  The solutions of Einstein's equations admitting one non-null Killing vector
field are best studied with the projection formalism of Geroch. When the
Killing vector is lightlike, the projection onto the orbit space still exists
and one expects a covariant theory with degenerate contravariant metric to
appear, its geometry is presented here. Despite the complications of
indecomposable representations of the local Euclidean subgroup, one obtains an
absolute time and a canonical, Galilean and so-called Newtonian, torsionless
connection. The quasi-Maxwell field (Kaluza Klein one-form) that appears in the
dimensional reduction is a non-separable part of this affine connection, in
contrast to the reduction with a non-null Killing vector. One may define the
Kaluza Klein scalar (dilaton) together with the absolute time coordinate after
having imposed one of the equations of motion in order to prevent the emergence
of torsion. We present a detailed analysis of the dimensional reduction using
moving frames, we derive the complete equations of motion and propose an action
whose variation gives rise to all but one of them. Hidden symmetries are shown
to act on the space of solutions.
