Evidence that highly non-uniform black strings have a conical waist

TL;DR

This work tests the conjecture that highly non-uniform vacuum black strings possess a conical waist near the minimal horizon sphere by numerically constructing non-uniform 6D strings and comparing the local geometry to a Euclidean cone over S^2 × S^3. The authors perform precise metric and curvature tests, using the Kretschmann invariant and full metric components, and find strong evidence for cone-like waist structure away from the apex within their resolution limits. They also argue that the cone description persists for weak electric charge, deriving the leading behavior of the electrostatic potential near the tip. Together, these results reinforce Kol’s cone-based picture of the black string/hole transition and offer a practical geometric framework for understanding highly non-uniform string geometries in Kaluza-Klein spacetimes.

Abstract

Numerical methods have allowed the construction of vacuum non-uniform strings. For sufficient non-uniformity, the local geometry about the minimal horizon sphere (the "waist") was conjectured to be a cone metric. We are able to test this conjecture explicitly giving strong evidence in favour of it. We also show how to extend the conjecture to weakly charged strings.

Figures (6)

• Figure 1: Plot of $\rho$ measured from the $\lambda = 3.9$ string solution.
• Figure 2: Plot showing $K$, the scalar curvature invariant, measured from the string ($\lambda = 3.9$) compared to the cone prediction. The ratio is also shown, and we see good agreement slightly away from the apex, where finite $\lambda$ 'resolves' the cone. Moving far from the cone ie. $z<2.1$ or $r>0.4$ the cone is no longer a good approximation, the actual curvature becoming more homogeneous in $z$.
• Figure 3: Plots of $\sin{\chi}$ on the $z = L_n$ axis measured from string solutions with $\lambda = 1.5, 3.9$. For the larger $\lambda$ we see the curve become more 'constant', and furthermore, the value is remarkably consistent with one, the cone prediction.
• Figure 4: Plot of $\sin{\chi}$ computed from $\lambda = 3.9$ string, including normalisation correction to ensure $\sin{\chi} = 1$ at $z = L_n$, so that $\chi$ can be meaningfully extracted.
• Figure 5: Plots of remaining metric components $c1, c2$ computed from the numerical solution with $\lambda = 3.9$ using the 'corrected' $\sin{\chi}$. The cone predicts that $c1 = c2 = e^{2 B}$ which is also plotted for comparison. Very good agreement is seen, given that the underlying data is on a lattice of only approximately $20*20$ and resolution effects may give significant numerical errors.