Enhanced asymptotic BMS_3 algebra of the flat spacetime solutions of generalized minimal massive gravity
aa r X i v : . [ h e p - t h ] F e b Enhanced asymptotic
BM S algebra of theflat spacetime solutions of generalizedminimal massive gravity M. R. Setare , H. Adami Department of Science, University of Kurdistan, Sanandaj, Iran.
Abstract
We apply the new fall of conditions presented in the paper [1] onasymptotically flat spacetime solutions of Chern-Simons-like theories ofgravity. We show that the considered fall of conditions asymptoticallysolve equations of motion of generalized minimal massive gravity. Wedemonstrate that there exist two type of solutions, one of those is trivialand the others are non-trivial. By looking at non-trivial solutions, forasymptotically flat spacetimes in the generalized minimal massive gravity,in contrast to Einstein gravity, cosmological parameter can be non-zero.We obtain the conserved charges of the asymptotically flat spacetimes ingeneralized minimal massive gravity, and by introducing Fourier modes weshow that the asymptotic symmetry algebra is a semidirect product of a
BM S algebra and two U (1) current algebras. Also we verify that the BM S algebra can be obtained by a contraction of the AdS asymptoticsymmetry algebra when the AdS radius tends to infinity in the flat-spacelimit. Finally we find energy, angular momentum and entropy for aparticular case and deduce that these quantities satisfy the first law of flatspace cosmologies. It is well known that the group of asymptotic symmetries of asymptoticallyflat space-times at future null infinity is the BMS group [2, 3, 4]. The BMSsymmetry algebra in n space-time dimension consists of the semi-direct sumof the conformal Killing vectors of a ( n − E-mail: [email protected] E-mail: [email protected]
1o propagating physical degrees of freedom [19, 20]. So choosing appropri-ate conditions at the boundary is crucial in this theory. Depending on thechosen boundary conditions, this theory can lead to completely differentboundary theories. Recently Detournay and Riegler have introduced a newasymptotic boundary conditions for pure Einstein gravity in 2 + 1 dimen-sions [1]. In fact these boundary conditions are the flat space counterpartof the enhanced asymptotic symmetry algebra of
AdS spacetimes whichhave been introduced by Troessaert previously in [21]. They have shownthat the resulting asymptotic symmetry algebra is generated by a BM S algebra and two affine U (1) current algebras. Then they have applied theirboundary conditions to Topologically Massive Gravity (TMG) [22] and haveshown that the presence of the gravitational Chern-Simons term lead to thecentral extensions of the asymptotic symmetry algebra. In the other handTMG has a bulk-boundary unitarity conflict. Either the bulk or the bound-ary theory is non-unitary, so there is a clash between the positivity of thetwo Brown-Henneaux boundary central charges and the bulk energies. In or-der to overcome on this problem, Bergshoeff et.al, have introduced MinimalMassive Gravity (MMG) [23], which has the same minimal local structureas TMG. The MMG model has the same gravitational degree of freedom asthe TMG. It seems that the single massive degree of freedom of MMG isunitary in the bulk and gives rise to a unitary CFT on the boundary. Fol-lowing this work Generalized Minimal Massive Gravity (GMMG) introduced[24]. This model is realized by adding higher-derivative deformation termto the Lagrangian of MMG. As has been shown in [24], GMMG also avoidsthe aforementioned “bulk-boundary unitarity clash”. Hamiltonian analysisshow that the GMMG model has no Boulware-Deser ghosts and this modelpropagate only two physical modes. So this model is viable candidate forsemi-classical limit of a unitary quantum 3 D massive gravity.In this paper we extend the work of [1] and apply the boundary condi-tions introduced there to the Chern-Simons-like theories of gravity (CSLTG)[25, 26], and as a example we consider the GMMG model. It is one of theinteresting extensions of [1] which have been mentioned in the conclusion of[1]. 2 Quasi-local conserved charges in Chern-Simons-like theories of gravity
The Lagrangian 3-form of the Chern-Simons-like theories of gravity (CSLTG)is given by [25] L = 12 ˜ g rs a r · da s + 16 ˜ f rst a r · a s × a t . (1)In the above Lagrangian a ra = a raµ dx µ are Lorentz vector valued one-formswhere, r = 1 , ..., N and a indices refer to flavour and Lorentz indices, respec-tively. We should mention that, here, the wedge products of Lorentz-vectorvalued one-form fields are implicit. Also, ˜ g rs is a symmetric constant metricon the flavour space and ˜ f rst is a totally symmetric ”flavour tensor” whichare interpreted as the coupling constants. We use a 3D-vector algebra nota-tion for Lorentz vectors in which contractions with η ab and ε abc are denotedby dots and crosses, respectively . It is worth saying that a ra is a collec-tion of the dreibein e a , the dualized spin-connection ω a , the auxiliary field h aµ = e aν h νµ and so on. Also for all interesting CSLTG we have ˜ f ωrs = ˜ g rs [26].The total variation of a ra due to a diffeomorphism generator ξ is [27] δ ξ a ra = L ξ a ra − δ rω dχ aξ , (2)where χ aξ = ε abc λ bcξ and λ bcξ is generator of the Lorentz gauge transfor-mations SO (2 , δ rs denotes the ordinary Kronecker delta and theLorentz-Lie derivative along a vector field ξ is denoted by L ξ . We assumethat ξ may be a function of dynamical fields. In the paper [28], we haveshown that quasi-local conserved charge perturbation associated with a fielddependent vector field ξ is given by ˆ δQ ( ξ ) = 18 πG Z Σ (˜ g rs i ξ a r − ˜ g ωs χ ξ ) · ˆ δa s , (3)where G denotes the Newtonian gravitational constant and Σ is a spacelikecodimension two surface. We can take an integration from (3) over theone-parameter path on the solution space [29, 30] and then we find that Q ( ξ ) = 18 πG Z ds Z Σ (˜ g rs i ξ a r − ˜ g ωs χ ξ ) · ˆ δa s , (4) Here we consider the notation used in [25]. We denote variation with respect to dynamical fields by ˆ δ . S = 14 G Z Horizon dφ √ g φφ ˜ g ωs a sφφ , (5)where φ is angular coordinate and g φφ denotes the φ - φ component of space-time metric g µν . Generalized minimal massive gravity (GMMG) is an example of the Chern-Simons-like theories of gravity [24]. In the GMMG, there are four flavoursof one-form, a r = { e, ω, h, f } , and the non-zero components of the flavourmetric and the flavour tensor are˜ g eω = − σ, ˜ g eh = 1 , ˜ g ωf = − m , ˜ g ωω = 1 µ , ˜ f eωω = − σ, ˜ f ehω = 1 , ˜ f fωω = − m , ˜ f ωωω = 1 µ , ˜ f eff = − m , ˜ f eee = Λ , ˜ f ehh = α. (6)where σ , Λ , µ , m and α are a sign, cosmological parameter with dimensionof mass squared, mass parameter of Lorentz Chern-Simons term, mass pa-rameter of New Massive Gravity [32] term and a dimensionless parameter,respectively. The equations of motion of GMMG are [24, 33] − σR (Ω)+ (1+ σα ) D (Ω) h − α (1+ σα ) h × h + Λ e × e − m f × f = 0 , (7) − e × f + µ (1 + σα ) e × h − µm D (Ω) f + µαm h × f = 0 , (8) R (Ω) − αD (Ω) h + 12 α h × h + e × f = 0 , (9) T (Ω) = 0 , (10)where Ω = ω − αh (11)4s ordinary torsion-free dualized spin-connection. Also, R (Ω) = d Ω + Ω × Ω is curvature 2-form, T (Ω) = D (Ω) e is torsion 2-form, and D (Ω) de-notes exterior covariant derivative with respect to torsion-free dualized spin-connection. In this section, we consider the following fall of conditions for asymptoticallyflat spacetimes in 3D g uu = M ( φ ) + O ( r − ) g ur = − e A ( φ ) + O ( r − ) g uφ = N ( u, φ ) + O ( r − ) g rr = O ( r − ) g rφ = − e A ( φ ) E ( u, φ ) + O ( r − ) g φφ = e A ( φ ) r + E ( u, φ ) [2 N ( u, φ ) − M ( φ ) E ( u, φ )] + O ( r − ) (12)with N ( u, φ ) = L ( φ ) + u ∂ φ M ( φ ) , E ( u, φ ) = B ( φ ) + u∂ φ A ( φ ) (13)which have been introduced in the paper [1]. In the above metric M ( φ ), A ( φ ), B ( φ ) and L ( φ ) are arbitrary functions. The metric, under transfor-mation generated by vector field ξ , transforms as δ ξ g µν = £ ξ g µν . The vari-ation generated by the following Killing vector field preserves the boundaryconditions ξ u = α ( u, φ ) − r e −A ( φ ) E ( u, φ ) β ( u, φ ) + O ( r − ) ,ξ r = rX ( φ ) + e −A ( φ ) [ E ( u, φ ) ∂ φ X ( φ ) − ∂ φ β ( u, φ )] , + 1 r e − A ( φ ) β ( u, φ ) [ N ( u, φ ) − M ( φ ) E ( u, φ )] + O ( r − ) ,ξ φ = Y ( φ ) + 1 r e −A ( φ ) β ( u, φ ) + O ( r − ) , (14)with α ( u, φ ) = T ( φ ) + u∂ φ Y ( φ ) , β ( u, φ ) = Z ( φ ) + u∂ φ X ( φ ) , (15) Where £ ξ denotes usual Lie derivative along ξ . T ( φ ), X ( φ ), Y ( φ ) and Z ( φ ) are arbitrary functions of φ . Since ξ depends on the dynamical fields so we need to introduce a modified versionof the Lie brackets. Let’s consider a modified version of the Lie brackets [15](see also [34]) [ ξ , ξ ] = £ ξ ξ − δ ( g ) ξ ξ + δ ( g ) ξ ξ , (16)where ξ = ξ ( T , X , Y , Z ) and ξ = ξ ( T , X , Y , Z ). In the equation(16), δ ( g ) ξ ξ denotes the change induced in ξ due to the variation of metric δ ξ g µν = £ ξ g µν . By substituting Eq.(14) into Eq.(16), one finds[ ξ , ξ ] = ξ , (17)where ξ = ξ ( T , X , Y , Z ), with T = Y ∂ φ T − Y ∂ φ T + T ∂ φ Y − T ∂ φ Y ,X = Y ∂ φ X − Y ∂ φ X ,Y = Y ∂ φ Y − Y ∂ φ Y ,Z = Y ∂ φ Z − Y ∂ φ Z + T ∂ φ X − T ∂ φ X . (18)Under transformation generated by the Killing vector fields (14), the arbi-trary functions M ( φ ), A ( φ ), B ( φ ) and L ( φ ) , which have appeared in themetric, transform as δ ξ M ( φ ) = − ∂ φ X ( φ ) ∂ φ A ( φ ) + 2 ∂ φ Y ( φ ) M ( φ ) + Y ( φ ) ∂ φ M ( φ )+ 2 ∂ φ X ( φ ) , (19) δ ξ A ( φ ) = Y ( φ ) ∂ φ A ( φ ) + ∂ φ Y ( φ ) + X ( φ ) , (20) δ ξ B ( φ ) = T ( φ ) ∂ φ A ( φ ) + Y ( φ ) ∂ φ B ( φ ) + Z ( φ ) + ∂ φ T ( φ ) , (21) δ ξ L ( φ ) = ∂ φ T ( φ ) M ( φ ) + Y ( φ ) ∂ φ L ( φ ) + 2 ∂ φ Y ( φ ) L ( φ ) + 12 T ( φ ) ∂ φ M ( φ ) − ∂ φ Z ( φ ) ∂ φ A ( φ ) − ∂ φ X ( φ ) ∂ φ B ( φ ) + ∂ φ Z ( φ ) . (22)By introducing Fourier modes ξ ( T ) m = ξ ( e imφ , , , ,ξ ( X ) m = ξ (0 , e imφ , , ,ξ ( Y ) m = ξ (0 , , e imφ , ,ξ ( Z ) m = ξ (0 , , , e imφ ) , (23)6e will have i h ξ ( T ) m , ξ ( T ) n i = 0 , i h ξ ( T ) m , ξ ( Z ) n i = 0 , i h ξ ( X ) m , ξ ( X ) n i = 0 ,i h ξ ( X ) m , ξ ( Z ) n i = 0 , i h ξ ( Z ) m , ξ ( Z ) n i = 0 , i h ξ ( T ) m , ξ ( X ) n i = − nξ ( Z ) m + n ,i h ξ ( X ) m , ξ ( Y ) n i = mξ ( X ) m + n , i h ξ ( Y ) m , ξ ( Z ) n i = − nξ ( Z ) m + n ,i h ξ ( T ) m , ξ ( Y ) n i = ( m − n ) ξ ( T ) m + n , i h ξ ( Y ) m , ξ ( Y ) n i = ( m − n ) ξ ( Y ) m + n . (24)Now we introduce the following dreibein e u = r − r M ( φ ) + O ( r − ) , e r = 12 r e A ( φ ) + O ( r − ) ,e φ = r E ( u, φ ) − r [2 N ( u, φ ) − M ( φ ) E ( u, φ )] + O ( r − ) ,e u = O ( r − ) , e r = O ( r − ) , e φ = re A ( φ ) + O ( r − ) ,e u = − r − r M ( φ ) + O ( r − ) , e r = 12 r e A ( φ ) + O ( r − ) ,e φ = − r E ( u, φ ) − r [2 N ( u, φ ) − M ( φ ) E ( u, φ )] + O ( r − ) . (25)One can use the equation g µν = η ab e aµ e bν , where η ab = diag( − , ,
1) is justthe Minkowski metric, to obtain metric (12) from the dreibein (25). Sincethe Riemann curvature tensor R αβµν is related to the torsion-free curvature2-form as R a (Ω) = 12 e aλ ǫ λαβ R αβµν dx µ ∧ dx ν , (26)therefore, for the given spacetime, we have R (Ω) = O ( r − ) . (27)Now, in the context of GMMG, we consider following ansatz f = F e, h = He, (28)where F and H are just two constant parameters. By substituting Eq.(27)and Eq.(28) into the equations of motion of GMMG (7)-(10), we find thatΛ = α (1 + ασ ) H + F m ,F = µ (1 + ασ ) H + µαm HF,F + 12 α H = 0 . (29)7hus, the metric (12) solves equations of motion of GMMG asymptoticallyif Λ , F and H satisfy equations (29). Equations (29) admit the followingtrivial solution Λ = F = H = 0 . (30)Now we consider the case in which α = 0. In that case we have two non-trivial solutions H ± = m µα ± (cid:20) m µ α + m α (1 + ασ ) (cid:21) ,F ± = − α H ± , Λ ± = αH ± (cid:20) (1 + ασ ) + α m H ± (cid:21) . (31)We mention that, if one consider the case in which α = 0 then one againgets the trivial solution (30). Thus, for asymptotically flat spacetimes inthe GMMG model, in contrast to Einstein gravity, cosmological parametercould be non-zero. One can use equations (6), (28), (29) to simplify expression (3) for thatquasi-local conserved charge perturbation associated with a field dependentvector field ξ in the GMMG model for asymptotically flat spacetimesˆ δQ ( ξ ) = 18 πG lim r →∞ Z π (cid:26) − (cid:18) σ + αHµ + Fm (cid:19) h i ξ e · ˆ δ Ω φ + ( i ξ Ω − χ ξ ) · ˆ δe φ i + 1 µ ( i ξ Ω − χ ξ ) · ˆ δ Ω φ (cid:27) dφ. (32)By demanding that the Lie-Lorentz derivative of e a becomes zero explicitlywhen ξ is a Killing vector field, we find the following expression for χ ξ [31, 35] χ aξ = i ξ ω a + 12 ε abc e νb ( i ξ T c ) ν + 12 ε abc e bµ e cν ∇ µ ξ ν , (33)and one can show that this expression can be rewritten as [36] i ξ Ω − χ ξ = − ε abc e bµ e cν ∇ µ ξ ν . (34)8lso we remind that the torsion free spin-connection is given byΩ aµ = 12 ε abc e αb ∇ µ e cα . (35)As we mentioned in section 2, one can take an integration from (32) overthe one-parameter path on the solution space to find the conserved chargecorresponds to the Killing vector field (14) for dreibein (25), then Q ( T, X, Y, Z ) = M ( T ) + J ( X ) + L ( Y ) + P ( Z ) , (36)with M ( T ) = − πG (cid:18) σ + αHµ + Fm (cid:19) Z π T ( φ ) M ( φ ) dφ, (37) J ( X ) = 18 πG Z π X ( φ ) (cid:20)(cid:18) σ + αHµ + Fm (cid:19) ∂ φ B ( φ ) − µ ∂ φ A ( φ ) (cid:21) dφ, (38) L ( Y ) = − πG Z π Y ( φ ) (cid:26) (cid:18) σ + αHµ + Fm (cid:19) L ( φ ) − µ h M ( φ ) + ( ∂ φ A ( φ )) − ∂ φ A ( φ ) i(cid:27) dφ, (39) P ( Z ) = 18 πG (cid:18) σ + αHµ + Fm (cid:19) Z π Z ( φ ) ∂ φ A ( φ ) dφ. (40)The above surface charges display the universal property of 3D gravity thatthe space of solutions is dual to the asymptotic symmetry algebra. Thealgebra of conserved charges can be written as [37, 38] { Q ( ξ ) , Q ( ξ ) } = Q ([ ξ , ξ ]) + C ( ξ , ξ ) (41)where C ( ξ , ξ ) is the central extension term. Also, the left hand side of theequation (41) can be defined by { Q ( ξ ) , Q ( ξ ) } = ˆ δ ξ Q ( ξ ) . (42)Therefore one can find the central extension term by using the followingformula C ( ξ , ξ ) = ˆ δ ξ Q ( ξ ) − Q ([ ξ , ξ ]) . (43)9y substituting Eq.(17), Eqs.(19)-(22) and Eq.(36) into Eq.(43) we obtainthe central extension term C ( ξ , ξ ) = − πG (cid:18) σ + αHµ + Fm (cid:19) Z π (cid:26)(cid:0) T ∂ φ X − T ∂ φ X (cid:1) + (cid:0) Y ∂ φ Z − Y ∂ φ Z (cid:1) − ( X ∂ φ Z − X ∂ φ Z ) (cid:27) dφ + 116 πGµ Z π (cid:8)(cid:0) Y ∂ φ X − Y ∂ φ X (cid:1) − X ∂ φ X − Y ∂ φ Y (cid:9) dφ. (44)By introducing Fourier modes M m = Q ( e imφ , , ,
0) = M ( e imφ ) ,J m = Q (0 , e imφ , ,
0) = J ( e imφ ) ,L m = Q (0 , , e imφ ,
0) = L ( e imφ ) ,P m = Q (0 , , , e imφ ) = P ( e imφ ) , (45)we find that i { M m , M n } = 0 , i { M m , P n } = 0 , i { P m , P n } = 0 ,i { J m , J n } = k J nδ m + n, , i { J m , P n } = k P nδ m + n, ,i { M m , J n } = − nP m + n − ik P n δ m + n, ,i { J m , L n } = mJ m + n + ik J m δ m + n, ,i { L m , P n } = − nP m + n − ik P n δ m + n, ,i { M m , L n } = ( m − n ) M m + n ,i { L m , L n } = ( m − n ) L m + n − k J n δ m + n, , (46)where k P and k J are given as k P = − G (cid:18) σ + αHµ + Fm (cid:19) , k J = 18 Gµ . (47)Now we set ˆ M m ≡ M m , ˆ J m ≡ J m , ˆ L m ≡ L m and ˆ P m ≡ P m , also we replacethe Dirac brackets by commutators i { , } → [ , ], therefore we can rewrittenequations (46) as following[ ˜ L m , ˜ L n ] = ( m − n ) ˜ L m + n + c L m δ m + n, [ ˜ M m , ˜ L n ] = ( m − n ) ˜ M m + n + c M m δ m + n, , [ ˜ M m , ˜ M n ] = 0 , (48)10 ˜ M m , ˆ J n ] = − n ˆ P m + n , [ ˜ M m , ˆ P n ] = 0 , [ ˜ L m , ˆ J n ] = − n ˆ J m + n , [ ˜ L m , ˆ P n ] = − n ˆ P m + n , [ ˆ P m , ˆ P n ] = 0 , [ ˆ J m , ˆ J n ] = k J nδ m + n, , [ ˆ J m , ˆ P n ] = k p nδ m + n, , (49)with c L = 24 k J = 3 Gµ , c M = 12 k P = − G (cid:18) σ + αHµ + Fm (cid:19) , (50)where we have performed a shift as˜ M m = ˆ M m − im ˆ P m , ˜ L m = ˆ L m − im ˆ J m . (51)The resulting asymptotic symmetry algebra (48) and (49) is a semidirectproduct of a bms algebra ,with central charges c L and c M , and two u (1)current algebras [1]. If we set σ = − α = 0 and m → ∞ the algebra (48)and (49) will be reduced to the one presented in [1] for topologically massivegravity.The algebra among the asymptotic conserved charges of asymptoticallyAdS spacetimes in the context of GMMG is isomorphic to two copies ofthe Virasoro algebra [39] (cid:2) L ± m , L ± n (cid:3) = ( m − n ) L ± m + n + c ± m δ m + n, , (cid:2) L + m , L − n (cid:3) = 0 , (52)where c ± are central charges and they are given by c ± = − l G (cid:18) σ + αHµ + Fm ∓ µl (cid:19) . (53)The BM S algebra (48) can be obtained by a contraction of the AdS asymptotic symmetry algebra˜ L m = L + m − L −− m , ˜ M m = 1 l (cid:0) L + m + L −− m (cid:1) , (54)when the AdS radius tends to infinity in the flat-space limit [40, 41]. Thencorresponding BM S central charges in the algebra (48) become c M = lim l →∞ l ( c + + c − ) , c L = lim l →∞ ( c + − c − ) , (55)and it can be readily checked. In Eq.(53), l is AdS radius. Thermodynamics
We know that energy and angular momentum are conserved charges corre-spond to two asymptotic Killing vector fields ∂ u and − ∂ φ , respectively. Itcan be seen that ∂ u and − ∂ φ are asymptotic Killing vector fields admitted byspactimes which behave asymptotically like (12) when we have M ( φ ) = M , A ( φ ) = A , L ( φ ) = L and B ( φ ) = B , where M , A , L and B are constants.Hence, with this assumption, one can use Eq.(32) to find energy and angularmomentum as following E = Q ( ∂ u ) = − G (cid:18) σ + αHµ + Fm (cid:19) M , (56) J = Q ( − ∂ φ ) = 14 G (cid:20)(cid:18) σ + αHµ + Fm (cid:19) L − µ M (cid:21) , (57)respectively. We know that cosmological horizon is located at where therewe have g uu g φφ − ( g uφ ) = 0 , (58)and then one can deduced that cosmological horizon is located at r H = e −A √M |L − MB| . (59)One can associate an angular velocity to the cosmological horizon asΩ H = − g uφ g φφ (cid:12)(cid:12)(cid:12)(cid:12) r = r H = − ML . (60)Since the norm of Killing vector ζ = ∂ u + Ω H ∂ φ vanishes on the cosmolog-ical horizon, it seems sensible that one can associate a temperature to thecosmological horizon as T H = κ H π (61)where κ H = (cid:20) − ∇ µ ζ ν ∇ µ ζ ν (cid:21) r = r H , (62)therefore we have T H = M π L . (63)12s we have mentioned in section 2, one can obtain entropy by using Eq.(5).Thus, we use Eq.(28) and Eq.(29) to simplify Eq.(5) for asymptotically flatspacetimes (12) in the context of GMMG S = 14 G Z r = r H dφ √ g φφ (cid:20) − (cid:18) σ + αHµ + Fm (cid:19) g φφ + 12 µ Ω φφ (cid:21) . (64)Since on the cosmological horizon we have g φφ (cid:12)(cid:12) r = r H = L M , Ω φφ (cid:12)(cid:12) r = r H = L , (65)then Eq.(64) becomes S = π G (cid:20) − (cid:18) σ + αHµ + Fm (cid:19) L√M + 1 µ √M (cid:21) . (66)One can easily check that the quantities appear in Eq.(56), Eq.(57), Eq.(60),Eq.(63) and Eq.(66) satisfy the first law of thermodynamics of flat spacecosmologies [42] which is given by δ E = − T H δ S + Ω H δ J . (67)It is easy to see that the obtained results (56), (57) and (66) will be reducedto the corresponding results in topologically massive gravity case [1] whenwe set σ = − α = 0 and m → ∞ . In this paper we have applied the fall of conditions presented in the paper[1] on asymptotically flat spacetime solutions of Chern-Simons-like theoriesof gravity. In section 2 we have reviewed the method of obtaining quasi-local conserved charges in Chern-Simons-like theories of gravity. In section3 we have considered generalized minimal massive gravity model as an ex-ample of Chern-Simons-like theories of gravity. The equations of motion ofGMMG are given by (7)-(10). In section 4, we have considered the fall ofconditions (12) for the asymptotically flat spacetimes in three dimensions.The considered fall of conditions have preserved by the variation generatedby the asymptotic Killing vector field (14). Since the asymptotic Killingvector field (14) depends on the dynamical fields, the algebra among theasymptotic Killing vectors is closed in the modified version of the Lie brack-ets (16). We have considered the ansatz (28) and hence we have showed13hat the fall of conditions (12) asymptotically solve equations of motion ofGMMG. We have obtained two types of solutions, one of those is trivial (30)and the others are non-trivial (31). By looking at non-trivial solutions (31),one can see that, for asymptotically flat spacetimes in the GMMG model,in contrast to Einstein gravity, cosmological parameter could be non-zero.In section 5, we have calculated conserved charge (36), of asymptoticallyflat spacetimes, corresponds to the asymptotic Killing vector field (14). Byintroducing Fourier modes (45), we showed that asymptotic symmetry al-gebra, (see Eq.(48) and Eq.(49)) is a semidirect product of a bms algebra,with central charges c L and c M , and two U (1) current algebras. Also weverified that the BM S algebra (48) can be obtained by a contraction ofthe AdS asymptotic symmetry algebra (52) when the AdS radius tends toinfinity in the flat-space limit. In section 6, we found energy, angular mo-mentum and entropy for a particular case and we showed that they satisfythe first law of flat space cosmologies. All the obtained results in this paperwill be reduced to the corresponding results in topologically massive gravitycase [1] when we set σ = − α = 0 and m → ∞ . M. R. Setare thanks Max Riegler and Blagoje Oblak for reading the manuscript,helpful comments and discussions.
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