Is there a novel Einstein–Gauss–Bonnet theory in four dimensions?

Abstract

No! We show that the field equations of Einstein–Gauss–Bonnet theory defined in generic D> 4 dimensions split into two parts one of which always remains higher dimensional, and hence the theory does not have a non-trivial limit to D= 4. Therefore, the recently introduced four-dimensional, novel, Einstein–Gauss–Bonnet theory does not admit an intrinsically four-dimensional definition, in terms of metric only, as such it does not exist in four dimensions. The solutions (the spacetime, the metric) always remain D> 4 dimensional. As there is no canonical choice of 4 spacetime dimensions out of D dimensions for generic metrics, the theory is not well defined in four dimensions.