Fields of Lorentz transformations on space-time

Abstract

Fields of Lorentz transformations on a space--time 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 can be expressed, at least locally, as the exponential $e^F$ where $F$ 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 very 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 very interesting basis for the four by four complex matrices, and we also observe that there are two different kinds of two dimensional complex null subspaces.