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On the evolution equations for a self-gravitating charged scalar field. (English) Zbl 1266.83031
Summary: 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.
MSC:
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C22 Einstein-Maxwell equations
83C15 Exact solutions to problems in general relativity and gravitational theory
78A25 Electromagnetic theory, general
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