On self-dual gravity

We study the Ashtekar-Jacobson-Smolin equations that characterize four-dimensional complex metrics with self-dual Riemann tensor. We find that we can characterize any self-dual metric by a function that satisfies a nonlinear evolution equation, to which the general solution can be found iteratively. This formal solution depends on two arbitrary functions of three coordinates. We study the symmetry algebra of these equations and find that they admit a generalized $W_{\infty }\oplus W_{\infty }$ algebra. We then find the associated conserved quantities which are found to have vanishing Poisson brackets (up to surface terms). We construct explicitly some families of solutions that depend on two free functions of two coordinates, included in which are the multi-center metrics of Gibbons and Hawking. Finally, in an appendix, we show how our formulation of self-dual gravity is equivalent to that of Plebañski.