8.962 - General Relativity SP'18

Lecturer: Prof. Alan Guth
Recitation Instructor: Nikhil Raghuram
TA: Reginald Caginalp
Lecture: MW11-12.30 (56-154)

Errata To Sean Carroll'S Spacetime and Geometry

Announcements:

FRI 04/20/2018: DROP DATE IS COMING
Dear 8.962 Students,
As you probably know, this coming Thursday (April 26) is Drop Date --- the last day to cancel subjects from Registration, and the last day to change a subject from Credit to Listener.
To give you a better idea of how grading is done in this class, I have posted the histogram of last year's grades, under the General Info tab on the website.
I suspect that most of you are not concerned about Drop Date, but a few of you are. I have to apologize that the grading of problem sets has gone so slowly. We started the course with too few graders, we then found more graders to hire, but they are still working on getting caught up. By Drop Date, I am told that Problem Sets 3 and 4 will be graded. If you are concerned about a Drop Date decision, please let us (nikhilr@mit.edu, caginalp@mit.edu, guth@ctp.mit.edu) know, and we expect that we can arrange to have your Problem Set 5 graded before Drop Date. If you would like us to do this, please send an email by late this afternoon.
Best wishes, and let me know if you have any further questions.
— Alan Guth —

THU 04/19/2018: PRELIMINARY PROBLEM SET 10 POSTED; INCLUDES A BIT OF QUANTUM
Dear 8.962 Students,
I have posted a preliminary version of Problem Set 10, which will be due next Thursday, April 26, at 5:00 pm. So far it includes one problem, in which you are asked to numerically solve the Tolman-Oppenheimer-Volkoff (TOV) equation for a degenerate non-interacting gas of neutrons. In part (a) of the problem, you are asked to find expressions for the energy density and pressure of this gas. The TOV equation is discussed in Chapter 5 of Carroll, and we will discuss it in class next Monday.
To help you with part (a), Nikhil Raghuram will review these methods in recitation tomorrow (Friday) and Monday. In addition, I will be posting, with the problem set, 5 scanned pages from David J. Griffiths, "Introduction to Quantum Mechanics," 2nd edition, about this subject. Griffith's treatment, however, is nonrelativistic, while this problem requires a relativistic treatment. The only difference, however, is that you need to use the fact that the energy of a particle is E=Sqrt[p^2+m^2], where p is the magnitude of the 3-momentum and m is the rest mass, while Griffiths uses E=p^2/2m.
The final problem set will probably include one other problem.
— Alan Guth —

WED 04/18/2018: POSTPONEMENT OF PROBLEM SET 9 DUE DATE
Dear 8.962 Students,
Since Monday and Tuesday were holidays, I have decided to delay the due date for Problem Set 9 until Friday at 5:00 pm. The main goal is to give you more access to office hours, if you would like help on the problem set. Nikhil will be having his usual office hour today from 5:00 to 6:00 pm in Room 8-320, and Reggie will be moving his usual Tuesday office hour (3:00-4:00 pm, Roomn 8-320) to tomorrow, Thursday, at 3:00-4:00 pm. We still need to confirm the room, but it will probably be Room 8-320. If it is someplace else, there will be a note on the door of 8-320 to tell you where to find Reggie.
I hope this relieves some pressure.
— Alan Guth —

MON 04/16/2018: PROBLEM SET 8 SOLUTIONS ARE POSTED
Dear 8.962 Students,
This is just to let you know that Problem Set 8 Solutions are now posted. I am hoping to be fully caught up with solutions before we resume classes on Wed.
— Alan Guth —

FRI 04/13/2018: PROBLEM SET 9 IS POSTED
Dear 8.962 Students,
Problem Set 9 is now posted. It is a fairly short, one problem 35-point problem set, that will be due next Thursday, April 19.
— Alan Guth —

SUN 04/08/2018: REVISED VERSION OF PROBLEM SET 8
Dear 8.962 Students,
I just posted a new version of Problem Set 8, with a few revisions that are itemized in a footnote on the first page. The only change that affects the actual questions is in part (e), where I am now asking you to consider both timelike and lightlike cases, to be consistent with the rest of the problem. The other changes are minor clarifications.
Best wishes,
— Alan Guth —

THU 04/05/2018: MINOR TYPO IN PROBLEM SET 7, PROBLEM 3
Dear 8.962 Students,
In Problem Set 7, Problem 3(a), where it says "Using the same definitions as in Problem 1," it really meant Problem 2. Sorry for any confusion caused by this error.
— Alan Guth —

WED 04/04/2018: 8.962, CURVATURE TENSORS FOR PROBLEM SET 7
Hi Everyone,
I've attached a document with the Christoffel symbols and curvature tensors for Problem Set 7. Of course, if you find any typos or issues with what I've sent, let me know and I'll be more than happy to fix things.
Thanks,
Nikhil

FRI 03/23/2018: PROBLEM SET 7 IS POSTED
Dear 8.962 Students,
Problem Set 7 is now posted.
— Alan Guth —

WED 03/21/2018: ON THE USE OF COMPUTER ALGEBRA PROGRAMS
Dear 8.962 Students,
Several of you have asked me what the policy is concerning the use of computer algebra programs, such as Mathematica, in doing the problem sets.
I am perfectly happy with your using computer algebra programs to help with algebra and calculus, and I would even encourage you to use such programs. You should still hand in solutions that are human-readable. What you hand in could be in the form of annotated computer code, or a handwritten solution with occasional sentences such as "Mathematica expanded and simplified this expression, resulting in ..."
There is a limitation, however, that I would like to impose on the use of computer algebra programs. There are programs out there that would essentially do some of these problems for you. That is, there are programs for which you can enter the metric, and the program will then compute, on its own, the Christoffel symbols and the Riemann curvature tensor. That is going too far, since it means that you will not be implementing the definitions of these quantities on your own. So the use of a program of this sort is not acceptable on the homework (unless you wrote it yourself). If my announcing this limitation makes it hard for you to hand in the problem set tomorrow, please ask me for an extension.
I hope that it's more or less clear what sort of boundaries I have in mind, and that you will find these limitations reasonable. If in doubt, just ask.
— Alan Guth —

WED 03/20/2018: VISUALIZING SEVEN-MANIFOLDS
Dear 8.962 Students,
In class yesterday, we talked a little about seven-dimensional manifolds, and I mentioned a fact that I learned from my son Larry: John Milnor showed in the 1950s that there are 28 different manifolds which are homeomorphic to S^7. One of these is the standard 7-sphere, while the other 27 are homeomorphic, but not diffeomorphic, to S^7. In parsing this statement, it is worth remembering that homeomorphism is a statement about continuous maps, while diffeomorphism requires C^(infinity) maps. I hope I got across the fact that this was intended as interesting cultural background, but is not really part of what you are expected to learn in 8.962.
Miguel Young de la Sota mentioned that he knew a website that illustrated these constructions, and I promised to forward the link to the class. Here it is:
https://nilesjohnson.net/seven-manifolds.html
The animation was constructed by Niles Johnson of the Department of Mathematics at Ohio State University. The web page says that the animation is best viewed in a larger format available on YouTube:
https://youtu.be/II-maE5HEj0
The animation is certainly lovely, but if you want somebody to explain it to you, you should be sure to ask anybody but me.
— Alan Guth —

FRI 03/16/2018: PROBLEM SET 6 IS POSTED
Dear 8.962 Students,
Problem Set 6 is now posted. For those attending today's recitation section, you might want to look at them first.
— Alan Guth —

FRI 03/09/2018: PROBLEM SET 5 IS POSTED
Dear 8.962 Students,
Problem Set 5 is now posted, and I have also posted (on the Lecture Notes tab) some notes about the "Geometry for General Relativity" that were written in 2016 by Sam B. Johnson, who was the recitation instructor that year. These notes are a good summary of the definitions and key concepts concerning manifolds and structures defined on them.
— Alan Guth —

SAT 03/03/2018: FINAL VERSION OF PROBLEM SET 4 IS POSTED
Dear 8.962 Students,
The final version of Problem Set 4 is now posted, replacing the preliminary version posted yesterday. The new version has an additional problem, Problem 5, which is a guided derivation of the geodesic equation from the equivalence principle.
— Alan Guth —

FRI 03/02/2018: PRELIMINARY VERSION OF PROBLEM SET 4 IS POSTED
Dear 8.962 Students,
I have posted a preliminary version of Problem Set 4, consisting of 4 problems. The final version, which I expect to post by tomorrow morning, will include one more problem, which will concern the derivation of the geodesic equation from the equivalence principle.
Best wishes,
— Alan Guth —

MON 02/26/2018: TIME CHANGE FOR REGGIE CAGINALP'S OFFICE HOUR TOMORROW (TUES)
Dear 8.962 Students,
Reggie Caginalp's office hour for tomorrow (Tues 2/27/18) will be held from 4:45 to 5:45 pm, in the usual room, 8-320.
Next week Reggie's office hour will return to its regular time of 3:00-4:00pm on Tuesday.
Best wishes,
— Alan Guth —

SUN 02/25/2018: PROBLEM SET 2 SOLUTIONS ARE POSTED
Dear 8.962 Students,
This is just to let you know that Problem Set 2 Solutions are posted.
— Alan Guth —

SAT 02/17/2018: PROBLEM SET 1 SOLUTIONS ARE POSTED
Dear 8.962 Students,
The solutions to Problem Set 1 are now posted, on the Problem Sets tab of the course web page.
For those of you with extensions, you are of course on your honor not to look at the solutions, or to discuss the problem set with anyone who has.
It may be helpful for you to know that the direct-access URL for problem sets in this course will be
http://web.mit.edu/8.962/www/probsets/psNN-grs18.pdf ,
where NN is the two-digit problem set number (e.g., 01). The solutions will be accessible at
http://web.mit.edu/8.962/www/probsets/psolNN-grs18.pdf .
Best wishes,
— Alan Guth —

FRI 02/16/2018: ABOUT PROBLEM SET 2
Dear 8.962 Students,
Problem Set 2 has been posted since about 11:00 am, on the course website, http://web.mit.edu/8.962/www, on the Problem Sets tab. In the future I will try to get them posted by some time on Thurs evening, so you will be able to look them over and ask relevant questions in recitation on Friday.
The problem set is entirely about geodesics. In lecture on Wed I almost got far enough for you to do the problem set, but ... not quite. I will certainly finish discussing this material at the next lecture, but that is not until Tues. For those of you who want to work on the problem set before then, I would recommend looking at last year's lecture notes, posted on the website. If you read the notes from Lecture 3 of last year, you will be all set for the problem set. It is also in the textbook, pp. 106-108, although Carroll does not write what I am calling the "primitive" form of the geodesic equation, Eq. (3.27) of the 2017 Lecture notes.
Best wishes,
— Alan Guth —

THU 02/15/2018: COMMENT ON PROBLEM 4, PROBLEM SET 1
Dear 8.962 Students,
Several of you have asked me about Problem 4, "Ordering of Events", on Problem Set 1. In particular, what does it mean to support your answer by drawing a spacetime diagram? The wording is Carroll's, and if I give the problem again I will modify it a bit.
It is possible to achieve all three orderings mentioned in the problem --- ABC, CBA, and ACB --- while using only 1 spatial dimension. That is probably the solution that Carroll had in mind, and for that solution a spacetime diagram of the x-t plane is very useful.
There are other solutions that use more than one spatial dimension. Since the problem does not specify anything about using only one spatial dimension, these solutions are certainly also correct. Spacetime diagrams with more than 2 spacetime dimensions are harder to draw on a 2-dimensional piece of paper. But since Carroll asked for a spacetime diagram, you should do your best. Maybe a perspective diagram will work, or maybe you can draw projections into different planes. If you can't find a way of diagramming your answer, maybe you should look for a simpler answer.
Incidentally, with 4 spacetime dimensions, it is possible to identify 4 events which can appear to be in any of the 4! possible orderings, depending on the frame of reference. It's not part of the problem, but if you have some spare time and enjoy puzzling, you might want to think about it. The solution will be included with the Problem Set 1 Solutions.
— Alan Guth —

THU 02/15/2018: A FEW ERRORS IN CARROLL'S EQUATIONS
Dear 8.962 Students,
Jason Gross pointed out that Eq. (1.106) of Carroll's book, on p. 32, has the wrong sign. You don't necessarily need this equation for Problem 6 of this week's problem set, but you might want to use it, so if you do you should correct the sign.
It turns out that Sean Carroll has a whole long list of errata to the book, including Eq. (1.106), posted at
https://www.preposterousuniverse.com/spacetimeandgeometry/
Best wishes,
— Alan Guth —