Mathematical issues in a fully constrained formulation of the Einstein equations

Bonazzola, Gourgoulhon, Grandcl\'ement, and Novak [Phys. Rev. D 70, 104007 (2004)] proposed a new formulation for $3+1$ numerical relativity. Einstein equations result, according to that formalism, in a coupled elliptic-hyperbolic system. We have carried out a preliminary analysis of the mathematical structure of that system, in particular, focusing on the equations governing the evolution for the deviation of a conformal metric from a flat fiducial one. The choice of a Dirac gauge for the spatial coordinates guarantees the mathematical characterization of that system as a (strongly) hyperbolic system of conservation laws. In the presence of boundaries, this characterization also depends on the boundary conditions for the shift vector in the elliptic subsystem. This interplay between the hyperbolic and elliptic parts of the complete evolution system is used to assess the prescription of inner boundary conditions for the hyperbolic part when using an excision approach to black hole space-time evolutions.