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

Masashi Kimura; Hideki Ishihara; Ken Matsuno; Takahiro Tanaka

doi:10.1088/0264-9381/32/1/015005

paper

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

arXiv:1407.6224 · doi:10.1088/0264-9381/32/1/015005

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.

arXiv comment26 pages, 2 figures, v2: discussions improved, v3: minor revisions, to appear in Classical and Quantum Gravity

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

Abstract

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