# The problem of reconstruction for static spherically-symmetric $4D$ metrics in scalar-Einstein-Gauss-Bonnet model

Source: [The problem of reconstruction for static spherically-symmetric $4D$ metrics in scalar-Einstein-Gauss-Bonnet model](<https://www.sota2.com/research/paper/68fa4adb1e209131ce941b58>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T08:02:44.939Z

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

Published: 2025-02-28T23:21:46.000Z

## Abstract

We consider the $4D$ gravitational model with a scalar field $\\varphi$, Einstein and Gauss-Bonnet terms. The action of the model contains a potential term $U(\\varphi)$, Gauss-Bonnet coupling function $f(\\varphi)$ and a parameter $\\varepsilon = \\pm1 $, where $\\varepsilon = 1$ corresponds to ordinary scalar field and $\\varepsilon = -1 $ - to phantom one. Inspired by the recent works of Nojiri and Nashed, we explore a reconstruction procedure for a generic static spherically symmetric metric written in the Buchdal parametrization: $ds^2 = \\left(A(u)\\right)^{-1}du^2 - A(u)dt^2 + C(u)d\\Omega^2$, with given $A(u) \> 0$ and $C(u) \> 0$. The procedure gives the relations for $U(\\varphi(u))$, $f(\\varphi(u))$ and $d\\varphi/du$, which lead to exact solutions to equations of motion with a given metric. A key role in this approach is played by the solutions to a second order linear differential equation for the function $f(\\varphi(u))$. The formalism is illustrated by two examples when: a) the Schwarzschild metric and b) the Ellis wormhole metric, are chosen as a starting point. For the first case a) the black hole solution with a \`\`trapped ghost'' is found which describes an ordinary scalar field outside the photon sphere and phantom scalar field inside the photon sphere. For the second case b) the sEGB-extension of the Ellis wormhole solution is found when the coupling function reads: $f(\\varphi) = c\_1 + c\_0 ( \\tan ( \\varphi) + \\frac{1}{3} (\\tan ( \\varphi))^3)$, where $c\_1$ and $c\_0$ are constants.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2503.00244>)

[PDF](<https://arxiv.org/pdf/2503.00244.pdf>)