Connecting black holes and black strings

TL;DR

The paper advances numerical construction of static, localized black holes in 5D and 6D Kaluza-Klein gravity, aiming to connect them to uniform and non-uniform black strings. It solves the axisymmetric Einstein equations via a relaxation method for a metric ansatz with a compact circle, then extracts mass $M$ and tension $n$ from the first law and Smarr relation with fixed circle length $L$. The 6D results provide evidence that the localised BH branch merges with the non-uniform string branch at a topology-changing cone transition near $n/n_{crit} \simeq 0.55$, with horizon geometry and thermodynamics supporting the merger scenario; 5D data show analogous behavior, extending the conjectured unified phase structure. This work informs holographic phase structure and the broader understanding of higher-dimensional gravity by illustrating how distinct KK black object branches may be topologically connected.

Abstract

Static vacuum spacetimes with one compact dimension include black holes with localised horizons but also uniform and non-uniform black strings where the horizon wraps over the compact dimension. We present new numerical solutions for these localised black holes in 5 and 6-dimensions. Combined with previous 6-d non-uniform string results, these provide evidence that the black hole and non-uniform string branches join at a topology changing solution.

Figures (7)

• Figure 1: Embeddings of the spatial horizon geometry of various 6-d BHs and NUSs in 5-d Euclidean space (suitably projected onto the page). For BH solutions we include the exposed symmetry axis in the embedding. The red vertical lines are to be periodically identified, generating the compact dimension.
• Figure 2: Plot of $X_{max}$ for 6-d NUSs and BHs, consistent with merger of the branches at $n/n_{crit} \simeq 0.55$.
• Figure 3: Plot of horizon geometric quantities for 6-d solutions. Branches are consistent with a topology changing merger where both $L_{axis}$ and $R_{min}$ go to zero, and $R_{eq}$ tends to $R_{max}$. All solutions have $L = 1$.
• Figure 4: Entropy and temperature for 6-d solutions.
• Figure 5: Mass against $n$ for 6-d solutions. The highlighting indicates which branch is entropically favoured for a given mass (see figure \ref{['fig:SM']}).