Black hole solutions in the warped DGP braneworld

We study the static, analytical solution of black holes in the warped DGP braneworld scenario. We show that the linearized field equations and matching conditions lead to solutions that are not compatible with Schwarzschild-(A)dS$_{(4)}$ solutions on the brane. This incompatibility is similar to vDVZ discontinuity in massive gravity theory. Following the standard procedure to remove this discontinuity, which firstly was proposed by Vainshtein, we keep some appropriate nonlinear terms in the field equations. This strategy has its origin in the fact that the spatial extrinsic curvature of the brane plays a crucial role in the nonlinear nature of the solutions and also in recovering the well-measured predictions of General Relativity (GR) at small scales. Using this feature, we obtained an interesting black string solution in the bulk when it is compatible with 4D GR solutions on the brane.