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

TL;DR

This work embeds the Yilmaz–Rosen and Janis–Newman–Winicour spacetimes into the four-dimensional scalar-Einstein-Gauss-Bonnet (sEGB) gravity framework and applies a reconstruction method to determine the scalar potential $U(\varphi)$ and Gauss–Bonnet coupling $f(\varphi)$ from static, spherically symmetric ansatzes. It finds that in the Yilmaz–Rosen background the potential vanishes and the scalar is phantom, with all standard energy conditions violated, and proves a no-go result: for any nonzero reconstruction constant $C_0$ the sign of $\varepsilon\dot{\varphi}^2$ cannot be constant across all $u>0$; the trivial case $C_0=0$ yields $U=0$ and a pure ghost. For the JNW spacetime, similar reconstructions yield explicit $f(u)$ and $U(u)$ depending on the parameter $s$, and the YR metric emerges as the $s\to\infty$ limit of JNW. The paper also provides exact scalar-tensor minimal-coupling solutions (with $U=0$) in both YR and JNW cases and analyzes special values such as $s=2$ (BBMB-like black hole) and $s=1/2$, illustrating how the scalar field can be ordinary or phantom depending on integration constants. Overall, the results extend the sEGB reconstruction program to classic scalar-gravity spacetimes, clarify energy-condition violations, and illuminate how scalar dynamics shape exotic compact objects in 4d gravity.

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 Yılmaz-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 Yılmaz-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 Yılmaz-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.

Figures (2)

• Figure 1: The function $h\left(u\right)$ for $\mu = 1$ and $C_0 = -10$. A vertical red dashed line crosses point $u_{\ast}=\frac{1}{3}$. A) The function $h\left(u\right)$ is positive in the interval $\left(0, u_{\ast}=\frac{2}{3}\right)$. This means that in this interval a scalar field is ordinary one. B) The function $h\left(u\right)$ is negative in the interval $\left(u_{\ast}=\frac{1}{3}, u_1=3.4686\right)$ and positive in the interval $\left(u_1=3.4686, +\infty\right)$. Therefore in the interval $\left(u_{\ast}=\frac{1}{3}, u_1=3.4686\right)$ a scalar field is ghost one and in the interval $\left(u_1=3.4686, +\infty\right)$ we obtain a solution with an ordinary field.
• Figure 2: The function $h\left(u\right)$ for $\mu = 1$ and $C_0 = 10$. A vertical red dashed line crosses point $u_{\ast}=\frac{1}{3}$. A) The function $h\left(u\right) < 0$ in the interval $\left(0, u_{\ast}=\frac{1}{3}\right)$. This means that in this interval a scalar field is ghost one. B) The function $h\left(u\right)$ is positive in the interval $\left(u_{\ast}=\frac{1}{3}, u_1=2.1496\right)$ and negative in the interval $\left(u_1=2.1496, +\infty\right)$. Therefore only in the interval $\left(u_{\ast}=\frac{1}{3}, u_1=2.1496\right)$ we obtain a solution with an ordinary field.