On the Smoothness of the Horizons of Multi-Black Hole Solutions

TL;DR

This paper investigates the smoothness properties of multi-black hole solutions in higher dimensions, revealing that their horizons are not infinitely differentiable but only finitely smooth, despite bounded curvature.

Contribution

It demonstrates that certain multi-black hole solutions have horizons with limited smoothness, specifically showing the metric is only finitely differentiable, not infinitely smooth.

Findings

Curvature remains bounded near the horizon.

Some derivatives of curvature are unbounded, indicating limited smoothness.

The solutions are static, ruling out radiation as a cause of non-smoothness.

Abstract

In a recent paper it was suggested that some multi-black hole solutions in five or more dimensions have horizons that are not smooth. These black hole configurations are solutions to $d$-dimensional Einstein gravity (with no dilaton) and are extremally charged with a magnetic type $(d-2)$-form. In this work these solutions will be investigated further. It will be shown that although the curvature is bounded as the horizon of one of the black holes is approached, some derivatives of the curvature are not. This shows that the metric is not $C^{\infty },$ but rather it is only $C^k$ with $k$ finite. These solutions are static so their lack of smoothness cannot be attributed to the presence of radiation.