Lorentz transformations in terms of initial and final vectors

Abstract

Given arbitrary initial vector(s) and their final vector(s) in a Lorentz transformation, the problem is to determine the operator of the transformation. The solution presented here consists of expressing the tensor contained in the Lorentz operator in terms of the given vectors and then the operator itself follows by the exponentiation of a generating matrix. This is possible in certain special cases. In the general case a number of simultaneous nonlinear equations have to be solved. The analytical solution of these equations is elusive while attempts at numerical solution indicate that supplementary information is required. The corresponding procedure for a singular operator is also presented.