Massive graviton in topologically new massive gravity

Yong-Wan Kim,1,* Yun Soo Myung,2,† and Young-Jai Park3,‡

1Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea2Institute of Basic Science and School of Computer Aided Science, Inje University, Gimhae 621-749, Korea

3Department of Physics and Department of Global Service Management, Sogang University, Seoul 121-742, Korea(Received 8 January 2013; published 15 March 2013)

We investigate the topologically new massive gravity in three dimensions. It turns out that a single

massive mode is propagating in the flat spacetime, comparing to the conformal Chern-Simons gravity,

which has no physically propagating degrees of freedom. Also we discuss the realization of the Bondi-

Metzner-Sachs/Galilean conformal algebras correspondence.

DOI: 10.1103/PhysRevD.87.064020 PACS numbers: 04.60.Rt, 04.20.Ha, 11.25.Tq

I. INTRODUCTION

It is well known that the AdS/CFT correspondence [1]was supported by the observation that the asymptoticsymmetry group of AdS3 spacetime is a two-dimensionalconformal symmetry group (two Virasoro algebras) on theboundary [2]. Similarly, the asymptotic symmetry group offlat spacetime is the infinite dimensional Bondi-Metzner-Sachs (BMS) group whose dual CFT is described by theGalilean conformal algebras (GCA). The latter was calledthe BMS/GCA correspondence [3]. The centrally extendedBMS (or GCA) algebra is generated by two kinds ofgenerators Ln and Mn:

½Lm; Ln� ¼ ðm� nÞLnþm þ c112

ðn3 � nÞ�nþm;0;

½Lm;Mn� ¼ ðm� nÞMnþm þ c212

ðn3 � nÞ�nþm;0;

½Mm;Mn� ¼ 0:

(1)

It is very important to establish the BMS/GCA correspon-dence by choosing a concrete model. Recently, a holographiccorrespondence between a conformal Chern-Simonsgravity (CSG) in flat spacetime and a chiral conformalfield theory was reported in Ref. [4]. Choosing the CSGas the flat-spacetime limit of the topologically massivegravity in the scaling limit of � ! 0 and G ! 1 whileG� is fixed, the BMS central charges are determined to bec1 ¼ 24kð¼ 3=G�Þ and c2 ¼ 0. This implies that the CSGis dual to a chiral half of a CFTwith c ¼ 24k. On the otherhand, c1 ¼ 0 and c2 ¼ 3=G were predicted when using theEinstein gravity without taking the scaling limit [5].

Considering the flat spacetime expressed in terms ofoutgoing Eddington-Finkelstein (EF) coordinates, thelinearized equation of the CSG leads to the third orderequation ðDhÞ3 ¼ 0. The solution to the first order equationðDh�Þ ¼ 0 is given by [4]

h��� ¼ e�ið�þ2Þ�r�ð�þ2Þðm1 �m1Þ; (2)

where � is the eigenvalue of L0 and a Killing vector ofm1 ¼ iei�ð@u � @r � i

r @�Þ. Furthermore, one solution to

ðDhÞ3 ¼ 0 is given by hlog�� ¼ �iðuþ rÞh���, while the

other is hlog 2

�� ¼ � 12 ðuþ rÞ2h���. These are the flat-space

analogues of log and log 2 solutions on theAdS3 spacetime.At this stage, we wish to point out that the solutions

fh�; hlog ; hlog 2g could not represent any physical modespropagating on the flat spacetime background becausethe CSG has no physical degrees of freedom. Actually,these all belong to the gauge degrees of freedom. Hence, iturges one to find a relevant action that has a physicallymassive mode propagating on the Minkowski spacetime.This might be found when including a curvature squarecombination K, leading to the topologically new massivegravity (TNMG) [6]. The TNMG is also obtained from thegeneralized massive gravity (GMG) with two differentmassive modes [7,8] when turning off the Einstein-Hilbert term and cosmological constant. If the Einstein-Hilbert term is omitted, it is called the cosmologicalTNMG [9]. It turned out that the linearized TNMG pro-

vides a single spin-2 mode with mass m2

� in the Minkowski

spacetime, which becomes a massless mode of masslessNMG in the limit of � ! 1 [6,10]. Very recently, it wasargued that this reduction (2 ! 1) of local degrees offreedom is an artifact of the linearized approximation byusing the Hamiltonian formulation where the nonlineareffect is not ignored [11]. We note that the linearizedTNMG has a linearized Weyl (conformal) invariance asthe CSG does show [6].In this paper, we explicitly show that a massive spin-2

mode is propagating on the flat spacetime when introduc-ing the TNMG. Furthermore, we observe how the BMS/GCA correspondence is realized in the TNMG.

II. TNMG IN FLAT SPACETIME

We start with the TNMG action

ITNMG ¼ ICSG þ IK; (3)

*ywkim65@gmail.com†ysmyung@inje.ac.kr‡yjpark@sogang.ac.kr

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ICSG ¼ 1

2�2�

Zd3x

ffiffiffiffiffiffiffi�gp

������

�@��

� þ 2

3����

��

�;

(4)

IK ¼ 1

�2m2

Zd3x

ffiffiffiffiffiffiffi�gp �

R��R�� � 3

8R2

�; (5)

where �2 ¼ 16�G, G is the Newton constant, � is theChern-Simons coupling, and m2 is a mass parameter. Wenote that the GMG action is given by [7,8]

IGMG ¼ 1

16�G

Zd3x

ffiffiffiffiffiffiffi�gp ðR� 2�0Þ þ ITNMG; (6)

where the TNMG is recovered in the limits of ! 0 and�0 ¼ �1=‘2 ! 0. The equation of motion of the TNMGaction is given by

1

�C�� þ 1

2m2K�� ¼ 0; (7)

where the Cotton tensor C�� takes the form

C�� ¼ �� �r

�R�� � 1

4g��R

�; (8)

and the tensor K�� is given by

K�� ¼ 2r2R�� � 1

2r�r�R� 1

2r2Rg�� þ 4R��R

� 3

2RR�� � g��RR

þ 3

8R2g��: (9)

As a solution to Eq. (7), let us choose the Minkowskispacetime expressed in terms of the outgoing EFcoordinates

ds2EF ¼ �g��dx�dx� ¼ �du2 � 2drduþ r2d�2; (10)

where u ¼ t� r is a retarded time. Considering theperturbation h�� around the EF background �g��

g�� ¼ �g�� þ h��; (11)

the linearized equation of Eq. (7) takes the form

1

��C�� þ 1

2m2�K�� ¼ 0: (12)

Now, we consider the transverse and traceless conditionsto select a massive mode propagating on the EF back-ground as

�r�h�� ¼ 0; h�� ¼ 0: (13)

Then, we have the linearized fourth-order equation ofmotion as

�� � �r

�r2

��� þ �

m2��

�r

�h� ¼ 0; (14)

where the mass of the graviton is identified withM ¼ m2=�. Furthermore, this equation can be expressedcompactly as

ðD3DMhÞ�� ¼ 0 (15)

by introducing two mutually commuting operators as

D�� ¼ ��

� �r ; ðDMÞ�� ¼ ��� þ �

m2��

� �r : (16)

Now, let us solve the first-order massive equation

ðDMhMÞ�� ¼ hM�� þ �

m2��

� �r hM�� � ðEOMÞð��Þ ¼ 0

(17)

directly. This will be done by assuming a proper ansatz

hM��ðu; r; �Þ ¼ fð�ÞGðu; rÞ0 0 0

0 FrrðrÞ Fr�ðrÞ0 Fr�ðrÞ F��ðrÞ

0BB@

1CCA: (18)

Then, the traceless condition of h�� ¼ 0 takes the form

r2Frr þ F�� ¼ 0; (19)

while the transverse conditions �r�h�� ¼ 0 lead to

0¼ F��Gf0 þ rf½rGF0

r� þFr�ðGþ r@rG� r@uGÞ�;0¼ Fr�Gf

0 þ f

�G

rðr3F0

rr �F��Þ

þ rFrrðGþ r@rG� r@uGÞ�; (20)

for � ¼ �, r, respectively, and for � ¼ u, it vanishes. Herethe prime (0) denotes the differentiation with respect to itsargument.The nine equations of motion take the following forms:

0 ¼ ðEOMÞð11Þ ¼ ðEOMÞð21Þ ¼ ðEOMÞð31Þ;0 ¼ ðEOMÞð12Þ¼ �rFrrGf

0 þ f½rGF0r� þ Fr�ðGþ r@rG� r@uGÞ�;

(21)

0 ¼ ðEOMÞð13Þ ¼ rFr�Gf0 þ f½r2GFrr � rGF0

��

þ F��ðG� r@rGþ r@uGÞ�; (22)

0 ¼ ðEOMÞð22Þ ¼ m2r2FrrGfþ�G½�rFrrf0

þ fðFr� þ rF0r�Þ þ�rfFr�@rGÞ�; (23)

0 ¼ ðEOMÞð23Þ ¼ �rfF��@rG � G½�rFr�f0

þ fð�r2Frr þ �F��

� r2ðm2rFr� þ �F0��ÞÞ�; (24)

0 ¼ ðEOMÞð32Þ ¼ Fr�G��r

m2Frr@uG; (25)

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0 ¼ ðEOMÞð33Þ ¼ F��G��r

m2Fr�@uG; (26)

with ðu; r; �Þ ¼ ð1; 2; 3Þ. From Eq. (25), one finds therelation

Fr� ¼ �r

m2

@uG

GFrr: (27)

Also, from Eq. (26), one obtains the relation

F�� ¼ �r

m2

@uG

GFr� ¼ �2r2

m4

ð@uGÞ2G2

Frr: (28)

Comparing this with the traceless condition (19), we have

�@uG

G

�2 ¼ �

�m2

�

�2; (29)

which could be solved to give

Gðu; rÞ ¼ C1ðrÞe�im2

� u: (30)

Choosing the ‘‘�’’ sign, we obtain

Gðu; rÞ ¼ C1ðrÞe�im2

� u; Fr� ¼ �irFrr;

F�� ¼ �r2Frr: (31)

Using these relations, Eqs. (21)–(24) reduce to a singleequation

0 ¼ �rfC1ðrÞF0rr � ½i�C1ðrÞf0 � fð2�þ im2rÞC1ðrÞ

þ�rC01ðrÞgf�Frr; (32)

which has a solution

Frr ¼ e�im2

� rrif0f �2

C1ðrÞ : (33)

Again, using this, Eqs. (21)–(24) become a single equationfor fð�Þ

½f0ð�Þ�2 ¼ fð�Þf00ð�Þ; (34)

whose solution is given by

fð�Þ ¼ eC2� (35)

with an undermined constant C2.As a result, we arrive at a solution

hM��ðu;r;�Þ¼e�im2

� ðuþrÞeC2�riC2�2

0 0 0

0 1 �ir

0 �ir �r2

0BB@

1CCA (36)

with uþ r ¼ t. We note that C1ðrÞ disappears in Eq. (36),suggesting that one may choose GðuÞ initially, instead ofGðu; rÞ in Eq. (18).

Importantly, when solving the first-order massive equa-tion (17), one could not determine C2. However, if one

introduces the 2D GCA representations, it could be fixed tobe C2 ¼ �i�, implying that the integration constant C2

can be interpreted as the eigenvalue of L0 of the asymptoticBMS algebra in Minkowski spacetime. This may be takenas a hint that the TNMG provides a possible realization ofthe BMS algebra in three dimensions. Making the choiceof C2 ¼ �i�, we have the solution

hM��ðu;r;�Þ¼e�im2

� ðuþrÞe�i��r��2

0 0 0

0 1 �ir

0 �ir �r2

0BB@

1CCA; (37)

which is regarded as our main result.To confirm that hM�� satisfies the full equation (15), we

apply the massless operator D n times on hM�� as

ðDnhMÞ�� ¼��m2

�

�nhM��: (38)

Using Eq. (38), it is easy to check that hM�� satisfies the

linearized fourth-order equation (15) as

ðD3DMhMÞ�� ¼ ðD3hMÞ�� þ �

m2ðD4hMÞ�� ¼ 0: (39)

III. BMA/GCA CORRESPONDENCE

We need more works to confirm that the TNMG providesa possible realization of the BMS algebra in three dimen-sions. In this direction, we may show that the BMS centralcharges are defined in the TNMG. We might interpret (37)to be one-parameter deformation of the CSG representationof the BMS algebra, parametrized by M ¼ m2=�. This is

because hM��ðu; r; �Þ � e�iMðuþrÞh���ðr; �Þ where h���ðr; �Þis the CSG wave function in (2). Hence, we have to look forthe BMS central charges c1 and c2, eigenvalue � of operatorL0 and eigenvalue � of operator M0.To see what is going on in the BMS/GCA correspon-

dence in the TNMG, we first consider ‘‘the AdS/CFTcorrespondence on the AdS3 and its boundary’’ withinthe GMG (6). Two central charges of the GMG on theboundary are given by [12–14]

cL ¼ 3‘

2G

�þ 1

2m2‘2� 1

�‘

�; (40)

cR ¼ 3‘

2G

�þ 1

2m2‘2þ 1

�‘

�: (41)

Taking two limits of ! 0 and ‘ ! 1 to obtain theTNMG, the corresponding BMS central charges are de-fined to be

c1 ¼ lim!0;‘!1

ðcR � cLÞ ¼ 3

G�; (42)

c2 ¼ lim!0;‘!1

cR þ cL‘

¼ 0: (43)

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Here we observe the disappearance of m2 in c2 (43), whichmight not be good news. In defining c1;2, we have used the

convention of the ultrarelativistic limit in Refs. [4,15,16],which is opposite to c1 and c2 in the original convention ofthe nonrelativistic limit [17,18]. The former convention isbetter to take the flat-spacetime limit from the AdS3 space-time. Equations (42) and (43) show clearly that the BMScentral charges are determined by the CSG (4) solely, imply-ing that the central extensions are unaffected by the presenceof the IK term (5). This explains why we have chosenC2 ¼ �i� in deriving the massive wave solution (37).

Now let us determine which one of the rigidity (weight)� and scaling dimension 4 is related to the deformedparameter, mass M ¼ m2=� of the graviton. Since theseare eigenvalues as shown in

L0j4; �i ¼ �j4; �i; M0j4; �i ¼ 4j4; �i; (44)

they are defined by

� ¼ lim‘!1;!0

ðh� �hÞ; 4 ¼ lim‘!1;!0

hþ �h

‘: (45)

Note that Eq. (44) can be understood as the two actingoperators L0 ¼ i@� and M0 ¼ i@t on the solution (37),defined in Ref. [3]. Here two weights h and �h are definedas the highest weight conditions of the GMG on the AdS3:

L0jc ��i ¼ hjc ��i and �L0jc ��i ¼ �hjc i. Then, h and �h

are determined to be [19]

ðh; �hÞ ¼�3þ ‘m1

2;�1þ ‘m1

2

�; (46)

where the mass is given by

m1 ¼ m2

2�þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2‘2� m2 þ m4

4�2

s: (47)

According to the ultrarelativistic convention [15,16],the connection between the GCA and the Virasoro algebrasis given by

Ln ¼ Ln � �L�n; Mn ¼ Ln þ �L�n

‘: (48)

After a computation, one finds that

� ¼ 2; 4 ¼ m2

�: (49)

The eigenvalue � ¼ 2 arises because it represents spin-2excitations. In the limit of � ! 1, 4 ! 0 as recoveringthe massless NMG. Using these, the massive wave solution(37) respects that of the GMG on the AdS3 as

~hM��ðu; r; �Þ ¼ e�im2

� ðuþrÞ�2i�

0 0 0

0 1 �ir

0 �ir �r2

0BB@

1CCA: (50)

At this stage, we note again that the central charges c1and c2 in Eqs. (42) and (43) remain intact as comparedto the CSG, but the scaling dimension4was changed from

0 to m2=�. Thinking that the TNMG is a one-parameterdeformation of the CSG representation of the BMS alge-bra, one might expect that their central charges are alsodeformed. To explore this idea, we observe cL=R in (40) and(41) carefully. Considering the flat-spacetime limit fromthe AdS3 spacetime, one possibility is to consider the case

~c2 ¼ lim!0;‘!1

‘ðcL þ cRÞ ¼ 3

2Gm2; (51)

while c1 remains the same as in Eq. (42). This also requiresa modification of the generator ~Mn as

~Mn ¼ ‘ðLn þ �L�nÞ (52)

instead of Mn in (48). In this case, the flat-spacetimedefinition of the scaling dimension is changed to be

~4 ¼ lim‘!1;!0

‘ðhþ �hÞ; (53)

while the rigidity � remains unchanged. For the TNMG,this leads to the infinity as

~4 ¼ lim‘!1;!0

‘ð1þ ‘m1Þ ! 1; (54)

which cannot be acceptable. As a result, the BMS charge ~c2seems to be unphysical, even though it has a finite value(51) in the flat-spacetime limit.

IV. GMG SOLUTION IN THEFLAT-SPACETIME LIMIT

Now, it is very important to know what waveform ofthe GMG [12] provides (50) in the flat-spacetime limit.Directly, this task will determine which one between

(c2 ¼ 0, � ¼ m2=�) and (~c2 ¼ 32Gm2 , ~� ¼ 1) is correct.

In particular, the GMG wave solution for the left-movingmassive graviton is described by

c L��ð; �þ; ��Þ ¼ fð; �þ; ��Þ

1 0 2isinh ð2Þ

0 0 02i

sinh ð2Þ 0 � 4sinh 2ð2Þ

0BBB@

1CCCA

(55)

in the light-cone coordinates of (, �� ¼ ���) for theAdS3 spacetime. Here the amplitude f is given by

fð; �þ; ��Þ ¼ e�ih�þ�i �h��ðcoshÞ�ðhþ �hÞsinh 2; (56)

where two weights h and �h are already given by (46).We note that c L

�� satisfies the traceless and transverse

conditions: c L�� ¼ 0, �r�c

L�� ¼ 0. As is suggested in

Ref. [4], we express the EF coordinates in terms of globalcoordinates

u ¼ ‘ð�� Þ; r ¼ ‘; � ¼ �: (57)

Then, we have a transformed tensor mode

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c L��ðu; r; �Þ ¼ fðu; r; �Þ

1 1þ 2isinh ð2r‘ Þ

‘

1þ 2isinh ð2r‘ Þ

1þ 4isinh ð2r‘ Þ

� 4sinh 2ð2r‘ Þ

�1þ 2i

sinh ð2r‘ Þ

�‘

‘

�1þ 2i

sinh ð2r‘ Þ

�‘ ‘2

0BBBBBBB@

1CCCCCCCA; (58)

where the transformed amplitude takes the form

fðu; r; �Þ ¼ e�iðhþ �h‘ ÞðuþrÞ�iðh� �hÞ�

�cosh

�r

‘

���ðhþ �hÞsinh 2

�r

‘

�:

(59)

Thus, taking the flat-spacetime limit of ‘ ! 1 while keep-ing u and r finite, and making use of� in (45) [but not ~� in(53)], we arrive at

c L��ðu; r; �Þ ’ e�im

2

� ðuþrÞ�2i�

0 0 0

0 1 �ir

0 �ir �r2

0BB@

1CCA; (60)

which is exactly the same form of (50). This proves that themassive wave solution (50) represents a truly massivegraviton mode propagating in the Minkowski spacetimebackground. In this case, c2 ¼ 0ð� ¼ m2=�Þ is a correctBMS representation for the TNMG. Finally, we wish tostress that the TNMG provides one-parameter (m2=�) de-formation of the CSG representation of the BMS algebra.However, the central charges ðc1; c2Þ are not affected by thisdeformation, but the scaling dimension � is changed.

V. DISCUSSIONS

Our work was inspired by the observation that eventhough the CSG has no local degrees of freedom, it pro-vides the first evidence for a holographic correspondence(the BMS/CFT correspondence) [4]. Its dual field theory isconsidered as a chiral CFTwith a central charge of c ¼ 24.

To see what happens for the holographic properties of agravitational theory with a local degree of freedom, wehave investigated the TNMG in the Minkowski spacetime.Solving the first-order massive equation (17) together withthe traceless and transverse conditions, we have found amassive wave solution (37). Concerning the BMS/GCAcorrespondence in the TNMG, we have c1 ¼ 3

G� and

c2 ¼ 0 as in the CSG. This means that the NMGterm (IK) does not contribute to the central charge of theboundary field theory. Also we have the same rigidity

� ¼ 2 as in the CSG [20] where ðh; �hÞ ¼ ð3þ‘�2 ;�1þ‘�

2 Þ,

but a different scaling dimension � ¼ m2=� from � ¼ �of the CSG. Here, some difference arises in defining �: inRef. [3], � ¼ 0 for the CSG because they have taken thescaling limit of � ! 0. However, in this work, we did notrequire the scaling limit of� ! 0, G ! 1, but use the flatspacetime limit of ! 0, ‘ ! 1 to get the TNMG.Importantly, we have obtained the massive graviton wavesolution (50), which is recovered from the GMG-wavesolution when taking the flat spacetime limit and using� ¼ 2 and � ¼ m2=�.We discuss the asymptotically flat boundary condition

on the wave solution (50). Actually, there is a differencebetween the CSG and the TNMG because there is a change

in the radial solution between h��� (2) and ~hM�� (50): ~hM�� is

regular in the interior but incompatible with the asymptoti-cally flat boundary condition (3) in Ref. [4]. Therefore,there is a little improvement on the radial boundary con-dition of a massive graviton mode.Consequently, we have shown that the TNMG has a

single massive mode propagating on the flat spacetime,whereas there is no physically propagating degrees offreedom from the CSG. This means that the TNMG pro-vides one-parameter deformation of the CSG representa-tion of the BMS algebra, parametrized bym2=�. However,their central charges ðc1; c2Þ are unaffected by this defor-mation, but the scaling dimension � is changed.

ACKNOWLEDGMENTS

We would like to thank D. Grumiller for helpful dis-cussions. This work was supported by the NationalResearch Foundation of Korea (NRF) grant funded bythe Korea government (MEST) through the Center forQuantum Spacetime (CQUeST) of Sogang Universitywith Grant No. 2005-0049409. Y. S.M. was also supportedby the National Research Foundation of Korea (NRF) grantfunded by the Korea government (MEST) (No. 2012-040499). Y.-J. P. was also supported by World ClassUniversity program funded by the Ministry of Education,Science and Technology through the National ResearchFoundation of Korea (Grant No. R31-20002).

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