The relationship between the proper, orthochronous Lorentz group, the special linear group of 2×2 matrices with complex entries and unit determinant, and the group of fractional linear transformations of the extended complex plane is explicitly described with examples. The existence of non-singular and singular Lorentz transformations is explained as are infinitesimal Lorentz transformations and their close similarity to electromagnetic fields. This chapter demonstrates how a natural generalization of Lorentz transformations leads to a construction of an impulsive gravitational wave.
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