Black Saturn

TL;DR

Black Saturn constructs an exact 4+1D vacuum solution describing a rotating spherical black hole surrounded by a rotating black ring, balanced by angular momentum. The solution, obtained via the inverse scattering method with a three-soliton transformation, reveals rich physics including continuous non-uniqueness for fixed mass and total angular momentum, and rotational frame-dragging where a ring can induce BH rotation even when the BH has vanishing intrinsic spin. Analyses of the parameter space, rod structure, horizons, and Komar integrals show that both horizon components contribute to a nontrivial phase diagram with co- and counter-rotating branches, and that the total ADM mass and angular momentum decompose consistently into horizon charges. Thework demonstrates striking phenomena in higher-dimensional gravity and sets the stage for generalizations to multiple rings, doubly spinning saturns, and charged configurations, with implications for black hole thermodynamics and stability in higher dimensions.

Abstract

Using the inverse scattering method we construct an exact stationary asymptotically flat 4+1-dimensional vacuum solution describing Black Saturn: a spherical black hole surrounded by a black ring. Angular momentum keeps the configuration in equilibrium. Black saturn reveals a number of interesting gravitational phenomena: (1) The balanced solution exhibits 2-fold continuous non-uniqueness for fixed mass and angular momentum; (2) Remarkably, the 4+1d Schwarzschild black hole is not unique, since the black ring and black hole of black saturn can counter-rotate to give zero total angular momentum at infinity, while maintaining balance; (3) The system cleanly demonstrates rotational frame-dragging when a black hole with vanishing Komar angular momentum is rotating as the black ring drags the surrounding spacetime. Possible generalizations include multiple rings of saturn as well as doubly spinning black saturn configurations.

Figures (14)

• Figure 1: Sources for the seed metric $G_0$. The solid rods have positive density and the dashed rod has negative density. The rods are located at the $z$-axis with $\rho=0$ and add up to an infinite rod with uniform density such that $\det G_0 = -\rho^2$. The labeling of the rod endpoints is a little untraditional, but is simply motivated by the fact that we are going to use the inverse scattering method to add solitons at $z=a_1$, $a_2$ and $a_3$.
• Figure 2: Rod structure of the black saturn solution. Note that the rods are placed on the $\bar{z}$-axis, see section \ref{['s:rods']} for the definition of $\bar{z}$. The dots in figure \ref{['fig:saturnrod']}(a) denote singularities at $\bar{z}=0$, which are removed by the fixing $c_1$ according to (\ref{['eqn:c1']}) (figure \ref{['fig:saturnrod']}(b)). This choice makes the $\rho=0$ metric smooth across $\bar{z}=0$. Figure \ref{['fig:saturnrod']}(b) also shows the directions of the rods.
• Figure 3: Behavior of the reduced physical parameters for the Myers-Perry black hole (light gray) and the black ring (dark gray). Note that we are using a logarithmic scale for the temperature.
• Figure 4: For fixed total mass and some representative values of the $a_\textrm{H}^\textrm{BR}$, the various reduced quantities are plotted vs. $j^2$. The gray curves correspond to the Myers-Perry black hole (light gray) and the black ring (darker gray) respectively.
• Figure 5: Plots of $a_\textrm{H}$ vs. $j^2$ for different representative values of $(a_\textrm{H}^\textrm{BH})^2=6,3,\frac{3}{2}, \frac{1}{10}$ (black solid curves). The dotted curve corresponds to a Myers-Perry black hole surrounded by a nakedly singular ring. Again, the gray curves correspond to the Myers-Perry hole (gray) and the black ring (darker gray). The smaller plot zooms in on the small $j$ part of the $a_\textrm{H}^\textrm{BH}=\sqrt{6}$ curve.