Connection between the self-dual Yang-Mills and general relativity einstein equations

Abstract

We give a manifestly gauge invariant formulation of the self-dual gravitational field analogous to that given by Yang for the self-dual Yang-Mills field. This formulation connects in a direct and explicit way the self-dual Yang-Mills and general relativity equations. We give the “ R gauge” parametrisation of the self-dual gravitational field which corresponds to modified Yang's type and Ernst equations and analyze the correspondence between their respective types of solutions. No assumption about the existence of symmetries in the space-time is needed.