Connections with the Lorentz Group of Special Relativity
The Lorentz transformations and their origin in physics at the beginning of this century are described. It is shown that there is an isomorphism between the direct isometry group of the hyperbolic plane and the restricted Lorentz group in two space variables and time. A peculiar feature of the Lorentz group is shown to be connected with a property of the hyperbolic isometry group discussed in Section 8.3. We then generalize to one more dimension and show an isomorphism between the direct isometry group of three-dimensional hyperbolic space and the Lorentz group in three space variables and time. We then describe the so-called relativistic velocity space and show that it has the same geometry as the three-dimensional hyperbolic space.
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