Without knowing it, during the last three posts (Our first field expedition, Our second field expedition, The Third Expedition) we became hyperbolic. Hyperbolic and conformal. Conformal and relativistic, relativistic and non-Euclidean.
It will take us a while to get accustomed to these terms and, moreover, there is no good systematic way of progressing. So, we will progress in a somewhat haphazard way – that is the natural way.
The three streams that we had seen in three recent notes have one thing in common: they all map the unit disk of the complex plane into itself. Here are again these streams, but now, with the unit disk marked up:
I do like disks. Throwing a disk so that it would fly really far,and in the intended direction, is an art.
The three one-parameter transformation groups and their vector fields that we have met are
Looking at the pictures of flows it is not completely clear that and indeed map the disk into itself. For it is not clear from the picture what happens to at the bottom of the disk. and for it is not clear what happens to at the right. But we can easily calculate from the equations that for all and, similarly, for all
So is the fixed point of and is a fixed point of (The disk origin is a fixed point of ) Yet the fact that and map the unit disk onto itself (and the unit circle onto itself) is not completely evident and needs a proof, that is if one wants to know rather than to believe. We will return to this problem later on, in the future posts. For now there are more urgent questions:
What it has to do with hyperbolic, conformal, relativistic and non-Euclidean? Hyperbolic – we can partly guess: there are hyperbolic functions in the formula. But what they have to do with hyperbolas? Conformal? Our transformations preserve angles, but that is not evident. Relativistic? Once we know why hyperbolic, we will start to understand about being relativistic and non-Euclidean.
To start with, let me tell you right away how I got these formulas. First, I will tell it without explaining and/or elaborating. I will explain the details and steps later on.
I do like the groups SU(2,2) and its baby version SU(1,1). Here I am using SU(1,1) – the special unitary group for signature . This is the group of all complex matrices of determinant one, that have the property that
and star is Hermitian conjugated matrix to
If is a path in SU(1,1) through identity, then, by differentiating the defining equation
then, from the condition it follows that that is Then for we get
and from Eq. (10) we get and Thus is of the form
where is real and is complex. This is the Lie algebra of the group SU(1,1). I first take its generator
I exponentiate (using, for instance, Mathematica, but by hand it is also easy, once you notice that :
My prescription for acting with complex matrices on complex numbers is via linear fractional transformations. Thus if is of the form
then it acts on complex numbers as
That is how I arrived at Doing the same for I got , and with I obtained
That was a short summary. With time it will all become clear. Little by little it will also becoming clear what is, or can be, the relation of these
games with matrices to spinning tops and Dzhanibekov’s T-handles. In short: I am interested in relativistic T-handles. That is when not only space but also time are set in motion. It is kinda futuristic, but that is why it is fun to do it.