Localized five-dimensional rotating brane-world black hole Analytically Connected to an to an AdS$_5$ boundary

Abstract: We provide a method to describe the geometry of an analytic, exponentially localized $5D$ rotating braneworld black hole, using the $5D$ Janis Newman algorithm in Hopf coordinates. The induced metric on the brane matches the standard $4D$ Kerr spacetime. Two curvature singularities arise: one confined to the $3$-brane at $z = r = 0$, and another that, on the brane, reproduces the Kerr singularity at $r = 0$, $\bar{\theta} = \pi/2$. The inner and event horizons, together with the stationary limit hypersurfaces, extend into the extra dimension in a pancake-like shape. We describe their behavior in the bulk. The energy momentum tensor represents a source transitioning from an anisotropic, non diagonal structure to a vacuum with negative cosmological constant. Thus, the localized black hole connects to an AdS$_5$ boundary. The geometry is supported by a non-diagonal anisotropic fluid in the bulk, requiring no matter on the brane. To evaluate the energy conditions, we use a one form from the dual basis that yields a diagonal energy momentum tensor.