# Small Hairy Black Holes in

###### Abstract:

We study small hairy black holes in a consistent truncation of gauged supergravity that consists of a single charged scalar field interacting with the metric and a gauge field. Small very near extremal RNAdS black holes in this system are unstable to decay by superradiant emission. The end point of this instability is a small hairy black hole that we construct analytically in a perturbative expansion in its charge. Unlike their RNAdS counterparts, hairy black hole solutions exist all the way down to the BPS bound, demonstrating that Yang Mills theory has an entropy at all energies above supersymmetry. At the BPS bound these black holes reduce to previously discussed regular, supersymmetric horizon free solitons. We use numerical methods to continue the construction of these solitons to large charges and find that the line of soliton solutions terminates at a singular solution at a finite charge. We conjecture that a one parameter family of singular supersymmetric solutions, which emerges out of , constitutes the BPS limit of hairy black holes at larger values of the charge. We analytically determine the near singularity behaviour of , demonstrate that both the regular and singular solutions exhibit an infinite set of damped ‘self similar’ oscillations around and analytically compute the frequency of these oscillations. At leading order in their charge, the thermodynamics of the small hairy black holes constructed in this paper turns out to be correctly reproduced by modeling these objects as a non interacting mix of an RNAdS black hole and the supersymmetric soliton in thermal equilibrium. Assuming that a similar non interacting model continues to apply upon turning on angular momentum, we also predict a rich family of rotating hairy black holes, including new hairy supersymmetric black holes. This analysis suggests interesting structure for the space of (yet to be constructed) hairy charged rotating black holes in , particularly in the near BPS limit.

^{†}

^{†}preprint: TIFR/TH/

ITFA10-13

## 1 Introduction

Black hole solutions of IIB theory on constitute the thermodynamic saddle points of Yang Mills theory on via the AdS/CFT correspondence. A complete understanding of the space of stationary black hole solutions in is consequently essential for a satisfactory understanding of the state space of Yang Mills theory at energies of order . While the Kerr RNAdS black hole solutions are well known [1, 2, 3, 4, 5, 6, 7], it seems likely that several additional yet to be determined families of black hole solutions will play an important role in the dynamics and thermodynamics of Yang Mills theory.

In this paper we will construct a new class of asymptotically
black hole solutions. The black holes we construct
are small, charged, and have parametrically low temperatures; our
construction is perturbative in the black hole charge. Our solutions
are hairy, in the sense that they include condensates of charged
scalar fields^{4}^{4}4See [8, 9, 10, 11, 12]
and references therein for reviews of recent work - sparked by an
observation by Gubser [13] -on hairy black branes in
spaces..
In the BPS limit these hairy black holes reduce to regular horizon free
solitons.
We also use numerical techniques to continue our perturbative construction
of these solitons to charges of order unity, and uncover an intricate self
similar behaviour in the space of solitons in the neighborhood of
a finite critical value of the charge. The solutions presented in this paper
suggest a qualitatively new picture of the near BPS spectrum of
Yang Mills.

Our perturbative construction of hairy black holes is close in spirit and technique to the constructions presented in the recent paper [14], which may be regarded as an immediate precursor to the current work. For this reason we first present a brief review of [14] before turning to a description and discussions of the new black hole solutions constructed in this paper.

It was demonstrated in [14] that small charged black holes in global spaces are sometimes unstable to the condensation of charged matter fields. More precisely any system governed by the Lagrangian

(1) |

possesses small RNAdS black holes that are unstable to decay by superradiant discharge of the scalar field whenever . The end point of this superradiant tachyon condensation process is a hairy black hole. The authors of [14] constructed these hairy black hole solutions (working with a particular toy model Lagrangian of the form (1)) in a perturbation expansion in their mass and charge. At leading order in this expansion, the hairy black holes of [14] are well approximated by a non interacting mix of a small RNAdS black hole and a weak static solitonic scalar condensate. In particular, it was shown in [14] that the leading order thermodynamics of small hairy black holes could be reproduced simply by modeling them as a non interacting mix of an RNAdS black hole and a regular charged scalar soliton.

The results of [14] suggest that the density of states of certain field theories with a gravity dual description might be dominated in certain regimes by previously unexplored phases consisting of an approximately non interacting mix of a normal charged phase and a Bose condensate. In order to make definitive statements about the actual behaviour of Yang Mills theory, however, it is necessary to perform the relevant calculations in IIB supergravity on rather than a simple toy model Lagrangian; this is the subject of the current paper. As the IIB theory on is a very special system, the reader might anticipate that hairy black holes in this theory have some distinctive special properties not shared by equivalent objects in the toy model studied in [14] at least at generic values of parameters. As we will see below, this indeed turns out to be the case.

In order to avoid having to deal with the full complexity of IIB
SUGRA, in this paper we identify^{5}^{5}5In unpublished work,
S. Gubser, C. Herzog and S. Pufu have independently identified this
consistent truncation, and have numerically investigated hairy black branes
in this set up. We thank C. Herzog for informing us of this. work with
a consistent truncation
of gauged supergravity (itself a consistent truncation of
IIB SUGRA on ). Of the complicated spectrum of
supergravity, our truncation retains only a single
charged scalar field , a gauge field and the
metric.^{6}^{6}6Under the correspondence, is dual to
the operator while is dual to the
conserved current . Here , and denote the three complex
chiral scalars in the Lagrangian. The Lagrangian for
our system is given by

(2) |

and is of the general structure (1) with and . As , small very near extremal black holes in this system lie at the precipice of the super radiant instability discussed in [14]. The question of whether they fall over this precipice requires a detailed calculation. We perform the necessary computation in this paper. Our results demonstrate that very near extremal small black holes do suffer from the super radiant instability; we proceed to construct the hairy black hole that constitutes the end point of this instability.

The calculations presented in this paper employ techniques that are similar to those used in [14]. In particular we work in a perturbation expansion in the scalar amplitude using very near extremal vacuum RNAdS black holes as the starting point of our expansion. This expansion is justified by the smallness of the charge of our solutions. We implement our perturbative procedure by matching solutions in a near (horizon) range, intermediate range and far field range. This matching procedure is justified by the parametrically large separation of scales between the horizon radius of the black holes we construct and AdS curvature radius. We refer the reader to the introduction of [14] for a more detailed explanation of physical motivation for this perturbative expansion and its formal structure. Actual details of our calculations may be found in sections 3 and 4 below. In the rest of this introduction we simply present our results and comment on their significance.

We study black holes of charge ^{7}^{7}7 is the charge of the black
hole solutions under each of the three diagonal U(1) Cartan’s of SO(6).
The consistent truncation of this paper forces these three U(1) charges to
be equal. (normalized so that each of the
three complex Yang Mills scalars has unit charge) and mass
(normalized to match the scaling dimensions of dual operators). We find
it convenient to deal with the ‘intensive’ mass and
charge, and , given by

As in [14], in this paper we are primarily interested in small black holes for which and . We now recall some facts about RNAdS black holes in this system. First, the masses of such black holes obey the inequality

Black holes that saturate this inequality are extremal, regular and have finite entropy (see subsection 6.1 for more details). The chemical potential of extremal black holes is given by

and approaches unity in the limit of small charge. Note also that extremal black holes lie above the BPS bound

and in particular that

We will now investigate potential superradiant instabilities (see the introduction of [14] for an explanation of this term) of these black holes. Recall that a mode of charge and energy scatters off a black hole of chemical potential in a superradiant manner whenever . The various modes of the scalar field in (2) have energies and all carry charge . As the chemical potential of a small near extremal black hole is approximately unity, it follows that only the ground state of (with ) could possibly scatter of a near extremal RNAdS black hole in a superradiant manner. This mode barely satisfies the condition for superradiant scattering; as a consequence we will show in this paper that small RNAdS black holes in (2) suffer from a superradiant instability into this ground state mode only very near to extremality, i.e. when

Unstable black holes eventually settle down into a new branch of stable hairy black hole solutions. We have constructed these hairy black holes in a perturbative expansion in their charge in Section 4; we now proceed to present a qualitative description of these solutions and their thermodynamics.

Recall that the zero mode of the scalar field obeys the BPS bound (and so is supersymmetric) at linear order in an expansion about global . It has been demonstrated in [15, 16, 17] (and we reconfirm in section 3 below) that this linearized BPS solution continues into a nonlinear BPS solution upon increasing its amplitude. In this paper we will refer to this regular solution as the supersymmetric soliton. The hairy black holes of this paper may approximately be thought of as a small, very near extremal RNAdS black hole located in the center of one of these solitons. Although the soliton is supersymmetric, the black hole at its center is not, and so hairy black holes are not BPS in general. These solutions exist in the mass range

(3) |

At the lower bound of this range (136) hairy black holes reduce to the supersymmetric soliton. At the upper bound (which is also the instability curve for RNAdS black holes) they reduce to RNAdS black holes.

In Fig. 1 below we have plotted the near extremal micro canonical ‘phase diagram’ for our system. As is apparent from Fig. 1 our system undergoes a phase transition from an RNAdS phase to a hairy black hole phase upon lowering the energy at fixed charge. This phase transition occurs at the upper end of the range (3). Note that the phase diagram of Fig. 1 has several similarities with the phase diagram depicted in Fig. 1 in [14]; however there is also one important difference. The temperature of the hairy black holes of this paper decreases with decreasing mass at fixed charge, and reaches the value zero at the BPS bound. In contrast the temperature of the hairy black holes of [14] increases with decreasing mass (at fixed charge), approaching infinity in the vicinity of the lower bound.

As we have emphasized, the phase diagram depicted in Fig. 1
applies only in the limit of small charges and masses. We would now
like to inquire as to how this phase diagram continues to large
charges and masses. In order to address this question we first focus
on solitonic solutions. These solutions may be determined much more
simply than the generic hairy solution, as they obey the constraints
of supersymmetry rather than simply the equations of motion. It turns
out that spherically symmetric supersymmetric solutions are given as
solutions to a single nonlinear, second order ordinary differential
equation [15, 16, 17]. The solitons
constitute the unique one parameter set of regular solutions to this
equation. It is easy to continue our perturbative construction of the
solitonic solutions to large charges by solving this equation
numerically: in fact this exercise was already carried out in
[16]. This numerical solution reveals that the
solitonic branch of solutions terminates at a finite charge . For there are no supersymmetric spherically
symmetric solutions to the equations of motion of
(2)^{8}^{8}8To be more precise, there are smooth solitonic
solutions up to a slightly higher value , but in a sense
that will be explained in section 5, the point
marks the boundary between regular solitonic solutions and
singular ones..

Recall that solitons constitute the lower edge of the space of hairy black hole solutions of Fig. 1. The non existence of regular supersymmetric solutions for might, at first, suggest that at these charges the space of hairy black hole solutions terminates at a mass greater than (i.e. does not extend all the way down to supersymmetry). While this is a logical possibility, we think it is likely that the truth lies elsewhere. As we will explain in section 5 the solitonic branch of supersymmetric solutions terminates in a distinguished singular solution . It turns out that is also the end point (or origin) of a one parameter set of supersymmetric solutions that are all singular at the origin. The charges of these solutions increase without bound (indeed we have found an explicit analytic solution for the singular supersymmetric solution in the limit of arbitrarily large charge). The two one parameter families of solutions, regular and singular ones, are joined at the special solution . We conjecture that smooth hairy black hole solutions exist in our system at every and for . Upon taking the limit , these smooth solutions reduce to the smooth soliton for but reduce to the singular supersymmetric solutions described above when . In summary, we conjecture that the phase diagram of our system takes the form displayed in Fig. 2 below.

The distinguished solution clearly plays a special role in the space
of spherically symmetric supersymmetric solutions. In subsection 5.2
we analytically determine the near singularity behaviour of this solution.
Viewing the 2nd order differential equation that determines supersymmetric
solutions as a dynamical system in the ‘time’ variable , we demonstrate
that the solution is a stable fixed point of this system, and analytically
compute the eigenvalues that characterize the approach to this fixed point.
This eigenvalue has an imaginary part (which damps fluctuations) and a real
part (that results in oscillations). Solitonic - and singular -
solutions in the neighborhood of may be thought of as configurations
that that flow to at large . The oscillations^{9}^{9}9We are
extremely grateful to M. Rangamani for suggesting that we look for this
‘self similar’ structure in the space of solitons in the neighborhood of
. The results reported in this paragraph are the outcome of
investigations that were spurred directly by this suggestion.
referred to above result in the following
phenomenon: the system develops a multiplicity of supersymmetric solitonic
(or singular) at charges when comes near enough to . The
number of solutions diverges as . The space of solitonic
and singular supersymmetric solutions are usefully plotted as a curve
on a plane parametrized by the charge and the expectation value of the
operator dual to the scalar . On this plane supersymmetric solitons
spiral into the point , while the singular solutions spiral out of
the same point (see Fig. 15); the two spirals are non self
intersecting ^{10}^{10}10We are very grateful to V. Hubeny for suggesting the
possibility of a spiral structure for these curves.. We find this extremely
intricate structure of supersymmetric solutions quite fascinating, and
feel that its implications for Yang Mills physics certainly
merits further investigation.

In this paper we have so far only considered charged black holes with
vanishing angular momentum. Such solutions are spherically symmetric;
i.e. they preserve the rotational isometry
group. As all known supersymmetric black holes in
possess angular momentum
[18, 19, 20], it is of
interest to generalize the study of this paper to black holes with
angular momentum. Let us first consider a spinning Kerr RNAdS black
hole that preserves only . Perturbations about such a solution are functions of an angle and
a radius and are given by solutions to partial rather than ordinary
differential equations. However there exist RNAdS black holes with
self dual angular momentum. The angular momentum of such a black hole
lie entirely within one of the two above, and so preserve a
subgroup of the rotation group. Perturbations
around these solutions may be organized in representations of
and so obey ordinary rather than partial differential
equations.^{11}^{11}11We thank A. Strominger for pointing this out to
us. Consequently, the generalization of the hairy black hole
solutions determined in this paper to solutions with self dual spin
appears to be a plausibly tractable project; which we however leave to
future work.

Even though we do not embark on a serious analysis of charged rotating
black holes in this paper, we do present a speculative appetizer for
this problem. In more detail we present a guess (or a prediction) for
the leading order thermodynamics of these spinning hairy
solutions. Our guess is based on the observation that the
thermodynamics of the hairy solutions constructed in this paper can be
reproduced, at leading order, by modeling the hairy black hole as a
non interacting mix of a RNAdS black hole and the soliton. In section
7 we simply assume that a similar model works for charged
spinning black holes, and use this model to compute the thermodynamics
of a certain class of spinning hairy black holes. The most interesting
aspect of our
results concern the BPS limit. Our non interacting model predicts that
extremal hairy black holes are BPS at every value of the angular
momentum and charge. This is in stark contrast with Kerr RNAdS black
holes that are BPS only on a co dimension one surface of the space of
extremal black holes. According to our non interacting model, BPS
hairy black holes are a non interacting mix of Gutowski Reall black
holes [18, 19, 20] and
supersymmetric solitons. Such a mix is thermally equlibrated at all
values of charge and angular momentum because of an important property
of Gutowski Reall black holes; their chemical potential is exactly
unity^{12}^{12}12We are very grateful to S. Kim for explaining this to us..
We find this result both intriguing and puzzling (see
e.g. [21]). We emphasize that our prediction
is based on the non interacting superposition model, which may or may
apply to actual black hole solutions. We leave further investigation of
this extremely interesting issue to future work.

This paper has been devoted largely to the study of small very near
extremal charged black holes in that are smeared over
. As uncharged small smeared black holes are well known to suffer
from Gregory-Laflamme type instabilities [22], the
reader may wonder whether the black holes studied in this paper might
suffer from similar instabilities. We believe that this is not the
case. Recall that the likely end point of a Gregory Laflamme type
instability is a small black hole of proper horizon radius localized
on the
. In order
that this black hole be near extremal, it has to zip around the
at near the speed of light, i.e. at with . The charge of such a black hole is given by while its energy above extremality of such
a black hole is given by .^{13}^{13}13To
see this let the sphere be given by equations
where are the three complex
embedding coordinates. A black hole we study is located at
, and moves with speed
in each of the three orthogonal
planes. Let the proper mass of the black hole be . Its angular momentum in each plane, , is given by

We also note that the Gubser Mitra instability
[23, 24] afflicts three equal charge black
holes only when the black holes in question have large enough charge.
It follows that the small black holes primarily studied in this paper
do not suffer from Gubser Mitra type instabilities^{16}^{16}16We thank
M. Rangamani for a discussion on this point..

It is conceivable that the solutions presented in this paper might
suffer from further superradiant instabilities, once embedded in IIB
theory on . In order to see why this might be the
case, let us recall once again why the field - dual to the
chiral Yang Mills operator condensed in the
presence of very near extremal charged RNAdS black hole. The reason is
simply that the energy of this field is equal to its charge
. As a consequence the Boltzmann suppression factor,
of this mode exceeds unity when
causing this mode to Bose condense. However exactly the same
reasoning applies to, for instance, the field dual to the chiral
operator all of which have ^{17}^{17}17
This statement is true more generally of every operator in the
chiral ring of the theory..
It seems likely that there exist other
hairy solutions in which some linear combination of (rather
than simply ) condense ^{18}^{18}18In the BPS limit any linear
combination of can condense and we have an infinite
dimensional moduli space of solutions (see
[25, 17]). We expect the introduction of a
black hole to lift this moduli space, to a discrete set of
solutions.. It is important to know whether any solution of this
form has higher entropy than the black holes with pure
condensate presented in this paper. If this is the case then the hairy
black holes of our paper would likely suffer from superradiant
instabilities towards the condensation to the entropically dominant
black hole. On the other hand the black holes of this paper, with
, the lightest chiral scalar operator that preserves all
discrete symmetries of the problem, as the only condensate,
are quite special. It seems quite plausible to us that the solution
presented in this paper has the largest entropy of all the hairy
solutions with condensates. If this is indeed the case then
the hairy black hole solutions presented in this paper constitute the
thermodynamically dominant saddle point of Yang Mills
very near to supersymmetry; and the entropy of Yang Mills
very near to the BPS bound is given the formula (122) below.

To end this introduction we would like to emphasize that the black
hole solutions of this paper give a qualitatively different picture of
the density of states of Yang Mills theory at finite
charge compared to a picture suggested by RNAdS black holes. As we
have seen above, there exist no RNAdS black holes with masses between
and , a fact had previously been taken to
suggest that, for some mysterious reason, there are less than states in Yang Mills theory between and
. The new black hole solutions of this paper establish, on
the other hand, that Yang Mills theory has states all the way down to the BPS bound at least at small
charge, and plausibly at all values of the charge (see
Fig. 2).^{19}^{19}19This difference is starkest in the limit of large charge,
i.e. in the Poincare Patch limit. The energy density, , of
RNAdS black branes is bounded from below by
where is the charge density. Fig. 2 , on the other
hand predicts that the energy density of a charged black brane can
be arbitrarily small at any given value of the charge density. The
saddle point that governs near BPS behaviour is a
mix of a charged Bose condensate and a normal charged
fluid. It would be fascinating to find some (even qualitative)
confirmation of this picture from a direct field theory analysis.

## 2 A Consistent Truncation and its Equations of Motion

### 2.1 A Consistent Truncation of Gauged Supergravity

gauged supergravity constitutes a consistent truncation of IIB theory on . In addition to the metric, the bosonic spectrum of this theory consists of 42 scalar fields, 15 gauge fields and 12 two form fields. The scalars transform in the + + + of SO(6), the gauge fields transform in the 15 dimensional adjoint representation, while the two form fields transform in the representation of .

It has been shown [26] that gauged supergravity admits a further consistent truncation that retains only the scalars in the 20 and the vector fields in the together with the metric, setting all other fields to zero. The action for this consistent truncation is given by [26]

(4) |

where

(5) |

Here denote the vector indices and are the space time indices. are symmetric unimodular (i.e. is a matrix of unit determinant) tensors. Further is the rank of the gauge group of the dual Yang Mills theory, and we work in units in which the with unit radius solves (4).

We will now describe a further consistent truncation of (4). For this purpose we find it useful to move to a complex basis for the vector indices that appear summed in (4). Let denote Cartesian directions. We define the complex coordinates

We will now argue that the restriction

(6) |

constitutes a consistent truncation of (4). To see this is the case note that the permutations of labels , as also separate rotations by in the , and planes, can each be generated by separate gauge transformations. It follows that these discrete transformations are symmetries of (4). Now it is easy to convince oneself that (6) is the most general field configuration of (4) that is invariant separately under each of these four discrete symmetries. It follows that (6) is a consistent truncation of the system (4).

The consistent truncation (6) is governed by the Lagrangian

(7) |

Note that has charge 2 and .
Under the AdS/CFT dictionary this field maps to an operator
of dimension Note also that the kinetic term of the gauge
field has the factor prefactor rather than the (more usual)
as employed, for instance, in [14].
^{20}^{20}20Consequently gauge fields and chemical potentials
in this paper and [14] are related by

### 2.2 Equations of Motion

We now list the equations of motion that follow from varying (7). We find the Einstein equation

(8) |

where

(9) |

the Maxwell equation

(10) |

and the scalar equation

(11) |

In this paper we study static spherically symmetric configurations of the system (7) We adopt a Schwarzschild like gauge and set

(12) |

The four unknown functions , , and are constrained by Einstein’s equations, the Maxwell equations and the scalar equations. It is possible to demonstrate that are solutions to the equations of motion if and only if

(13) |

where

The equations and are derived from the and components of the Einstein equations, is the component of the Maxwell equation and is the equation of the scalar field.

As in [14] the equations (13) contain only first derivatives of and , but depend on derivatives up to the second order for and . It follows that (13) admit a 6 parameter set of solutions. One of these solutions is empty AdS space, given by , , . We are interested in those solutions to (13) that asymptote to AdS space time, i.e. solutions whose large behaviour is given by

(14) |

As in [14] it turns out that these conditions effectively impose two conditions on the solutions of (13), so that the system of equations admits a four parameter set of asymptotically AdS solutions. We usually also be interested only in solutions that are regular (in a suitable sense) in the interior. This requirement will usually cut down solution space to distinct classes of two parameter space of solutions; the parameters may be thought of as the mass and charge of the solutions.

### 2.3 RNAdS Black Holes

The AdS-Reissner-Nordstrom black holes constitute a very well known two parameter set of solutions to the equations (13). These solutions are given by

(15) |

where is the chemical potential of the RNAdS black hole. The function in (15) vanishes at and consequently this solution has a horizon at . In fact, it can be shown that is the outer event horizon provided

(16) |

As explained in [14] and in the introduction, (15) is unstable to superradiant decay provided in the presence of field of charge and minimum energy provided . Now our field has and . Moreover, in the limit , RNAdS black holes have (this inequality is saturated at extremality). It follows that small extremal black holes lie at the edge of instability, as mentioned in the introduction. We show below that very near extremal RNAdS black holes do in fact suffer from super radiant instabilities.

## 3 The Supersymmetric Soliton in Perturbation Theory

In this section we will construct the analogue of the ground state soliton in [14]. The new feature in here is that the soliton turns out to be supersymmetric (this is obvious at linearized order).

In this section we generate the solitonic solution in perturbation theory. We use only the equations of motion without imposing the constraints of supersymmetry, but check that our final solution is supersymmetric (by verifying the BPS bound order by order in perturbation theory). This method has the advantage that it generalizes in a straightforward manner to the construction of non supersymmetric hairy black holes in subsequent sections.

In Section 5 we will revisit this solitonic solution; we will rederive it by imposing the constraints of supersymmetry from the start. That method has the advantage that it permits a relatively simple extrapolation of supersymmetric solutions to large charge.

### 3.1 Setting up the perturbative expansion

We now turn to the description of our perturbative construction. To initiate the perturbative construction of the supersymmetric soliton we set

(17) |

and plug these expansions into (13). We then expand out and solve these equations order by order in . All equations are automatically solved up to . At order the last equation in (13) is trivial while the first three take the form

(18) |

On the other hand, at order the first three equations in (13) is trivial while the last equation reduces to

(19) |

Here the source terms

### 3.2 The Soliton up to

The perturbative procedure outlined in this subsection is very easily implemented to arbitrary order in perturbation theory. In fact, by automating the procedure described above, we have implemented this perturbative series to 17th order in a Mathematica programme. In the rest of this subsection we content ourselves with a presentation of our results to .

(21) |

(22) |

(23) |

(24) |

The soliton obeys the BPS relation to the order to which we have carried out our computation (we present more details of the thermodynamics in Section 6).

## 4 The Hairy Black Hole in Perturbation Theory

### 4.1 Basic Perturbative strategy

We will now present our perturbative construction of hairy black hole solutions. In order to set up the perturbative expansion we expand the metric gauge field and the scalar fields as

(25) |

where the unperturbed solution is taken to be the RNAdS black hole

(26) |

The chemical potential of our final solution will be given by an expression of the form

(27) |

Note that, at the leading order in the perturbative expansion, .

Our basic strategy is to plug the expansion (25) into the equations of motion and then to recursively solve the later in a power series in . We expand our equations in a power series in . At each order in we have a set of linear differential equations (see below for the explicit form of the equations), which we solve subject to the requirements of the normalisability of and at infinity together with the regularity of and the metric at the horizon. These four physical requirements turn out to automatically imply that i.e. the gauge field vanishes at the horizon, as we would expect of a stationary solution. These four physical requirements determine 4 of the six integration constants in the differential equation, yielding a two parameter set of solutions. We fix the remaining two integration constants by adopting the following conventions to label our solutions: we require that fall off at infinity like (definition of ) and that the horizon area of our solution is (definition of ). This procedure completely determines our solution as a function of and . We can then read of the value of in (27) on our solution from the value of the gauge field at infinity.

As in [14], the linear differential equations that arise in perturbation theory are difficult to solve exactly, but are easily solved in a power series expansion in , by matching near field, intermediate field and far field solutions. At every order in we thus have a solution as an expansion in . Our final solutions are, then presented in a double power series expansion in and .

### 4.2 Perturbation Theory at

In this section we present a detailed description of the implementation of our perturbative expansion at . The procedure described in this subsection applies, with minor modifications, to the perturbative construction at for all .

Of course all equations are automatically obeyed at . The only nontrivial equation at is where is the linearized gauge covariantised derivative about the background (26). We will now solve this equation subject to the constraints of normalisability at infinity, regularity at the horizon, and the requirement that

at large .

#### 4.2.1 Far Field Region ()

Let us first focus on the region . In this region the black hole (26)

(28) |

is a small perturbation about global AdS space. For this reason we expand

(29) |

where the superscript out emphasises that this expansion is good at large . In the limit , (28) reduces to global AdS space time with . A stationary linearised fluctuation about this background is gauge equivalent to a linearised fluctuation with time dependence about global AdS space with ( is the temporal component of the gauge field). The required solution is simply the ground state excitation of an minimally coupled scalar field about global AdS and is given by

(30) |

The overall normalisation of the mode is set by our definition of which implies

We now plug (29) into the equations of motion and expand to to solve for . Here is the gauge covariant Laplacian about the background (28). Now

where is the gauge covariant Laplacian about global AdS space time with background gauge field . It follows that, at ,

This equation is easily integrated and we find