Comment on Cosmological Topological Massive Gravitons and Ph

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Comment on “Cosmological Topological Massive Gravitons and Photons”Wei Li,Wei Song 1and Andrew Strominger Center for the Fundamental Laws of Nature Je?erson Physical Laboratory,Harvard University,Cambridge MA 02138,USA Abstract In a recent paper [1]it was shown that all global energy eigen-states of asymptotically AdS 3chiral gravity have nonnegative energy at the linearized level.This result was questioned [2]by Carlip,Deser,

Waldron and Wise (CDWW),who work on the Poincare patch.They exhibit a linearized solution of chiral gravity and claim that it has negative energy and is smooth at the boundary.We show that the solution of CDWW is smooth only on that part of the boundary of AdS 3included in the Poincare patch.Extended to global AdS 3,it is pergent at the boundary point not included in the Poincare patch.Hence it is consistent with the results of [1].

We consider the theory of TMG(topologically massive gravity)with a negative cosmological constant.This is described by[3][4][5]2

I=1

?g(R+216πGμI CS(1)

where I cs is the Chern-Simons term

I cs=?1

?g?λμνΓρλσ[?μΓσρν+2

2G (1?1

μ?

)and hence is purely chiral atμ?=1.

Solutions of the linearized equations which are energy eigenstates(where energy is de?ned as in[10][11]using the global timeτ)fall into SL(2,R)L×SL(2,R)R representations which can be characterized by the(L0,ˉL0)eigen-values(h L,h R)of the highest weight states.In[1]it was shown that there are three representations labelled by the highest weights(h L,h R)

(2,0),(0,2),(4) corresponding to massless left and right moving boundary gravitons and

(3

2

,?

1

2

),(5)

corresponding to the so-called massive gravitons.Forμ?>1,despite the positivity of the weights,the massive gravitons have negative energy because the kinetic term has an overall negative prefactor of(1

.(6)

z2

They?nd a continuum(rather than a discrete set)of solutions of the lin-earized wave equations,work out the linearized Einstein tensor Hρσ(CDWW equation(44)).They?nd that e.g.H??is given by(CDWW equation(48) and page12)

H??～zJ3(z)→z4,z→0(7) where J is a Bessel function.This indeed vanishes at the boundary z=0of the Poincare patch.However this does not cover all of the boundary points of global AdS3.The missing boundary point can be approached by?xing Poincare time x+?x?=constant and taking z→∞.From Bessel function asymptotics it is then easily seen that the curvature perges as

H??～√

3Actually to be more precise CDWW use a negative Newton’s constant so the energy is positive,but with our sign choice it is negative.

2

AdS3.This is fully consistent with the observation of[1]that all asymptoti-cally AdS3(L0,ˉL0)eigensolutions of linearized chiral gravity have nonnega-tive energy.

Acknowledgements

We thank S.Carlip,D.Grumiller and A.Maloney for helpful discussions. The work is supported by DOE grant DE-FG02-91ER40654.WS is also supported by CSC.WS would like to thank the hospitality of the physics department of Harvard University.

References

[1]W.Li,W.Song and A.Strominger,“Chiral Gravity in Three Dimen-

sions,”JHEP0804,082(2008)[arXiv:0801.4566[hep-th]].

[2]S.Carlip,S.Deser,A.Waldron and D.K.Wise,“Cosmological Topolog-

ically Massive Gravitons and Photons,”arXiv:0803.3998[hep-th]. [3]S.Deser,R.Jackiw and S.Templeton,Annals Phys.140,372(1982)

[Erratum-ibid.185,406.1988APNYA,281,409(1988APNYA,281,409-449.2000)]

[4]S.Deser,R.Jackiw and S.Templeton,“Three-Dimensional Massive

Gauge Theories,”Phys.Rev.Lett.48,975(1982).

[5]S.Deser and B.Tekin,“Massive,topologically massive,models,”Class.

Quant.Grav.19,L97(2002)[arXiv:hep-th/0203273].

[6]J.D.Brown and M.Henneaux,“Central charges in the canonical real-

ization of asymptotic symmetries:an example from three-dimensional gravity,”Commun.Math.Phys.104,207(1986).

[7]K.Hotta,Y.Hyakutake,T.Kubota and H.Tanida,“Brown-Henneaux’s

Canonical Approach to Topologically Massive Gravity,”arXiv:0805.2005 [hep-th].

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[8]P.Kraus and 3d0771d176a20029bd642d8crsen,“Holographic gravitational anomalies,”JHEP

0601,022(2006)[arXiv:hep-th/0508218].

[9]P.Kraus,“Lectures on black holes and the AdS(3)/CFT(2)correspon-

dence,”arXiv:hep-th/0609074.

[10]S.Deser and B.Tekin,“Energy in topologically massive gravity,”Class.

Quant.Grav.20(2003)L259[arXiv:gr-qc/0307073].

[11]S.Olmez,O.Sarioglu and B.Tekin,“Mass and angular momentum of

asymptotically AdS or?at solutions in the topologically massive gravity,”

Class.Quant.Grav.22(2005)4355[arXiv:gr-qc/0507003].

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