature tensors and to provide a pedagogically accessible
tool for doing so. As a consequence, it's easy to read and
understand, but slow. Connor has very kindly allowed me to
post the Mathematica notebooks he created for doing such
calculations, which speed things up by several orders of
magnitude. (No exaggeration: Computing curvature tensors in
certain higher dimensional spacetimes changes from hours of
waiting before you get frustrated and kill the process to a
few seconds.) They are posted under the handouts link. Enjoy!
03.10.09
Holonomy in action: Look here for
an illustration of vectors rotating as they parallel transport
around a closed loop in a curved manifold.
03.08.09
Solution set #3 posted,
along with output from
Mathematica to organize solution problem 5.
03.04.09
Question from recitation: I was asked in recitation
today how, when one represents a two-index tensor as a matrix,
to choose row and column. I so rarely do this that my initial
answer was little better than a guess, but fortunately I
guessed correctly: First index labels row, second index labels
column. Note, it's a good idea to try to step away from
explicit matrix multiplication notation as much as possible
since it's a route to slow madness when we start building many
index objects. (E.g., we will use a 4-index object to
describe curvature, and will at one point want to take its
derivative --- making a 5-index object!) Take a look at the
mathematica notebook GRTool.nb to see how one can manipulate
these things without doing explicit matrix multiplication.
03.04.09
Solution to Problem Set #3 will be a touch late. My
apologies, but I am desperately trying to finish writing a
talk and then travelling to give said talk in Toronto on
Friday. I will probably polish the solution and post it on
Saturday.
03.04.09
Problem Set #4 posted.
02.26.09
Problem Set #3 posted.
02.25.09
New handouts: On that page, you will find 3 new useful
handouts. GRTool.nb is a mathematica notebook that we will
occasionally use to aid certain tedious tensor manipulations;
it can save you a lot of pain! I've also linked a copy
of a set of notes by Ed Bertschinger on some of the
mathematics which underlies working in an orthonormal basis.
Since most undergraduate physics is done in orthonormal bases,
but most of our manipulations will be in a coordinate basis,
you may find the discussion here interesting. Finally, I've
added some brief notes explaining why (as I stated without
careful discussion in lecture yesterday) partial derivatives
are not tensorial in general.
02.25.09
Handouts link added to webpage. Thanks to a minor
technical glitch, a link to a page which gathers together
various handouts for this class was not properly included
on this page. Now added.
02.21.09
Please check your grades for pset #1 in Stellar via the
gradelink to the left. Everything should be working
correctly, but it would be useful to find any problems or
errors early so that we can make corrections.
02.21.09
Solution set #2 posted.
02.19.09
Solution set #1 posted.
Note, in general I aim to post solutions very shortly after
everyone hands in their problem sets. I was a bit late on
this first round because a few people had extensions, and
because of my travel. (I've temporarily got a very strong
wireless signal in my hotel, so I'm catching up now.)
02.12.09
REMINDER: There is no class on Tuesday, Feb 17 since MIT
follows the Monday schedule that day.
02.12.09
Two notes on the problem set: In discussion following
today's lecture, I realized that the limit I ask you to take
in problem 2 of pset 2 is not very well defined. I have added
a bit of text to the pset explaining this limit a bit better;
if you have already downloaded the problem set, please get the
new version (page 1 is slightly different).
In addition, the identity I ask you to use in problem 4 is one
that I did not have time to introduce in lecture. The main
concepts are sufficiently simple that I have decided to
provide supplementary
notes providing some background to this problem. Please
read these notes
carefully since, aside from their utility for this problem,
they are important material for the class.
02.11.09
Problem Set #2 posted.
02.11.09
More on 1-forms: As I've emphasized, we are not going
to need the geometrical viewpoint of how to visualize a 1-form
too much in this class. Nonetheless, quite a few people are
interested in understanding this a bit better. My lectures
were based on Sec. 3.3 of Schutz's textbook; at a similar
level, you may also benefit from Chaps. 2.5 and 2.6 of MTW,
and Chap. 4 of MTW.
02.11.09
Addendum
clarifying Problem 2, part (h) posted.
02.04.09
Problem Set #1 posted.
02.04.09
Stellar access: Grades for this course are going to be
managed using MIT's Stellar database system. If you are an MIT
student, your certificates should allow you to access this system
with no problems. If you are not an MIT student, I am told that
you will be prompted for some registration information the first
time that you log in. Please just click the "Grades" link on the
left-hand side of the page.
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