Feynman finally gets around to discussing tensors 376 pages into volume II in “The Feynman Lectures on Physics” and a magnificent help it is (to me at least). Tensors must be understood to have a prayer of following the math of General Relativity (a 10 year goal, since meeting classmate Jim Hartle who wrote a book “Gravity” on the subject).
There are so many ways to describe what a tensor is (particularly by mathematicians and physicists) that it isn’t obvious that they are talking about the same thing. I’ve written many posts about tensors, as the best way to learn something it to try to explain it to someone else (a set of links to the posts will be found at the end).
So why is Feynman so helpful to me? After plowing through 370 pages of Callahan’s excellent book we get to something called the ‘energy-momentum tensor’, aka the stress-energy tensor. This floored me as it appeared to have little to do with gravity, talking about flows of energy and momentum. However it is only 5 pages away from the relativistic field equations so it must be understood.
Back in the day, I started reading books about tensors such as the tensor of inertia, the stress tensor etc. These were usually presented as if you knew why they were developed, and just given in a mathematical form which left my intuition about them empty.
Tensors were developed years before Einstein came up with special relativity (1905) or general relativity (1915).
This is where Feynman is so good. He starts with the problem of electrical polarizability (which is familiar if you’ve plowed this far through volume II) and shows exactly why a tensor is needed to describe it, e.g. he derives the tensor from known facts about electromagnetism. Then on to the tensor of inertia (another derivation). This allows you to see where all that notation comes from. That’s all very nice, but you are dealing with just matrices. Then on to tensors over 4 vector spaces (a rank 4 tensor) not the same thing as a 4 tensor which is over a 4 dimensional vector space.
Then finally we get to the 4 tensor (a tensor over a 4 dimensional vector space) of electromagnetic momentum. Here are the 16 components of Callahan’s energy momentum tensor, clearly explained. The circle is finally closed.
He briefly goes into the way tensors transform under a change of coordinates, which for many authors is the most important thing about them. So his discussion doesn’t contain the usual blizzard of superscripts and subscript. Covariant and contravariant are blessedly absent. Here the best explanation of how they transform is in Jeevanjee “An introduction to Tensors and Group Theory for Physicists” chapter 3 pp. 51 – 74.
Here are a few of the posts I’ve written about tensors trying to explain them to myself (and hopefully you)
How formal tensor mathematics and the postulates of quantum mechanics give rise to entanglement
Chemiotics II 