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

TL;DR

The paper introduces a straightforward method to identify non-smoothness and curvature singularities at black hole horizons by examining the divergence of derivatives of the Riemann tensor, revealing singularities even when scalar invariants are finite.

Contribution

It provides a new, simple diagnostic technique to detect horizon singularities through covariant derivatives of the Riemann tensor, applied to higher-dimensional Kaluza-Klein black holes.

Findings

Curvature singularities exist at horizons in certain higher-dimensional black holes.

Scalar invariants can be finite even when curvature singularities are present.

The method confirms singularities in Myers' Kaluza-Klein black holes beyond five dimensions.

Abstract

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 covariant derivative of Riemann tensor diverge at the horizon in the coordinates of the $C^1$ 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.