On the evolution equations for a self-gravitating charged scalar field

TL;DR

This paper develops a first order symmetric hyperbolic evolution system for a complex scalar field coupled to gravity and electromagnetism, using a 1+3 tetrad formalism with Weyl tensor components.

Contribution

It introduces a novel hyperbolic formulation of the Einstein-Maxwell-Klein-Gordon system incorporating new tensor variables for derivatives, ensuring symmetric hyperbolicity.

Findings

Constructed a well-posed evolution system for the coupled fields.

Utilized a 1+3 tetrad formalism with Weyl tensor components.

Ensured hyperbolicity through additional tensor variables.

Abstract

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.