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

Source: [On a reconstruction procedure for special spherically symmetric metrics in the scalar-Einstein-Gauss-Bonnet model: the Schwarzschild metric test](<https://www.sota2.com/research/paper/68fa498c1e209131ce937e12>)

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Data updated: 2025-10-25T07:23:41.571Z

Authors: K. K. Ernazarov, V. D. Ivashchuk

Published: 2024-06-03T13:10:08.000Z

## 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2406.01301>)

[PDF](<https://arxiv.org/pdf/2406.01301.pdf>)