Bimetric Gravity From Adjoint Frame Field In Four Dimensions
aa r X i v : . [ h e p - t h ] D ec Bimetric Gravity From Adjoint Frame Field In Four Dimensions
Zhi-Qiang Guo ∗ and Iv´an Schmidt † Departamento de F´ısica y Centro Cient´ıfico Tecnol´ogico de Valpara´ıso,Universidad T´ecnica Federico Santa Mar´ıa,Casilla 110-V, Valpara´ıso, Chile
We provide a novel model of gravity by using adjoint frame fields in four dimensions. It has anatural interpretation as a gravitational theory of a complex metric field, which describes interactionsbetween two real metrics. The classical solutions establish three appealing features. The sphericalsymmetric black hole solution has an additional hair, which includes the Schwarzschild solutionas a special case. The de Sitter solution is realized without introducing a cosmological constant.The constant flat background breaks the Lorentz invariance spontaneously, although the Lorentzbreaking effect can be localized to the second metric while the first metric still respects the Lorentzinvariance.
PACS numbers: 04.20.Jb, 04.50.Kd, 04.70.Bw, 11.30.Qc
I. INTRODUCTION
Since its construction in 1915, Einstein’s gravitationaltheory has passed numerous experimental tests in thelast century. However, the observation of cosmologicalacceleration at later times provided us with the hint thatEinstein’s gravity could need to be modified at large dis-tances, and in fact several modifications have been pro-posed along different directions [1–8]. One intriguing pos-sibility is that there could exist another metric field f µν besides the conventional metric field g µν . This kind ofbimetric gravity was proposed by Rosen in 1940 [9], al-though in general it has been shown to present a ghostproblem [10]. Nevertheless, the ghost-free bimetric grav-ity has recently been constructed [11, 12] in the frame-work of massive gravity [13], which is viable to explainthe cosmological acceleration [14].In this paper we propose a novel model of bimetricgravity, which is not within those traditionally consid-ered. In Einstein’s theory, gravity is described by themetric tensor g µν , which can be recast into an equiva-lent formulation by using the frame field e Iµ . Here e Iµ transforms as the fundamental representation of the localLorentz group SO (1 , e IJµ . An immediate consequence ofthis adjoint frame field is that we can obtain two gaugeinvariant metric fields [15] g µν = 12 η IM η JN e IJµ e MNν (1)and f µν = 14 ǫ IJMN e IJµ e MNν . (2)Hence a theory of e IJµ intrinsically describes interactionsbetween two metrics. In contrast with the massive bi- ∗ [email protected] † [email protected] metric gravity, which retains intact the Lorentz invari-ance [11, 12], the bimetric theory of e IJµ has to breakLorentz covariance spontaneously. One way to see this isby counting degrees of freedom (dofs): e IJµ has 18 gaugeinvariant dofs, but g µν and f µν have a total of 20 su-perficial dofs. So g µν and f µν are not independent, andactually they are constrained by the complex conditiondet( g µν + if µν ) = 0 , (3)which eliminates two dofs. This constraint also forbids g µν and f µν to be proportional to the Lorentz metric si-multaneously, hence at least one of them is required tobreak Lorentz invariance. Nevertheless, as demonstratedin [16–20], the spontaneous breaking of Lorentz invari-ance is not problematic, but it is useful in order to estab-lish the consistence of bimetric gravity.In section II, we construct the first order Lagrangian,using the frame fields. We show that a concise formula-tion of the Lagrangian exists when we use the SO (3 , C )variables instead of the SO (1 ,
3) variables, and that thespin connection can be uniquely determined thoroughthe variation principle. In section III, we provide themetric-like formulation, using the complex metric fields g µν + if µν . In section IV, we obtain the spherical blackhole solution and the time-dependent solution. II. FRAME-LIKE FORMULATION
We begin with the Frame-like formulation. Considerthe frame field e IJµ , where
I, J = 0 , , , SO (1 ,
3) adjoint representation, and e IJµ is antisym-metric with respect to I and J . The field strength isbuilt from the spin connection F IJµν = ∂ µ ω IJν − ∂ ν ω IJµ + ω IµN ω NJν − ω IνN ω NJµ . (4)Besides g µν and f µν in Eqs. (1) and (2), we also have twoother gauge invariant quantities u µ = 16 ǫ µνρσ η IK η JL η MN e IMν e NJρ e KLσ , (5) v µ = − ǫ µνρσ ǫ IJMN η KL e IJν e MKρ e NLσ . (6)From e IJµ and F IJµν , we can obtain the following gaugeinvariant operators O = ∗ F µνIJ e IMµ e NJν η MN , (7) O = ∗ F µνIJ e KMµ e NLν ǫ IJKL η MN . (8)We also have the operators associated to u α O = ∗ F µνIJ e IJν u ρ g ρµ , O = ∗ F µνIJ e νKL ǫ IJKL u ρ g ρµ , (9) O = ∗ F µνIJ e IJν u ρ f ρµ , O = ∗ F µνIJ e νKL ǫ IJKL u ρ f ρµ , and the operators associated to v α O = ∗ F µνIJ e IJν v ρ g ρµ , O = ∗ F µνIJ e νKL ǫ IJKL v ρ g ρµ , (10) O = ∗ F µνIJ e IJν v ρ f ρµ , O = ∗ F µνIJ e νKL ǫ IJKL v ρ f ρµ , where ∗ F µνIJ = ǫ µναβ F αβIJ is the dual field strength. Inorder to construct the Lagrangian, we also need the gaugeinvariant pseudo scalars and scalars g = det( g µν ) , f = det( f µν ) , (11) ̺ = f µν g µν , ς = g µν f µν . (12)As defined in Eqs. (1) and (2), g µν and f µν are in gen-eral invertible, and g µν and f µν are their inverses respec-tively. Then a general Lagrangian can be constructedas L = a O + a O + X i =1 o i χ i O i , (13)where a i and o i are constant coefficients, and χ i ( g, f, ̺, ς )are some pseudo-scalar functions, to ensure that L has the proper transformation properties. The La-grangian (13) is constructed using the SO (1 ,
3) variables e IJµ and ω IJµ , and a general Lagrangian can depend on 10constant coefficients and 8 pseudoscalar functions. How-ever, in the following we shall show that a simplified ver-sion can be constructed if we use the SO (3 , C ) variables.We define the complex variables e kµ = − ǫ kmn e mnµ + ie kµ , (14) A kµ = − ǫ kmn ω mnµ + iω kµ , (15)where the small latin letters take values 1 , ,
3. Then e kµ and A kµ are the SO (3 , C ) variables. We can also definethe complex field strength F kµν = − ǫ kmn F mnµν + iF kµν , (16) which can be rewritten in terms of the complex connec-tion as F kµν = ∂ µ A kν − ∂ ν A kµ + ǫ kmn A mµ A nν . (17)Using the complex variables, we have the complex met-ric g µν = e kµ e kν = g µν + if µν . (18)and the complex pseudo-vector u µ = 16 ǫ µνρσ ǫ kmn e kν e mρ e nσ = u µ + iv µ . (19)From Eqs. (18) and (19), we see the two facets of g µν and u µ . They can be expressed either using the SO (3 , C )variable in Eq. (14) or using the SO (1 ,
3) variables inEqs. (1), (2), (5) and (6). From Eq. (18), we know thatthe determinant of g µν is zero, and we can also verifythat u µ is its eigenvector with eigenvalue 0. Actually, g µν and u µ satisfy the identities g µρ u ρ = 0 , (20) u µ u ν = 16 ǫ µαρτ ǫ νβσθ g αβ g ρσ g τθ . (21)For a given g µν , we can obtain u µ by using Eq. (21). So u µ is not independent of g µν , but it can be completelydetermined by g µν . We also define a complex pseudo-scalar Φ = 14 ¯g µν u µ u ν = g − f + igς, (22)where ¯g µν is the complex conjugate of the complex metric g µν . Using Φ, we define the covariant pseudo-vector v µ = − ¯g µρ u ρ , (23)which satisfies u α v α = − v µ , we can define E αk = 12 ǫ αθρσ ǫ kmn v θ e mρ e nσ , (25)then we have E αj e iα = δ ij , (26)and we also have n αβ = E αk e kβ . (27) E αi can be regarded as the left inverse of e iα , and n αβ playsthe role of the projection tensor. Note that g µν is notinvertible. We can obtain its generalized inverse as g µν = E µk E νk = 12 ǫ µαρτ ǫ νβσθ v α v β g ρσ g τθ , (28)which satisfies g µρ g ρσ g σν = g µν , g µρ g ρσ g σν = g µν . (29)And we also have the relation n αβ = g αρ g ρβ = δ αβ + u α v β . (30)Using the above complex variables, we propose the La-grangian L = κ ǫ µναβ ǫ ijk e iµ e jν F kαβ (31)+ λ √− Φ ǫ µναβ v µ e kν F kαβ + c . c , where κ and λ are complex constants, and c . c is the com-plex conjugate. Compared with Eq. (13), Eq. (31) has 4real constant coefficients. It is a special case of Eq. (13),and it can be rewritten into the formulation (13) in termsof the SO (1 ,
3) variables. In the following, we shall con-sider the concise formulation (31) in terms of SO (3 , C )variables. Varying Eq. (31) with respect to A iµ , we obtain κǫ µναβ ǫ ijk T jµα e kβ = 2 λ √− Φ ǫ νµαβ v α T iµβ + 2 λǫ νµαβ ∂ µ ( v α √− Φ) e iβ , (32)where T iµν = 12 ( ∂ µ e iν + ǫ ijk A jµ e kν ) − ( µ ↔ ν ) (33)is the torsion tensor. Variations of Eq. (31) with respectto e iα yield − κ ǫ µναθ ǫ ijk e jθ F kµν + λ √− Φ ǫ µναβ v β F iµν + λ √− Φ ǫ µνρσ e kσ F kµν ¯g ρθ g θτ u α e iτ (34)= ¯ λ p − Φ ǫ µνρσ ¯e kσ ¯F kµν (cid:18) ¯u α ¯u θ ¯v ρ + ¯u α δ θρ + ¯u θ δ αρ (cid:19) e iθ . In the above, ¯ z always means the complex conjugate ofthe complex variable z . Eqs. (32) and (34) give the firstorder formulation of the equations of motion. III. METRIC-LIKE FORMULATION
In the above, we have obtained the first order formu-lation. In this section, we show that the second orderformulation gives simpler expressions in terms of metricvariables. First we need to solve Eq. (32) to obtain theconnection. We define the affine connection Γ iµν = ∂ µ e iν + ǫ ijk A jµ e kν . (35)The metric-like formulation of this connection is Γ ρµν = E ρi Γ iµν , (36)which satisfy the condition Γ ρµν = n ρτ Γ τµν (37)according to Eqs. (26) and (27). From Eq. (36), we have ∂ µ e iν + ǫ ijk A jµ e kν = Γ ρµν e iρ . (38) From this equation, we obtain the metric compatibilitycondition ∂ µ g αβ = Γ ρµα g ρβ + Γ ρµβ g ρα . (39)This equation can be solved as Γ ρµν = g ρσ Γ σ,µν − g ρσ ( T θµσ g θν + T θνσ g θµ ) + T ρµν , (40)where T ρµν = 12 ( Γ ρµν − Γ ρνµ ) , (41)is the torsion tensor, and Γ τ,ρσ = 12 ( ∂ ρ g τσ + ∂ σ g τρ − ∂ τ g ρσ ) (42)is the Christoffel symbol of the first kind. In order tosatisfy Eq. (39), the torsion tensor need to satisfy u τ Γ τ,αβ = u τ ( T ρατ g ρβ + T ρβτ g ρα ) . (43)Using the torsion tensor, Eq. (32) can be rewritten as κ ( u τ T αβτ + u τ T θθτ δ αβ + u α T θθβ ) (44)= λ √− Φ ǫ ατρσ v ρ T θτσ g θβ + λǫ ατρσ ∂ τ ( v ρ √− Φ) g σβ . Eqs. (37), (43) and (44) uniquely determine the torsiontensor as T µαβ = 12 g µσ u τ ( v α Γ τ,σβ − v β Γ τ,σα ) (45)+ v α K µβ − v β K µα + λ κ √− Φ g µτ g θσ ( ∂ τ v θ − ∂ θ v τ ) u ν ǫ νσαβ + κ λ √− Φ ( g µρ g σθ − g µσ g ρθ ) u τ Γ τ,ρθ u ν ǫ νσαβ . where K αβ = λκ ǫ αθστ v θ g σβ ∂ τ √− Φ (46)+ λκ √− Φ n αρ g σβ ǫ ρστθ ∂ τ v θ + λ κ Φ g ατ n θβ ( ∂ τ v θ − ∂ θ v τ ) . From Eq. (38), we can express the field strength in termsof the curvature F kµν = 12 ǫ kmn R ρσµν e nρ E σm , (47)and R ρσµν = ∂ µ Γ ρσν − ∂ ν Γ ρσµ + Γ ρµτ Γ τσν − Γ ρντ Γ τσµ (48)is the definition of Riemann tensor. The Lagrangian (31)can be rewritten as L = − κ ǫ αβµν R αβµν − λ √− Φ R + c . c , (49)where R = g αβ R ταβθ n θτ (50)is a scalar curvature, and R αβµν = g ατ R τβµν (51)is the covariant Riemann tensor. The second term ofEq. (49) is similar to the Einstein-Hilbert Lagrangian.The first term of Eq. (49) vanishes in Einstein’s gravity,due to the first Bianchi identity, although it contributesin our case because the connection (40) has the torsionpiece (45). We define the contracted tensor R αβ = g τθ R ατθβ . (52)Then the equations of motion in Eq. (34) can be rewrit-ten as 2 λ √− Φ( n ασ R σβ − R δ αβ ) − κ ǫ ασµν R βσµν = ( S αβ − c . c) , (53)where S αβ = λ √− Φ R u α v β − λ √− Φ u τ R ρτ n αρ v β − λ √− Φ u τ R στ n ρσ ¯g ρβ u α . (54)The left hand of Eq. (53) is similar to the Einstein’s grav-itational equation, although it is complex. Note that onlythe imaginary part of S αβ contributes to the equations ofmotion. IV. CLASSICAL SOLUTIONS
We consider classical solutions of Eq. (53). For sta-tionary solutions with spherical symmetry, we use thefollowing ansatz for g µν g µν dx µ dx ν = − p ( r ) dt + q ( r ) (cid:0) dr + r d Ω (cid:1) (55)and for f µν f µν dx µ dx ν = − p ( r ) q ( r ) dtdr + β q ( r ) r d Ω . (56)where d Ω = dθ + sin θdφ , and β is a real con-stant. The ansatz in Eqs. (55) and (56) satisfy the con-straint (3). Eq. (53) is a complex equation, whose realpart yields two independent equationsRe( a ) q ( r ) − b ) r dqdr = 4Re( b ) r d qdr , (57)1 Q dQdr − q dqdr = 1 p dpdr , (58)where Q ( r ) = q ( r ) + 2 rq ( r ) dqdr , and b = 2(1 + iβ )( κ − λ )¯ λ, (59) a = − b − ¯ λλ . (60)The solutions for q ( r ) and p ( r ) are q ( r ) = c r α − + c r − α − , (61) p ( r ) = − αc c − c r α c + c r α , (62) where α = b ) p Re( b )Re( a + b ), and c , c and c are three integral constants. We can check that theabove solutions also solve the imaginary part of Eq. (53),hence p ( r ) and q ( r ) in Eqs. (61) and (62) are solutionsof Eq. (53). When α is real, c and c are required tobe real; in this case, c and c can be absorbed into theredefinition of r and t . When α is imaginary, c and c are required to be conjugate complex numbers; in thiscase, the real (or imaginary) part of c and c can beabsorbed into the redefinition of r and t . In both cases,only one effective parameter is left. With α , the metricin Eq. (61) has two hairs. An interesting case happenswhen α = 12 , c = 1 , c = 14 r s , c = 1 . (63)We obtain, from Eq. (55) − (cid:18) − r s r r s r (cid:19) dt + (cid:18) r s r (cid:19) (cid:0) dr + r d Ω (cid:1) = g µν dx µ dx ν . (64)This is the Schwarzschild metric in isotropic coordinates,where r s is the Schwarzschild radius. For α = , we needRe( a ) = 0, which determines ββ = Re (cid:0) κ ¯ λ + 3 λ ¯ λ (cid:1) Im (cid:0) κ ¯ λ − λ ¯ λ (cid:1) . (65)Now we consider time-dependent solutions. The ansatzfor g µν is g µν dx µ dx ν = − dt + a ( t ) dx i dx i , (66)and f µν is f µν dx µ dx ν = f dt + 2 f i dtdx i + f ij dx i dx j , (67) f µν = f f f f f f f f f f f f f f f f , in which f = 12 (cid:0) ω − ω (cid:1) , f = 12 (cid:0) ω − ω (cid:1) a ( t ) , (68) f = ω a ( t ) , f = 12 √ (cid:0) ω + 1 ω (cid:1) a ( t ) . Here ω is a real constant. The left hand of Eq. (53) yieldstwo independent expressions − iω λω (cid:0) ( − i + ω ) κ + ( i + ω ) λ (cid:1) a ( ˙ a + 2 a ¨ a ) , (69)14 √ λω (1 + ω )(1 + 2 iω ) (cid:0) κ + λ (cid:1) ( ˙ a − a ¨ a ) , (70)where ˙ a = dadt . There are 4 independent expressions forthe right hand of Eq. (53). Eq. (53) can be satisfied if( − i + ω ) κ + ( i + ω ) λ = 0 , (71)˙ a − a ¨ a = 0 . (72)The solution of Eq. (72) is a ( t ) = e Ht , (73)where H is a integral constant. Eq. (71) is a complexequation, whose solution yields λ ¯ λ = κ ¯ κ, (74) ω = κ ¯ λ + λ ¯ κ | κ + iλ | , or ω = − κ ¯ λ + λ ¯ κ | κ − iλ | , (75)where | z | is the module of the complex number z . For thissolution, g µν is the de Sitter metric. For the existence ofthis solution, the couplings λ and κ are required to satisfythe constraint (74).Now we consider the solution of constant background.Obviously, any constant value of g µν and f µν which sat-isfy the constraint (3) are solutions of Eq. (53). For thesolution in Eqs. (61) and (62), in the case that the pa-rameters are given by Eq. (63), p ( r ) and q ( r ) are constantwhen r → ∞ . We see that g µν is the Lorentz metric inthis limit, but f µν breaks Lorentz invariance. For thesolution in Eqs. (66) and (67), when a ( t ) = 1, the met-rics are constant, and they solve Eq. (53). We see that g µν is the Lorentz metric, but f µν breaks Lorentz in-variance. Hence the constant background breaks Lorentzinvariance spontaneously. V. DISCUSSIONS
In this paper, we have provided a gravitational modelin terms of the adjoint frame field e IJµ . This model de-scribes interactions between two metrics. In section II,using the SO (3 , C ) variables, we construct a concise La-grangian with 2 complex coupling constants. In sec-tion III, we give the metric-like formulations of the La- grangian and equations of motion. We also obtain theSchwarzschild solution and the de Sitter solution in sec-tion IV.The black hole solution in Eq. (61) has two effectivehairs, which reduces to the Schwarzschild solution in aspecial case. The stability, uniqueness, and thermody-namical properties of this black hole solution are of the-oretical interest. The α hair of this solution shall correctthe geodesic equations, and its value can be restrictedby the experimental data from the perihelion precessionof Mercury, the deflection of light by the sun and thegravitational redshift.The coupling constants κ and λ is required to satisfythe constraint (74) for the existence of de Sitter solution.It remains to be answered wether this constraint dependson our ansatz (66) and (67). The existence of de Sittersolution without cosmological constant supports that ourbimetric gravity model could have the capability to in-terpret the cosmological acceleration. To obtain a real-istic cosmological model, the matter energy-momentumtensor is required to be plugged into Eq. (53). BecauseEq. (53) is a complex equation, an additional energy-momentum tensor besides the conventional one could berequired to ensure the consistency of the equation.Bimetric gravity generally suffers from a ghost prob-lem [10] and the vDVZ discontinuity problem [21, 22]. Adetailed analysis is required in order to see wether ourbimetric gravity model is free from these problems. ACKNOWLEDGMENTS
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