Metastable Black Saturns

TL;DR

The paper analyzes the thermodynamic stability of five-dimensional black Saturns with multiple horizons by extremizing the total entropy in a 2D moduli space at fixed $M$ and $J$. It reproduces isothermal equilibria with $T_H=T_R$ and $Ω_H=Ω_R$ and uncovers nonisothermal equilibria that can be metastable when the entropy Hessian is negative definite. Metastable equilibria appear on the thin ring branch within a narrow window $0.92457<j<0.92463$, consistent with classical stability expectations and suggesting a link between thermal and classical stability in the spirit of the Gubser-Mitra conjecture. The work provides a quantitative framework, discusses potential AdS dual plasma configurations, and sheds light on multihorizon black hole thermodynamics.

Abstract

Black Saturns have multiple horizons and so offer a testing ground for the ideas of black hole thermodynamics. In this note, we numerically scan for phases that are in equilibrium by extremizing total entropy in the 2-dimensional moduli space of stationary, singly rotating black Saturns with fixed total mass and angular momentum. On top of the known T_H=T_R, Omega_H=Omega_R configurations, we find phases that do not balance the temperature and angular velocity of the ring and the hole. But these (and most of the balanced Saturns) go away when we demand that the system is metastable, by imposing that the Hessian of the entropy is negative definite. Metastablity occurs when the dimensionless total angular momentum lies in a narrow window 0.92457<j<0.92463 of the thin ring branch. This is consistent with the expected range of classical stability of black Saturns and therefore may imply that thermal stability is tied to classical stability, in analogy with Gubser-Mitra in the translationally-invariant case. We also comment on the possibility of constructing plasma configurations that are dual to black Saturns in AdS.

Figures (4)

• Figure 1: The moduli space of black Saturns that we consider is 2-dimensional. At regular points $A_H$, the area of the event horizon of the black hole, and $J_H$, the Komar angular momentum of the black hole, are good coordinates. However for some values of $(J_H,A_H)$ there are multiple states. The $(J_H,\ A_H)$ coordinate system breaks down on critical curves. When a state is on such a curve, all radiation between the hole and the ring carries a fixed angular momentum to entropy ratio from the hole. All nonisothermal equilibria lie on such curves.
• Figure 2: The moduli space of equilibria is dominated by the isothermal equilibria, shown on the left. On the right one sees all of the equilibria, the nonisothermals are localized at small angular momentum $j$.
• Figure 3: The phase diagram of equilibria (most of which are unstable) is much richer than previously thought. Here are the lowest angular momentum equilibria, where the structure is the most intricate. The maximum entropy is attained on the upper-left where it asymptotes to the minimally rotating black ring. On this branch the hole is counter-rotating. It continues to the lower-right, where the hole becomes so counter-rotating that its area eventually goes to zero at $j=1$. The two isothermal branches go off to the right slightly higher. They meet at the triple point $j=.92457$ and form a nonisothermal branch whose continuation yields the rest of the equilibria. It ends when it becomes degenerate as its second derivatives vanish.
• Figure 4: The metastable equilibria are all on the thin ring isothermal branch, in a narrow window that ends at the minimal angular momentum isothermal equilibrium, a triple point in the phase diagram where one may expect a second order phase transition.