A note on the novel 4D Einstein-Gauss-Bonnet gravity

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)].