  The dual problem of optimal transportation in Lorentz-Finsler geometry is
studied. It is shown that in general no solution exists even in the presence of
an optimal coupling. Under natural assumptions dual solutions are established.
It is further shown that the existence of a dual solution implies that the
optimal transport is timelike on a set of full measure. In the second part the
persistence of absolute continuity along an optimal transportation under
obvious assumptions is proven and a solution to the relativistic Monge problem
is provided.
