Localized Five-Dimensional Rotating Brane-World Black Hole Analytically Connected to an to an AdS5 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 $r=0$, $\overline{\theta }=\pi /2$. 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, nondiagonal structure to a vacuum with a negative cosmological constant. Thus, the localized black hole connects to an AdS₅ boundary. The geometry is supported by a nondiagonal 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. The energy conditions are satisfied close to the brane, while they are violated at a location outside the brane but within the extension of the event horizon. The latter is required to support the rotating geometry.