Lorentz Transformations from ReflectionsSome Applications.pdf

Lorentz Transformations from Reflections: Some Applications 2 0 0 H.K. Urbantke 2 c Institut f¨ur Theoretische Physik, Universit¨at Wien e D Boltzmanngasse 5, A-1090 Vienna, Austria 2 3 December 2002 1 1 v 8 Abstract 3 0 We point out, by exhibiting two examples and mentioning a third one, 2 that it is sometimes useful to consider Lorentz transformations as gener- 1 2 ated from hyperplane or line reflections. One example concerns the con- 0 struction of boosts linking two given 4-vectors, the other one concerns the / h Minkowski geometric understanding of V. Moretti’s polar decomposition p of orthochronous Lorentz matrices. - h t a m : v i X r a 0 1 Introduction 1 The Cartan-Dieudonn´e theorem , to the effect that the orthogonal groups O(n,C), O(p, q ) are generated by hyperplane reflections—each group element being a prod- uct of at most n = p +q of them—is usually invoked only to study the structure of those groups in general and to establish their relation to Clifford algebras. How- ever, the possibility of generating orthogonal transformations from reflections may sometimes be of some practical value in solving simple concrete problems involving orthogonal transformations. In this note we want to illustrate this point by discussing two problems involving Lorentz boosts from a geometrical rather than matrix calculational point of view. One concerns a geometric understand- ing of the construction of the boost matrix that carries two