Is the Bianchi identity always hyperbolic?

TL;DR

This paper demonstrates that in Einstein's equations, when the hypersurfaces are Riemannian, the Hamiltonian and momentum constraints form a symmetric hyperbolic system, ensuring solutions to the reduced equations extend to the full set.

Contribution

It shows that the subsidiary system governing Einstein's constraints is symmetric hyperbolic in Riemannian foliations, linking the solutions of reduced and full Einstein equations.

Findings

Constraints form a symmetric hyperbolic system when hypersurfaces are Riemannian.

Solutions to reduced Einstein equations extend to full solutions if constraints hold initially.

The hyperbolic structure ensures well-posedness of the initial value problem in this setting.

Abstract

We consider $n+1$ dimensional smooth Riemannian and Lorentzian spaces satisfying Einstein's equations. The base manifold is assumed to be smoothly foliated by a one-parameter family of hypersurfaces. In both cases---likewise it is usually done in the Lorentzian case---Einstein's equations may be split into `Hamiltonian' and `momentum' constraints and a `reduced' set of field equations. It is shown that regardless whether the primary space is Riemannian or Lorentzian whenever the foliating hypersurfaces are Riemannian the `Hamiltonian' and `momentum' type expressions are subject to a subsidiary first order symmetric hyperbolic system. Since this subsidiary system is linear and homogeneous in the `Hamiltonian' and `momentum' type expressions the hyperbolicity of the system implies that in both cases the solutions to the `reduced' set of field equations are also solutions to the full set of equations provided that the constraints hold on one of the hypersurfaces foliating the base manifold.