Eur. Phys. J. C (2025) 85: 756
https://doi.org/10.1140/epjc/s10052-025-14481-7

Regular Article - Theoretical Physics

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

¹ Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, 117198, Moscow, Russian Federation
² Center for Gravitation and Fundamental Metrology, Scientific Research Center of Applied Metrology Rostest, 46 Ozyornaya Street, 119361, Moscow, Russian Federation

^a ivashchuk@mail.ru

Received: 5 March 2025
Accepted: 30 June 2025
Published online: 9 July 2025

Abstract

We consider the 4D gravitational model with a scalar field $\phi$, Einstein and Gauss–Bonnet terms. The action of the model contains a potential term $U(\phi )$, Gauss–Bonnet coupling function $f(\phi )$ and a parameter $\epsilon =\pm 1$, where $\epsilon =1$ corresponds to ordinary scalar field and $\epsilon =-1$ - 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: $d{s}^2={(A(u))}^{-1}d{u}^2-A(u)d{t}^2+C(u)d{\mathrm{\Omega }}^2$, with given $A(u)>0$ and $C(u)>0$. The procedure gives the relations for $U(\phi (u))$, $f(\phi (u))$ and $d\phi /du$, 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 $f(\phi (u))$. 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: $f(\phi )=c_1+c_0(tan(\phi )+\frac{1}{3}{(tan(\phi ))}^3)$, where $c_1$ and $c_0$ are constants.