The problem of classifying line fields or, equivalently, Lorentz metrics up to
homotopy is studied. Complete solutions are obtained in many cases, e.g. for all closed
smooth manifolds N, orientable or not, of dimension n ≡ 0(4) and, in particular, in
the classical space-time dimension 4.
Our approach is based on the singularity method which allows us to classify the
monomorphisms u from a given (abstract) line bundle α over N into the tangent
bundle. The analysis of the transition to the image line field u(α) then centers around
the notion of ‘antipodality’.
We express our classification results in terms of standard (co-)homology and
characteristic classes. Moreover, we illustrate them for large families of concrete sample
manifolds by explicit bijections or by calculating the number of line fields.