Twistor action for general relativity

We reformulate Euclidean general relativity without cosmological constant as an action governing the complex structure of twistor space. Extending Penrose's non-linear graviton construction, we find a correspondence between twistor spaces with partially integrable almost complex structures and four-dimensional space-times with off-shell metrics. Using this, we prove that our twistor action reduces to Plebanski's action for general relativity via the Penrose transform. This should lead to new insights into the geometry of graviton scattering as well as to the derivation of computational tools like gravitational MHV rules.