A novel four-dimensional Einstein-Gauss-Bonnet gravity was formulated by Glavan and Lin (Phys. Rev. Lett. 124:081301, 2020), which is intended to bypass the Lovelock’s theorem and to yield a non-trivial contribution to the four-dimensional gravitational dynamics. However, the validity and consistency of this theory has been called into question recently. We study a static and spherically symmetric black hole charged by a Born–Infeld electric field in the novel four-dimensional Einstein–Gauss–Bonnet gravity. It is found that the black hole solution still suffers the singularity problem, since particles incident from infinity can reach the singularity. It is also demonstrated that the Born-Infeld charged black hole may be superior to the Maxwell charged black hole to be a charged extension of the Schwarzschild-AdS-like black hole in this new gravitational theory. Some basic thermodynamics of the black hole solution is also analyzed. Besides, we regain the black hole solution in the regularized four-dimensional Einstein–Gauss–Bonnet gravity proposed by Lü and Pang (arXiv:2003.11552).

We analyze the tensor perturbations of flat thick domain wall branes in f (R) gravity. Our results indicate that under the transverse and traceless gauge, the metric perturbations decouple from the perturbation of the scalar field. Besides, the perturbed equation reduces to the familiar Klein-Gordon equation for massless spin-2 particles only when the bulk curvature is a constant or when f (R) = R. As an application of our results, we consider the possibility of localizing gravity on some flat thick branes. The stability of these brane solutions is also shortly discussed.

The stability and localization of the gravitational perturbations for a special brane system in Eddingtoninspired Born-Infeld gravity were studied in Liu et al. [Phys. Rev. D 85, 124053 (2012)]. In this paper, we show that the gravitational perturbations for a general brane system are stable, the four-dimensional graviton (massless KK graviton) can be localized on the brane, and the mass spectra of massive KK gravitons are gapless and continuous. Two models are constructed as examples. In the first model, which is a generalization of Liu et al. [Phys. Rev. D 85, 124053 (2012)], the brane has no inner structure and there is no gravitational resonance (quasilocalized KK gravitons). In the second one, the background scalar field is a double kink when the parameter in the model approaches its critical value. Correspondingly, the brane has inner structure and some gravitational resonances appear.