Exploring the gravity sector of emergent higher-spin gravity: effective action and a solution
UUWThPh 2021-1
Exploring the gravity sector of emergenthigher-spin gravity: effective action and a solution
Stefan Fredenhagen a,b and Harold C. Steinacker a a Faculty of Physics, University of ViennaBoltzmanngasse 5, 1090 Vienna, Austria b Erwin Schr¨odinger International Institute for Mathematics and Physics,University of Vienna, Boltzmanngasse 9, 1090 Vienna, Austria
Abstract
We elaborate the description of the semi-classical gravity sector of Yang-Mills ma-trix models on a covariant quantum FLRW background. The basic geometricstructure is a frame, which arises from the Poisson structure on an underlying S bundle over space-time. The equations of motion for the associated Weitzenb¨ocktorsion obtained in [1] are rewritten in the form of Yang-Mills-type equations forthe frame. An effective action is found which reproduces these equations of mo-tion, which contains an Einstein-Hilbert term coupled to a dilaton, an axion anda Maxwell-type term for the dynamical frame. An explicit rotationally invariantsolution is found, which describes a gravitational field coupled to the dilaton. E-mail: [email protected] , [email protected] a r X i v : . [ h e p - t h ] J a n ontents Z ˙ α
218 Discussion 24A Conventions and useful formulas 25B Divergence constraint and antisymmetric torsion 26
B.1 Divergence constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26B.2 The totally antisymmetric torsion T ( AS ) . . . . . . . . . . . . . . . . . . . . . 28 C 6-dimensional configuration space and constraints 29
C.1 General setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29C.2 SO (3)-invariant functions and Poisson brackets . . . . . . . . . . . . . . . . . 31 D Geometric energy-momentum tensor 31E Geometric actions and identities 33
E.1 Einstein-Hilbert term from torsion . . . . . . . . . . . . . . . . . . . . . . . . . 33E.2 Axion identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34E.3 Variation of the action S T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Classical gravity is well described by general relativity (GR) whose dynamics is given bythe Einstein-Hilbert action. However, this formulation is not well suited for quantization.Moreover, there is no straightforward way to generalize the Einstein-Hilbert action to non-commutative spaces, which are expected in a quantum theory of gravity [2]. This suggeststhat the Einstein-Hilbert action should only be considered as an effective action, rather thana fundamental starting point. This is indeed what happens in string theory which does leadto a similar effective gravity action, but only in critical dimensions 10 or 26. This motivates1o consider matrix models such as the IIB or IKKT matrix model [3] as fundamental startingpoint. While this model is known to be related to critical IIB string theory, it opens up thepossibility to study novel ways and mechanisms to obtain space-time and gravity on suitablenoncommutative spaces or brane solutions, cf. [4–13].In this paper, we study the non-linear dynamics of the effective gravity sector whichemerges from the IKKT or IIB matrix model on a certain type of 3+1-dimensional covariantquantum space [14] (cf. [15–18]). The effective metric arises from a dynamical frame E ˙ α = { Z ˙ α , . } , which arises from the basic matrices Z ˙ α of the matrix model via the Poisson structurein the semi-classical limit. The semi-classical equations of the matrix model were recast in [1]as non-linear geometric equations, in terms of the torsion of the Weitzenb¨ock connectionassociated to this frame. Covariance under volume-preserving diffeomorphisms is manifest inthis formalism, which originates from the gauge invariance of the matrix model. The frameand all derived objects are in general higher-spin valued, somewhat reminiscent of Vasiliev-type higher-spin gauge theory [19, 20]. The higher-spin modes arise here due to the internalstructure of the brane solution as twisted S bundle over space-time. The resulting 3+1-dimensional gravity theory was shown to be free of ghosts in [21], and to reproduce the Ricci-flat linearized metric perturbations of general relativity including a linearized Schwarzschildsolution [22].In the present paper, we rewrite these geometric equations for the frame in a more familiarform using the standard Levi-Civita connection. It turns out that the totally antisymmetricsector of the torsion reduces on-shell to a scalar field identified as axion, while the contractionof the torsion determines the dilaton. We obtain covariant equations of motion for all thesefields, including generalized Maxwell-type equations for the frame.Moreover, we find an effective action for these geometric quantities, in a generalized sense:The effective action reproduces these equations of motion, provided the frame, the dilatonand the axion are considered as independent quantities. This action takes a fairly familiarform involving an Einstein-Hilbert term and kinetic terms for the frame, the dilaton and theaxion. This action, however, becomes a trivial identity once all constraints of the frameworkare used. Nevertheless, the action is expected to be useful, at least as a device to recover thegeometric equations of motion in a transparent way.The resulting gravity theory is clearly richer than GR, notably due to the presence of adilaton and axion field, which is determined by the frame. In particular, there is no manifestlocal Lorentz (gauge) invariance acting on the frame. That is not a problem per se, andinvariance under volume-preserving diffeos is manifest [1]. The vacuum solutions are notguaranteed to be Ricci-flat; however, Ricci-flatness does hold at the linearized level [22].As a first step to understand the resulting physics at the non-linear level, we obtain inthe second part of this paper an explicit solution of the non-linear geometric equations for aspherically symmetric static geometry centered at some point in space. The resulting geometrycoincides with the linearized Schwarzschild geometry at the linearized level (as expected),but it deviates from it at the non-linear level, with a non-vanishing dilaton contribution.In particular there is no horizon, and the singularity at the origin is mild and integrable.Therefore this solution should presumably be interpreted as vacuum solution without matter,and its physical significance is not clear at this point; however, it illustrates how gravity isextended in the present framework. From a structural point of view, it also illustrates howthe higher-spin contributions may cancel in the effective metric.We expect that the present paper should be a useful starting point for finding further2olutions of the model, and exploring the resulting theory in more depth. The paper isorganized as follows: after a brief summary of the required background in section 2, the mainresults are the covariant equations of motion in section 3, the effective action in section 4, andthe new solution in sections 6 and 7. In addition, the appendices contain a number of newidentities and structural results. The model underlying the present paper is the IKKT or IIB matrix model [3] with mass term, S [ Z, Ψ] = Tr (cid:0) [ Z ˙ α , Z ˙ β ][ Z ˙ α , Z ˙ β ] + 2 m Z ˙ α Z ˙ α + ΨΓ ˙ α [ Z ˙ α , Ψ] (cid:1) . (2.1)We will ignore the fermionic matrices Ψ. Solutions of this model are then given in termsof some “matrix configurations” consisting of 9+1 hermitian matrices Z ˙ α ∈ End ( H ) for˙ α = 0 , ...,
9, which satisfy the equations of motion (e.o.m.)[ Z ˙ α , [ Z ˙ α , Z ˙ β ]] = m Z ˙ β . (2.2)Dotted indices transform under a global SO (9 , η ˙ α ˙ β . Fluctuations Z ˙ α → Z ˙ α + A ˙ α of this background are then governed by a(non-commutative) gauge theory, where the fluctuations A are typically viewed as functionson a background “brane” M (6) defined by the Z ˙ α .In the present paper, we focus on one such solution Z ˙ α given in [14], which can be inter-preted in terms of a cosmological Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) spacetime;for an introductory review see [23]. In this solution only the Z ˙ α for ˙ α = 0 , ..., Z ˙ α = 0 for ˙ α = 4 , ...,
9; we will thus consider only the indices ˙ α = 0 , ..., M (6) loc ∼ = M , × S (2.3)has the structure of a twisted S bundle over space-time M , , i.e. the local stabilizer of apoint p ∈ M , acts non-trivially on the fiber S . Non-trivial harmonics on S then lead tohigher-spin modes on M , . Using the correspondence of matrices with functions, End ( H ) ∼ = C ( M (6) ) , (2.4)one can interpret the Z ˙ α and A ˙ α as (quantized) functions on M (6) . We will work in thesemi-classical limit, where commutators [ ., . ] ∼ i { ., . } are replaced by Poisson brackets. Thenthe e.o.m. (2.2) reduces to { Z ˙ α , ˆΘ ˙ α ˙ β } = m Z ˙ β , ˆΘ ˙ α ˙ β = −{ Z ˙ α , Z ˙ β } . (2.5)At the linearized level, the resulting higher-spin gauge theory was elaborated in [14, 21]. Asuitable formalism to understand the gravity sector at the non-linear level was developed in [1],based on the Weitzenb¨ock connection associated to the frame E ˙ α = { Z ˙ α , . } = E µ ˙ α ∂ µ , E µ ˙ α = { Z ˙ α , x µ } , (2.6)where x µ are local coordinates on M , . In the present paper, we will develop this descriptionfurther, and find a non-trivial solution at the non-linear level. More precisely, the bundle is SO (3 ,
1) equivariant. .1 The background geometry Without giving details about the specific solution of the matrix model we are considering,we start by reviewing the background geometry to which it corresponds, more details can befound in [14]. The background spacetime M , is (a double cover of) the segment of R , with Cartesian coordinates x µ satisfying η µν x µ x ν =: − R cosh ( η ) ≤ − R (2.7)thereby defining the cosmic time variable η . Here, R is related to the mass parameter inthe matrix model e.o.m. (2.2) by ( mR ) = 3. The space-like 3-hyperboloids H defined by η = const will be recognized below as equal-time slices of a k = − x µ transform as a vector of SO (3 , H rather than as (local) Lorentz group.As indicated above, M , is the base manifold of an S bundle M (6) . The fiber S isdescribed by 4 further functions t µ on M (6) which also transform as vector of SO (3 , t µ t µ = ˜ R − cosh ( η ) , (2.8) t µ x µ = 0 , (2.9)where ˜ R is a parameter that characterizes the solution. At the reference point x = ( x , , , t = 0 and t i t i = ˜ R − cosh ( η ). Hence the x µ , t µ should be viewed asfunctions on M (6) which characterize the bundle structure. The space of functions C ( M (6) ) = (cid:77) s ≥ C s (2.10)accordingly decomposes into higher-spin modules C s , which can be viewed as functions on M , taking value in the spin s representation of the local stabilizer SO (3), given by irreducible rank s polynomials in t µ . The bundle space M (6) carries a symplectic form ω which is invariantunder SO (4 , { x µ , x ν } = θ µν = − ˜ R R { t µ , t ν } (2.11) { t µ , x ν } = η µν sinh( η ) . (2.12)The Poisson tensor θ µν satisfies the constraints t µ θ µν = − sinh( η ) x ν , (2.13a) x µ θ µν = − ˜ R R sinh( η ) t ν , (2.13b) η µν θ µκ θ νλ = R ˜ R η κλ − R ˜ R t κ t λ + ˜ R x κ x λ . (2.13c)Explicitly, θ µν = ˜ R cosh ( η ) (cid:16) sinh( η )( x µ t ν − x ν t µ ) + (cid:15) µνκλ x κ t λ (cid:17) . (2.14)In this geometric regime, the matrix model solution is given by ¯ Z ˙ α = t ˙ α . For more details werefer to [14]. The two sheets of M , describe the universe before and after the Big Bounce. We will only consider thelate-time era, where the global structure is irrelevant. The matrix model solution is realized on a doubleton representation space H n with an integer parameter n . For large n the parameter ˜ R that characterizes the size of the fiber is given by ˜ R = n R . rame and metric. The background solution ¯ Z ˙ α = t ˙ α gives rise to the background frame(2.6) ¯ E ˙ α = { ¯ Z ˙ α , . } = ¯ E µ ˙ α ∂ µ , ¯ E µ ˙ α = { ¯ Z ˙ α , x µ } = sinh( η ) δ µ ˙ α (2.15)which defines the auxiliary metric¯ γ µν = η ˙ α ˙ β ¯ E µ ˙ α ¯ E ν ˙ β = sinh ( η ) η µν . (2.16)It turns out that the effective metric ¯ G which governs the kinetic term of all propagatingmodes is a conformal rescaling of the auxiliary metric ¯ γ µν , given by [14]¯ G µν = 1¯ ρ ¯ γ µν , ¯ ρ = ρ M (cid:112) | ¯ γ | − = sinh ( η ) . (2.17)Here, | ¯ γ | is the absolute value of the determinant of (¯ γ µν ), and ρ M = sinh( η ) (2.18)(in Cartesian coordinates x µ ) is the density arising from the SO (4 , ρ M d x on M , , which originates from the symplectic volume form on M (6) . ¯ G µν is a(hyperbolic, k = −
1) FLRW metric, and can be written in terms of the cosmic scale factor a ( t ) and the comoving time t as ds G = ¯ G µν dx µ dx ν = − dt + a ( t ) d Σ , (2.19)where d Σ is the SO (3 , H . The cosmic scale parameter a ( t ) is determined as a ( t ) = R sinh( η ) cosh ( η ) η →∞ ≈ R sinh ( η ) (2.20) dt = R sinh( η ) dη . (2.21)Note that a ( t ) also sets the curvature scale of the background. In the present paper, we willfocus on the asymptotic regime a ( t ) → ∞ i.e. η → ∞ , considering only perturbations of thegeometry on scales far below the cosmic scale. Then the space-like metric d Σ on H ≈ R can be approximated by a flat metric near x i = 0, neglecting the cosmic curvature. We canthen rewrite the Cartesian coordinates around the reference point ξ = ( ξ , , ,
0) in terms oflocal spherical coordinates with radius r := x i x j δ ij (cid:28) x ,η µν x µ x ν = − x + r = − R cosh ( η ) . (2.22)In this regime, the internal sphere and the Poisson tensor are characterized by [1] | t | ≈ ˜ R − cosh( η ) θ i ξ ≈ ˜ R R t i (cid:29) θ ij ξ ≈ ˜ R R sinh( η ) (cid:15) ijk t k η →∞ ∼ const . (2.23)5 Geometric description of the non-linear regime
We recall the geometric formalism based on (a higher-spin generalization of) the Weitzenb¨ockconnection and torsion. We will focus on local perturbations of the geometry in the asymptoticregime η → ∞ , where the dominant contributions arise from the derivations along M , ratherthan the internal directions; for a more detailed discussion see [1]. The fundamental degrees of freedom of the matrix model are given by matrix configurations Z ˙ α and the associated vielbein E ˙ α = { Z ˙ α , . } , E µ ˙ α = { Z ˙ α , y µ } (3.1)where y µ are any local coordinate functions on M , . The inverse vielbein is defined as usual E ˙ αµ E µ ˙ β = δ ˙ α ˙ β , γ µν = η ˙ α ˙ β E µ ˙ α E ν ˙ β ,E ˙ αν E µ ˙ α = δ µν , η ˙ α ˙ β = E µ ˙ α E ν ˙ β γ µν . (3.2)It is natural to define a Weitzenb¨ock connection on M , which respects this vielbein0 = ∇ ν E µ ˙ α = ∂ ν E µ ˙ α + Γ µνρ E ρ ˙ α (3.3)cf. [24]. This connection is automatically compatible with the metric ∇ γ µν = 0. For anyvector field V µ on M , (possibly with a dependence on the fiber parameter t µ correspondingto higher-spin modes), the (Weitzenb¨ock) covariant derivative is then ∇ µ V ν = ∂ µ V ν + Γ νµρ V ρ . (3.4)This connection is flat since the frame is parallel, ∇ E ˙ β = 0. However, it typically has torsion, T [ X, Y ] = ∇ X Y − ∇ Y X − [ X, Y ] (3.5)which can be computed as [1] T ρµν = Γ ρµν − Γ ρνµ T ˙ αµν = T ρµν E ˙ αρ = ∂ µ E ˙ αν − ∂ ν E ˙ αµ . (3.6)The torsion satisfies a Bianchi identity,0 = ∇ σ T µλρ + ∇ λ T µρσ + ∇ ρ T µσλ + T νλρ T µνσ + T νρσ T µνλ + T νσλ T µνρ (3.7)which follows from the first Bianchi identity for a connection with zero curvature [25], or fromthe Jacobi identity in the matrix model [1]. Its contraction gives the identity ∇ µ T µλρ = 0 . (3.8)In terms of the frame-valued torsion T ˙ αµν the Bianchi identity reads0 = ∂ σ T ˙ αµν + ∂ µ T ˙ ανσ + ∂ ν T ˙ ασµ . (3.9)6iewing T ˙ αµν as components of a two-form T ˙ α , we see from (3.6) that it is the exteriorderivative of the vielbein, and the Bianchi identity simply states that T ˙ α is closed, T ˙ α = dE ˙ α = 12 T ˙ αµν dx µ ∧ dx ν , dT ˙ α = 0 . (3.10)The Levi-Civita connection ∇ ( γ ) for the metric γ µν is related to the Weitzenb¨ock connectionvia Γ ρµν = Γ ( γ ) ρµν + K ρµν ∇ µ V ν = ∇ ( γ ) µ V ν + K νµρ V ρ . (3.11)Here K σµν = 12 ( T σµν + T σµν − T σν µ ) = − K σµ ν (3.12)is the contorsion of the Weitzenb¨ock connection, which is antisymmetric in the last 2 indices. Itcarries the same information as the torsion, which can be reconstructed as T σµν = K σµν − K σνµ .Note that all these quantities can in general take values in the higher-spin algebra C of functionson M (6) . In particular, the torsion of the cosmic background is given by (see (7.26) in [1])¯ T µρσ ≈ a ( t ) (cid:0) δ µσ τ ρ − δ µρ τ σ (cid:1) (3.13)where τ µ = ¯ G µν τ ν and τ = x µ ∂ µ = a ( t ) ∂ t (3.14)is the time-like vector field on the FLRW background. Effective metric.
Similarly to the discussion of the background geometry, the form of thekinetic term for fluctuations around a matrix model solution leads us to introduce an effectivemetric G µν (see (2.17)) that differs from γ µν by a conformal factor (see eq. (4.10) in [1]): G µν := 1 ρ γ µν , ρ = ρ M (cid:112) | γ | − (3.15)where ρ M d y is the symplectic volume form. We shall therefore denote the scalar field ρ as dilaton . The Levi-Civita connection ∇ ( G ) for the effective metric G µν is thenΓ ( G ) σµν = Γ σµν + δ σν ρ − ∂ µ ρ − K σµν . (3.16)Here K σµν = K σµν + (cid:16) G µν ρ − ∂ σ ρ − δ σµ ρ − ∂ ν ρ (cid:17) = −K σµ ν (3.17) T σµν = T σµν + ρ − (cid:0) δ σν ∂ µ ρ − δ σµ ∂ ν ρ (cid:1) (3.18)is the Weitzenb¨ock contorsion and torsion tensor of the effective frame E ˙ αµ = ρE ˙ αµ . (3.19)7his allows us to rewrite the effective Levi-Civita connection in terms of the Weitzenb¨ockconnection and the contorsion: ∇ ( G ) µ V σ = ∇ µ V σ − K σµν V ν + ρ − ∂ µ ρ V σ . (3.20)Accordingly, its indices should be raised and lowered with G µν . To avoid any confusion, it issafer to write all connection and (con)torsion symbols with two lower and one upper index,where no ambiguity arises. Calligraphic fonts indicate the effective frame.The Jacobi identity in the matrix model implies a relation between the trace of the torsionand the dilaton [1, Lemma 5.2], T µµσ = K µµσ = 2 ρ ∂ σ ρ . (3.21)For the rescaled frame resp. effective metric, we analogously have T µµσ = K µµσ = − ρ − ∂ σ ρ . (3.22) Starting from the semi-classical e.o.m. (2.5), the following e.o.m. for the torsion in vacuumwas obtained in [1] ∇ ν T νρµ + T σν µ T νσρ = m γ ρµ . (3.23)The non-linear equation of motion encodes the non-linear structure of the Yang-Mills equationsof motion (2.5). It can be rewritten in terms of the Levi-Civita connection using (3.20)and (3.22), m γ ρµ = ∇ ( G ) ν T νρµ − K σνρ T νσµ − K σνµ T νρσ + T σν µ T νσρ = ∇ ( G ) ν T νρµ + 12 ( − T λνρ T νλµ − T λρ ν T νλµ − T νρλ T λνµ + T νρλ T λµ ν ) − ρ − ∂ σ ρ ( T σρ µ + T σµρ ) + 2 ρ − ∂ µ ρ∂ ρ ρ . (3.24)Together with the Bianchi identity (3.7) (and the flatness of ∇ ), this captures the dynamicalcontent of the model. The antisymmetric part of the e.o.m.
When we consider the e.o.m. (3.23), we observethat the right hand side is proportional to the metric and hence is a symmetric tensor. The lefthand side is not automatically symmetric, and the e.o.m. can be split into two components:one that requires that the antisymmetric part of the left hand side vanishes, and the remainingsymmetric part.The antisymmetric part of (3.23) reads ∇ ν (cid:0) T νρµ − T νµρ (cid:1) + 2 T σν [ µ | T νσ | ρ ] = 0 (3.25)where the square brackets denote anti-symmetrization. When we introduce the totally anti-symmetric component of the torsion as (cf. appendix B.2) T ( AS ) νρµ = T νρµ + T νµ ρ + T νρµ , (3.26)8e can rewrite the antisymmetric e.o.m. (3.25) as ∇ ν T ( AS ) νρµ = T σν [ ρ T ( AS ) νµ ] σ , (3.27)where the contracted Bianchi identity (3.8) has been used.The antisymmetric e.o.m. can be written in a more convenient form using the Levi-Civitacovariant derivative with respect to the effective metric G . Using the relation (3.20) between ∇ ( G ) and ∇ , as well as the result on the trace of the contorsion (3.22) we find ρ − ∇ ( G ) ν (cid:0) ρ T ( AS ) νρµ (cid:1) = ∇ ν T ( AS ) νρµ + 2 K σν [ ρ | T ( AS ) νσ | µ ] (3.28)= ∇ ν T ( AS ) νρµ − T σν [ ρ | T ( AS ) ν | µ ] σ , (3.29)which vanishes due to the equation of motion. Hence ρ − ∇ ( G ) ν ( ρ T ( AS ) ν ρµ ) = 0 . (3.30)This means that it is consistent to set T ( AS ) ρσµ = 0, which holds for the background solution ¯ T in (3.13). We can interpret the antisymmetric part T ( AS ) ν ρµ as the components of a 3-form,13! G νν (cid:48) T ( AS ) ν (cid:48) ρµ dx ν ∧ dx ρ ∧ dx µ = ρ T ˙ α ∧ E ˙ α . (3.31)The equation of motion (3.30) of T ( AS ) can then be rewritten – via the Hodge star with respectto G – as ∗ d ∗ ( ρ T ˙ α ∧ E ˙ α ) = 0 . (3.32)Expressing T ( AS ) ν ρµ in terms of the ∗ -dual 1-form T = T σ dx σ , T ( AS ) νρµ =: (cid:112) | G | G νν (cid:48) ε ν (cid:48) ρµσ G σσ (cid:48) T σ (cid:48) ⇐⇒ ρ T ˙ α ∧ E ˙ α = ∗ T , (3.33)the equation of motion (3.30) for T ( AS ) becomes0 = ρ − ∇ ( G ) ν ( (cid:112) | G | G νν (cid:48) ε ν (cid:48) ρµσ ρ G σσ (cid:48) T σ )= ρ − (cid:112) | G | G σσ (cid:48) G νν (cid:48) ε νρµσ ∂ ν (cid:48) ( ρ T σ ) . (3.34)In terms of differential forms this relation simply reads d ( ρ T ) = 0 . (3.35)This in turn means that T µ can be written on-shell as ρ T µ = ∂ µ ˜ ρ (3.36)in terms of a scalar field ˜ ρ , which will be denoted as axion , for reasons that will become clearbelow. Hence the anti-symmetric part of the e.o.m. for the torsion reduces the 4 dof of T ( AS ) to the scalar field ˜ ρ , while the remaining 3 dof of the general frame disappear on-shell.The Bianchi identity (3.10) for T ˙ α implies ∗ d ∗ ( ρ − T ) = ∗ d ( T ˙ α ∧ E ˙ α ) = ∗ ( T ˙ α ∧ T ˙ α ) . (3.37)9xpressing T µ on-shell in terms of ˜ ρ via (3.36) results in ∗ d ∗ ( ρ − d ˜ ρ ) = ∗ ( T ˙ α ∧ T ˙ α ) , (3.38)or, in components (for an explicit calculation see (B.22)), ∇ µ ( G ) ( ρ − ∂ µ ˜ ρ ) = 14 (cid:112) | G | − ε νρµκ T ˙ ανρ T κµ ˙ α . (3.39)Thus ˜ ρ is recognized as an axion-like field. Eom for ρ . Taking the trace of the equation of motion (3.23) for T yields γ νσ ∇ σ T ρνρ + γ µρ T σν µ T νσρ = 4 m . (3.40)We now rewrite γ in terms of the effective metric G via (3.15), and express ∇ in terms of ∇ ( G ) using (3.20). Using the results (3.21) and (3.22) for the trace of the (con-)torsion, we obtain − ρ G µν ∇ ( G ) µ ( ρ − ∂ ν ρ ) + ρ G µρ T σν µ T νσρ = 4 m . (3.41)Finally, rewriting the quadratic term in the torsion using (B.19) and replacing T µ by itson-shell value (3.36), we arrive at (cf. (5.50) in [1]) −∇ µ ( G ) (cid:0) ρ − ∂ µ ρ ) = 2 ρ − m + 14 T σµ ρ T ρνσ G µν − ρ − G µν ∂ µ ˜ ρ∂ ν ˜ ρ . (3.42)We have thus identified two scalar fields ρ and ˜ ρ which encode the trace and the totally anti-symmetric components of the torsion tensor, respectively. Both satisfy second-order equationsof motion sourced by the torsion. This reflects the fact that in contrast to general relativity,there is no local Lorentz invariance for the frame. In the present theory, the frame is a physicalobject which satisfies the divergence constraint ∇ ( G ) ν ( ρ − E ν ˙ α ) = 0 (3.43)which is a consequence of the relation (3.21) between the contraction of the torsion and thedilaton (as shown in appendix B.1), and gives rise to extra physical degrees of freedom encodedin ρ and ˜ ρ . These have no counterpart in general relativity. Similar fields are known to arisein various generalizations of general relativity [26–28]. Equation for frame-valued torsion.
The equation of motion (3.23) for the torsion canbe rewritten in terms of the frame-valued torsion T ˙ αµν = T σµν E ˙ ασ , γ νν (cid:48) ∇ ν (cid:48) T ˙ ανρ + T νσρ T σ ˙ αν = m E ˙ αρ , (3.44)where one has used that the vielbein is covariantly constant.This equation can be expressed in terms of the Levi-Civita connection with respect to theeffective metric as follows γ νν (cid:48) ( ∇ ( G ) ν (cid:48) T ˙ ανρ − K σν (cid:48) ρ T ˙ ανσ + ρ − ∂ ν (cid:48) ρT ˙ ανρ ) + T νσρ T σ ˙ αν = m E ˙ αρ (3.45)10sing (3.22). The second term can be rewritten using − γ νν (cid:48) K σν (cid:48) ρ T ˙ ανσ + T σ νρ T ˙ ανσ = − ρ G νν (cid:48) T ( AS ) σν (cid:48) ρ T ˙ ανσ − ρG σσ (cid:48) ∂ σ (cid:48) ρT ˙ αρσ (3.46)where G νν (cid:48) = ρ − γ νν (cid:48) , which results in ρ − ∇ ν ( G ) ( ρ T ˙ ανρ ) = 12 G νν (cid:48) T ( AS ) σν (cid:48) ρ T ˙ ανσ + ρ − m E ˙ αρ . (3.47)Expressing T ( AS ) in terms of the dual 1-form T (see (B.15)), we obtain (using (A.2)) ∇ ν ( G ) ( ρ T ˙ ανρ ) = 12 ρ (cid:112) | G | − ε νρ (cid:48) σµ G ρρ (cid:48) T µ T ˙ ανσ (cid:48) + m E ˙ αρ , (3.48)or, using the on-shell relation (3.36) for T µ , ∇ ν ( G ) ( ρ T ˙ ανρ ) = 12 (cid:112) | G | − ε νρ (cid:48) σµ G ρρ (cid:48) ∂ µ ˜ ρ T ˙ ανσ (cid:48) + m E ˙ αρ . (3.49)In terms of differential forms (see (3.10)), this equation can be written concisely as d ( ρ (cid:63) T ˙ α ) = d ˜ ρ ∧ T ˙ α + m (cid:63) E ˙ α . (3.50)This can easily be verified for the cosmic background solution. If the derivatives ∂ρ ≈ ≈ ∂ ˜ ρ vanish or are negligible, this reduces to the linear equation for the torsion ∇ ν ( G ) T ˙ ανµ = m ρ E ˙ αµ (3.51)which has the structure of Maxwell equations for each frame index ˙ α . Note that m ¯ ρ = a ( t ) isthe cosmic curvature scale, so that the rhs is often negligible. This equation can be rewrittenas harmonic equation for the frame E ˙ αν ∆ ( G ) E ˙ αν − ρ − R νσ E ˙ ασ = m ρ E ˙ αν , (3.52)where ∆ ( G ) = ∇ µ ( G ) ∇ ( G ) µ , dropping again ∂ρ ≈ ≈ ∂ ˜ ρ and using the divergence constraint(3.43). These are reminiscent of Maxwell equations for the vector potential. The Ricci tensor for the effective metric G µν can be expressed in terms of the torsion and thedilaton. Using the e.o.m. (3.23) for the torsion, the following relation was derived in [1] R µν − G µν R = T µν (3.53)11absorbing a factor 8 π for convenience) where T µν is the effective energy-momentum tensorassociated to the torsion, T µν = −
12 ( T δρ ν T ρµδ + T δρ µ T ρνδ ) − K ρδ µ K δρ ν + 2 ρ − ∂ µ ρ∂ ν ρ + G µν (cid:16) − T σδρ T δρσ + 18 T δσρ T δσρ − ρ − ∂ρ · ∂ρ − R − ρ − (cid:17) . (3.54)As shown in appendix D (see (D.6)), this can be written more succinctly as T µν = T µν [ E ˙ α ] + T µν [ ρ ] + T µν [ ˜ ρ ] − ρ − m G µν (3.55)in terms of contributions of the dilaton ρ , the axion ˜ ρ , and a Maxwell-like contribution fromthe frame fields E ˙ α . This shows again that the frame is physical, and acts – together withthe dilaton and the axion – as source of the Einstein equations. The contributions T µν [ E ˙ α ]and T µν [ ˜ ρ ] of the frame and the axion turn out to have slightly non-standard form and sign,and Ricci-flat geometries may arise if the various contributions cancel. While this may seemunlikely at first sight, it was shown previously that the standard Ricci-flat solutions ariseindeed in the linearized regime [14, 22], and they are expected to arise more generally in theabsence of axions and dilatons in view of (3.52). Even though E ˙ α , ρ and the metric G µν are mutually related, let us consider them as inde-pendent quantities for the moment. Then all solutions of the above equations of motion arecritical points of the following effective action S eff [ E, G, ρ ] = c R S R + c E S E + c T S T + c ρ S ρ + c m S m + c ˜ m S ˜ m (4.1)where S R = (cid:90) d x (cid:112) | G |R [ G ] S E = (cid:90) d x (cid:112) | G | ρ G νν (cid:48) G σσ (cid:48) T ˙ ανσ T ν (cid:48) σ (cid:48) ˙ α = 2 (cid:90) ρ dE ˙ α ∧ (cid:63)dE ˙ α S T = (cid:90) d x (cid:112) | G | G µν T µ T ν S ρ = (cid:90) d x (cid:112) | G | G µν ρ − ∂ µ ρ∂ ν ρS m = (cid:90) d x (cid:112) | G | m E ˙ ακ E ˙ ακ (cid:48) G κκ (cid:48) S ˜ m = (cid:90) d x (cid:112) | G | m ρ − . (4.2)12ere G, E and ρ are considered as independent objects which define the torsion T ˙ αµν , and T µ is defined as T = (cid:63)T ( AS ) (B.14). The variations of these terms are as follows δS R = (cid:90) d x (cid:112) | G | (cid:16) R µν − G µν R (cid:17) δG µν δS E = (cid:90) d x (cid:112) | G | (cid:0) − δG µν T µν [ E ˙ α ] − ∇ ν ( ρ G κκ (cid:48) T ˙ ανκ ) δE ˙ ακ (cid:48) + 2 G νν (cid:48) T ˙ ανσ T σν (cid:48) ˙ α ρ − δρ (cid:1) δS T = (cid:90) d x (cid:112) | G | (cid:0) δG µν T ( AS ) µν [ T ] + 4 T · T ρ − δρ (cid:1) − (cid:90) d x ρ δE ˙ αµ (cid:0) T κ ε νµσκ T ˙ ανσ + E ˙ ασ ε νµσκ ρ − ∂ ν ( ρ T κ ) (cid:1) δS ρ = (cid:90) d x (cid:112) | G | (cid:16) δG µν T µν [ ρ ] − ∇ µ ( G ) ( ρ − ∂ µ ρ ) ρ − δρ (cid:17) δS m = (cid:90) d x (cid:112) | G | (cid:0) − m ρ − G µν δG µν + 2 m G µν E ˙ αµ δE ˙ αν (cid:1) δS ˜ m = (cid:90) d x (cid:112) | G | (cid:16) − m ρ − G µν δG µν − m ρ − δρ (cid:17) (4.3)using the results of Appendix D. The coefficient of δρ coincides with the e.o.m. (3.42) if c ρ = − c E , c T = − c E , c ˜ m = − c E . (4.4)The coefficient of δG µν coincides with the e.o.m. (3.53) and using (3.55) if c E = 12 c R , c T = − c R , c ρ = − c R , c m = c R . (4.5)Finally, the coefficient of δE ˙ α vanishes as a consequence of the e.o.m. (3.48) if c T = − c E . (4.6)Remarkably, all these conditions are compatible. Therefore all critical points of the semi-classical matrix model are critical points of S eff [ E, G, ρ ] = 2 S R + S E − S T − S ρ + 2 S m − S ˜ m (4.7) for independent variations of E, G and ρ . This result should be useful to understand betterthe present gravity theory. For example, it provides an “explanation” for (or at least a book-keeping of) the effective Einstein equation (3.53) and the explicit form of the contributions(3.55) on the rhs.It should be noted that the sign of S E is non-standard, which is reflected in the negativesign in the energy-momentum tensor associated to the frame (D.10). This looks “wrong”at first sight, but remember that the E ˙ α does not play the role of a vector field coupled togravity, it rather defines the metric. This is very different from the usual role of a vector fieldin gravity, and it was shown in [21] that no ghosts arise in this theory, at least at the linearizedlevel. 13 xtra gauge invariance. Note that the unrestricted action S eff [ E, G, ρ ] enjoys the gaugeinvariance E ˙ αµ → E ˙ αµ + ∂ µ Λ ˙ α (4.8)subject to the constraint 0 = E ˙ αµ ∂ µ Λ ˙ α (4.9)(sum over ˙ α !), which guarantees that the mass term S m is invariant. We shall not pursue thisobservation any further here. Alternative action using the axion.
Now we impose the e.o.m. (3.34) for the totallyantisymmetric part of the torsion, replacing T µ → ρ − ∂ µ ˜ ρ according to (3.36). If we consider˜ ρ as independent quantity, its e.o.m. (3.39) is recovered if we replace the term S T by S T → c ˜ ρ S ˜ ρ + c ˜ E S ˜ E (4.10)where S ˜ ρ = (cid:90) d x (cid:112) | G | G νν ρ − ∂ µ ˜ ρ∂ ν ˜ ρS ˜ E = (cid:90) d x ˜ ρ ε νσµκ T ˙ ανσ T µκ ˙ α = 4 (cid:90) ˜ ρ dE ˙ α ∧ dE ˙ α . (4.11)Hence ˜ ρ is recognized as axion associated to the frame field. The variations are δS ˜ ρ = (cid:90) d x (cid:112) | G | (cid:0) − ∇ µ ( G ) ( ρ − ∂ µ ˜ ρ ) δ ˜ ρ − G µν ρ − ∂ µ ˜ ρ∂ ν ˜ ρδρ − T µν [ ˜ ρ ] δG µν (cid:1) δS ˜ E = (cid:90) d x (cid:0) − ε νµσκ T ˙ ανσ ∂ κ ˜ ρ δE µ ˙ α + ε νκσµ T ˙ ανσ T µκ ˙ α δ ˜ ρ (cid:1) . (4.12)The δ ˜ ρ terms reproduce the e.o.m. (3.39) for ˜ ρ if c ˜ E = − c ˜ ρ . The coefficient of δρ agrees withthat of S T if c ˜ ρ = − c T , and the remaining equations of motion are also satisfied if c T = 2 c ˜ E .Therefore all critical points of the semi-classical matrix model are critical points of S eff [ E, G, ρ, ˜ ρ ] = 2 S R + S E + S ˜ ρ − S ˜ E − S ρ + 2 S m − S ˜ m (4.13) for independent variations of E, G, ρ and ˜ ρ . This is somewhat reminiscent of axion-dilatongravity, cf. [26–28], though the kinetic term for the frame is distinct. Reduced action and triviality.
Now we impose the constraint (3.15) G µν = ρ − E ˙ αµ E ν ˙ α . (4.14)Imposing also the divergence constraint, we can use the identity (E.4) S E = 4( S R − S ρ ) . (4.15)14hen S eff [ E ] = c R S R + c E S R − S ρ ) + c T S T + c ρ S ρ + c m S m = c R (cid:16) S R − S T − S ρ (cid:17) + c m S m . (4.16)This action is trivial: the first term vanishes due to the identity (E.6)3 S R = 14 S ρ + 12 S T , (4.17)and the mass term is topological in view of (A.8). This means that the effective action becomestrivial if all constraints are imposed. Even though this may be disappointing, it is in line withthe notorious difficulty of obtaining a geometric action for higher-spin gravity. The presentmatrix-model framework does provide such an action, where the frame is obtained as derivative of the (semi-classical) matrix degrees of freedom. The geometric actions (4.7) and (4.13) onlyserve as effective actions for an extended configuration space where the geometrical quantitiesare viewed as independent. It is well-known that for any frame E ˙ αν associated to a given metric G µν , one can act with alocal Lorentz transformation on the frame as follows E ˙ αν ( x ) → Λ ˙ β ˙ α ( x ) E ˙ βν ( x ) (5.1)leading to the same metric. In this sense the frames form an SO (3 ,
1) bundle over space-time,leading to the Cartan formalism with a spin connection as discussed further below. However,in the present setting, the frame E ˙ β = ρ − E ˙ β satisfies the divergence constraint (3.43), as wellas the on-shell relation (3.34). This means that the above local Lorentz (gauge) symmetryis broken, and the extra degrees of freedom of the frame encode the dilaton ρ and the axion˜ ρ . This is not a problem, but it exhibits a fundamental difference between general relativityand the present theory, where the frame is a fundamental object. The dilaton ρ is also relatedto the invariant symplectic volume via (A.8), leading to a reduction of the diffeomorphisminvariance to the volume-preserving diffeos (B.12).We should therefore ask the following question: Given some metric G µν , can we always finda gauge (i.e. a representative in the frame bundle) and functions ρ and ˜ ρ such that E ˙ β = ρ − E ˙ β satisfies the divergence constraint (3.43) as well as the conditions (3.36)? By counting degreesof freedom (d.o.f.), it is plausible that the answer should be generically yes. Indeed, if weconsider ρ and ˜ ρ as independent fields, then the 6 d.o.f. of Λ ˙ β ˙ α ( x ) together with the 2 d.o.f. ρ and ˜ ρ should allow to satisfy the 4+4 equations (3.36) and (3.43). But this means that boththe dilaton ρ and the axion ˜ ρ are determined by the given metric G µν . This is reflected inrelations such as (D.5). In particular they do not add any propagating degrees of freedom.This is consistent with the results in [14,21] for the linearized case, where it was shown that theonly physical, propagating modes are those encoded in the degrees of freedom of a (massive)graviton. A more detailed understanding of ρ , ˜ ρ and their relation to the geometry should bedeveloped elsewhere. 15 elation with the Cartan structure equations. To make contact with the standardCartan formalism of general relativity, we consider the (co-)frame (3.19) E ˙ β = ρE ˙ β = E ˙ βµ dx µ .Then the spin connection one-form ω ˙ α ˙ β = ω µ ˙ α ˙ β dx µ = − ω ˙ β ˙ α satisfies the first Cartan structureequations d E ˙ α = − ω ˙ β ˙ α ∧ E ˙ β . (5.2)Clearly this is closely related to the (con)torsion of the Weitzenb¨ock connection. Indeed, d E ˙ α = 12 ( ∂ µ E ˙ αν − ∂ ν E ˙ αµ ) dx µ ∧ dx ν = − ω ˙ βµ ˙ α E ˙ βν dx µ ∧ dx ν = 12 T µν ˙ α dx µ ∧ dx ν (5.3)so that T µν ˙ α = − ω ˙ βµ ˙ α E ˙ βν + ω ˙ βν ˙ α E ˙ βµ = ω ν ˙ αµ − ω µ ˙ αν . (5.4)This provides the relation of the spin connection ω of the frame bundle to the torsion tensor T of the Weitzenb¨ock connection. Even though the spin connection is not a tensor in theCartan formalism due to the local Lorentz transformations, T µν ˙ α is a tensor in the presentformalism, where the frame is physical. Hence T µν ˙ α can be viewed as a physical manifestationof the spin connection in the Cartan formulation of Riemannian geometry. In this section, we will find a simple rotationally invariant static solution of the nonlinearequations of motion for the frame. SO (3) -invariant frames. We are interested in local perturbations of the background whichare centered at x i = 0, hence at r = 0, and invariant under (global) SO (3) rotations. Wefirst observe that the background frame ¯ E µ ˙ α = sinh( η ) δ µ ˙ α (2.15) is SO (3)-invariant if theframe index ˙ α is transformed as a vector. This is the manifest global SO (3) symmetry of thematrix model, and it seems natural to keep this SO (3) symmetry manifest for the perturbedrotationally-invariant geometry. This is easily achieved in Cartesian coordinates x µ , adoptingthe notation t = x , r = x i x j δ ij . (6.1)Then the most general spherically symmetric ansatz for the frame E ˙ αµ is E = A ,E i = Ex i ,E i = Dx i ,E ij = F x i x j + δ ij B + S(cid:15) ijm x m , (6.2) The t defined here should not be confused with the comoving time that was introduced in section 2.1 andwhich will not be used in the current section. A, B, D, E, F and S are assumed to be functions of r only. In particular we consideronly static configurations. We can eliminate D and F using a simple change of coordinates t → f ( r ) + t and x i → g ( r ) x i , which is understood from now on. In terms of differential forms E ˙ α = E ˙ αµ dx µ , the (co)frame is then E = Adt , E i = Bdx i + Ex i dt + S(cid:15) ijm x m dx j (6.3)and the associated torsion 2-form T ˙ α = dE ˙ α is obtained as T = dE = A (cid:48) dr ∧ dt ,T i = dE i = B (cid:48) dr ∧ dx i + Edx i ∧ dt + x i E (cid:48) dr ∧ dt + S(cid:15) ijm dx m ∧ dx j + S (cid:48) (cid:15) ijm x m dr ∧ dx j . (6.4)It is easy to see that the totally antisymmetric part T ( AS ) of the torsion vanishes if S = 0,which we assume henceforth. Then T µ = 0 , ˜ ρ = const , (6.5)and the frame is given in matrix form by E ˙ αµ = A Ex B Ex B Ex B . (6.6)The inverse frame is E µ ˙ α = A − − EAB x − EAB x − EAB x B − B −
00 0 0 B − (6.7)and the metric obtained from this frame reads γ = − A + r E ,γ i = BEx i ,γ ij = δ ij B (6.8)in Cartesian coordinates. Now we can compute the dilaton ρ . The condition (3.21) yields − ρ ∂ µ ρ = T µν ˙ α E ˙ αν = ∂ µ E E + ∂ µ E ij E ji − ∂ i E µ E i − ∂ j E iµ E ji , (6.9)taking into account the above form for the (co)frame. For the space-like components µ = k ,this gives − ρ ∂ k ρ = A − ∂ k A + 2 B − ∂ k B (6.10) More precisely, the configurations are static on scales much shorter than the cosmic expansion rate.
17o that ρ is determined by AB ρ = c (6.11)where c is a numerical constant. For the time-like components µ = 0, (6.9) leads to − ρ ∂ ρ = 0 = 1 B (cid:0) rEA − A (cid:48) − ( rE (cid:48) + 3 E ) (cid:1) (6.12)which is solved by c Ar − = E . (6.13)In particular, we can set E = 0 for c = 0. Furthermore, one finds for the determinantsdet( γ µν ) = − A B , (cid:112) | G | = c AB (6.14)where E drops out. In particular, we observe (cid:112) | G | ρ − = c B . (6.15)It is straightforward to check that these relations also imply the divergence constraint (3.43).The components of the effective metric G µν = ρ γ µν are then obtained explicitly as G = c (cid:16) − AB + r E AB (cid:17) = c ( − c r − ) AB ,G i = c EAB x i = c c AB r − x i ,G ij = c A δ ij . (6.16)We will focus on the case E = 0 (and S = 0) in this paper for simplicity, so that c = 0. Then ds G = − c AB dt + c A δ ij dx i dx j . (6.17) Equations of motion and solution.
For the spherically invariant frames as above, it isconvenient to use the formulation (3.50) of the equations of motion. Using (cid:63)T ˙0 = r − A (cid:48) x i (cid:63) ( dx i ∧ dt ) = r − A (cid:48) (cid:112) | G | G ii (cid:48) G ε i (cid:48) kl x i dx k ∧ dx l (cid:63)T ˙ k = r − B (cid:48) x i (cid:63) ( dx i ∧ dx k ) = r − B (cid:48) (cid:112) | G | G ii (cid:48) G kk (cid:48) ε i (cid:48) k (cid:48) l x i dt ∧ dx l (6.18)in Cartesian coordinates (sum over i is understood), we obtain d ( (cid:63)ρ T ˙0 ) = r − A (cid:48) ρ (cid:112) | G | G ii (cid:48) G ε i (cid:48) kl dx i ∧ dx k ∧ dx l + ( r − A (cid:48) ρ (cid:112) | G | G ii (cid:48) G ) (cid:48) ε i (cid:48) kl x i dr ∧ dx k ∧ dx l d ( (cid:63)ρ T ˙ k ) = r − B (cid:48) ρ (cid:112) | G | G ii (cid:48) G kk (cid:48) ε i (cid:48) k (cid:48) l dx i ∧ dx l ∧ dt + r − ( r − B (cid:48) ρ (cid:112) | G | G ii (cid:48) G kk (cid:48) ) (cid:48) ε i (cid:48) k (cid:48) l x i x j dx j ∧ dx l ∧ dt (6.19)18here (cid:48) denotes radial derivative, and G ij ≡ A ( r ) δ ij . Assuming m = 0 for simplicity (as wellas S = 0 = E ) and using ε ikl x i rdr ∧ dx k ∧ dx l = 2 r d x and ε ikl dx i ∧ dx k ∧ dx l = 6 d x , theequations of motion for ˙ α = 0 reduce to0 = 2 r − ( A (cid:48) ρ (cid:112) | G | AG ) + ( A (cid:48) ρ (cid:112) | G | AG ) (cid:48) ddr (cid:0) r (cid:112) | G | ρ AG A (cid:48) (cid:1) . (6.20)This gives ddr (cid:0) r B − ( A − ) (cid:48) (cid:1) = 0 . (6.21)On the other hand, the space-like equations for ˙ α = 1 , , B (cid:48) = 0;the same conclusion is reached using the formulation (3.49). Thus assume B ( r ) = b = const .Then we obtain ( A − ) (cid:48) = − Mr , A − = 1 + Mr (6.22)for some constant M . Here we imposed the asymptotic behavior A ( r ) → r → ∞ (6.23)corresponding to an asymptotically constant frame E ˙0 = A ( r ) dt = 11 + Mr dt , E ˙ k = b dx k (6.24)i.e. E ˙00 = A ( r ) , E ˙ kk = b . This leads to the following non-trivial metric and dilaton ds G = G µν dx µ dx ν = − c b − (1 + Mr ) dt + c (cid:16) Mr (cid:17) (cid:88) i ( dx i ) = − c b − (1 + Mr ) dt + c (cid:16) Mr (cid:17) ( dr + r d Ω ) ρ = c b − (cid:16) Mr (cid:17) . (6.25)This reproduces the linearized Schwarzschild metric (cf. [22]), but it deviates from the fullSchwarzschild metric at the non-linear level; this should be expected due to the dilaton. Thesame result can also be obtained from (3.49) using Gauss’ theorem which yields (6.21), ordirectly from (3.49) using the Christoffel symbols. The apparent singularity of the metric atthe origin is a coordinate artifact, and the metric is seen (e.g. using u = √ r ) to be regularat the origin. We can compute the associated Ricci tensor R µν and the Einstein tensor G µν :using the relations ρ G = − α , α := c b − (6.26) Strictly speaking we should introduce another normalization constant to recover the cosmic backgroundframe for r → ∞ , which we drop for simplicity.
19e find the following result in spherical coordinates for c = 1 R µν dx µ dx µ = M M + r ) d Ω + αM M + r ) dt G µν dx µ dx µ = − M r ( M + r ) dr + M M + r ) d Ω + 3 αM M + r ) dt . (6.27)The Einstein tensor is non-vanishing, but decays like M r − as r → ∞ , reflecting the linearizedSchwarzschild geometry. This is clearly integrable at the origin, so that the solution shouldbe viewed as a vacuum or “remnant” solution, rather than representing the gravitationalfield of matter at the origin. This is possible because the present gravity theory is richerthan GR, and the present solution has a non-trivial dilaton. Indeed (D.5) implies that Ricci-flat solutions can arise (in the limit m →
0) only if both ρ and ˜ ρ vanish or cancel eachother. It is therefore not surprising that the present solution differs from the Schwarzschildsolution. However, we expect that there are other SO (3)-invariant solutions with differentcharacteristics. In particular, such solutions might contribute to the physics underlying theapparent “dark matter”.Finally, it is interesting to note that the total “apparent mass” is proportional to M , (cid:90) ∞ dr πr G = πM . (6.28) Effective energy-momentum tensor (3.54) and Einstein equations.
The above framehas a simple diagonal structure, similar to the background frame. It is related to the dilatonvia E ˙00 = A ( r ) = α ρ − (6.29)using (6.11) and (6.26). This leads to the following expressions for the frame-valued torsion T ˙ k = 0 T ˙0 = A (cid:48) ( r ) dr ∧ dt = − α ρ − dρ ∧ dt = 1(1 + Mr ) Mr dr ∧ dt , (6.30)so that the non-vanishing components are T ˙0 i = − T ˙00 i = − α ρ − ∂ i ρ . (6.31)We note that only the time component E ˙0 of the frame contributes to the torsion, withasymptotic behavior T ˙0 ∼ r → , r → ∞ T ˙0 → M (cid:54) = 0 , r → . (6.32) Note that we must set α = 1 if the frame should reduce to the background frame for r → ∞ . T µν [ ρ ] = 2 ρ − (cid:16) ∂ ν ρ∂ µ ρ − G νµ ∂ σ ρ∂ σ ρ (cid:17) , T [ ρ ] = − ρ − ∂ σ ρ∂ σ ρ = − ρ − G rr ∂ r ρ∂ r ρ , T [ ρ ] = − G ρ − ∂ σ ρ∂ σ ρ = 12 G T [ ρ ] . (6.33)Hence the contribution of the frame to the energy-momentum tensor (D.10) is T µν [ E ˙0 ] = ρ (cid:0) T ˙0 µσ T ˙0 νρ G ρσ − G νµ ( T ˙0 σκ T ˙0 σ (cid:48) κ (cid:48) G σσ (cid:48) G κκ (cid:48) ) (cid:1) = 4 α ρ − (cid:16) G (cid:0) ∂ µ ρ∂ ν ρ − G νµ (cid:0) ∂ σ ρ∂ σ (cid:48) ρG σσ (cid:48) (cid:1)(cid:1) + δ µ δ ν ( ∂ σ ρ∂ σ (cid:48) ρG σσ (cid:48) ) (cid:17) = 2 α ρ − (cid:0) G T µν [ ρ ] − δ µ δ ν T [ ρ ] (cid:1) = − T µν [ ρ ] + 4 δ µ δ ν T [ ρ ] (6.34)using (6.26), (6.33) and ρ = ρ ( r ). Note that the frame and the dilaton contribute the sameterms with opposite sign, and the total contribution to the Einstein equations is given by T µν = T µν [ E ˙0 ] + T µν [ ρ ] = − T µν [ ρ ] + 4 δ µ δ ν T [ ρ ] . (6.35)Explicitly, for the above solution with c = 1 we obtain T [ ρ ] = − ρ − G rr ∂ r ρ ∂ r ρ = −
12 1(1 + Mr ) M r (6.36)hence T = 3 T [ ρ ] = 32 G T [ ρ ] = 34 αM ( r + M ) T rr = − T rr [ ρ ] = 12 G rr T [ ρ ] = − M ( r + M ) r T ϑϑ = − T ϑϑ [ ρ ] = − G ϑϑ T [ ρ ] = 14 M ( r + M ) . (6.37)This agrees precisely with the above results for the Einstein tensor (6.27), as it should. Notethat T µν ∼ M r − as r → ∞ . We will give an explicit solution Z ˙ α below, which realizes thisframe via the Poisson brackets. Z ˙ α Given a classical solution E µ ˙ α of the above equations for the frame, we must finally find acorresponding solution Z ˙ α of the matrix model such that { Z ˙ α , x µ } = E µ ˙ α . (7.1)21his problem was partially solved in [21] sections 6.3 (point 1) and 9.2, where it was shownthat the divergence constraint ∂ µ ( ρ M E µ ) = 0 (7.2)is satisfied for all { x µ , Z (1) } ≡ A ( − ) µ [ Z (1) ] modes with Z (1) ∈ C (in the notation of [21],cf. (2.10)), and moreover these modes are complete in C ⊗ R (together with the pure gaugemodes). Here ( ) denotes the projection to the spin 0 sector C , i.e. to functions on M , .This means there is always some Z ˙ α ∈ C such that { Z ˙ α , x µ } = E µ ˙ α as desired, but thismay be accompanied by higher spin contributions { Z ˙ α , x µ } ∈ C . Whether or not thesehigher-spin contributions can be canceled by suitable higher-spin “counterterms” or coordinatere-definitions ˜ x µ is an open question in general.We shall illustrate this reconstruction explicitly for the spherically invariant solution (6.24),by providing explicit functions Z ˙ α on the bundle space. In that case it is indeed possible tocancel the higher-spin contributions (for the dominant terms at late times), but this requiresan infinite tower of higher-spin modes from the point of view of the cosmic background. Semi-classical Z ˙ α for the spherical solution. For the specific solution (6.24), we shouldaccordingly find “potentials” Z ˙ α – and possibly new coordinates ˜ x µ – such that { Z , ˜ x } = − sinh( η ) f ( r ) , { Z , ˜ x i } ≈ { Z i , ˜ x } ≈ , { Z i , ˜ x j } ≈ sinh( η ) δ ij for f ( r ) = A − ( r ). Here we re-inserted a factor sinh( η ) to recover the background frame (2.15).The second line suggests to leave the space-like generators Z k = t k and the x i coordinatesunchanged. Then x should not be modified either, in order to preserve { Z i , x } = 0. Thefirst ansatz one might try is Z = t g ( r ) for some function g ( r ). This would give { g ( r ) t , x } = g ( r ) { t , x } + t { g ( r ) , x } = − sinh( η ) (cid:16) g ( r ) + 2 g (cid:48) ( r ) p (cid:17) ∈ C ⊕ C (7.3)using (C.12). However, the component in C is not subleading and hence cannot be neglectedsince | p | = | p · xx | = r , cf. section C.2. Therefore we consider the more general ansatz Z := t g ( r, χ ) , Z k = t k ˜ x = x , ˜ x i = x i where χ is the central generator in the SO (3)-invariant sub-algebra given in (C.13), whichallows for higher spin corrections. Then { t g ( r , χ ) , x } = − sinh( η ) g ( r , χ ) − Rx ∂ r g ( r , χ ) t p = − sinh( η ) g ( r , χ ) − sinh( η ) ∂ r g ( r , χ )2( p ) = − sinh( η ) (cid:0) r − χ ) ∂ r (cid:1) g ( r , χ ) (7.4)so that we need to solve (cid:0) r − χ ) ∂ r (cid:1) g ( r , χ ) = f ( r ) . (7.5)22he solution is g ( r , χ ) = 1 p (cid:16) a ( χ ) + 12 (cid:90) r χ f ( u ) √ u − χ du (cid:17) . (7.6)The term with a ( χ ) satisfies the homogeneous equation, hence we can simply drop a ( χ ), oruse it to impose boundary conditions. For the above solution with f ( r ) = A − = 1 + Mr (7.7)(assuming sinh( η ) ≈ const), the above integral can be evaluated explicitly, (cid:90) r χ M √ u √ u − χ du = 2 (cid:16) M log (cid:16) r + (cid:112) r − χ (cid:17) + (cid:112) r − χ (cid:17) − M log( χ )= 2 ( M log ( r + p ) + p ) − M log( r − p )= 2 p + M log (cid:16) r + p r − p (cid:17) . (7.8)We recall that p r ≈ (cid:126)p · (cid:126)x | (cid:126)p || (cid:126)x | =: cos ϑ and therefore − r ≤ p ≤ r due to (C.10) and (C.8), so thatthe last term is well-defined up to a mild log-type singularity at antipodal points ϑ = 0 , π ofthe internal S fiber. Thus˜ Z = t + M R ˜ R log (cid:16) r + p r − p (cid:17) = t + M R ˜ R (cid:16) p r + 23 p r + ... (cid:17) = t A − ( r ) + O (cid:16) p r (cid:17) , (7.9)which has indeed the structure in (7.3) up to higher-spin corrections. One can check directlythat it satisfies the desired relation { ˜ Z , x } = − sinh( η ) (cid:0) Mr (cid:1) = − sinh( η ) A − ( r ) . (7.10)Finally, note that { ˜ Z , x i } = p { g, x i } (cid:28) sinh( η ) is negligible at late times compared to { ˜ Z , x } , because p , r , χ and θ ij are all bounded i.e. they do not grow with η . Therefore the Z ˙ α indeed reconstruct the frame at late times η (cid:29) but we should requirethat ˜ Z exists as a function on the bundle space C P , . As such, the above solution ˜ Z exhibitsa logarithmic singularity for p = ± r , which arises at the antipodes of the internal S where (cid:126)p (cid:107) (cid:126)x . One of them can be canceled by choosing a suitable a ( χ ) in (7.6), but not simultaneouslyfor both singularities. However, this singularity is integrable. This strongly suggests that thesolution is acceptable, since the underlying framework of quantized symplectic spaces is basedon the relation End ( H ) ∼ L ( C P , ) between Hilbert-Schmid operators and square-integrablefunctions on C P , . At the operator level, the internal S is actually a fuzzy sphere [18] and Here p = ± (cid:112) r − χ (C.16) is understood. The explicit p cancels in ˜ Z , so that the sign ambiguitydrops out. Note that p has a clean definition as su (4 ,
2) generator at the matrix level, hence this is not aproblem. cf. the black hole solutions in Vasiliev higher spin theory [29, 30]. arises from a su (4 ,
2) generator P , where the semi-classical bound (C.8) is replaced by astrict operator bound. All this strongly suggests that there should be a well-defined underlyingmatrix solution, which should be studied in more detail elsewhere.The situation at r = 0 is more intricate. Then the location of the singularity on S becomes inconsistent, which signals a more serious singularity. This suggests that some extrastructure is present at the origin. On the other hand, the energy-momentum tensor (6.37) isintegrable at the origin. This suggests that the solution should be viewed as a purely geometricmatrix configuration with a “quantum” or matrix singularity at the origin. Such a matrixsolution might be considered as a “remnant”, which is precisely the type of structure wherethe underlying matrix framework may provide unique new insights. We have elaborated the non-linear dynamics of the effective gravity sector which emerges fromthe IKKT or IIB matrix model on a certain type of covariant quantum space. This should beviewed as a candidate for a modified gravity theory. The most convenient description seemsto be in terms of the frame E ˙ α , for which we find a simple covariant equation of motion. Thiscaptures the equation of motion derived previously for the torsion, avoiding the use of theWeitzenb¨ock connection.However, there are important differences to the frame formulation of general relativity.The main difference is that the local Lorentz invariance of the frame bundle is broken, andthe frame contains more information than just the metric. All frames which arise in the modelhave the form E ˙ α = { Z ˙ α , . } , which entails a divergence constraint. This is consistent with thefact that diffeomorphism invariance is reduced to a type of volume-preserving diffeomorphisms,which reflects the invariant symplectic volume form on the underlying quantum space. Anysuch frame leads to a rank 3 tensor which is identified with the torsion of the Weitzenb¨ockconnection. The trace of this tensor is related to the dilaton, and the antisymmetric partreduces on-shell to a scalar field identified as axion. These two scalar fields satisfy second-order equations, sourced by the torsion.Furthermore, we find a simple spherically invariant static solution for the equations ofmotion, which is localized at some point in space. This solution reproduces the linearizedSchwarzschild solution, but deviates from the full Schwarzschild solution at higher order.In particular, there is no horizon, and there is no singularity at the origin, more preciselythe singularity is mild and integrable. The effective energy-momentum tensor is smoothlydistributed, and can be attributed to the dilaton as well as the time component of the framefield. Therefore this solution is not associated to matter, but should be viewed as a vacuum orperhaps as ”remnant“ solution, with no counterpart in general relativity. Hence this gravitytheory is richer than general relativity, which is potentially very interesting.The extra degrees of freedom of the present theory comprise not only the dilaton andaxion fields, but also a tower of higher-spin fields, reminiscent of Vasiliev theory. This leads tomany open questions. The equations derived for the frame should be viewed as equations forhigher-spin valued frames, and it is not evident if the higher-spin contributions to the framecan always be eliminated to reproduce some given metric. For the present solution, we showthat this is indeed the case. However, in general the problem of reconstructing frames andthe role of the higher-spin fields remains to be clarified.24t is tempting to speculate about the feasibility of the present framework as a physicaltheory. From this point of view, the deviation from Ricci-flatness is perhaps the most inter-esting – and challenging – feature: On the one hand it opens up the possibility for geometricexplanations of the open problems such as dark energy and dark matter. On the other hand,it remains to be seen if the theory can meet the precision tests of gravity. In particular, theanalog of the full Schwarzschild solution arising from matter at the origin is still to be found.It may also be that a proper description of gravity in the presence of matter requires theinclusion of quantum effects, such as an induced Einstein-Hilbert action as discussed previ-ously [5, 14, 31]. In either case, the tools developed in this paper should provide a useful basisfor further work towards a more complete understanding of gravity in this remarkable model. Acknowledgments.
We would especially like to thank Sergio H¨ortner for collaboration onthis project. HS would also like to thank Yuhma Asano for useful discussions and a relatedcollaboration. The work of HS was supported by the Austrian Science Fund (FWF) grantP32086.
A Conventions and useful formulas
Levi-Civita symbol.
The Levi-Civita symbols will be used with the following convention ε = 1 = ε (A.1)so that ε νρσµ G νν (cid:48) G σσ (cid:48) G µµ (cid:48) G ρρ (cid:48) = | G µν | ε νρ (cid:48) σµ (cid:48) . (A.2) Contraction of contorsion.
We will also need the following contraction formulas whichwere obtained in (7.54) and (7.55) of [1] K σρµ K ρσν = 14 (cid:16) T µσρ ( T νρσ + T νσρ ) − T ρσµ T ρσν (cid:17) (A.3) K ρµσ K σρ µ = 14 T µσρ (2 T µρσ + T µσρ ) . (A.4) Dilaton and densities.
We elaborate the relation between the dilaton ρ (which is a scalarfunction) and the metric and symplectic densities. To this end, recall that ρ is defined by ρ = | γ | − ρ M = ρ | G | − ρ M , hence ρ − = (cid:112) | G | − ρ M . (A.5)Here ρ M is the SO (4 , H , which is related to the dilaton ¯ ρ of the cosmic background via the analogous relation( ¯ ρ ) − = (cid:113) | ¯ G | − ρ M = sinh − ( η ) (A.6)25f. (7.23) in [1]. Taking the ratio gives ¯ ρ ρ = (cid:112) | ¯ G | (cid:112) | G | (A.7)hence (cid:112) | G | ρ − = (cid:113) | ¯ G | ¯ ρ − = ρ M (= sinh − ( η )) (A.8)where the last equality holds only in Cartesian coordinates. B Divergence constraint and antisymmetric torsion
B.1 Divergence constraint
We start with the following identity T νµν = T µν ˙ α E ˙ αν = ( ∂ µ E ˙ αν − ∂ ν E ˙ αµ ) E ν ˙ α = E ν ˙ α ∂ µ E ˙ αν + E ˙ αµ ∂ ν E ν ˙ α − ρ ∂ µ ρ (B.1)where E is the effective frame (3.19). Together with the relation − ρ ∂ µ ρ = T νµν (3.21) for theframe in the matrix model (in the asymptotic regime), we obtain − ρ ∂ µ ρ = E ν ˙ α ∂ µ E ˙ αν + E ˙ αµ ∂ ν E ν ˙ α − ρ ∂ µ ρ . (B.2)The first term on the rhs can be rewritten as ∂ µ ln( (cid:112) | G | ) = det( E ˙ αν ) − ∂ µ det( E ˙ αν ) = E ν ˙ α ∂ µ E ˙ αν (B.3)so that 0 = ∂ µ ln (cid:112) | G | + E ˙ αµ ∂ ν E ν ˙ α − ρ ∂ µ ρ = 1 (cid:112) | G | ∂ µ (cid:112) | G | + ρ E ˙ αµ ∂ ν ( ρ − E ν ˙ α )= ρ (cid:112) | G | E ˙ αµ ∂ ν (cid:0)(cid:112) | G | ρ − E ν ˙ α (cid:1) (B.4)which results in the divergence constraint (3.43), ∂ ν (cid:0)(cid:112) | G | ρ − E ν ˙ α (cid:1) = 0 = ∇ ( G ) ν ( ρ − E ν ˙ α ) . (B.5)This means that the ρ − E µ ˙ α are volume-preserving vector fields. It can be viewed as a gauge-fixing relation analogous to a Lorentz gauge condition, which reflects the fact that there is nomanifest local Lorentz invariance in the present framework.The divergence constraint can also be formulated in terms of differential forms. Becauseof E ν ˙ α = ρ G νµ E ˙ αµ , we find, using the Hodge star operation with respect to G , d ( ∗ E ˙ α ) = 0 . (B.6) This was already shown in (7.18) in [1]. lternative derivation. The origin of the divergence constraint (3.43) from the Jacobiidentity can be seen as follows: E µ ˙ α ∼ − θ µν ∂ ν Z ˙ α ∂ µ ( ρ M E µ ˙ α ) ∼ − ρ M θ µν ∂ µ ∂ ν Z ˙ α = 0 (B.7)using ∂ µ ( ρ M θ µν ) = 0, which holds in the asymptotic regime. Together with (A.8), we recover ∂ µ (cid:0)(cid:112) | G | ρ − E µ ˙ α (cid:1) = 0 . (B.8)Yet another derivation is obtained using (6.24) in [23], which implies that E µ ˙ α = { Z ˙ α , x µ } for Z ˙ α ∈ C satisfies0 = ¯ ∇ ν (sinh − ( η ) E µ ˙ α ) = 1 (cid:112) | ¯ G | ∂ ν (cid:16) sinh − ( η ) (cid:113) | ¯ G | E ν ˙ α (cid:17) = 1 (cid:112) | ¯ G | ∂ ν (cid:16) sinh − ( η ) E ν ˙ α (cid:17) = 1 (cid:112) | ¯ G | ∂ ν (cid:16)(cid:112) | G | ρ − E ν ˙ α (cid:17) (B.9)where ¯ G is the effective metric of the FLRW solution M , , and x µ are Cartesian coordinates. Divergence constraint and diffeomorphisms.
As a consistency check, we verify thatthe divergence constraint is preserved by the diffeomorphisms which arise from gauge trans-formations { Λ , . } in the matrix model. Indeed, the frame transforms under a diffeomorphismgenerated by the vector field ξ µ = { Λ , x µ } as δ Λ E µ ˙ α = L ξ E µ ˙ α = ξ ν ∂ ν E µ ˙ α − E ν ˙ α ∂ ν ξ µ (B.10)so that the divergence constraint transforms as δ Λ ∂ µ (cid:0)(cid:112) | G | ρ − E µ ˙ α (cid:1) = ∂ µ (cid:0)(cid:112) | G | ρ − ξ ν ∂ ν E µ ˙ α (cid:1) − ∂ µ (cid:0)(cid:112) | G | ρ − E ν ˙ α ∂ ν ξ µ (cid:1) = ∂ µ ∂ ν (cid:0)(cid:112) | G | ρ − ξ ν E µ ˙ α (cid:1) − ∂ µ ∂ ν (cid:0)(cid:112) | G | ρ − E ν ˙ α ξ µ (cid:1) − ∂ µ (cid:0) ∂ ν (cid:0)(cid:112) | G | ρ − ξ ν (cid:1) E µ ˙ α (cid:1) = − ∂ µ (cid:0) ∂ ν (cid:0)(cid:112) | G | ρ − ξ ν (cid:1) E µ ˙ α (cid:1) (B.11)using (B.5). This reduces precisely to the constraint for the diffeomorphisms (7.3) in [1] (cid:112) | G |∇ ( G ) ν (cid:0) ρ − ξ ν (cid:1) = ∂ ν (cid:0)(cid:112) | G | ρ − ξ ν (cid:1) = ∂ ν (cid:0) ρ M ξ ν (cid:1) = ∂ ν (cid:0)(cid:113) | ¯ G | ¯ ρ − ξ ν (cid:1) = (cid:113) | ¯ G |∇ ( ¯ G ) ν (cid:0) ¯ ρ − ξ ν (cid:1) = (cid:113) | ¯ G |∇ ( ¯ G ) ν (cid:0) β ξ ν (cid:1) = 0 (B.12)noting that (cid:112) | G | ρ − = ρ M , where ¯ G is the cosmic background metric and β = sinh − η = ¯ ρ − .Therefore the divergence constraint is consistent with the volume-preserving diffeos arising inthe model. One can also check that it is compatible with the e.o.m. for the frame (3.51).27 .2 The totally antisymmetric torsion T ( AS ) The totally antisymmetric torsion is defined by (3.26) T ( AS ) νρµ = T νρµ + T νµ ρ + T νρµ . (B.13)Then G νν (cid:48) T ( AS ) ν (cid:48) ρµ is totally antisymmetric in the indices ν, ρ, µ and can naturally be inter-preted as a 3-form T ( AS ) . It is related by the Hodge star (cid:63) corresponding to G µν to a 1-form T σ dx σ via T ( AS ) := 16 G νν (cid:48) T ( AS ) ν (cid:48) ρµ dx ν ∧ dx ρ ∧ dx µ = (cid:63) ( T σ dx σ ) . (B.14)In coordinates, this amounts to T ( AS ) νρµ = (cid:112) | G | G νν (cid:48) ε ν (cid:48) ρµσ G σσ (cid:48) T σ (cid:48) ,ε ν (cid:48) ρµκ G ν (cid:48) ν T ( AS ) νρµ = 6 (cid:112) | G | G κσ (cid:48) T σ (cid:48) (B.15)using the conventions (A.1). This implies T ( AS ) σρµ T ( AS ) ρσ ν = | G | G σσ (cid:48) ε σ (cid:48) ρµκ G ρρ (cid:48) ε σρ (cid:48) νη G κκ (cid:48) T κ (cid:48) G ηη (cid:48) T η (cid:48) = ε σρµ (cid:48) κ ε σρνη G µ (cid:48) µ T κ G ηη (cid:48) T η (cid:48) = − T ν T µ − G νµ G κη T κ T η ) (B.16)using (A.2), and the contraction gives T ( AS ) σρµ T ( AS ) ρσ ν G µν = 6 T ν T µ G µν . (B.17)We also note the following identity T ( AS ) σρµ T ( AS ) ρσ ν = − T σµ ρ T ρσν + T σν ρ T ρσµ ) + 2( T σµ ρ T ρνσ − T σµ ρ T ρν σ ) + T σρµ T ρσ ν . (B.18)Contracting this with G µν gives T ( AS ) σρµ T ( AS ) ρσ ν G µν = 3 (cid:16) T σµ ρ T ρνσ G µν − T σµρ T ρσ ν G µν (cid:17) T σµρ T ρσ ν G µν = 12 T σµ ρ T ρνσ G µν − T ( AS ) σρµ T ( AS ) ρσ ν G µν = 12 T σµ ρ T ρνσ G µν − T ν T µ G µν . (B.19)We will also need the formulas T νµσ T σν ρ − T νρσ T σν µ = −
12 ( T ( AS ) νσµ T σν ρ − ( µ ↔ ρ )) (B.20)and 2 K σνρ T νσµ + 2 K σνµ T νρσ = −
12 ( T ( AS ) νσµ T σν ρ − ( µ ↔ ρ )) . (B.21)28 he divergence of T µ . The divergence of T σ satisfies the identity G κσ ∇ ( G ) κ T σ = 16 (cid:112) | G | − ε ν (cid:48) ρµκ G ν (cid:48) ν ∇ ( G ) κ T ( AS ) νρµ = 12 (cid:112) | G | − ε νρµ (cid:48) κ G µ (cid:48) µ ∇ ( G ) κ T µνρ = 12 (cid:112) | G | − ε νρµ (cid:48) κ G µ (cid:48) µ ∇ ( G ) κ (cid:16) ( ∇ ( G ) ν E ˙ αρ − ∇ ( G ) ρ E ˙ αν ) E µ ˙ α (cid:17) = 12 (cid:112) | G | − ε νρµ (cid:48) κ G µ (cid:48) µ (cid:16) R σκν ρ E ˙ ασ − R σκρ ν E ˙ ασ + ( ∇ ( G ) ν E ˙ αρ − ∇ ( G ) ρ E ˙ αν ) ∇ ( G ) κ E µ ˙ α (cid:17) = 12 (cid:112) | G | − ε νρµκ (cid:16) ( ∇ ( G ) ν E ˙ αρ − ∇ ( G ) ρ E ˙ αν ) ∇ ( G ) κ ( ρ E ˙ αµ ) (cid:17) = 14 ρ (cid:112) | G | − ε νρµκ T ˙ ανρ T κµ ˙ α + 12 ( ∂ κ ρ ) (cid:112) | G | − ε νρµκ T νρµ = 14 ρ (cid:112) | G | − ε νρµκ T ˙ ανρ T κµ ˙ α + 16 ρ − ∂ κ ρ (cid:112) | G | − ε νρµκ G µµ (cid:48) T ( AS ) µ (cid:48) νρ = 14 ρ (cid:112) | G | − ε νρµκ T ˙ ανρ T κµ ˙ α + 2 ρ − ∂ κ ρG κκ (cid:48) T κ (cid:48) (B.22)which follows from (B.15) using the algebraic Bianchi identity for the Riemann tensor. Mul-tiplied by ρ − this leads to the identity˜ ρ ε νρµκ T ˙ ανρ T κµ ˙ α = 4 (cid:112) | G | ˜ ρ G µν ∇ ( G ) µ ( ρ − T ν ) , (B.23)and using the e.o.m. (3.36) for T µ this implies˜ ρ ε νρµκ T ˙ ανρ T κµ ˙ α = 4 (cid:112) | G | ˜ ρ G µν ∇ ( G ) µ ( ρ − ∂ ν ˜ ρ ) . (B.24)As a by-product, we obtain the following interesting identity S ˜ E = − (cid:90) d x ˜ ρ ε νρµκ T ˙ ανρ T κµ ˙ α = 4 (cid:90) d x (cid:112) | G | ρ − G µν ∂ µ ˜ ρ∂ ν ˜ ρ = 4 S ˜ ρ . (B.25)Since the first term has the structure (cid:82) ˜ ρ dE ∧ dE , this suggests to view ˜ ρ as an axion associatedto the frame. C 6-dimensional configuration space and constraints
C.1 General setup
We want to describe the 6-dimensional symplectic space M (6) (2.3) underlying the cosmicbackground solution more explicitly. It is described through the 8 functions x µ and t µ subjectto the constraints (2.9), which transform covariantly under the global SO (3 ,
1) isometry groupof the k = − M , . To exhibit the S bundle structure over M , , it isuseful to consider M (6) as Cartesian product M (6) ∼ = R x × R p (C.1) This description misses the double cover structure of M (6) ∼ = C P , [14], but it captures the structureafter the Big Bounce. x i , p j , while x and p are determined bythe constraints. This is most transparent in terms of the re-scaled generators p α = ˜ RRt α (C.2)with Poisson brackets { p µ , x ν } = ˜ Rx η µν , x = (cid:112) x µ x µ + R . (C.3)Then the constraints (2.9) take the form p x = p k x k , − ( p ) + p i p i = ( x ) − r (C.4)(sum over i is understood) where r = x i x i . (C.5)The constraints can be written as( p + x ) = p i p i + r + 2 p k x k = | (cid:126)x + (cid:126)p | ( p − x ) = p i p i + r − p k x k = | (cid:126)x − (cid:126)p | (C.6)which is solved by x = 12 ( | (cid:126)x + (cid:126)p | + | (cid:126)x − (cid:126)p | ) > p = 12 ( | (cid:126)x + (cid:126)p | − | (cid:126)x − (cid:126)p | ) (C.7)which reproduces the cosmic background (after the Big Bounce). Note that p can take eithersign, consistent with (C.4), and the triangle inequality applied to (C.7) implies the bound | p | ≤ | (cid:126)x | . (C.8)For a reference point ξ ∈ M , with (cid:126)x = 0, this gives x = | p | , p = 0 . (C.9)More generally in a local region near ξ with | p | = x (cid:29) r , we have x = 12 (cid:0)(cid:112) p + r + 2 px + (cid:112) p + r − px (cid:1) ≈ | p | p = 12 (cid:0)(cid:112) p + r + 2 px − (cid:112) p + r − px (cid:1) ≈ px | p | ≈ pxx (C.10)up to corrections suppressed by rx . Therefore | x | ≈ | p | ≈ const is essentially the cosmictime, which completes the spacetime coordinates on M , x µ = ( x ≈ | p | , x i ) , (C.11)while the S fiber is described by the two transversal p i .30 .2 SO (3) -invariant functions and Poisson brackets Consider the subalgebra of SO (3)-invariant functions of r , x , p using the notation (C.2).Here r , x are viewed as elements of the algebra C describing the spin 0 sector of functionson M , , while p ∈ C is a higher spin generator (since it transforms non-trivially under thelocal SO (3)). The Poisson brackets are { r , p } = 0 { r , x } = − Rx p { p , x } = − ˜ Rx . (C.12)They span a space of functions in 3 variables. Therefore there must be a central generator,and the symplectic leaves are 2-dimensional. This generator is found to be { χ, . } = 0 , χ := r − ( p ) = x − | (cid:126)p | (C.13)which is central within the SO (3)-invariant subalgebra. Indeed, { χ, r } = − p { p , r } = 0 { χ, x } = { r , x } − p { p , x } = − Rx p + 2 ˜ Rx p = 0 . (C.14)We also note that (C.8) implies r ≥ χ ≥ p = ± (cid:112) r − χ . (C.16) D Geometric energy-momentum tensor
It was shown in (eq. (5.70) in [1]) that the Ricci tensor in vacuum satisfies the followingvacuum equation of motion R νµ = −
12 ( T δρ µ T ρνδ + T δρ ν T ρµδ ) − K ρδ ν K δρ µ + 2 ρ − ∂ ν ρ∂ µ ρ + G νµ (cid:0) ρ − m − T σν δ T νσρ G δρ (cid:1) . (D.1)The first three terms can be rewritten using (A.3) and (B.18) as − T ρµσ T σρ ν − T ρνσ T σρ µ − K ρσ µ K σρ ν = 14 T ( AS ) σρµ T ( AS ) ρσ ν − T σρµ T νσρ . (D.2)Therefore the on-shell Ricci tensor (D.1) can be written as R νµ = 14 T ( AS ) σρµ T ( AS ) ρσ ν − T ρµσ T σν ρ + 2 ρ − ∂ ν ρ∂ µ ρ + 14 G νµ (cid:0) ρ − m + T σνδ T νσ ρ G δρ − T ( AS ) σρµ T ( AS ) ρσ ν G µν (cid:1) (D.3)31nd R = − T ( AS ) σρµ T ( AS ) ρσ ν G µν + 2 ρ − ∂ µ ρ∂ µ ρ + 4 ρ − m . (D.4)Using (B.17) and the on-shell relation (3.36) for T µ , we obtain the on-shell relation R = − ρ − G µν ∂ µ ˜ ρ∂ ν ˜ ρ + 2 ρ − ∂ µ ρ∂ µ ρ + 4 ρ − m . (D.5)This implies that solutions with R = 0 can arise (in the limit m →
0) only if both ρ and ˜ ρ vanish or cancel each other. Effective energy-momentum tensor.
We now define the effective energy-momentum (e-m) tensor due to torsion in terms of the Einstein equations (absorbing a factor 8 π in the e-mtensor), which thus decomposes into different contributions T µν := R νµ − G µν R = 14 T ( AS ) ρσ µ T ( AS ) σρν − T ρµσ T σν ρ + 2 ρ − ∂ ν ρ∂ µ ρ + 14 G νµ (cid:0) − ρ − m + T σνδ T νσ ρ G δρ − T ( AS ) ρσ µ T ( AS ) σρν G µν − ρ − ∂ µ ρ∂ µ ρ )= T µν [ E ˙ α ] + T µν [ ρ ] + T ( AS ) µν [ T ] − ρ − m G νµ . (D.6)Here the energy-momentum tensor of the dilaton is T µν [ ρ ] = 2 ρ − (cid:16) ∂ ν ρ∂ µ ρ − G νµ ∂ σ ρ∂ σ ρ (cid:17) = 2 (cid:0) ∂ ν σ∂ µ σ − G νµ ∂ κ σ∂ κ σ (cid:1) , ρ = e − σ . (D.7)For the contribution of T ( AS ) we can use (B.16) and (B.17), so that T ( AS ) µν [ T ] = 14 (cid:0) T ( AS ) σρµ T ( AS ) ρσ ν − G νµ ( T ( AS ) σρµ T ( AS ) ρσ ν G µν ) (cid:1) = − (cid:16) T µ T ν − G νµ ( T ρ T ρ ) (cid:17) . (D.8)Using the e.o.m. (3.36) for T µ , this reduces to the e-m tensor of the axion, T ( AS ) µν [ T ] = − ρ − (cid:16) ∂ µ ˜ ρ∂ ν ˜ ρ − G νµ ( G σσ (cid:48) ∂ σ ˜ ρ∂ σ (cid:48) ˜ ρ ) (cid:17) =: T µν [ ˜ ρ ] (D.9)which has an explicit (and non-standard) minus sign. Finally, the e-m tensor of the frame is T µν [ E ˙ α ] := − T ρµσ T σν ρ + 14 G νµ T σνδ T νσ ρ G δρ = − ρ (cid:0) F µσ [ E ˙ α ] F νρ [ E ˙ α ] G ρσ − G νµ ( F ρκ [ E ˙ α ] F σκ (cid:48) [ E ˙ α ] G ρσ G κκ (cid:48) ) (cid:1) (D.10)32here F µν [ E ˙ α ] := ∂ µ E ˙ αν − ∂ ν E ˙ αµ = T ˙ αµν (D.11)is the field strength of the frame vector field. Thus T µν [ E α ] has the structure of a negative Maxwell-like e-m tensor T µν [ A ] = F µρ F νσ G ρσ − G µν ( F ρσ F ρσ ) , F = dA (D.12)from each frame component E ˙ α , with opposite sign for space- and time-like components. Henceit is possible in principle to obtain Ricci-flat solutions, provided the various contributions withdifferent signs cancel. E Geometric actions and identities
E.1 Einstein-Hilbert term from torsion
We start from the identity (5.61) in [1] and obtain the Ricci scalar as R [ G ] = 2 ∇ µ ( G ) K ννµ − K µµρ K ρνσ G νσ + K µνρ K ρµσ G νσ = K µνρ K ρµσ G νσ − K µµρ K ρνσ G νσ − ∇ µ ( G ) ( ρ − ∂ µ ρ )= K µνρ K ρµσ G νσ + 2 ρ − G µν ∂ µ ρ∂ ν ρ − ∇ µ ( G ) ( ρ − ∂ µ ρ )= − T µσρ T ρµσ (cid:48) G σσ (cid:48) − T µσρ T ρµ σ (cid:48) G σσ (cid:48) + 2 ρ − G µν ∂ µ ρ∂ ν ρ − ∇ µ ( G ) ( ρ − ∂ µ ρ ) (E.1)using (3.17), the contraction relation (3.21), and (A.4) in the last step. This is an identity, noequations of motion are used. Upon integration, we can rewrite the second term as − (cid:90) d x (cid:112) | G | T σµρ T ρσ ν G µν = 2 (cid:90) d x (cid:112) | G | ∇ ( G ) µ E ˙ ασ E ρ ˙ α ( ∇ ( G ) σ (cid:48) E ˙ βρ − ∇ ( G ) ρ E ˙ βσ (cid:48) ) E µ ˙ β G σσ (cid:48) = 4 (cid:90) d x (cid:112) | G | E µ ˙ β ∇ ( G ) µ E ˙ ασ ∇ ( G ) σ (cid:48) E ˙ βρ E ρ ˙ α G σσ (cid:48) = − (cid:90) d x (cid:112) | G | ∇ ( G ) µ E ˙ ασ ∇ ( G ) σ (cid:48) E µ ˙ α G σσ (cid:48) = 4 (cid:90) d x (cid:112) | G | ∇ ( G ) σ (cid:48) ∇ ( G ) µ E ˙ ασ E µ ˙ α G σσ (cid:48) = 4 (cid:90) d x (cid:112) | G | (cid:0) ∇ ( G ) µ ∇ σ ( G ) E ˙ ασ E µ ˙ α + G σσ (cid:48) R σ (cid:48) µ ; σκ G κµ (cid:1) = 4 (cid:90) d x (cid:112) | G | (cid:0) − ∇ ( G ) σ (cid:48) E ˙ ασ ∇ ( G ) µ E µ ˙ α + R (cid:1) = 4 (cid:90) d x (cid:112) | G | ( − ρ − ∂ µ ρ∂ µ ρ + R ) (E.2)using the divergence constraint (3.43). Then (E.1) gives (cid:90) d x (cid:112) | G | R [ G ] = (cid:90) d x (cid:112) | G | (cid:16) − T µσρ T ρµσ (cid:48) G σσ (cid:48) + 2( − ρ − ∂ µ ρ∂ µ ρ + R ) (cid:17) (E.3)33ence (cid:90) d x (cid:112) | G | R [ G ] = (cid:90) d x (cid:112) | G | (cid:16) T µσρ T ρµσ (cid:48) G σσ (cid:48) + 6 ρ − G µν ∂ µ ρ∂ ν ρ (cid:17) . (E.4)This is an identity which applies to any off-shell configuration. Thus all action terms quadraticin the torsion can be written in terms of the Einstein-Hilbert term and the dilaton. Similaridentities are used in the teleparallel reformulation of general relativity [24], but the role ofthe dilaton and the divergence constraint is specific to the present framework. E.2 Axion identity
The identities (E.2), (B.19) and (E.4) can be combined as follows4 (cid:90) d x (cid:112) | G | ( − ρ − ∂ µ ρ∂ µ ρ + R ) = − (cid:90) d x (cid:112) | G | T σµρ T ρσ ν G µν = (cid:90) d x (cid:112) | G | (cid:0) − T σµ ρ T ρνσ G µν + T ν T µ G µν (cid:1) = (cid:90) d x (cid:112) | G | (cid:0) − R [ G ] + 12 ρ − G µν ∂ µ ρ∂ ν ρ + T ν T µ G µν (cid:1) . (E.5)Therefore all configurations of the matrix model satisfy0 = (cid:90) d x (cid:112) | G | (cid:16) − R + 14 ρ − G µν ∂ µ ρ∂ ν ρ + 12 T ν T µ G µν (cid:17) . (E.6)This allows us to express (cid:82) d x (cid:112) | G |R in terms of the totally antisymmetric part of the torsion.Using the on-shell relation for T µ (3.36), this reduces to0 = (cid:90) d x (cid:112) | G | (cid:16) − R + 14 ρ − G µν ∂ µ ρ∂ ν ρ + 12 ρ − G µν ∂ µ ˜ ρ∂ ν ˜ ρ (cid:17) . (E.7)This illustrates the fact that dilaton and axion are not independent degrees of freedom, asdiscussed in section 5. E.3 Variation of the action S T Consider the action term for the totally antisymmetric torsion S T = (cid:90) d x (cid:112) | G | G µν T µ T ν . (E.8)Here T µ is defined in (B.15), or explicitly T ρ = 12 (cid:112) | G | − ρ G ρκ ε νσµκ T νσµ . (E.9)Its variation can be written as2 δT ρ = (cid:112) | G | − ρ (cid:16) G ρκ ε νσµκ δT νσµ + δG ρκ ε νσµκ T νσµ (cid:17) + (cid:16) ρ − δρ + 12 G αβ δG αβ (cid:17) T ρ . (E.10)34sing ε νσµκ δT νσµ = ε νσµκ (cid:16) ( ∂ ν E ˙ ασ − ∂ σ E ˙ αν ) δE ˙ αµ + ( ∂ ν δE ˙ ασ − ∂ σ δE ˙ αν ) E ˙ αµ (cid:17) = ε νσµκ (cid:16) T ˙ ανσ δE ˙ αµ + 2 ∂ ν δE ˙ ασ E ˙ αµ (cid:17) (E.11)and partial integration we obtain (cid:90) d x ρ T κ ε νµσκ δT νσµ = (cid:90) d x (cid:16) ρ T κ ε νµσκ T ˙ ανσ δE ˙ αµ − ε νµσκ ∂ ν ( ρ T κ ) E ˙ αµ δE ˙ ασ − ρ T κ ε νµσκ T ˙ ανµ δE ˙ ασ (cid:17) = 2 (cid:90) d x ρ δE ˙ αµ (cid:16) T κ ε νµσκ T ˙ ανσ + E ˙ ασ ε νµσκ ρ − ∂ ν ( ρ T κ ) (cid:17) . (E.12)Therefore the variation of S T can be written as δS T = (cid:90) d x (cid:112) | G | δG µν (cid:16) T µ T ν − G µν T · T (cid:17) + (cid:90) d x (cid:112) | G | G µν T µ δT ν = (cid:90) d x (cid:112) | G | δG µν (cid:16) T µ T ν + 12 G µν T · T (cid:17) + (cid:90) d x (cid:0) ρ T κ ε νσµκ δT νσµ + ρ T η G ηρ δG ρκ ε νσµκ T νσµ (cid:1) + 2 (cid:90) d xρ − δρ T · T = (cid:90) d x (cid:112) | G | (cid:16) − δG µν ( T µ T ν − G µν T · T ) + 2 ρ − δρ T · T (cid:17) + (cid:90) d xρ T κ ε νσµκ δT νσµ = (cid:90) d x (cid:112) | G | (cid:16) − δG µν ( T µ T ν − G µν T · T ) + 2 ρ − δρ T · T (cid:17) − (cid:90) d xρ δE ˙ αµ (cid:16) T κ ε νµσκ T ˙ ανσ + E ˙ ασ ε νµσκ ρ − ∂ ν ( ρ T κ ) (cid:17) (E.13)where T · T = G µν T µ T ν . This is used in (4.3). References [1] H. C. Steinacker,
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