Noether–Wald energy in Critical Gravity

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Physics

Letters

B

www.elsevier.com/locate/physletb

Noether–Wald

energy

in

Critical

Gravity

Giorgos Anastasiou

∗

,

Rodrigo Olea,

David Rivera-Betancour

DepartamentodeCienciasFísicas,UniversidadAndresBello,Sazié2212,Piso7,Santiago,Chile

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Articlehistory:

Received20September2018 Accepted7November2018 Availableonline15November2018 Editor:M.Cvetiˇc

Criticalityrepresentsaspecificpointintheparameterspaceofahigher-derivativegravitytheory,where thelinearizedfieldequationsbecomedegenerate.In4DCriticalGravity,theLagrangiancontainsa Weyl-squaredterm,whichdoesnotmodifytheasymptoticformofthecurvature.TheWeyl2_{couplingischosen}

such thatiteliminatesthe massive scalarmodeand itrendersthe massive spin-2mode massless.In doingso,thetheoryturnsconsistentaroundthecriticalpoint.

Here,weemploytheNoether–WaldmethodtoderivetheconservedquantitiesfortheactionofCritical Gravity.Itismanifestfromthisenergydefinitionthat,atthecriticalpoint,themassisidenticallyzero forEinsteinspacetimes,whatisadefiningpropertyofthetheory.Astheentropyisobtainedfromthe Noether–Waldchargesatthehorizon,itisevidentthatitalsovanishesforanyEinsteinblackhole.

©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

General Relativity (GR) is a successful theory of gravity at a classical level but it lacks of consistency in a quantum regime because it is not renormalizable. On the other hand, in the low energylimitofStringTheory,whichshouldbefinitetoallorders, thereappearcontributionsthat arequadraticinthecurvature.As aconsequence,highercurvatureextensionsofEinsteingravityare expectedtogive rise toa gravity theorywitha better ultraviolet behavior.Earlyworkonthesubjecthassuggestedthatthisclassof theoriesshouldberenormalizable[1].

Lower-dimensionalexamples have beenextensively studiedin recentliterature.Theyareregardedasinsightfultoymodelswhich capture essential features of 4D gravity. One of them is New Massive Gravity (NMG) [2], a parity-even three-dimensional the-ory which describes two propagating massive spin-2 modes, in contrast to 3D Einstein gravity which is topological. Picking up the conventional sign of the Einstein–Hilbert action, the energy of the massive excitations is negative (ghost modes), while the mass of the Banados–Teitelboim–Zanelli (BTZ) black hole is pos-itive. Clearly, this inconsistency persists even if one reverses the signofthekineticterm.Aphysicallyreasonable theoryarisesata specificpointofparametricspace, wherethemassivespin-2field turnsmassless[3].Atthisparticularpoint,boththeenergyofthe gravitonandthemassoftheBTZblackholevanishidentically[4].

*

Correspondingauthor.

E-mailaddresses:georgios.anastasiou@unab.cl(G. Anastasiou), rodrigo.olea@unab.cl(R. Olea),d.riverabe@gmail.com(D. Rivera-Betancour).

Furthermore,both centralchargesturnintozero,what leadsto a vanishing entropy [4]. Another featureof the theory is the pres-enceofnewmodeswithlogarithmicbehavioratthecriticalpoint [5].Thesemodesareeliminated whenstandardBrown–Henneaux boundaryconditionsareconsidered.Relaxing theasymptotic con-ditionsto includelogtermsswitchesonnewholographicsources attheboundary[6].

Another theory in three dimensions sharing similar features withNMG is Topologically MassiveGravity (TMG) [7]. The corre-spondingcriticalpointdefinestheconceptofChiralGravity. How-ever,inthiscase,thecentralchargesaredifferentfromeachother due to a parity-violating term in the action. As a consequence, neither mass nor entropy vanish for BTZblack holes atthe chi-ralpoint.

Thegeneralizationoftheconceptofcriticality,presentinthese models, to four dimensions is given by theories which include quadratictermsinthe curvaturewithparticularcouplings ontop oftheEinstein–Hilbertaction.Themostgeneralformofa gravity actionwithquadratic-curvaturecorrectionsin4D isgivenby

I

=

1

16

π

G

M

d4x

√

−

g

R

−

2

+

α

RμνRμν

+

β

R2

,

(1)

where

α

and

β

are arbitrary couplings, and

= −

3/2 _{is the} cosmologicalconstantintermsoftheAdSradius

.

The Riemann-squared term is not present, as it can be always traded off by theGauss–Bonnet(GB)invariantplusthecurvature-squaredterms present in the action (1). The GBterm does not affect the field equationsinthebulkbutitdoesmodifytheboundarydynamics.

https://doi.org/10.1016/j.physletb.2018.11.021

Thisclassoftheoriesleadstoequationsofmotion(EOM) with up to four derivatives in the metric. Generically, they describe modesthatrepresentamasslessspin-2graviton,amassivespin-2 field anda massive scalar. For a quadratic-curvaturegravity the-orywitharbitrarycouplingconstants,perturbationsaroundagiven backgroundwouldgiverise toghosts.The problemwiththesign oftheenergyofthesemodescanbecircumventedbyasignflipof theconstantinfront ofEinsteinkinetic term. Ontheotherhand, EinsteinblackholesaresolutionstothetheorydefinedbyEq. (1). Therefore,the changeinthe signmentioned abovewouldleadto anegativemassforSchwarzschild-AdSblackhole.Needlesstosay, thispictureisclearlyunphysicalastheenergyoftheperturbations aroundabackgroundandthemassofa blackholecarry opposite signs.

Inviewofthisgeneralobstructiontoobtainafour-dimensional gravitytheorywhichisfreeoftheinconsistenciesdiscussedabove, itwasquitesurprisingwhentheauthorsofRef.[8] pointedoutthe fact that, forthe particular couplings

α

= −

3β and

β

= −

1/2, the massive scalar is eliminated and the massive spin-2 mode turnsmassless. Thischoicerendersthe theoryphysicallysensible aroundthecriticalpoint.Thisfactisconfirmedbyusingthe Ostro-gradskymethodforLagrangianswithderivatives ofhigherorder: theenergyforthemassivemodevanishesforthecriticalvalueof thecouplings.From the pointof view oftheenergyof theblack holesofthetheory,onecanusetheAbbott–Deser–Tekin(ADT) for-mula [9,10] to evaluate the mass of Schwarzschild-AdS solution, whatresultsin

M

=

m

(

1

+

2

(

α

+

4

β)) ,

(2)

wheremisthemassparameterinthesolution.

ThegeneralformulaEq. (2),makesevidentthat,forthecritical conditionmentionedabove,themassforSchwarzschild-AdSblack holevanishes.

Inthepresentwork,asanalternativetoDeser–Tekinprocedure, weemployNoether–Waldmethod[11,12] tocomputethecharges inCriticalGravity. Thisfull(non-linearized)expression derivedin this wayhas a remarkable property: the energy of any Einstein spaceisidenticallyzero,aslonganticipatedinRef. [13].

2. Deser–Tekinenergyin4Dquadratic-curvaturegravity

As mentioned in the previous section, in Refs. [9,10], the authors provide a generic definition of energy for an arbitrary curvature-squared gravitytheory. Thatdefinitionofthe energyis obtainedasanextensionoftheAbbott–Desermethod[14].

Inorder to obtain the ADT mass for a generalasymptotically AdS (AAdS) solution, we need to write down the metric of the spacetimeintheformofgμν

= ¯

gμν

+

hμν,whereg

¯

μν isthemetric

ofthe backgroundandhμν is theperturbation tensor. Such

con-structionleavesthefirst-ordervariationoffieldequationsas

δ

Gμν

+

Eμν

=

[1

+

2

(

α

+

4

β)

]GLμν

+

α

¯

−

2

3 G L

μν

−

2

3 R L_g

_¯

μν

+

+

(

α

+

2

β)

− ¯∇μ

¯∇ν

+ ¯

gμν

¯

+

¯

gμν

RL

,

(3)

whereGL_μν andRL arethelinearizedexpressionofEinsteintensor andRicciscalar,respectively.ThetensorEμν isthecontributionof

fourthorderinthederivativestothefieldequations.Theequation (3) hastobeequaltoan effectiveenergy–momentumtensor Tμν,

whichiscovariantlyconserved.Onecanwriteaconservedcurrent, forasetofKillingfields

{¯

ξ

μ

}

thatrepresentstheisometriesofthe background

Jμ_{A D T}

=

8

π

G Tμν

ξ

¯

ν

.

(4)

Inordertoevaluatethemassofagravitationalobject,theKilling vectorneedstobetimelike,atleast,atinfinity.

Wheneverthereisacurrentwhichisconserved,oneisableto writedown Jμasthedivergenceofa2-formprepotential,i.e.,

Jμ_{A D T}

= ∇ν

F

μν

.

(5)

Onecan consideraspacetime foliatedbya normal(radial) direc-tionz

ds2

=

N2

(z)dz

2

+

hi j

(z,

x)dxidxj

,

(6)

wherehi j

(

z

,

x

)

istheinducedmetricon

∂

M,anditsradial evolu-tionisdefinedbytheunitvectornν

=

N

(

z

)δ

νz.

Inthiscoordinateframe,theconservedchargecanbeexpressed asanintegralontheco-dimensiontwosurface

Qμ_{A D T}

[¯

ξ

] =

dSν

F

μν

.

(7)

Here,dSν

=

d2x

√

−

h nν isasurfacenormalvectorthatdefinesthe

integrationforafixedtime andradius.Forthecaseof curvature-squaredgravityinfourdimensions,theconservedquantityadopts theform1

8

π

G Q_{A D T}μ

[¯

ξ

] =

[1

+

2

(

α

+

4

β)

]

∂M

d3x Gμ_Lλ

ξ

¯

λ

+

(

α

+

2

β)

dSν

2

ξ

¯

[μ

¯∇

ν]RL

+

RL

¯∇

μ

ξ

¯

ν

−

α

dSν

2

ξ

¯

λ

¯∇

[μGν_L]λ

+

2Gλ_L[μ

¯∇

ν]

ξ

¯

λ

.

(8)

3. CriticalGravity

In Ref. [8], the energy of the graviton modes in quadratic-curvature gravity was studied. These excitations come from the linearizedEOM (3). The choice

α

= −

3β leads toa traceless per-turbation(h

=

0)whicheliminates themassivescalarmode. Con-sequently,theequationforthepropagatingmodetakestheform

¯

2

−

2

3

¯

2

−

2

3

−

2

β

+

1

3

β

hμν

=

0

.

(9)

The first factor of the equation describes the propagation of a masslessgravitoninanAdSbackgroundwhilethesecondone rep-resentsamassivespin-2field.

It is clear that the latter becomes massless by imposing the critical value

β

= −

1/2. This particular coupling produces the fourthorderequation

¯

2

−

2

3

2

hμν

=

0

,

(10)

which reflects the appearance of both massless and logarithmic modes[8].

In order to obtain the energy of the excitations, the authors inRef. [16] followedaHamiltonianapproach. Foranunrestricted valueof

β

,theactionuptoquadraticorderinhμνis

I

= −

1

Using theOstrogradskymethod forhigher-derivative Lagrangians, oneobtainsthefollowingconjugatemomenta

π

₍μν₁₎

=

1

Due to the fact that the Lagrangian is time independent, the Hamiltoniancanbewrittenasitstimeaverage,thatis

H

=

1

Evaluating for the case of massless and massive propagating modes,one obtainsthe followingexpressionsforthe correspond-ingon-shellenergies

where the subscriptsm and M stand for massless graviton and massivespin-2field,respectively.

Inagravitytheorywithquadratictermsinthecurvature,where thecouplingsarerelatedas

α

= −

3β,thereisonlyaspecificvalue of

β

that killsthe negative energystates.More specifically, from Eqs. (15),(16) itisshownthatfor

β

= −

1/2,theenergyofboth themasslessandthemassivemodesiszero.Hence,alltheghosts disappearleadingtoaconsistenttheoryofgravity.

Therefore,theactionofCriticalGravityreads

Icritical

=

On the other hand,the generic expression forthe energy of the blackholesinthisgravitytheoryisgivenbyEq. (8).Foranystatic black hole, the only nonvanishing contribution comes from the first term on the right hand side of Eq. (8). In particular, for a Schwarzschild-AdSblackhole, theADT charge leads tothe result inEq. (2). It iseasy tonoticethat,forthe particularvalueofthe couplingswhichdefineCriticalGravity(

α

= −

3β,

β

= −

1/2),the massoftheblackholevanish.

Inwhatfollows,weprovideanalternativeformulaofconserved chargesinCriticalGravity,whichmakesmanifestthefactthatthe energyforEinsteinblackholesisidenticallyzero.

4. Noether–WaldchargesinCriticalGravity

Ageneralprescription todefine conservedcharges inan arbi-trarytheory ofgravity was giveninRefs. [11,12,17]. Forthe pur-poseofthediscussionbelow,wewillrestrictourselvestothecase

whereLagrangiandensityisafunctionalonlyofthemetricandthe curvature,

L

(

gμν

,

Rμναβ

).

Fora givenset ofKillingvectors

{

ξ

μ

}

,

theNoethercurrentiswrittendownas

√

For simplicity,we assume that the surface term

μ isseparable into a partthat contains variations ofthe Christoffelsymbol and anotherpartthat contains variationsofthe metric.Aswe are in-terestedindiffeomorphicchargesforgravity,allthevariations are replacedbyaLiederivativealongthevector

{

ξ

μ

}

.

Using theKillingequation,

δ

ξgμν

= ∇

μ

ξ

ν

+ ∇

ν

ξ

ν

=

0,one can

noticethat firstterminEq. (18) vanishes.The samerelation,this timefortheLiederivativeoftheChristoffelconnection,would pro-duceacombinationofdoublecovariantderivativesandcurvatures. Thiscaststhecurrent,foragenericgravitytheory,intheform

Jμ

=

2Eμν_αβ

Here,thetensorEμν_αβ isthefunctionalderivativeof

L

withrespect tothespacetimeRiemanntensor Rμν_αβ,thatis,

Eμν_αβ

=

δ

L

δR

αμνβ

.

(20)

It canbeshown, bymeans ofthe generalformofthefield equa-tions for these class of gravity theories, that the last two terms ontherighthandsideof(19) formtheEOM contractedwiththe Killingfield.

Thus, on-shell, the first term on the right side of (19) is the onlynonvanishingpart.

Asthetensor Eμν_αβ satisfiesBianchiidentity,theconserved cur-rentturnsintoatotalderivative

Jμ

=

2

∇ν

Eμν_αβ

∇

α

ξ

β

.

(21)

AstheNoethercurrent Jμcanbewrittenas Jμ

= ∇

νqμν,the

con-servedchargeisexpressedasanintegralontheco-dimensiontwo surface

Qμ

[

ξ

] =

dSνqμν (22)

asmentionedpreviouslyinSection2.Finallytheconservedcharge iswrittenas

Qμ

[

ξ

] =

2

dSνEαμνβ

∇

α

ξ

β

.

(23)

AnalternativeformfortheactionofCriticalGravityconsidersthe differencebetweenWeyl2 andtheGBterm

E

4,astheGBinvariant termdoesnotalterthebulkdynamics[18]

Icritical

=

Using the Noether–Wald formula for the current(21) for the first part, the functional derivative with respect to the Riemann tensoroftheLagrangianin IM M produces

Eμν_αβ

=

whereas,fortheConformalGravitypart IC G,weget

˜

UsingtheNoether–Waldformula(23),thetotalchargeforthe the-ory

Bydefinition,theWeyltensoris

W_σγ_λδ

=

Rγ_σδ_λ

−

1

wheretherighthandside,isreferred toasAdScurvature2 _Using theabovefact,theconservedquantityinCriticalGravityis identi-callyzeroforEinsteinspaces.

5. ElectricpartoftheWeyltensorandEinsteinmodesin ConformalGravity

CGinfourdimensionsisinvariant underlocalWeyl rescalings ofthemetric(gμν

→ ˜

gμν

=

e2ωgμν).SolutionstoCGareBach-flat

geometries,whichincludeEinsteinspacetimes.

Fromaholographic viewpoint, asymptoticallyAdS spaceinCG areendowedwithnewsourcesattheconformalboundary.Indeed, wecansetanyAAdSspacetimeinFefferman–Graham(FG)formof themetric

Here,the ellipsis denotes higher-orderterms whichdo not enter intotheholographicdescriptionof4D AAdSspaces.

Thepresenceofthetermzg(1)i j reflectsthefactthespace con-tainsanon-Einsteinpart.Bydemandingthevanishingofthelinear termonz,onerecoverstheEinsteinbranch,withonlyeven pow-ersofzintheexpansion.ThisisachievedbyimposingaNeumann boundaryconditiononthemetric,

∂

zg

|

z=0

=

0 [21].

On the other hand, the Noether–Wald charge for Conformal Gravityis proportional tothe Weyltensor, asshownby Eq. (26).

2 _{ThefieldstrengthfortheAdSgroupalsocontainsthetorsionalongthe} gen-eratorsofAdStranslationsinRiemann–Cartan theory.ForRiemanniangeometry, Eq. (29) istheonlynonvanishingpartofthecurvatureoftheAdSgroup.

However, itisnot obviouswhether,forEinsteinspaces,the holo-graphicmodesofCGattheboundaryarecontainedintheelectric partoftheWeyltensor

Ei_j

=

Wi_jμ_νnμnν

=

Wizjz

,

(32)

asitisthecaseinEinsteingravity.

AsEinsteinspacesare solutionsoftheEOMofCGinthebulk, we restrict the discussionto the surface terminthe variation of IC G,thatis,

Thesecondtermintheaboverelationcanbeeliminatedusing theBianchi identityofsecond kind. Aprojection ofall indicesto theboundarycanbeperformedbytakingtheexplicitformofthe normalvectornμ inGaussiancoordinates.Then,thesurface term

takestheform

In Gauss normal frame (6), the relevant components of the Christoffelsymbolare withthisresult,thevariationoftheactioniswrittenas

δI

C G

=

aftersomealgebraicmanipulationandindexrelabeling.

The restof the proof relieson a power-countingargument in theradialcoordinatez.Inordertodoso,itisrequiredtoexpand thetensorialquantitieswhichappearatthesurfaceterm.

First, we consider the FG expansion for Einstein spacetimes, whereN

(

z

)

=

/

zandhi j

(

z

,

x

)

=

gi j

(

z

,

x

)/

z2withthemetricatthe conformalboundarygivenby

gi j

(z,

x)

=

g(0)i j

+

z2g(2)i j

+

z3g(3)i j

+

... .

(37)

where Si_j isthe Schoutentensordefinedfortheboundarymetric

In a similar fashion, one can compute the fall-off of the differ-entcomponentsofthespacetimeWeyltensor.Here,wejustwrite downtheoneswhichareofrelevanceforthisholographic discus-sion

where

W

ik_jm correspond tothe boundaryWeyltensorandthe in-dicesof g(3)areraisedandloweredwiththemetricg(0).

Replacing all the above quantities in Eq. (36), we realizethat thefirsttermandthirdtermsintheintegrandareoforder

O

z2

_. Thatimpliesthatthesetermsdonotcontributeinthelimitz

→

0. Inturn,theonlynonvanishingcontributioncomesfromthesecond terminEq. (36) as

expressedintermsoftheholographicEinsteinmodes.

Onecantakeafewstepsbackintheexpansionoftheboundary quantitiesandappropriatelycovariantizethelastresult,inorderto expressitintermsofthesubtraceofthespacetimeWeyltensor

δI

C G

=

canbetradedoffbytheelectricpartoftheWeyltensor

W_ij

= −

W_izjz

.

(46)

Asaconsequence,thevariationoftheConformalGravityactionis

δI

C G

= −

fortheEinsteinmodesofthetheory.Atthesametime,thismeans thatthedefinitionofconservedquantitiesforthatsectorofCGcan bemappedtothenotionofConformalMassin4D [22].

6. Conclusions

In the presentwork, we have shown that, in Critical Gravity, theenergyofanyEinsteinsolutionvanishesidentically.Thisproof doesnot make use of anyparticular Einstein black hole, nor re-liesonchargeformulasobtainedfromthelinearizationofthefield equations.Inthisrespect,chargeexpression(27) providesthe ex-plicitrealizationofaclaimoriginallystatedinRef. [13].

The holographicderivation inSection 5 confirms thefact that the boundary stress tensor forthe total action (24) is zero,in a similarwayasinRef.[23].

Whenone goesbeyondEinstein spaces,theexpression(27) is abletocapturetheeffectsduetothepresenceofhigher-derivative termsinthecurvature.Indeed,asitwas showninRef. [24],only thenon-Einstein modesoftheWeyl tensorsurviveinthesurface termformthevariationoftheCriticalGravityaction.Asamatter offact,theboundarycontributionsareexpressibleintermsofthe

Bach tensor, what enormouslysimplify the computation of holo-graphiccorrelationfunctionsatthecriticalpoint[25].

Noether–Wald charges provides the black hole entropy in a givengravitytheory,whenevaluatedatthehorizonr

=

rh,

S

= −

2

h

dSνE00να

∇

α

ξ

0

.

(48)

As the condition in the Weyl tensor (29) holds throughout the spacetime forEinstein solutions,it isevidentfromtheabove for-mula that the entropy vanishes in Critical Gravity. The addition of topological invariants to the four-dimensional AdS gravity ac-tionhasledtoenergydefinitionswhicharefinite[26,27],butalso hasprovidedinsightontheproblemofholographyfor asymptoti-callyAdSspacesinEinsteingravity[20].Theresultpresentedhere indicatesthattheGauss–Bonnettermalsoplaysaroleinthe holo-graphicdescriptionofgravitybeyondEinsteintheory.

Acknowledgements

We thankO. Miskovicforhelpfuldiscussions. G.A.isa Univer-sidad AndresBello (UNAB)Ph.D.Scholarshipholder,andhiswork is supported by Dirección General de Investigación (DGI-UNAB). D.R.B.is a UNABM.Sc. Scholarshipholder. Thiswork was funded inpartbyFONDECYTGrantNo.1170765;CONICYTGrantNo. DPI 20140115andUNABGrantNo.DI-1336-16/R.

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