The unit disk in the complex plane, together with geometry defined by invariants of fractional linear SU(1,1) action, known as the Poincaré disk, that is the arena of hyperbolic geometry. But why “hyperbolic”? It is time for us to learn, and to use. In principle the answer is given in Wikipedia, under the subject “Poincaré disk model”. There we find the following picture
We want to derive these formulas ourselves. Let us first introduce our private notation. The hyperboloid will live in a three-dimensional space with coordinates This is the space-time of Special Relativity Theory, but in a baby version, with coordinate suppressed.
The light cone in our space-time has the equation Of course we assume the constant speed of light Inside the future light cone (the part with ) there is the hyperboloid defined by The coordinates of events on this hyperboloid satisfy the equation
As can be seen from the picture above, every straight line passing through the point with coordinates and a point with coordinates on the hyperboloid, intersects the unit disk at the plane at a point with coordinates . We want to find the relation between and
Given any two points, , the points on the line joining and have coordinates parametrized by a real parameter as follows:
For we are at for we are at , and for other values of we are somewhere on the joining line. Our has coordinates our has coordinates on the hyperboloid, and we are seeking the middle point with coordinate So we need to solve equations
From the last equation we find immediately that and the first two equations give us
We need to find the inverse transformation. First we notice that
Therefore and so
Using Eqs. (4) we now finally get
Thus we have derived the formulas used in Wikipedia. Wikipedia mentions also that the straight lines on the disk, that we were discussing in a couple of recent posts, are projections of sections of the hyperboloid by planes. We will not need this in the future. But we will use the derived formulas for obtaining the relation between SU(1,1) matrices and special Lorentz transformations of space-time events coordinates. This is the job for the devil of the algebra!