The Yilmaz-Rosen and Janis-Newman-Winicour metric solutions in the scalar-Einstein-Gauss-Bonnet $4d$ gravitational model

Abstract

We consider the scalar-Einstein-Gauss-Bonnet (sEGB) $4d$ gravitational model with a scalar field $\varphi\left(u\right)$, Einstein and Gauss-Bonnet terms. The model action contains a potential term $U\left(\varphi\right)$, a Gauss-Bonnet coupling function $f\left(\varphi\right)$ and a parameter $\varepsilon = \pm 1$, where $\varepsilon = 1$ corresponds to the ordinary scalar field, and $\varepsilon = -1$ to the phantom field. In this paper we applied the sEGB reconstruction procedure from our previous work \cite{Er_Ivash} to the Y{\i}lmaz-Rosen metric, a solution potentially describing a quasi-black hole without an event horizon. Within this framework, we also derived analytical solutions based on scalar-tensor theory with minimal coupling. Our results indicate that for this configuration, the potential $U$ vanishes and the scalar field is phantom-like. Furthermore, an analysis of the Einstein equations in the Y{\i}lmaz-Rosen metric reveals that all energy conditions are violated. The corresponding energy-momentum tensor suggests the presence of exotic matter with negative pressure, as indicated by the negative value of $T_u^u$. This could originate from a scalar field (such as the Higgs field or another nonlinear field), or from phenomena like dark energy or quintessence. In addition, we considered the application of our reconstruction method in the sEGB model in the Janis-Newman-Winicour (JNW) metric. As noted in this paper, the Y{\i}lmaz-Rosen metric is a limiting case of the Janus metric (as $s \to +\infty$). Furthermore, we obtained some exact solutions of scalar-tensor theory with minimal coupling in the JNW metric.