The Einstein constraints and differential forms

TL;DR

This paper reformulates the vacuum Einstein constraints using differential forms, demonstrating that for real-analytic metrics, these constraints can be expressed as a local system of first-order PDEs through a special coframe choice.

Contribution

It introduces a novel formulation of Einstein constraints with differential forms and shows their reduction to first-order PDEs for real-analytic metrics.

Findings

Constraints can be expressed as first-order PDEs for real-analytic metrics.

A special coframe reduces second-order terms in the scalar constraint.

The formulation provides a new perspective on Einstein constraints.

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 the coframe, which reduces to zero all second order terms in the scalar constraint.