Singularities in Static Spherically Symmetric Configurations of General Relativity with Strongly Nonlinear Scalar Fields

Oleksandr Stashko,Valery I Zhdanov

doi:10.3390/galaxies9040072

Abstract

There are a number of publications on relativistic objects dealing either with black holes or naked singularities in the center. Here we show that there exist static spherically symmetric solutions of Einstein equations with a strongly nonlinear scalar field, which allow the appearance of singularities of a new type (“spherical singularities”) outside the center of curvature coordinates. As the example, we consider a scalar field potential ∼sinh(ϕ2n),n>2, which grows rapidly for large field values. The space-time is assumed to be asymptotically flat. We fulfill a numerical investigation of solutions with different n for different parameters, which define asymptotic properties at spatial infinity. Depending on the configuration parameters, we show that the distribution of the stable circular orbits of test bodies around the configuration is either similar to that in the case of the Schwarzschild solution (thus mimicking an ordinary black hole), or it contains additional rings of unstable orbits.

Highlights

Nowadays, the concept of a black hole has become a common element of astrophysical research [1,2,3]
We have shown that General Relativity allows the existence of SS in static spherically symmetric configurations in the case of the scalar field (SF) potential V(φ) = sinh(φ2n), which represents potentials with fast growth rate
These SS are “physical” singularities that cannot be removed by a coordinate transformation: this can be seen from the behavior of the Kretschmann invariant

Summary

Introduction

The concept of a black hole has become a common element of astrophysical research [1,2,3]. A number of models are based on static solutions of the Einstein equations with scalar field (SF) [29,30,31,32,33,36,41] These papers use analytically defined metrics, in particular the Fisher–Yanis–Newman–Winicour solution [42,43,44,45]. Papers [29,30,31,32,33,36] deal with singularities in the center, the solutions used do not have other “physical” singularities outside the center It was proved [46] that this is a fairly general situation, at least when considering an isolated spherically symmetric configuration of General Relativity with a scalar field. We estimate the radiation flux from the accretion disk within the Page–Thorne model [47]

Initial Relations

Asymptotic Behavior Near Singularity

Numerical Solutions

Test Particle Motion

Discussion