Astronomy 260, Spring 2007
General Relativity
Vanderbilt University
Department of Physics & Astronomy
Syllabus
http://brahms.phy.vanderbilt.edu/a260
Tu-Th, 1:10-2:25, Stevenson Center 1120
(Wherever the heck that room is... I'll have to find it.)
- Course Goals
- Instructor
- What You Need To Know To Take This Course
- Textbooks
- Course Description
- Homework, Tests, and Grading
- Course Schedule
Course Goals
- To throw people into black holes
- To blow up the Universe
Instructor
Prof. Robert Knop
Web page: http://brahms.phy.vandrebilt.edu/~rknop
E-mail: robert.a.knop@vanderbilt.edu rknop@pobox.com
Jabber: rknop@jabber.org (preferred)
AIM: rbrtknop
ICQ: 39159347
Office: Stevenson Center 6912
Office Phone: (615)322-6165
I won't have regular office hours scheduled for this class; you may drop by and talk to me any time I'm in my office; I'll let you know if it's not a good time to talk. However, I'm frequently off at meetings and doing various other things. If you want to be assured of talking to me about something related to the course, please E-mail me and we can set up an appointment.
What You Need To Know To Take This Course
General Relativity is a conceptually challenging (although extremely elegant and beautiful) subject, and also can lead one into some pretty serious math. Be not afraid.
This course is designed for junior and senior Physics majors. In particular: you do not need to know any tensor algebra or calculus, or even what a tensor is. You do need to be comfortable with Math 175-level calculus, and it would also be best if you were free of any deep phobia of linear algebra. I strongly recommend that you have taken Physics 227a. Although this course doesn't build too much on that particular course, having gone that far in Physics indicates that you have built some familiarity in thinking about and working with abstract concepts and differential equations, and then connecting them to the real world. We will make some reference to the Principle of Least Action, but if you haven't seen that, it won't be a serious handicap.
It would be nice if you have seen Special Relativity before coming into this class, but that is not essential. I know at the moment that there is no "official" place for SR in the Vandy Physics curriculum. We will spend a fair amount of time at the beginning of the class just talking about SR, so that we might more deeply understand it, and so that we might learn the notation and mathematics we'll be using for GR in a hopefully somewhat more familiar environment.
If you haven't taken Physics 227a, but have taken Physics 225a, talk to me; you might still be OK taking this course. If you have only taken Physics 121 (or another calculus-based Physics course), but not yet Physics 225, I strongly recommend that you hold off on taking this course until the next time it is offered (which should be Spring 2009).
Textbooks
- Gravity, by James B. Hartle (Addison Wesley, 2003)
This is the primary text for the course. Except for the very beginning of the course (see below), all readings will come out of this book. We won't cover every chapter in the book, but we will be going through it approximately in order. This book is designed for an undergraduate course in General Relativity like this one. While Physics 360 (the graduate course) sometimes uses the same book, they go through it in a different order (jumping directly to the more formal treatment towards the back that graduate courses generally start with).
- Flatland, by Edwin A. Abbott
A short novel written in the last century. (Er, the century before the last century, but near the end.) A quick read, a fun book, and something every math/physics nerd aficionado should have read. From our hyperdimensional perspective as three-dimensional creatures, understanding the geometrical plight of A. Square will help us understand our own plight as creatures in a four-dimensional spacetime which is intrinsically curved. (As we get into that class ,that term will become more meaningful.)
And, besides, by reading a novel in an advanced physics course, you can participate in the full liberal-arts experience.
Other useful/interesting references
- The Elegant Universe by Brian Greene
A popular treatment of string theory. Includes near the beginning one of the best popular-level descriptions of how the principle of equivalence combined with special relativity leads you to the inevitable conclusion that spacetime must be curved in the presence of mass.
- Warped Passages by Lisa Randall
Is this book good? I dunno. I haven't read it yet. It's on my list. Randall is a very well-regarded Harvard theorist, however. This doesn't mean she writes a readable book, but it does suggest that she knows what she's talking about.
- Black Holes and Time Warps: Einstein's Outrageous Legacy by Kip Thorne
A fun and readable popular level treatment of some of bizarre things that General Relativity leads us to, written by one of the leading theorists working in the area. Some have criticized this book as being a bit too much of "a celebration of Kip Throne," but if you can put up with that, it's a great book.
- It's About Time, by N. David Mermin
This is a new book about Special Relativity. It starts out with an excellent treatment of Galilean relativity, or relativity where speeds are slow compared to the speed of light. I found this part excellent, and indeed at my advanced age felt that I gained some nice insight into Physics reading it. By and large the book is excellent, although I think it makes the whole thing needlessly complicated when it gets into my favorite part of SR, spacetime diagrams. A good read. The book claims to be designed for a high-school class, but it would be an advanced high-school class. In any event, there is no math beyond high-school algebra in here, but the book does assume some facility in thinking back and forth between equations, pictures, and words. (Physics majors all have some experience with that sort of thing!)
- A First Course in General Relativity by Bernard F. Schutz
A good undergraduate level GR text that takes a more formal, approach, developing the mathematics of differential geometry more fully before getting to the fun stuff that we will get to a little sooner in the course.
- Spacetime and Geometry, by Sean Carroll
A more advanced book, designed for GR at the graduate level. This is the book I'd use were I to teach a graduate GR course.
Course Description
Spacetime is curved.
No, really.
The 20th Century saw two fundamental revolutions in our view of the Universe, perhaps more mind-blowing than the Copernican revolution centuries previously. One was quantum mechanics, which taught us that on the smallest scales, the Universe is fundamentally stochastic. The second was relativity, which taught us that there is no such thing as absolute time, and that gravity is in fact the very geometry of space.
It's just too bad the two theories don't play nice together.
This course will explore relativity at a level approachable by the motivated upper-level undergraduate physics student. We will spend some time reviewing Special Relativity in depth. We will discuss how gravity can be thought of as geometry. We will explore some of the more interesting astrophysical applications of General Relativity, including black holes and cosmology. We will do our best to wrap our brains around the idea of curved spacetime, and to understand what that really means.
Homework, Tests, and Grading
Your grade in the course will be based on the following:
| Reading Questions | 15% |
|---|---|
| Homework | 40% |
| Midterm | 20% |
| Final | 25% |
- Reading Questions
On days when there is a reading assignment due, there will be a set of two to four short questions on the reading. The purpose of this is threefold. First, it is to get you to actually do the reading.... Second, it is to get you thinking about the material. Third, it is so I can see what parts of the reading you are most comfortable which, and which you are having the most trouble with, so I know what needs to be most directly addressed in lecture.
These reading questions will be answered online, and will be due by 10:00 AM on the day of class. (The computer will no longer accept answers after 10:00 AM on the day of class.)
- Homework
Every other week I will give you a homework assignment. Some problems may be marked "solo" problems. Those problems you must do on your own, without consulting anybody other than the professor. (You may, of course, use your notes, books, and any other references.) You are allowed and encouraged to work with other students on any problem that is not a "solo" problem. However, make sure that the final solution you turn in is your own, not a copy of somebody else's which you don't fully understand or couldn't do yourself.
- Midterm
There will be a single in-class midterm given part-way through the semester. I will strive to avoid any problems that require long calculations on the midterm, because you won't have time to complete such problems....
- Final
The final will take one of two forms; I will decide later in the course which form to use. Either I will administer an oral final, or I will give you a take-home open-book final. (In the latter case, the final would be just like an additional homework assignment composed entirely of "solo" problems.) Last time I offered this course, I ended up with the open-book final... although if memory serves, most students ended up taking the final during the scheduled exam time with me watching them.
Course Schedule
Under Construction; everything below is wrong.
Subject to change! This course schedule, including all reading assignments, may be updated later in the semester. Be sure to look at current versions of the syllabus on the web for the most up-to-date course schedule.
Reading assignments should be completed before class on the day that they are assigned. When a reading assignment is listed for a given day, there will be Reading Questions due by 10:00AM that day. Except where noted, all reading assignments are in Hartle.
When doing readings from Hartle, also read the inset "boxes".
| Date | Topic | Assignment |
|---|---|---|
| 01/14 | Flatland1 | — |
| 01/16 | Flatland1 | Chap. 1 (p. 3-12) |
| 01/17 | Overview; Geometry and Curvature | Read Flatland |
| 01/19 | Wonky Geometry | Chap. 2 (p. 13-29) |
| 01/21 | Group Problems | — |
| 01/24 | Physics 121 | Chap. 3.1-3.3 (p. 31-41) |
| 01/26 | Physics 227 | Chap. 3.4-3.5 (p. 41-45) |
| 01/28 | Group Problems | Homework #1 Due |
| 01/31 | Spacetime Diagrams | Chap. 4.1-4.3 (p. 47-60) |
| 02/02 | Spacetime Diagrams | Chap. 4.4-4.6 (p. 60-73) |
| 02/04 | Group Problems | — |
| 02/07 | Fun with Four-Vectors | Chap. 5.1-5.3 (p. 77-89) |
| 02/09 | Physics 227 at High Speed | Chap. 5.4-5.6 (p. 89-99) |
| 02/11 | Catchup and/or Group Problems | — |
| 02/14 | Comprehension-Independent Notation in Special Relativity |
Chapters 4 and 5 Homework #2 Due |
| 02/16 | Getting Down and Funky with Special Relativity |
Chapters 4 and 5 |
| 02/18 | Group Problems and/or Catchup | — |
| 02/21 | The Principle of Equivalence | Chap. 6-6.5 (p. 107-126) |
| 02/23 | Physics 121, Warped | Chap. 6.6 (p. 126-131) |
| 02/25 | Group Problems | — |
| 02/28 | Index Gymnastics | Chap. 7-7.6 (p. 135-148) Homework #3 Due |
| 03/02 | Boldly Going with Vectors | Chap. 7.7-7.9 (p. 148-163) |
| 03/04 | Group Problems | — |
| 03/07 | Spring Break | eat |
| 03/09 | Spring Break | drink |
| 03/11 | Spring Break | be merry (or pippin) |
| 03/14 | The Shortest Distance Between Two Points (goes through Chicago O'Hare) |
Chap. 8-8.1 (p. 169-174) |
| 03/16 | Mirror, Mirror | Chap. 8.2-8.3 (p. 175-183) |
| 03/18 | Catchup and/or Group Problems | — |
| 03/21 | The Schwarzchild Metric | Chap. 9-9.3 (p. 186-204) Homework #4 Due |
| 03/23 | Orbiting a Star | Chap. 10 (p. 219-232); focus on Sec. 10.4 |
| 03/25 | Group Problems | — |
| 03/28 | Black Holes | Chap 12-12.2 (p. 255-269) |
| 03/30 | Black Holes | Chap 12.3-12.4 (p. 269-276) |
| 04/01 | Catchup and/or Group Problems... Really! | — |
| 04/04 | Class Cancelled | |
| 04/06 | Class Cancelled | |
| 04/08 | Class Cancelled | |
| 04/11 | Astronomy 175 | Chap. 17 (p. 347-364) |
| 04/13 | The FRW Metric | Chap. 18-18.3 (p. 366-376) |
| 04/15 | Cosmology | Chap. 18.4-18.7 (p. 376-395) |
| 04/16 | Saturday, 2-5PM Blowing up the Universe |
Chap 18 |
| 04/18 | Real Black Holes | Chap. 14 (p. 296-308) |
| 04/20 | Active Galactic Nuclei | Chap. 15 (p. 310-328) |
| 04/22 | Group Problems | — |
| 04/25 | Culture: The Stress-Energy Tensor, Curvature, and the Einstein Field Equation |
Chap. 20-24 (not really)2 Homework #5 Due |
Notes
I will be at the American Astronomical Society in San Diego the first two days of class. Read Flatland and answer the reading questions for both Friday and Monday; class will not meet these first two days. If you don't have a password to access the reading questions, please E-mail me before Thursday. The first regular meeting of the class will be Monday, January 17.
To make up for the missed two classes, we will schedule at some point during the semester a "pizza and movie" night at which we will watch The Elegant Universe (the Nova special), which has some great stuff for visualizing GR as well as presenting the problem of quantum gravity.
There is a reason why GR is sometimes taught as a two-semester graduate course. We've dug into some of the fun stuff that comes out of GR in this course, but have only scratched the surface of the full formalism. (And you thought the math was hard already!) There are a few things that are core to the full description of GR that we haven't talked about; take a GR class in grad school to get into the full depth. On the last day of class, I'll tell you a little about them so you know what they are. You do not really need to do this reading assignment, but if you want to learn more, Part III of the textbook is not a bad place to start! (Indeed, it's what Physics 360a used last time it was taught at Vanderbilt.) This stuff won't be on the final.