Conformal gravity, the Einstein equations and spaces of complex null geodesics

The aim of this work is to give a twistorial characterisation of the field equations of conformal gravity and of Einstein spacetimes. The authors provide strong evidence for a particularly concise characterisation of these equations in terms of 'formal neighbourhoods' of the space of complex null geodesics. They consider second-order perturbations of the metric of complexified Minkowski space. These correspond to certain infinitesimal deformations of its space of complex null geodesics, Pn. Pn has a natural codimension one, embedding into a larger space (the product of twistor space and its dual). They show that deformations extend automatically to the fourth-order embedding (that is, the fourth formal neighbourhood). They extend to the fifth formal neighbourhood if and only if the corresponding perturbation in the metric has vanishing Bach tensor (these are the equations of conformal gravity). Finally, deformations which extend to the sixth formal neighbourhood correspond to perturbations in the metric that are conformally related to ones satisfying the Einstein equations, at least when the Weyl curvature is sufficiently algebraically general. One can attempt to construct such formal neighbourhoods in the fully curved case. They present arguments which suggest that the results will also hold when spacetime is fully curved.