Fields of Lorentz transformations on space-time

TL;DR

This paper explores the mathematical structure of Lorentz transformation fields on space-time, revealing obstructions to their global definition and connecting these ideas to electromagnetic fields and complex matrix algebra.

Contribution

It introduces a detailed analysis of the singularities and obstructions in Lorentz transformation fields, including the role of null transformations and their algebraic properties.

Findings

Identifies homotopy and differential obstructions to global Lorentz fields.

Connects null transformations to electromagnetic fields of moving charges.

Develops a new basis for 4x4 complex matrices and explores null subspaces.

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.