The Einstein constraints and differential forms

Abstract

We express the vacuum Einstein constraints in terms of differential forms—the forms include one-forms constituting an orthonormal coframe of the spatial metric. We show that if the metric is real-analytic, then the constraints can be always expressed locally as a system of first order PDE’s—this system is obtained by a special choice of coframe, which reduces to zero all second order terms in the scalar constraint. We also present a general formula for coframes, which allow for this simplification of the constraint and give it additionally a certain symmetric form.