I will describe joint work with Kirill Krasnov and Michael Singer. The Einstein equations for a Riemannian metric are elliptic (modulo diffeomorphisms) and index zero. This means that naively one might hope that solutions are isolated, or “locally rigid”. I will explain a curvature condition in dimension 4 which ensures that this is the case. For context, note that Koiso proved that an Einstein metric of negative curvature is locally rigid (in dimensions at least 3). Our theorem is a chiral version of Koiso’s with roughly “one half” the hypotheses. The proof is completely different to Koiso’s. It uses a new variational characterisation of Einstein metrics as critical points of something called the “pure connection action” S, which makes sense only in dimension 4. We prove that, when our chiral curvature condition is satisfied, the Hessian of S at a critical point is strictly positive (modulo gauge). It follows that the critical points are isolated as claimed.