ads in stability lessons from the scalar field

Cvetiˇc Keywords: Resonant instability AdS WearguedinarXiv:1408.0624thatthequarticscalarfieldinAdShasfeaturesthatcouldbeinstructive foransweringthegravitationalstabilityquestionofAdS.Inde

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aInternational Center for Theoretical Sciences, Indian Institute of Science Campus, Bangalore – 560012, India

bCenter for High Energy Physics, Indian Institute of Science, Bangalore – 560012, India

Article history:

Received 2 March 2015

Received in revised form 21 April 2015

Accepted 6 May 2015

Available online 8 May 2015

Editor: M Cvetiˇc

Keywords:

Resonant instability

AdS

WearguedinarXiv:1408.0624thatthequarticscalarfieldinAdShasfeaturesthatcouldbeinstructive foransweringthegravitationalstabilityquestionofAdS.Indeed,theconservedchargesidentifiedthere haverecentlybeenobservedinthefullgravitytheoryaswell.Inthispaper,wecontinueourinvestigation

of the scalar field in AdS and provide evidence that in the Two-Time Formalism (TTF), even for initial conditionsthat are far fromquasi-periodicity, the energyin the highermodes atlate timesis exponentiallysuppressedinthemodenumber Basedonthisandsomerelatedobservations,weargue thatthereisnothermalizationinthescalarTTF modelwithintime-scalesthatgoas∼12,where

measurestheinitialamplitude(withonlylow-lyingmodesexcited).Itistemptingtospeculatethatthe resultholdsalsoforAdScollapse

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense

(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3

1 Overview

ThequestionofwhetherAdSspaceisstable[1,2]against

turbu-lentthermalizationandtheformationofblackholesundergeneric

(non-linear) perturbations has received much attention recently

AdSspacewithconventionalboundaryconditionsislikeabox,and

thereforeperturbations that were weak tobeginwith canreflect

multiple times from the boundary, potentially resulting in

suffi-cient localizationofenergy tocreate blackholes Aside fromthe

factthat blackholeformationisaquestion offundamental

inter-estin(quantum)gravity,thisproblemacquiresanotherinteresting

facet via the AdS/CFT correspondence: it captures the physics of

thermalizationinstronglycoupledquantumfieldtheories

Atthemomenthowever,itisfairtosay thattheevidencefor

andagainsttheinstabilityofAdSwhenexcitedbylow-lying,

low-amplitudemodesismixed[3–11].Inaneffortto(partially)clarify

thissituation,inthispaperwe will makesome commentsabout

twolooselyinter-relatedquestions:

•Does“most”initialdataleadtothermalization?

•Can one argue that within a time-scale of order O(1/ 2),

where  captures the amplitude of the initial perturbation,

thermalization does (not?) happen? This is an interesting

* Corresponding author.

E-mail addresses:pallabbasu@gmail.com (P Basu), chethan.krishnan@gmail.com

(C Krishnan), pnbalasubramanian@gmail.com (P.N Bala Subramanian).

question because the statement of[1] isthat blackhole for-mationhappensinAdSwithinthistimescale

We will ask thesequestions, which are inspired by gravitational (in)stability in AdS, in the context of a simpler problem: a self-interactingφ4scalarfieldinAdS.Theworksof[6–10]suggestthat thesesystemshaveclosesimilarities,sowebelievethatthiseffort willbeinstructiveandworthwhile

One ofour main toolswill be the Two-Time Formalism (TTF) developedin[4](wewilldescribethisapproachinSection2 We willarguewhythisapproachhasvariousadvantages,andwhywe believe it capturesthe essential physics ofresonances in thefull (i.e.,non-TTF)model.Butweemphasizethatthiswillshedlighton theinstability question onlyiftheinstability, ifit exits,iscaused

by resonances (which seems plausibleto us) If theinstability is causedbysomeother(possiblylongertime-scale)dynamics,TTFin theleadingordercanmissthatphysics.Butweexpectthatphysics

intheO(1/ 2)time-scaleshouldbecapturedbyTTF

Furthermore,forconcreteness,wewilltakethefollowingasthe definition of the absence of thermalization: the presence of expo-nentiallydistributed energies in the higher modes, asa function

ofthemodenumber.1 Thatis,ifthesystemhas A j∼e−j β atlate

1 Note that the definition of thermalization is somewhat ambiguous We are adopting this as asufficient but not necessary conditionfor the absence of thermal-ization as we will make more precise at the beginning of Section 3 One source of ambiguity is that our system is classical and suffers from a UV catastrophe: so once http://dx.doi.org/10.1016/j.physletb.2015.05.009

0370-2693/©2015 The Authors Published by Elsevier B.V This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ) Funded by 3

timesforsome positiveβ wewillsaythatitdoesnotthermalize

(atleastforaverylongtime).Loosely,onecouldalsoadopta

defi-nitionthatthesystemhasinstabilitytowardsthermalizationifthe

late-timebehaviorofthe j’th modegoesasA j∼j−α , where α is

apositivequantity–itispossiblehoweverthat thisisnota

nec-essarynor a sufficientcondition [8], andwe will not usethisin

ourpaper

Withinthecontextofthesethreelimitations(namely,working

withthescalarfield andintheTTFapproachandwithina

partic-ular definitionof thermalization) ourresults implythe following

answersforthetwoquestions(combinedintoone):

• Initialdatawithonlylowlyingmodesdo not lead to

thermal-izationforthequarticscalarfieldintheTTFformalismwithin

a time-scale of O(1/ 2) This suggeststhat ifat all there is

thermalization in the full theory, it should be coming from

non-resonanttransferofenergy

Together, we believe that these observations present fairly

strong evidence that thermalization (as defined above) does not

happenforinitialvaluedatawhichhaveonlythelow-lyingmodes

excited.Ourresults,asalreadyemphasized, arefortheφ4-scalar:

butwe believe similar statements apply forAdS gravity as well

Wemakevarious furthercommentsofvaryingdegreesof

techni-cality inlatersections

For completeness, lets also state that our results are still not

quiteconclusive.Apartfromthepointsemphasizedabove,thereis

alsotheperversepossibilitythatcollapsehappens, butnotdueto

resonances–butnotehoweverthatthetime-scaleforthiswillbe

biggerthan∼1/ 2

2 TTF formalism

Theactionforthescalarfieldtheoryisgivenby



−g



1

2∇μ φ ∇μφ +V(φ)



(2.1)

wherethepotentialisgivenby

Themetricforthespaceisgivenby



−dt2+dx2+sin2x d2



(2.3)

Theequationsofmotionforthescalarfieldaregivenby

φ(2,0)+ sφ ≡ φ(2,0)− φ(0,2)− 2

sin x cos xφ

(0,1)

6 cos2xφ

wherethesrepresentsthespatialLaplacianoperator This

oper-atorhasaneigenfunctionbasisgivenby

e j(x) =4



(j+1)(j+2)

3x2F1



−j,j+3;3

2;sin2x



(2.6)

ω2j= (2 j +3)2 j=0,1,2, (2.7)

the system has fully thermalized, the average energy per state would be zero, if we

don’t truncate it In particular, the distribution of energies shouldnot becompared

to a canonical ensemble distribution, rather it should be thought of as capturing the

efficiency of energy transfer to higher modes.

Theinnerproductinthisbasisisdefinedas

(f,g) =



In the Two-Time Framework (TTF), we havethe slow-movingtime definedas τ = 2t, whichrequiresthetimederivativestobe rede-fined as∂t→ ∂t+ 2∂τ Thescalarfieldiswrittenasanexpansion

inthesmall-parameter as

φ =  φ(1)(t, τ ,x) + 3φ(3)(t, τ ,x) + O ( 5) (2.9)

Notethattheratiobetweentheslowandfasttimes(τ and t) also controlstheoverallscaleoftheamplitude.Puttingthisexpansion

inthescalarfieldequation ofmotionEq.(2.4)weget

order : ∂2

tφ(1)(t, τ ,x) − ∂2

xφ(1)(t, τ ,x)

sin x cos x∂xφ(1)(t, τ ,x) =0 (2.10) order3: ∂2

tφ(3)(t, τ ,x) +2∂t∂τφ(1)(t, τ ,x) − ∂2

xφ(3)(t, τ ,x)

sin x cos x∂xφ(3)(t, τ ,x)

6 cos2xφ

3

Theorder equationhasthegeneralrealsolution

φ(1)(t, τ ,x) =

∞



j=0



A j( τ )e−i ω j t+A j( τ )e i ω j t

e j(x) (2.12)

Notethattheintroductionoftheslowtimesgivesanextravariable that we cantune –we will usethisatorder 3 to cancelofthe resonant terms.Theequationsthataccomplish thisarecalledthe TTF equations.Substituting the above first order results into the order 3 equationsweget

∂t2φ(3)(t, τ ,x) −2i

∞



k=0

ωk

∂τ A j( τ )e−i ω j t− ∂τA j( τ )e i ω j t

e j(x)

+ sφ(3)(t, τ ,x)

6 cos2x

∞



j , k , l=0



A j( τ )e−i ω j t+A j( τ )e i ω j t

× A k( τ )e−i ω k t+A k( τ )e i ω k t

× A l( τ )e−i ω l t+A l( τ )e i ω l t

e j(x)e k(x)e l(x) (2.13)

Projectingonthebasissolutionsgive



e j(x), [∂2

t + ω2j]φ(3)(t, τ ,x)



−2iωj

∂τ A j( τ )e−i ω j t− ∂τA j( τ )e i ω j t

= − λ 6

∞



k , l , m=0

A k( τ )e−i ω k t+A k( τ )e i ω k t

× A l( τ )e−i ω l t+A l( τ )e i ω l t

× A m( τ )e−i ω m t+A m( τ )e i ω m t (2.14)

π /2



0

dx tan2(x)sec2(x)e j(x)e k(x)e l(x)e m(x) (2.15)

By direct computation (using properties of Jacobi polynomials –

which are an alternate way to describe the basis functions, see

Appendix A), onecanshowthatthenecessaryandsufficient

con-ditionforresonancesis

The absence of other combinations for the resonances for the

scalar theory was recognized and used in [6] (see footnote 3 of

[7]fora simple proof) Theyare also absent in the gravity case,

butthecomputationrequiredtoshowthisinthatcaseis

substan-tiallymorelengthy[5].Thecloseparallelbetweenthestructureof

the resonances in the two cases is evidently one of the reasons

whytheyexhibitsimilaritiesintheirthermalizationdynamics[6]

Inanyevent, atthisstage wehavethefreedom tochoosethe

A j( τ ) as mentioned above so that the resonances on both sides

arecanceled.Thisisaccomplishedbysolvingthe A jaccordingto

−2iωj∂τ A j= − λ

6

∞



k , l , m=0

anditscomplexconjugate.Bydoingarescalingofthemodesas

A i→ √ ωi A i,and C i jkl→ C jklm

√ ω

jωkωlωm

weget

−2i∂τ A j= − λ

6

∞



k , l , m=0

These are the TTF equations that we will use extensively in the

next section Once the resonances are canceled, the coupling to

thehighermodesisexpectedtobeweakandwebelieveitis

un-likelythatthere willbeefficient channelsforthermalization:but

thisisa prejudice,andpossibly farfromproof In anyevent, we

can systematically solve for φ(3)(t τ ,x) at this stage if we wish,

withoutbeingbotheredbyresonances

Note that the simplicity of the quartic scalar arises from the

fact that the C i jkl have a (relatively) simple expression We will

commentmoreaboutthisinAppendix A

Beforeconcludingthissectionwequotesomepertinentresults

from[6]forourscalarTTFsystem.Firstly,wecangettheTTF

equa-tionsusinganeffectiveLagrangian

i

(A i A˙¯i− ¯A i A˙i)

+ C i jkl A¯i( τ ) ¯A j( τ )A k( τ )A l( τ ), (2.19)

wheresummationintheinteractiontermisover i , j ,k,l such that

ωi+ ωj− ωk− ωl =0 In writing the expression in this form,

we havedone an appropriate scaling ofeach mode by ωk andλ

foreasy comparisonwiththenotation of[6]: A k are therescaled

modes.Thesystemhasadilatationsymmetry: A k( τ ) → A k(12τ )

SoifthermalizationhappensintheTTF theoryitshouldscale

in-verselyas the square ofthe amplitude: theassumption that TTF

theory captures the relevant physics is the assumption that the

system has such ascalingregime

However, the system hasthe following conserved charges[6]

arisingfromacorrespondingsetofsymmetries:

Fig 1 Thelog-plot of j α j vs. j forthe quasi-periodic solutions The linear fit is indicative of exponential suppression ofA jwith j.

Q0= A k A¯k,symmetry: A k→e i θ A k, (2.20)

Q1= k A k A¯k,symmetry: A k→e ik θ A k, (2.21)

ω i+ω j−ω k−ω l=0

C i jkl A¯i( τ ) ¯A j( τ )A k( τ )A l( τ ),

Various pieces of evidence indicating that the evolution of the quarticscalarinAdShassomecloseconnectionstocollapseinAdS gravity were presentedin[6].The above conservedchargeswere identifiedforthefullgravitysystemin[7](seealso[8])

3 Results

Inthissection,we willstudyvariousaspects oftheTTF equa-tions for the quartic scalar in some detail As mentioned in the introduction,we willtake theexponential decayof A j( τ )with j

asanindication thatthermalizationissuppressed.In[8]some ar-guments were madethat a powerlaw A j∼j−a for positive a is

indicative of thermalization/black hole formation We will make this somewhat more precise as follows The basic object that is takenasanindicatorofcollapsein[1,4]isthequantity t 0) |2, the unbounded growthof whoseprofile is takenasthe onset of collapse Theanalogueofthisquantity inourscalarTTF casecan

be taken as | ˙φ(1)(t 0) |2 (compare Fig 1(A) and the accompany-ing discussion in [6]to Fig 3 in [4]) At this point, using(2.12),

(2.6)and(2.7)wecanseethatthisquantitycanbeestimatedand boundedvia

| ˙φ(1)(t,0) |2∼ | j2A j|2 j2|A j|2. (3.1)

Now,itis evidentthat when A j∼e−j α the last quantity isfinite and therefore the LHS can never diverge, which is what we set outtoshow.Thisindicatesthatexponential suppressionofhigher modesisasufficientconditionfor absence of thermalization.Note howeverthatwe aresilent aboutwhatconstitutesthermalization

atthelevel ofmodes– fortunately,we willnever needaprecise definitionofthatforthepurposesofthispaper

The TTFtheory hasquasi-periodic solutions(see [4]fora dis-cussionofanalogoussolutionsinthegravitysystem)oftheform

A j( τ ) = αjexp( −iβjτ ),whereβj= β0+j(β1− β0). (3.2)

Onecanchoose α0, α1(orβ0, β1)anddeterminetherestofthe αj

viatheTTFequations(2.18),2 ifonetruncatesthesystematsome

2 For some initial conditions we see more than one quasi-periodic solution.

Fig 2 Evolution plots of perturbations around quasi-periodic solutions.

Fig 3 Plot of C i jkltogether with an inverse linear fit.

j = j max anddemandstability ofthe solutionagainstvariationin

j max.Wehavedonethis, andtheresultingmodesdecaywiththe

modenumber j as ∼exp(−c j )

j forsome positive c, seeFig 1 This

isobviouslyconsistentwithourdefinitionof(non-)thermalization

In[4]the j maxwastakentobe∼50,inourcaseweareabletogo

upto j max=150

Ifweperturba quasi-periodic solutionwe expecttoget

oscil-lationsofthe A j’saround αj.SeeFig.2aforsolutionswherethe

initialvalue ofthe A j areclose3 to theirquasi-periodic values If

on the other hand, theinitial A j values are sufficiently far from

theirquasi-periodic values,weexpectthatthesolutionstransition

tochaos.ThisexpectationisqualitativelyverifiedinFig.2 where

welaunchthe A jfarawayfromquasi-periodicity.Inwhatfollows

wewillshow that even in thesefar-fromquasi-periodic solutions,

themaximumvalueattainedby the A j asweevolve thesolution

isexponentiallysuppressedin j Thisisan indicationthat energy

3 In order to make these statements precise, we will need a notion of closeness

between solutions in terms of modes A convenient way to define a dimensionless

measure of the “distance” between two solutions (say 1 and 2) is to consider

(1)

j A (2)∗

j

k|A ( k1)|2

l|A ( l2)|2

(3.3)

 ∼1 is close The summation is only up to mode numberj .

transfertothehighermodesissuppressedeveninthesesolutions – ifthis behavior holdsalso in gravity, it could be an indication thatthesesolutionsgenericallydonotcollapse

One of the ways in which one might try to understand the efficiency of energy transfer to higher modes is by studying the coefficients4 C i jkl whichsignifythe couplingbetweenthemodes

Tounderstand thebehavior ofTTFequationsatlarge j, welookat various kinds of limitswe may consider for C i jkl as the i, j ,k,l

are sent to ∞ One is a simple scaling of indices, i,j,k,l→

ai,aj,ak,al By fitting the plot (see Fig 3b), we see that in this case C i jkl goes as O(1

a) as a → ∞ Another case is where we keep two modesfixed andtake another two to infinity: i ∼ j ∼ approximately fixed, but withk ∼l ∼a and we take a → ∞ We find that they also havea O(1

a) fall off It is important to note that becauseoftheresonancecondition, thesearetheonly possi-blecouplingsavailableforahighmode–onecannot(forexample) holdthreeindicessmallwhilesendingtheforthonetoinfinity.So progressivelyhighermodesareweaklycoupled,bothtoeachother

aswellastothelow-lyingmodes

Finally, we consider the evolution of the modes when we launch the system both near and far from quasi-periodic initial conditions The way we do thisis by calculating the coefficients

4 Note that the coefficients C i jkl can be determined via (2.15) analytically, but using Mathematica Some comments on this are given in Appendix A

Fig 4 Plotof log[Max[A n ( τ )]]for j∈ [0,150]and a linear fit The fit has been done

in the region j∈ [40,150] The last figure corresponds to quasi-periodic initial data.

C i jkl analytically(seeAppendix Aforsomecommentsonthis)and

then integrating the resulting TTF equations numerically for the

variousinitial conditions.In allcases we plotmaximumvalue of

A j thatisattainedduringtheentireperiodofevolutionagainst j,

andwefindthatthisMax[A j( τ ) ] exponentiallydecayswith j for

all initial data Weseean exponentialdecaywithrespectto j, not

just for solutions close to quasi-periodic solutions, but also for

thosethatarefarfromit:seeFig.4.Thisistrueeventhoughfor

someinitialconditions(whereinitial valuesofmodeenergies are

ofthesame order)we see an approximatelypowerlawdecayof

modesup tosome intermediate frequency.These statementscan

beverified usingthenorm (3.3) withthe understandingthat the

summationover j has toberestrictedtobeabovesome

appropri-atelychosen j min(andofcoursebelow j max)whenwearetalking abouthighmodes

Acknowledgements

We thankOlegEvninfordiscussions, andPiotr Bizonand An-drzejRostworowskiforcorrespondence

Appendix A Comments onC i jkland Jacobi polynomials

Thedeterminationofthe Ci jkl isinprinciplestraightforwardby directevaluationof(2.15).Thisisananalyticallytractableproblem becausethebasisfunctions ej(x)canbewrittenintermsofJacobi polynomials as

e j(x) =4



(j+1)(j+2)

π

× (j+1)(3 2)

(j+3 2) cos

3x P ( j1/2,3/2)(cos 2x). (A.1)

Jacobi polynomials are (orthogonal) polynomials in their argu-mentsandthereforeinourcasetheymerelyinvolveonly(afinite number of)powers ofsinusoids.5 Therefore the integral for C i jkl, whichisintherange[0, π /2]can,againinprinciple,be straight-forwardlyevaluated.It turnsoutthat theresultcanbeexpressed

in terms of finite sums of finite products of Gamma functions and such, but simplifying them on Mathematica becomes time-consuming Onecouldinprinciple trytosimplifytheexpressions manually, butwe have adopted a more pragmatic approach: we evaluatetheintegralsanalyticallyonMathematicabyre-expressing thepowersofsinusoidsintermsofproductformulas.Sincethe in-tegralsareover[0, π /2]thismakesthemsubstantiallyless inten-sive asfarastimerequirements areconsidered Thiswayweare abletoalgorithmizethe(analytic)computationof Ci jkl on Mathe-matica, afterwhich weuse themin theTTF equations todo our numericalevolutions

References

[1] P Bizo ´n, A Rostworowski, On weakly turbulent instability of anti-de Sitter space, Phys Rev Lett 107 (2011) 031102, arXiv:1104.3702.

[2] P Bizon, Is AdS stable?, Gen Relativ Gravit 46 (2014) 1724, arXiv:1312.5544 [3] H.P de Oliveira, L.A Pando Zayas, C.A Terrero-Escalante, Turbulence and chaos

in anti-de Sitter gravity, Int J Mod Phys D 21 (2012) 1242013, arXiv: 1205.3232 [hep-th].

[4] V Balasubramanian, A Buchel, S.R Green, L Lehner, S.L Liebling, Holographic thermalization, stability of AdS, and the Fermi–Pasta–Ulam–Tsingou paradox, Phys Rev Lett 113 (2014) 071601, arXiv:1403.6471.

[5] B Craps, O Evnin, J Vanhoof, Renormalization group, secular term resumma-tion and AdS (in)stability, J High Energy Phys 10 (2014) 48, arXiv:1104.3702 [6] P Basu, C Krishnan, A Saurabh, A stochasticity threshold in holography and the instability of AdS, arXiv:1408.0624.

[7] B Craps, O Evnin, J Vanhoof, Renormalization, averaging, conservation laws and AdS (in)stability, arXiv:1412.3249.

[8] A Buchel, S.R Green, L Lehner, S.L Liebling, Conserved quantities and dual turbulent cascades in anti-de Sitter spacetime, arXiv:1412.4761.

[9] I.-S Yang, The missing top of AdS resonance structure, arXiv:1501.00998 [10] F.V Dimitrakopoulos, B Freivogel, M Lippert, I.-S Yang, Instability corners in AdS space, arXiv:1410.1880.

[11] P Bizo ´n, A Rostworowski, Comment on “Holographic Thermalization, stability

of AdS, and the Fermi–Pasta–Ulam–Tsingou paradox” by V Balasubramanian et al., arXiv:1410.2631.

5 Explicit expressions for Jacobi polynomials can be found in numerous places ranging from Wikipedia to Abramowitz and Stegun.

SoifthermalizationhappensintheTTF theoryitshouldscale

in- verselyas the square ofthe amplitude: theassumption that TTF

theory captures the relevant physics is the assumption that the. .. polynomials in their argu-mentsandthereforeinourcasetheymerelyinvolveonly(afinite number of)powers ofsinusoids.5 Therefore the integral for C i jkl, whichisintherange[0,