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 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(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.