This World Wide Web page written by
Oberlin College Physics Department;
last updated 24 February 2011.
In graduate school I studied relativity many times and from many different perspectives. There I was introduced to an extremely elegant and beautiful formulation of relativity called the tensor formulation. Don't think that the beauty of this formulation implies that it is flimsy or delicate -- on the contrary it is a robust and powerful tool for problem solving. Seduced by both the beauty and the power of tensor techniques, I gladly spent many hours mastering the tools, until I could astound my classmates (and sometimes my teachers) with my technical skills. I felt thoroughly at home with the tools of tensor analysis, and convinced myself that I had finally grown to understand relativity.
[[I particularly recall a day when one of my fellow grad students, Susan, came to me and said "Have you tried the second problem in the Relativistic Quantum Mechanics problem set? I've worked on it for 18 hours and I've made no progress whatsoever." I said I hadn't worked seriously on the problem, but I had a hunch. I took out a sheet of paper and showed her how to solve the problem in 40 seconds using just three lines of algebra. She looked at me with a face full of awe. No woman, before or after, has ever looked at me that way.]]
After graduate school I was hired by Oberlin College. I successfully taught relativity several times in technical courses, using a style that introduced my beloved tensor methods. Then I began teaching Einstein and Relativity, a course not for physicists but for a general audience of thoughtful and critical people with a general but not a technical interest in relativity. I thought it would be easy. I would just teach in the manner to which I had grown accustom, but work each derivation slowly and in detail with full attention to even the smallest mathematical step.
This was a disaster. It was not that my students couldn't follow the mathematics; it was that they didn't find the mathematics to be convincing. Once a student asked in class why a derivation worked, so I worked through the derivation again, but more slowly. He replied that he had followed the mathematics the first time, but instead of a convincing argument, it was ``just a bunch of mathematical gobblygook." My heart was sore at this insult to my beloved tensor methods, but I had to admit that my questioner was right. The tensor methods are indeed powerful, but they are powerful at answering the kinds of questions that physicists tend to ask. I was being stumped in class by simple questions -- reasonable questions -- that didn't happen to lend themselves to the tensor formulation. I grew to realize that the tensor methods, lovely and powerful though they still were, were divorced from the physical word in a sort of otherworldly mathematical beauty.
So I went back to the beginning and worked through my entire understanding of relativity again, from a more concrete and less ethereal point of view. (I also rethought my understanding of what it means to say "I understand".) David Mermin's book Space and Time in Special Relativity was a great help. (I had known about this book for many years, but had always denigrated it as unsophisticated because it used so few equations. Now I think of my former equation-ladened approach as being mechanical and unsophisticated.) Also very helpful were questions and objections from my students, and the discipline of sitting down and typing up lecture notes. One year I had a blind student in my course, and the time I spend refining and clarifying my arguments for his benefit refined and clarified them for me and for you as well.
I went down a lot of blind alleys as I rebuilt my understanding, and I'm not finished yet. Hopefully I never will be. I am no longer confident, which means that I'm open to correction and improvement.
I am still fond of tensor methods and I use them whenever I have a suitable problem, but I am not bound to them. I do not confuse my adeptness at manipulating the equations with true understanding. I am no longer comfortable with relativity, but at least I do not delude myself that I am comfortable. Every time I teach Einstein and Relativity I find new holes in my knowledge. That's why it's so much fun to teach.