ANNOUNCEMENTS
MTH 437/537 — Spring 2019

6/17/19
The exams have been graded, and course grades assigned, although you may not be able to see them until tomorrow.
The median score was 65 out of 80; the mean was 59.
You can collect your exam from me (and look at exam solutions) in my office. I should be available most of this afternoon. After that, check with me or stop by and take your chances — I do not keep a regular schedule during the summer.
6/10/19
The final exam will be available for download here after 12 PM today.
You will need to provide a username and password, which will be sent to your ONID email address this afternoon. If you have any difficulty accessing the exam, please let me know.
6/5/19
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 does not seem able to compute the components of the Riemann tensor in a reasonable amount of time using this method. It can, however, still compute the connection, and verify that the Ricci tensor vanishes.
6/4/19
My office hours next week are listed below.
In most cases, the ending times will be extended if necessary to accommodate those already present.
Other times may be possible, and I will also respond to email inquiries.
Please remember that you may not discuss any aspect of the exam with anyone but me while you are waiting your turn.
6/3/19
Additional discussion of the Standard Models ($\Lambda=0$) can be found in the text.
(Contrary to what was stated in class, these models do not necessarily have $p=0$.)
We only discussed the first example ($p=0=k$) in class, known as the Einstein–de Sitter cosmology.
A discussion of cosmological redshift can also be found in the text.
5/30/19
My office hour tomorrow afternoon is again "preponed" by half an hour, and will therefore be 1–1:45 PM.
5/29/19
The take-home final will be distributed online on Monday afternoon, 6/10/19.
It will be due in my office at noon on Thursday, 6/13/19.
5/26/19
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.
Several examples have been implemented using this package, including
(There is also a partial verification of Birkhoff's Theorem, which provides a glimpse of the further capabilities of Sage.)
5/25/19
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 $\GG$.
5/24/19
I may be slightly late for my office hour this afternoon.
I am also available from 10–11 AM.
5/22/19
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.)
5/21/19
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.
5/20/19
In addition to my usual WF office hours this week, I should be in my office hour this afternoon from roughly 1:30–2:30 PM.
I am also available this morning, from roughly 10:30–11:30 AM.
5/19/19
In question 1b of HW 6, it is not necessary to give an explicit formula for $g^{ij}$, although you may wish to do so in the case of two dimensions. For the general case, a correct statement of the system of equations, together with an argument as to why you expect a unique solution, is sufficient, possibly accompanied by an interpretation of these equations in terms of linear algebra.
5/18/19
As mentioned in class yesterday, I didn't get all the signs right for expressions involving $\TT$.
(The energy density $\rho$ should be positive; the question in each case is whether a given component is positive or negative.)
To the best of my knowledge, the signs in Chapter 8 of the text (which is Chapter 7 of the wiki) are correct.
In particular, the correct expression for observed energy density is $\rho = +g(\TT,\vv\cdot d\rr)\cdot \vv = +T^i{}_j v_i v^j$.
5/17/19
My office hour this afternoon is "preponed" by half an hour, and is now 1–1:45 PM.
I am also available from 9–10 AM.
5/8/19
Here are the answers to the midterm questions:
  1. (a) $-1$ (b) $-\left(1-\frac{2m}{r}\right)$ (c) $-1$
  2. (a) $e=h(r)\dot{t}$ (b) $-\sqrt{\frac{2m}{r}-\frac{q^2}{r^2}}$
  3. (a) Many answers possible, including $\rho=e^{\pm\alpha}$
  4. (a) No (b) $r=m\pm\sqrt{m^2-q^2}$
Worked solutions can be seen in my office, and will be discussed in class on Friday.
5/7/19
The midterm has been moved to Bexl 417.
I should be in my office tomorrow (Wed) morning (roughly 10:30–11:45 AM) in addition to my office hour at 1:30 PM.
5/6/19
As discussed in class today, there are 10 independent Killing vectors in 4-dimensional Minkowsk space, namely 4 translations: $\xhat$, $\yhat$, $\zhat$, $\Hat{t}$; 3 rotations: $r\,\phat=x\,\yhat-y\,\xhat$, $y\,\zhat-z\,\yhat$, $z\,\xhat-x\,\zhat$; and 3 boosts: $x\,\Hat{t}+t\,\xhat$, $y\,\Hat{t}+t\,\yhat$, $z\,\Hat{t}+t\,\zhat$.
Each of these Killing vectors can be realized as coordinate symmetries of the line element in appropriate coordinates, e.g. by switching to round (spherical or cylindrical) or Rindler-like coordinates.
It is straightforward to show that each of the above vectors satisfies Killing's equation, namely $d\XX\cdot d\rr=0$. Less obvious (but not difficult to show) is that these are the only independent solutions of that equation.
The collection of all Killing vectors forms a Lie algebra under the operation of commutation, where vector fields act on each other by differentiation. Lie algebras are infinitesimal versions of Lie groups, representing continuous symmetries. For (some) further information, see Chapter 10 of our octonions book.
5/4/19
A formula sheet will be available on the midterm. You can find a copy here.
The current version is tentative; feel free to propose additions during the review session on Monday.
I should be in my office most of Monday morning (roughly 9:30–11:45 AM) as well as before (roughly 12:45–2 PM) and after class. I will also be in my office on Tuesday (roughly 11 AM–2 PM, with a break for lunch); (roughly 12:30–2 PM); if you want to talk to me before 10 AM or after 2 PM you should contact me in advance.
5/3/19
Further information about charged and rotating black holes and their Penrose diagrams can be found in the undergraduate textbook by d'Inverno, which is on reserve.
A more advanced treatment can be found in the book The Large Scale Structure of Space-Time by Hawking & Ellis, available in the library.
5/1/19
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.
4/29/19
The midterm will be Wednesday 5/8/19 in class. The main topics to be covered on the midterm are:
Further information:
I will hold extra office hours next week, most likely MW morning; other times will be available by appointment.
4/28/19
Figure 3.9 on page 38 (also available as Figure 8 in this section), showing the relationship between shell coordinates and rain coordinates, is correct but misleading.
This figure shows the relationships between certain differential forms, using the geometric description of §13.8, but without displaying the stacks. However, it is not easy in such diagrams to read off the magnitudes of the differential forms, which do not correspond directly to the lengths of the sides.
A more traditional figure, using the language of infinitesimal displacement, is shown at the right.
Note added: Spacetime diagrams implicitly show relationships between vectors. For example, the figure at the right shows that $$d\rr = \sqrt{1-\frac{2m}{r}}\,dt\,\Hat{t} + \frac{dr\,\Hat{r}}{\sqrt{1-\frac{2m}{r}}} = dT\,\Hat{T}$$ if $dR=0$, that is, for a freely-falling object. Figure 3.9 shows instead a relationship between 1-forms, namely that $$1\,\sigma^T = \frac{\sigma^t}{\sqrt{1-\frac{2m}{r}}} + \frac{\sqrt{\frac{2m}{r}}\,\sigma^r}{\sqrt{1-\frac{2m}{r}}}$$ which is always true. (The explicit inclusion of orthonormal basis 1-forms restores the proper scaling to the triangle.)
4/15/19
The embedding diagram shown in class for the Schwarzschild geometry can be constructed as follows.
Thus, $r$ is a quadratic function of $z$; the embedding diagram is a sideways parabola, as displayed in class.
Do not forget that "$z$" does not exist! We can embed Schwarzschild geometry in a higher-dimensional flat space, but we do not need to.
4/10/19
Breaking news: See the first actual image of a black hole here.
Also take a look here at today's xkcd, showing just how large this black hole is.
4/10/19
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.
4/9/19
There will be no class on Friday, 4/12/19.
I encourage you to use this time – and the classroom – to work jointly on the homework assignment due on Monday.
4/8/19
We didn't quite finish deriving the geodesic equation on the sphere. The answer can be found in §19.3.
4/5/19
The schedule has been updated.
Among other things, the links for Wednesday and Friday of this week were incorrect.
Figures for length contraction, which we discussed in class, can be found here.
The analogous figure for time dilation, which we did not discuss, can be found here.
4/3/19
As mentioned in class today, 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.
4/2/19
My office hours have now been posted on the course homepage.
I am often in my office MWF mornings from roughly 10–11:30 AM, and am also usually available after class. Feel free to drop in at those times — or to contact me to arrange an appointment at these or other times.
My rough schedule can be found here.
4/1/19
As discussed in class today, I propose an extra, optional meeting to go over the final from last term.
This session is tentatively scheduled for this Friday, 4/5/19, at 2 PM in our regular classroom (Bexl 323) Bexl 320.
3/30/19
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.