Is the Bianchi identity always hyperbolic?

Rácz, István (2014) Is the Bianchi identity always hyperbolic? CLASSICAL AND QUANTUM GRAVITY, 31 (15). ISSN 0264-9381

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Abstract

We consider n + 1 dimensional smooth Riemannian and Lorentzian spaces satisfying Einsteinʼs 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—Einsteinʼs equations may be split into ‘Hamiltonian’ and ‘momentum’ constraints and a ‘reduced’ set of field equations. It is shown that regardless of 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.

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