  A dual foliation treatment of General Relativity is presented. The basic idea
of the construction is to consider two foliations of a spacetime by spacelike
hypersurfaces and relate the two geometries. The treatment is expected to be
useful in various situations, and in particular whenever one would like to
compare objects represented in different coordinates. Potential examples
include the construction of initial data and the study of trapped tubes. It is
common for studies in mathematical relativity to employ a double-null gauge. In
such studies local well-posedness is treated by referring back, for example, to
the generalized harmonic formulation, global properties of solutions being
treated in a separate formalism. As a first application of the dual foliation
formulation we find that one can in fact obtain local well-posedness in the
double-null coordinates directly, which should allow their use in numerical
relativity with standard methods. With due care it is expected that practically
any coordinates can be used with this approach.
