  We consider a complex scalar field minimally coupled to gravity and to a U(1)
gauge symmetry and we construct of a first order symmetric hyperbolic evolution
system for the Einstein-Maxwell-Klein-Gordon system. Our analysis is based on a
1+3 tetrad formalism which makes use of the components of the Weyl tensor as
one of the unknowns. In order to ensure the symmetric hyperbolicity of the
evolution equations, implied by the Bianchi identity, we introduce a tensor of
rank 3 corresponding to the covariant derivative of the Faraday tensor, and two
tensors of rank 2 for the covariant derivative of the vector potential and the
scalar field.
