On a Reconstruction Procedure for Special Spherically Symmetric Metrics in the Scalar-Einstein–Gauss–Bonnet Model: the Schwarzschild Metric Test

Abstract

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 \(2\mu<u<3\mu\) (\(3\mu\) is the radius of the photon sphere). In both cases the potential \(U(\varphi)\) is negative.