Dual Foliation Formulations of General Relativity

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.