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Course announcements
5.20.08
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+
90% < score < 98%: A
86% < score < 90%: A-
83% < score < 86%: B+
79% < score < 83%: B
76% < score < 79%: B-
73% < score < 76%: C+
63% < score < 68%: C
60% < score < 63%: C-
50% < score < 60%: D
score < 50%: F
Contact me if you have any questions.
5.19.08
Last psets: Please hand in your psets by 6 PM
today. Put them in the mailbox on my door; do not slip
them under my door or the graders may not be able to get
them!!! Grades will be available tomorrow.
5.15.08
Course evaluations: For those of you who didn't have
time to fill out the eval at the end of class, please fill it
out and hand it in to 4-315. If you wish to pick one up, I'll
leave a batch in the mailbox on my office door.
5.13.08
Equations describing the ergosphere: Mike Matejek asked
whether I had a sign error (or switched min/max) in the
equation defining the orbital frequencies that helped map out
the ergosphere. It turns out my equation was correct, but
there is a subtlety there that required some explaining. The
formula in question is
Ωmax/min = [-gtφ +/-
sqrt(gtφ2 -
gttgφφ)]/gφφ.
I claimed that when gtt = 0, Ωmin
= 0. A key reason that this works is that gtφ
< 0: From the May 8 lecture, we have
gtφ = -2 G M a r
sin2θ/(r2 +
a2cos2θ).
This is a negative definite quantity (note that a >= 0; we
could allow a < 0, but this would just switch the roles of
Ωmin and Ωmax --- same
result, but with a few different labels). Given this, when
gtt = 0, we have
Ωmin = [-gtφ -
abs(gtφ)]/gφφ.
This is a positive quantity minus an equal and positive
quantity --- zero. So the result stated in lecture was
correct, but my statement was far too glib. Thanks to Mike
for forcing me to think this through and clarify!
5.13.08
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).
5.13.08
Typo on pset 11: There is a missing factor of π in
the result for periastron precession; it should read
δφ = 6πGM/[a(1 - e2)]
A correction has been posted; sorry about that!
5.08.08
End of semester, check your grades!! Now is a perfect
time to check whether the grades database is up to date. Let
us know if anything is missing. People who had extensions and
got the psets in late should check particularly carefully ---
those psets may have been rushed, which is when we tend to
forget to update the database.
5.08.08
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).
5.08.08
Problem set #11 posted.
Note, solutions to this pset will not be posted, but will
be available upon request.
5.06.08
Kruskal-Szekeres coordinates: A much cleaner version
of the diagram than the one I attempted to draw in lecture is
in Carroll, Fig 5.12. That picture and associated discussion
makes a nice addendum to today's discussion of black holes and
white holes.
5.06.08
Minor error on pset 10: In problem 2(c), the definition
of redshift z is off by an overall minus sign.
5.06.08
Solution #9 posted.
5.01.08
Buchdahl's theorem: I mentioned in class that the
result R* > (9/4) GM holds more
generally than just for the rho = constant equation of
state. I thought this was proven in Weinberg's textbook,
but this isn't correct; the source is a 1959 paper:
General
Relativistic Fluid Spheres, by H. A. Buchdahl, Phys.
Rev. 116, 1027 (1959).
5.01.08
Problem set
10 posted, along with a Mathematica notebook illustrating
their tool for solving systems of ordinary differential
equations.
4.30.08
Numerical typo on Pset 9, Problem 4: In part (a) of
problem 4, I give an incorrect conversion of MeV into gm
cm2sec-2. The conversion given is for
eV --- there's a factor 106 error there.
The correct conversion is
1 MeV = 1.6 x 10-6gm cm2sec-2
Sorry about that! (Note that, given the exponential
expansion in the inflationary epoch, an error of few orders of
magnitude is quickly dealt with by just changing the number of
e-foldings slightly.)
4.29.08
Warning: I'm pretty easy about giving extensions as
long as I get advanced notice. Unfortunately, I can't be very
generous as we head into the last week of classes. Grades are
due at the registrar shortly after the class ends (flip side
of having no final exam). I don't know quite what the
deadline will be, but it will probably be around May
20th. I'll put up more detailed info once I know for sure.
4.24.08
Solution #8 posted.
4.22.08
Problem set
#9 posted. This requires some stuff that we will cover in
lecture on April 24, but if you are reading you may be able
to get a jump on it.
4.22.08
Solution #7 posted.
4.16.08
Correction to pset 8, problem 2: In this problem, the
members of the binary have a separation R; their
orbit does not have a "radius" R. When one maps this
to the reduced problem, this means that the radius of the orbit
in the reduced description has a radius R. I'm
very sorry for this error; the problem can become a mess if
you try to do it as worded. I am happy to give an extension
to anyone who needs time to clean up their solution based on
this correction.
4.15.08
United Airlines hates me. They hate you too, which is
why I was forced to cancel class on April 15th. I'm very
sorry about that; I actually got up at 4 am in order to catch
a 6 am flight, but ended up sitting on the runway until just
after 7 am while we waited for a mechanic to come fix the
co-pilot's seat belt. As a consequence, I missed my
connection in Chicago and didn't arrive in Boston until after
class was scheduled to start.
This shouldn't have any impact on the main topics we cover in
the course; we actually have an "extra" lecture, which I was
planning to use to cover cosmological perturbation theory.
(Last year, Eddie Farhi gave a lecture on time travel.)
Unfortunately, the cosmological perturbations lecture won't
fit anymore. I may post my notes on the topic to the course
website for anyone who is interested.
I am adjusting the dates for pset 9 to accomodate this
unplanned schedule change: It will be assigned April 24th, and
due May 1st.
4.14.08
A giant
has fallen.
4.11.08
I'm presently trapped at the airport waiting to
find out if my plane (which needs minor repairs) is going to
get me to the APS meeting today, or whether I'm going to need
to reschedule for tomorrow. Clearly this is karmic
retribution for teasing everyone who is flying American to the
conference. In the meantime, here is a clear illustration of
the meaning of the
principle of equivalence. (It's worth noting that the
little popup that you get when you put your mouse over the
cartoon makes no sense: Changing over to an inverse linear or
constant law rather than to an inverse cube law is far more
logical if one has a highly extended mass distribution. It's
also worth noting that I'm very bored.)
4.10.08
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.
4.10.08
Pset 7 will come back late. Thanks to the APS meeting,
the schedule for the upcoming week is going to be a bit messy.
Will will not be able to completely grade pset 7 on time this
week; our plan is to have everything back by Tuesday, April
16th. Note, since we can't hold to our usual schedule, we are
going to be a little more lenient with extension requests,
especially since several students are also travelling to this
conference.
4.10.08
Problem set #8 posted.
4.08.08
I've created a new index link on the left-hand
side of the page that links us to all the various supplemental
material and readings that have been posted during the
semester.
4.08.08
Solution #6
posted. This solution refers to some notes by Ed Bertschinger on the
Einstein-Hilbert action. Given our various discussions on
this issue, people may be interested in these notes on more general grounds.
4.08.08
Thanks to Adrian Liu, a minor but potentially
confusing sign error has been found on Problem set 7, number 2. Here's
a synopsis:
The definition h0i = -betai given
on the pset is correct. This definition implies
h0i = +betai. Since the spatial
metric is the identity, betai = betai.
Putting all these facts together, we find that the sign of the
beta term in the line element ds2 is wrong:
it should be + rather than -.
If you just worked with the definition provided, everything
would have worked out fine. However, if you tried to
reconcile the definition with the line element provided, you
would have become mightily confused. Sorry for the error!
4.03.08
Article that
describes the gauge invariant decomposition of gravity,
linearized around flat spacetime. Section 2.2 is most
relevant.
4.03.08
Solution #5 posted; sorry
for the delay, no good excuse except absentmindedness.
4.03.08
Problem set #7 posted.
3.31.08
Geometrized luminosity: A lot of people
have asked why I don't give units to convert luminosity into
on pset 6, problem 5(e). Suffice it to say that this is a
feature, not a bug: Luminosity is energy per unit time. Think
what the correct units would be if both energy and time are
consistently converted to geometrized units.
3.20.08
An interesting link
on the history of Noether's theorem and its relation to
general relativity --- further useful food for thought on
issues of Lagrangians and the field equations. Thanks to Ari
Le for finding this!
3.20.08
Problem set #6 posted.
3.20.08
Action principle issues: Today's lecture was
sufficiently confused that I wanted to flag clarifying points.
(Sorry about the confusion --- I was preparing material for
the grad open house until the last minute, and didn't have a
chance today to review my notes carefully before lecture.)
1. Carroll Section 4.3 fills in a lot of the calculational
details that I glossed over in lecture. For questions related
to details in the variations that I quoted without
justification, see that section.
2. The very end of Carroll Section 4.3 addresses the fact that
the association of the stress energy tensor with the metric
variation is *not* what people normally encounter in a field
theory class. Instead, one normally applies Noether's theorem
with respect to the 4 spacetime directions and constructs a
symmetric object from the various variations. So what gives?
--- why our more complicated formulation? A key issue is that
the Noether's theorem method doesn't work in curved
spacetime. This technique is only good if there is a
symmetry associated with spacetime translations; in curved
spacetime, that won't always be the case. The methods are
only equivalent in the flat spacetime limit; the method we
outlined in class is actually more general.
3. I've been trying for a while to find an accessible
presentation of the Palatini variation. I'm still searching,
but very motivated students may be interested in the
discussion in Appendix E of Wald. The notation is quite a bit
different from what we use in our class, but the essential
content is there.
3.20.08
Lateness policy: A lot of people are making last
minute requests for extensions on homeworks. I don't mind the
requests; I do mind their last minute nature. Many of the
requests amount to "I'm extremely busy and didn't appreciate
how much time the pset would take." I understand that you are
busy, but you should assume that the pset will be time
consuming and plan your time accordingly. If you didn't
realize until 4 am on the morning the pset is due that you
needed more time, something broke down!
I believe you should have a good idea whether you can finish a
pset on time or not with two days notice. Please let me know
by Tuesday whether you will need an extension;
extension requests after this will be assessed a 20% penalty.
(Emergencies and other surprises will of course be handled
much more gently.)
3.18.08
Typo and omitted text on pset 5: On problem 1, there's
an omitted factor of r in the conversion to Cartesian
coordinates (hopefully this was obvious!!). On problem 4(c),
I neglected to mention explicitly that the solid angle in
question must be on the sphere --- we can't apply the result
of part (b) to any solid angle, despite what the
original wording may have implied! Corrected pset has been
posted.
3.18.08
Solution #4
posted, plus Mathematica
notebook used for part of the solution (PDF version). Posting again was
delayed a few days to wait for students who had extensions.
3.16.08
Office hours: After a query this morning, I realized I
never posted my detailed office hour policy for this term. In
the past, I've found that office hours don't work well for
this class; people can rarely make the specified hours, so we
just get together whenever works anyway. So, the "official"
policy is office hours by appointment. Just send email
and we'll find a time to get together.
3.16.08
Something cool: To get some insight into parallel
transport and triangles on a spherical surface, click over to
this java
applet. Thanks to Adrian Liu for finding this!
3.16.08
Bug in GRTool.nb!!
Adrian Liu found that the nonlinear connection terms in GRTool.nb had the wrong signs.
After verifying this, I have reposted a corrected version of
the notebook.
3.13.08
Problem set #5 posted.
3.13.08
CHEATING: We have accumulated some
disturbing evidence that people are unfairly taking advantage
of resources that may be available. Many of the problems that
are assigned (indeed, on some psets, ALL of the problems) have
appeared in previous versions of this class. This reflects
the fact that making challenging but doable GR problems is
difficult, and that we are short on time to develop and debug
new problems (especially so now that the class is roughly
twice its historical size).
It is thus disturbing and frankly disappointing to find a
handful of people who have handed in solutions that clearly
took advantage of familiarity with a solution set from a
previous year. The point of a graduate physics course is to
study something that interests you; if you are such a
grade grubber that you feel you have to get a high score at
the expense of your personal integrity, I would rather you
simply drop the class.
We are going to give people the benefit of the doubt for pset
3, but with an asterisk. We have noted the names of all the
people who appear to have used a solution set. We are
currently recording the grade that assumes you did not use the
solutions, but we are keeping a "shadow grade" in which you
are given zero points for every problem for which we
suspect you of cheating. If we find future evidence of
cheating from you, you will be given zero points on the
problems in question, and the shadow grade for pset 3 will be
restored.
You have been warned. If these conditions are not ones that
you can live with, please drop the class now. And for those
of you who are doing things fairly and correctly: thanks! You
folks remind me why I like doing this.
3.13.08
Curvature coupling and motion: People
seemed quite intrigued by the (schematic) equations I wrote
down showing how a body that is extended or has structure
beyond a simple point-like monopole can couple to the
curvature tensor, changing the worldline that it follows. A
very general discussion of such issues is an excellent and
comprehensive paper by Kip Thorne and Jim Hartle, available in
Phys Rev D:
Kip S. Thorne and James B. Hartle, Laws of
motion and precession for black holes and other bodies,
Phys. Rev. D 31, 1815 (1985).
For the specific case of a pointlike body endowed with a spin
angular momentum (the so-called "pole-dipole" approximation),
the equations of motion were originally derived by Achilles
Papapetrou. I'm having some trouble tracking down exactly the
correct reference; there are several candidates, but not all
are available electronically, and not all are written in
languages I can read (Papapetrou published in English, French,
and German!). Most people agree, however, that William Dixon
did an excellent job re-deriving and explaining what is now
called the Papapetrou equation:
W. G. Dixon, Dynamics
of extended bodies in general relativity I. Momentum and
angular momentum, Proc. Roy. Soc. London A 314, 499
(1970).
As an example of how this equation is used, the following
paper by Michael Hartl (no relation to Jim Hartle)
examines the practical impact of spin-curvature coupling upon
the dynamics of orbits around black holes:
M. D. Hartl, Dynamics of
spinning test particles in Kerr spacetime, Phys. Rev. D
67, 024005 (2003).
3.12.08
Minor clarification on pset 4, #2: In part
2(b), I've instructed you to compute a quantity written with
two adjacent spatial indices, both in the downstairs position.
I was a bit sloppy here: I was taking advantage of the fact
that, in the limit being considered and in inertial
coordinates, the spatial metric is diag(1,1,1); as such, up
versus down for index placement is immaterial. As such, we
are allowed to extend the summation convention so that
repeated indices are taken to be summed over even if they are
both in the same position. (This "extended" summation
convention is commonly used in the linearized approximation to
general relativity, which we will discuss in depth following
spring break.)
3.11.08
A few notes on the holonomy: A very nice visualization
of vector rotation under the impact of the holonomy. For
further discussion and some pretty good references, the wikipedia
article is actually very nice. Your textbook and the
supplementary texts (particularly Wald) have deeper
discussion.
3.11.08
Posted a Mathematica notebook, GRTool.nb, to facilitate computing
connection coefficients and curvature tensors. This notebook
is your friend!! Please email Scott or Will if you have any
trouble using it.
3.11.08
Solution #3 posted, plus Mathematica notebook used
for part of the solution (PDF version). Note, this
solution has been ready for a few days, but I delayed posting
it to wait for some students who had extensions to hand in
their assignments.
3.11.08
Pset 4 is due March 13th, not March 15th as is
indicated on a version that I posted earlier. Sorry about
that; I edited a previous year's pset, and failed to update
the post and due date information.
3.07.08
I have been forgetting to update the list of suggested
readings on the syllabus; sorry
about that! I just went through and caught up the list with
where we should be at this point.
3.05.08
Problem set #4 posted.
2.27.08
Notes by Ed Bertschinger on
tensor analysis using orthonormal coordinates.
2.27.08
Problem set #3 posted.
2.27.08
Solution #2 posted.
2.15.08
Solution #1 posted.
2.14.08
Problem set #2 posted.
2.05.08
Problem set #1 posted.
2.01.08
Updated course info and syllabus loaded onto the website.
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