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$C^{N + 2}$ extension across the horizon if some components of $N$-th covariantderivative of Riemann tensor diverge at the horizon in the coordinates of the$C^1$ extension. In particular, the divergence of a component of the Riemanntensor at the horizon directly indicates the presence of a curvaturesingularity. By using this method, we can confirm the existence of a curvaturesingularity for several cases where the scalar invariants constructed from theRiemann tensor, e.g., the Ricci scalar and the Kretschmann invariant, takefinite values at the horizon. As a concrete example of the application, we showthat the Kaluza-Klein black holes constructed by Myers have a curvaturesingularity at the horizon if the spacetime dimension is higher than five.