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.