  On spacetimes that are not time orientable we construct a U(1) bundle to
measure the twisting of the time axis. This single assumption, and simple
construction, gives rise to Maxwell's equations of electromagnetism, the
Lorentz force law and the Einstein-Maxwell equations for electromagnetism
coupled to General relativity. The derivations follow the Kaluza Klein theory,
but with the constraints required for connections on a U(1) bundle rather than
five spacetime dimensions. The non time orientability is seen to justify and
constrain Kaluza Klein theories exactly as required to unify gravitation with
electromagnetism. Unlike any other schemes, apparent net electric charges arise
naturally because the direction of the electric field reverses along a time
reversing path. The boundary of a time reversing region can therefore have a
net electric flux and appear exactly as a region containing an electric charge.
The treatment is purely classical, but motivated by links between acausal
structures and quantum theory.
