Nonuniform black strings in various dimensions

TL;DR

This work numerically constructs nonuniform black strings (NUBS) in D=6–11, extending the GL branch into the strongly non-linear regime. Using a relaxation-based solver in conformal coordinates, the authors test the near-merger geometry against a cone and provide evidence for a power-law scaling of the minimal horizon radius with a critical exponent, while observing no wiggles within their data. They find that NUBS are more massive and, in the examined range, exhibit entropy and temperature trends that depend on dimension, with cone-like waist geometry confirmed to within ~10% near the waist. The results reinforce the proposed phase diagram and offer quantitative estimates of asymptotic thermodynamic and geometric quantities, though numerical convergence becomes challenging beyond D≈11 and at extreme deformations.

Abstract

The nonuniform black strings branch, which emerges from the critical Gregory-Laflamme string, is numerically constructed in dimensions 6 <= D <= 11 and extended into the strongly non-linear regime. All the solutions are more massive and less entropic than the marginal string. We find the asymptotic values of the mass, the entropy and other physical variables in the limit of large horizon deformations. By explicit metric comparison we verify that the local geometry around the ``waist'' of our most nonuniform solutions is cone-like with less than 10% deviation. We find evidence that in this regime the characteristic length scale has a power-law dependence on a parameter along the branch of the solutions, and estimate the critical exponent.

Figures (17)

• Figure 1: A suggested phase diagram. The vertical axis is the dimensionless mass density and the horizontal axis is related to the scalar charge. Shown is the GL point where the uniform string becomes marginally unstable and from which a new branch of non-uniform strings emerges. This branch extends until it meets the caged black hole branch at the "merger point". In $D>13$ the NUBSs branch is less massive than the critical string indicating that the order of the phase transition, triggered by appearance of the GL-mode, changes from first to second.
• Figure 2: The grid hierarchy, 3 meshes are shown. While the coarsest mesh covers the entire domain, the finer grids extend only to half size of the previous coarser mesh. The displayed field $B$ (as well as $A$ and $C$) is most variable near the horizon, justifying this "AMR"-type construction.
• Figure 3: The normalized entropy (\ref{['area']}) of NUBSs in 6D as a function of $\lambda$. Shown are the solutions obtained using three mesh resolutions: low (pluses), medium (crosses) and high (stars), obtained by halving the grid-spacings. In the region where all three overlap we observe nearly second order convergence. The NUBSs are more entropic than that of the critical uniform string. In the limit of large $\lambda$ the entropy reaches saturation. The bending that shows up beyond certain $\lambda$ at lower resolution, at a higher resolution moves to larger $\lambda$'s in the manner that doesn't form clearly convergent sequence.
• Figure 4:
• Figure 5: The mass as a function of $\lambda$ in 6D, same conventions as in figure \ref{['fig_S_6D']}. The markers designate the mass computed from (\ref{['relative_charges']}), the solid lines indicate the result obtained by integration of the first law, see (\ref{['m_1law']}). Both methods agree well and the agreement improves at higher resolution.