Constraints as evolutionary systems

TL;DR

This paper reformulates Einstein constraint equations as evolutionary systems, demonstrating their structure as hyperbolic and parabolic systems and establishing local existence and uniqueness of solutions.

Contribution

It shows that Einstein constraints can be expressed as evolutionary systems with hyperbolic and parabolic components, applicable to spaces of any signature.

Findings

Constraints can be formulated as hyperbolic and parabolic systems.

Existence and uniqueness of solutions are established.

Applicable to both Riemannian and Lorentzian spaces.

Abstract

The constraint equations for smooth $[n+1]$-dimensional (with $n\geq 3$) Riemannian or Lorentzian spaces satisfying the Einstein field equations are considered. It is shown, regardless of the signature of the primary space, that the constraints can be put into the form of an evolutionary system comprised either by a first order symmetric hyperbolic system and a parabolic equation or, alternatively, by a symmetrizable hyperbolic system and a subsidiary algebraic relation. In both cases the (local) existence and uniqueness of solutions are also discussed.