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8.962 :: General Relativity

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.