ANNOUNCEMENTS
MTH 437/537 — Spring 2015

5/28/15

Another minor comment on the exam:

• The "lifetime" in the last problem is the lifetime of the universe — which is also the lifetime of the lightbeam.

5/26/15

Two minor corrections/comments on the exam:

• "Locally equivalent" can be interpreted as "equal to".
  ("Local" just means that you don't have to worry about matching up the coordinate domains.)
• The extra credit problem should refer to the second problem, not the first (which wasn't there when originally written).

5/25/15

Here are my office hours this week:

• Monday: 3–4 PM. (The building is probably locked; call ahead, or yell from the north door.)
• Tuesday: 10 AM – 3:30 PM, except for a lunch break (most likely 12–1 PM).
• Wednesday: 9–11:30 AM & 1:30–2:30 PM 2–2:45 PM. (I will try to be available by 1:45 PM.)
• Thursday: 10 AM – 3:30 PM, except for a lunch break (most likely 12–1 PM).
• Friday: 9–10 AM & 1–2:30 PM. (You can probably also catch me in Weniger 495 from 11–11:30 AM.)

Other times may be possible if these don't work for you.

5/23/15

The final has been sent out to the ONID addresses I have on file.

If you didn't receive a copy, let me know ASAP!

5/22/15

The final covers Chapters 1–9 in the text.

We didn't cover all of Chapter 5, and we've only just started Chapter 9. It is fair to assume that all exam questions can be reasonably answered based on mastery of the material we have covered in class.

There is a nice discussion of Olber's paradox in §22.2 of d'Inverno's text.

Olber's proposed resolution was to postulate the existence of an interstellar gas that would absorb radiation; this argument fails. The paradox holds regardless of whether the universe is Euclidean, or whether it is infinite. One possible resolution with a static universe is to assume that stars did not start radiating until recently, but some observed stars are too old. The accepted resolution is that an expanding universe causes a redshift in the observed frequency, which reduces the energy that reaches us.

5/21/15

My office hour tomorrow (Friday) afternoon will be in Weniger 495.

5/20/15

The take-home final will be distributed no later than Sunday morning, 5/24/15, and will be due in class 5/29/15.

• The exam will be emailed to your ONID address in PDF format.
• I will hold extensive office hours Tuesday through Thursday; details will be posted here later.
• Detailed ground rules will be posted later.

5/15/15

As announced in class, 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.
(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.

5/14/15

Tomorrow afternoon's office hour is canceled.

I will likely be in my office from 11–11:30 AM and from 1–1:20 PM, but if you want to be sure of catching me please contact me via email beforehand. I should also be available before class.

5/13/15

There were three points of confusion during today's lecture:

1. What is the correct scaling for u?
   We resolved this one in class, after initially making algebraic errors.
   The correct scaling is given by dr = vdτ+uds, which is equivalent to requiring u·∇f=df/ds.
2. Why is dv/ds=du/dτ?
   The informal argument given in class is basically correct:
   u and v correspond (in the sense above) to partial derivative operators, which commute.
   A better argument is to compute 0=d²r=dv∧dτ+du∧ds, which vanishes since the connection is torsion free.
3. Why isn't d²u/dτ² = d/dτ(dv/ds) = d/ds(dv/dτ) = 0, since dv/dτ=0 if v is geodesic?
   It is not true that the exterior derivative of a vector can be expanded in terms of partial derivative operators!
   (If it were, then d²v would be zero for any v, and the curvature would vanish!)
   Thus, "partial derivatives" acting on vectors do not commute (because they involve the connection).

A corrected derivation of the relationship between curvature and geodesic deviation can be found here.

5/7/15

There is a good chance I will be in my office and able to answer questions from 1:45–2:30 PM.

5/5/15

A formula sheet will be available on the midterm. You can find a copy here.

(My apologies for not posting these formulas sooner. Any questions or comments, please let me know.)

4/30/15

The midterm will be Wednesday 5/6/15 in class. The main topics to be covered on the midterm are:

• Line elements;
• Spacetime diagrams;
• Geodesics and their properties;
• Schwarzschild geometry.

Further information:

• The exam is closed book;
• Monday's class will be devoted to review. Come prepared to ask questions!
• A formula sheet will be provided, and will be discussed at the review.

4/29/15

The slides from my colloquium are available here.

4/28/15

Based on yesterday's discussion in class, we will likely have a midterm next Wednesday, 5/6/15.

• There is a homework assignment due Monday, 5/4/15.
• Monday 5/4/15 would become a review day. In particular, we could go over that assignment (and others).
• There will also be a take-home final exam during Week 9.

Please be prepared to express your opinion regarding the timing of a midterm (and the pros and cons of having one) during class tomorrow. In the absence of a consensus to the contrary, we will proceed as outlined above.

4/24/15

Happy birthday to the Hubble Space Telescope!

4/17/15

The figures I showed in class today are from §3.5 of the text, and can also be found online.

4/15/15

A nice example of an Einstein ring due to gravitational lensing can be found here.

Further pictures can be found via Google, and more information is available on Wikipedia.

4/7/15

My special relativity text is finally available online as an ebook through the OSU library.

4/6/15

There will be no class next Monday, 4/13/15.

My office hour that afternoon is also canceled.

I will be away at two conferences:

The PNWMAA/NUMS meeting, where I am an invited speaker,

and the April APS meeting celebrating the 100th anniversary of general relativity.

You are strongly encouraged to attend my physics colloquium immediately after class on Monday, 4/27/15.

This talk will be quite similar to the PNWMAA talk.

4/3/15

The slides from today's lecture are available here.

3/30/15

Due to lack of enthusiasm, there will not be any review sessions.

Material from a review session in a previous year can be found here.
(All of this content is contained in the textbook.)

3/21/15

I propose offering two optional "review" sessions, at times to be arranged. The first will go over the final from MTH 434/534, and the second is intended as an intensive summary of the use of differential forms.

Please be prepared to discuss times for these sessions on the first day of class.

(We will probably not use differential forms before the end of Week 2.)

3/20/15

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 the prepublication copy available online should be sufficient.

You may also wish to purchase a more traditional text, in which case I recommend 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).

• EBH uses only basic calculus to manipulate line elements, and only discusses black holes, but does so in great detail.
• Relativity discusses the math first, then the physics.
• Gravity begins essentially the same way, starting from a given line element to discuss applications, including both black holes and other topics. This is followed by a full treatment of tensor calculus, including a derivation of Einstein's equation. This book is the most advanced of the three, and is aimed at advanced undergraduate physics majors.

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.

• If you are seriously interested in the physics of general relativity, Gravity is worth having.
• If you are primarily interested in the mathematics, you may find Relativity easier to read. It covers more topics more quickly than Gravity.
• However, we will use the language of differential forms wherever we can, which is not extensively covered in any of these other books. We will therefore take a somewhat more sophisticated approach than EBH, while trying to avoid most of the tensor analysis 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.