From SU(1,1) to Lorentz
Space-time variables X,Y,T
Disk variables: z=x+iy

From X,Y,T to x,y

hyperboliatodisk3f_1.gif

hyperboliatodisk3f_2.png

The SU(1,1) matrix
A=hyperboliatodisk3f_3.png
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

hyperboliatodisk3f_4.gif

From z=x+iy to z1=x1+iy1 using fractional linear transformation
hyperboliatodisk3f_5.png

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

hyperboliatodisk3f_7.gif

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.

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

hyperboliatodisk3f_14.gif

Here is the matrix M in matrix form

hyperboliatodisk3f_15.png

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We verify that is has determinant 1

hyperboliatodisk3f_17.png

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We define Minkowski space-time metric G

hyperboliatodisk3f_19.png

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 hyperboliatodisk3f_20.png. One way of doing it is below

hyperboliatodisk3f_21.png

hyperboliatodisk3f_22.png

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

hyperboliatodisk3f_31.png

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

hyperboliatodisk3f_32.gif

hyperboliatodisk3f_33.png

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hyperboliatodisk3f_38.png

hyperboliatodisk3f_39.png

hyperboliatodisk3f_40.png

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Created with the Wolfram Language