Constructing hyperbolic systems in the Ashtekar formulation of general relativity

TL;DR

This paper develops three types of hyperbolic formulations of the Ashtekar equations in general relativity, crucial for well-posedness and stable numerical evolution of Lorentzian vacuum spacetimes.

Contribution

It introduces weakly, diagonalizable, and symmetric hyperbolic systems within the Ashtekar formulation, detailing their eigenvalues and the constraints involved.

Findings

Ashtekar's original equations are weakly hyperbolic.

Construction of symmetric hyperbolic systems involves gauge and reality condition constraints.

The paper provides explicit forms and eigenvalues for each hyperbolic system type.

Abstract

Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system.