Are tangent vectors contravariant or covariant?

Correct but annoying answer I: Neither! They simply are.

Correct but annoying answer II: Both! It depends on the person you ask.

More to come on this once I find the time to write up stuff in LaTeX.

UPDATE:
No, I haven’t found time to write things up in LaTeX yet. But I did come up with a great analogy to explain why CBAA-II says, “it depends.”

The core idea underlying the difference between the two is that of active transformations versus passive transformations. What that means is something I’ll get into later. For now, just the analogy:

If I’m wearing jeans that are too short for my legs, there are two ways to understand why this might be the case (other than the possibility that I have very poor taste in clothing, of course):

1. I may have had a growth spurt overnight. (This would be the active transformation as far as my legs are concerned. It would also be the passive transformation as far as the jeans are concerned.)
2. My jeans might have shrunk after going through the washer-dryer. (This would be the passive transformation, as far as my legs are concerned. It would also be the active transformation as far as the jeans are concerned.)

My legs thus covary with my height (assuming I grow evenly, of course), and contravary with my jeans’ length. Are my legs covariant or contravariant? It depends on whether the transformation is active or passive!

Strings, branes, and all that

 I came across a few articles today on Plus, an online mathematics magazine, which presented some of the basics of theoretical physics—string theory and M-branes in particular—in a simple, accessible way. If you’re interested, a good way to do this would be to start “From Newton to Einstein and beyond” and to try “Tying it all up” before you “Meet the mother theory”. 


Of course, the hard math underlying all these theories really is essential if you want to grok this stuff. I would recommend Sir Roger Penrose’s The Road to Reality, about two-thirds of which is spent in actually explaining the math behind such things as general relativity. Stephen Hawking wrote in A Brief History of Time that his editor warned him that every equation he put in would scare away half of his readers. By this metric, Sir Roger’s editor was probably content to have an audience of five. Including himself. (I’ve worked through less than a third of the book so far!) The Road to Reality tries to cover, quite literally, everything. Reading it will do to your mind what diving into a P90X training program will do to an obese Indian bureaucrat’s body.