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

Recently, a novel 4 D Einstein–Gauss–Bonnet gravity has been proposed by Glavan and Lin(2020 Phys. Rev. Lett. 124 081301) by rescaling the coupling α→α(D-4) and taking the limit D→ 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→4 in the higher Ddimensional equations of motion because the tensor indices depend on the spacetime dimension and behave discretely. On the other hand, if one works with 4 D 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 4 D 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 with2 D Einstein gravity, we show several subtleties when applying the method used in(2020 Phys.Rev. Lett. 124 081301).