Grey Galaxies in $AdS_5$

TL;DR

This work extends the Grey Galaxy program to AdS$_5$ by identifying rank-2 and rank-4 Grey Galaxy phases arising from two angular velocities $ω_1$ and $ω_2$, and by constructing a bulk rank-4 solution supported by a dilaton gas. The boundary stress tensor is shown to be the sum of a central Kerr–AdS$_5$ black hole contribution and a bulk conformal-fluid contribution, with the gas part taking a universal hydrodynamic form $T_{μν} \propto γ^4(θ)(4u_{μ}u_{ν}+g_{μν})$. A detailed bulk computation in AdS$_5\times S^5$ demonstrates that the bulk gas behaves like an equilibrated perfect fluid and that the backreaction can be consistently captured by an improved, conserved bulk stress tensor, which reduces to a fluid form consistent with the partition function framework. The authors conjecture a general boundary stress tensor for Grey Galaxies with arbitrary bulk matter, and they develop a microcanonical phase diagram for ${\cal N}=4$ SYM that exhibits phase transitions between regular black holes and the various Grey Galaxy phases, along with discussions of higher-dimensional generalizations and potential instabilities.

Abstract

It has recently been conjectured \cite{Kim:2023sig} that the end point of the rotational superradiant instability of black holes in $AdS_4$ is a Grey Galaxy: an $ω=1$ black hole sitting at the centre of $AdS_4$, surrounded by a large disk of rapidly rotating gravitons and other bulk fields. In this paper we study Grey Galaxies in $AdS_5$. In this case, the rotational group is of rank 2, and so has two distinct angular velocities $ω_1$ and $ω_2$. We demonstrate that $AdS_5$ hosts two qualitatively distinct Grey Galaxy phases: the first with either $ω_1\approx 1$ or $ω_2\approx 1$, and the second with both angular velocities $\approx 1$. We use these results to present a conjecture for a part of the phase diagram of ${\cal N}=4$ Yang-Mills (as a function of energy and the two angular momenta) that displays several phase transitions between regular black holes and various Grey Galaxy phases. We present an explicit gravitational construction of the phases in which $ω_1$ and $ω_2$ are both parametrically close to unity, and demonstrate that the corresponding boundary stress tensor is the sum of two pieces. The first is the stress tensor of the central black hole. The second - the contribution of the bulk gas - takes the form of the stress tensor of an equilibrated boundary conformal fluid, rotating at the given angular speeds $ω_i$. We also briefly comment on the structure of Grey Galaxies in $AdS_D$ for $D > 5$.

Figures (4)

• Figure 1: The plot shows that extremal BHs are below the superradiant BHs. While the plots above are presented for $\mathbf{b=0.1}$, the analogous plots at all values of $b \in (0,1)$ are qualitatively similar.
• Figure 2: Here the red curve is for $b \rightarrow 0.99,a\rightarrow0.01$ and blue curve is for $a\rightarrow0.99,b\rightarrow0.01$ , finally the orange curve is for $a\rightarrow0.99,b\rightarrow0.99$.
• Figure 3: A constant $J_2$ ($J_2=2$) slice of the phase diagram. We have plotted $E$ on the $y$ axis and $J_1$ on the $x$ axis. The black line is the unitarity bound $E=2+J_1$. The blue curve is the line $\omega_1=1$. When $J_1 >2$, this curve represents a phase boundary. Above this curve, the dominant phase is the usual vacuum black hole. Below this curve (shaded yellow region) we have a rank 2 Grey Galaxy. All points on the green 45-degree line (in this phase) have the same entropy, given by the entropy of the vacuum $\omega_1=1$ black hole at the lower end of the green line. The rank 2 Grey Galaxy phase ends on the red 45-degree line, which meets the $\omega_1=1$ curve at $J_1=2$. Below the red curve (shaded blue region) the dominant phase is the rank four Grey Galaxy all the way down to the unitarity bound. We have not attempted to sketch the phase diagram for $J_1<2$; part of the phase diagram is best visualized in constant $J_2$ slices.
• Figure 4: Here we plot the line diagram, which shows the transitions of the different phases as we lower the energy $E$ (Keeping the $J_1$ and $J_2$ fixed) starting from the vacuum black hole phase. These plots are presented for three different cases; namely $J_1> J_2$, $J_2>J_1$, and $J_1 = J_2$.