Recently, a novel 4D Einstein-Gauss-Bonnet gravity has been proposed by Glavan and Lin [Phys. Rev. Lett. 124, 081301 (2020)] by rescaling the coupling $\alpha \rightarrow \alpha/(D-4)$ and taking the limit $D\rightarrow 4$ at the level of equations of motion. This prescription, though was shown to bring non-trivial effects for some spacetimes with particular symmetries, remains mysterious and calls for scrutiny. Indeed, there is no continuous way to take the limit $D\rightarrow 4$ in the higher $D$-dimensional equations of motion because the tensor indices depend on the spacetime dimension and behave discretely. On the other hand, if one works with four-dimensional spacetime indices the contribution corresponding to the Gauss-Bonnet term vanishes identically in the equations of motion. A necessary condition (but may not be sufficient) for this procedure to work is that there is an embedding of the four-dimensional spacetime into the higher $D$-dimensional spacetime so that the equations in the latter can be properly interpreted after taking the limit. In this note, working with 2D Einstein gravity, we show several subtleties when applying the method used in [Phys. Rev. Lett. 124, 081301 (2020)].

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关于小说 4D 爱因斯坦-高斯-博内引力的注释

最近，Glavan 和 Lin 提出了一种新颖的 4D Einstein-Gauss-Bonnet 引力 [Phys. 牧师莱特。124, 081301 (2020)] 通过重新缩放耦合 $\alpha \rightarrow \alpha/(D-4)$ 并在运动方程的水平上取极限 $D\rightarrow 4$。这个处方虽然被证明对某些具有特定对称性的时空带来了非平凡的影响，但仍然神秘并需要仔细研究。事实上，没有连续的方法在更高的 $D$ 维运动方程中取极限 $D\rightarrow 4$，因为张量指数取决于时空维度并且表现离散。另一方面，如果使用四维时空指数，则对应于高斯-博内项的贡献在运动方程中完全消失。这个过程起作用的一个必要条件（但可能不充分）是四维时空嵌入到更高的 $D$ 维时空中，这样后者中的方程可以在取限制。在本说明中，使用 2D 爱因斯坦引力，我们展示了在应用 [Phys. 牧师莱特。124, 081301 (2020)]。