In this note we give two new simple derivations of the “good-cut” equation, the equation which governs (complex) self-dual asymptotically flat gravitational fields. One of these derivations is remarkably simple, involving only a few lines. Our main point of interest, however, is in the second derivation. Though it is slightly more complicated, this method of derivation is almost certainly generalizable to cover real asymptotically flat space-times and thus lead to a generalization of the good-cut equation.

A detailed outline is presented of several convergent points of view connecting the self-dual and anti-self-dual fields with their free data. This is done for the Maxwell and for linearized gravity as exemplifying the approaches. The Sparling equation provides one tool of great power and characterizes one approach. The twistor theory of Penrose yields another equally powerful point of view. The links between these two basic approaches given in this paper provide a unification that allows workers and others with interest (...) in this area to proceed more readily toward the goal of understanding the full nonlinear Einstein equations. (shrink)