Contents lists available atScienceDirect
Physics
Letters
B
www.elsevier.com/locate/physletb
AdS
5
magnetized
solutions
in
minimal
gauged
supergravity
Jose
Luis Blázquez-Salcedo
a,
∗
,
Jutta Kunz
a,
Francisco Navarro-Lérida
b,
Eugen Radu
c aInstitutfürPhysik,UniversitätOldenburg,Postfach2503,D-26111 Oldenburg,GermanybDept.deFísicaAtómica,MolecularyNuclear,CienciasFísicasUniversidadComplutensedeMadrid,E-28040Madrid,Spain cDepartamentodeFísicadaUniversidadedeAveiroandCIDMA,CampusdeSantiago,3810-183Aveiro,Portugal
a
r
t
i
c
l
e
i
n
f
o
a
b
s
t
r
a
c
t
Articlehistory: Received27March2017 Accepted5May2017 Availableonline10May2017 Editor:M.Cvetiˇc
WeconstructageneralizationoftheAdSchargedrotatingblackholeswithtwoequalmagnitudeangular momentainfive-dimensionalminimalgaugedsupergravity.Inadditiontothemass,electricchargeand angularmomentum,thenewsolutionspossessanextra-parameterassociatedwithanon-zeromagnitude ofthemagneticpotentialatinfinity.Incontrastwiththeknowncases,thesenewblackholespossessa non-trivialzero-horizonsizelimitwhichdescribesaoneparameterfamilyofspinningchargedsolitons. AllconfigurationsreportedinthisworkapproachasymptoticallyanAdS5spacetimeinglobalcoordinates andarefreeofpathologies.
©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
1. Introductionandmotivation
Thesolutionsofthefive-dimensionalgaugedsupergravity mod-elsplayacentralroleintheAdS/CFTcorrespondence
[1,2]
, provid-ingadualdescriptionofstrongly-coupledconformalfieldtheories (CFTs)onthefour-dimensionalboundaryoffive-dimensional anti-deSitter(AdS)spacetime.In the minimal case, the bosonic sector of the gauged super-gravitymodelconsistsonlyofthegraviton andan Abelianvector field.However,despiteitssimplicity,constructingsolutionsofthis theoryisanontrivialtask,sincetheknowngenerationtechniques donotworkinthepresenceofacosmologicalconstant.Thusone hastoresorttotrialanderrorortonumericalcalculations,starting fromanappropriate Ansatz.Restrictingtostationarysolutions ap-proachingasymptoticallyagloballyAdS5spacetime,onenotesthat theproblemgreatlysimplifiesforthespecialcasewherethetwo independentangularmomentaofthegenericconfigurationsareset equal.Thisfactorizesthe dependenceontheangularcoordinates, leading toa cohomogeneity-1 problem,with ordinarydifferential equations.Subjecttotheseassumptions,ageneralblackhole(BH) solutionhasbeenfoundinclosedformin
[3,4]
byCvetiˇc, Lüand Pope(CLP).Thissolutionischaracterized by threenon-trivial pa-rameters,namelythemass,theelectriccharge,andone indepen-dent angular momentum. These parameters are subjectto some constraints,suchthat closedtimelike curvesandnakedsingulari-*
Correspondingauthor.E-mailaddress:jose.blazquez.salcedo@uni-oldenburg.de(J.L. Blázquez-Salcedo).
tiesareavoided.Moreover,theCLPsolutionpossessesanextremal limitwhichpreservessomeamountofsupersymmetry
[5]
.AsimpleinspectionoftheBHin
[3]
showsthatitdoesnot pos-sess aglobally regularsolitoniclimit whichcouldbe viewedasa deformationoftheAdSbackground,whilethemagneticfield van-ishes asymptotically. However, a number of recent studies [6–9]have provided evidence that the previously known solutions of the Einstein–Maxwellsysteminaglobally AdS4 background, rep-resent only ‘thetip ofthe iceberg’,being insome sense the AdS counterpartsofthe(well-known)MinkowskispacetimeBHs.A va-rietyofnewconfigurationswereshowntoexist.Instrongcontrast to theasymptotically flatcase, thisincludes particle-likesolitonic configurations [6,7] and even BHs withno spatial isometries
[9]
. Their existence can be traced back to the “box”-like behavior of theAdSspacetime,whichallowstheexistenceofelectric(or mag-netic)multipoles,astestfields,whichareeverywhereregular.However,this“box”-likebehaviorisnotspecifictoAdS4 space-time. It hasbeenshownrecentlythat cohomogeneity-1 solutions of Einstein–Maxwell theory in odd D dimensions can be ob-tained witha non-vanishing magnetic field at the AdSD bound-ary1 [10]. These represent new families of static solitons and black holes with rather different properties as compared to the well-known Reissner–Nordström–AdS solutions. This result sug-gests that similar solutions should exist also for D
=
5 dimen-sionswithintheminimalgaugedsupergravitymodel.However,the1 Seealsothemoregeneralresultsin[11,12].
http://dx.doi.org/10.1016/j.physletb.2017.05.014
0370-2693/©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.
Einstein–Maxwell–Chern–Simonscaseismorecomplex;apartfrom theabsenceoftheelectric-magnetic duality,wenote thatthe so-lutionswitha magneticfield necessarily rotateandalsothat the signoftheelectricchargebecomesrelevant
[13,14]
.This paperpresents the results ofa preliminary investigation in this direction, by focusing on the simplest case of configura-tionswithequalmagnitudeangularmomenta.The newsolutions reported here provide an extension of the CLP BHs which con-tains an additional parameter associated with the magnitude of themagneticpotential atinfinity. Ourresults showthe existence ofa variety ofnew propertiesof thesolutions. For example,the BHspossessa nontrivialparticle-likelimit describing charged ro-tating solitons. Also, one finds solutions which rotate locallybut havevanishingtotalangularmomentum.
2. Themodel
2.1.Theactionandequations
TheactionforD
=
5 minimalgaugedsupergravityisgivenby I=
1 16π
M d5x√
−
g(
R+
12 L2−
FμνF μν+
2λ
3√
3ε
μναβγA μFναFβγ)
+
Ib,
(1)whereR isthecurvaturescalar,L istheAdSlengthscale,Aμ isthe gaugepotentialwiththefieldstrengthtensor Fμν
= ∂
μ Aν− ∂
ν Aμ andε
μναβγ is the Levi-Civita tensor. Also,λ
=
1 is the Chern–Simons(CS)couplingconstant.However,
λ
willbekeptgeneralin allrelationsbelow,(suchthat(1)
willdescribeagenericEinstein– Maxwell–Chern–Simons (EMCS) model), although the numerical resultswillcovertheSUGRAcaseonly.Inaddition,
(1)
containsaboundarytermwhichisrequiredfor aconsistent variationalprinciple anda properrenormalizationof variousphysicalquantities,Ib
= −
1 8π
∂M d4x−
h K−
3 L(
1+
L2 12R)
−
L 2log(
L r)
FabFab.
(2)Here,hab isthemetricinducedby gμν ontheboundary(R being the corresponding Ricci scalar), andK is the trace (with respect toh) of theextrinsic curvature ofthe boundary. Also, Fab is the electromagnetictensorinducedontheboundarybythebulkfield, whiler isanormalcoordinate.
ThefieldequationsofthismodelconsistoftheEinstein equa-tions Gμν
=
6 L2gμν+
2 FμρFρν−
1 4F 2,
(3)togetherwiththeMaxwell–Chern–Simons(MCS)equations
∇
νFμν+
λ
2
√
3ε
μναβγF
ναFβγ
=
0.
(4)2.2.Theprobelimit:Maxwell–Chern–SimonssolutionsinafixedAdS background
Beforeapproachingthe fullmodel,itisinteresting toconsider theprobe limit, i.e. aU(1) field in a fixed AdS spacetime witha line-element ds2
= −
N(
r)
dt2+
dr 2 N(
r)
+
1 4r 2(
σ
2 1+
σ
22+
σ
32),
with N(
r)
=
1+
r 2 L2.
(5)Intheabovelineelement,the(round) S3 sphere iswrittenasan S1-fibration over S2
≡ CP
1, withσ
i theleft invariant one-forms,σ
1=
cosψ
dθ
+
sinψ
sinθ
dφ
,σ
2= −
sinψ
dθ
+
cosψ
sinθ
dφ
,σ
3=
dψ
+
cosθ
dφ
; also,the coordinatesθ
,φ
,ψ
are the Euler angles on S3,withtheusualrange.Thegaugefield Ansatzcontainsan electricpotential,a0,anda magneticone,aϕ [3]
A
=
a0(
r)
dt+
aϕ(
r)
1
2
σ
3,
(6)whichresultsinthefieldstrengthtensor
F
=
a0(
r)
dr∧
dt+
aϕ(
r)
12dr
∧
σ
3+
12aϕ
(
r)
σ
2∧
σ
1.
(7)ThenonewritesthefollowingMCSequations
rNaϕ=
1+
8 3λ
2 r2a 2 ϕ4 raϕ
,
a 0= −
4λ
a3ϕ√
3r3.
(8) Forλ
=
0 (i.e. apure Maxwell field in AdS spacetime) the elec-tricpotentialcanbesettozeroandonefindsthefollowingexact solution aϕ(
r)
=
cm 1−
L 2 r2log(
1+
r2 L2)
,
(9)withcmanarbitrary(nonzero)constant.Unfortunately,theEqs.(8) cannot be solved in closedform2 for
λ
=
0. However, such a so-lution exists; its small-r expansion reads (with u,
v0,
cm andμ
ˆ
nonzeroconstants): aϕ(
r)
=
ur2+ (−
2 3 u L2+
8u3λ
2 9)
r 4+ . . . ,
a0(
r)
=
v0−
2uλ
√
3r 2+ . . . ,
(10)whileitsformforr
→ ∞
is aϕ(
r)
=
cm+
ˆ
μ
+
2cmL2log(
L r)
1 r2
+ . . . ,
a0(
r)
= −
2cm2λ
√
3 1 r2+ . . . .
(11)Thisimpliestheexistenceofanonvanishing asymptoticmagnetic field, Fθ φ
→ −
12cmsinθ
,such thattheparametercm canbe iden-tifiedwiththemagneticfluxatinfinitythroughthebasespace S2 ofthe S1fibration,m
=
1 4π
S2 ∞ F= −
1 2cm.
(12)Thesmoothprofilesconnectingtheasymptotics
(10)
,(11)
are con-structednumerically,atypicalexamplebeingshowninFig. 1
.Also,onecanshowthatbothaϕ
(
r)
anda0(
r)
arenodeless func-tions.OtherpropertiesoftheMCSsolutionsaresimilartothoseof their gravitatinggeneralizationsdiscussedin Section3.Moreover, rathersimilarconfigurationsarefoundwhenconsideringinsteada Schwarzschild–AdSBHbackground.2 Althoughonecan constructaperturbativesolutionaround(9),this involves complicatedspecialfunctions,beingnotsouseful.
Fig. 1. TheprofileofatypicalsolutionoftheMaxwell–Chern–Simonsequationsin afixedAdSbackground.
2.3. Thebackreactingcase
When taking into account the backreaction on the geometry, the solutionsabove should result inEMCS solitons andBHs. Un-fortunately, itseemsthat noanalyticaltechniquescanbe usedto construct these solutions in closed form.3 As such, in this work weapproachthisproblembysolvingtheEMCS equations numer-ically,subjecttoasetofboundaryconditionscompatiblewithan approximateexpansion atthe boundariesof the domainof inte-gration.4
The corresponding metric Ansatz is found by supplementing
(5)withfourundeterminedfunctionswhichtakeintoaccountthe deformationofthe AdSbackgroundandfactorizetheangular de-pendenceallowingforconfigurationswithtwoequalangular mo-menta ds2
= −
f(
r)
N(
r)
dt2+
1 f(
r)
m(
r)
N(
r)
dr 2+
1 4r 2 m(
r)(
σ
12+
σ
22)
+
n(
r)
σ
3−
2ω
(
r)
r dt2
,
(13)while the gauge field Ansatz5 is still given by (6). This frame-work can be proven to be consistent, and, asa result, the EMCS equationsreducetoasetofsixsecondorderordinarydifferential equationsplusafirstorderconstraintequation,whoseexpression can befound inRef. [14].Also, theseequationspossesstwo first integrals a0
+
ω
ra ϕ−
4λ
√
3 f3/2a2ϕ r3√
mn=
2 f3/2π
√
mnr3c1,
16λ
3√
3a 3 ϕ−
n3/2√
mr3 f5/2(
rω
−
ω
)
=
c 2−
8 c1π
aϕ,
(14)wherec1 andc2aretwoconstants.
3 Some partial results can be found, however,in the Einstein–Maxwell case (λ=0).Anapproximateformofthestaticsolitons(a0(r)=ω(r)=0)canbe con-structedtherebyconsideringaperturbativeexpansionofthesolutionsintermsof theparametercm.
4 Tointegrate the equations, weused the differentialequationsolver COLSYS whichinvolvesaNewton–Raphsonmethod[15].
5 A rathersimilarframework has been usedin[16] toconstruct magnetized squashedBHsinD=5 Kaluza–Kleintheory.However,thepropertiesofthose solu-tionsareverydifferent.
2.4. Theasymptotics
In deriving the far field expression of the solutions, we im-posethat,asymptotically,i
)
thegeometrybecomesAdSinastatic frame, ii)
the electricpotential vanishes, a0→
0; and, asa new feature as compared to the CLP case, iii)
the magneticpotential approachesaconstantnonzerovalue,aϕ→
cm.Thenafar-field ex-pression of asolution compatiblewiththeseassumptions canbe constructedinasystematicway.Thefirstfewtermsinthis expan-sionread f(
r)
=
1+
ˆ
α
+
12 5 c 2 mL2log(
L r)
1 r4
+ . . . ,
ω
(
r)
=
ˆ
J r3−
4q 3ˆ
μ
−
1 3cmL 2(
1−
6 log(
L r))
1 r5
+ . . . ,
m(
r)
=
1+
ˆβ +
4 5c 2 mL2log(
L r)
1 r4
+ . . . ,
n(
r)
=
1+
3(
α
ˆ
− ˆβ) +
4 15c 2 mL2+
24 5 c 2 mL2log(
L r)
1 r4
+ . . . ,
aϕ(
r)
=
cm+
ˆ
μ
+
2cmL2log(
L r)
1 r2
+ . . . ,
a0(
r)
= −
q r2+
cmλ
√
3 2μ
ˆ
+
cmL2−
1+
4 log(
L r)
1 r4
+ . . . ,
(15)with
{ ˆ
α
,
ˆβ,
ˆ
J;
cm,
μ
ˆ
,
q}
undeterminedparameters.Concerning the solitons, one can also construct a small-r ap-proximateformofthesolutionsasapowerseriesinr,compatible withtheassumptionofregularityatr
=
0.Thefirsttermsinthis expansionare6 f(
r)
=
f0+
m0−
f0 L2+
4u2f02 3m0 r2+ . . . ,
m(
r)
=
m0+
m2r2+ . . . ,
ω
(
r)
=
w1r−
8u3f5/2 0λ
3√
3m20 r 2+ . . . ,
n(
r)
=
m0+
3m0(
m0−
f0)
f0L2−
m2+
4u2f0 3r2
+ . . . ,
a0(
r)
=
v0−
2u2f03/2λ
√
3m0+
u w1 r2+ . . . ,
aϕ(
r)
=
ur2+
u 9 f0L2m0 4u2f02L2(
1+
2λ
2)
+
3(
4m20−
3 f0(
2m0+
L2M2))
r4
+ . . . ,
(16)withthefreeparameters
{
f0,
m0,
m2,
w1;
u,
v0}
.However,whengravitatingsolitonsexistinagivenmodel, nor-mally one can also construct bound states of such solitons with an event horizon [17,18]. These BHs have a horizon which is a squashed S3 sphereandresidesataconstant valueof the
quasi-6 Itisinterestingtocontrasttheseasymptoticswiththosesatisfiedbythe topo-logicalsolitonsin[23]which,however,possessavanishingmagneticfieldat infin-ity,cm=0.Fortopologicalsolitons,thepropersizeoftheψ-circlegoestozeroas
r→0,whilethecoefficientoftheroundS2-partin(13)ispositive(thisholdsalso forgrrand−gtt).
isotropicradialcoordinater
=
rH>
0.Thenon-extremalsolutions7 havethefollowingexpansionvalidasr→
rHf
(
r)
=
f2(
r−
rH)
2+
O(
r−
rH)
3,
m(
r)
=
m2(
r−
rH)
2+
O(
r−
rH)
3,
n(
r)
=
n2(
r−
rH)
2+
O(
r−
rH)
3,
ω
(
r)
=
ω
0+
O(
r−
rH) ,
a0(
r)
=
a0(0)+
O(
r−
rH)
2,
aϕ(
r)
=
a(ϕ0)+
O(
r−
rH)
2,
(17) with{
f2,
m2,
n2,
ω
0;
a(00),
a (0)ϕ
}
undetermined parameters. Also, note that the behavior of solutions inside the horizon (r<
rH) isnotdiscussedinthiswork.2.5.Physicalparameters
In the next Section we give numerical evidence forthe exis-tenceofsmoothEMCSsolutionsinterpolatingbetweenthe asymp-toticsabove.Mostofthephysicalpropertiescanbereadofffrom theasymptoticdatanearthehorizon/originandatinfinity.
The mass M and angular momentum J of these solutions is computedby usingthe quasilocal formalism [19], witha bound-arystresstensorTab
=
√2−hδhδIab.Then M and J aretheconservedchargesassociatedwithKilling symmetries
∂
t,∂
ψ ofthe inducedboundarymetrich,foundforalarge constantvalue ofr.This re-sultsin8 M
= −
π
8(
3α
ˆ
+ ˆβ)
L2+
c2 mπ
30+
3π
32L 2,
J=
π
4ˆ
J.
(18)Theelectriccharge Q ,ascomputedfromtheusualdefinition,is
Q
= −
1 2 S3∞˜
F=
π
q,
(19)with Fμ
˜
1μ2μ3≡
μ1μ2μ3ρσ Fρσ .However, thisquantity is not re-latedtoanyconservationlawifcm
=
0.Amoreappropriate defini-tionisnowthePagecharge[20,21]
,Q(P)
= −
1 2 S3 ∞˜
F+
√
λ
3A∧
F=
Q−
2√
π
3λ
c 2 m≡
c1,
(20)being related to the total derivative structure of the Maxwell– Chern–Simons equations (with c1 the integration parameter we introducedinthefirstintegral
(14)
).Anotherphysicallyrelevantparameteronecandefineisthe R-charge, associated with the conservation of the R-current of the dualtheoryattheAdSboundary
[22]
:Q(R)
= −
1 2 S3∞˜
F+
2λ
3√
3A∧
F=
Q−
4π
3√
3λ
c 2 m.
(21)Notethatthesethreechargescoincideintheabsenceofa bound-arymagneticfield,cm
=
0.Also,thefirstintegral(14)
impliesthat 12cmQ(R)
+
J= −
16πc2.7 WehavefoundnumericalevidencefortheexistenceofextremalBHsaswell. Suchsolutionspossessadifferentnearhorizonexpression,whilethefarfield ex-pansion(15)holdsalsointhatcase.TheextremalBHspossessanumberofdistinct featuresandwillbereportedelsewhere(however,somepropertiesofthe TH=0
limitcanbeseeninFig. 5).
8 NotethatM and J areevaluatedrelativetoaframewhichisnonrotatingat infinity.
In the above relations,
α
,β
,ˆ
J and q are parameters which enter the far field expansion (15). Also, we remark that the in-terpretationproposedforcm intheprobelimit,asamagneticflux atinfinity,stillholdsinthebackreactingcase.Turningnow toBH quantitiesdefinedinterms ofthehorizon boundarydatain
(16)
,wenotethatthesolutions’horizonangular velocityisH
=
ω
0 rH,
(22)while the area of thehorizon AH andthe Hawking temperature TH ofthesolutionsaregivenby
AH
=
2π
2r3H m2 f2 n2 f2,
TH=
1 2π
1+
r 2 H L2 f2√
m 2.
(23)The horizon electrostatic potential
H as measured in a co-rotatingframeonthehorizonis
H
=
a(00)+
Ha(ϕ0).
(24)Also,tohaveameasureofthesquashingofthehorizon,we intro-ducethedeformationparameter
ε
=
n(
r)
m(
r)
r=rH=
n2 m2,
(25)which gives the ratio of the S1 and the round S2 parts of the (squashed S3)horizonmetric,respectively.
Finally, let us remark that all configurations reported in this work have f
,
m,
n strictly positive functionsfor r>
rH (or r≥
0 forsolitons).Assuch,t isaglobaltimecoordinateandthemetric isfreeofcausalpathologies[23]
.WehavealsomonitoredtheRicci andtheKretschmannscalarsofthesolutionsanddidnotfindany indicationforasingularbehavior.3. Thesolutions
3.1. Solitons
Thenumericalresults indicatetheexistence ofafamilyof ev-erywhere regular solutions with finite mass, charge and angular momentum.Suchconfigurationscanbeviewedasdeformationsof the(globally)AdSbackground,correspondingtocharged, spinning EMCS solitons. They possess no horizon, while the size of both partsofthe S3-sectorofthemetricshrinkstozeroasr
→
0.The profileofatypicalsolutionisexhibitedinFig. 2
(left).Thesolitonshaveratherspecialproperties.The onlyinput pa-rameter here is the constant cm which fixes the magnitude at infinityofthemagneticpotential.Forany
λ
=
0,theelectriccharge andangularmomentumaregivenby9J
= −
λ
π
3√
3c 3 m,
Q=
3Q(R)=
2λ
π
√
3c 2 m,
(26)suchthatthefollowinguniversalrelationissatisfied
J
=
mQ(R),
(27)with
mcomputedfrom
(12)
.The M
(
cm)
dependencecan be found only numerically, being displayedinFig. 3
.Agoodfituptoarelativelyhighvaluecm∼
4 reads9 Therelations(26)arefoundbyevaluatingthefirstintegrals(14)forthe asymp-toticexpansions(15),(16)(notethatthesolitonshavec1=c2=0).
Fig. 2. The profiles of a typical soliton with cm=1, L=1 (left) and a typical black hole with rH=2, J=0, Q=10, cm=10 and L=15 (right).
Fig. 3. Themass, angularmomentumandelectric chargeofgravitatingspinning solitonsareshownasafunctionofthemagneticparametercm.
M L2
=
3π
32+
a2 c2m L2+
a4 c4m L4+
a6 c6m L6,
(28)witha2
= −
1.
52,a4=
0.
175,a6= −
0.
0025,andvarianceof resid-uals of 3×
10−4. Note that no upper bound on|
cm
|
seems to exist;however,thenumerics becomes difficultforlargevalues of it.Also, similartotheprobelimit,wecouldnotfindexcited solu-tions(whichwouldpossessamagneticpotentialwithnodes[14]
); thisholds alsoin theBH case. However, we conjecture the exis-tenceofsuchsolutionsforlargeenoughvaluesoftheCS coupling constantλ
.3.2. Blackholes
As expected, thesesolutions possess BH generalizations. They canbe constructedstarting withany CLP solutionandslowly in-creasingthevalueoftheparametercm.TheprofileofatypicalBH isshownin
Fig. 2
(right).Finding the domain of existence of these BHs together with theirgeneralpropertiesisa considerabletaskwhichisnotaimed atin thispaper. Instead, we analyze severalparticular classes of solutions,lookingforspecialproperties.
In Fig. 4 we display the results for configurations with fixed valuesofboth Q(R) and J andseveralvaluesofcm.Thefirst
fea-turewe noticeisthat cm
=
0 leadsto some differencesforsmall valuesofTH only,whilethesolutionswithlargetemperaturesare essentiallyCLP BHs.Also, asexpected, thequalitative behavior of solutionswithsmall|
cm|
resemblesthatoftheunmagnetizedcase. However,thischangesforlargeenoughvaluesofcm andonefinds e.g. amonotonicbehaviorofmassandhorizonareaasafunction of temperature. In particular, this means that for large valuesof|
cm|
,theBHs becomethermodynamicallystableforthe fullrange ofTH,withtheexistenceofonebranchofsolutionsonly.Also,the signofcm isrelevantforsmallvaluesof|
cm|
,only.More unusualfeatures occur aswell. Forexample, incontrast with the CLPcase, one finds BHs which have J
=
0 but still ro-tateinthebulk.10SomeresultsinthiscaseareshowninFig. 5
forsolutionswithafixedvalueoftheelectriccharge Q(R)
= −
0.
044.These 3D plots exhibit the temperatureas a function of cm and horizonarea,massandhorizonangularvelocity,respectively. The
(
TH;
cm,
R(
rH))
-diagram is also included there (with R(
rH)
the Ricciscalarevaluatedatthehorizon),toshowthattheseBHs pos-sessaregularhorizon.114. Furtherremarks
The main purpose of this paper was to report a generaliza-tion of theknown Cvetiˇc, Lü and Pope(CLP) BH solutions [3]of the D
=
5 minimal gaugedsupergravity, whichcontainsan extra-parameterinadditiontomass,electricchargeandangular momen-tum. This extra-parameter can be identified with the magnitude ofthemagneticpotentialatinfinity.12Assuch,thesolutionshere can be viewedasthesimplestAdS5 generalizationsofthe D=
4 Einstein–MaxwellsolitonsandBHsrecentlyreportedinthe litera-ture[6–9]
.Thusonecanpredicttheexistenceofavarietyofother D=
5 solutions with the U(1) potentials satisfyingnon-standard farfieldboundaryconditions.Themostinterestingnewfeatureascomparedto theCLPcase isperhapstheexistenceofaoneparameterfamilyofglobally reg-ular,smoothsolitonicconfigurations.Differentfromthepreviously known EMCS solitons with cm
=
0 which are supported by the nontrivial topology of spacetime [23], the solutions here can be considered as deformations of the globally AdS background and requirea non-vanishing magneticfield onthe boundary.We also remarkthatboththeBHsandthesolitonscanbeupliftedtotype IIBortoeleven-dimensionalsupergravitybyusingthestandard re-sultsintheliterature(seee.g.[24–27]).The study of these magnetized solutions in an AdS/CFT con-text isaninteresting openquestion. Forexample,thebackground metric uponwhichthe dualfield theoryresidesis a D
=
4 static10 Thisfeaturehasbeennoticedin[14]forc
m=0 BHsinEMCStheorywithλ>1.
11 Theonlyexceptionistheextremalconfigurationwith A
H=0,forwhichthe
Ricciscalardivergesatthehorizonataparticularvalueofcm.Notethatthis
con-figurationmarkstheseparationoftwodifferentbranchesofextremalBHsandhas =0.
12 Solutionsoftheminimalgaugedsupergravitywithanon-vanishingmagnetic fieldontheboundaryhavebeenconsideredin[28].However,thosesolutions pos-sessaRicciflathorizon,beingasymptotictoPoincaréAdS5,andhaveverydifferent propertiesascomparedtotheBHsinthiswork.
Fig. 4. AreaAH,massM,angularvelocityH andhorizondeformationε,asafunctionofthehorizontemperatureTHandforseveralvaluesofthemagneticfluxonthe
boundary.ThesesolutionshaveAdSlengthL=1,angularmomentum J=0.003,andtheR-chargeQ(R)=0.044.
Fig. 5. TheareaAH,massM,angularvelocityHandRicciscalarR(rH)ofmagnetizedblackholesareshownasafunctionof(TH,cm).ThesesolutionshaveL=1,afixed
electricchargeQ(R)= −0
Einstein universe with a line element ds2
=
γ
abdxadxb
= −
dt2+
14L 2
(
σ
21
+
σ
22+
σ
32)
. However, differentfrom the solution in [3], inthiscasethetheoryisformulated inabackgroundU(
1)
gauge field, withF(0)=
12cmσ
2∧
σ
1.The expectationvalue of thestress tensorofthedual theorycan becomputedby usingtheAdS/CFT “dictionary”,with√
−
γ γ
ab<
τ
bc
>
=
limr→∞√
−
hhabTbc. Thenonvanishingcomponentsof<
τ
ab>
are<
τ
θθ>
=<
τ
φφ>
=
1 8π
L 1 8−
5(
α
ˆ
− ˆβ)
2L4−
32cm2 15L2,
<
τ
ψψ>
=
1 8π
L 1 8−
7α
ˆ
−
11ˆβ
2L4+
2cm2 5L2,
<
τ
φψ>
=
cosθ
<
τ
ψψ>
− <
τ
φφ>
,
<
τ
ψt>
=
1 cosθ
<
τ
t φ>
= −
1 4L 2<
τ
ψ t>
=
ˆ
J 8π
L3,
<
τ
tt>
=
1 8π
L−
3 8+
3α
ˆ
+ ˆβ
2L4−
2c2 m 15L2.
Thetraceofthistensorisnonzero,with
<
τ
aa>
= −
c 2 m 2π
L3= −
L 64π
F 2 (0),
(29)resulting from the coupling of the dual theory to a background gaugefield
[29]
.Aninterestingquestionhereconcernsthepossible existence, within theproposed framework, ofconfigurations pos-sessingaKillingspinor.However,theresultsin[30]
showthatthis isnotthecase:asupersymmetricsolutionwithanonzero bound-ary magnetic field is not compatible with the far field asymp-totics(15),requiringasquashed S3 sphereatinfinity.Asavenuesforfutureresearch,we remarkthattheframework andthepreliminaryresults proposedin thiswork mayprovidea fertile ground for the further studyof charged rotating configu-rations in the D
=
5 gauged supergravity model.For example,it would be interesting to studyin a systematic way their domain ofexistence, together withthe extremallimit. Moreover, one ex-pectssomeofthesolutions’propertiestobegenericwhenadding scalarsortakingunequalspins.Wehopetoreturnelsewherewith adiscussionofsomeoftheseaspects.Acknowledgements
WegratefullyacknowledgesupportbytheDFGResearch Train-ing Group 1620 “Models of Gravity” and by the Spanish Min-isterio de Ciencia e Innovación, research project FIS2011-28013. E.R. acknowledgesfundingfromtheFCT-IFprogramme.Thiswork wasalsopartiallysupportedbytheH2020-MSCA-RISE-2015Grant No. StronGrHEP-690904, and by the CIDMA project UID/MAT/ 04106/2013.J.L.B.S.andJ.K.gratefullyacknowledgesupportbythe grantFP7,MarieCurieActions,People,InternationalResearchStaff Exchange Scheme (IRSES-606096). F.N.-L. acknowledges funding fromComplutenseUniversityunderProjectNo. PR26/16-20312.
References
[1]J.M.Maldacena,Int.J.Theor.Phys.38(1999)1113,Adv.Theor.Math.Phys.2 (1998)231,arXiv:hep-th/9711200.
[2]E.Witten,Adv.Theor.Math.Phys.2(1998)253,arXiv:hep-th/9802150. [3]M.Cvetic,H.Lü,C.N.Pope,Phys.Lett.B598(2004)273,arXiv:hep-th/0406196. [4]Z.W.Chong, M.Cvetic,H. Lü,C.N.Pope,Phys. Rev.Lett.95(2005) 161301,
arXiv:hep-th/0506029.
[5]J.B. Gutowski,H.S.Reall,J. High Energy Phys. 0402(2004) 006, arXiv:hep-th/0401042.
[6]C.Herdeiro,E.Radu,Phys.Lett.B749(2015)393,arXiv:1507.04370[gr-qc]. [7]M.S.Costa,L.Greenspan,M.Oliveira,J.Penedones,J.E.Santos,Class.Quantum
Gravity33(2016)115011,arXiv:1511.08505[hep-th].
[8]C.Herdeiro,E.Radu,Phys.Lett.B757(2016)268,arXiv:1602.06990[gr-qc]. [9]C.A.R.Herdeiro,E.Radu,Phys.Rev.Lett.117(2016)221102,arXiv:1606.02302
[gr-qc].
[10]J.L.Blázquez-Salcedo,J.Kunz,F.Navarro-Lérida,E.Radu,Entropy18(2016)438, arXiv:1612.03747[gr-qc].
[11] P. Chrusciel, E. Delay, Lett. Math. Phys. (2017), http://dx.doi.org/10.1007/ s11005-017-0955-x,arXiv:1612.00281[math.DG].
[12]P.T.Chru´sciel,E.Delay,P.Klinger,arXiv:1701.03718[gr-qc].
[13]J.L.Blázquez-Salcedo,J.Kunz,F.Navarro-Lérida,E.Radu,Phys.Rev.D92(2015) 044025,arXiv:1506.07802[gr-qc].
[14] J.L.Blázquez-Salcedo,J.Kunz,F.Navarro-Lérida,E.Radu,Phys.Rev.D95(2016) 064018, http://dx.doi.org/10.1103/PhysRevD.95.064018, arXiv:1610.05282 [gr-qc].
[15]U.Ascher,J.Christiansen,R.D.Russell,Math.Comput.33(1979)659; U.Ascher,J.Christiansen,R.D.Russell,ACMTrans.Math.Softw.7(1981)209. [16]P.G.Nedkova,S.S.Yazadjiev,Phys.Rev.D85(2012)064021,arXiv:1112.3326
[hep-th];
P.G.Nedkova, S.S.Yazadjiev,Eur.Phys. J.C73(2013)2377,arXiv:1211.5249 [hep-th].
[17]D.Kastor,J.H.Traschen,Phys.Rev.D46(1992)5399,arXiv:hep-th/9207070. [18]M.S.Volkov,D.V.Gal’tsov,Phys.Rep.319(1999)1,arXiv:hep-th/9810070. [19]V.Balasubramanian,P.Kraus,Commun.Math.Phys.208(1999)413,
arXiv:hep-th/9902121.
[20]D.N.Page,Phys.Rev.D28(1983)2976.
[21]D.Marolf,Chern–Simonstermsandthethreenotionsofcharge,arXiv:hep-th/ 0006117.
[22]O.Aharony,S.S.Gubser,J.M.Maldacena,H.Ooguri,Y.Oz,Phys.Rep.323(2000) 183,arXiv:hep-th/9905111.
[23]M.Cvetic,G.W.Gibbons,H.Lü,C.N.Pope,Rotatingblackholesingauged su-pergravities:thermodynamics,supersymmetriclimits,topologicalsolitonsand timemachines,arXiv:hep-th/0504080.
[24]A.Chamblin,R.Emparan,C.V.Johnson,R.C. Myers,Phys. Rev.D60 (1999) 064018,arXiv:hep-th/9902170.
[25]J.P.Gauntlett,D.Martelli,J.Sparks, D.Waldram,Class. QuantumGravity21 (2004)4335,arXiv:hep-th/0402153.
[26]J.P.Gauntlett,E.O.Colgain,O.Varela,J.HighEnergyPhys.0702(2007)049, arXiv:hep-th/0611219.
[27]J.P.Gauntlett,O.Varela,Phys.Rev.D76(2007)126007,arXiv:0707.2315 [hep-th].
[28]E.D’Hoker, P.Kraus,J. HighEnergyPhys.1003(2010)095,arXiv:0911.4518 [hep-th];
E.D’Hoker, P.Kraus,J.HighEnergyPhys. 0910(2009)088,arXiv:0908.3875 [hep-th].
[29]M.Taylor,Moreoncounterterms inthegravitational actionand anomalies, arXiv:hep-th/0002125.
[30]D.Cassani,D.Martelli,J.HighEnergyPhys.1408(2014)044,arXiv:1402.2278 [hep-th].
