On a reconstruction procedure for special spherically symmetric metrics in the scalar-Einstein-Gauss-Bonnet model: the Schwarzschild metric test

The 4D gravitational model with a real scalar field $\phi$, Einstein and Gauss-Bonnet terms is considered. The action contains the potential $U(\phi )$ and the Gauss-Bonnet coupling function $f(\phi )$. For a special static spherically symmetric metric $d{s}^2=(A(u){)}^{-1}d{u}^2-A(u)d{t}^2+u^{2}d{\mathrm{\Omega }}^2$, with $A(u)>0$ ($u>0$ is a radial coordinate), we verify the so-called reconstruction procedure suggested by Nojiri and Nashed. This procedure presents certain implicit relations for $U(\phi )$ and $f(\phi )$ which lead to exact solutions to the equations of motion for a given metric governed by $A(u)$. We confirm that all relations in the approach of Nojiri and Nashed for $f(\phi (u))$ and $\phi (u)$ are correct, but the relation for $U(\phi (u))$ contains a typo which is eliminated in this paper. Here we apply the procedure to the (external) Schwarzschild metric with the gravitational radius $2\mu$ and $u>2\mu$. Using the ``no-ghost’’ restriction (i.e., reality of $\phi (u)$), we find two families of $(U(\phi ),f(\phi ))$. The first one gives us the Schwarzschild metric defined for $u>3\mu$, while the second one describes the Schwarzschild metric defined for $2\mu <u<3\mu$ ($3\mu$ is the radius of the photon sphere). In both cases the potential $U(\phi )$ is negative.