柱对称时空中四维新理论与正则化EGB理论解的等价性
Equivalence of solutions between the four-dimensional novel and regularized EGB theories in a cylindrically symmetric spacetime
摘要：
最近，一个新的四维Einstein-Gauss-Bonnet（EGB）理论被提出来绕过Lovelock定理，并给出四维局部引力的非平凡效应。主要机制是引入重新定义$α\rightarrowα/(D-4)$并取极限$D\rightarrow4$。然而，该理论没有标准的四维场方程。然后提出了一些正则化过程来解决这个问题[arXiv：2003.11552，arXiv：2003.12771，arXiv：2004.08362，arXiv：2004.09472，arXiv：2004.10716]。由此得到的正则化四维EGB理论具有与原理论相同的壳作用量。因此，可以预期，新的四维EGB理论是等价于它的正则化版本。然而，这两个理论的等价性是依赖于时间的。在本文中，我们测试的等价性在一个圆柱对称时空。得到了这两种理论的严格场方程，并进一步分析了它们在这类时空中的等价性。圆柱形宇宙弦被认为是度规的具体例子。得到了三组解，并计算了相应的弦质量密度。结果揭示了四维高斯-邦纳项如何对新理论中的弦几何作出贡献。
英文摘要：
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 $α\rightarrowα/(D-4)$ 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.
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