A simple diagnosis of non-smoothness of black hole horizon: curvature singularity at horizons in extremal Kaluza–Klein black holes

We propose a simple method to prove non-smoothness of a black hole horizon. The existence of a C¹ extension across the horizon implies that there is no C^N+2 extension across the horizon if some components of the Nth covariant derivative of the Riemann tensor diverge at the horizon in the coordinates of the C¹ extension. In particular, the divergence of a component of the Riemann tensor at the horizon directly indicates the presence of a curvature singularity. By using this method, we can confirm the existence of a curvature singularity for several cases where the scalar invariants constructed from the Riemann tensor, e.g., the Ricci scalar and the Kretschmann invariant, take finite values at the horizon. As a concrete example of the application, we show that the Kaluza–Klein black holes constructed by Myers have a curvature singularity at the horizon if the spacetime dimension is higher than five. (paper)