Being an undergraduate (I've recently completed my Bachelor's degree and am currently starting with Magister degree studies) in physics I tend to take interest in things where math is concerned, even when it is just a snapshot of a math test I notice during watching anime... A few moments ago I came across this:

The title roughly (my Japanese does not go much further beyond being able to look up kanji and some basic everyday-life phrases) translates to "First year, first semester, midterm exams, math I.", the first problem is something like "Collect similar terms and put them in order in the following expressions." and the fourth problem goes like "Factorize the following expressions." Well, from this I can guess that the math level on first year of 高等学校 (high school, students are around 15 years old) in Japan is roughly the same as in the first year of střední škola (high school, students are as well around 15 years old) in Czech, although I've never seen such answers to these types of math problems... Well, it's from anime and it's quite usual there to go a bit overboard, but nevertheless it gave me a good laugh.

On a similar note, over the past week from time to time I've been trying to find various derivations of Lorentz boosts in general direction in hope to come up with an elegant one. Today I've finally managed to find one that is purely mathematical and based on the assumptions that a) general Lorentz boosts does not contain space coordinate rotation and b) Lorentz boosts in a given direction form a one-dimensional sub-group of the Lorentz group (which is isomorphic to O(1,3)), whose another sub-group is the group of rotations O(3). Lorentz group is a group of (linear) transformations that preserve the Minkowski metric

Limiting only to transformations that do not contain time or space inversions (and thus in matrix representation they have determinant equal to 1 and the "time-time" component greater than 0) I get a continuous subgroup SO(1,3) and can further derive limitations imposed on the matrices forming it's Lie algebra (so(1,3)). It turned out, as expected, that the 4x4 matric has a 3x3 antisymmetric submatrix (the space rotations), has zero trace (because determinant of the group representing matrices is 1) and the remaining elements are symmetric – which also leads to (already known) fact that the dimension of this group is 6 (three space rotations and Lorentz boosts in three mutually orthogonal directions). Note: I use geometrized units in which c=1 and otherwise the remaining elements wouldn't be symmetric, but differed by a factor of c^2.

From there, finding the matrix describing the Lorentz boost in general direction is simply a matter of filling the symmetric elements of the matrix representing the so(1,3) algebra (and leave the other elements 0) and making an exponential out of it. Further one can replace the coefficients describing the direction of the boost and hyperbolic sines and cosines by the usual gamma factor and three-velocity components to get the usual from of the transformation.

I really like this derivation – even though it's rather long compared to other derivations I've found or came across during past week, it's purely mathematical in nature and does not require knowledge of the Lorentz transformation for frames in standard configuration.

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## 7 comments:

Now you know you MUST tell us the title of this anime, Yiu is a common Japanese name so googling for it is not really useful :p

Well, I was hoping you'll guess from the art style ;-) No, really, it's from K-ON!.

Your idea of obtaining that matrix belonging to the proper Lorentz group sounds interesting, perhaps it could be published...And what can you say of the spread of use of free software in your bachelor campus? As students of physics, we are encouraged to get along with the Mathematica-Kile-GFortran combo.

Well, there is one computer lab where we have linux installed on the computers alongside windows. It used to be gentoo but last year they switched to fedora. As for other software, well in the group of people I hang with, the most spread are either Mathematica (mostly quantum physicists) or Maple (usually relativistic physicists) and TeX, while most people are "lazy" and are using LaTeX (not sure if anyone uses some specific frontends like Kile), I use plain (and use gedit to create the sources). Throughout my studies I haven't had to use any nonfree software other than Maple. When we were learning Pascal and C in first year, it was OK to submit works compilable by either free-pascal or gcc, I've used gfortran only once to play with fractals.

It is also interesting to find someone in Planet Fedora who study physics. I find it really amazing!

@arcarlos: Martin is a really cool geek, he is studying advanced physics for Ph.D., working with Free software graphics and learning Japanese. What can you get more than that?

@nicu,arcarlos: while I ultimately aim at PhD in theoretical physics (with specialization in Relativistic Physics), I need to finish the Magister (analogue to NAm Masters) degree first ;-)

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