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

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.

26 pages, 2 figures, v2: discussions improved, v3: minor revisions, to appear in Classical and Quantum Gravity

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Cited by in corpus (2)

• All higher-dimensional Majumdar-Papapetrou black holes
• Vacuum-dual static perfect fluid obeying $p=-(n-3)\rho /(n+1)$ in $n(\ge 4)$ dimensions