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8 Nov 2002
Question about the exam. Seeing as this is the
first year that you are taking this course, is there any point in
checking (petros') past exam papers for an idea of what will come up
in June? Or does your course bear no relation to his (structure wise)?
If not, will you be giving homework so we have some idea of what we
will be up against?
Reply. I will start giving you problem sheets soon and that
will give you some idea of what the exam will be like. Nearer the time
I will also set a sample paper and go through Petros's exams and point
out which of his questions are relevant to my exam. I will try not to
stray too far from the style and standard of Petros's questions, but
some change in style is unavoidable and the material is slightly
different.
12 Nov 2002
Finds calculations badly outlined.
Just a quick note to say I find all the calculations very confusing.
While usually I understand the way it progresses from step to step I
would much rather that an explination was fully given of the rules
behind the calculations were given. Plus all the mistakes are slightly
offputting especiallyas the class seem to like to keep it to
themselves when they spot one. I would consider myself one of the
weaker students in the class and I find that I learn better not from
individual calculations but from overall views of how to do the
calculations. Plus Simms courses would have set me and everyone else
up nicely for this course if I hadnt crammed for the last 3 weeks to
pass his exams, so through none but my own fault, i am at quite a
disadvantage. I just thought Id let you know how i, and others, doing
your course are getting on. I find you are quite an interesting
lecturer though! It keeps me awake.
Reply. Thanks for your message. I think the problem is that the
calculations we are doing in class aren't intended to teach a general
method for doing similar calculations, there are important
calculations in develouping the theory, each of which has its own set
of rules and assumptions. The thing to do is to go through the
calculations step by step and it will become clearer and clearer what
is going on in them. You have to think of them as being like the big
theorems in a more mathematical subject, rather than being like
examples in a methods course. I am sorry about the mistakes, it is
annoying when there are mistakes, but, considering the number and
complexity of the calculations, I think I am doing okay. Of course, it
is a problem if you feel you can't trust your notes, if that is the
case for some part of course, let me know and I will try and produce a
corresponding note for the web. I actually find the class pretty good
at pointing out mistakes, and pretty good at asking useful questions
when I put something too vaguely. As for David's course, well you were
foolish not to follow it, I am sure it was excellant, but I don't
think it should hold you back too much here, the main point of contact
was tensors and I introduce these in more simple minded way than he
does. Ray d'Inverno has quite a long discussion of tensors which you
should look at if you are confused.
12 Nov 2002
Error in note 2
I think there's a small error in Note 2. On Page 2, eqn. 11 has a
mistake in the last term of the first bracket; the indices should read cdab
instead of bcab.
Reply. Yup, thanks a mill and well done for spotting that, I've
corrected the note. Of course, this formula should be usefull in
question 12 on problem sheet 1.
15 Nov 2002
Question about exercises I am one of the
(few!) maths students taking your course and find that some of the
physics is challenging. Nonetheless, I really enjoy it but (like a
previous email you received) I would certainly consider myself one of
the weeker students. Would it be possible for you to post on the web,
at some stage, solutions to your exercises? Perhaps walking myself
through them might help? Thanks.
Reply. I am intending to spend one or maybe two lectures
doing the exercises and afterwards, I intend to put the solutions on
the web afterwards. I am glad you are enjoying the course.
10 Feb 2002
serious general relativity questionWhat's
that picture on your homepage of 3 guys beating someone with sticks?
You're not the guy getting the beating are you?
Reply.
Thanks for your concern. In fact, that's a picture from an early production of
_Waiting for Godot_, the guy being beaten is Lucky. I used to have
long hair and look a bit like him, that's why I have the picture
there. I don't look so much like him anymore, but I can't find another picture of someone who looks like me.
24 Feb 2003
J.E. Lidsey Would it therefore be fair to say
that this J.E. Lidsey was not too impressed by your request. Perhaps
we should all send him a barrage of abusive e-mails. It probably
wouldn't encourage him to grant access to his notes but at least we'd
have our fun.
Reply. Well, to be fair we don't know why he
didn't reply, maybe he isn't coming in at the moment and we don't know
who password protected the notes. I was a bit suprised by the whole
thing. Anyway, best not to send abusive emails, you never know what is
going to happen in the future so it is best to try and act with good
will. It is already bad enough me saying what I said in the textbook
section, it is childish of me.
3 Mar 2003
Mach's principal I was just trying to sort out
the historical development of general relativity in my head, and in
particular the reasoning which followed from Einstein's belief in
Mach's principle. Einstein used Mach's principle as a justification
of the effect which matter has in determining the geometry of
spacetime. However relativity does not at the same time address Mach's
principle since within the structure of relativity, it is more the
action of close-up bodies which affects the inertial properties of a
body than those distant in the universe. My question is, have any
theories (or perhaps furtherings of Einstein's theories) been
developed which give some reason for Mach's principle?
Reply I am sorry I haven't replied to this, I don't really
think I know the answer. I have now written to a few people who might
be more knowledgeable than me and will put up their answer if they
give me one.
The great Ian Drummond has replied, he says "I too have never seen why Mach's Priciple was so important to Einstein. It's modern significance seems to be subsumed by issues relating to the initial value problem in GR. A source which might be the beginning of a thread leading to wisdom is the discussion of Mach's Principle in Misner, Thorne and Wheeler - it's pp453-455 I think. Somehow Mach's ideas helped Einstein in his efforts to formulate GR and to see that a local inertial frame could be sensibly influenced by the distribution of even distant matter. In my mind the idea is rather a fugitive one. This may not satisfy your students or yourself. If you get a clearer picture I would be glad to hear about it." That doesn't really answer your question, but it leaves me feeling better.
29 Apr 2003
ExamIf you ask about the bending of light by the sun, do we have to derive the schwarzschild solution?
Are you going to set a problem sheet/ sample question on inflation?
Reply It will be clear from the question weather you need to derive Schwarzwchild or not, with schwarzchild it would be a very long question. I will put a problemsheet on the web before friday with inflation and the special topics. I also have corrects to do on the solutions and I will put up a few remarks about the exam, sample questions next week when the externs get back to me.
1 Jun 2003
Correction to formula sheet
Just to let you know, I think there is a typo on the list of formulae
you have posted "parameters and equations of cosmology". Just before you write
eqn (11) you give the equation of state for lamda. You write rho sub-lamda =
lamda over 3. Should the three be an eight pi? I think that's what's in your
notes.
Reply You are right of course, I will make appropriate arrangements.
1 Jun 2003
Exam, PS4 Hi there! Just a couple of things:
1. Are there any constants or integrals that I should memorize for
the exam, since I can never remember any of them?
2. Where can I get a solution to set 4 problem 3 or is it unnecessary
for the exam?,br>
Thanks,
ReplyI don't think there are any constants you need to know or any integrals
beyond the standard ones, eg the hardest will be the sort of sine
substitution integrals you use for calculating the age of the universe,
I have tried to do Q3 PS3 a few times and I always get depressed, it is
too long.
1 Jun 2003
Geodesics
I was just having a look at the 442 sample paper, and in particular the
question on geodesics, and I just have a couple of questions about it if
you have the time to answer.
The definition of a geodesic is a curve that parallel transports it's own
tangent vector. Then the question asks to show that this is equivalent to
the distance being a minimum (show this be stationary rather than a
minimum).
Now in the notes this is done not by doing Euler- Lagrange, not on ds, but
rather on ds/dt.ds. In the notes this is explained away to minimising the
square length, and I haven't been able to align this with what you've
done.
However I do know (I think) that any parameter parametrising a
geodesic must be affine ( a linear function of s), when the geodesic is
not null, so if we presume that the parameter is an affine variable, then
changing to ds/dt is allowed, because it will just be a constant. And when
the geodesic is null, the ds are zero anyway so the change is allowed.
Also, I was wondering if the parameters must be affine for timelike and
spacelike geodesics is there any similar restrictions on the lightlike
case. It would seem that allowing any parametrisation would be slightly
too lax.
It's quite possible that I've been thinking far too much about this and
made a mountain out of a molehill, but any help clearing up the confusion
would be very helpful.
ReplyYes, it should be stationary, not minimum, the obvious example being the
great circle.
A action with squared integrand has the same extrema as the original
action so if we find the paths extremizing the square it extremizes the
original distance, this simplifies the Euler-Lagrange equations. This is
discussed in for example Weinberg, but is a common trick.
Why does the geodesic parameter have to be affine?
1 Jun 2003
Palatini
Would it be possible for you to answer a couple of quick questions ahead
of the 442 exam on Tuesday? here goes: On the sample paper, in question
4, you give the Palintini identity . 2 questions: in the notes, the first
term had a delta preceding the connection coefficient; but none there. Is
this a typo? Secondly, and more importantly i cannot derive the identity,
even in geodesic coordinates. I use the Riemann tensor calculated in "
the explicit formula for the Riemann tensor". I throw away the gamma
squared terms (geodesic coordinates). I let b=d to get Rac. Is this
correct? If it is, the problem is that in "the explicit formula" the
second term is differentiated w.r.t. a (the first index on the ricci
tensor), whereas in palintini, the second term is differentiated w.r.t. b
These things always confuse me.
Reply There is of course a delta missing. The Ricci tensor
is symmetric in its two indices, you can use that fact to change which
of the indices is doing the differenciating, it is actually just a
question of which indices you contract over, ie if R_{abcd} is the
Riemann tensor, I defined R_{ac}=g^{bd}R_{abcd} but I could have
R_{bd}=g^{ac}R_{abcd}, its the same because of the symmetries of the
Riemann tensor.
Further queriesThanks for that.
Just to be a hundred percent sure:
I think my problem was that I was using R_{abc}^d =
\Gamma^d_{ac,b} - \Gamma^d_{bc,a} (goedesic coordinates).
I then
contracted b and d to get R_{ac} == R_{abc}^b (*)
This gave R_{ac} = \Gamma^b_{ac,b} - \Gamma^b_{bc,a} (#)
This is basically just equations (25) and (26) of solutions to problem
sheet 2.
This is surely not symmetric in a and c.
So how can the Ricci tensor be symmetric in a and c?
From what you have written, it seems that the correct approach is to write
the Riemmann tensor with all indices downstairs before starting the
problem.
Am I correct in saying that all my problems stem from using this (*)
incorrect definition of the Ricci tensor?
If I am being stupid and (#) is symmetric under a and c swapping, how?
I really want (*) to be right - it makes curvature questions a fair bit
quicker.
Is it just a case of i) We know R_ab is symmetric, from writing it with
all indices downstairs, so (#) must be
symmetric, and also (*) holds, or ii) is it something else?
Reply So, the point is that the Ricci tensor is symmetric in its indices, though
this isn't obvious from the expression in terms of connection
coefficients, in the same way, all the symmetries of the Riemann tensor
aren't obvious from the explicit formula. Remember that R_{abcd}=R_{cdab},
now, your definition of the Ricci tensor is correct:
R_{ac}=R_{abc}{}^b=g^{bd}R_{abcd}
now apply the symmetry
R_{ac}=g^{bd}R_{abcd}=g^{bd}R_{cdab}=g^{db}R_{cdab}=R_{ca}
and there it is R_{ac}=R_{ca}. So, the point is that the
explicit expression you have is not manifestly symmetric, but it is
invariant, so you can switch the a and the c in the second term. You don't
need to do to geodesic coordinates by the way, if you look at the full
variation you can get rid of the quadratic terms by changing the partial
derivatives to full derivatives.