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The Einstein constraints and differential forms

Andrzej Okolow, Jakub Szymankiewicz

TL;DR

This work recasts the vacuum Einstein constraints in terms of differential forms within the Teleparallel Equivalent of General Relativity (TEGR), replacing the spatial metric by an orthonormal coframe $\theta^I$ and the extrinsic curvature by momentum forms $r_I$. In the convenient gauge $\xi^I=0$, the TEGR constraints are shown to be equivalent to the standard Einstein constraints for the spatial metric $q$ and extrinsic curvature $K$, establishing a full equivalence between the formulations. For real-analytic spatial metrics, Bryant's theorem guarantees a local coclosed coframe, which eliminates second-order terms in the scalar constraint and reduces the system to a first-order PDE set for $(\theta^I,r_I)$. This teleparallel perspective offers a new avenue for constructing exact solutions and alternative initial-value formulations, with future work addressing extensions to smooth metrics and broader gauge choices.

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.

The Einstein constraints and differential forms

TL;DR

This work recasts the vacuum Einstein constraints in terms of differential forms within the Teleparallel Equivalent of General Relativity (TEGR), replacing the spatial metric by an orthonormal coframe and the extrinsic curvature by momentum forms . In the convenient gauge , the TEGR constraints are shown to be equivalent to the standard Einstein constraints for the spatial metric and extrinsic curvature , establishing a full equivalence between the formulations. For real-analytic spatial metrics, Bryant's theorem guarantees a local coclosed coframe, which eliminates second-order terms in the scalar constraint and reduces the system to a first-order PDE set for . This teleparallel perspective offers a new avenue for constructing exact solutions and alternative initial-value formulations, with future work addressing extensions to smooth metrics and broader gauge choices.

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.

Paper Structure

This paper contains 17 sections, 2 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Phase space of TEGR
  4. The TEGR constraints
  5. The TEGR constraints versus the vacuum Einstein constraints
  6. The Einstein constraints as a system of first order PDE's
  7. Antisymmetric part of $(d\theta^I)$
  8. Metrics admitting coclosed orthonormal coframes
  9. The system of first order PDE's
  10. Dual frames
  11. Other gauges
  12. Darboux gauge
  13. Nester gauge
  14. Summary and outlook
  15. Acknowledgments
  16. ...and 2 more sections

Key Result

Theorem 4.1

Let $q$ be a real-analytic Riemannian metric defined on a three-dimen-sio-nal manifold. Then there exists locally an orthonormal coframe of the metric that is coclosed.

Theorems & Definitions (2)

  • Theorem 4.1
  • Theorem 5.1