Abundant stable gauge field hair for black holes in anti-de Sitter space

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by  J. E. Baxter

Abundant stable gauge field hair for black holes in anti-de Sitter space
J. E. Baxter,1 Marc Helbling,2 and Elizabeth Winstanley1,∗
1Department of Applied Mathematics, The University of Sheffield,
Hicks Building, Hounsfield Road, Sheffield, S3 7RH, United Kingdom.
2INSA de Rouen, Laboratoire de Math´ematiques (LMI),
Place Emile Blondel BP 08, 76131 Mont Saint Aignan Cedex, France.
(Dated: February 1, 2008)
Wepresentnewhairyblackholesolutionsofsu(N)Einstein-Yang-Millstheory(EYM)inasymp-
totically anti-de Sitter (adS) space. These black holes are described by N +1 independent param-
eters, and have N −1 independent gauge field degrees of freedom. Solutions in which all gauge
field functions have no zeros exist for all N, and for sufficiently large (and negative) cosmological
constant. Atleast some ofthesesolutions areshown tobestableunderclassical, linear, spherically
symmetric perturbations. Therefore there is no upper bound on the amount of stable gauge field
8
hair with which a black hole in adS can beendowed.
0
0 PACSnumbers: 04.20.Jb,04.40.Nr,04.70.Bw
2

n
The “no-hair” conjecture [1] states that black hole dialco-ordinater only. Here,andthroughoutthis letter,
a
J equilibrium states possess extremely simple geometries, the metric has signature (−,+,+,+) and we use units

8 determined completely by the mass, angular momentum in which 4πG = c = 1. In the presence of a negative
and charge of the black hole. While hairy black hole cosmological constant Λ, we write the metric function µ

]
solutionsofthe Einsteinequationshavebeendiscovered, as
c
q particularlyinEinstein-Yang-Mills(EYM)theoryandits 2m(r) Λr2
- variants (see [2] for a review), many of the plethora of µ(r)=1− − . (2)
r r 3
g new black hole solutions found in the literature are clas-
[ sically unstable. Those hairy black holes which are sta- The most general, spherically symmetric, ansatz for the

2 ble (such as the su(2) EYM black holes in anti-de Sitter su(N) gauge potential has been given in [8]. Here, we
v space (adS) [3, 4]) have, at least to date, been described assume that the gauge potential is purely magnetic and
6 by only a small number of parameters additional to the has the gauge-fixed form:
5
mass, angular momentum and charge of the black hole.
3 This means that the “spirit” if not the “letter” of the A= 1 C−CH dθ− i C+CH sinθ+Dcosθ dφ,
2 2 2
. no-hair conjecture is maintained. (cid:0) (cid:1) (cid:2)(cid:0) (cid:1) (cid:3) (3)
8
In recent years there has been an explosion of interest where C is an (N × N) upper-triangular matrix with
0
7 in hairy black holes in adS, partly because at least some non-zero entries immediately above the diagonal:
0 oftheseconfigurationsarestable,butalsobecauseofthe
: importance of the adS/CFT correspondence[5] in string Cj,j+1 =ωj(r), (4)
v
theory. Inparticular,ithasbeensuggested[6]thatthere
Xi shouldbeobservablesinthedual(deformed)CFTwhich for j = 1,...,N −1, with CH the Hermitian conjugate
of C, and D is a constant diagonal matrix:
r are sensitive to the presence of black hole hair (see also
a
[7] for an adS/CFT interpretation of some stable seven-
D =Diag(N −1,N −3,...,−N +3,−N +1). (5)
dimensional black holes with so(5) gauge fields). Our
purpose in this letter is to present new stable, asymp- The (N −1) Yang-Mills equations take the form
totically adS, hairy black hole solutions of su(N) EYM
2Λr3
for sufficiently large |Λ| which are described by an un- r2µω′′+ 2m−2r3p − ω′ +W ω =0 (6)
bounded number of parameters. The existence of these j (cid:18) θ 3 (cid:19) j j j
solutionscaststhestatusofthe“no-hair”conjectureina ′
for j =1,...,N −1, where a prime denotes d/dr, and
completely new light: equilibrium black holes in adS are
no longer simple objects, but rather require an infinite N
number of parameters in order to fully determine their pθ = 41r4 ωj2−ωj2−1−N −1+2j 2 , (7)
geometry. Xj=1h(cid:0) (cid:1) i
We consider static, spherically symmetric, four- 1
dimensional black holes with metric Wj = 1−ωj2+ 2 ωj2−1+ωj2+1 , (8)
(cid:0) (cid:1)
ds2 =−µS2dt2+µ−1dr2+r2dθ2+r2sin2θdφ2, (1) withω0 =ωN =0. TheEinsteinequationstaketheform
′
S 2G
where the metric functions µ and S depend on the ra- m′ =µG+r2pθ, = , (9)
S r
2
where todivergeortheyhaveconvergedtotheasymptoticform
at infinity.
N−1
G= ω′2. (10) As in the su(2) case [3], we find black hole solutions
j in open subsets of the N-dimensional parameter space
Xj=1
(ω (r ),r ) for fixed Λ. For sufficiently large |Λ| (where
j h h
The field equations (6,9) have the following trivial so- how large “sufficiently large” is depends on the radius
lutions. Setting ωj(r) ≡ ± j(N −j) for all j gives the of the event horizon rh), we find that the gauge field
Schwarzschild-adSblackhoplewithm(r)=M =constant functions ωj(r) all have no zeros. In figure 1 we show a
(which can be set to zero to give pure adS space). Set- typical nodeless solution, for su(4) EYM. It can be seen
tingω (r)≡0forallj givestheReissner-Nordstro¨m-adS
j
black hole with magnetic charge. There is an additional
special class of solutions, given by setting
ω (r)=± j(N −j)ω(r) ∀j =1,...,N −1. (11)
j
p
In this case, it is possible to show, using a rescaling
method along the lines of that in [9], that the field vari-
ables ω(r), m(r) and S(r) satisfy the su(2) EYM field
equations with a negative cosmological constant. Fur-
thermore, the boundary conditions (as discussed below)
are also preserved. Therefore any su(2), asymptotically
adS, EYM black hole solution can be embedded into
su(N) EYM to give another asymptotically adS black
hole.
In this letter we study black hole solutions of the field
equations (6,9), returning to soliton solutions elsewhere
[10]. We assume there is a regular, non-extremal, black
hole event horizon at r = rh. The field variables ωj(r), FIG. 1: A typical black hole solution of su(4) EYM in which
m(r) andS(r) will haveregularTaylorseries expansions all the gauge field functions ωj(r) are nodeless. For this so-
aNbo+u1t rqu=anrthit.ieTshωes(ere)x,pran,sSio(nrs)afroerdfiexteedrmcionsemdobloygitchael lfuutnicotnio,nΛso=n−th1e0evaenndthrhor=izo1n.arTeh:eωv1(arluhe)s=o2f.t3h,eω2g(aruhg)e=fie2l.d6
constant Λ. Since jthehfieldh equahtions (6,9) are invariant and ω3(rh)=2.2.
under the transformation ω (r) → −ω (r) (for any j in-
j j
thatthemetricfunctionsm(r)andS(r)haveverysimilar
dependently), we may consider ω (r ) > 0 without loss
j h behaviourtothesu(2)case,andthat,since|Λ|issolarge,
of generality. For the event horizon to be non-extremal,
the gauge field functions do not vary significantly from
it must be the case that
their values at the event horizon.
2m′(r )=2r2p (r )<1−Λr2, (12) The phase space of black hole solutions in the su(3)
h h θ h h
case, with Λ=−10 and r =1 is shown in figure 2, and
h
which constrains the possible values of the gauge field is typical of the phase space for large values of |Λ|. In
functions ω (r ) at the event horizon. At infinity, the figure 2 we have examined, for Λ = −10 and r = 1, all
j h h
boundary conditionsare considerablyless stringentthan values of the ω (r ) and ω (r ) which satisfy the con-
1 h 2 h
intheasymptoticallyflatcase. Inorderforthemetric(1) straint (12). The inequality in (12) is saturated on the
tobeasymptoticallyadS,wesimplyrequirethatthefield outer-mostcurveinfigure2. Itcanbe seenfromfigure2
variables ω (r), m(r) and S(r) converge to constant val- that not all values of (ω (r ),ω (r )) give black hole so-
j 1 h 2 h
ues as r →∞, andhave regularTaylorseries expansions lutions;thosevaluesforwhichnoregularblackholesolu-
inr−1 nearinfinity. SinceΛ<0,thereisnocosmological tion satisfying the boundary conditions at infinity could
horizon. be found lie in the narrow band on the outside of the
Thefieldequations(6,9)areintegratednumericallyus- plot. The region between this narrow band and the co-
ing standard ‘shooting’ techniques [11]. The equation ordinateaxescontainsblackholesolutionsinwhichboth
for S(r) decouples from the other Einstein equation and gauge field functions ω (r) and ω (r) have no zeros. We
1 2
the Yang-Millsequationssocanbe integratedseparately havealso plotted in figure 2 the line ω (r )=ω (r ), on
1 h 2 h
if required. We start integrating just outside the event which lie embedded su(2) solutions given by (11). The
horizon, using as our shooting parameters the N vari- significance of the shaded region in figure 2 will be de-
ables ω (r ) andr , subjectto the weakconstraint(12). scribed shortly. More detailed properties of the phase
j h h
The field equations are then integrated outwards in the space of black hole solutions will be discussed elsewhere
radial co-ordinate r until either the field variables start [10].
3
gauge field hair, are stable. We consider linear, spher-
ically symmetric perturbations only for simplicity. Even
for spherically symmetric perturbations, the analysis is
highly involved in the su(N) case and the details will be
presentedelsewhere. Here we briefly outline just the key
features.
Firstly we consider spherically symmetric perturba-
tions of the gauge potential (3), fixing the gauge so that
the perturbed potential is purely magnetic and has the
form [8]
1
A = Bdr+ C−CH dθ
2
(cid:0) (cid:1)
i
− C+CH sinθ+Dcosθ dφ. (13)
2
(cid:2)(cid:0) (cid:1) (cid:3)
Here,the matricesB andC dependonbotht andr, and
matrix D is still constant and given by (5). The matrix
FIG. 2: Phase space of black hole solutions in su(3) EYM
with Λ = −10 and r = 1. The shaded region shows where B(t,r) is traceless, diagonal and has purely imaginary
h
solutionsexistwhichsatisfy theinequalities(16)attheevent entries. The only non-zero entries of the matrix C(t,r)
horizon. are:
C (t,r)=ω (t,r)exp(iγ (t,r)). (14)
j,j+1 j j
In [3], the existence of black hole solutions for which
As usual, the metric retains the form (1) but now the
the gauge function ω(r) had no zeros was proven ana-
functions m and S depend on both t and r. With this
lytically in the su(2) case. Since su(2) solutions can be
choiceofgaugepotential(13),theperturbationequations
embeddedassu(N)solutionsvia(11),wehaveautomat-
decouple into two sectors:
icallyananalyticproofoftheexistenceofnodelesssu(N)
EYMblackholes inadS.However,these embeddedsolu-
• thesphaleronic sectorconsistingofentriesofB and
tions are ‘trivial’ in the sense that they are described by
the functions γ ;
j
just three parameters: r , Λ and ω(r ). An important
h h
questioniswhetherthe existenceof‘non-trivial’(thatis, • the gravitational sector which consists of the per-
genuinely su(N)) solutions in which all the gauge field turbations of the metric functions δm and δS and
functions ωj(r) haveno zeroscanbe provenanalytically. the perturbations of the gauge field functions δωj.
Theanswertothisquestionisaffirmative,andinvolvesa
generalization to su(N) of the continuity-type argument Theformoftheperturbationequationsinthesphaleronic
usedin[3]. Thedetailsarelengthyandwillbepresented sector is little changed from the asymptotically flat case
elsewhere. However, the main thrust of the argument [12]. Itconsistsof2N−1coupledequationsforthe2N−1
can be simply stated. We firstly prove (generalizing the variables (N diagonalentries of the matrix B and N −1
analysisof[9]to include Λ)thatthe fieldequations(6,9) functions γ ). In addition, there is the Gauss constraint,
j
and initial conditions at the event horizon possess, lo- which gives N coupled consistency conditions. After
cally in a neighborhood of the horizon, solutions which much algebra (along the lines of [12]), the sphaleronic
are analytic in r, r , Λ and the parameters ω (r ). This sector perturbation equations can be cast in the form
h j h
enablesus to provethat, inasufficiently smallneighbor-
hood of any embedded su(2) solution in which ω(r) has −Ψ¨ =UΨ, (15)
no nodes, there exists (at least in a neighborhood of the
event horizon) an su(N) solution in which all the ω (r) whereadotdenotes ∂/∂t,the (2N−1)-dimensionalvec-
j
have no nodes. The key part of the proof lies in then tor Ψconsists of combinationsof perturbationsand U is
showing that these su(N) solutions can be extended out a self-adjoint,secondorder,differentialoperator(involv-
to r →∞ and that they satisfy the boundary conditions ing derivatives with respect to r but not t), depending
at infinity. This gives genuinely su(N) black hole solu- on the equilibrium functions ωj(r), m(r) and S(r). It
tionsinwhichallthe gaugefieldfunctionshavenozeros, canbe shownthatthe operatorU is regularandpositive
andwhicharecharacterizedbytheN+1parametersrh, providedthe unperturbed gaugefunctions ωj(r) haveno
Λ and ω (r ). zeros and satisfy the N −1 inequalities
j h
The other outstanding question is whether these new 1
black holes, with potentially unbounded amounts of ωj2 >1+ 2 ωj2+1+ωj2−1 (16)
(cid:0) (cid:1)
4
for all j = 1,...N −1. These inequalities define a non- above, our analytic work ensures the existence of gen-
empty subset of the parameter space, which is shown in uinely su(N) solutions in a sufficiently small neighbor-
the su(3) case in figure 2. hood of these embedded su(2) solutions. These su(N)
Theshadedregioninfigure2showswheretheinequal- solutions are such that the inequalities (16) are satisfied
ities (16) aresatisfiedfor the gaugefieldfunctions atthe for all r ≥r (and therefore the solutions are stable un-
h
eventhorizon. However,therequirementsof(16)arecon- der sphaleronic perturbations). The positivity of M can
siderablystronger,astheinequalitieshavetobesatisfied then be extended to these genuinely su(N) solutions us-
for all r ≥r . Our analytic work shows that, in fact, for ingananalyticityargument,basedonthenodaltheorem
h
sufficiently large |Λ|, there do exist solutions to the field of [13]. The technical details of this argument will be
equations for which the inequalities (16) are indeed sat- presented elsewhere. Therefore at least some of our so-
isfied for all r (an example of such a solution is shown lutions are linearly stable in both the gravitational and
in figure 3). This involves proving that for at least some sphaleronic perturbation sectors.
For sufficiently large |Λ| (for each fixed r ), we have
h
shown the existence of su(N) EYM black holes in adS,
which are described by N +1 parameters and are stable
underlinear,sphericallysymmetric perturbations. Ifthe
cosmological constant is very large and negative, there
are potentially a very large number of possible gauge
field configurations giving the same mass and magnetic
charge at infinity. As explained in the introduction, we
anticipate that these solutions may well have interesting
consequences for the adS/CFT correspondence [5]. We
hope to return to these questions in the near future.
We thank Eugen Radu for many informative discus-
sions. The work of JEB is supported by UK EPSRC,
and the work of EW is supported by UK PPARC, grant
reference number PPA/G/S/2003/00082.
FIG. 3: An example of an su(3) solution for which the in-
equalities (16) are satisfied for all r ≥ r . In this example,
h
Λat=th−e1ev0e,nrthh=or1izoanndarteheω1v(arlhu)es=o2f,tωh2e(rgha)ug=e1fi.e9l5d. functions [1∗] REl.eRcturffionniciaandddrJe.sAs:.EW.Wheienlsetra,[email protected]ffidaeyld2.a4c.3u0k(1971).
[2] M. S. Volkov and D. V. Gal’tsov, Phys. Rept. 319 1
solutions for which the gauge field function values at the (1999).
eventhorizonlie withinthe regionwherethe inequalities [3] E. Winstanley,Class. Quant.Grav. 16 1963 (1999).
(16)aresatisfied,thegaugefieldfunctionsremainwithin [4] J. Bjoraker and Y. Hosotani, Phys. Rev. Lett. 84, 1853
this open region. (2000); Phys.Rev.D62 043513 (2000).
[5] J. Maldacena, Adv. Theor. Math. Phys. 2 231 (1998);
For the gravitational sector, the metric perturbations
E. Witten, Adv. Theor. Math. Phys. 2 253 (1998); ibid
can be eliminated to yield a set of N −1, coupled per- 2 505 (1998).
turbation equations of the form [6] T. Hertog and K. Maeda, JHEP 07 (2004) 051.
[7] J. P. Gauntlett, N. Kim and D. Waldram, Phys. Rev.
−δω¨ =Mδω, (17) D63 126001 (2001).
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D76 104017 (2007).
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[11] W. Vetterling, W. Press, S. Teukolsky and B. Flannery,
rium functions ω (r), m(r) and S(r). The operator M
j Numerical RecipesinFORTRAN(CambridgeUniversity
is more difficult to analyze than the operator U. For
Press, 1992).
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