On taking the D → 4 limit of Gauss-Bonnet gravity: theory and solutions

Abstract

We comment on the recently introduced Gauss-Bonnet gravity in four dimensions. We argue that it does not make sense to consider this theory to be defined by a set of D → 4 solutions of the higher-dimensional Gauss-Bonnet gravity. We show that a well-defined D → 4 limit of Gauss-Bonnet Gravity is obtained generalizing a method employed by Mann and Ross to obtain a limit of the Einstein gravity in D = 2 dimensions. This is a scalar-tensor theory of the Horndeski type obtained by dimensional reduction methods. By considering simple spacetimes beyond spherical symmetry (Taub-NUT spaces) we show that the naive limit of the higher-dimensional theory to D = 4 is not well defined and contrast the resultant metrics with the actual solutions of the new theory.

4D Einstein-Gauss-Bonnet gravity: Massless particles and absorption of planar spin-0 waves

“…[24] using the ADM decomposition, and it was pointed out that the theory either breaks the diffeomorphism invariance or has additional degrees of freedom, thus in accordance with the Lovelock's theorem [20]. Nevertheless, the 4D-EGB theory can also be obtained as a particular case of the Horndeski theory [23]. In any case, the original solution proposed in [19] is also solution for the consistent formulations of the 4D-EGB theory [23,24].…”

“…However, a number of authors pointed out that this dimensional limit taking method is ill-defined [21][22][23]. For instance, divergent contributions to the field equations in four dimensions were observed [22], as well as the lack of covariant description [21].…”

“…Nevertheless, the 4D-EGB theory can also be obtained as a particular case of the Horndeski theory [23]. In any case, the original solution proposed in [19] is also solution for the consistent formulations of the 4D-EGB theory [23,24].…”

Born–Infeld black holes in 4D Einstein–Gauss–Bonnet gravity

“…However, there are also some debates on whether the novel gravity is a consistent and well-defined theory in four dimensions [45][46][47][48][49][50][51][52][53]. Such as, Gurses et al pointed out that the novel 4D EGB gravity does not admit a description in terms of a covariantly-conserved rank-2 tensor in four dimensions and the dimensional regularization procedure is ill-defined, since one part of the GB tensor, the Lanczos-Bach tensor, always remains higher dimensional [46].…”

“…Based on studying the evolution of the homogeneous but anisotropic universe described by the Bianchi type I metric, Tian and Zhu found that the novel 4D EGB gravity with the dimensional-regularization approach is not a complete theory [49]. On the other hand, some regularized versions of 4D EGB gravity were also proposed by [51][52][53], and the resulting theories belong to the family of Horndeski gravity.…”