So we're down to two more assignments, period, for the semester, and one of them is due today. We're doing relativity in E&M, and it's been a real treat. For the first time, I feel like I'm getting a really solid foundation in this most beautiful of subjects, and the index gymnastics of Minkowski space tensor analysis is even starting to feel natural to me!

The first problem was to find the general Lorentz transformation for constant velocity in an arbitrary direction. I knew it had to reduce to the simple (quasi)block-diagonal form for velocity along one of the axes, and I was hoping for some kind of beautiful symmetry. My plan of attack hinged on realizing that regular rotations are Lorentz transformations, too!

So the idea is this: if the velocity vector makes a polar angle theta with the z-axis, and if its projection on the xy-plane makes an azimuthal angle phi with the x-axis, then we do a rotation about the z-axis by an angle phi, then rotate about the new y-axis by an angle theta. (The point of this is to line up the z-axis with the velocity vector.) Now we know how to do a Lorentz transformation along one of the axes; it's really easy! So we do that along our new z-axis, and then we make the inverse of the (composite) rotation we did to align our z-axis (which puts us back into the old coordinates).

Any symmetry I could have hoped for, I got. The first thing that jumped out at me was that when all was said and done, there was not the slightest clue in the final matrix that it was the z-axis I had chosen to align. (This is necessary for it to be correct, of course, but still very aesthetically pleasing!) Even better was when I subtracted off the identity matrix: the space-space components all had the same factor of (gamma - 1), and they all paired up perfectly: the xy-component had the product of the x- and y-projections of the velocity, same for the yz-component, the xx-component, etc. Simply beautiful!

The second problem was a more practical one: given that 2/3 of charged pions are observed to survive a distance 30 metres from a collision site (in a particle accelerator), and given the half-life of pions in their own rest frame, what is the energy of the pions in the collision? It wasn't particularly difficult, but it was a fun problem, particularly because if you use lab time instead of proper time, you'll come to the (erroneous) conclusion that they're zipping along faster than light. :)

The third problem was the hardest to do, but the easiest to explain to a general audience. Essentially, it's a more practical realization of the famous twin paradox. Usually, the twin paradox goes something like this: suppose of two twins on Earth, one leaves on a spaceship and travels near the speed of light for 20-odd years, and returns to Earth. How much time will have passed on Earth during that journey? (Answer: a lot more!) But if any reference frame is as good as any other, then the twin on the spaceship can surely claim to have been stationary while the Earth moved far away and then back, right? Well, the difference lies in acceleration: in order to get to the speed of light in the first place, the spaceship must accelerate, which is an effect that can be detected on the ship but not on Earth. Remember, not all reference frames are created equal, just all inertial reference frames.

But that's still a bit sketchy, becuase when the spaceship turns around, going from near the speed of light one way to near the speed of light the other way is a huge amount of acceleration, and our spacefaring twin has much bigger worries than a now-long-dead sibling. Jackson poses a more satisfying question: let the spaceship accelerate instead constantly, for 5 years, at a rate equal to the gravitational acceleration near the Earth's surface. (This will make the ride much more comfortable!) For the next 5 years it decelerates at the same rate, then turns around and does the same thing. 20 years total, from the spaceship's reference frame. How many years have passed on Earth in the intervening time?

Three hundred thirty-eight.

That's right, 338 years pass on Earth during that spaceship's trip. And believe me, that answer was not easy (for me) to come by! Consider this: the acceleration is constant in the spaceship's own reference frame. But it's speed in its own reference frame -- i.e., its speed with respect to itself -- is always zero by definition! Trying to figure out how it can be always increasing yet always zero is just one conceptual hurdle that had to be cleared. Really, all the difficulties stem from the fact that the velocity isn't constant: acceleration complicates things in special relativity.

I feel like I'm learning a lot in this class, and this semester overall. Now, time for exams and then Christmas break! (When I'll finally be able to respond to emails again... :P)