Fields of Lorentz transformations on space-time

Abstract

Fields of Lorentz transformations on a space-time M are related to tangent bundle self isometries. In other words, a gauge transformation with respect to the −+++ Minkowski metric on each fibre. Any such isometry L :T(M)→T(M) can be expressed, at least locally, as L= e F where F :T(M)→T(M) is antisymmetric with respect to the metric. We find there is a homotopy obstruction and a differential obstruction for a global F. We completely study the structure of the singularity which is the heart of the differential obstruction and we find it is generated by “null” F which are “orthogonal” to infinitesimal rotations F with specific eigenvalues. We find that the classical electromagnetic field of a moving charged particle is naturally expressed using these ideas. The methods of this paper involve complexifying the F bundle maps which leads to a interesting algebraic situation. We use this not only to state and prove the singularity theorems, but to investigate the interaction of the “generic” and “null” F, and we obtain, as a byproduct of our calculus, a interesting basis for the 4×4 complex matrices, and we also observe that there are two different kinds of two-dimensional complex null subspaces.