J
H
E
P06(2020)104
Published for SISSA by Springer
Received: April 17, 2020
Accepted: May 24, 2020
Published: June 16, 2020
To go or not to go with the 
ow: Hawking radiation
at strong coupling
Jorge E. Santos
Department of Applied Mathematics and Theoretical Physics, University of Cambridge,
Wilberforce Road, Cambridge, CB3 0WA, U.K.
Institute for Advanced Study,
Princeton, NJ 08540, U.S.A.
E-mail: jss55@cam.ac.uk
Abstract: We construct the gravitational dual of a one-parameter class of states of
strongly coupled SU(N) N = 4 SYM at innite N and asymptotic temperature T1, on
a xed Schwarzschild black hole background with temperature TBH. The resulting bulk
geometry is of the 
owing type and allow us to measure Hawking radiation at strong cou-
pling. The outgoing Hawking 
ux is a function of the dimensionless ratio   T1=TBH and
appears to be non-monotonic in  . At present, we have no eld theory understanding for
this behaviour.
Keywords: AdS-CFT Correspondence, Black Holes, Nonperturbative Eects
ArXiv ePrint: 2003.05454
Open Access, c
 The Authors.
Article funded by SCOAP3.
https://doi.org/10.1007/JHEP06(2020)104
J
H
E
P06(2020)104
Contents
1 Introduction 1
2 Constructing the holographic dual 3
3 The horizon and its properties 7
4 Extracting the holographic-stress energy tensor 9
5 Results 11
6 Discussion 17
A Convergence tests 18
1 Introduction
Understanding the behaviour of Quantum Field Theories (QFTs) in curved spacetime is
an important problem, not least because we know that the universe does contain regions
of very large curvature. A key discovery was Hawking's calculation demonstrating particle
production in black hole backgrounds [1, 2]. These particles have a thermal spectrum,
conrming that black holes should properly be thought of as thermodynamic objects. A
general argument based on the Euclidean time formalism shows that for any QFT, an
equilibrium state on a black hole background (the so-called Hartle-Hawking state) should
be thermal [3]. However most of what is known about QFTs in curved spacetime comes
from calculations involving free or weakly interacting theories. Little is known about the
case when the QFT is strongly coupled. In fact, even in the weakly interacting regime
there are interesting open puzzles, as for instance those reported in [4].
Gauge/Gravity Duality provides a new way of probing the behaviour of certain strongly
coupled QFTs in curved backgrounds. In its most precise and well motivated form, it is the
claim that Type IIB Superstring theory on AdS5 S5 is equivalent to N = 4 Super Yang-
Mills (SYM) theory on the (3 + 1) dimensional conformal boundary [5{8]. In the large
N strong coupling limit of the boundary gauge theory, the bulk string theory becomes
weakly coupled and the string length scale becomes small. In principle this should allow
us to study quantum eects in the strongly coupled theory, such as Hawking radiation, by
solving classical gravitational equations of motion in the bulk. This technique was explored
in [9{12], and reviewed rather beautifully in [13].
In order to probe the Hawking eect with this technique, we need to take a xed
Schwarzschild boundary geometry. This leads to two classes of bulk gravitational duals:
\Black Funnels" and \Black Droplets" [9{12]. Black Funnels have a connected horizon
{ 1 {
J
H
E
P06(2020)104
extending from the boundary black hole into the bulk and out to an asymptotic region. In
contrast, the horizon in a Black Droplet solution has two disconnected parts. One extends
from and surrounds the boundary black hole (the \droplet"), and the other is a deformed
planar horizon which is not connected to the boundary at all.
Given the xed black hole geometry on the boundary with temperature TBH, the
bulk solutions are characterized by two free parameters: T1, the temperature of the bulk
horizon in the asymptotic region, and TH , the temperature of the bulk horizon where it
meets the boundary black hole. If TH 6= TBH then the Euclidean boundary geometry
exhibits a conical singularity and the stress tensor diverges at the horizon. These solutions
are important (the Boulware vacuum state is described by such a solution [14]) but we will
not consider them in this paper. From this point on we shall always take TH = TBH .
Given our remaining freedom in xing the parameters, there are two special cases of
particular interest. The rst is the choice of parameters T1 = TBH. This gives an equilib-
rium solution and corresponds to the Hartle-Hawking state of the eld theory. Physically,
a thermal state of the eld theory is in equilibrium with a Schwarzschild black hole which
emits Hawking radiation at the same temperature. The second case of particular interest is
when T1 = 0. This is an out of equilibrium solution corresponding to the so called Unruh
state of the eld theory. Physically, the eld theory approaches its natural vacuum state
in the asymptotically 
at region (where a natural choice of vacuum state exists) but we
now expect to see an outgoing energy 
ux coming from the Hawking radiation emitted by
the higher temperature black hole. This is a good approximation to the state of the eld
theory that would be obtained after gravitational collapse, but before the resulting black
hole has had time to evaporate.
Bulk duals corresponding to the Hartle-Hawking state and to the Unruh state have
been constructed previously. For the Hartle-Hawking T1 = TBH choice of parameters only
black funnel solutions exist [15, 16]. These solutions can be said to describe Hawking
radiation in that they contain a black hole in equilibrium with a plasma, but there is no
net energy 
ux. For the Unruh T1 = 0 choice of parameters, the only bulk solutions
constructed so far have been of the droplet type [17]. Surprisingly, this means that they
also exhibit no energy 
ux, at least at the leading order in N in which a classical description
of the bulk spacetime is valid. It is expected that Hawking radiation will be present at
next leading order in N , but the goal of observing something which can be interpreted as
Hawking radiation in a classical solution to the Einstein equation has not yet been realised
in the case of the Unruh state.
In this paper we construct for the rst time numerical solutions of the black funnel type,
containing a Schwarzschild black hole at the boundary, in which T1 6= TBH. Such states
have been considered previously in the weak coupling context by Frolov and Page in [18].
We will study the properties of the system as we vary the ratio T1=TBH. We will start with
T1=TBH > 1 and decrease it passing through T1=TBH = 1 into the region T1=TBH < 1.
While we are unable to take T1 all the way to zero, we can lower it signicantly, and observe
an energy 
ux corresponding to outgoing Hawking radiation from the black hole. This also
means that our bulk horizon is not Killing, where we evade the zeroth law of black hole
mechanics [19{21] due to the fact that the horizon is non-compact. Other \
owing funnel"
{ 2 {
J
H
E
P06(2020)104
Figure 1. An illustration of the coordinate transformation mapping the original black funnel
triangular domain (left panel), into a square domain (right panel).
solutions containing two boundary black holes have been obtained previously in the context
of AdS boundary geometries [22, 23] (see also [24{31] for closely related geometries).
The outline of the paper is as follows. Section 2 explains how to formulate the con-
struction of 
owing funnels with arbitrary T1=TBH. In section 3 we detail how to compute
some of the horizon properties of 
owing geometries, such as its expansion and shear. Sec-
tion 4 shows how one can extract the holographic stress energy tensor from the numerical
solutions, and in section 5 we present our numerical results. We close with some nal
discussions in section 6.
2 Constructing the holographic dual
We will work in ve bulk spacetime dimensions, and thus have a four-dimensional bound-
ary spacetime. We expect that appropriate generalisations exist in higher dimensions.
The solution we seek to construct necessarily exhibits temperature gradients across the
horizon, the horizon generators cannot be associated with the integral curves of a Killing
eld [23, 24], i.e. the horizon is not Killing. So we expect our solution to follow under the
class of 
owing geometries constructed in [24{31].
We begin by reviewing a coordinate system that is well adapted to the construction of
black funnels [15, 23]. In essence, a black funnel contains a single component horizon which
connects the boundary black hole to an asymptotic region that is located innitely far away
from the boundary black hole (see left panel of gure 1). In this asymptotic region, the
geometry should approach that of a standard ve-dimensional planar black hole, which we
denote on the left panel of gure 1 by P. Thus, black funnels admit a natural triangular
integration domain with three boundaries: a horizon, the planar black hole metric, and the
conformal boundary.
Working with triangular integration domains can be tricky, so in [15, 23] a new coordi-
nate system was introduced, such that the point where the bulk future horizon H+ meets
the boundary, was blown up into a line. By a careful inspection of the region where the
bulk horizon meets the boundary horizon, one can show that the metric approaches that
of a hyperbolic black hole. In fact this has to be the case for any horizon that meets the
boundary, the reason being close to the conformal boundary the geometry is hyperbolic,
up to subleading terms that are related to the holographic stress energy tensor. Moreover,
{ 3 {
J
H
E
P06(2020)104
the most general cohomogeneity one line element that is static and manifestly exhibits
hyperbolic symmetry for each constant time hyperslice is that of a hyperbolic black hole.
Each of these hyperbolic black holes is then determined by a choice of boundary black hole
temperature, and bulk horizon temperature at the hyperbolic black hole horizon. We are
interested in the situation where the boundary black hole has the same temperature as the
bulk horizon temperature at the hyperbolic black hole horizon. The only such black hole
is the zero energy hyperbolic black hole (which is isometric to pure AdS), and is given by
ds2H =
L2
z2

 1 z2
 dt^2 + dz2
1 z2 + d
2 + sinh2  d
22

; (2.1)
where here henceforth d
22 will denote the unit radius round metric on S
2.
We want to nd coordinates (x; y) where at x = 0 we approach the zero energy hyper-
bolic black hole with the same temperature as the boundary black hole TBH, at x = 1 we
want to impose that we approach a Schwarzschild black brane with some temperature T1
and nally we want y = 0 to denote the conformal boundary. Ideally the horizon would
be an hypersurface with y = 1, which was achieved in [23] by a careful choice of gauge. In
this work we decided to follow more closely the approach devised in [24], but with some
important dierences which we will detail below.
Finally, at the conformal boundary we want to consider a geometry which is conformal
to Schwarzschild, since this is the spacetime we want our eld theory to live on. This
means that close to y = 0 the solution has to approach a Schwarschild black string
ds2 =
L2
y2
2641 2M
r

dt2 +
dr2
1 2M
r
+ r2d
22 + dy
2
375 ; (2.2)
with temperature
TBH =
1
8M
=
1
4r0
; (2.3)
where we dened the Schwarzschild radius r0 = 2M . Note that the temperature of the
black string matches that of the Hawking temperature of a Schwarzschild black hole.
The general idea in both [23] and [24] was to use the so called DeTurck method, which
was rst introduced in [32], and reviewed in detail in [33, 34]. Recall we seek to construct
solutions of the ve-dimensional Einstein equation
Rab +
4
L2
gab = 0 ; (2.4)
where L is the AdS length scale. We will take latin indices to run over bulk spacetime
dimensions, and greek indices to run over boundary spacetime dimensions. In order to
apply the DeTurck method, we modify eq. (2.4) and consider instead
Rab +
4
L2
gab r(ab) = 0 ; (2.5)
where a  gbc[abc(g) abc(g)], with g being a reference metric which should obey to the
same Dirichlet boundary conditions as the physical metric g we wish to nd, abc(g) is the
{ 4 {
J
H
E
P06(2020)104
Christoel connection associated with a metric g. Of course, solutions of eq. (2.5) will
only coincide with solutions of eq. (2.4) if r(ab) = 0. Under certain special circumstances
one can show that solutions with a 6= 0 cannot exist (see for instance [17, 35]), and as
such solutions of eq. (2.5) will necessarily coincide with solutions of eq. (2.4). However,
the solutions we seek do not satisfy the conditions of such theorems, so we need to check
a posteriori if  approaches 0 in the continuum limit.
There are many other circumstances where one cannot show that  is necessarily zero
on solutions of (2.5) (see for instance [36{40]). However, in many of these cases, one can
show that the resulting system of equations is Elliptic, and as such one can trust local
uniqueness of solutions to determine whether  vanishes or not. The situation here is,
however, more delicate. One can show that the system of equations we wish to solve does
not appear Elliptic, instead the system appears to be of the mixed Elliptic-Hyperbolic type.
This in turn means that we cannot use local uniqueness to distinguish between solutions
with  6= 0 and solutions with  = 0. We note however that this problem has an alternative
formulation [41] which does appear to have an Elliptic character.
We will now introduce our line element, and show that it reproduces all of the Dirichlet
conditions detailed above. Our line element reads
ds2 =
L2
xy2H(x)2
(
xq1dv22xq2dvdy+q5dy2 (2.6)
+
q3
4x(1x)4

dx+x(1x)2 q6dy+x(1x)2 q7dv
2
+
q4
4(1x)2d

2
2
)
For the reference metric, we will use the line element above with
q1 = (1 y2)(1 + x y2)
q2 = H(x)
q3 = q4 = 1
qi = 0 for i 2 f5; 6; 7g:
Finally, we will also choose H(x) = 1 + $x
p
2 x2. We shall see below that $ will
control the temperature of the black brane innitely far way from the hole, and thus x
the temperature of the eld theory reservoir.
We now discuss the thorny issue of boundary conditions. At y = 0 we demand q1 =
q3 = q4 = 1, q2 = H(x) and qi = 0 for i 2 f5; 6; 7g. The metric then reads
ds2y=0 =
L2
y2H(x)2
"
 x dv2  2xH(x) dv dy + dx
2
4x(1 x)4 +
d
22
4 (1 x)2
#
: (2.7)
We now take the following coordinate transformations
v =
t
2 r0
H(x) y (2.8a)
r =
r0
1 x (2.8b)
z = 2 r0 y H(x) (2.8c)
{ 5 {
J
H
E
P06(2020)104
which reveals
ds2z=0 =
L2
z2
241 r0
r

dt2 +
dr2
1 r0
r
+ r2d
22 + dz
2 +O(z)
35 : (2.9)
The line element above is that of a Schwarzschild black string in AdS (2.2). That is to say,
the boundary metric is conformal to that of a Schwarzschild black hole. This is precisely
what we want if the eld theory is to live on a xed Schwarzschild black hole background.
At x = 0, we demand q1 = 1  y2, q2 = q3 = q4 = 1 and qi = 0 for i 2 f5; 6; 7g. This
transforms (2.6) to
ds2x=0 =
L2
y2

(1 y2)dv2  2dv dy + 1
4x2
dx2 +
1
4x
d
22

: (2.10)
We now dene the following coordinate transformations
x = e2  ; dv = dt^ dy
1 y2 and y = z ;
which brings eq. (2.10) to coincide with the large  limit of eq. (2.1). We also recall that
such black hole has temperature
TH =
1
4 r0
= TBH ; (2.11)
measure by the time dened in eq. (2.8a).
Finally, we come to the more delicate boundary located at x = 1. At this boundary
we want the impose the metric of a planar Schwarzschild black brane with a dierent
temperature than the line element (2.10). At x = 1, we demand q1 = 1  y4, q2 = H(1)
and q3 = q4 = 1 and qi = 0 for i 2 f5; 6; 7g, which brings the line element (2.6) to
ds2x=1 =
L2
H (1)2 y2

(1 y4)dv2  2H(1)dv dy + 1
4 (1 x)4dx
2 +
1
4 (1 x)2d

2
2

:
(2.12)
Now take
x = 1 1
2H(1)r
(2.13a)
dv = dt H(1)dy
1 y4 (2.13b)
which transforms the metric given in eq. (2.12) to
ds2x=1 =
L2
y2

(1 y4) dt
2
H(1)2
+
dy2
1 y4 + dr
2 + r2d
22

: (2.14)
This metric describes a ve-dimensional planar black hole with temperature
T1 =
1
2 r0H(1)
; (2.15)
measure by the time dened in eq. (2.8a).
{ 6 {
J
H
E
P06(2020)104
Note that if we compare eq. (2.11) and eq. (2.15) we see that there is a gradient of
temperatures between the two horizons unless, H(1) = 2, i.e. $ = 1. We will be interested
in situations where $ 6= 1, and in particular
  T1=TBH = 2
1 +$
6= 1 : (2.16)
One might wonder whether we are still missing a boundary condition in the interior.
However, that is not the case. One can understand this in the following manner: the
solution we are seeking is regular in ingoing coordinates, but badly singular in outgoing
coordinates. This is enough to show that there are two possible solutions in the interior:
one that blows up and one that does not. By working with a Chebyshev-Lobatto grid we
automatically assume enough smoothness to guarantee that we only capture the smooth
solution. The price we pay of not imposing a boundary condition in the interior is that we
do not know a priori where the horizon is. All we know is that it must join y = 1 at both
x = 0 and x = 1, due to the Dirichlet boundary conditions detailed above.
3 The horizon and its properties
Horizons in general relativity are null-hypersurfaces, and can thus be written as the zero
level sets of a function h^, which we choose to take the simple form h^(x; y)  y  P (x) [24]
gabrah^rbh^ = 0 ; (3.1)
i.e. dh^ is null. Eq. (3.1) yields and ODE for P , which one can readily solve. Furthermore, we
know that P (0) = P (1) = 1, either of which we can use as boundary conditions. Since the
horizons we seek to construct are not Killing horizons, they will have unusual properties.
For example, the expansion  and shear  of the horizon generators will be non-vanishing.
Note, however, that since the horizon is a null-hypersurface, the rotation of the horizon
generators can always be chosen to vanish even for non-Killing horizons.
In principle, computing  and  can be a relatively daunting task. However, in [23] a
series of tricks where used to determine  and , as well as a choice of ane parameter  for
the horizon generators. Since the properties of such quantities will play an important test
of our numerics, we will review (and generalise slightly) the construction detailed in [23].
The class of spacetimes we seek to nd have a time translation symmetry and an S2
symmetry. We shall use coordinates fv; x; y; ; g, where @=@v is the Killing vector eld
associated with time translations,  and  parametrise the S2, and y = P (x) on H+, the
future event horizon. For the sake of notation, let us dene the shorthand notation @I , so
that @v  @=@v, @  @=@ and @  @=@, and thus I = v; ; .
H+ is 4-dimensional, with a 3-dimensional space of generators. We want to x an x
coordinate on H+, and then uniquely label all horizon generators using the coordinates
(v; ; ) of their intersection with this surface. This should be possible provided that H+ is
not Killing. We choose one horizon generator and pick an ane parameter .  can then
be extended to a scalar function on H+ by requiring it to be independent of v,  and , so
that  = (x). By symmetry,  will serve as an ane parameter for each geodesic.
{ 7 {
J
H
E
P06(2020)104
Let ka be the tangent vector to the horizon generators with ane parameter . Since
@I are all parallel to H+ we have k ? @I , for all I.
@I are deviations vectors for the geodesic congruence, since k and @I commute. There-
fore we have
kcrc(@I)a = Bac(@I)c (3.2)
where Bab = rakb is symmetric, since k is hypersurface orthogonal.
Now dene hIJ = @I 
 @J for I; J = v; ; . In the coordinates fv; x; y; ; g, we can
identify hIJ with the IJth component of the metric tensor through
hIJ = gIJ : (3.3)
The following is then true
d
d
hIJ = k
crc(@I 
 @J)
= (@J)aB
a
c(@I)
c + (@I)
aB ca (@J)c
= 2Bac(@J)
a(@I)
c
= 2BIJ ; (3.4)
where BIJ denotes the IJth component of B.
To obtain the expansion and shear of the congruence, it is necessary to work in the
3 = 5  2 dimensional vector space V^ obtained by rst restricting to vectors orthogonal
to k, and then quotienting by an equivalence relation where two vectors are equivalent if
they dier by a multiple of k. Tensors in the spacetime give rise to natural tensors in V^
if they obey the property that contracting any one index with ka or k
a and the remainder
with vectors or dual vectors having natural realisations in V^ , gives zero. B, g and @I all
have this property. Furthermore, since k and all @I are linearly independent, we have that
all @^I are linearly independent on V^ .
We can thus pick f@^Ig as a basis for V^ . For any tensor Tab naturally giving rise to a
tensor in V^ , i.e.
Tab(@I)
a(@J)
b = T^ab(@^I)
a(@^J)
b ; (3.5)
we can identify T^ in V^ by reading o the components fv; ; g of T . This is the case for
the tensors g and B.
The expansion and shear can now be determined using the usual formulae in terms of
the quantities
g^IJ = gIJ ; (3.6a)
B^IJ = BIJ =
1
2
d
d
gIJ ; (3.6b)
where the nal term means the derivative of the IJth component of the metric along a
geodesic, not the IJthe component of the covariant derivative of the metric.
It remains to nd  as a function of x, but this can be achieved by solving Raychaud-
huri's equation written using the above expressions for g^ and B^ [23]. This gives an equation
{ 8 {
J
H
E
P06(2020)104
for  as a function of x on H+:
d
dx
log

d
dx


=
1
(hIJDhIJ)

D

hIJDhIJ


+
1
2
hIKhJL (DhIJ) (DhKL)

; (3.7a)
where
D  @
@x
+
dP
dx
@
@y
: (3.7b)
It is then possible to obtain B^ as
B^IJ =
1
2
dx
d
DgIJ : (3.7c)
To compute the expansion and shear, we simply look at the trace and trace-free parts of
B^, namely
 = hIJ B^IJ and IJ = B^IJ  hIJ
3
 : (3.8)
4 Extracting the holographic-stress energy tensor
One of the most interesting quantities to extract from these solutions is the associated holo-
graphic stress energy tensor hFPjT jFPi, where FP stands for Frolov-Page states [18]. We
follow closely [42, 43] whose starting point is to cast our solutions into Feerman-Graham
coordinates [44{47]. Fortunately, we only need to perform this coordinate transformation
asymptotically.
We start by determining the behaviour of all q~I , with
~I 2 f1; : : : ; 7g, close to y = 0
by solving eq. (2.5) asymptotically. A careful analysis reveals the following asymptotic
expansion
q~I(x; y) = 1 +
5X
i=1
q
(i)
~I
(x)yi + eq~I(x)y5 log y + bq~I(x)y2+2p3 + o(y2+2p3) ; (4.1)
with fq(4)1 (x); bq1(x); q(5)2 (x); q(4)3 (x); q(4)5 (x); q(5)5 (x); bq7(x)g not being xed by any local anal-
ysis of the Einstein-DeTurck equation, i.e. these coecients correspond to data that can
only be determined once regularity deep in the bulk is imposed and the corresponding
equations of motion are solved. The remaining coecients are all determined as functions
of fq(4)1 (x); bq1(x); q(5)2 (x); q(4)3 (x); q(4)5 (x); q(5)5 (x); bq7(x); H(x)g and their derivatives along x.
Imposing  = 0 asymptotically imposes further constraints, which in turn imply the
tracelessness and transversality of the holographic stress energy tensor. In addition, it
imposes a linear relation between bq1(x) and bq7(x), which can be used to show that the
terms proportional to y2+2
p
3 are pure gauge. Note, however, that in the DeTurck gauge
they are present, and change the expected convergence of the spectral collocation methods
we used to solve the Einstein-DeTurck equation from exponential to power law.
{ 9 {
J
H
E
P06(2020)104
To determine the holographic stress energy tensor, we dene the following asymptotic
change of coordinates
v =
V  r?(w)
2 r0
+
5X
i=1
i(w)z
i + e(x)z5 log z + b(w)z2+2p3 + o(z2+2p3) ;
x = w +
5X
i=1
i(w)z
i + e(x)z5 log z + b(w)z2+2p3 + o(z2+2p3) ; (4.2)
y =
5X
i=1

i(w)z
i + e
(x)z5 log z + b
(w)z2+2p3 + o(z2+2p3) ;
where V is the boundary Eddington-Finkelstein coordinate and r? is the standard
Schwarzschild tortoise coordinate dened as
r?(w) =
Z w
w1
r0
~w(1 ~w)2d ~w ; (4.3)
with w1 2 (0; 1) a constant. The coecients i, i, 
i, e, e, e
, b, b and b
 are then
determined by changing to Feerman-Graham coordinates, where the ve-dimensional line
element takes a particularly simple form
ds2 =
L2
z2

g(x
; z)dxdx + dz2

: (4.4)
The metric components g(x
; z) themselves admit an expansion around z = 0 of the form
g(x
; z) = g(x
) + z2A(x
) + z4B(x
) + z4 log z C (x
) + o(z4) : (4.5)
We choose our conformal boundary metric, i.e. g(x), to be Schwarzchild
ds2@  g(x)dxdx = 

1 r0
r

dV 2 + 2dV dr + r2d
22 ; and r =
r0
1 w : (4.6)
Since our choice of boundary metric is Ricci 
at, one can show that C = 0 [43]. Indeed,
all the log terms in eq. (4.1) and eq. (4.2) conspire to give a net C = 0. Furthermore, as
we have commented above, the terms proportional to z2+2
p
3 in eq. (4.2) can be adjusted
to show that the terms proportional to y2+2
p
3 in eq. (4.1) are pure gauge, justifying why
no such irrational powers appear in the Feerman-Graham expansion. Following [43], the
holographic stress energy tensor can then be uniquely reconstructed from g, A and B [43]
hFPjTjFPi = L
3
4G5
"
B +
1
8

AA
 A2
 g  1
2
AA

 +
1
4
AA
#
; (4.7)
where A  gA . Since the boundary metric is Ricci 
at, there are no ambiguities in
dening hFPjTjFPi. Finally, following the standard AdS/CFT dictionary [5] we identify
G5 =

2
L3
N2
: (4.8)
{ 10 {
J
H
E
P06(2020)104
One quantity of interest to compute, once the holographic stress energy tensor is de-
termined, is the heat 
ux . This quantity is interpreted here as the energy 
ux integrated
over a two-sphere of constant r, that is to say
 = 
Z
S2r
hFPjT jFPin
ph d2x ; (4.9)
where  = (@=@V ),
ph is the volume element induced on a constant r slice and n
an outward unit normal to S2r . By virtue of the conservation of the stress energy tensor
and since  is a Killing vector,  is independent of r. With our conventions, an outgoing

ux of radiation corresponds to  > 0 and an ingoing 
ux to  < 0. We expect  to be
positive if T1 < TBH, i.e. $ > 1, to vanish for the Hartle-Hawking state $ = 1, and to be
negative for $ < 1. In the subsequent section we will conrm this behaviour numerically.
5 Results
We start by studying the geometry of the future horizon H+. To help us gain intuition,
we construct isometric embeddings of spatial cross sections of H+ into hyperbolic space.
These are specially useful if we want to visualize where the horizon bulges out. Since spatial
cross-sections of H+ are three-dimensional, we seek to construct embeddings into H4. We
foliate four-dimensional hyperbolic space using three-dimensional 
at space (i.e. Poincare
slicing of Euclidean AdS)
ds2H4 =
~L2
Z2

dZ2 + dR2 +R2d
22


: (5.1)
One then searches for an embedding of the form (Z(x); R(x)), that is to say the pull-back of
eq. (5.1) to a parametrised surface (Z(x); R(x)), which gives the following induced metric
ds^2H4 =
~L2
Z(x)2

Z 0(x)2 +R0(x)2

dx2 +R(x)2d
22
	
: (5.2)
We can now compare this line element with the pull back of the metric for the 
owing
funnel, i.e. eq. (2.6), induced on the intersection of the future event horizon H+, identied
via solving eq. (3.1), with a partial Cauchy surface of constant v and read o a nonlinear
rst order equation for Z(x). We x the boundary conditions by demanding Z(1) = 1. In
making the identication between line elements we also set ~L = L. The advantage of this
embedding is that a black string, see eq. (2.2), appears as a line of constant R, whereas a
ve-dimensional planar black hole, see eq. (2.14), appears as a line of constant Z. Thus, a
black funnel should naturally interpolate between these two curves.
In addition, we shall also be interested in constructing isometric embeddings of the
ergosurfaces of our solutions. These surfaces are dened as surfaces for which the norm
k@=@vk2 becomes spacelike. Inside the ergosurface, that is to say in the ergoregion, the
character of eq. (2.5) changes from elliptic to hyperbolic. This is the reason why the
existence of these surfaces is at the core of the diculties in trying to prove that our
numerical method ensures the absence of DeTurck solitons. We shall follow the same
strategy and also embed the ergosurfaces into four-dimensional hyperbolic space.
{ 11 {
J
H
E
P06(2020)104
Figure 2. Isometric embeddings of the spatial cross section of H+ (red disks) and the ergosurface
(blue squares). These curves were generated with T1=TBH = 2:5. The right panel shows a zoom of
the shaded region on the left, so that one can more easily identify the ergoregion. The red shaded
region corresponds to the interior of the black funnel, and the blue shaded region to the ergoregion.
In gure 2 we can observe the embedding of a partial Cauchy surface of constant v
with H+ (blue disks) and of the ergosurface (orange squares). The ergoregion remains very
narrow even when T1=TBH = 2:5. The same is true in the opposite limit, i.e. T1=TBH 
0:6. As expected, the ergoregion shrinks down to zero size at the boundary (y = 0) and
at the planar black hole asymptotic region (x = 1). The existence of the bulk ergoregion
also raises questions about the stability of the solutions we have just found, specially in
light of [48]. However, we note that in [48] it was essential that the spatial cross section
of the boundary metric was (metrically) a round sphere. A comprehensive analysis of the
linear-mode stability of our solutions is outside the scope of this manuscript, though we
plan to return to it in the near future.
Perhaps the most important quantity to extract is the stress energy tensor and its as-
sociated Hawking 
ux  dened in eq. (4.9). Within our symmetry class, the holographic
stress energy tensor has four non-zero independent components. In addition, it should be
traceless and covariantly conserved, which gives three constraints amongst these four com-
ponents. Thus, the full stress energy tensor is determined by, say, hTV V i  hFPjTV V jFPi.
Using conservation of the holographic stress energy tensor, it is also simple to show that if
hFPjTV wjFPi 6= 0 on H+, and the holographic stress energy tensor is smooth on H+, then
hTV V ijH+ 6= 0. This is what we expect to happen when T1=TBH 6= 1.
In gure 3 we plot hTV V i for several values of T1=TBH. On the left panel of gure 3 blue
disks, orange squares and green diamonds correspond to T1=TBH = 0:557103 ; 0:844773 ; 1,
respectively. The right panel of gure 3 shows hTV V ijH+ and, as anticipated, it is non-zero
except when T1=TBH = 1 (the Hartle-Hawking state). For the Hartle-Hawking state, we
recover the results of [14, 15].
{ 12 {
J
H
E
P06(2020)104
� � �� �� �� �� ��
��������
����-�
��������
��������
��������
��� ��� ��� ��� ���
�
����-�
����-�
�����-�
����-�
Figure 3. Left: The null-null component of the holographic stress energy tensor, hTV V i, as a
function of r=M . Each curve corresponds to a dierent value of T1=TBH. From bottom to top, we
have T1=TBH = 0:557103 ; 0:844773 ; 1. The horizontal dashed lines indicate the asymptotic value
of hTV V i computed for a planar black hole at temperature T1. Right: The null-null component of
the holographic stress energy tensor evaluated on H+, hTV V ijH+ , as a function of several values of
T1=TBH. As expected, hTV V ijH+ is non-zero except for the Hartle-Hawking state corresponding
to T1=TBH = 1. In both plots we have used r0 = 2M .
We now turn our attention to the Hawking 
ux  dened in eq. (4.9), which we plot
in gure 4. The right panel shows a zoom of the left panel in the region T1=TBH  1.
As expected  < 0 for T1=TBH > 1, meaning that there is an in
ux of radiation into the
black hole. The temperature of the heat bath provides an energy reservoir that sources
the geometry. For the Hartle-Hawking state, T1=TBH = 1, there is no net 
ux of Hawking
particles, so that  = 0. This is marked as a black disk in gure 4. Finally, we see a
somehow surprising result when T1=TBH < 1. This is indeed when we expect the black hole
to radiate Hawking quanta, and as such  > 0. However,  does not seem to be monotonic
in T1=TBH. Instead, it initially grows with decreasing T1=TBH, but it reaches a maximum
at T1=TBH  max  0:698282. This value is marked in gure 4 as a vertical dashed
black line. At the moment, we have no eld theory understanding for this non-monotonic
behaviour of . Using the results of [49], we have some preliminary results indicating that
the 
owing geometry is Gregory-La
amme unstable [50] in the region T1=TBH < max. It
is natural to expect that the endpoint of such instability is one of the black droplets found
in [16]. Such evolution will necessarily involve a violation of the weak cosmic censorship
conjecture [51], alike the non-linear evolution of the Gregory-La
amme instability [52].
This is not the rst time that violations of weak cosmic censorship conjecture play a role
in the AdS context, see for instance [53{57], although it seems one of the easiest scenarios
to actually sort out the evolution numerically using the methods outlined in [58, 59].
We now discuss the properties of this novel class of horizons. For simplicity, we restrict
our discussion to the region of moduli space T1=TBH < 1. We begin by studying the
behaviour of hIJ itself, and in particular its determinant h  dethIJ . We plot this quantity
{ 13 {
J
H
E
P06(2020)104
��� ��� ��� ��� ���-����
-����
-����
-����
-����
-����
����
��� ��� ��� ��� ����������
�������
�������
�������
�������
Figure 4. Hawking 
ux  plotted as a function of T1=TBH. The right panel shows a zoom of
the left panel in the region T1=TBH  1.  does not appear to behave monotonically in T1=TBH,
and in particular it has a maximum at T1=TBH  max  0:698282, which is marked as a vertical
dashed line in both left and right panels. The red disks indicate when  > 0, and the blue squares
when  < 0. The Hartle-Hawking state is represented by a black disk.
��� ��� ��� ��� ��� ���
�������
�������
�������
�������
�������
�������
�������
�������
��� ��� ��� ��� ��� ���
�����
�����
�����
�����
�����
Figure 5. Left: The determinant h as a function of x on H+ plotted for xed T1=TBH  0:557103.
Right: The coordinate velocity 
(x) on H+ plotted as a function of x for xed T1=TBH  0:557103.
on the left panel of gure 5 for T1=TBH  0:557103. h is clearly a monotonically increasing
function of x, showing that x increases towards the future along the future horizon, and
thus that the past horizon lies at x = 0. We note that in this region the temperature of
the black hole is higher than that of the heat reservoir.
To understand the direction of the 
ow, we can also determine the horizon coordinate
velocity 
(x) along H+. To do so, we can look at the pull-back of our line element (2.6)
{ 14 {
J
H
E
P06(2020)104
to the future horizon H+ (see eq. (3.1))
ds2H+ =
L2
xP 2H2
(
x ~q1dv22x ~q2P 0dvdx+~q5P 02dx2 (5.3)
+
~q3
4x(1x)4

dx+x(1x)2 ~q6P 0dx+x(1x)2 ~q7dv
2
+
~q4
4(1x)2d

2
2
)
=
L2
4xP 2H2
("
~q3

1+(1x)2x ~q5P 0

2
(1x)4x +4x ~q7P
02
#
(dx
(x)dv)2+ ~q4
(1x)2d

2
2
)
;
where the last equality follows from the fact that H+ is null, ~q~I = q~I(x; P (x)) and 0
denotes dierentiation with respect to x. The last equality in eq. (5.3) implicitly denes the
coordinate velocity 
(x). We have plotted 
(x) for T1=TBH  0:557103 on the right panel
of gure 5, and we nd 
(x)  0. Indeed, we nd that 
(x)  0 so long as T1=TBH  1,
and negative otherwise. Both the sign of 
 and the monotonicity properties of h, indicate
that the past horizon is located at the black hole horizon (the hotter reservoir). This is to
contrast with the coordinate choice made in [22, 23], in which the cooler horizon appears
to be closer to the past horizon H.
Next, we look at the behaviour of the horizon expansion  as a function of . According
to Raychaudhuri's equation and the denition of , it better be that  > 0, d=d < 0
and that  approaches zero for large . On the left panel of gure 6 we plot the ane
parameter  as a function of x on the future event horizon. Recall that x is one of the
coordinates introduced in eq. (2.6). As expected,  is a monotonically increasing function
of x, with x = 0 locating the past horizon. On the right panel, we show the expansion
 as a function of the ane parameter , and we conrm that d=d < 0. In order to
show that this is the case using Raychaudhuri's equation, one needs to use the equations of
motion. We thus see this as the most solid test of our numerical procedures. On the right
panel of gure 6 we also show, as a dashed red line, a linear t in the (log10 ; log10 )
plane, which yields log10  = 0:00195019 log10 . The t seems to work for all ranges of
x, so there is a clear indication that  is diverging as 1 at the past horizon (located at
 = 0), signalling the presence of a caustic. This caustic, in turn, gives rise to a curvature
singularity. As noted in [23], the easiest way to see this is to note that for any Killing eld,
the Ricci identify for vectors implies that
rarbKc = RcbadKd : (5.4)
In particular, this will be true for any Killing vector on the two-sphere, say K = @=@. It
is easy to see that for any such Killing vector kKk2 diverges at x = 0, i.e. when  = 0.
But since K obeys the second order partial dierential equation (5.4), it can only diverge
at nite ane parameter  = 0 if Rabcd diverges at  = 0 in all orthonormal frames.
Finally, we end with a comment regarding the behaviour of our solutions close to
the minimal temperature ratio T1=TBH = 0:557103 that we have reached. In gure 7
we monitor the behaviour of C2  maxC
CabcdCabcd, as a function of T1=TBH, where
C stands for domain of outer communications and C is the ve-dimensional Weyl tensor.
{ 15 {
J
H
E
P06(2020)104
��� ��� ��� ��� ��� ��� ���
����-��
����
�����
�����
-�� -�� � �� �� �� ��
-��
-��
-��
-��
�
��
��
Figure 6. Left: Logarithmic plot of the ane parameter  as a function of x on H+. Right:
log10 log10 plot of the expansion  as a function of . The dashed red line indicates a best
t consistent with log10  = 0:00195019  log10 . Both panels were generated with T1=TBH 
0:729262.
��� ��� ��� ��� ���
�
���
���
���
���
���
���
Figure 7. C2  maxC
CabcdCabcd, where C stands for domain of outer communications, as a
function of T1=TBH. We see no apparent divergent behaviour as we lower the temperature ratio
T1=TBH.
Throughout moduli space, we nd no evidence for a divergent behaviour. We suspect the
reason why we cannot reach lower temperatures is purely numerical, and is related to the
existence of large gradients inside the future event horizon.
{ 16 {
J
H
E
P06(2020)104
6 Discussion
We constructed the holographic dual of the Frolov-Page states [18] for N = 4 SYM with
gauge group SU(N) and asymptotic temperature T1, at large 't Hooft coupling and innite
N on a Schwarzschild black hole background with temperature TBH. When TBH = T1,
the Frolov-Page states reduce to the Hartle-Hawking state, and when T1 ! 0 one recovers
the Unruh state. The resulting bulk geometries are of the 
owing type, and thus similar
in nature to the ones rst uncovered in [23, 24]. They possess spherical symmetry and
a stationary Killing eld @=@v as well as an horizon H+ whose spatial cross sections are
non-compact. However, the horizon is not a Killing horizon and in particular it is not
generated by any linear combination of Killing elds. This is not in contradiction with
the standard rigidity theorems [19{21], since the latter assume horizons with compactly
generated spatial cross sections.
When TBH 6= T1 we nd a net 
ux of Hawking radition, being outgoing when-
ever TBH > T1, and ingoing otherwise. For the Hartle-Hawking state, corresponding
to TBH = T1, the 
ux vanishes identically, as expected, and the results of [15] are re-
covered. These novel 
owing horizons have unfamiliar properties, such as a non-vanishing
expansion . We have studied how  varies as a function of TBH=T1, and nd it is posi-
tive and extends to the future along each null generator. The horizon generators extend to
innite ane parameter  in the far future, but reach a caustic (at nite ane parameter
. We have computed CabcdC
abcd and it seems to remain bounded in the domain of outer
communications, indicating that the caustic located at  = 0 is likely to be tidal singularity.
We can now merge the results of [16] to the ones found in this manuscript, to infer
a complete phase diagram for N = 4 SYM with gauge group SU(N) and asymptotic
temperature T1, at large 't Hooft coupling and innite N on a Schwarzschild black hole
background with temperature TBH. We attempt to draw such diagram in gure 8 as a
function of   T1=TBH. The black droplets of [16] were found to exist in the window
 2 [0; D], with D  0:93. The solution with  = 0, corresponding to the Unruh vacuum,
was found previously in [17]. In the droplet phase, there is no O(N2) Hawking radiation,
and the corresponding state can be best understood in terms of a `jammed phase', though
no underlying understanding of this phenomena exists to date on the eld theory side. The
funnel solutions we found in this manuscript exist at least in the range  > F  0:557103,
though we expect such solutions to exist for even lower values of  , which we are unable to
probe with current numerical techniques. Most notably, the Hawking 
ux  in the 
owing
funnel phase does not appear to be monotonic with decreasing  , reaching a maximum
at  = max  0:698282. Borrowing intuition from the Stefan-Boltzman law, this would
suggest that the eective number of degrees of freedom in the eld theory acquires a non-
trivial temperature dependence. We conjectured that 
owing funnels with  < max are
linearly unstable to the Gregory-La
amme instability, and that the concomitant endpoint
is one of the droplets found in [16]. Using results from [49], one can give numerical evidence
in favour of this conjecture, which will be presented elsewhere. It is clear that our results
will largely be stable to corrections in the 't Hooft coupling, which manifest themselves as
higher derivative corrections in the bulk. However, this is not the case for nite N eects.
{ 17 {
J
H
E
P06(2020)104
Figure 8. Phase diagram of N = 4 SYM with gauge group SU(N) and asymptotic temperature
T1, at large 't Hooft coupling and innite N on a Schwarzschild black hole background with
temperature TBH. In this phase diagram   T1=TBH.
At nite N , we expect Hawking radiation in the bulk, and thus a non-trivial 
ux of Hawking
radiation even in the droplet phase. We stress, however, that this eect is subleading in
N , perhaps being O(1), instead of the O(N2) eect of the 
owing funnel phase.
The most resounding mystery of this work still pertains the existence of the droplet
phase [16, 17], in which Hawking radiation seems trapped by the local geometry and does
not escape to innity, perhaps due to some form of local connement yet to be explored
on the quantum eld theory side.
Acknowledgments
J. E. S. thanks Sebastian Fischetti, Juan Maldacena, Don Marolf, Benson Way and Ed-
ward Witten for many discussions and O. J. C. Dias, B. Ganchev and J. F. Melo for reading
an earlier version of this manuscript. J. E. S. also thanks Toby Crisford for many discus-
sions and collaboration in the initial stages of this work. J. E. S. is supported in part by
STFC grants PHY-1504541 and ST/P000681/1. J. E. S. also acknowledges support from a
J. Robert Oppenheimer Visiting Professorship. This work used the DIRAC Shared Mem-
ory Processing system at the University of Cambridge, operated by the COSMOS Project
at the Department of Applied Mathematics and Theoretical Physics on behalf of the STFC
DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-
infrastructure capital grant ST/J005673/1, STFC capital grant ST/H008586/1, and STFC
DiRAC Operations grant ST/K00333X/1. DiRAC is part of the National e-Infrastructure.
A Convergence tests
We monitored all the components of a as a function of the number of grid points in
the x and y directions, Nx and Ny, respectively. For simplicity, we present results for
Nx = Ny = N. Note that since the problem is mixed Elliptic-Hyperbolic, it does not
suce to monitor the norm aa, as 
a can be null in certain regions of our integration
domain. Independently of the temperature ratio   T1=TBH, we found that the approach
to the continuum limit was consistent with power law convergence, with the details of the
convergence depending slightly on which component of  one looks at. This convergence
properties is to be expected, since non-analytic terms were identied in an expansion o the
{ 18 {
J
H
E
P06(2020)104
�� �� �� �� �� ���
��-�
��-�
��-�
�����
�� �� �� �� �� ���
����-�����
-�
����-������
�����
�����
�� �� �� �� �� ���
����-�����-�
����-�����-�
����-�����-�
Figure 9. Convergence properties of the non-vanishing components of DeTurck vector a, as a
function of the number of grid points N. From left to right, we plot in a logarithmic scale maxC jtj,
maxC jxj, maxC jyj, with C being the domain of outer communications. All plots were generated
using  = 0:665557.
�� �� �� �� �� ���
����-�
����-�
����-�
����-�
Figure 10. N as a function of N computed for xed  = 0:7293. Convergence is now consistent
with N / N5:85.
conformal boundary (see eq. (4.2)). In gure 9 we show a logarithmic plot of maxC jtj (left),
maxC jxj (middle) and maxC jyj (right) as a function of N, where C stands for domain of
outer communications. For  = 0:665557, we nd maxC jtj / N12:22, maxC jxj / N9:89
and maxC jyj / N9:83.
Finally, we also investigated the convergence properties of , by dening
N 
1 NN+10
 ; (A.1)
where N stands for  computed on a mesh with Nx = Ny = N grid points. The results
can be seen in gure 10 for  = 0:7293 where we now nd N / N5:85, which is very
close to the non-analytic power reported in the expansion (4.2).
{ 19 {
J
H
E
P06(2020)104
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
References
[1] S.W. Hawking, Black hole explosions, Nature 248 (1974) 30 [INSPIRE].
[2] S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199
[Erratum ibid. 46 (1976) 206] [INSPIRE].
[3] G.W. Gibbons and M.J. Perry, Black Holes in Thermal Equilibrium, Phys. Rev. Lett. 36
(1976) 985 [INSPIRE].
[4] E.T. Akhmedov, H. Godazgar and F.K. Popov, Hawking radiation and secularly growing loop
corrections, Phys. Rev. D 93 (2016) 024029 [arXiv:1508.07500] [INSPIRE].
[5] J.M. Maldacena, The large N limit of superconformal eld theories and supergravity, Int. J.
Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
[6] S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical
string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
[7] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253
[hep-th/9802150] [INSPIRE].
[8] O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large N eld theories,
string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
[9] V.E. Hubeny, D. Marolf and M. Rangamani, Hawking radiation in large N strongly-coupled
eld theories, Class. Quant. Grav. 27 (2010) 095015 [arXiv:0908.2270] [INSPIRE].
[10] V.E. Hubeny, D. Marolf and M. Rangamani, Black funnels and droplets from the AdS
C-metrics, Class. Quant. Grav. 27 (2010) 025001 [arXiv:0909.0005] [INSPIRE].
[11] V.E. Hubeny, D. Marolf and M. Rangamani, Hawking radiation from AdS black holes, Class.
Quant. Grav. 27 (2010) 095018 [arXiv:0911.4144] [INSPIRE].
[12] M.M. Caldarelli, O.J.C. Dias, R. Monteiro and J.E. Santos, Black funnels and droplets in
thermal equilibrium, JHEP 05 (2011) 116 [arXiv:1102.4337] [INSPIRE].
[13] D. Marolf, M. Rangamani and T. Wiseman, Holographic thermal eld theory on curved
spacetimes, Class. Quant. Grav. 31 (2014) 063001 [arXiv:1312.0612] [INSPIRE].
[14] S. Fischetti, J.E. Santos and B. Way, Dissonant Black Droplets and Black Funnels, Class.
Quant. Grav. 34 (2017) 155001 [arXiv:1611.09363] [INSPIRE].
[15] J.E. Santos and B. Way, Black Funnels, JHEP 12 (2012) 060 [arXiv:1208.6291] [INSPIRE].
[16] J.E. Santos and B. Way, Black Droplets, JHEP 08 (2014) 072 [arXiv:1405.2078] [INSPIRE].
[17] P. Figueras, J. Lucietti and T. Wiseman, Ricci solitons, Ricci 
ow and strongly coupled CFT
in the Schwarzschild Unruh or Boulware vacua, Class. Quant. Grav. 28 (2011) 215018
[arXiv:1104.4489] [INSPIRE].
[18] V.P. Frolov and D.N. Page, Proof of the generalized second law for quasistationary
semiclassical black holes, Phys. Rev. Lett. 71 (1993) 3902 [gr-qc/9302017] [INSPIRE].
{ 20 {
J
H
E
P06(2020)104
[19] S.W. Hawking, Black holes in general relativity, Commun. Math. Phys. 25 (1972) 152
[INSPIRE].
[20] S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge
Monographs on Mathematical Physics, Cambridge University Press, (2011).
[21] S. Hollands, A. Ishibashi and R.M. Wald, A higher dimensional stationary rotating black hole
must be axisymmetric, Commun. Math. Phys. 271 (2007) 699 [gr-qc/0605106] [INSPIRE].
[22] S. Fischetti and D. Marolf, Flowing Funnels: Heat sources for eld theories and the AdS3
dual of CFT2 Hawking radiation, Class. Quant. Grav. 29 (2012) 105004 [arXiv:1202.5069]
[INSPIRE].
[23] S. Fischetti, D. Marolf and J.E. Santos, AdS 
owing black funnels: Stationary AdS black
holes with non-Killing horizons and heat transport in the dual CFT, Class. Quant. Grav. 30
(2013) 075001 [arXiv:1212.4820] [INSPIRE].
[24] P. Figueras and T. Wiseman, Stationary holographic plasma quenches and numerical methods
for non-Killing horizons, Phys. Rev. Lett. 110 (2013) 171602 [arXiv:1212.4498] [INSPIRE].
[25] R. Emparan and M. Martinez, Black String Flow, JHEP 09 (2013) 068 [arXiv:1307.2276]
[INSPIRE].
[26] M. Sun and Y.-C. Huang, Kerr Black string 
ow, Nucl. Phys. B 897 (2015) 98
[arXiv:1405.6906] [INSPIRE].
[27] I. Amado and A. Yarom, Black brane steady states, JHEP 10 (2015) 015
[arXiv:1501.01627] [INSPIRE].
[28] E. Megias, Out-of-equilibrium energy 
ow and steady state congurations in AdS/CFT, PoS
EPS-HEP2015 (2015) 366 [arXiv:1510.04219] [INSPIRE].
[29] C.P. Herzog, M. Spillane and A. Yarom, The holographic dual of a Riemann problem in a
large number of dimensions, JHEP 08 (2016) 120 [arXiv:1605.01404] [INSPIRE].
[30] E. Megias, Far-from-equilibrium energy 
ow and entanglement entropy, EPJ Web Conf. 164
(2017) 01010 [arXiv:1701.00098] [INSPIRE].
[31] J. Sonner and B. Withers, Universal spatial structure of nonequilibrium steady states, Phys.
Rev. Lett. 119 (2017) 161603 [arXiv:1705.01950] [INSPIRE].
[32] M. Headrick, S. Kitchen and T. Wiseman, A new approach to static numerical relativity and
its application to Kaluza-Klein black holes, Class. Quant. Grav. 27 (2010) 035002
[arXiv:0905.1822] [INSPIRE].
[33] T. Wiseman, Numerical construction of static and stationary black holes, in Black Holes in
Higher Dimensions, Cambridge University Press, (2012), pp. 233{279.
[34] O.J.C. Dias, J.E. Santos and B. Way, Numerical Methods for Finding Stationary
Gravitational Solutions, Class. Quant. Grav. 33 (2016) 133001 [arXiv:1510.02804]
[INSPIRE].
[35] P. Figueras and T. Wiseman, On the existence of stationary Ricci solitons, Class. Quant.
Grav. 34 (2017) 145007 [arXiv:1610.06178] [INSPIRE].
[36] A. Donos and J.P. Gauntlett, The thermoelectric properties of inhomogeneous holographic
lattices, JHEP 01 (2015) 035 [arXiv:1409.6875] [INSPIRE].
[37] A. Donos and J.P. Gauntlett, Minimally packed phases in holography, JHEP 03 (2016) 148
[arXiv:1512.06861] [INSPIRE].
{ 21 {
J
H
E
P06(2020)104
[38] M.S. Costa, L. Greenspan, J. Penedones and J.E. Santos, Polarised Black Holes in ABJM,
JHEP 06 (2017) 024 [arXiv:1702.04353] [INSPIRE].
[39] T. Crisford, G.T. Horowitz and J.E. Santos, Testing the Weak Gravity-Cosmic Censorship
Connection, Phys. Rev. D 97 (2018) 066005 [arXiv:1709.07880] [INSPIRE].
[40] G.T. Horowitz and J.E. Santos, Further evidence for the weak gravity | cosmic censorship
connection, JHEP 06 (2019) 122 [arXiv:1901.11096] [INSPIRE].
[41] J.E. Santos, in preparation.
[42] V. Balasubramanian, P. Kraus, A.E. Lawrence and S.P. Trivedi, Holographic probes of
anti-de Sitter space-times, Phys. Rev. D 59 (1999) 104021 [hep-th/9808017] [INSPIRE].
[43] S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and
renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595
[hep-th/0002230] [INSPIRE].
[44] C. Feerman and C.R. Graham, Conformal invariants, in The Mathematical Heritage of Elie
Cartan (Lyon, 1984), Asterisque 98 (1985) 95.
[45] C.R. Graham, Volume and area renormalizations for conformally compact Einstein metrics,
in Proceedings, 19th Winter School on Geometry and Physics, (1999), math/9909042.
[46] M.T. Anderson, Geometric aspects of the AdS/CFT correspondence, IRMA Lect. Math.
Theor. Phys. 8 (2005) 1 [hep-th/0403087] [INSPIRE].
[47] C. Feerman and C.R. Graham, The ambient metric, Ann. Math. Stud. 178 (2011) 1
[arXiv:0710.0919] [INSPIRE].
[48] S.R. Green, S. Hollands, A. Ishibashi and R.M. Wald, Superradiant instabilities of
asymptotically anti-de Sitter black holes, Class. Quant. Grav. 33 (2016) 125022
[arXiv:1512.02644] [INSPIRE].
[49] P. Figueras, K. Murata and H.S. Reall, Black hole instabilities and local Penrose inequalities,
Class. Quant. Grav. 28 (2011) 225030 [arXiv:1107.5785] [INSPIRE].
[50] R. Gregory and R. La
amme, Black strings and p-branes are unstable, Phys. Rev. Lett. 70
(1993) 2837 [hep-th/9301052] [INSPIRE].
[51] R. Penrose, Gravitational collapse: The role of general relativity, Riv. Nuovo Cim. 1 (1969)
252 [INSPIRE].
[52] L. Lehner and F. Pretorius, Black Strings, Low Viscosity Fluids and Violation of Cosmic
Censorship, Phys. Rev. Lett. 105 (2010) 101102 [arXiv:1006.5960] [INSPIRE].
[53] B.E. Nieho, J.E. Santos and B. Way, Towards a violation of cosmic censorship, Class.
Quant. Grav. 33 (2016) 185012 [arXiv:1510.00709] [INSPIRE].
[54] O.J.C. Dias, J.E. Santos and B. Way, Localised AdS5  S5 Black Holes, Phys. Rev. Lett. 117
(2016) 151101 [arXiv:1605.04911] [INSPIRE].
[55] G.T. Horowitz, J.E. Santos and B. Way, Evidence for an Electrifying Violation of Cosmic
Censorship, Class. Quant. Grav. 33 (2016) 195007 [arXiv:1604.06465] [INSPIRE].
[56] T. Crisford and J.E. Santos, Violating the Weak Cosmic Censorship Conjecture in
Four-Dimensional Anti-de Sitter Space, Phys. Rev. Lett. 118 (2017) 181101
[arXiv:1702.05490] [INSPIRE].
{ 22 {
J
H
E
P06(2020)104
[57] D. Marolf and J.E. Santos, Phases of Holographic Hawking Radiation on spatially compact
spacetimes, JHEP 10 (2019) 250 [arXiv:1906.07681] [INSPIRE].
[58] P.M. Chesler and L.G. Yae, Numerical solution of gravitational dynamics in asymptotically
anti-de Sitter spacetimes, JHEP 07 (2014) 086 [arXiv:1309.1439] [INSPIRE].
[59] K. Balasubramanian and C.P. Herzog, Losing Forward Momentum Holographically, Class.
Quant. Grav. 31 (2014) 125010 [arXiv:1312.4953] [INSPIRE].
{ 23 {