Flat-Space Energy-Momentum Tensor from BMS/GCA Correspondence
aa r X i v : . [ h e p - t h ] M a r Flat-Space Energy-Momentum Tensorfrom BMS/GCA Correspondence
Reza Fareghbal, Ali Naseh
School of Particles and Accelerators,Institute for Research in Fundamental Sciences (IPM)P.O. Box 19395-5531, Tehran, Iran
E-mails: [email protected], [email protected]
Abstract
Flat-space limit is well-defined for asymptotically AdS spacetimes written in co-ordinates called the BMS gauge. For the three-dimensional Einstein gravity with anegative cosmological constant, we calculate the quasi-local energy momentum tensorin the BMS gauge and take its flat-space limit. In defining the flat-space limit, we usethe BMS/GCA correspondence which is a duality between gravity in flat-spacetimeand a field theory with Galilean conformal symmetry. The resulting stress tensor re-produces correct values for conserved charges of three dimensional asymptotically flatsolutions. We show that the conservation relation of the flat-space energy-momentumtensor is given by an ultra-relativistic contraction of its relativistic counterpart. Theconservation equations correspond to Einstein equation for the flat metric written inthe BMS gauge. Our results provide further checks for the proposal that the holo-graphic dual of asymptotically flat spacetimes is a field theory with Galilean conformalsymmetry.
Introduction
One of the outstanding questions in modern theoretical physics is understanding the holo-graphic principle. For this, it is of interest to explore whether holography exists beyond theknown example of AdS/CFT and for spacetimes other than AdS. The study of holographyfor asymptotically flat spacetimes may be the first step in this direction due to simplicity offlat space and its close relation with the real physical world. During the past two decades,there have been several attempts at trying to formulate flat space holography, but there havebeen several hurdles to this apparently simple problem.A new method for studying this problem was proposed recently in [1, 2]. This was basedon the fact that the asymptotic symmetry group (ASG) of asymptotically flat spacetimesat null infinity is the Bondi, van der Burg, Metzner and Sachs (BMS) group [3]. In threeand four dimensions, BMS group is infinite dimensional and the corresponding algebra canbe centrally extended [4]- [6]. Thus it is probable that the theory dual to three or fourdimensional flat space is given by a (two or three dimensional) field theory living on its nullboundary which has exactly the symmetries of the BMS group.The observation of [1] was that BMS algebra is isomorphic Galilean conformal algebra(GCA) which is a result of contraction of conformal algebra [7]. This connection dubbed theBMS/GCA correspondence was studied carefully in [2] where it was shown that in order tohave a well-defined correspondence at the level of centrally extended algebras, at the levelof spacetime, the time direction must be contracted in the CFT and the resultant theoryis an ultra-relativistic field theory. In other words, flat space limit of AdS which in thebulk side implemented by taking the large radius limit of asymptotically AdS spacetimes isequivalent to the contraction of time on CFT in the boundary side. In this way one canstudy holography of flat spacetimes just by starting from AdS/CFT and taking appropriatelimit: large radius limit in the bulk and contraction in the boundary [2].This method has met with some very interesting recent successes. It was shown in [8] thata Cardy-like formula for field theories with Galilean conformal symmetry in two dimensions(resulting from time-contraction of two dimensional CFT on the cylinder) produces thesame entropy as that of three-dimensional cosmological solutions of flat gravity. Thesesolutions of Einstein gravity in three dimensions were first studied in [9] and are shift-boost orbifolds of three dimensional Minkowski spacetime. They can be viewed as the largeradius limit of non-extremal BTZ black holes in AdS . Moreover, it was shown in [10] thatthere is a Hawking-page like transition between Minkowski and flat cosmological solutions.The limiting procedure has been repeated for the topologically massive gravity in three1imensions and the corresponding two dimensional unitary theory has been recognized [11].A link between GCA and the symmetries of the tensionless limit of closed bosonic stringtheory has been shown in [12] and finally a similar contraction has been found for the W algebra which extends the BMS algebra to include spin-three fields in gravity threedimensional flat spacetimes [13].A subtle point in using proposal of [1, 2] for flat-holography is that the flat space limitin the bulk is not well-defined for all coordinates of asymptotically AdS spacetimes, e.g.the Fefferman-Graham coordinate system. However, [14] proposed a new coordinate systemfor asymptotically AdS spacetimes where taking flat-space limit is well-defined. In thesecoordinates, called BMS gauge, one can define appropriate fall-off conditions for the threedimensional metric components and find the asymptotic symmetry group in a manner similarto the seminal method of [15].In this paper we show that writing asymptotically AdS spacetimes in BMS gauge givesmore insight about holography of asymptotically flat spacetimes. A natural question to askis whether bulk computations in flat space can yield the correlations of the boundary theorywith the symmetries of a 2d GCA. The AdS/CFT correspondence provides a beautiful recipefor finding such correlators. If we follow methods of [1] and [2] for flat-space holography, itis plausible that taking flat limit of bulk calculations in the AdS case gives correlators whichcorresponding to the contracted correlators in the CFT side. As the first step in this directionwe apply the method of [16] to compute the one-point function of energy-momentum tensorand by writing it in the BMS gauge we define an appropriate flat limit for the quasi-localenergy-momentum tensor. Our definition of stress tensor for flat holography is consistentwith the results of a field theory with Galilean conformal symmetry.Our final answer for the one point function of energy-momentum tensor can be usedin the calculation of conserved charges of bulk solutions. In three dimensional flat gravitythe most important solutions are cosmological solutions with a cosmological horizon andfinite entropy, mass and angular momentum. Our energy- momentum tensor reproduces thecorrect values for the mass and angular momentum of these solutions.This paper is organized as follows. In the next section we start from asymptotically AdS spacetimes written in the BMS gauge and introduce its generic solution. The proper flatlimit of the BMS gauge is also discussed in this section. A brief introduction to BMS/GCAcorrespondence is given in section 3. The quasi-local energy momentum tensor in the BMSgauge together with a well-defined limit of it are developed in section 4. The derivation ofconserved charges of solutions of three dimensional flat gravity and also the relation withone point functions of field theory with Galilean conformal symmetry are also presented in2ection 4. Note added:
While this paper was being prepared for submission, the work [21] ap-peared on arXiv which has some overlap with our work. In [21], the author has performeda holographic renormalization of AdS gravity in the BMS gauge and taken its flat limit.We believe that the two pieces of work agree where there is overlap and in our paper theconnection of flat limit with field theory with Galilean conformal symmetry is highlighted.This is novel and has not been addressed in [21]. in BMS gauge and its flat limit Instead of Fefferman-Graham gauge which is a common way of defining asymptotically AdSspace-times one can write a generic solution of three dimensional Einstein gravity withcosmological constant, S = 116 πG Z d x √− g ( R + 2 ℓ ) , (2.1)in a new coordinate system as [14] ds = e β Vr du − e β dudr + r ( dφ − U du ) , (2.2)where β, V, U are arbitrary functions of coordinates and u is a null coordinate known as retarded time . This coordinate-system is known as BMS coordinate and it is possible to findthe appropriate coordinate transformation for writing all asymptotically AdS solutions inthis form. For AdS in the global coordinate ds = − (1 + r ℓ ) dt + dr r ℓ + r dφ , (2.3) u is given by u = t − ℓ arctan rℓ . It is clear that the boundary of AdS is at r = ∞ . Forasymptotically AdS spacetimes β = U = o (1). Thus one can solve the equations of motionsand find ds = (cid:18) − r ℓ + M (cid:19) du − dudr + 2 N dudφ + r dφ , (2.4)where M ( u, φ ) = 2 (cid:0) Ξ( x + ) + ¯Ξ( x − ) (cid:1) , N ( u, φ ) = ℓ (cid:0) Ξ( x + ) − ¯Ξ( x − ) (cid:1) , (2.5)and Ξ , ¯Ξ are arbitrary functions of x ± = uℓ ± φ .3he Killing vectors which preserve the form of line element (2.4) are ξ u = f, ξ φ = y − r ∂ φ f, ξ r = − r∂ φ y + ∂ φ f − r N ∂ φ f, (2.6)where f = ℓ Y + ( x + ) + Y − ( x − )) , y = 12 ( Y + ( x + ) − Y − ( x − )) , (2.7) Y + and Y − are arbitrary functions of x + and x − . At the leading term the generators L n = ξ (cid:0) Y + = − i exp( inx + ) , Y − = 0 (cid:1) , ¯ L n = ξ (cid:0) Y + = 0 , Y − = − i exp( inx − ) (cid:1) , (2.8)satisfy the algebra[ L m , L n ] = ( m − n ) L m + n , [ ¯ L m , ¯ L n ] = ( m − n ) ¯ L m + n , [ L m , ¯ L n ] = 0 . (2.9)In the level of conserved charges, the above algebra has a central extension with centralcharges c = ¯ c = 3 ℓ/ G [14].The flat limit of asymptotically AdS spacetimes written in the BMS gauge is well-defined.This is done by taking ℓ → ∞ limit of the AdS case. Precisely, we will take G/ℓ → G fixed. The starting point is the metric (2.4) and generic forms for functions M and N in (2.5). The Fourier expansion of functions Ξ and ¯Ξ areΞ( x + ) = ∞ X n = −∞ f n (cid:18) Gℓ (cid:19) e inx + , ¯Ξ( x − ) = ∞ X n = −∞ ¯ f n (cid:18) Gℓ (cid:19) e inx − . (2.10)We assume that functions f n and ¯ f n have well-defined behaviour at ℓ → ∞ . Thus one canwrite an expansion for them in terms of positive powers of G/ℓ . If we demand that M = lim Gℓ → M , N = lim Gℓ → N , (2.11)are well-defined, the coefficients of ( G/ℓ ) in f n and ¯ f − n must be the same. Then, it is notdifficult to check that M = θ ( φ ) , N = χ ( φ ) + u θ ′ ( φ ) , (2.12)where θ and χ are arbitrary functions. Thus the asymptotically flat solution is ds = M du − dudr + 2 N dudφ + r dφ . (2.13)4he line element (2.13) with condition (2.12) is exactly the generic result which one may findfor solutions of Einstein gravity (without cosmological constant) in the BMS gauge (2.2) [6].We should emphasize that we don’t use the modified Penrose limit of [14] which needs thescaling of the Newton constant G and results in an ambiguity in the definition of conservedcharges .Similarly, the Killing vectors of flat case can be given by taking G/ℓ → Y + and Y − . The final result is ξ u = F, ξ φ = Y − r ∂ φ F, ξ r = − r∂ φ Y + ∂ φ F − r N ∂ φ F, (2.14)where Y = lim Gℓ → y = Y ( φ ) , F = lim Gℓ → f = T ( φ ) + uY ′ ( φ ) . (2.15) Y and T are arbitrary functions. This result again coincides with direct calculations of [6].Using (2.14), one can find asymptotic symmetry group of flat spacetimes. The importantpoint which we should emphasize here is that the u-coordinate of BMS gauge for flat andAdS spacetimes characterizes different hyperspaces from the causal structure point of view.As mentioned earlier, ( u, φ ) for AdS determine spatial boundary of spacetime, however( u, φ ) for the flat case are the coordinates of future null infinity I + . This fact is easily seenby the definition of u = t − ℓ arctan rℓ in the AdS case which is transformed to u = t − r bytaking the ℓ → ∞ limit. Defining generators as L n = ξ ( Y = − i exp( inφ ) , T = 0) , M n = ξ ( Y = 0 , T = − i exp( inφ )) , (2.16)one can easily check that in the leading term we have[ L m , L n ] = ( m − n ) L m + n , [ L m , M n ] = ( m − n ) M m + n , [ M m , M n ] = 0 . (2.17)This algebra is known as BMS algebra [18]. The interesting point is that the algebras (2.9)and (2.17) are related by L n = L n − ¯ L − n , M n = Gℓ (cid:0) L n + ¯ L − n (cid:1) , (2.18)in the limit G/ℓ →
0. In fact the relation (2.18) is also applicable for (2.9) with central A new Grassmannian method for mapping gravity in AdS to gravity in flat spacetime was introducedrecently in [22]. c = ¯ c = 3 ℓ/ G , then thealgebra (2.17) has a non-zero central extension in the [ L m , M n ] part. Interestingly, the algebra (2.17) also appears in another branch of higher energy physics. Ifone considers a two dimensional CFT and takes its non-relativistic limit by contracting one ofthe coordinates, the contracted algebra is exactly (2.17) [7]. In this context the algebra (2.17)is called Galilean conformal algebra due to appearance of Galilean subalgebra. Because ofthis similarity, it was proposed in [1] that the dual theory for the asymptotically flat spacetimes is a non-relativistic conformal filed theory and this duality was coined as BMS/GCAcorrespondence.The parent CFT in the study of [1] is on the plane and the contracted coordinate was the x -coordinate. This assumptions dictated some restriction on the centrally extended algebrain such a way that in order to reproduce the known results of flat-space gravity the parentCFT must be non-unitary. This problem was reconsidered in the paper [2] and the correctway of constructing the dual theory of flat-space gravity was proposed.The main idea of [2] is that the CFT must be on a cylinder and time coordinate is theappropriate coordinate for contraction. The importance of cylinder versus plane is clear fromthe fact that in the bulk side the flat limit is only well-defined for the global AdS. Anotherway of introducing the contracted coordinate is that we should contract the non-compactcoordinate of the boundary theory.The states for the field theory with Galilean conformal symmetry is given by | h L , h M i where h L and h M are the labels of the states and are eigenvalues of L and M . There existsa notion of primary states in this theory and the representations are built by acting withraising operators L − n and M − n on these primary states. Similar to usual 2d CFTs, one canfind a Cardy-like formula for the entropy of states | h L , h M i . This calculation was done in [8]and the final entropy for large h L and h M is S = ln d ( h L , h M ) = π (cid:18) C LL r h M C LM + 2 h L r C LM h M (cid:19) . (3.1)where C LL and C LM are the central charges which appear respectively in the [ L m , L n ] and[ L m , M n ] parts of the algebra (2.17) . It was shown in [8] that (3.1) gives the correct entropy The leading correction to this formula was introduced recently in [23].
The fact that the flat space limit is well-defined in the BMS gauge tempts us to repeatall holographic calculations in these coordinates and take its flat limit. As a first step, wewould like to study the method of holographic renormalization and find correlation functionsof energy momentum tensor of the dual theory [17]. The warm up calculation is findingone-point function of energy momentum tensor which can be used in the bulk side for thecalculation of conserved charges. The simple method for this calculation is given by [16]and [17] which uses Brown and York’s proposal [19] for the definition of quasi-local stresstensor. According to [19], the Brown and York’s quasi-local energy-momentum tensor isgiven by T µν = 2 √− γ δSδγ µν , (4.1)where S = S grav ( γ µν ) is the gravitational action viewed as a functional of boundary metric γ µν . To be more precise, the gravitational action is S = 116 πG Z M d x √− g (cid:18) R − ℓ (cid:19) − πG Z ∂ M d x √− γ K + 18 πG S ct ( γ µν ) , (4.2)where the second term is known as Gibbons-Hawking term and is required for a well definedvariational principle. The S ct is the counterterm action that we must add in order to obtaina finite stress tensor. Moreover, K is trace of the extrinsic curvature of the boundary K = γ µν K µν = γ µν γ ρµ ∇ ρ n ν , (4.3)where γ µν = g µν − n µ n ν is the boundary metric and n ν is the outward pointing normal vectorto the boundary ∂ M . This is also the standard formula for the stress tensor of field theory with action S that lives on abackground with metric γ µν . T µν = − πG (cid:18) K µν − K γ µν + 2 √− γ δS ct δγ µν (cid:19) , (4.4)where the counterterm action in three dimensional bulk is [16, 17] S ct = − ℓ Z ∂ M √− γ. (4.5)Let us start from (2.4) and by using (2.5) write it in the light-cone coordinates x ± = ul ± φ , ds = r G (cid:26) − G dx + dx − + ℓ G r (cid:2) Ξ( dx + ) + ¯Ξ( dx − ) + (Ξ + ¯Ξ) dx + dx − (cid:3)(cid:27) − ℓ ( dx + + dx − ) dr. (4.6)Using (4.5) we have T ctµν = − πGℓ γ µν , (4.7)therefore (4.4) reduces to T rr = O ( 1 r ) , T r + = O ( 1 r ) , T r − = O ( 1 r ) , T + − = O ( 1 r ) T ++ = ℓ πG Ξ( x + ) + O ( 1 r ) , T −− = ℓ πG ¯Ξ( x − ) + O ( 1 r ) . (4.8)Thus we find the energy-momentum one-point function of dual CFT as h T ++ i = ℓ Ξ8 πG , h T −− i = ℓ ¯Ξ8 πG , h T + − i = 0 . (4.9)In the coordinate { u, φ } we have h T uu i = Ξ + ¯Ξ8 πGℓ = M πGℓ , h T uφ i = Ξ − ¯Ξ8 πG = N πGℓ , h T φφ i = ℓ (Ξ + ¯Ξ)8 πG = ℓ M πG . (4.10)Since the ℓ → ∞ limit of M and N is well-defined, it is obvious from (4.10) that definingenergy-momentum tensor of flat-space holography by taking the limit from the AdS case8irectly, does not make sense. However, the combinations similar to (2.18), i.e. T = lim G/ℓ → Gℓ ( T ++ + T −− ) , T = lim G/ℓ → ( T ++ − T −− ) , (4.11)are finite in the flat limit. We define the energy-momentum tensor of flat-space , ˜ T ij , by T = ( ˜ T ++ + ˜ T −− ) , T = ( ˜ T ++ − ˜ T −− ) , ˜ T + − = 0 , (4.12)where light-cone coordinates for flat spacetime, ˜ x ± , are given by ˜ x ± = u/G ± φ . Using (4.12)we see that ˜ T uu = M πG , ˜ T uφ = N πG , ˜ T φφ = M π , (4.13)are finite where M and N are given by (2.12). One can check that (4.13) is given by thefollowing scaling from the energy-momentum tensor of AdS case:˜ T uu = lim Gℓ → ℓG T uu , ˜ T uφ = lim Gℓ → ℓG T uφ , ˜ T φφ = lim Gℓ → Gℓ T φφ . (4.14)As mentioned in the section (3), the holographic dual of asymptotically flat spacetimesis a field theory with Galilean symmetry. Thus the energy-momentum tensor which wedefine in this section must be consistent with what one expects to see for such a field theory.The first observation is that the combination (4.11) is the same as what people define asthe energy-momentum tensor of non-relativistic CFTs which are deduced by contracting arelativistic one( see for example [20] ) .In fact the relation with an ultra-relativistic theory imposes that we change the conser-vation relation of energy-momentum tensor. To find the correct expression of conservationwhich our energy-momentum tensor must satisfy, let us indicate the coordinates of relativis-tic CFT on the cylinder with ( t, x ) where x is periodic and the radius of cylinder is absorbedin its periodicity. The light-cone coordinates are x ± = t ± x . The field theory with Galileansymmetry is deducible from the relativistic one by contracting time as t → ǫt and takingthe ǫ → T ij , we have T = ˜ T ++ + ˜ T −− = lim ǫ → ǫ ( T ++ + T −− ) , T = ˜ T ++ − ˜ T −− = lim ǫ → ( T ++ − T −− ) , (4.15) In the definition of [20] the ǫ factor (reminsent of our G/ℓ ) appears in the subtraction of components ofenergy-momentum tensor. In fact in that paper GCA arises by contracting x -coordinate which is differentfrom our case. , − ) in ˜ T ij are defined by contracted time. Now one can check that the relativisticconservation of energy momentum tensor i.e. ∇ i T ij = 0 reduces to ∂ t T = 0 , ∂ x T − ∂ t T = 0 , (4.16)where t is the contracted time. In the flat bulk side, u corresponds to t and Gφ correspondsto x . Thus using (4.13) we can translate the ultra relativistic conservation relation (4.16) interms of parameters of the bulk geometry as ∂ u M = 0 , ∂ φ M − ∂ u N = 0 , (4.17)which are satisfied by using values of M and N in (2.12). In fact (4.17) are exactly theequations which one finds by putting the ansatz (2.13) in the Einstein equations.The proposal for dual boundary theory of asymptotically flat spacetimes as an ultra rela-tivistic CFT, can be used for finding conserved charges. According to (4.6), the coordinates( t, x ) of parent CFT corresponds to ( Gℓ u, Gφ ) in the bulk side.It is obvious from this identi-fication that ultra-relativistic contraction t → ǫt corresponds to the flat limit Gℓ → t, ˜ x ) the coordinates of ultra-relativistic theory, they corre-spond to ( u, Gφ ) in the bulk side. We deduce that the ultra-relativistic field theory must liveon a flat spacetime ∂M with line element ds = − du + G dφ . Now we can use componentsof our energy-momentum tensor (4.13) for finding conserved charges of symmetry generators(2.14) by making use of Brown and York definition [19] Q ξ = Z Σ dφ √ σv µ ξ ν ˜ T µν , (4.18)where Σ is the spacelike surface of ∂M ( u = constant surface), σ ab is metric of Σ i.e. σ ab dx a dx b = G dφ and v µ is the unit timelike vector normal to Σ. Putting every thingstogether we find Q T,Y = 116 πG Z π dφ ( θT + 2 χY ) , (4.19)which is consistent with the result of [6]. 10 Conclusion
BMS/GCA is a correspondence between a theory of gravity in asymptotically flat spacetimesand a field theory with Galilean conformal symmetry. Finding a field theory with thissymmetry may seem somewhat difficult at first but a route to this problem was found in [2]where it was proposed that the field theory would be an ultra relativistic limit of a usualrelativistic CFT. It was also shown how the flat limit in the bulk corresponds to this ultrarelativistic contraction of the boundary CFT.In this paper we showed that this proposal can be used in defining the energy-momentumtensor of asymptotically flat spacetimes. Taking the flat limit from energy-momentum tensorof AdS case is not trivial in general. However, on the boundary side we know the relationbetween generators of conformal symmetry and GCA. This gives a good hint for definingone point function of energy-momentum tensor of Galilean CFT. In this paper we usedsuch a relation and found reasonable expressions for the energy-momentum tensor of flatgravity. As expected, the corresponding energy-momentum tensor satisfies a conservationrelation which is the ultra-relativistic version of relativistic equations. The fact that theseconservation equations are similar to Einstein equations shows that our definition in the bulkside is consistent. We also used our energy-momentum tensor to calculate conserved chargesof asymptotically flat spacetimes. The results are consistent with other methods. The pointwhich we used in this calculation is that we implicitly assumed that the ultra-relativistictheory lives on a cylinder rather than null infinity. In other words, if we indicate metric ofboundary by ds = − dt + dx for the relativistic theory and contract time as t = ǫt ′ , thenon-relativistic theory lives on a hyperspace with line element ds = − dt ′ + dx .We believe that this work is extendible to higher dimensions. For example one can findan energy-momentum tensor for producing mass and angular momentum of AdS-Kerr blackholes. The point is that one should write it in the BMS gauge and take its limit. Ourlesson from three dimensional case is that in order to find the energy-momentum tensor inthe context of flat holography one should start from BMS gauge and define the flat limit byconstructing a dimensionless parameter from ℓ and G and scale all components of T ij writtenin the BMS-coordinate in such a way that ℓ disappears. In fact this way of constructingenergy-momentum tensor has roots in the dual ultra-relativistic conformal field theory. Wehope to address this problem in future work.11 cknowledgement The authors would like to specially thank A. E. Mosaffa and A. Bagchi for their commentson the manuscript and revised version. We also would like to thank M. Alishahiha, D.Allahbakhshi and A. Vahedi for valuable discussions and useful comments during the courseof this work.
References [1] A. Bagchi, “Correspondence between Asymptotically Flat Spacetimes and NonrelativisticConformal Field Theories,” Phys. Rev. Lett. , 171601 (2010).A. Bagchi, “The BMS/GCA correspondence,” arXiv:1006.3354 [hep-th].[2] A. Bagchi and R. Fareghbal, “BMS/GCA Redux: Towards Flatspace Holography fromNon-Relativistic Symmetries,” JHEP , 092 (2012) [arXiv:1203.5795 [hep-th]].[3] H. Bondi, M. G. van der Burg, and A. W. Metzner, “Gravitational waves in generalrelativity. 7. Waves from axisymmetric isolated systems,”
Proc. Roy. Soc. Lond. A (1962) 21.R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flatspace-times,”
Proc. Roy. Soc. Lond. A (1962) 103.R. K. Sachs, “Asymptotic symmetries in gravitational theory,”
Phys. Rev. (1962)2851.[4] G. Barnich and G. Compere, “Classical central extension for asymptotic symmetries atnull infinity in three spacetime dimensions,” Class. Quant. Grav. , F15 (2007) [Erratum-ibid. , 3139 (2007)] [arXiv:gr-qc/0610130].[5] G. Barnich and C. Troessaert, “Symmetries of asymptotically flat 4 dimensional space-times at null infinity revisited,” arXiv:0909.2617 [gr-qc].[6] G. Barnich and C. Troessaert, “Aspects of the BMS/CFT correspondence,” JHEP ,062 (2010) [arXiv:1001.1541 [hep-th]].[7] A. Bagchi and R. Gopakumar, “Galilean Conformal Algebras and AdS/CFT,” JHEP , 037 (2009) [arXiv:0902.1385 [hep-th]]12. Alishahiha, A. Davody and A. Vahedi, “On AdS/CFT of Galilean Conformal FieldTheories,” JHEP , 022 (2009) [arXiv:0903.3953 [hep-th]].A. Bagchi and I. Mandal, “On Representations and Correlation Functions of GalileanConformal Algebras,” Phys. Lett. B , 393 (2009) [arXiv:0903.4524 [hep-th]]A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, “GCA in 2d,” JHEP , 004 (2010)[arXiv:0912.1090 [hep-th]].[8] A. Bagchi, S. Detournay, R. Fareghbal and J. Simon, “Holography of 3d Flat Cosmolog-ical Horizons,” Phys. Rev. Lett. , 141302 (2013) [arXiv:1208.4372 [hep-th]].[9] L. Cornalba and M. S. Costa, “A New cosmological scenario in string theory,” Phys.Rev. D , 066001 (2002) [hep-th/0203031].[10] A. Bagchi, S. Detournay, D. Grumiller and J. Simon, “Cosmic evolution from phase tran-sition of 3-dimensional flat space,” Phys. Rev. Lett. , 181301 (2013) [arXiv:1305.2919[hep-th]].[11] A. Bagchi, S. Detournay and D. Grumiller, “Flat-Space Chiral Gravity,” Phys. Rev.Lett. , 151301 (2012) [arXiv:1208.1658 [hep-th]].[12] A. Bagchi, “Tensionless Strings and Galilean Conformal Algebra,” JHEP , 141(2013) [arXiv:1303.0291 [hep-th]].[13] H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel, “Higher spin theory in3-dimensional flat space,” Phys. Rev. Lett. (2013) 121603 [arXiv:1307.4768 [hep-th]].[14] G. Barnich, A. Gomberoff and H. A. Gonzalez, “The Flat limit of three dimen-sional asymptotically anti-de Sitter spacetimes,” Phys. Rev. D , 024020 (2012)[arXiv:1204.3288 [gr-qc]].[15] J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization ofAsymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun. Math.Phys. , 207 (1986).[16] V. Balasubramanian and P. Kraus, “A Stress tensor for Anti-de Sitter gravity,” Com-mun. Math. Phys. , 413 (1999) [hep-th/9902121].[17] S. de Haro, S. N. Solodukhin and K. Skenderis, “Holographic reconstruction of space-time and renormalization in the AdS / CFT correspondence,” Commun. Math. Phys. ,595 (2001) [hep-th/0002230]. 1318] A. Ashtekar, J. Bicak and B. G. Schmidt, “Asymptotic structure of symmetry reducedgeneral relativity,” Phys. Rev. D , 669 (1997) [gr-qc/9608042].[19] J. D. Brown and J. W. York, Jr., “Quasilocal energy and conserved charges derivedfrom the gravitational action,” Phys. Rev. D , 1407 (1993) [gr-qc/9209012].[20] A. Bagchi, “Topologically Massive Gravity and Galilean Conformal Algebra: A Studyof Correlation Functions,” JHEP , 091 (2011) [arXiv:1012.3316 [hep-th]].[21] R. N. C. Costa, “Aspects of the zero Λ limit in the AdS/CFT correspondence,”arXiv:1311.7339 [hep-th].[22] C. Krishnan, A. Raju and S. Roy, “A Grassmann Path From AdS3