Generalized 2D dilaton gravity and KGB
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Generalized 2D dilaton gravity and kinetic gravity braiding
Kazufumi Takahashi ∗ and Tsutomu Kobayashi † Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan
We show explicitly that the nonminimal coupling between the scalar field and the Ricci scalarin 2D dilaton gravity can be recast in the form of kinetic gravity braiding (KGB). This is as itshould be, because KGB is the 2D version of the Horndeski theory. We also determine all the staticsolutions with a linearly time-dependent scalar configuration in the shift-symmetric KGB theoriesin 2D.
I. INTRODUCTION
Dilaton gravity in 2D allows us to study aspects ofquantum gravity while avoiding technical complexity. Itarises from spherical or hyperbolic reduction of generalrelativity in 4D (or, more generically, Lovelock gravityin higher dimensions [1]) as well as from string theory(see Ref. [2] for a review). It would be very useful if onecould have the most general action involving the met-ric and the scalar field with second-order Euler-Lagrange(EL) equations and thus generalize the framework of 2Ddilaton gravity.In fact, such generalized 2D dilaton gravity has alreadybeen obtained earlier. In Ref. [3], Horndeski derived themost general second-order EL equations derived from ascalar-tensor theory in four or less dimensions, togetherwith the action whose variation yields those EL equa-tions. The action of the Horndeski theory in 2D space-time takes the form of kinetic gravity braiding (KGB) [4]: S KGB = Z d x √− g [ K ( φ, X ) − G ( φ, X ) ✷ φ ] , (1)where K and G are arbitrary functions of φ and X := − g µν ∂ µ φ∂ ν φ/ ξ ( φ ) R , is absent in the ac-tion (1), though it has been often considered in the lit-erature. For example, Jackiw-Teitelboim gravity [5, 6] isdescribed by the action of the form S = Z d x √− g φ ( R − Λ) , (2)which can be embedded into Einstein-Maxwell-dilatongravity in higher dimensions [7]. More recently, the fol-lowing action was studied in Ref. [8]: S = Z d x √− g (cid:2) ξ ( φ ) R + k ( φ, X )+ C ( φ, X ) ∇ µ φ ∇ ν φ ∇ µ ∇ ν φ (cid:3) , (3)which can describe the effective dynamics of sphericallysymmetric spacetime in Lovelock gravity [9, 10] and was ∗ Email: takahashik“at”rikkyo.ac.jp † Email: tsutomu“at”rikkyo.ac.jp conjectured to be the most general 2D action havingsecond-order EL equations for the metric and the scalarfield. The second-order nature of the EL equations mayimply the theory (3) can be recast in the KGB form (1)as the latter was shown to yield the most general second-order EL equations, but this is far from obvious becausewe do not have a dictionary to translate ( ξ, k, C ) in (3) to(
K, G ) in (1). Thus, there is an apparent gap between ex-isting 2D dilaton gravity and the Horndeski theory in 2D,concerning the nonminimal coupling to the Ricci scalarand the terms involving a second derivative of φ .The main purpose of this short note is to fill in thisgap. We demonstrate that ξ ( φ ) R [as well as the thirdterm in (3)] can be recast in the KGB form (1). We alsoderive a new 2D black hole solution with a linearly time-dependent scalar field in the presence of shift symmetry. II. NONMINIMAL COUPLING AND KGB
In what follows, we use the notation φ µ := ∇ µ φ and φ µν := ∇ µ ∇ ν φ . The EL equations in the 2D KGB theoryare obtained by taking variations of the action (1) withrespect to g µν and φ : E µν := 1 √− g δS KGB δg µν = 0 , E φ := 1 √− g δS KGB δφ = 0 . (4)The explicit form for E µν is E µν = 12 K X φ µ φ ν + 12 Kg µν − φ ( µ ∇ ν ) G + 12 g µν φ λ ∇ λ G − G X φ µ φ ν ✷ φ = XG X ( g µν ✷ φ − φ µν ) + 12 ( K X − G φ ) φ µ φ ν + 12 ( K − XG φ ) g µν , (5)where subscripts φ and X represent partial derivativeswith respect to φ and X , respectively. Here, we haveused the identity g µν φ λ φ σ φ λσ + φ µ φ ν ✷ φ − φ λ φ ( µλ φ ν ) = 2 X ( φ µν − g µν ✷ φ ) , (6)which holds in two spacetime dimensions. This identitycan be checked on a component-by-component basis bychoosing a coordinate system where the metric is confor-mally flat, which is always possible in 2D. Regarding thescalar-field equation of motion, E φ satisfies E φ φ ν = 2 ∇ µ E µν , (7)which is nothing but the Noether identity associated withgeneral covariance. Therefore, E φ = 0 is automaticallysatisfied for a configuration ( g µν , φ ) that satisfies E µν =0. Now we consider a nonminimal coupling between thescalar field and the Ricci scalar in 2D spacetime: S NMC = Z d x √− gξ ( φ ) R. (8)It is obvious that the action yields second-order EL equa-tions and thus should be recast in the form of the KGBaction (1). We verify this by comparing the EL equationfor the metric derived from the action (8) with Eq. (5).Varying the action (8) with respect to g µν , we have1 √− g δS NMC δg µν = ∇ µ ∇ ν ξ ( φ ) − g µν ✷ ξ ( φ )= ξ ′ ( φ ) φ µν − ξ ′ ( φ ) g µν ✷ φ + ξ ′′ ( φ ) φ µ φ ν + 2 ξ ′′ ( φ ) Xg µν , (9)where a prime stands for differentiation with respect to φ . The right-hand side is reproduced from Eq. (5) bytaking K = 2 ξ ′′ ( φ ) X (2 − ln X ) , G = − ξ ′ ( φ ) ln X. (10)Note that this choice also reproduces the correspondingscalar-field EL equation thanks to the identity (7). Thus,we have established that the nonminimal coupling to theRicci scalar described by the action (8) is written in theKGB form, and hence is redundant in 2D.The discussion so far may remind us of the nonminimalcoupling to the Gauss-Bonnet term in 4D, ξ ( φ ) × (Gauss-Bonnet term). Though we know that the Horndeski the-ory gives the most general second-order scalar-tensor the-ory in 4D, it is far from trivial that this term is indeedincluded in the Horndeski action. However, by comparingthe EL equations directly, Ref. [11] showed how one canexpress the nonminimal coupling to the Gauss-Bonnetterm in terms of the four functions of ( φ, X ) in the Horn-deski action in 4D. Notice that the Gauss-Bonnet term istopological in 4D, and so is the Ricci scalar in 2D. Thus,the situation in 2D dilaton gravity is analogous to thatin 4D gravity involving the Gauss-Bonnet term.Note in passing that the third term in the action (3)can also be rewritten in the KGB form in the followingway. For C ( φ, X ) in (3) let us define D ( φ, X ) := Z X C ( φ, X ′ )d X ′ . (11)Here, D is determined only up to an arbitrary functionof φ , but this ambiguity is irrelevant in the following argument. Then, we have φ µ ∇ µ D = − D φ X − Cφ µ φ ν φ µν . (12)Integration by parts leads to Z d x √− gCφ µ φ ν φ µν = Z d x √− g ( − D φ X + D ✷ φ ) , (13)and now the right-hand side is of the KGB form. Notethat the formula (13) holds in any dimensions.We have thus seen that the action (3) studied in Ref. [8]reduces to the most general second-order scalar-tensortheory in 2D derived by Horndeski [3]. Summarizingthe above results, the former characterized by k ( φ, X ), C ( φ, X ), and ξ ( φ ) is equivalent to the latter with K = k − D φ X + 2 ξ ′′ ( φ ) X (2 − ln X ) ,G = − D − ξ ′ ( φ ) ln X, (14)where D is defined so that D X = C .More generically, the quartic Galileon Lagrangian [12]in 2D, S Gal = Z d x √− g (cid:8) F ( φ, X ) R + F X (cid:2) ( ✷ φ ) − φ µν φ µν (cid:3)(cid:9) , (15)can be recast in the KGB form (1). To see this, thefollowing identity is useful: F R + F X (cid:2) ( ✷ φ ) − φ µν φ µν (cid:3) + ∇ µ (cid:20) FX ( φ ν φ µν − φ µ ✷ φ ) (cid:21) = F φ (cid:18) ✷ φ + 1 X φ µ φ ν φ µν (cid:19) , (16)which can be verified by using Eq. (6) and R µν = Rg µν .Thus, the action (15) is written as S Gal = Z d x √− gF φ (cid:18) ✷ φ + 1 X φ µ φ ν φ µν (cid:19) , (17)and this reduces to the KGB form using the formula (13). III. ALL STATIC SPACETIMES WITH ALINEARLY TIME-DEPENDENT SCALAR FIELDIN SHIFT-SYMMETRIC THEORIES
Now let us derive static spacetimes in the 2D KGBtheory in the presence of the shift symmetry, φ → φ + c ,where c is a constant. In this case, K and G are functionsof X only, which greatly simplifies the analysis. Notethat the nonminimal coupling of the form φR respectsthe shift symmetry, because R is a total divergence in2D. Indeed, φR corresponds to K = 0 and G = − ln X inthe KGB language. Therefore, the simplest theory in theshift-symmetric class with the term φR would be givenby S = Z d x √− g ( φR + X − Z d x √− g [ X −
2Λ + (ln X ) ✷ φ ] , (18)where Λ is a constant. A black hole solution in this par-ticular theory was obtained in Refs. [13, 14]. In this sec-tion, we extend the previous results to derive the staticmetric in the most general shift-symmetric KGB the-ory while allowing for the linear time-dependence of thescalar field.The most general form of the metric of 2D static space-time is given byd s = − f ( x )d t + d x f ( x ) . (19)In shift-symmetric theories, the scalar field in staticspacetime may depend linearly on time because the ELequations contain φ µ and φ µν but not φ without deriva-tives. We therefore look for solutions of the form φ = qt + ψ ( x ) , (20)where q is a nonvanishing constant. We have X = 12 (cid:20) q f − ( ψ ′ ) f (cid:21) , (21)where a prime here denotes differentiation with respectto x .The EL equations read E µν = 0, where E tt = 12 (cid:2) K X q − Kf − G X Xf (2 f ψ ′′ + f ′ ψ ′ ) (cid:3) , (22) E tx = q (cid:18) K X ψ ′ + G X X f ′ f (cid:19) , (23) E xx = 12 (cid:20) K X ( ψ ′ ) + Kf + G X X f ′ f ψ ′ (cid:21) . (24)Since the equation of motion for φ is automatically sat-isfied if E µν = 0 is satisfied, we do not need the ex-plicit expression for E φ . Interestingly, one can integrateEqs. (22)–(24) to obtain the most general solution of theform (19) and (20) without assuming the concrete formof the functions K ( X ) and G ( X ).From E tx = 0 and E xx = 0, we obtain K ( X ) f = 0 . (25)This means that X must be a constant, X = X , (26)where X is a real root of K ( X ) = 0 (if it exists) andhence is determined from the Lagrangian under consid-eration. Then, using E tx = 0 we get ψ ′ = λ f ′ f , (27) where λ := − X G X ( X ) K X ( X ) (28)is a constant. Here and hereafter we assume that K X ( X ) = 0 and λ = 0. Equation (27) immediatelygives φ = qt + λ ln f + const , (29)where the integration constant may be absorbed into atrivial shift of the origin of t .Using Eqs. (25) and (27), the EL equation E tt = 0reduces to q + λ (cid:2) f f ′′ − ( f ′ ) (cid:3) = 0 . (30)This equation can be solved to give f = C − µx − q − λ µ λ C x , (31)where C ( = 0) and µ are integration constants. One canverify that X is indeed a constant for the above configu-rations of φ and f : X = q − λ µ C (= X ) , (32)which leads to q = ± (cid:0) λ µ + 2 CX (cid:1) / . (33)In order for this to be a real number, the two integrationconstants must satisfy λ µ ≥ − CX .We have thus obtained the general static solution witha linearly time-dependent scalar field, (29) and (31),which is characterized by the two integration constants, C and µ . Interestingly, the structure of (31) is the sameas the metric function of a 2D black hole with a cosmo-logical constant found in the literature (see Ref. [14]).As we will see shortly, µ is related to the mass of theblack hole. The effective cosmological constant is givenby Λ eff = ( q − λ µ ) / (4 λ C ) = X / (2 λ ). Note thatΛ eff is determined from the functions in the Lagrangianand it is independent of the integration constants. Forthe simplest model (18), one has Λ eff = Λ. Note alsothat the above solution has a curvature singularity at x = 0 unless µ = 0. It is possible to obtain a nonsingularblack hole by introducing appropriate terms that breakthe shift symmetry (see, e.g., Refs. [8, 15]).Let us now consider a point mass M placed at x =0. The EL equation is then given by E µν = − T µν / − ( M/ δ ( x ) δ µ δ ν . The gravitational field sourced by thispoint mass is described by f = C − µ | x | − Λ eff x . (34)Indeed, one finds that E tt = − K X ( X ) λ Cµ δ ( x ) , (35)showing that µ is proportional to the mass. See Ref. [16]for the causal structure of spacetime described by (34).So far we have assumed that q = 0. If we start with theansatz φ = ψ ( x ), the ( tx ) equation becomes trivial andas a result the EL equations would admit more varietyof solutions, depending on the concrete form of the func-tions K ( X ) and G ( X ). Note that even in that case oursolution remains a solution, and therefore taking q = 0in our solution does make sense: f = C (cid:18) − µ | x | C (cid:19) , φ = λ ln f, C = − λ µ X . (36)We see that for q = 0 the two integration constants arerelated to each other, and the solution has a degeneratehorizon (when X < µ > IV. CONCLUSIONS
In this short note, we have clarified how the action (3)with the nonminimal coupling to the Ricci scalar, ξ ( φ ) R ,can be expressed in the form of kinetic gravity braid-ing (KGB) in 2D, i.e., the most general 2D scalar-tensortheory having second-order EL equations. The relationbetween these two theories is explicitly given by (14).We have also obtained all static spacetimes with a lin-early time-dependent scalar field in the shift-symmetric2D KGB theory without assuming any specific form ofthe functions in the action. For a black hole solution, itwould be intriguing to study its thermodynamical prop-erties, but this point is beyond the scope of the presentnote. Moreover, it would also be interesting to constructnonsingular black holes in the 2D KGB theory, whichcould give us some hints on nonsingular black holes in4D.It is interesting to mention another characteristic ofthe 2D scalar-tensor theory. In 4D, by means of metricredefinition of the form g µν → A ( φ, X ) g µν + B ( φ, X ) φ µ φ ν (called disformal transformation [17]), one can promote the Horndeski class to a broader one [18–20]. However,this is not the case in 2D: The 2D Horndeski theory (i.e.,the KGB theory) is closed under the disformal transfor-mation. This fact makes the KGB theory a firm groundfor studying 2D dilaton gravity.Yet another is that the Horndeski theory can be ap-plied to the AdS/CFT correspondence in higher dimen-sions [21, 22]. Thus, it may also be interesting to explorean application of the KGB theory to the AdS /CFT correspondence.Finally, we would like to comment on 2D dilaton grav-ity as an effective theory to describe spherical (or hyper-bolic) spacetime in higher dimensions. As mentioned ear-lier, a spherical reduction of Lovelock gravity leads to theaction of the form (3), whose EL equations are at most ofsecond order. This is as expected because Lovelock grav-ity is the most general metric theory having second-orderEL equations in arbitrary dimensions. However, thereare some examples of theories having higher derivativesin their EL equations but still having only up to secondderivatives under the spherically symmetric ansatz [23–25]. From these observations, it would be intriguing toidentify all such higher derivative theories whose spher-ical reduction yields second-order field equations. Thisissue will be addressed in a future publication. ACKNOWLEDGMENTS
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