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 $\varphi$, Einstein and Gauss–Bonnet terms is considered. The action contains the potential $U(\varphi)$ and the Gauss–Bonnet coupling function $f(\varphi)$. For a special static spherically symmetric metric $ds^{2}=(A(u))^{-1}du^{2}-A(u)dt^{2}+u^{2}d\Omega^{2}$, 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 $U(\varphi)$ and $f(\varphi)$ 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 $f(\varphi(u))$ and $\varphi(u)$ are correct, but the relation for $U(\varphi(u))$ 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 $\varphi(u)$), we find two families of $(U(\varphi),f(\varphi))$. 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