Generalized Ellis-Bronnikov wormhole solution in the scalar-Einstein-Gauss-Bonnet $4d$ gravitational model

TL;DR

The paper investigates a generalized Ellis-Bronnikov wormhole within the four-dimensional scalar-Einstein-Gauss-Bonnet framework, introducing a scalar field with potential $U(\varphi)$ and a Gauss-Bonnet coupling $f(\varphi)$. Using the Buchdal parameterization and a reconstruction approach, it derives the master equation for $f(\varphi(u))$, along with expressions for $U(\varphi(u))$ and $h(u)=\varepsilon \dot{\varphi}^2$, and analyzes the sign structure of $h(u)$. A central no-go theorem is established: for any non-trivial reconstruction constant $C_0\neq0$, $h(u)$ cannot maintain a constant sign for all $u$, excluding the possibility of a purely ordinary ($\varepsilon=+1$) or purely phantom ($\varepsilon=-1$) scalar field on the entire spacetime; when $C_0=0$ the Ellis-Bronnikov background reduces to a phantom solution with $U=0$. The work also shows that the asymptotic behavior of $h(u)$ forces either ordinary or ghost behavior in different regions, and energy conditions are violated for this wormhole, consistent with traversable wormhole physics. Overall, the results highlight limitations of sEGB reconstructions for Ellis-Bronnikov geometries and point to the necessity of mixed-field configurations or alternative models to realize such wormholes.

Abstract

We consider the 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 usual scalar field, and $\varepsilon = -1$ to the phantom field. In this paper, the sEGB reconstruction procedure considered in our previous paper is applied to the metric of the Ellis-Bronnikov solution, which describes a massive wormhole in the model with a phantom field (and zero potential). For this metric, written in the Buchdal parameterization with a radial variable $u$, we find a solution of the master equation for $ f\left(\varphi\left(u\right)\right)$ with the integration (reconstruction) parameter $C_0$. We also find expressions for $U\left(\varphi\left(u\right)\right)$ and $\varepsilon \dot{\varphi}^2 = h\left(u\right)$ for $\varepsilon = \pm 1$. We prove that for all non-trivial values of the parameter $C_0 \neq 0$ the function $h\left(u\right)$ is not of constant sign for all admissible $u \in \left(-\infty , +\infty\right)$. This means that for a fixed value of the parameter $\varepsilon = \pm 1$ there is no non-trivial sEGB reconstruction in which the scalar field is a purely ordinary field ($\varepsilon = 1$) or a purely phantom field ($\varepsilon = - 1$).

Figures (2)

• Figure 1: The function $h\left(u\right)$ for $\mu = L = 1$ 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 $\left(-\infty, u_{\ast}=\frac{1}{3}\right)$. 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 $\mu = L = 1$ 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 $\left(-\infty, u_{\ast}=\frac{1}{3}\right)$. 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.