Critical gravity in the Chern-Simons modified gravity
aa r X i v : . [ h e p - t h ] O c t Critical gravity in the Chern-Simons modified gravity
Taeyoon Moon a ∗ and Yun Soo Myung b † , a Center for Quantum Space-time, Sogang University, Seoul, 121-742, Korea b Institute of Basic Sciences and School of Computer Aided Science, Inje UniversityGimhae 621-749, Korea
Abstract
We perform the perturbation analysis of the Chern-Simons modified gravity aroundthe AdS spacetimes (its curvature radius ℓ ) to obtain the critical gravity. In gen-eral, we could not obtain an explicit form of perturbed Einstein equation whichshows a massive graviton propagation clearly, but for the Kerr-Schild perturbationand Chern-Simons coupling θ = kx/y , we find the AdS wave as a single massivesolution to the perturbed Einstein equation. Its mass squared is given by M =[ − ℓ /k − ] / ℓ . At the critical point of M = 0( k = ℓ / PACS numbers:
Typeset Using L A TEX ∗ e-mail address: [email protected] † e-mail address: [email protected] Introduction
The search for a consistent quantum gravity is mainly being suffered from obtaining arenormalizable and unitary quantum field theory. Stelle has first introduced curvaturesquared terms of a ( R µν − R /
3) + bR in addition to the Einstein-Hilbert term of R/ κ [1].If ab = 0, the renormalizability was achieved, but the unitarity was violated unless a = 0.This clearly shows that the renormalizability and unitarity exclude to each other. In otherwords, the renormalizability requires 8 DOF (2 massless graviton, 5 massive graviton from a -term, and 1 massive scalar from b -term), whereas the unitarity imposes 3 DOF (2 masslessgraviton and 1 massive scalar). Although the a -term of providing massive graviton improvesthe ultraviolet divergence, it induces ghost excitations which jeopardize the unitarirty. Inthis sense, a first test for the quantum gravity is to require the unitarity, which means thatthere are no tachyon and ghost in its particle contents.To this end, we would like to comment that the critical gravities as candidates forquantum gravity were recently investigated in the AdS spacetimes [2, 3, 4, 5, 6, 7, 8].At the critical point, a degeneracy takes place and massive gravitons coincide with eithermassless gravitons ( D >
3) or pure gauge modes ( D = 3). Instead of massive gravitons,an equal amount of logarithmic modes appears in the theory [9]: 1 DOF for topologicallymassive gravity (TMG) [10], 2 DOF for new massive gravity [11, 12, 13], 5 DOF for highercurvature gravity in 4D [5]. In general, we have D ( D + 1) / − ( D + 1) DOF for massivegraviton. However, the non-unitarity issue of the log-gravity is not still resolved [2, 6],indicating that any log-gravity suffers from the ghost problem. Furthermore, the criticalgravity on the Schwarzschild-AdS black hole has suffered from the ghost problem when thecross term E cross is non-vanishing [14].In this work, we introduce a Lorentz-violating theory of Cherns-Simons modified grav-ity [15]. A silent feature of this theory is the presence of a constant vector v c which spoilsthe isotropy of spacetime (CPT-symmetry) and is coupled to the Pontryagin density of ∗ RR . Motivation of considering Cherns-Simons modified gravity is twofold in Minkowskispacetimes: one is its close connection to the TMG which accommodates a single massivegraviton in three dimensions [16] and the other is the crucial dependence of massive gravitonon a choice of constant vector v c . It was shown that a timelike vector of v c = ( µ, ~
0) did notprovide any massive mode, leaving massless graviton with 2 DOF, while a spacelike vector v c = (0 , ~v ) yielded a massive graviton with 5 DOF [17]. However, the authors [18] have2hown that the only tachyon- and ghost-free model is the case of timelike vector v c = ( µ, ~ spacetimes to obtain the critical gravity, instead of Minkowski spactimes.Under the transverse and traceless gauge, we could not obtain a compactly third-orderperturbed equation which shows a massive graviton with 5 DOF, but for the Kerr-Schildperturbation with spacelike vector v c = k (0 , , /y, − x/y ), we find the AdS wave as asingle massive graviton propagating on AdS spacetimes. This was found as a solution tothe Einstein equation [19]. This (1 DOF) contrasts to propagating DOF of graviton inMinkowski spacetimes. At the critical point of k = ℓ /
2, the solutions takes the log-formand the linearized excitation energies vanish, which indicates a feature of critical gravity.
Let us first consider the Chern-Simons modified gravity in four dimensions with a cosmo-logical constant (Λ) whose action is given by S = 116 πG Z d x √− g n R −
2Λ + θ ∗ RR o (2.1)where θ is an external function of spacetime and ∗ RR = ∗ R a cdb R bacd is the Pontryagindensity with ∗ R a cdb = 12 ǫ cdef R abef . (2.2) θ is a diffeomorphism breaking parameter and it will be fixed by the equation of motion. Therefore, itis hard to be considered as a Lagrange multiplier. In the Chern-Simons modified Maxwell theory, θ can befixed as µt which yields the modified Ampere’s law [15]. At this stage, one may ask the question “can wecall any model without diffeomorphism as gravity?”. In order to answer it, we remind the feature of thegravitational Chern-Simons modified theory [15]. Here the diffeomorphism breaking is being realized fromthe fact that the covariant divergence of the four-dimensional Cotton tensor is non-zero [see Eq.(2.5)], incontrast to the case in three dimensions. Therefore, a consistency condition on this theory is that ∗ RR = 0for ∇ b θ = 0 (the theory reduces to the general relativity for ∇ b θ = 0 because of C ab = 0). In this sense,diffeomorphism symmetry breaking is suppressed dynamically for the case of ∗ RR = 0 (e.g., Schwarzschildblack hole or AdS spacetimes), even if it may occur at the action level.
3n this expression, ǫ cdef denotes the four-dimensional Levi-Civita tensor. Varying for g ab onthe action (2.1) leads to the Einstein equation which takes the form R ab − g ab R + Λ g ab + C ab = 0 (2.3)where C ab is the four-dimensional Cotton tensor given by C ab = ∇ c θ ǫ cde ( a ∇ | e | R b ) d + 12 ∇ c ∇ d θ ǫ cef ( b R d a ) ef . (2.4)Note that C ab is a traceless and symmetric tensor. As a result of applying Bianchi identityto (2.3), one has ∇ a C ab = h ∇ b θ i ∗ R acdf R acdf . (2.5)On the other hand, one finds that Eq.(2.3) has an AdS solution in which the Riemanntensor, Ricci tensor and Ricci scalar of the AdS spacetimes are given by¯ R abcd = Λ3 (¯ g ac ¯ g bd − ¯ g ad ¯ g bc ) , ¯ R ab = Λ¯ g ab , ¯ R = 4Λ . (2.6)Here “overbar” denotes the background AdS -metric ¯ g ab .In order to obtain perturbation equations, we introduce the perturbation around thethe background metric as g ab = ¯ g ab + h ab . (2.7)The linearized equation to (2.3) can be written by δR ab ( h ) − g ab δR ( h ) − Λ h ab + δC ab ( h ) = 0 , (2.8)where the linearized tensor δR ab ( h ) , δR ( h ) , and δC ab ( h ) take the form δR ab ( h ) = 12 (cid:0) ¯ ∇ c ¯ ∇ a h bc + ¯ ∇ c ¯ ∇ b h ac − ¯ ∇ h ab − ¯ ∇ a ¯ ∇ b h (cid:1) δR ( h ) = ¯ ∇ a ¯ ∇ b h ab − ¯ ∇ h − Λ hδC ab ( h ) = " v c ǫ cdea (cid:0) ¯ ∇ e δR bd − Λ ¯ ∇ e h bd (cid:1) + 18 v cd ǫ cefb (cid:16) ¯ ∇ e ¯ ∇ f h da + ¯ ∇ e ¯ ∇ a h df − ¯ ∇ e ¯ ∇ d h af − ¯ ∇ f ¯ ∇ e h da − ¯ ∇ f ¯ ∇ a h de + ¯ ∇ f ¯ ∇ d h ae (cid:17) + " a ↔ b (2.9)4ith v c = ¯ ∇ c θ and v cd = ¯ ∇ c ¯ ∇ d θ . Imposing the transverse and traceless (TT) gaugecondition as ¯ ∇ a h ab = 0 , h = ¯ g ab h ab = 0 (2.10)which takes into account the diffeomorphism [20] δ ξ h ab = ¯ ∇ a ξ b + ¯ ∇ b ξ a , (2.11)the perturbation equation (2.8) takes a simpler form −
12 ¯ ∇ h ab + Λ3 h ab + δC ab = 0 . (2.12)Here the linearized tensor δC ab ( h ) is given by δC ab ( h ) = h − v c ǫ cdea ¯ ∇ e ¯ ∇ h bd + Λ6 v c ǫ cdea ¯ ∇ e h bd + 14 v cd ǫ cefb (cid:16) ¯ ∇ e ¯ ∇ a h df − ¯ ∇ e ¯ ∇ d h af (cid:17)i + h a ↔ b i . (2.13)We observe that δC ab ( h ) takes still a complicated form, depending v c and v cd . It is important to note that the perturbation equation (2.12) has the dependency of θ .For a choice of θ = t/µ [15], the Cotton tensor (2.4) reduces to the TMG when choosingthe Schwarzschild coordinates. However, in the AdS spacetimes, such a choice is notguaranteed since the second term v ab survives. In the AdS spacetimes, there exists aparticular choice of θ [19] which makes v cd vanish. This choice of θ could be made bychoosing the Poincare coordinates ( u, v, x, y ) for the AdS spacetimes: θ = k xy , ¯ g ab = φ − η ab = ℓ y η ab , (3.1)where k ( >
0) has the dimension of [mass] − , ℓ is the AdS curvature radius ( ℓ = − / Λ)and η ab is η ab dx a dx b = 2 dudv + dx + dy . (3.2)5onsidering Eq.(3.1), Eq.(2.12) becomes (cid:16) ¯ ∇ −
23 Λ (cid:17)(cid:16) h ab + v c ǫ cde ( a ¯ ∇ | e | h b ) d (cid:17) = 0 . (3.3)Alternatively, it leads to (cid:16) δ a ′ ( a δ db ) + δ a ′ ( a v | c | ǫ cdeb ) ¯ ∇ e (cid:17)(cid:16) ¯ ∇ −
23 Λ (cid:17) h a ′ d = 0 (3.4)which is found by using commutation between two operations in Eq.(3.3). Here, v c is givenby v c = k , , y , − xy ! (3.5)which may generate the mass. In this case, v c is not a constant vector but a vector field. Wewish to comment that Eq.(3.3) is an extended version in four dimensions when comparingwith the TMG [21]. In three dimensions, one analyzes the perturbation equation by using D -operator (cid:16) D µ/ ˜ µ (cid:17) βα = δ βα ± µ ǫ γβα ¯ ∇ γ . (3.6)However, it is not easy to apply D -operator directly to Eq. (3.3) because v c is not aconstant vector. In order to see this case explicitly, we introduce ˆ D ˜ M -operator in the AdS spacetimes (cid:16) ˆ D M/ ˜ M (cid:17) ff ′ ad = δ f ( a ′ δ f ′ d ) ± δ f ( a ′ v | c | ǫ cf ′ ed ) ¯ ∇ e . (3.7)Then, Eq.(3.4) can be rewritten as (cid:16) ˆ D M (cid:17) a ′ dab (cid:16) ¯ ∇ −
23 Λ (cid:17) h a ′ d = 0 . (3.8)Now we use ˆ D ˜ M ˆ D M -operation to find (cid:16) δ f ( a ′ δ f ′ d ) − δ f ( a ′ v | c ′ | ǫ c ′ f ′ e ′ d ) ¯ ∇ ′ e (cid:17)(cid:16) δ a ′ ( a δ db ) + δ a ′ ( a v | c | ǫ cde b ) ¯ ∇ e (cid:17) h ff ′ = − v (cid:16) ¯ ∇ −
23 Λ − v (cid:17) h ab + 4 v e ′ v e ¯ ∇ e ′ ¯ ∇ e h ab − θv e ¯ ∇ e h ab − v e v e ′ ¯ ∇ h ee ′ g ab + 83 Λ v e v e ′ h ee ′ g ab + h − v e ′ v e ¯ ∇ a ¯ ∇ e h e ′ b + 3 v e v a ¯ ∇ h eb − Λ3 θv e ¯ ∇ a h eb + v e ′ v e ¯ ∇ a ¯ ∇ b h ee ′ − v e v e ′ ¯ ∇ e ′ ¯ ∇ a h eb −
83 Λ v e v b h ea + ( a ↔ b ) i = 0 (3.9)6ith v ≡ v e v e . In obtaining this, we have used the gauge condition (2.10). At this stage,it is very difficult to derive the massive second-order equation , (cid:16) ¯ ∇ −
23 Λ − M (cid:17) h ab = 0 (3.10)unless we choose a simple form of the metric perturbation h ab .In order to analyze Eq.(3.3), we consider the AdS wave as the Kerr-Schild form h ab = 2 ϕλ a λ b , (3.11)where λ a is a null and geodesic vector whose form is given by λ a = (1 , , ,
0) and ϕ is anarbitrary function of coordinates ( u, v, x, y ). To maintain the TT gauge condition (2.10),one confines ϕ to ϕ ( u, x, y ) by requiring the condition of λ a ¯ ∇ a ϕ = 0 ( → ∂ v ϕ = 0). Plugging h ab = 2 ϕλ a λ b into Eq.(3.3) leads to λ a ′ λ d h δ a ′ ( a δ db ) + δ a ′ ( a v | c | ǫ cdeb ) (cid:16) ∂ e φφ + ¯ ∇ e (cid:17)ih ¯ ∇ + 23 Λ + 4 φ ∂ f φ ∂ f i ϕ = 0 . (3.12)At this stage, we introduce the separation of variables by considering ϕ ( u, x, y ) = U ( u ) X ( x ) Y ( y ) . (3.13)Taking into account λ a , v c , and φ , Eq.(3.12) can be reduced to h y∂ y + x∂ x + 1 − ℓ k ih y ( ∂ y + ∂ x ) + 2 y∂ y − i XY = 0 . (3.14)Note that the right bracket in Eq.(3.14) represents the perturbation equation of the masslessscalar which corresponds to the right parenthesis of massless tensor in Eq.(3.4). On theother hand, we expect that the left bracket in Eq.(3.14) is related to the massive-modeequation as was suggested in three dimensions [21]. In order to obtain the massive-mode(scalar) equation from the left bracket in Eq.(3.14), we introduce an operator of y∂ y + A with A an arbitrary constant. Furthermore, we assume that X ( x )=constant. Then, wecheck that the quadratic perturbation equation yields( y∂ y + A )( y∂ y + 1 − ℓ /k ) Y = 0 → h y ∂ y + y (2 − ℓ /k + A ) ∂ y + A (1 − ℓ /k ) i Y = 0 , (3.15) Assuming that all terms except the first term of − v (cid:16) ¯ ∇ − Λ − v (cid:17) h ab vanish, it has still a problemto derive Eq.(3.10). This is because v is not a constant scalar as k ℓ = µ in the TMG, but it is a scalarfunction given by v = k ℓ x + y y . h ¯ ∇ + 23 Λ + 4 φ ∂ a φ∂ a − M i ϕ = 0 → ℓ h y ∂ y + 2 y∂ y − − M ℓ i Y = 0 . (3.16)Comparing Eq.(3.15) with Eq.(3.16), we find A = ℓ k , ℓ k = 12 (cid:16) √ ℓ M (cid:17) . (3.17)It is worth noting that for real ℓ /k , the allowed region of M is given by M = 14 ℓ h − (cid:18) ℓ k − (cid:19) i ≥ M = − ℓ , (3.18)where M corresponds to the Breitenlohner-Freedman (BF) bound for a massive scalar inAdS spacetimes [23]. This occurs also for k = 2 ℓ . Importantly, in the critical limit of M →
0, we obtain k = ℓ / k > ℓ /
2, wehave an allowed bound for negative M (see Fig.1) M ≤ M < y " y∂ y + 12 − ℓ r M + 94 ℓ ! y∂ y − ih y∂ y + 2 i Y = 0 . (3.20)Now we solve Eq.(3.20) for two cases:(i) k = ℓ / M = 0) ϕ ( u, y ) = U ( u ) Y ( y ) = c ( u ) y [ − ℓ √ M +9 / ℓ ] + c ( u ) 1 y + c ( u ) y, (3.21)which is a single massive solution in AdS spacetimes. There also exists the solution of (1 − √ ℓ M ) /
2. However, it violates the allowed region of M , M < M BF for k >
0. This induces the tachyon instability. Hence, we ignore this solution for theChern-Simons coupling k > k - - M BF = Figure 1: M graph as function of k with ℓ = 1. For k > / M < M ≥ M BF = − /
4. Hence we havethe stable region for positive k ( k > k = 1 / M = 0. Also, in the limit of k → ∞ , it approaches M = − k = ℓ / M = 0)In this case, Eq.(3.20) degenerates as( y∂ y + 2)( y∂ y − Y = 0 . (3.22)We obtain the solution as ϕ ( u, y ) = U ( u ) Y ( y ) = c ( u ) y ln( y ) + c ( u ) 1 y + c ( u ) y. (3.23)In this approach, c i ( u ) as functions of u remain undetermined.We note that the solution (3.23) will be a half of the solution obtained from higher cur-vature gravity which gives the fourth-order perturbation equation at the critical point [24].To see this more closely, we construct the fourth-order equation instead of the third-orderequation (3.20) by considering h ( y∂ y + ℓ /k )( y∂ y + 1 − ℓ /k ) ih ( y∂ y − y∂ y + 2) i Y = 0 . (3.24)9or k = ℓ /
2, the solution to Eq.(3.24) is given by Y = Y + Y where Y and Y satisfy thefollowing second-order equations, respectively: h ( y∂ y + ℓ /k )( y∂ y + 1 − ℓ /k ) i Y = 0 , h ( y∂ y − y∂ y + 2) i Y = 0 . (3.25)The corresponding solutions and combined solution are Y ( y ) = d y [ − ℓ √ M +9 / ℓ ] + d y − [1+2 ℓ √ M +9 / ℓ ] , (3.26) Y ( y ) = d y − + d y, (3.27) → Y (= Y + Y ) = d y [ − ℓ √ M +9 / ℓ ] + d y − [1+2 ℓ √ M +9 / ℓ ] + d y − + d y, (3.28)where M appeared in (3.18) and d i are undetermined constants. We note that althoughthe solution form is the same as found in the higher curvature gravity [24], the mass squared M in (3.18) is different from that [(8) in [24]] in the higher curvature gravity. At the criticalpoint of M = 0 ( k = ℓ / h(cid:16) y∂ y − (cid:17)(cid:16) y∂ y + 2 (cid:17)i Y = 0 , (3.29)whose solution is given by Y ( y ) = d y ln( y ) + d y + d y + d y ln( y ) (3.30)which shows that the last term is absent in (3.23). This solution is exactly the same foundin the higher curvature gravity [24]. In the perturbation analysis, it is important to check whether the ghost mode exists or not.For this purpose, we construct the Hamiltonian of the action. Firstly, the quadratic actionof h ab takes the form S (2) = − πG Z d x √− gh ab h δ (cid:16) R ab − g ab R + Λ g ab (cid:17) + δC ab i = − πG Z d x √− g h
12 ( ¯ ∇ c h ab )( ¯ ∇ c h ab ) + Λ3 h ab h ab + 12 ǫ cdea (cid:16) v ce h ab ¯ ∇ h bd + v c ¯ ∇ e h ab ¯ ∇ h bd + 23 Λ v c h ab ¯ ∇ e h bd (cid:17) − ǫ cefb (cid:16) v cde h ab ¯ ∇ a h df + v cd ¯ ∇ e h ab ¯ ∇ a h df − v cde h ab ¯ ∇ d h af − v cd ¯ ∇ e h ab ¯ ∇ d h af (cid:17)i . (4.1)10rom the action (4.1), we define the conjugate momentum given byΠ ab (1) = − ¯ ∇ h ab − ǫ cd a v c ¯ ∇ h bd − Λ3 v c ǫ ca d h db + 12 ǫ bc f v cd ¯ ∇ a h df + 12 ǫ cebf v ac ¯ ∇ e h f − ǫ cebf v c e h af − ǫ bc f v cd ¯ ∇ d h af − ǫ cebf v c ¯ ∇ e h af + 12 ¯ ∇ ( ǫ caed v c ¯ ∇ e h db ¯ g ) . (4.2)Using the method of Ostrogradsky, we find the conjugate momentum for the second-timederivative as Π ab (2) = −
12 ¯ ∇ ( ǫ caed v c ¯ ∇ e h db ¯ g ) . (4.3)Then the Hamiltonian can be written by H = Z d x (cid:16) ˙ h ab Π ab (1) + ˙ K ai Π ai (2) (cid:17) − S (2) (4.4)with K ai = ¯ ∇ h ai . Considering (3.1) and (3.11), one finds that the Hamiltonian (4.4) isidentically zero ( H = 0), irrespective of any solution form ϕ . This means that there is noghost for AdS waves. In the Minkowski spacetimes, the ghost- and tachyon-free mode of Chern-Simons modifiedgravity is just a massless graviton with 2 DOF [18]. This amounts to the choice of a timelikevector v c = ( µ, ~ spacetimes because its linearized equation is a very complicated form, compared to theTMG, showing a single massive scalar [21]. However, choosing v c as a vector field (3.5) whichmakes the perturbation equation simple and then, the Kerr-Schild perturbation (3.11), wehave a single massive scalar ϕ propagating on the AdS spacetimes. This is ghost-freeand tachyon-free if the mass squared (3.18) satisfies the BF bound M ≥ M . Even forthe negative bound of − M ≤ M <
0, there is no tachyon instability (no exponentiallygrowing modes) [22]. At the critical point of M = 0( k = ℓ / spacetimes from the Chern-Simons modified gravity, compared to the higher curvaturegravity [9]. This is mainly because massive excitations depend critically on the choice ofcoupling field v c ( θ ). Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) grantfunded by the Korea government (MEST) through the Center for Quantum Spacetime(CQUeST) of Sogang University with grant number 2005-0049409. Y. Myung was partlysupported by the National Research Foundation of Korea (NRF) grant funded by the Koreagovernment (MEST) (No.2011-0027293).