# 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:

**SAT 06/02/2018: FINAL GRADES AND HISTOGRAM ARE POSTED**

Dear 8.962 Students,

Your final numerical averages and final letter
grades are now posted on the Gradebook website. (I believe that
your letter grades have already been available through WebSIS.)
There is a histogram of class grades posted under the General
Info tab on the course website.

If you would like to get back any problem sets
that you have not already gotten back, please see Nikhil in Room
6-314. If you have any questions about the grading of any
particular problems, or about the determination of your final
grade, please don't hesitate to ask. It is possible that we made
mistakes, and if so, we would like to correct them.

All of the problem set solutions are now
posted.

Have a great summer!

— Alan Guth —

**THU 05/24/2018: POST-TERM LECTURE TODAY AT 1:30 PM: GRAVITATIONAL RADIATION**

Dear 8.962 Students,

Today at 1:30 pm I will be giving a post-term
--- totally, totally optional --- lecture about gravitational
radiation. It will be in our usual lecture room, 56-154.

If you want to come, I will see you then!

— Alan Guth —

**MON 05/21/2018: PROBLEM SET 12 SOLUTIONS POSTED**

Dear 8.962 Students,

I just thought I would let you know that
Problem Set 12 Solutions are now posted. The solutions to
Problem Set 13 should be finished in a few days.

— Alan Guth —

**SAT 05/19/2018: POST-TERM LECTURE; EVALUATIONS**

Dear 8.962 Students,

First, I want to remind you that course
evaluations are now open, but will close on Monday at 9:00 am.
Please give us your feedback, because it is very useful in
planning for future years.

Second, I want to remind you that I will be
giving a post-term lecture on gravitational radiation on Thursday
(5/24/18), at 1:30 pm. I still need to arrange a room, so I'll
get back to you about that. The lecture is of course totally
optional --- Problem Set 13 was the last, so there will be no
further problem sets on this material!

Best wishes, and good luck on your final
exams.

— Alan Guth —

**TUE 05/15/2018: CORRECTION TO PROBLEM 3(B) OF PROBLEM SET 13**

Dear 8.962 Students,

In Problem 3(b) of Problem Set 13, the metric
shown in Eq. (3.10) includes only the dr and dt parts of the
Schwarzschild metric. The full metric would also include the
usual r^2 d Omega^2 term. The r^2 d Omega^2 term is not relevant
to the problem, but is of course part of the full metric.

Sorry for any confusion.

— Alan Guth —

**TUE 05/15/2018: OFFICE HOURS TODAY**

Dear 8.962 Students,

Reminder:
As I announced yesterday, Nikhil Raghuram has
moved his office hour this week, since the problem set is due on
Wednesday. So today there will be two office hours:

3:00-4:00 pm: Reggie Caginalp, Room 8-320

4:00-5:00 pm: Nikhil Raghuram, Room 8-320

Best wishes,

— Alan Guth —

**SUN 05/13/2018: PROBLEM SET 11 SOLUTIONS, FINAL PROBLEM SET 13 POSTED**

Dear 8.962 Students,

Problem Set 11 Solutions were just posted, and
the final version of Problem Set 13 was posted early yesterday
morning.

The Problem Set 11 Solutions are quite long,
18 pages, but that is in part because I included 5 pages of extra
discussion about the Killing vectors of the gravitational wave
spacetime, after giving the actual solution.

The final version of Problem Set 13 includes
one additional problem, Problem 3, which is a guided calculation
of the Hawking temperature of a Schwarzschild black hole, and the
Gibbons-Hawking temperature of de Sitter space.

I hope you find these problems interesting.

— Alan Guth —

**FRI 05/11/2018: A PRELIMINARY VERSION OF PROBLEM SET 13 IS POSTED**

Dear 8.962 Students,

The subject line says it all.

— Alan Guth —

**SAT 05/05/2018: PROBLEM SET 12 IS POSTED**

Dear 8.962 Students,

Sorry for the delay, but Problem Set 12 is now
posted.

— Alan Guth —

**MON 04/30/2018: MY OFFICE HOUR TODAY CHANGED TO 5:00-6:00 PM**

Dear 8.962 Students,

Tonight there is an event at the Central
Square Theater that I will be going to, which starts at 6:30 pm.
So I will have my office hour today from 5:00 to 6:00 pm, instead
of the usual time of 5:30 to 6:30. It will be in the usual
place, my office, Room 6-322.

I hope to see some of you soon.

— Alan Guth —

**FRI 04/27/2018: PROBLEM SET 11 IS POSTED**

Dear 8.962 Students,

Problem Set 11 is now posted.

— Alan Guth —

**SAT 04/21/2018: FINAL VERSION OF PROBLEM SET 10 IS POSTED**

Dear 8.962 Students,

The final version of Problem Set 10 is now
posted, replacing the preliminary version posted Thursday. The
new version has an additional problem, Problem 2, which is a
calculation of the geodesic equations in Kruskal coordinates.
Problem 1 has not be changed in any way.

— Alan Guth —

**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 —