From Black Strings to Black Holes

TL;DR

The paper investigates neutral compactified non-uniform black strings near the Gregory-Laflamme critical point using elliptic numerical methods within a Harmark-Obers framework. It tests Kol's conjecture that the GL branch connects to a black hole branch through a topology-changing transition by analyzing horizon geometry as the non-uniformity parameter $\lambda$ grows. The results show $R_{min}\to 0$ with finite $R_{max}$, horizon length, and temperature, while the Kretschmann invariant diverges at the waist, indicating a naked singularity at the topology-change point; it also demonstrates the GL strings can be expressed in HO form, suggesting HO strings are not a distinct branch. These findings provide quantitative support for the conjectured connection and offer a pathway to construct and test the topology-changing solution, with predictions for the maximal black hole geometry.

Abstract

Using recently developed numerical methods, we examine neutral compactified non-uniform black strings which connect to the Gregory-Laflamme critical point. By studying the geometry of the horizon we give evidence that this branch of solutions may connect to the black hole solutions, as conjectured by Kol. We find the geometry of the topology changing solution is likely to be nakedly singular at the point where the horizon radius is zero. We show that these solutions can all be expressed in the coordinate system discussed by Harmark and Obers.

Figures (5)

• Figure 1: Horizon geometry of non-uniform string found with $\lambda \simeq 4$. For the same asymptotic $S^1$ radius, the critical uniform string has unit horizon radius. This indicates that the large value of $\lambda$ is mainly due to a cycle shrinking, suggesting the possibility of a topology change for $\lambda \rightarrow \infty$.
• Figure 2: Plot of mass, normalised by the critical uniform string mass, against $\lambda$ for fixed asymptotic compactification radius. The data points indicate actual solutions. Three resolutions are shown superposed, the highest resolution allowing solutions to be found with $\lambda \simeq 4$.
• Figure 3: Plot of maximum and minimum horizon radius, normalised by the critical uniform string radius, against $1 / (1 + \lambda)$, for fixed asymptotic $S^1$ size. We see $R_{min} \propto 1 / \lambda$ for $\lambda \rightarrow \infty$ and $R_{max}$ remains finite.
• Figure 4: Plot of proper length of horizon against $R_{min}$, normalised by the critical uniform string value, for fixed asymptotic $S^1$ radius.
• Figure 5: Plot of horizon temperature against $R_{min}$, normalised by same the value for the critical uniform string, for fixed asymptotic $S^1$ size.