Static spherically symmetric configurations with N non-linear scalar fields: global and asymptotic properties

TL;DR

This paper investigates static, spherically symmetric solutions with multiple non-linear scalar fields in General Relativity, demonstrating that gravity prevents spherical singularities outside the center and analyzing the structure of such configurations.

Contribution

It provides a proof that solutions with multiple non-linear scalar fields are regular outside the center, under certain conditions, and explores their asymptotic and geodesic properties.

Findings

Solutions are regular for all r > 0, except at the center where naked singularities may occur.

Asymptotic relations near the center resemble Fisher solutions for free scalar fields.

Accretion disk images can resemble black hole shadows with bright rings and dark centers.

Abstract

In case of a spherically symmetric non-linear scalar field (SF) in flat space, besides singularity at the center, spherical singularities can occur for non-zero values of radial variable $r>0$. We show that in the General Relativity the gravitational field suppresses the occurrence of the spherical singularities under some generic conditions. Our consideration deals with asymptotically flat space-times around static spherically symmetric configurations in presence of $N$ non-linear SFs, which are minimally coupled to gravity. Constraints are imposed on the SF potentials, which guarantee a monotonicity of the fields as functions of radial variable; also the potentials are assumed to be exponentially bounded. We give direct proof that solutions of the joint system of Einstein -- SF equations satisfying the conditions of asymptotic flatness are regular for all values of $r$, except for naked singularities in the center $r=0$ in the Schwarzschild (curvature) coordinates. Asymptotic relations for SF and metric near the center are derived, which appear to be remarkably similar to the case of the Fisher solution for free SF. These relations determine two main types of the corresponding geodesic structure when photons can be captured by the singularity or not depending on the existence of the photon sphere. To illustrate, the case of one SF with monomial potential is analyzed in detail numerically. We show that the image of the accretion disk around the singularity, observed from infinity, can take the form of a bright ring with a dark spot in the center, like the case of an ordinary black hole.