Combining Everything (1) Becomes Angular Momentum Hole Area (a ) Projection of Moduli onto Plane Extremal Curve a Vs J Plane

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 TH = TR, ΩH = ΩR 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 suggests à la Gubser-Mitra that thermal stability is tied to classical stability. We also comment on the possibility of constructing plasma configurations that are dual to black Saturns in AdS.