Homogeneous Solutions of Minimal Massive 3D Gravity
aa r X i v : . [ h e p - t h ] M a r Homogeneous Solutions of Minimal Massive 3D Gravity
Jumageldi Charyyev Courant Institute of Mathematical Sciences,New York University, NY 10012, USANihat Sadik Deger Department of Mathematics, Bogazici University,Bebek, 34342, Istanbul-TurkeyFeza Gursey Center for Physics and Mathematics,Bogazici University, Kandilli, 34684, Istanbul-Turkey
ABSTRACT
In this paper we systematically construct simply transitive homogeneous spacetimesolutions of the three-dimensional Minimal Massive Gravity (MMG) model. In additionto those that have analogs in Topologically Massive Gravity, such as warped AdS and pp-waves, there are several solutions genuine to MMG. Among them, there is a stationaryLifshitz metric with the dynamical exponent z = − [email protected] [email protected] ontents SL (2 , R ) z ¯ z − type metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 SU (2)
85 Solutions on A ∞
96 Solutions on A B -type metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.2 B -type metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.3 B -type metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.4 B -type metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 ISO (2; θ ) B -type metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127.2 B -type metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 ISO (1 , θ ) B -type metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.2 B -type metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Minimal Massive Gravity (MMG) is a pure 3-dimensional gravity model proposed in[1], which attracted much attention during the last three years. It is an extension ofanother widely studied theory known as ‘Topologically Massive Gravity’ (TMG) [2] with1 particular curvature squared term in the field equation. It is ‘minimal’ in the sense thatthere is only one propagating spin-2 mode in the bulk like TMG. However, unlike TMG,it avoids the bulk-boundary unitarity clash since in a certain range of its parameters it ispossible to have central charge of the dual CFT and energy of the bulk graviton positivesimultaneously [1, 3] (however, see [4]) which makes it a potentially useful toy model forunderstanding quantum gravity in 4-dimensions.In this paper we make a systematic investigation of homogeneous spacetime solutions ofMMG with Lorentzian signature and obtain a large number of new ones. We will focus onsimply transitive Lie groups where any two points can be related by an isometry and sta-bility (isotropy) group of any point is trivial. Since a homogeneous (pseudo)Riemannianmanifold M has the form of a quotient G/H , where G is its group of isometries, whichis a Lie group, and H is a closed subgroup of G, this means that we take H to be justthe identity. In this case M and G can be identified and considering left action of such aLie group on itself one can construct its left-invariant metric up to automorphisms usingleft-invariant 1-forms. This results in a metric with constant coefficients and all curva-ture calculations become algebraic. In three dimensions, classification of Lie algebras wasdone by Bianchi [5] and one can systematically check whether corresponding metrics aresolutions of a particular 3-dimensional model. This method was successfully applied toTMG with vanishing cosmological constant in [6] and [7], and more recently for non-zerocosmological constant in [8]. This was also carried out in [9]-[10] (see also [11, 12]) foranother extension of TMG called ‘New (or General) Massive Gravity’ (NMG) [13].Since MMG is closely related to the (cosmological) TMG we will follow analysis in [8]closely which will make identification of most of the solutions we obtain straightforward.In [14] it was shown that solutions of TMG which have Segre-Petrov-types N and D arealso solutions of MMG after a redefinition of parameters. We find that, as should beexpected, MMG inherits such homogeneous solutions from TMG which include warped(A)dS and ( A ) dS × S solutions obtained in [15] and pp-wave spacetimes [16]. Some ofthe remaining solutions turn out to be solutions of TMG as well only if the cosmologicalconstant is zero which implies that their scalar curvatures vanish, which is not the casein MMG. Finally, we show that there are several homogeneous solutions that are genuineto MMG one of which is a stationary Lifshitz spacetime (86) with a dynamical exponent z = − In this section we will give a brief introduction to MMG [1] and describe our method forconstructing its homogeneous solutions. The theory is defined by the field equation G µν + ag µν + bC µν + cJ µν = 0 , (1)where G µν is the Einstein tensor and the Cotton tensor C µν , which is symmetric, tracelessand covariantly conserved, is related to the Schouten tensor S σν as C µν ≡ √− g ε µρσ ∇ ρ S σν , S σν ≡ R σν − Rg σν , (2)with ε = +1. The J -tensor is given as J µν ≡ R µρ R νρ − R µν R − g µν ( R ρσ R ρσ − R ) . (3)It is not covariantly conserved, but instead one finds [1]: √− g ∇ µ J µν = ε νρσ S τρ C στ , (4)which is not automatically zero. It follows that the MMG field equation (1) cannot bederived from an action that contains only the metric field [1]. However, for any solutionof the field equation (1) one can show that the right hand side of (4) vanishes whichestablishes the consistency of the model in a novel way. Moreover, it is still possible tocouple matter [15] and calculate charges of its solutions [21].Finally, the coefficients a , b and c in terms of physical parameters are a = ¯Λ ¯ σ , b = 1 µ ¯ σ , c = γµ ¯ σ . (5)When γ = 0 (i.e., c = 0) the model reduces to the (cosmological) TMG model [2], wheresuch solutions were studied before [6] -[10].There are two special points in the parameter space of the MMG theory [1]. The firstis called the ’chiral point’ at which one of the central charges vanish and is given by:¯ σ + γ σ − γ ¯Λ µ ) ± s ¯ σ − γ ¯Λ µ = 0 or 1 + c b (1 − ac ) ± √ − ac = 0 . (6)The second one is called the ’merger point’ where two possible values of the cosmologicalconstant coincide: ¯Λ = µ ¯ σ γ or ac = 1 . (7)3n order to find homogeneous solutions of MMG we will follow the method of [8] thatwas successfully used for the (cosmological) TMG [2] model which can be summarizedas follows: First a Lie algebra basis is fixed for each three-dimensional Lie algebra g which induces left-invariant Maurer-Cartan 1-forms. A left-invariant metric for the Liegroup at the identity is identified by a non-degenerate metric on the Lie algebra upto automorphism group of this Lie algebra. Starting from an arbitrary left-invariantmetric on the algebra, it is put into a simple form using automorphisms. The metricis expressed in terms of left-invariant 1-forms with constant coefficients which impliesthat all curvature calculations, and hence the MMG field equation (1), become algebraic.This method is different but equivalent to the one used in [6, 7, 9, 10] where instead offixing the Lie algebra basis, an orthonormal frame is chosen [22]. Then, SO (1 ,
2) Lorentztransformations are used to simplify the structure constants. We prefer the strategy of [8]since it enables us to compare our solutions with those of (cosmological) TMG [2] directly.Moreover, geometric identification of common solutions become trivial.Instead of solving algebraic equations for the constants in the metric { u, v, w, ... } in terms of the parameters { a, b, c } of the MMG theory (5) it is more convenient andilluminating to display the parameters in the theory in terms of the parameters of themetric. This reduces to solving a system of linear equations A · abc = V (8)for { a, b, c } where A is a matrix is of the dimension k × V is a k × k = 3 , k is determined by the number of independent components ofthe field equation (1). The rank of the matrix A can be at most three. When the rank of A is three, the linear equation (8) has a unique solution, provided that the solution exists.If the solution exists, the cases when the rank of A is less than three should be consideredseparately as in such situations new solutions may arise. If A is of the dimension 3 × A is enough to determine when the rank of A is lessthan three. When A is not a square matrix and for the cases when (8) does not have ageneral solution, a more careful analysis is required. For example, it may happen that fora particular relation among the parameters of the metric the system becomes consistent.To identify distinct Lie algebras one must determine sets of structure constants whichcannot be related by linear transformations. For three-dimensional Lie algebras thisclassification was done by Bianchi [5] but usually presented in a more modern approachdescribed in [23] (see also [24].) Besides the abelian R and the two familiar algebras sl and su , we also have the Lie algebras a ∞ and a , and two continuous families of Liealgebras: iso (1 , θ ) and iso (2; θ ) where parameter θ varies in (0 , π ]. In the first, θ values { , π } are special and should be considered separately which in total leads to 9 Bianchiclasses. Finally, iso (1 ,
1; 0) and iso (2; 0) are isomorphic to each other.We now begin constructing the homogeneous spacetime solutions of MMG goingthrough the above list of algebras. We assume that metrics are Lorentzian with mostlyplus signature and follow conventions and terminology of [8] to which we refer for details.4
Solutions on SL (2 , R ) For the Lie algebra sl of SL (2 , R ) a basis { τ , τ , τ } can be fixed with[ τ , τ ] = τ , [ τ , τ ] = τ , [ τ , τ ] = τ . (9)Let θ a be a dual basis of τ a . Elements of SL (2 , R ) can be parametrized by a grouprepresentative as (see for example [25]) V ( x ) = e t ( τ + τ ) e στ e ζτ . (10)It follows that the Maurer-Cartan one-forms are V − d V = ( e σ cosh ζ dt − sinh ζ dσ ) τ + (cosh ζ dσ − e σ sinh ζ dt ) τ + ( dζ + e σ dt ) τ . (11)There are 4 classes of left-invariant metrics on SL (2 , R ) that are given below (see [8]). g = uθ θ + vθ θ + wθ θ , (12)where uw < v > − u = v = w corresponds to the round AdS written as Hopf fibration over AdS spacetime in thePoincar´e coordinates. The general line element is ds = e σ ( u cosh ζ + v sinh ζ ) dt + ( u sinh ζ + v cosh ζ ) dσ − u + v ) e σ cosh ζ sinh ζ dtdσ + w ( dζ + e σ dt ) . (13)The coefficients a , b , c and the scalar curvature R in terms of u , v , and w are a = 1 Q · [( u + v + w ) − vw ] uvw ,b = − Q · √− uvw [ u − ( v − w ) ]( u + v + w ) ,c = 1 Q · uvw [( u + v + w ) − vw ] ,R = − ( u + v + w ) − vw uvw = − c Q uvw ) = − ( a Q ) / ( uvw ) / , (14)where Q = [( u + v + w ) − vw ] + 8[ u − ( v + w ) ][ u − ( v − w ) ] . (15)This solution in general represents a triaxially deformed AdS spacetime. Note that when c = 0, i.e. for TMG, R = 0. In this case, the cosmological constant a vanishes too.5lso, it can be shown that when R = 0 we have Q = 0. Therefore, for this solution Q isnon-zero, since Q and R cannot vanish at the same time.The matrix A in the equation (8) is a 3 × detA = − Q · ( u + v )( v − w )( u + w )8( − uvw ) / . (16)Thus the cases Q = 0, u = − v (or equivalently u = − w ) and v = w should be consideredseparately. We have checked that the case Q = 0 does not give rise to any solution to (1). u = − vu = − vu = − v : In this case the spacetime metric (13) becomes: ds = v [ − e σ dt + dσ ] + w ( dζ + e σ dt ) ≡ g (2) + w ( dζ + χ ) , (17)where w >
0. This solution was found before in [15] and is called the spacelike warped AdS. Note that dχ =vol g (2) . The coefficients a and b in terms of u , v , and w are a = 16 v ( w − v ) + c (4 v − w )(4 v − w )192 v , b = 8 v + c (4 v − w )12 √ v w . (18)The curvature scalar is as in (14). v = v = v = : In this case u < ds = u ( e σ cosh ζ dt − sinh ζ dσ ) + w [(cosh ζ dσ − e σ sinh ζ dt ) + ( dζ + e σ dt ) ] . (19)This metric was identified as timelike warped AdS in [8] and the coefficients a and b are: a = − w ( u + 4 w ) + c (3 u + 4 w )(7 u + 4 w )192 w , b = 8 w + c (3 u + 4 w )12 w √− u . (20)Again the curvature scalar is given in (14). g = v ( − θ θ + θ θ ) + wθ θ + z ( θ + θ ) , (21)with z = 0, v > w >
0. Here, z can be scaled to ±
1. Notice that, it is a z -deformation of the spacelike warped AdS metric (17).The coefficients a , b , c and the scalar curvature R in terms of v and w are a = 1 Q · ( w − v ) v ,b = 1 Q · v − w ) √ v w ,c = 1 Q · v ( w − v ) ,R = w − v v , (22) Constants v and w are related to the warping parameter ν in [15] as v = l ( ν +3) , w = l ν ( ν +3) . Thelimit ν → − u = v = w . Q = − ( w − v ) + 8(4 v − w ). Note that in the TMG limit, i.e., c = 0, both thescalar curvature and cosmological constant a vanish. Adapting the coordinate transfor-mations given in [8] to our case as: t = 12 x + y l , e σ = 2 x , ζ = ρkl + ln x , (23)we obtain ds = dρ + 2 dydx + ( R k l ) x dy + 2 kl xdρdy + zl e − kl ρ dy , (24)where v = l and w = k l with k >
0. This solution is Kundt type and corresponds to aspecial case found in [18], namely its equation (58) with some particular choices. A in equation (8) is a 4 × Q = 0 and v = w cases should be consideredseparately and the first does not provide any solution. v = v = v = : Here the coefficients a and b are equal to a = − w + c w , b = 8 w + c √ w . (25)In this case the coefficient of the third term in the metric (24) above vanishes since k = 1.By defining a new coordinate θ = xe ρ/l it becomes ds = dρ + 2 e − ρl dydθ + zl e − ρl dy , (26)which corresponds to the null warped AdS (Schr¨odinger) spacetime that was obtainedbefore in [15]. g = v ( − θ θ + θ θ + θ θ ) + z ( θ θ + θ θ ) , (27)with z = 0 and v >
0. Note that it is a z -deformation of the metric (12) with − u = v = w (i.e., round AdS). The constant z can be scaled to ± a , b , c and the scalar curvature R in terms of v are equal to a = − v , b = 8 √ v , c = − v , R = − v . (28)Note that the z -deformation has no affect on the scalar curvature which is the same asround AdS . Moreover, the solution is attained at the chiral point, i.e., the coefficientssatisfy the equality (6) with the plus sign.Using the coordinate transformations given above (23) with k = 1 we obtain ds = dρ + 2 dydx + ( 2 xl + zl e − ρ/l ) dydρ + zl e − ρ/l xdy , (29)where v = l . This is a particular case of a Kundt solution given in equation (47) of [18].7 .4 z ¯ z − type metric z ¯ z -type metric is of the form g = v ( − θ θ + θ θ ) + wθ θ + 2 zθ θ , (30)with vz = 0 and w >
0. Like (21) it is a deformation of the spacelike warped AdS metric(17). When v = w , then this solution is a deformation of round AdS. The line element is ds = [ v + z sinh 2 ζ ]( − e σ dt + dσ )+ w ( dζ + e σ dt ) +2 ze σ cosh 2 ζ dσdt − z sinh 2 ζ dσ (31)which was identified with type b) solution of [7] in [8].The coefficients a , b , c and the scalar curvature R are equal to a = 1 Q · (4 vw − w + 4 z ) w ( v + z ) ,b = 1 Q · w − v )( w + 4 z ) p w ( v + z ) ,c = 1 Q · w (4 vw − w + 4 z )( v + z ) ,R = − vw − w + 4 z w ( v + z ) , (32)where Q = (4 vw − w + 4 z ) + 8( w − v )( w + 4 z ). Notice that unlike TMG for which c = a = 0, in MMG the scalar curvature can be non-vanishing.Here the matrix A defined in (8) is a 4 × Q = 0, which does not produce any solution to (1). SU (2) We fix a basis { τ , τ , τ } and its dual basis θ a for the Lie algebra su with[ τ , τ ] = τ , [ τ , τ ] = τ , [ τ , τ ] = τ . (33)An element of SU (2) can be parametrized by (see [25]) V = e φτ e ξτ e ψτ . (34)The Maurer-Cartan one-forms are V − d V = (sin ψdξ − cos ψ sin ξdφ ) τ + (cos ψdξ + sin ψ sin ξdφ ) τ + ( dψ + cos ξdφ ) τ . (35)A left-invariant metric g on SU (2) can be written as: g = uθ θ + vθ θ + wθ θ , (36)8ith uvw <
0, not all negative. The coefficients a , b , c and the scalar curvature R are a = 1 Q · [( u − v − w ) − vw ] uvw ,b = 1 Q · √− uvw [ u − ( v − w ) ]( u − v − w ) ,c = 1 Q · uvw [( u − v − w ) − vw ] ,R = − ( u − v − w ) − vw uvw , (37)where Q = [( u − v − w ) − vw ] + 8[ u − ( v + w ) ][ u − ( v − w ) ] . (38)The line element is given by ds = ( u − v )(sin ψdξ − cos ψ sin ξdφ ) + v ( dξ + sin ξdφ ) + w ( dψ + cos ξdφ ) , (39)which corresponds to a triaxially deformed sphere and in MMG the scalar curvature givenin (37) is non-vanishing, unlike TMG.Moreover, A in equation (8) is a 3 × detA = Q uvw ) · √− uvw ( u − v )( u − w )( v − w ) . (40)Thus, the cases Q = 0 and u = v (is enough due to the symmetry) should be consideredseparately. Again the case Q = 0 does not give any solution. u = vu = vu = v : Note that in this case w <
0. The coefficients a and b are a = 16 v (4 v − w ) + c (4 v − w )(4 v − w )192 v , b = − v + c (3 w − v )12 √− v w . (41)In this case the metric (39) simplifies to a Hopf-fibration over S . Depending on whether | w | > | w | <
1, we have stretched or squashed warpings respectively. A ∞ The Lie algebra a ∞ of A ∞ is spanned by r , x , and y and has only one non-trivial bracket[ r, x ] = − y . (42)We denote the dual basis as { ˜ r, ˜ x, ˜ y } . The Baker-Campbell-Hausdorff formula allows usto write a representative as (see [25]) V = e sr e tx e ρy . (43)9he Maurer-Cartan one-forms are V − d V = ( ds ) r + ( dt ) x + ( dρ − tds ) y. (44)By the automorphism group, a left-invariant metric can be fixed as [8] g = u ˜ r ˜ r + v ˜ x ˜ x ± ˜ y ˜ y. (45)where uv = 0 and u or v can be scaled to ±
1. The line element reads as ds = uds + vdt ± ( dρ − tds ) , (46)which is a Hopf fibration over a flat space. We have a = 16 | uv | + 21 c uv ) , b = ± | uv | − c p | uv | , R = 12 | uv | . A a of A , spanned by r , x , and y , has non-vanishing brackets[ r, x ] = x, [ r, y ] = x + y. (47)We denote the dual basis as { ˜ r, ˜ x, ˜ y } . Again by the Baker-Campbell-Hausdorff formulawe can choose the representative V = e ξx + ρy e αr . (48)Then the Maurer-Cartan one-forms are V − d V = ( e − α dξ − αe − α dρ ) x + ( e − α dρ ) y + ( dα ) r. (49)The following 4 types of metrics are available [8]: B -type metric Metric is given by B = z ˜ r ± ˜ x + v ˜ y . (50)There is no solution for a , b , c . B -type metric Metric is given by B = z ˜ r ± x ˜ y , (51)10ith z >
0. The MMG field equation (1) is satisfied if a = − z + c z , b = ∓ z + c √ z , R = − z . (52)Note that the solution is attained at the chiral point (6).Under the coordinate transformations α → log( w ) , ρ → − lx + , ξ → lx − (53)where z = l the metric (51) becomes ds = ∓ l w [2 log( w )( dx + ) + 2 dx + dx − ∓ dw ] (54)which is the logarithmic pp-wave solution found in [16]. B -type metric Metric is given by B = z ˜ r + ˜ r ˜ x + v ˜ y . (55)In this case a = 0 is the necessary and sufficient condition to solve the equation (1). TheRicci, Cotton, and J -tensors are identically zero. Metric of this Ricci flat spacetime is ds = zdα + e − α ( dξdα − αdρdα ) + ve − α dρ . (56)As we discuss in section 9, it must be maximally symmetric based on a result of [26] andhence should locally be Minkowski spacetime. B -type metric Metric is given by B = z ˜ r + ˜ r ˜ y + u ˜ x , (57)where u >
0. The coefficients a , b , c and the scalar curvature R are found to be a = − u , b = − √ u , c = − u , R = 2 u. (58)The line element is ds = zdα + e − α dρdα + ue − α ( dξ − αdρ ) (59)Unfortunately, we could not determine which spacetime geometry this metric corresponds.Higher order curvature scalars are as follows R µν R µν = 12 u , R µρ R ρν R µν = 8 u . (60)11 Solutions on
I SO (2; θ ) Let the Lie algebra basis and the dual basis of iso (2; θ ) be { l, m , m } and { ˜ l, ˜ m , ˜ m } respectively. The non-vanishing brackets are[ l, m ] = 2 cos θm + 2 sin θm , [ l, m ] = 2 cos θm − θm , (61)where θ ∈ [0 , π/ V = e xm + ym e ρl . (62)Then the Maurer-Cartan one-forms are V − d V = ( dρ ) l + e − ρ cos θ [cos(2 ρ sin θ ) dx + sin(2 ρ sin θ ) dy ] m + e − ρ cos θ [ − sin(2 ρ sin θ ) dx + cos(2 ρ sin θ ) dy ] m . (63)There are two types of metrics as below [8]. The case θ = 0 should be analyzed separately. B -type metric Metric is given by B = u ˜ l ˜ l + v ˜ m ˜ m + w ˜ m ˜ m , (64)where uvw <
0, not all negative. The coefficient v or w can be rescaled freely.There is no general solution for a , b , and c . The scalar curvature is given by R = − vw cos θ + ( v − w ) sin θ ] uvw . (65)The matrix A in the equation (8) is 4 × θ = 0, θ = π , and v = w shouldbe considered separately. θ = 0 θ = 0 θ = 0: In this case due to the enlargement of the automorphism group, the metric (64)becomes B = | z | ( ± ˜ l ˜ l ± ˜ m ˜ m ± ˜ m ˜ m ) , (66)which in spacetime coordinates takes the form ds = | z | ( ± dρ + e − ρ [ ± dx + ± dy ]) , (67)which is either de Sitter for ( − , + , +) or AdS for (+ , + , − ) signs with R = ± / | z | .Cotton tensor (2) vanishes identically and we have the relation a = − ±| z | − c ) z . (68) θ = π θ = π θ = π : The coefficients a , b , c in terms of u , v , and w are equal to a = 1 Q · ( v − w ) uvw , b = 1 Q · √− uvw ( v + w ) , c = 1 Q · uvw , (69)12here Q = ( v − w ) + 8( v + w ) = 0. In this case the line element is ds = udρ + ( v − w )[cos(2 ρ ) dx + sin(2 ρ ) dy ] + w ( dx + dy ) , (70)which is not familiar to us. Higher order curvature invariants are as follows: R µν R µν = 4( v − w ) (3 v + 2 vw + 3 w )( uvw ) ,R µρ R ρν R µν = − v − w ) ( uvw ) . (71)A solution of this type also exists [11, 12] in NMG [13]. v = wv = wv = w : In this case Cotton tensor (2) vanishes identically and a = − θ ( u + c · cos θ ) u . (72)The line element is simply ds = udρ + ve − ρ cos θ [ dx + dy ] , (73)which is for u < v >
0, either de Sitter if θ = π/ θ = π/ B -type metric Metric is given by B = u ˜ l ˜ l + ˜ l ˜ m + w ˜ m ˜ m , (74)with u > w = 0.When θ = 0 the metric is Minkowski and the field equation (1) is solved only if a = 0.For θ = 0 the coefficients a , b , c and the scalar curvature R are equal to a = − w sin θ, b = 29 √ w sin θ , c = − w sin θ , R = 8 w sin θ. (75)The line element is ds = udρ + e − ρ cos θ [cos(2 ρ sin θ ) dxdρ + sin(2 ρ sin θ ) dydρ ]+ we − ρ cos θ [sin(2 ρ sin θ ) dx − cos(2 ρ sin θ ) dy ] . (76)We could not recognize the spacetime it corresponds. Higher order curvature scalars are R µν R µν = 192 w sin θ , R µρ R ρν R µν = 512 w sin θ . (77)13 Solutions on
I SO (1 , θ ) The basis { l, m , m } of iso (1 , θ ) has the brackets[ l, m ] = 2 cos θm + 2 sin θm , [ l, m ] = 2 cos θm + 2 sin θm , (78)and the dual basis is { ˜ l, ˜ m , ˜ m } . We choose the group representative as V = e xm + ym e ρl , (79)and the Maurer-Cartan one-forms are V − d V = ( dρ ) l + e − ρ cos θ [cosh(2 ρ sin θ ) dx − sinh(2 ρ sin θ ) dy ] m + e − ρ cos θ [cosh(2 ρ sin θ ) dy − sinh(2 ρ sin θ ) dx ] m , (80)with θ ∈ [0 , π/ θ = 0 the Lie algebras of ISO (1 ,
1; 0) and
ISO (2; 0) coincide.Hence, θ = 0 case is already covered in sections 7.1 and 7.2. From the automorphismgroup, two types of metrics can be fixed as below [8]. B -type metric The B -type metric is given by B = δ ˜ l ˜ l + u ( ˜ m + ˜ m ) + v ( ˜ m − ˜ m ) + 2 w ( ˜ m ˜ m − ˜ m ˜ m ) , (81)with w > uv and δ >
0. Two of the parameters ( u, v, w ) can be set to ± A in (8) is 6 × a , b , c . The scalarcurvature is R = − uv − w ) cos θ + uv sin θ ] δ ( uv − w ) . (82)However, the cases θ = π , θ = π , and uv = 0 should be considered separately. θ = π θ = π θ = π : For w = 0 the coefficients a and b are found as a = 2 δ ( w − uv )(4 uv − w ) + c (4 uv − w )(4 uv + 3 w )3 δ ( w − uv ) ,b = − δ ( w − uv ) + c (4 uv − w )3 w p δ ( w − uv ) . (83)The metric (81) simplifies to ds = δdρ + ue − √ ρ ( dx + dy ) + v ( dx − dy ) + 2 we − √ ρ ( dx − dy ) , (84)which was identified as timelike or spacelike warped AdS in [8] depending on signs.14hen w = 0 the Cotton tensor vanishes identically and we are at the merger point(7) with c = − δ/ /a = 4 /R . The metric (84) becomes ( A ) dS × S that was found in[15], which is clearly not a solution of TMG since c cannot be zero. Its absence is relatedto a no-go result on solutions of TMG with a hypersurface orthogonal Killing vector [27].However, it exists in NMG [13] as was found in [28]. θ = π θ = π θ = π : The coefficients a , b , c in terms of u , v , and w are equal to a = 2 u v δ ( uv + 8 w )( uv − w ) , b = − w p δ ( w − uv ) uv + 8 w , c = δ ( uv − w )2( uv + 8 w ) . (85)Its spacetime metric is: ds = δ dr r + dα r − r dt + 2 wdαdt , (86)where we set u = 1, v = − r = e ρ , α = x + y , t = x − y. (87)Notice that the following constant rescalings r → λr, α → λα, t → λ − t leave the metric(86) invariant. When w = 0 (which sets b = 0 and the merger point condition (7) issatisfied) this corresponds to the static Lifshitz spacetime with the dynamical exponent z = − w = 0 it is a stationary Lifshitz metric (see [29]). Note that the rotationparameter w is non-zero only when there is a contribution from the Cotton tensor. u = 0 u = 0 u = 0: The coefficients a and b are a = − θ ( δ + c · cos θ ) δ , b = w ( δ + 2 c · cos θ )2 √ w δ (cos θ − θ ) . (88)The spacetime metric (81) takes the form ds = δdρ + ve − ρ (cos θ − sin θ ) ( dx − dy ) + 2 we − ρ cos θ ( dx − dy ) , (89)which was identified in [8] as AdS pp-wave for θ = π/ θ = π/
2. When θ = π/ θ = π/ v = 0 v = 0 v = 0: The coefficients a and b are equal to a = − θ ( δ + c · cos θ ) δ , b = − w ( δ + 2 c · cos θ )2 √ w δ (cos θ + 2 sin θ ) . (90)The spacetime metric (81) becomes ds = δdρ + ue − ρ (cos θ +sin θ ) ( dx + dy ) + 2 we − ρ cos θ ( dx − dy ) , (91)which again corresponds to an AdS pp-wave in general [8]. But for θ = π/ θ = π/ .2 B -type metric For θ = 0 the metric is given by B = δ ˜ l ˜ l + ˜ l ˜ m + u ( ˜ m + ˜ m ) + v ( ˜ m − ˜ m ) + 2 w ( ˜ m ˜ m − ˜ m ˜ m ) , (92)with w = uv > u + v = 2 w . For w = 0, both Cotton and J -tensors vanish and a = 0 in (1), which locally corresponds to Minkowski spacetime as we discuss in section 9.Hence, we assume w = 0 which means that v = w is not allowed. One of the coefficients u or v can be set to ± ds = δ dr r + ur − n dα + vr − m dt +2 wr − ( n + m ) dαdt + 14 r − ( n +1) dαdr + 14 r − ( m +1) dtdr , (93)where n = (cos θ + sin θ ) and m = (cos θ − sin θ ). Note that the metric is invariant underthe scalings r → λr, α → λ n α, t → λ m t . Hence, the solution possesses a generalized(anisotropic) Lifshitz symmetry. For θ = π/
2, it becomes a stationary Lifshitz solutionwith dynamical exponent z = − a , b , c and the scalar curvature R are a = − vw sin θ v − w ) , b = − | v − w | w √ v sin θ ,c = − ( v − w ) vw sin θ , R = 128 vw sin θ ( v − w ) . (94)The only special case that should be considered separately is θ = π . θ = π θ = π θ = π : The coefficients a and b are given as a = 32 vw [( v − w ) + 168 c · vw ]3( v − w ) , b = − ( v − w ) − c · vw w | v − w |√ v . (95)Its metric is (93) with m = 0 and n = √ In this paper we constructed homogeneous solutions of MMG several of which are new.We summarize our results in Table 1 where we include only those that are non-trivial inthe sense that none of the terms in the MMG field equation (1) vanishes identically. In itslast column we give classification of our solutions with respect to the Segre-Petrov typeof their traceless Einstein tensor P ab ≡ R ab − Rδ ab , (96)as was proposed in [20] to which we refer for details.From the Table 1 we see that homogeneous solutions of MMG in comparison to TMGcan be grouped into three as follows: 16 omogeneous SolutionsGroups Metric MMG TMG Description Type SL (2; R ) 111-type (13) ✓ ( R =0) triaxially deformed AdS I R (17) ✓ spacelike warped AdS D(19) ✓ timelike warped AdS D12-type (24) ✓ ( R =0) Kundt II(26) ✓ null warped AdS N3-type (29), chiral pt. ✗ Kundt III1 z ¯ z -type (31) ✓ ( R =0) generic I C SU (2) (39) ✓ ( R =0) triaxially deformed sphere I R (41) ✓ stretched/squashed sphere D A ∞ (46) ✓ warped flat D A B -type (54), chiral pt. ✓ logarithmic pp-wave N B -type (59) ✗ generic II ISO (2; θ ) B -type (70), ( θ = π/ ✗ generic I R B -type (76), ( θ = 0) ✗ generic II ISO (1 , θ ) B -type (84), ( θ = π/ ✓ space/time-like warped AdS D(86), ( θ = π/ ✗ stationary Lifshitz I R (89) ✓ pp-wave N(91) ✓ pp-wave N B -type (93), ( θ = 0) ✗ generalized Lifshitz II(95), ( θ = π/ ✓ warped flat DTable 1: Comparison of non-trivial homogeneous solutions of MMG and TMG • Group 1: Solutions which are type N or D in the Segre-Petrov classification can beobtained from TMG solutions with a redefinition of constants as was shown in [14].Corresponding solutions have the same curvature.For Type D solutions, namely { (17), (19), (41), (46), (84), (95) } , we have a MMG = a TMG + c ·
148 ( R + 49 b )( R + 43 b ) , (97) b MMG = b TMG − c · b TMG R + 49 b ) . (98)For Type N solutions, that is { (26), (54), (89), (91) } , we have a MMG = a TMG − c · R a
TMG , (99) b MMG = b TMG − c · R b
TMG . (100) • Group 2: Solutions { (13), (24), (31), (39) } exist in TMG but only if the cosmologicalconstant vanishes. Hence, they have R = 0 in TMG. But for MMG, for these17olutions the cosmological constant is proportional to the MMG parameter c andtherefore R = 0 is possible. • Group 3: Solutions { (29), (59), (70), (76), (86), (93) } exist only in MMG.It is interesting to note that for all solutions in Group 2 and 3 we have R = 16 a/c .Moreover, three of the solutions in the third group, that is (59), (76) and (93), appearwhen ac = 1 /
81 and 9 b = − c . Whether this particular point in the parameter spaceof MMG has any physical significance like chiral (6) and merger (7) points remains to beseen. Also, more work is required to understand spacetimes that we found in (59), (70)and (76).Within the third group, Lifshitz type solutions, that is (86) and (93), are especiallyattractive due to their possible holographic applications (for a review see [30]). Moreover,only few exact Lifshitz solutions which are stationary are known [29]. In (86) the dynam-ical exponent is z = − b = 0 by choosing other parametersappropriately, after which remarkably one always ends up at the merger point (7). Thus,these are solutions of of the MMG theory without the Cotton tensor. The fundamentalequation of this specific model can be obtained from MMG (1) by taking the limit µ → ∞ , γ → ∞ while keeping γ/µ constant which was considered before in [16] and [19]. Forthis to be consistent, one should still make sure that Bianchi identity (4) is satisfied, i.e. V µ = ǫ µρσ S τρ C στ = 0 . (101)For our solutions it turns out that V µ is identically zero except for the following 3 cases:i) A spacetime with B -type metric (50): V µ = [ ∓ vz , , . (102)ii) ISO (2; θ ) spacetime with B -type metric (64): V µ = [ − v − w ) cos θ sin θu vw , , . (103)iii) ISO (1 , θ ) spacetime with B -type metric (81): V µ = [ 64 uv sin 4 θ sin θ ( w − uv ) z , , . (104)Recall from the section 6.1 that there is no B type solution for A which is in agree-ment with V µ being non-zero in (102). From our analysis in sections 7.1 and 8.1 one18an easily see that, the last two vectors become zero precisely at the solutions we found,independent of the value of b . Moreover, for all our solutions whenever b = 0 is possible,then the merger point condition (7) is satisfied. Exceptions appear only when the Cottontensor identically vanishes. Hence, we reach to the conclusion that: When the Cotton tensor is absent in the MMG equation (1), simply transitive homo-geneous solutions exist only at the merger point (7) provided that they are not conformallyflat. They satisfy the Bianchi condition (101).
For our solutions for which the Cotton tensor vanishes, it is useful to recall thatconformally flat spacetime solutions of MMG are locally maximally symmetric away fromthe merger point (7) which was proven in [26]. Indeed, (67) and (73) are (A)dS spacetimeswhich are in general away from the merger point but for a specific choice of the parameter c they also exist at the merger point. For (56) cosmological constant vanishes ( a = 0) andhence, we conclude that it must be locally Minkowski since the merger point condition, i.e. ac = 1, is impossible to satisfy. This applies also to (92) with w = 0. On the other handat the merger point, a conformally flat solution is not necessarily maximally symmetric.For example, the metric (84) with w = 0 corresponds to ( A ) dS × S that was found in[15] and it exists only at the merger point.We found that two of our solutions given in (29) and (52) exist at the the chiral point(6) of the parameter space. Only in the latter it is possible to set b = 0 in which case oneends up at the merger point (7) as we noted above.In this paper we focused on simply transitive homogeneous spacetimes. A naturalgeneralization would be to allow a non-trivial isotropy group as was studied in [34] forTMG, and in [10] for NMG. The following metric that was discussed both in [34] and [10]has 4-dimensional isometry group with no 3-dimensional simply transitive subgroup: ds = − dt + v ( dθ + sin θdφ ) . (105)This is a solution of the MMG field equation (1) at the merger point (7) with: a = 1 c = R v . (106)The Cotton tensor vanishes identically and its Segre-Petrov type is D.It would be interesting to repeat our investigation in models closely related to MMG[35, 36]. Finally, trying to classify all stationary axi-symmetric solutions of MMG as wasdone for TMG [37] using a method developed in [38] would be worth studying. We hopeto explore these issues in near future. Acknowledgements
We are grateful to George Moutsopoulos for many useful discussions. We would like tothank ¨Ozg¨ur Sarıo˘glu for a critical reading of an earlier version of this paper and to MarikaTaylor for comments about our Lifshitz solutions. JC and NSD are partially supported bythe Scientific and Technological Research Council of Turkey (T¨ubitak) project 113F034.19 eferences [1] E. Bergshoeff, O. Hohm, W. Merbis, A.J. Routh and P.K. Townsend,
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