The problem of reconstruction for static spherically-symmetric $4D$ metrics in scalar-Einstein-Gauss-Bonnet model

We consider the $4D$ gravitational model with a scalar field $\phi$, Einstein and Gauss-Bonnet terms. The action of the model contains a potential term $U(\phi )$, Gauss-Bonnet coupling function $f(\phi )$ and a parameter $\epsilon =\pm 1$, where $\epsilon =1$ corresponds to ordinary scalar field and $\epsilon =-1$ - to phantom one. Inspired by the recent works of Nojiri and Nashed, we explore a reconstruction procedure for a generic static spherically symmetric metric written in the Buchdal parametrization: $d{s}^2={(A(u))}^{-1}d{u}^2-A(u)d{t}^2+C(u)d{\mathrm{\Omega }}^2$, with given $A(u)>0$ and $C(u)>0$. The procedure gives the relations for $U(\phi (u))$, $f(\phi (u))$ and $d\phi \mathrm{/}du$, which lead to exact solutions to equations of motion with a given metric. A key role in this approach is played by the solutions to a second order linear differential equation for the function $f(\phi (u))$. The formalism is illustrated by two examples when: a) the Schwarzschild metric and b) the Ellis wormhole metric, are chosen as a starting point. For the first case a) the black hole solution with a ``trapped ghost’’ is found which describes an ordinary scalar field outside the photon sphere and phantom scalar field inside the photon sphere. For the second case b) the sEGB-extension of the Ellis wormhole solution is found when the coupling function reads: $f(\phi )=c_1+c_0(\tan (\phi )+\frac{1}{3}(\tan (\phi ){)}^3)$, where $c_1$ and $c_0$ are constants.