Rotating Circular Strings, and Infinite Non-Uniqueness of Black Rings

TL;DR

The paper constructs rotating circular strings (dipole black rings) in five-dimensional gravity with gauge dipoles, showing that regular-horizon solutions exist with only mass and angular momentum as conserved charges, but a continuous dipole parameter yields infinite non-uniqueness. It analyzes neutral and dipole rings, extremal limits, and the loop of fundamental strings, then extends to three-charge rings from brane intersections with explicit supergravity and entropy counting. A microscopic counting matches the Bekenstein–Hawking entropy of extremal rings to leading order, reinforcing a string-theory description of black rings as extended, momentum-carrying loops. The work demonstrates a rich landscape of non-unique, horizon-bearing solutions in higher-dimensional gravity and connects them to brane intersections and microstate counting in string/M-theory, with implications for black hole hair and stability.

Abstract

We present new self-gravitating solutions in five dimensions that describe circular strings, i.e., rings, electrically coupled to a two-form potential (as e.g., fundamental strings do), or to a dual magnetic one-form. The rings are prevented from collapsing by rotation, and they create a field analogous to a dipole, with no net charge measured at infinity. They can have a regular horizon, and we show that this implies the existence of an infinite number of black rings, labeled by a continuous parameter, with the same mass and angular momentum as neutral black rings and black holes. We also discuss the solution for a rotating loop of fundamental string. We show how more general rings arise from intersections of branes with a regular horizon (even at extremality), closely related to the configurations that yield the four-dimensional black hole with four charges. We reproduce the Bekenstein-Hawking entropy of a large extremal ring through a microscopic calculation. Finally, we discuss some qualitative ideas for a microscopic understanding of neutral and dipole black rings.

Figures (5)

• Figure 1: The non-topological winding number $n$ of the ring is proportional to the local charge ${\cal Q}$ measured from the electric flux of $H$ across an $S^2$ that encloses a section of the string. An azimuthal angle has been suppressed in the picture, so the $S^2$ is represented as a circle.
• Figure 2: Plot of horizon area vs (spin)$^2$, for given mass, for the neutral rotating black ring (solid) and black hole (dotted). The reduced variables $j^2$ and $a_H$ are defined in (\ref{['etadef']}), (\ref{['zetadef']}). There are two branches of black rings, which branch off from the cusp at $(j^2,a_H) = (27/32,1)$, and which are dubbed "large" and "small" according to their area. For spins in the range $27/32\leq j^2<1$ the black rings in the two branches coexist with a black hole of the same mass and spin, implying three-fold non-uniqueness. Other interesting features are: The black hole at $j^2=27/32$, i.e., with the same mass and spin as the minimally spinning ring, has ${a_H}=\sqrt{5}/2$. At $j^2={a_H}^2=8/9$ the curves intersect and we find a black hole and a (large) black ring both with the same mass, spin and area. The limiting solution at $(j^2, a_H)=(1,0)$ is a naked singularity. Fastly spinning black rings, $j^2\to\infty$, become thinner and their area decreases as ${a_H}\sim 1/(j\sqrt{2})$.
• Figure 3: $a_H$ vs $j^2$, for different values of $q$, for dipole rings with dilaton coupling $\alpha=2\sqrt{2/3}, \sqrt{2/3}, 0$ ($N=1,2,3$). $q$ varies continuously, but we have plotted the curves for only a few representative values, with $q$ increasing in the direction of the arrow: shorter dashing corresponds to larger $q$, and the solid curves correspond to the neutral ring ($q=0$). For any fixed $j$, i.e., fixed mass and spin, such that $j^2>27/32$, there are always rings with $q$ in a continuous range of values, implying continuous non-uniqueness. Observe also that for fixed $q\neq 0$, $j$ is bounded above and below so the range of allowed spins is finite, and becomes narrower as $q$ grows. When $q$ reaches its maximum value the curves degenerate to a point, which is at $(j^2,a_H)=(1,0)$, $(.95,0)$, $(.93,.37)$ for $N=1,2,3$ resp. (see fig. \ref{['fig:extr']} for the maximum values of $q$). For $N=3$, the endpoints of the curves lie on the thick grey line, which corresponds to extremal dipole rings of finite horizon area.
• Figure 4: Coordinate system for black ring metrics (adapted from ER2). The diagram sketches a section at constant $t$ and $\varphi$. Surfaces of constant $y$ are ring-shaped, while $x$ is a polar coordinate on the $S^2$ (roughly $x\sim\cos\theta$). $x=\pm 1$ and $y=-1$ are fixed-point sets ( i.e., axes) of $\partial_\varphi$ and $\partial_\psi$, resp. Infinity lies at $x=y=-1$.
• Figure 5: (Spin)$^2$ vs local charge, for extremal rings of fixed mass. Although we only plot the cases of integer $N$, the curves for all $1<N\leq 3$ are qualitatively similar to $N=2,3$, while for $0<N<1$ there is no upper bound for $q$, and the curve asymptotes to $j^2=1$ as $q\to \infty$.