Astronomy 260, Spring 2005
General Relativity
Syllabus
http://brahms.phy.vanderbilt.edu/a260
MWF, 12:10-1:00, Stevenson Center 6105
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.
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) used the same book last year, they went 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. We will be watching the 3-hour Nova program connected to this book later in the term.
- 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.
- 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.
- Gravitation, by Misner, Thorne, and Wheeler.
An imposing black tome, this is one of the standard graduate texts on General Relativity. Has alternatively been called "The Black Death" and "Misery, Torture, and Woe" by various graduate students.
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 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:
| Homework | 50% |
|---|---|
| Final | 20% |
| Group Problems | 15% |
| Reading Questions | 15% |
- 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.
- 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.)
- Group Problems
On most Fridays, I will bring in a sheet of problems which we will do in class. You will be divided into groups of two or three, and will work out the problems together. I will wander about the class answering questions and giving hints where they are needed.
- 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 9:00 AM on the day of class. (The computer will no longer accept answers after 9:00 AM on the day of class.)
Course Schedule
Updated 2005/02/09: Added another week of discussion of chapters 4 and 5 (Special Relativity), so that we can more fully digest that before moving on to all of this stuff in a curved spacetime.
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 9:00AM 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.