From SU(1,1) to Lorentz

Space-time variables X,Y,T

Disk variables: z=x+iy

From X,Y,T to x,y

The SU(1,1) matrix

A=

with λ and μ split into real and imaginary parts λ=λr + i λi, μ=μr + i μi, λb=λr - i λi, μb=μr - i μi

From z=x+iy to z1=x1+iy1 using fractional linear transformation

From the transformed disk variables x1,y1 to new space-time variables X1,Y1,T1

To get the final result we have to take into account the fact that A is in SU(1,1) that is |λ|^2-|μ|^2=1, and also the fact that X,Y,T is on hyperboloid T^2-X^2-Y^2=1.

We have obtained linear transformation> We now read the transformation matrix M

Here is the matrix M in matrix form

We verify that is has determinant 1

We define Minkowski space-time metric G

We need to check that M is Lorentz matrix, that is that MGM^T=G. This is tricky because we have to take into account the property . One way of doing it is below

Now we can define the map L from SU(1,1) to SO(1,2)

We calculate the resulting Lorentz matrices for three one-parameter subgroups of SU(1,1)