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

TL;DR

This paper verifies a reconstruction method for scalar-Einstein-Gauss-Bonnet gravity, corrects a typo in previous relations, and applies it to Schwarzschild metrics, finding two solution families with negative potentials.

Contribution

The paper confirms the validity of Nojiri and Nashed's reconstruction approach and corrects a typo, applying it to Schwarzschild metrics to find new solution families.

Findings

Two families of solutions for (U(φ), f(φ)) are found for Schwarzschild metrics.

The potential U(φ) in both solutions is negative.

The method confirms the correctness of the reconstruction procedure with a corrected relation.

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.