One of the things I thought I was going to do in 2012 was learn about relativity. For why see https://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/. Also my cousin’s new husband wrote a paper on a new way of looking at it. I’ve been putting him off as I thought I should know the old way first.
I knew that general relativity involved lots of math such as manifolds and the curvature of space-time. So rather than read verbal explanations, I thought I’d learn the math first. I started reading John M. Lee’s two books on manifolds. The first involves topological manifolds, the second involves manifolds with extra structure (smoothness) permitting calculus to be done on them. Distance is not a topological concept, but is absolutely required for calculus — that’s what the smoothness is about.
I started with “Introduction to Topological Manifolds” (2nd. Edition) by John M. Lee. I’ve got about 34 pages of notes on the first 95 pages (25% of the text), and made a list of the definitions I thought worth writing down — there are 170 of them. Eventually I got through a third of its 380 pages of text. I thought that might be enough to help me read his second book “Introduction to Smooth Manifolds” but I only got through 100 of its 600 pages before I could see that I really needed to go back and completely go through the first book.
This seemed endless, and would probably take 2 more years. This shouldn’t be taken as a criticism of Lee — his writing is clear as a bell. One of the few criticisms of his books is that they are so clear, you think you understand what you are reading when you don’t.
So what to do? A prof at one of the local colleges, James J. Callahan, wrote a book called “The Geometry of Spacetime” which concerns special and general relativity. I asked if I could audit the course on it he’d been teaching there for decades. Unfortunately he said “been there, done that” and had no plans ever to teach the course again.
Well, for the last month or so, I’ve been going through his book. It’s excellent, with lots of diagrams and pictures, and wide margins for taking notes. A symbol table would have been helpful, as would answers to the excellent (and fairly difficult) problems.
This also explains why there have been no posts in the past month.
The good news is that the only math you need for special relativity is calculus and linear algebra. Really nothing more. No manifolds. At the end of the first third of the book (about 145 pages) you will have a clear understanding of
l. time dilation — why time slows down for moving objects
2. length contraction — why moving objects shrink
3. why two observers looking at the same event can see it happening at different times.
4. the Michelson Morley experiment — but the explanation of it in the Feynman lectures on physics 15-3, 15-4 is much better
5. The Kludge Lorentz used to make Maxwell’s equations obey the Galilean principle of relativity (e.g. Newton’s first law)
6. How Einstein derived Lorentz’s kludge purely by assuming the velocity of light was constant for all observers, never mind how they were moving relative to each other. Reading how he did it, is like watching a master sculptor at work.
Well, I’ll never get through the rest of Callahan by the end of 2012, but I can see doing it in a few more months. You could conceivably learn linear algebra by reading his book, but it would be tough. I’ve written some fairly simplistic background linear algebra for quantum mechanics posts — you might find them useful. https://luysii.wordpress.com/category/linear-algebra-survival-guide-for-quantum-mechanics/
One of the nicest things was seeing clearly what it means for different matrices to represent the same transformation, and why you should care. I’d seen this many times in linear algebra, but seeing how simple reflection through an arbitrary line through the origin can be when you (1) rotate the line to the x axis by tan(y/x) radians (2) change the y coordinate to – y — by an incredibly simple matrix (3) rotate it back to the original angle .
That’s why any two n x n matrices X and Y represent the same linear transformation if they are related by the invertible matrix Z in the following way X = Z^-1 * Y * Z
Merry Christmas and Happy New Year (none of that Happy Holidays crap for me)