On rotations in a pseudo-Euclidean space and proper Lorentz transformations

Abstract

It is shown that in a general pseudo-Euclidean space Epn, 2-flats (planes) passing through the origin of the coordinate system may be classified into six invariant types and explicit formulas for ’’planar rotations’’ in these flats are obtained. In the physically important case of the Minkowski World E34, planar rotations are characterized as rotationlike, boostlike, and singular transformations and an invariant classification of proper Lorentz transformations into these types is given. It is shown that a general nonsingular proper Lorentz transformation may be resolved as a commuting product of two transformations one of which is rotationlike and the other boostlike while a singular transformation may be written as a product of two rotationlike transformations, each with a rotation angle π. Such a rotationlike transformation with angle π is called ’’exceptional’’ following Weyl’s terminology for similar transformations of SO(3). In all cases, explicit formulas for the angles and planes of rotations in terms of the elements of a given Lorentz matrix are obtained and the procedure yields in a natural manner an explicit formula for the image of L in the D10(D01) representation of SO(3,1) which in turn leads to two more classification schemes in terms of the character χ of L in the D10(D01) and the D (1)/(2) 0(D0 (1)/(2) ) representations.