Bicycling Black Rings

TL;DR

The paper constructs and analyzes two 4+1-dimensional vacuum black hole systems with spin in two planes: a novel bicycling black rings (bi-rings) configuration and the known doubly spinning black ring. It employs the inverse scattering method to build the bi-ring solution and provides a thorough analysis of its parameterization, balance, horizons, and absence of CTCs, exploring zero-temperature limits and phase structure. By revisiting the doubly spinning ring, the work details two extremal zero-temperature limits and clarifies how these limits relate to extremal Myers-Perry black holes, enriching the understanding of zero-temperature phases. Together, the results illuminate a rich landscape of extremal, non-unique black hole configurations in higher dimensions, including potential bi-ring saturns and generalized multi-ring systems, with implications for higher-dimensional gravity and microphysical entropy studies.

Abstract

We present detailed physics analyses of two different 4+1-dimensional asymptotically flat vacuum black hole solutions with spin in two independent planes: the doubly spinning black ring and the bicycling black ring system ("bi-rings"). The latter is a new solution describing two concentric orthogonal rotating black rings which we construct using the inverse scattering technique. We focus particularly on extremal zero-temperature limits of the solutions. We construct the phase diagram of currently known zero-temperature vacuum black hole solutions with a single event horizon, and discuss the non-uniqueness introduced by more exotic black hole configurations such as bi-rings and multi-ring saturns.

Figures (9)

• Figure 1: Bicycling black rings in orthogonal planes.
• Figure 2: Rod configuration representing the sources for the seed metric $G_0$. Solid black lines in the figure correspond to rod sources of uniform density $+1/2$ and the dashed rods to uniform densities $-1/2$.
• Figure 3: Rod structure of the regular bicycling black ring solution. Rod directions are shown over each rod.
• Figure 4: Visualization of the symmetric bicycling black ring system. Shown is the superposition of two identical singly spinning rings in orthogonal planes for six different values of the angular momentum $j_\psi=j_\phi=\sqrt{(1+\nu)^3/(8\nu)}$ (recall EEF that $0 <\nu < 1/2$ for the thin ring branch and $1/2 <\nu < 1$ on the fat black ring branch). The total mass is fixed to be the same for each plot. The embeddings are plotted on the same scale. Interactions between the two rings are ignored in this model, but in the real bi-ring solution interactions play an important role.
• Figure 5: (a) Area $a_\mathrm{H}$ vs. $j^2$ for solutions with equal angular momenta in the two planes of rotation, $j \equiv j_\psi = j_\phi$. The gray curve shows the Myers-Perry black hole phase and the black curve is the symmetric bi-ring configuration. Both curves have endpoints at $j^2 = 1/4$. The solutions at the endpoints have finite area and zero temperature. (b) Angular velocities vs. inner horizon radius $r_\mathrm{inner}$. The solid (dashed) curves corresponds to the angular velocity of the $S^2$ ($S^1$) of the horizon. The minimum inner radius is $r_\mathrm{m}\approx 0.95$.