ANNOUNCEMENTS
MTH 437/537 — Spring 2021

5/7/21
The exams have been graded. Answers to questions not asking you to show something are below.
  1. (b) $\frac{3m\pm\sqrt{9m^2-8q^2}}{2}$
  2. (a) all nonzero components are $\pm\frac{1}{a^2}$ (b) $-\frac{3}{a^2}d\rr$
IF your grade were determined only by your final exam, it would be (out of 90; extra credit included):
If you have any questions about these answers and/or how your final was graded, please contact me.
It may be possible to arrange a time to go over the solutions. Contact me if you are interested.
6/8/21
In problem 2, you may assume that the cosmological constant vanishes, that is, that $\Lambda=0$.
Please remember that there are known errors in some of the results in the appendix of the textbook; see the errata page.
6/7/21
The exam will be available on Gradescope starting at 2 PM this afternoon.
I will hold additional office hours at 4 PM both today and tomorrow, and at 10 AM both tomorrow and Wednesday.
Other times should be possible; contact me via email.
6/4/21
The Sage notebook I demonstrated in class can be found here.
This notebook uses the definition of the Kerr geometry that is built in to Sage using a coordinate basis. A version using an orthonormal frame can be found here, but Sage takes much longer to compute the Riemann tensor using this method (quite possibly due to inefficient programming on my part).
6/3/21
I will hold office hours on Monday, 6/7/21, from 10 AM–12 PM.
There will be roughly an hour of general review followed by roughly an hour of individual questions and consultation.
(I will record and post the first hour unless there are objections.)
5/30/21
The take-home final will be distributed online on Monday afternoon, 6/7/21.
It will be due on Gradescope at noon on Wednesday, 6/9/21.
5/29/21
Additional discussion of the Standard Models ($\Lambda=0$) can be found in the text.
Further discussion of cosmological redshift can also be found in the text.
5/28/21
Several examples have been posted using Sage, including:
(There is also a partial verification of Birkhoff's Theorem, which provides a glimpse of the further capabilities of Sage.)
5/26/21
The deadline for HW 7 has been extended until Friday, 5/28/21.
What do you know about the $R^i{}_{jkl}$? What is the relationship between $R_{ij}$ and $R^i{}_j$?
5/24/21
I have set up an experimental interface to Sage here that can in principle calculate the connection 1‑forms $\omega^i{}_j$, the curvature 2‑forms $\Omega^i{}_j$, and the Ricci and Einstein tensors for any line element in orthogonal coordinates.
Further examples will be posted soon. For help using this page, ask.
5/20/21
There are several computer algebra packages available for computing curvature components:
Printouts of (old!) sample computer algebra sessions are available for GRTensor and CLASSI.
Older versions of my instructions, that also include coordinate-based computations, are available for Maple and Mathematica packages, and for SHEEP/CLASSI.
You may use software to compute curvature on the homework! See me if you would like help getting started.
You must however document doing so, and it's up to you to ensure that you are working in an appropriate basis.
5/19/21
A discussion of Birkhoff's Theorem can be found in the Appendix.
(There are minor typos in the Schwarzschild curvature 2-forms as given in §A.3 of the text:
The coordinate expressions in the middle of Equations (A.52) and (A.53) are each missing a factor of 1/2.
Also, the initial minus sign should be removed from Equation (A.61).
(The final expressions in terms of an orthonormal frame are correct.)
The wiki version has been corrected, and a full list of errata can be found here.)
5/18/21
In question 1b of HW 6, it is not necessary to integrate your expression for arclength.
In question 2b, it is not necessary to give an explicit formula for the individual components $g^{ij}$ in terms of the components $g_{ij}$.
5/17/21
All indices can be raised and lowered with the metric.
For example, since $\TT=T^i{}_j\sigma^j\ee_i$, we have $\ee_k\cdot\TT = T^i{}_j\sigma^j\ee_k\cdot\ee_i = g_{ki}T^i{}_j\sigma^j = T_{kj}\sigma^j$ and similarly for $\RR$ and $\GG$.
5/13/21
WEBSITE RELOCATION:
OSU is shutting down all ONID user websites (people.oregonstate.edu) at the end of May.
Automatic redirection should take place, but the direct link to the new course website is
http://math.oregonstate.edu/~tevian/onid/MTH437
Both sites are active now, and should be identical.
5/12/21
Notes from today's class can be found here.
5/10/21
By popular request, today's class notes are available here.
Further details about geodesic deviation can be found in §7 and §A.2 of the text, both of which can also be found here.
5/8/21
Further information about charged and rotating black holes and their Penrose diagrams can be found in the undergraduate textbook by d'Inverno, which is available in the OSU library.
A more advanced treatment can be found in the book The Large Scale Structure of Space-Time by Hawking & Ellis, which is also available in the OSU library.
5/7/21
Solutions to the midterm questions were discussed in class today; the answers are below.
  1. (a) No (b) $1/\rho$ (c) $\boldsymbol{\hat\alpha}$ (d) $-1$ (e) $1/\rho^2$
  2. (a) $\rho\,\boldsymbol{\hat\alpha}$ (b) $\rho^2\dot\alpha$ (c) Many answers possible, including $\rho=e^{\pm\alpha}$
  3. (a) $r=m\pm\sqrt{m^2-q^2}$ (b) $-\sqrt{\frac{2m}{r}-\frac{q^2}{r^2}}$ (c) No
IF your grade were determined only by your midterm, it would be:
If you have any questions about these solutions and/or how your midterm was graded, please contact me.
5/3/21
Notes from today's review session can be found here.
The list of topics suggested via chat can be found here.
I will be available most of the day on Wednesday to answer questions about the midterm.
Contact me via email either to ask a question or to set up a Zoom session (using the Office Hours ID).
I will keep the Office Hours Zoom session open during the normal class time and subsequent office hour (3–5 PM).
At other times, I will respond to you as soon as possible.
4/28/21
The midterm will be Wednesday 5/5/21.
The exam will be available from 8 AM–midnight on Gradescope.
(You can access Gradescope through Canvas, but the exam will not show up as a Canvas assignment.)
The main topics to be covered on the midterm are:
Further information:
I will hold extra office hours next week, most likely Tuesday afternoon; other times will be available by appointment.
4/24/21
The basis 1-forms $\sigma^T=dT$, $\sigma^R$ in rain coordinates are defined in §3.9, but the rain cooordinate $R$ is not defined until §A.4.
4/23/21
There are two competing usages of "rain coordinates", due to the confusion between the two radial coordinates.
So "rain coordinates" should really refer to $(T,R,\theta,\phi)$, but is usually used as a synonym for Painlevé–Gullstrand coordinates.
Only the latter alternative corresponds to an orthogonal coordinate system, with line element $$ds^2 = -dT^2 + \frac{2m}{r}\,dR^2 + r^2 (d\theta^2+\sin^2\theta\,d\phi^2)$$ where $r$ is now an implicit function of $T$ and $R$.
Another way to see that surfaces with $\{T=\hbox{const}\}$ are flat is to note that $\sigma^R-\sqrt{\frac{2m}{r}}\,\sigma^T=dr$, so that $\sigma^R=dr$ if $dT=0$.
However, as pointed out in class, the $T$ and $R$ directions are orthogonal, but not the $T$ and $r$ directions.
Thus, the operator "$\frac{\partial}{\partial T}$" is ambiguous! Holding $R$ fixed, the resulting "$T$ direction" is not a symmetry, since the line element above depends on $T$ (through $r$). Holding $r$ fixed instead does lead to a symmetry; the Painlevé–Gullstrand line element has no $T$ dependence. However, the resulting "$T$ direction" is orthogonal to $r$, not $R$ — and is hence the same as the Schwarzschild $t$ direction, which we already know is a symmetry.
4/19/21
There will be an extra office hour tonight at 7 PM.
The due date for the assignment due today is postponed until 2 PM tomorrow.
You may resubmit your assignment on Gradescope until then without penalty.
4/14/21
The figures shown at the end of class today can be found in §3.5 of the text.
Here are some hints for this week's assignment that we didn't quite get to in class today:
4/12/21
Further information about the difference between the geometric radius and the physical radius can be found in my recent paper on embedding diagrams, which was published here.
A construction of the Schwarzschild embedding diagram is included, which should be accessible to students in this class.
4/8/21
One important theme in yesterday's derivation of the geodesic equation is that differentials are the numerators of derivatives. Thus, an equation involving 1-forms can be converted to one involving derivatives by dividing by a differential.
A more subtle message is that this doesn't work for second derivatives. In particular, the geodesic equation is a second order system of ODEs, but there is no way to take two derivatives with respect to the same parameter using differential forms...
4/7/21
Solutions of the geodesic equation in polar and spherical coordinates can be found in §19.5 and §19.6.
The latter section also discusses using vector analysis to describe arbitrary geodesics on the sphere.
You should check for yourself that $r\phat$ is indeed a Killing vector, that is, that $d(r\phat)\cdot d\rr=0$.
Details can be found in §2.2.
4/6/21
The software I used to draw lines is GeoGebra, which can be run online in a browser, or downloaded to most computers, tablets and smartphones. (You do not need to create an account in order to save files locally.)
The curves I drew in class were obtained by successively entering the following ordered pairs in the algebra panel:
$(2,y)$, $(x,2)$, $(x,x)$, $(3;\theta)$, $(x;\frac\pi3)$, $(\frac{1}{\cos\theta};\theta)$, $(\frac{1}{\cos\theta},\theta)$
Be sure to distinguish commas from semicolons when entering these expressions!
4/5/21
We didn't quite finish deriving the geodesic equation on the sphere. The answer can be found in §19.3.
4/2/21
A copy of my paper on the cylindrical twin paradox has been posted on canvas (as an ungraded reading assignment).
Today's office hour is replaced by the final exam review from last term.
The review will use the MTH 437/537 classroom Zoom ID, not the Office Hours link.
3/31/21
The sets of spacelike, timelike, and lightlike vectors do not close under addition (even with the zero vector included), and thus do not form a vector space.
Can you find counterexamples?
However, the set of future-pointing (or past-pointing) timelike vectors do close under addition (with the zero vector included), and therefore do form a vector space.
3/29/21
As discussed in class today, I propose an extra, optional meeting to go over the final exam from last term.
This session is tentatively scheduled for this Friday, 4/2/21, at 4 PM, using our regular classroom Zoom ID.
The session will be recorded, and posted on the Canvas pages for both courses.
3/24/21
Welcome to another term of remote teaching! Below is some information about how this course will be run.
Overview:
Details:
3/23/21
The primary text for this course will be my own book, which can be read online as an ebook through the OSU library.
There is also a freely accessible wiki version available, which is however not quite the same as the published version.
We will also refer briefly to my book on special relativity.
You may purchase this book if you wish, but it can also be read online as an ebook through the OSU library, and again there is also a wiki version.
You may also wish to purchase a more traditional text, in which case I recommend any of the first three optional texts listed on the books page. The level of this course will be somewhere between that of these books, henceforth referred to as EBH (Taylor & Wheeler), Relativity (d'Inverno), and Gravity (Hartle).
We will cover more material than EBH, but we will stop short of the full tensor treatment in Relativity or (the back of) Gravity. We will also cover some of the material on black holes from EBH which is not in Gravity or Relativity.
In short, none of these books is perfect, but all are valuable resources. In addition to the above books, OSU owns an electronic copy of Relativity Demystified, which summarizes many of the key aspects of relativity, but provides no derivations. By all means use it for reference, but I would not recommend using it as a primary text.