On the dynamics of Einstein's equations in the Ashtekar formulation

TL;DR

This paper demonstrates that Einstein's equations in Ashtekar's variables can be formulated as a symmetric hyperbolic system, ensuring well-posedness and stability of the evolution, by modifying the Hamiltonian flow outside the real solution sub-manifold.

Contribution

It extends previous results by showing how to obtain a symmetric hyperbolic system for Einstein's equations in Ashtekar variables without assuming real solutions a priori.

Findings

The evolution system is block diagonal in (,) variables.

Explicit eigenvector-eigenvalue formulas are provided.

The constraint equations form a symmetric hyperbolic system.

Abstract

We study the dynamics of Einstein's equations in Ashtekar's variables from the point of view of the theory of hyperbolic systems of evolution equations. We extend previous results and show that by a suitable modification of the Hamiltonian vector flow outside the sub-manifold of real and constrained solutions, a symmetric hyperbolic system is obtained for any fixed choice of lapse-shift pair, without assuming the solution to be a priori real. We notice that the evolution system is block diagonal in the pair $(\sigma^a,A_b)$, and provide explicit and very simple formulae for the eigenvector-eigenvalue pairs in terms of an orthonormal tetrad with one of its components pointing along the propagation direction. We also analyze the constraint equations and find that when viewed as functions of the extended phase space they form a symmetric hyperbolic system on their own. We also provide simple formulae for its eigenvectors-eigenvalues pairs.