Equivalence of solutions between the four-dimensional novel and regularized EGB theories in a cylindrically symmetric spacetime

TL;DR

This paper demonstrates the equivalence of the novel four-dimensional Einstein-Gauss-Bonnet (EGB) theory and its regularized version in cylindrically symmetric spacetimes, analyzing cosmic string solutions and their physical properties.

Contribution

It provides the first explicit verification of the equivalence between the two formulations of 4D EGB gravity in a cylindrically symmetric setting.

Findings

The two theories are equivalent in cylindrically symmetric spacetimes.

Cosmic string solutions are obtained in both theories.

Gauss-Bonnet term influences string geometry and mass density.

Abstract

Recently, a novel four-dimensional Einstein-Gauss-Bonnet (EGB) theory was presented to bypass the Lovelock's theorem and to give nontrivial effects on the four-dimensional local gravity. The main mechanism is to introduce a redefinition $\alpha\rightarrow\alpha/(D-4)$ and to take the limit $D\rightarrow4$. However, this theory does not have standard four-dimensional field equations. Some regularization procedures are then proposed to address this problem [arXiv:2003.11552, arXiv:2003.12771, arXiv:2004.08362, arXiv:2004.09472, arXiv:2004.10716]. The resultant regularized four-dimensional EGB theory has the same on-shell action as the original theory. Thus it is expected that the novel four-dimensional EGB theory is equivalent to its regularized version. However, the equivalence of these two theories is symmetry-dependent. In this paper, we test the equivalence in a cylindrically symmetric spacetime. The well-defined field equations of the two theories are obtained, with which our follow-up analysis shows that they are equivalent in such spacetime. Cylindrical cosmic strings are then considered as specific examples of the metric. Three sets of solutions are obtained and the corresponding string mass densities are evaluated. The results reveal how the Gauss-Bonnet term in four dimensions contributes to the string geometry in the new theory.