Ellis wormhole without a phantom scalar field

In this paper, we present an exact solution for ($3+1$)-dimensional Einstein-scalar-Gauss-Bonnet theory (EsGB) in electrovacuum. The solution is characterized by only one parameter, $Q$, which in general can be associated with the electromagnetic field and the scalar field. We show that the solution corresponds to a charged wormhole with a throat at the region $r=|Q|$ and is also supported by a real scalar field having a positive kinetic term. We show that the solution belongs to the most general class of solutions known as Ellis wormholes but without the need for “exotic matter” or a phantom scalar field.

Figure 1

The coupling functions $f(\varphi )$ for different EsGB theories that admit traversable wormhole solutions are illustrated, for the branch $n=0$ defined in (44). The parameters $f_{*}$ and ${\varphi }_{*}$, are given respectively by $f_{*}={10}^3(Q^2+Q_s^2)$ and ${\varphi }_{*}=2Q_s/Q$. Model I, correspond to $f(\varphi )$ given by (33); model II, $f(\varphi )/f_{*}=e^{-\varphi /{\varphi }_{*}}$; model III, $f(\varphi )/f_{*}=(\varphi /{\varphi }_{*}{)}^2$. Note that the range of the scalar field is ${\varphi }_0\le \varphi <{\varphi }_{\infty }$.

Figure 2

Similar as Fig. 1, with $n=1$ in the plot to the left; and $n=-1$ in the graphic to the right.