Grey galaxies in $AdS_5$

It has recently been conjectured [S. Kim et al., J. High Energy Phys. 11, 024 (2023)] that the end point of the rotational superradiant instability of black holes in

is a Grey Galaxy: an

\omega=1 ω = 1

black hole sitting at the centre of

AdS_4 A d S 4

, surrounded by a large disk of rapidly rotating gravitons and other bulk fields. In this paper we study Grey Galaxies in

AdS_5 A d S 5

. In this case, the rotational group is of rank 2, and so has two distinct angular velocities

\omega_1 ω 1

and

\omega_2 ω 2

. We demonstrate that

AdS_5 A d S 5

hosts two qualitatively distinct Grey Galaxy phases: the first with either

\omega_1≈ 1 ω 1 ≈ 1

or

\omega_2≈ 1 ω 2 ≈ 1

, and the second with both angular velocities

≈ 1 ≈ 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

\omega_1 ω 1

and

\omega_2 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 - has the spatial distribution of the stress tensor of an equilibrated boundary conformal fluid, rotating at the given angular speeds

\omega_i ω i

. We also briefly comment on the structure of Grey Galaxies in

AdS_D A d S D

for

D ;gt 5 D > 5

.