Boson star solutions of the Einstein–Klein–Gordon equations in the sense of Colombeau–Egorov's theory of generalized functions

Abstract

We present static, spherically symmetric solutions of the equations of motion of a scalar field interacting with gravity through the Einstein–Klein–Gordon (EKG) equations in the Colombeau–Egorov sense. The scalar fields are confined within the interior region of the solution, and the exterior fields are purely gravitational and coinciding with the Schwarzschild ones. The solution resembles the so‐called gravastars discussed in the literature. The presented solutions of the EKG equations open the possibility for the existence of static boson stars. The argument is based on defining an infinitely differentiable one‐parameter (ϵ‐dependent) family of radial dependencies of the metric and the scalar field. It is shown that in the limit ϵ → 0, the EKG equations are satisfied in the sense of the generalized functions. The solutions exhibit properties that qualitatively support their physical meaning. For example, close to the boundary at the interior, the scalar field energy density piles up toward the limit surface. On the other hand, also close to the separation surface but on the outside, the known “non‐hair” theorem clearly indicates that any scalar field perturbation also tends to be attracted to the boundary. The work also suggests the possibility for obtaining a regular gravastar, after using the singular configuration found as a first step in an iterative solution of the quantum EKG equations.