Robert H. Gowdy:   Web-page    e-mail

This course introduces the language of differential geometry that is needed to understand the current theories of gravitation. A thorough understanding of calculus is all that is really needed to follow this introduction. Special relativity, mechanics, and electromagnetism are used to show how this language can be used in physics. Similarly, the basic mathematical ideas of mappings, abstract algebra, and topology are used throughout. The notes are self-contained in these areas but it is helpful to have encountered these subjects in previous courses.

The material of this course straddles the border between mathematics and theoretical physics. Thus, I have attempted to make the course equally accessible to graduate students and advanced undergraduates majoring in both areas.

The course will be taught primarily from these web-notes and from the internet resources that are linked to these notes.

The basic topics that must be covered in an introductory course on gravitation theory have been firmly fixed ever since Einstein introduced his Field Equations of General Relativity in 1915 and a German soldier (and already well-known astrophysicist) named Karl Schwarzschild produced an elegant exact solution of those equations in 1916 (four months before his death on the Russian Front). The Field Equations are stated in the language of differential geometry, so we will spend much of the semester learning that language. As soon as we obtain the Field Equations, we will explore the solution that Schwarzschild found.

Schwarzschild's solution of the field equations describes the exterior gravitational field of any non-rotating spherical object. It is the foundation for most experimental tests of general relativity. The extension of the solution that is needed to follow the collapse of a star is usually called a "black hole" and is generally regarded as the most remarkable prediction of Einstein's theory of gravitation. We will study this solution in detail.

Since 1915, the tools of differential geometry have gradually spread to the rest of physics and have become the basic language of field theory. I will touch on just a few of these applications here.

General Relativity provides a window onto the global structure and history of space and time. Thus, we will not pass up discussing the cosmological solutions of Einstein's equations.

A number of extremely important topics will treated only briefly in this one-semester course. Many of these topics concern the nature of Einstein's equations as a dynamical system. The Initial Value Problem asks what quantities can be specified freely at a given time in order to determine the evolution of a spacetime. Gravitational waves are the dynamical degrees of freedom of the gravitational field. A variety of ways to simulate the evolution of a spacetime numerically on a computer are being explored. Alternative theories of Gravitation are extremely important since it is generally believed that Einstein's theory is not the final answer. Most of these topics must be left to a second semester course.

Homework will be distributed and collected as LaTeX files on the web. You will need either Scientific Notebook (SN) or Scientific Workplace (SW) to participate easily since the files are 'wrapped' using the protocol of these programs. The Homework Page presents a table with assignments and solutions. To do a homework assignment for credit, click on an assignment, download the file, and open it using SN or SW, and do the exercises. You will have the full power of a symbolic algebra program to perform or (more often) verify your calculations. When you are satisfied with your work, save it and e-mail the wrapped file to me. I will then insert grades and comments on your work, wrap the file, and e-mail it to you as an attachment.

Once a solution to an assignment has been posted, no credit can be given for that assignment. Each assignment has a due-date listed. The solution will be posted some time after the due-date. Please note that these assignments are short and are intended to be done immediately. I intend to have an assignment due at each class meeting time.

Each homework assignment will receive a grade between 0 and 100 points. The average of the best 75% of the assignments will become your homework score. Although you can skip an occasional assignment without direct penalty, you really need to work all of the homework problems in order to follow the material of this course.

Exams will be given after natural break-points in the material and a final exam will be given at the end of the course. All exams, including the final, will be distributed and collected as LaTeX files on the web. The Exam page presents a table with exams and solutions. The posting of a new exam will be announced on the course home page as "Exam in Progress". To take an exam, click on the latest exam link to download the file, open it using SN or SW and work the problems. Wrap the file, and e-mail it to me. I will then insert grades and comments and e-mail the resulting file back to you as an attachment.

Once a solution to an exam has been posted, no credit can be given for that exam.

Each exam will receive a grade between 0 and 100 points. Your exam average, with the final exam counted twice, becomes your exam score.

The scheduled meeting time for this course is Monday and Wednesday from 3:30pm to 4:45pm in room 2306 at 701 W. Grace St. Because the on-campus class is quite small, we may agree to change the time and place. Any such change will be announced on this web-page and in the course forum.

The class meetings provide motivation and communication. This course is set up so that you can do the whole thing on-line and never come to class. However, the course is not self-paced, so you have to keep up with it. You will find that attending class makes it much easier to realize when you are falling behind.

Because the class is small, I will assume that all on-campus students will attend lectures unless I hear otherwise. If, for any reason, you can't make it, I would appreciate an e-mail so that I do not end up preparing a lecture for an empty room.

One form of attendance does count for course credit, namely participation in the course forum. Questions, comments, and answers to other student's questions all count the same. Each contribution counts one point up to a maximum of 100. The total number of forum contributions becomes your class participation score.

Your grade will be determined from two final scores:

The first approximation to your final grade will be determined by the average of these two scores, but I will take both scores into account.

Note: Please contact the appropriate Coordinator of Services for Students with Disabilities to obtain an official memo detailing the academic adjustments or Accommodations which you need.