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

摘要

We propose a simple method to prove non-smoothness of a black hole horizon. The existence of a C-1 extension across the horizon implies that there is no CN+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.