ANNOUNCEMENTS
MTH 437/537 — Spring 2024

4/23/24
The due date for the assignment due today at 4 PM (HW #3) has been extended until 4 PM on Thursday, 4/25/24.
You may resubmit your assignment without penalty until then.
(Assignments will not be accepted after the extended due date.)
4/21/24
When working on this week's assignment (HW #3), the discussion of circular orbits in §3.6 of DFGGR may be helpful.
As discussed both in class and in this section, we argued that $\dot r=0$ implies that $V'=0$.
4/19/24
The figures shown at the end of class yesterday can be found in §3.5 of the text.
Here are some hints for this week's assignment:
4/18/24
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.
4/15/24
From the tagline of today's xkcd comic strip:
The standard North American NAD83 coordinate system is misaligned from the actual Earth, off-center by about 7 feet.
4/11/24
We didn't quite get to the derivation of the geodesic equation on the sphere...
The answer can be found in §19.3.
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 $d(r\phat)\cdot d\rr=0$ (in polar coordinates).
Details can be found in §2.2.
4/9/24
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...
4/8/24
I will be speaking about the mathematical side of my current research next Monday in the Geometry and Topology Seminar.
Octonions and the exceptional Lie algebras (and particle physics)
Tevian Dray
M 4/15/24 at 12 PM in Kidder 280
4/5/24
As pointed out in class yesterday, 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. [Still not a vector space since additive inverses are missing.]
As discussed in class yesterday, the resolution of the twin paradox is that only one of the twins is in an inertial frame. But what if the spatial topology were closed, that is, what if the universe looked like a cylinder, with $x=0$ identified with $x=\lambda$? Then the traveling twin doesn't need to accelerate...
Try to resolve this version of the paradox yourself before looking up the answer; a complete discussion can be found here.
4/4/24
A recording of a talk I gave a couple of years on the geometry of special relativity, covering essentially the same content as we did in class this week, can be found here.
There is also a poster describing my geometric approach to special relativity near Weniger 304.
The slides I showed in class today can be found here.
Several of these slides also appear in the video.
A few people haven't yet completed HW #0, which was due today.
There are no formal penalties for skipping this assignment, but submission would still be appreciated.
The optional meeting to go over the final from last term will indeed start at 10:30 AM next Tuesday, 4/9/24, in Weniger 328.
If you can't make it to this meeting, please feel free to ask about the exam during office hours.
4/2/24
As announced in class today, I will schedule an optional meeting to discuss the final exam from last term. We tentatively agreed that this meeting would be sometime on Tuesday morning, 4/9/24. I propose starting at 10:30 AM, but am willing to consider earlier times.
Please let me know if you would like to attend and this time will not work for you.
4/1/24
A schedule from a previous year can be found here.
This schedule is a good approximation to what we will cover when.
3/26/24
All assignments will be posted only on the homework page.
Assignments will not be posted on Canvas.
All assignments should be submitted via Gradescope.
Further information can be found on my own information page for Gradescope.
3/25/24
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.