The Phase Structure of Higher-Dimensional Black Rings and Black Holes

TL;DR

This work develops a semi-analytic, perturbative framework to map the phase structure of rotating black holes in D≥5, centering on thin black rings constructed by matching a bent boosted black string to a linearized exterior solution. A key result is the zero-tension equilibrium condition T_{zz}=0 that ensures horizon regularity, together with a near-horizon perturbation analysis that confirms the horizon remains regular and that global quantities M and J are unchanged at leading order in 1/R while the ring's geometry acquires latitude-dependent corrections. The authors show that, in the ultra-spinning regime for D≥6, black rings possess higher entropy than Myers–Perry holes, supporting a qualitative phase diagram in which rings, MP holes, and pinched black holes (and black Saturns) are interconnected through merger transitions. They further argue that the phase structure mirrors known KK phases on T^2, enabling a qualitative, semi-quantitative construction of the higher-dimensional phase diagram and predicting an infinite family of pinched black-hole branches emanating from the MP curve. The results provide a constructive scaffold for understanding black-hole phases in higher dimensions and motivate future numerical, charged, and AdS generalizations.

Abstract

We construct an approximate solution for an asymptotically flat, neutral, thin rotating black ring in any dimension D>=5 by matching the near-horizon solution for a bent boosted black string, to a linearized gravity solution away from the horizon. The rotating black ring solution has a regular horizon of topology S^1 x S^{D-3} and incorporates the balancing condition of the ring as a zero-tension condition. For D=5 our method reproduces the thin ring limit of the exact black ring solution. For D>=6 we show that the black ring has a higher entropy than the Myers-Perry black hole in the ultra-spinning regime. By exploiting the correspondence between ultra-spinning black holes and black membranes on a two-torus, we take steps towards qualitatively completing the phase diagram of rotating blackfolds with a single angular momentum. We are led to propose a connection between MP black holes and black rings, and between MP black holes and black Saturns, through merger transitions involving two kinds of `pinched' black holes. More generally, the analogy suggests an infinite number of pinched black holes of spherical topology leading to a complicated pattern of connections and mergers between phases.

Figures (6)

• Figure 1: Area vs spin for fixed mass, $a_H(j)$, in seven dimensions. The thin curve is our result for thin black rings, valid at large $j$ and extrapolated (dashed) down to $j\sim O(1)$. The thick curve is the exact result for the MP black hole. The vertical dotted line intersects this curve at the inflection point $j=j_{\rm mem}=2^{1/4}/\sqrt{3}$, $a_H=\sqrt{2}$. It signals the onset at larger $j$ of membrane-like behavior for MP black holes. Eq. (\ref{['7D']}) gives the asymptotic form of the curves. The same qualitative features appear for all $D\geq 6$.
• Figure 2: Correspondence between phases of black membranes wrapped on a $\mathbb{T}^2$ of side $L$ (left) and fastly-rotating MP black holes with rotation parameter $a\sim L\geq r_0$ (right: must be rotated along a vertical axis): (i) Uniform black membrane and MP black hole. (ii) Non-uniform black membrane and pinched black hole. (iii) Pinched-off membrane and black hole. (iv) Localized black string and black ring.
• Figure 3: $a_H (\ell)$ phase diagram in seven dimensions ($\mathcal{M}^5 \times \mathbb{T}^2$) for Kaluza-Klein black hole phases with one uniform direction. Shown are the uniform black membrane phase (dotted), the non-uniform black membrane phase (solid) and the localized black string phase (dashed). For the latter two phases, we have also shown their $k=2$ copy. The non-uniform black membrane phase emanates from the uniform black membrane phase at the GL point $\ell_{\rm GL} = 0.811$, while the $k=2$ copy starts at the 2-copied GL point $\ell_{\rm GL}^{(2)} = \sqrt{2} \ell_{\rm GL} =1.15$. The connection between the curves is shown in greater detail. The gap between the solid and dashed curves reflects the difficulty in getting numerical data close to the merger. This figure is representative for the phase diagram of phases on $\mathcal{M}^{D-2} \times \mathbb{T}^2$ for all $6 \leq D \leq 14$.
• Figure 4: Expected $a_H (\ell)$ phase diagram for KK black hole phases with one uniform direction on $\mathcal{M}^{D-2} \times \mathbb{T}^2$, when $D> 14$. We also show the $k=2$ copy of non-uniform phases, obtained from the main sequence using (\ref{['copy']}). The uniform black membrane curve (dotted) and the asymptotic form of the localized black string curves (dashed) are known exactly. Of the solid lines for non-uniform black membranes, we only know the position of the GL points where they begin, and the fact that they must be tangent to the uniform membrane curve at this point. The points of merger to the black string curves are unknown.
• Figure 5: Qualitative completion of fig. \ref{['fig:MPBR']} using fig. \ref{['fig:KKphases7']}. The gray line corresponds to the conjectured phase of pinched black holes, which branch off tangentially from the MP curve (thick) at a value $j_{\rm GL}> j_{\rm mem}$, and merge with the black ring curve (thin). At any given dimension, the phases may not display the swallowtail in fig. \ref{['fig:KKphases7']}, depicted here, but may instead be smoother like fig. \ref{['fig:abovecrit']}. Even if there is a cusp, the merger need not happen at it.