Criticality in Einstein-Gauss-Bonnet Gravity: Gravity without Graviton
aa r X i v : . [ h e p - t h ] S e p Criticality in Einstein-Gauss-Bonnet Gravity: Gravity without Graviton
Zhong-Ying Fan , Bin Chen , , and Hong L¨u Center for High Energy Physics, Peking University, No.5 Yiheyuan Rd,Beijing 100871, P. R. China Department of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University, No.5 Yiheyuan Rd, Beijing 100871, P.R. China Collaborative Innovation Center of Quantum Matter, No.5 Yiheyuan Rd,Beijing 100871, P. R. China Center for Advanced Quantum Studies, Department of Physics,Beijing Normal University, Beijing 100875, China
ABSTRACTGeneral Einstein-Gauss-Bonnet gravity with a cosmological constant allows two (A)dSspacetimes as its vacuum solutions. We find a critical point in the parameter space where thetwo (A)dS spacetimes coalesce into one and the linearized perturbations lack any bilinearkinetic terms. The vacuum perturbations hence loose their interpretation as linear gravitonmodes at the critical point. Nevertheless, the critical theory admits black hole solutions dueto the nonlinear effect. We also consider Einstein gravity extended with general quadraticcurvature invariants and obtain critical points where the theory has no bilinear kineticterms for either the scalar trace mode or the transverse modes. Such critical phenomenaare expected to occur frequently in general higher derivative gravities.
Emails: [email protected] [email protected] [email protected] ontents
Einstein gravity with a cosmological constant may be viewed as the simplest dynamical the-ory of the metric under the principle of general coordinate invariance. Owing to the existenceof a fundamental constant of length, namely the Planck length ℓ p = √ G N , where G N is theNewton’s constant, it is natural to consider higher-derivative extensions to Einstein gravity.This should be contrasted with quantum field theory where such a “minimum” length scaleis absent and hence it is “unnatural” to consider higher-derivative terms. String theory pre-dicts that such higher-derivative structures are inevitable in its low-energy effective theory.The explicit structure however is hard to determine.Introducing higher-derivative terms to Einstein gravity can have important advantages.It was shown that Einstein gravity extended with quadratic curvature invariants in fourdimensions can be renormalizable for appropriate coupling constants [1, 2]; however, thetheory contains ghost-like massive spin-2 modes. When a cosmological constant is included,2here exists a critical point of the parameter space [3, 4] of the coupling constants for whichthe ghost-like massive graviton becomes a logarithmic mode. In three dimensions, imposinga strong boundary condition to get rid of this mode may yield a consistent quantum theoryof gravity, whose degrees of freedom exist only in the boundary of the anti-de Sitter (AdS )spacetime [5, 6]. This procedure may not be possible in four or higher dimensions; ratherthe theories are expected to be dual to some logarithmic conformal field theories on theboundary of AdS. (See e.g. reviews [7, 8, 9].)In higher dimensions, there exist further special combinations of higher-order curvatureinvariants for which the linearized theories involve only two derivatives, and hence ghost ex-citations can be absent. These are Gauss-Bonnet or Lovelock gravities [10]. Einstein-Gauss-Bonnet (EGB) gravity contains two non-trivial parameters, namely the bare cosmologicalconstant Λ and the coupling constant γ of the Gauss-Bonnet term. In general, there existstwo (A)dS vacua in the EGB theory. In section 2, we consider linearized gravity in thesevacua and find that the linear modes have positive kinetic energy in one vacuum whilst theyhave negative energy in the other. At some critical point, the two (A)dS coalesce into oneand the effective coupling for the kinetic term vanishes. The theory at the critical point thusdoes not have propagators and hence the linear modes cannot be viewed as gravitons. Wethen derive the perturbative equations of motion at the quadratic order. Furthermore weobtain the most general static solutions with spherical/toric/hyperbolic topologies. Usingthe Wald formalism [11, 12], we find that these solutions have no negative mass, indicat-ing that the theory may not have nonlinear ghost modes. One of the solutions describesa black hole, which was obtained in [13], where general Lovelock gravities that addmit asingle (A)dS vacuum were classified and studied. We analyse its global structure and thethermodynamical phase transition.In section 3, we consider Einstein gravity extended with general quadratic curvatureinvariants, with additional αR + βR µν R µν terms. We find that there also exists a criticalpoint where the linearized equation of motion for the scalar trace mode is automaticallysatisfied. In this case the theory does not have a kinetic term for the scalar mode. We alsoobtain another critical point where the linearized equations of motion of the theory involveonly the trace scalar mode, whilst there is no kinetic term for any transverse mode. Notethat the critical case where all linear perturbations have no kinetic terms can only occurwhen α = 0 = β , in other words, in the EGB theory. We then consider the deviations fromthe critical points in section 4, which helps us to understand the integration constants ofthe solutions at the critical point in the more general setting. We conclude the paper in3ection 5. In this section, we consider Einstein-Gauss-Bonnet (EGB) gravity without matter fields.The Lagrangian density is given by L = κ √− g h(cid:0) R − (cid:1) + γ (cid:0) R − R µν R µν + R µνρσ R µνρσ (cid:1)i , (1)where κ = 1 / (16 π G N ) > G N being the Newton’s constant, and Λ is the barecosmological constant. The parameter γ has a dimension of length squared, and in stringtheory, it is related to the string tension or the coupling constant α ′ in string worldsheetaction.The equations of motion are E µν = 0 and E µν ≡ G µν + Λ g µν + 2 γ (cid:0) RR µν − R µσνρ R σρ + R µσρλ R νσρλ − R µρ R νρ (cid:1) − γg µν (cid:0) R − R µν R µν + R µνρσ R µνρσ (cid:1) , (2)where G µν = R µν − g µν R is the Einstein tensor. It is well known that there exist twodistinct (A)dS vacua for generic values of the coupling constants. The effective cosmologicalconstant Λ satisfies a quadratic algebraic equation κ (cid:2) (cid:0) Λ − Λ (cid:1) + ∆ Λ (cid:3) = 0 , with ∆ ≡ ( D − D − D − D − γ . (3)Thus the two effective cosmological constants are given byΛ ± = ±√ Λ − . (4)When γ = 0 and hence Λ = 0, one of the (A)dS spacetimes becomes Minkowski. WhenΛ = − / (8∆ ), the two effective cosmological constants Λ ± become the same, and the two(A)dS vacua degenerate into one, with the effective cosmological constantΛ + = Λ − = Λ ∗ ≡ . (5)Note that the reality condition for Λ ± requires that 8∆ Λ ≥ −
1. When this conditionis not satisfied, the theory then does not admit any maximally-symmetric spacetime as itsvacuum solution [14].General Lovelock gravities with only a single (A)dS vacuum were classified and studiedin [13]. 4 .1 Linearized gravity
We now study the linearized equations of motion of the metric perturbation g µν = ¯ g µν + h µν (6)around one of the (A)dS vacua for general parameters. They are simply κ eff G Lµν = 0 , κ eff = κ (1 + 4∆ Λ) . (7)The linearized Einstein tensor around the (A)dS vacuum is given by G Lµν = R Lµν − ¯ g µν R L − D − h µν , (8)where the linearized Ricci tensor R Lµν and scalar curvature R L ≡ (cid:0) g µν R µν (cid:1) L are respectively R Lµν = 12 (cid:16) ¯ ∇ σ ¯ ∇ µ h νσ + ¯ ∇ σ ¯ ∇ ν h µσ − ¯ (cid:3) h µν − ¯ ∇ µ ¯ ∇ ν h (cid:17) , R L = − ¯ (cid:3) h + ¯ ∇ σ ¯ ∇ µ h µσ − D − h . (9)For later purposes we also give the linearized Riemann tensors R L ρµλν = ¯ ∇ [ λ ¯ ∇ µ ] h ρν + ¯ ∇ [ λ ¯ ∇ | ν | h ρµ ] − ¯ ∇ [ λ ¯ ∇ ρ h µ ] ν = 12 h ¯ ∇ λ ¯ ∇ µ h ρν + ¯ ∇ λ ¯ ∇ ν h ρµ + ¯ ∇ µ ¯ ∇ ρ h λν − ¯ ∇ µ ¯ ∇ λ h ρν − ¯ ∇ µ ¯ ∇ ν h ρλ − ¯ ∇ λ ¯ ∇ ρ h µν i ,R Lµλνρ ≡ (cid:0) g ρσ R σµλν (cid:1) L = ¯ g ρσ R L σµλν + 2Λ( D − D − (cid:0) ¯ g µν h λρ − ¯ g νλ h µρ (cid:1) = (cid:16) ¯ ∇ [ λ ¯ ∇ | ν | h µ ] ρ − ¯ ∇ [ λ ¯ ∇ | ρ | h µ ] ν (cid:17) + 2Λ( D − D − (cid:16) ¯ g ν [ µ h λ ] ρ − ¯ g ρ [ µ h λ ] ν (cid:17) . (10)Taking the trace of (7) gives κ eff R L = 0 . (11)For the (A)dS background, it is advantageous to take the following gauge choice [5]¯ ∇ µ h µν = ¯ ∇ ν h . (12)It follows that R L = − D − h and R Lµν = h −
12 ¯ (cid:3) h µν + 2 D Λ( D − D − h µν i + h
12 ¯ ∇ µ ¯ ∇ ν h − D − D −
2) ¯ g µν h i , G Lµν = h −
12 ¯ (cid:3) h µν + 2Λ( D − D − h µν i + h
12 ¯ ∇ µ ¯ ∇ ν h + ( D − D − D −
2) ¯ g µν h i . (13)For generic parameters with κ eff = 0, the trace equation (11) implies the traceless condition h = 0 and hence the graviton mode is also transverse. The linearized equation of motionbecomes simply (cid:16) ¯ (cid:3) − D − D − (cid:17) h µν = 0 . (14)5his is the equation of motion satisfied by the massless graviton, in each of the two (A)dSvacua. It is worth pointing out however that the effective coupling constant κ eff on the twovacua has opposite signs, namely κ eff = ± κ p Λ . (15)Thus the linear graviton on the Λ + vacuum has the positive kinetic energy, whilst the oneon the Λ − vacuum has the negative kinetic energy, and hence is ghost-like.In string theory the bare cosmological constant Λ vanishes, and hence Λ + = 0 andΛ − = − / (2∆ ). It follows that the Minkowski vacuum remains ghost free under the α ′ correction. Note that γ is positive in string theory and hence the other vacuum is AdS,with ghost-like graviton modes. For a non-vanishing Λ , we have Λ + > Λ − for γ > + < Λ − for γ < When the parameters satisfy (5), i.e. the Gauss-Bonnet coupling constant γ and the barecosmological constant are related as follows∆ = − = − ∗ , (16)the two (A)dS vacua coalesce into one, with the effective cosmological constant being 2Λ .In this case, we have κ eff = 0 and hence the linearized equations of motion in the abovesubsection are automatically satisfied. The absence of the kinetic term for the fluctuation h µν at the quadratic order implies that the theory does not have any propagator, and henceit is no longer proper to take h µν as the usual graviton modes. We have thus a theory ofgravity without graviton.As we run the parameter 8∆ Λ + 1 → + , we have κ eff →
0. One might expect that κ eff becomes negative as one let 8∆ Λ + 1 be negative such that the theory has ghost-likemode. However this never happens. Instead, as 8∆ Λ + 1 becomes negative, the vacuumspacetime is no longer maximally-symmetric. Thus the critical point can be viewed asthe phase transition point, beyond which the maximally-symmetric spacetimes are not thesolutions of the theory.We arrived at the above critical point by studying the linearized equations of the EGBtheory. It happens that at the critical point the theory also admits only one (A)dS vacuum.In [13], it was shown that there exist such critical points where only a single (A)dS vacuumwas admitted in general Lovelock gravities. We may expect that the corresponding theoriesalso have no graviton at the linearized level. 6 .3 Quadratic-order equations at the critical point At the critical point, the linearized equations of motion are automatically satisfied, it isthus necessary to study the equations of motion at the quadratic order. It follows from (6)that we have g µν = ¯ g µν − h µν + h µσ h ν σ + O ( h ) . (17)Up to and including the quadratic order in h , we find that the Einstein equations E µν = 0become κ eff (cid:16) G Lµν + G Qµν (cid:17) − κ (cid:16) h µν R L − γ (cid:0) E (0) µν + E (1) µν + E (2) µν (cid:1)(cid:17) = 0 . (18)Here G Lµν is given by (8) and G Qµν = R Qµν − ¯ g µν R Q . The quantities R Qµν and R Q denote theRicci tensor and Ricci scalar at the quadratic order of h , respectively. It is clear that thedetails of these two quantities are irrelevant at the critical point κ eff = 0. The quantities E ( i ) µν are given by E (0) µν = 2 (cid:16) R L R Lµν − R Lµσ e R L σν − R Lµλνρ e R Lλρ + R Lµλρσ e R Lλρσν (cid:17) − ¯ g µν (cid:16) R L R L − R Lλρ e R Lλρ + R Lλρστ e R Lλρστ (cid:17) , E (1) µν = 8( D − D − D − (cid:16) R Lµλνρ h λρ + 2 R Lσ ( µ h σν ) − ¯ g µν R Lλρ h λρ − ( D − h µν R L (cid:17) , E (2) µν = 8Λ ( D − ( D − (cid:16) ¯ g µν h + ( D − D + 5) (cid:0) ¯ g µν h λρ h λρ − h µσ h σν (cid:1)(cid:17) . (19)Here the curvature tensors with tildes are defined by raising the indices from R Lµν and R Lµνρσ with the background metric ¯ g µν . All the untilded tensor or scalar quantities withthe superscript L are given in (9) and (10). Thus we see that the superscript of i in E ( i ) µν denotes the order of the bare h . It follows that the first term in the second bracket of (18)is similar to E (1) µν . We separate it out so that we can see clearly that it comes from theEinstein-Hilbert term rather than from the Gauss-bonnet term. At the critical point (16),only the second bracket in (18) survives. The general “spherically-symmetric” ansatz can be parameterized as ds = − hdt + dr f + r d Ω D − ,k , (20)where d Ω D − ,k with k = 0 , ± D − R ij = ( D − k δ ij . (Note that in this paper, we adopt the looseterminology “spherically-symmetric” to denote solutions for all k = 1 , , −
1, for the lacking7f a simple terminology for general topologies.) The Schwarzschild-like solutions for thegeneral EGB theory was obtained in [15, 16]. They become degenerate at the critical point.We find that at the critical point, there are two types of solutionstype 1 : h = f = g r + k − µr D − , Λ = − ( D − D − g , (21)type 2 : f = g r + k , and h is an arbitrary function of r . (22)We first examine the type-1 solution, which describes a black hole, with the outer horizonlocated at the largest r for which f ( r ) = 0. This solution was first constructed in [13, 17],where Lovelock gravities with single AdS vacuum were studied. The temperature and theentropy can be determined by the standard technique, given by T = ( D − g r + k ( D − πr , S = κωr D − (cid:16) ( D − k ( D − g r (cid:17) , (23)where ω is the volume of the metric d Ω D − ,k . One may determine the mass of the blackhole by the completion of the first law of black hole thermodynamics. However, the blackhole solution (21) has the unusual falloff, rather than the 1 /r D − falloff that is typicalof the condensation of the graviton modes. To derive the first law, we apply the Waldformalism. The explicit expressions of the Wald formalism [11, 12] for the spherically-symmetric solutions in gravity extended with quadratic curvature invariants were obtainedin [18]. It is given by δ H = ω κ π r D − s hf (cid:16) − D − r + 2( D − D − D − γ ( f − k ) r (cid:17) δf . (24)It is easy to verify that evaluating the above on the horizon yields δ H + = T δS ; whilstevaluating it at the asymptotic infinity gives δH ∞ = κω ( D − πg µδµ ≡ δM , (25)implying that the black hole mass is M = κω ( D − πg µ . (26)The quadratic dependence of the mass on the constant µ is a consequence that there areno linearized equations of motion in this theory. It follows that the first law of black holethermodynamics reads dM = T dS . One may also treat the effective cosmological constantΛ as a thermodynamical pressure [19, 20], and then the first law becomes dM = T dS + V dP ,where P = − Λ8 π , V = κωr D − D − D − g (cid:0) k − ( D − kg r − ( D − g r (cid:1) . (27)8he Smarr relation becomes M = D − D − T S − D − V P . (28)When k = 0, the solution becomes an AdS planar black hole. In this case,an extra scalingsymmetry emerges and leads to an additional Smarr relation that is independent of thepressure [21] M = D − D − T S . (29)It is also worth noting that the solution has a space-like curvature singularity at the origin r = 0. The singularity is milder than the usual Schwarzschild-like black hole. We consider D = 5 as an example, in which case, g tt is non-divergent at r = 0, and the Riemann tensorsquared, R ≡ R µνσρ R µνσρ , (30)has the 1 /r divergence rather than the 1 /r divergence as r →
0. The free energy of theblack hole F = M − T S is given by F = κωr D − π ( D − g (cid:0) ( D − k − g r k − ( D − g r (cid:1) . (31)Thus for k = 1, there is also a Hawking-Page-type phase transition [22]. The minimumtemperature for the black holes is T min = p ( D − D − g π , (32)under which only the thermal vacuum can exist. For any given temperature above T min ,there can exist both the thermal AdS vacuum and the black holes of both large and smallradii. There exists a phase transition temperature T phs . = g s ( D − D − D + 17) + D − D + 34 D + 12 D − D − D − π , (33)above which the black hole with the larger radius develops a negative free energy and hencethe thermal vacuum will collapse to form a black hole. It is worth pointing out however thatthe Euclidean action is divergent even after subtracting the background values, indicating apossibility of violating the quantum statistic relation (QSR). (In [13], Euclidean action wereconstructed by introducing boundary counterterms; however there is no covariant expressionfor such counterterms.) Such violations were reported in analysing the black holes in theHorndeski gravity [23, 24], and in other gravity theories with non-minimally coupled matter[25]. 9he second type of solutions in (22) are rather unusual, since h can be an arbitraryfunction of r . For k = 1 and h being regular for r ∈ [0 , ∞ ), the solution describe a smoothsoliton. It also allows Lifshitz-type spacetime [26, 27, 28] when h ∼ r z with a genericLifshitz exponent z . Since f contains no integration constant, it follows that δf = 0 whenevaluating asymptotically, and hence the solution has no mass or any non-trivial conservedcharges. These properties imply that the solutions describe the degenerate condensates ofthe non-dynamical linear modes.It is remarkable that the two types of solutions (21, 22) comprise the most generalspherically-symmetric solutions at the critical point, and all of them have the mass M ≥ In this section, we consider the Einstein gravity extended with the general three quadraticcurvature tensor invariants in general dimensions D . The Lagrangian is given by L = κ √− g (cid:16) ( R − ) + αR + β R µν R µν + γ (cid:0) R − R µν R µν + R µνρσ R µνρσ (cid:1)(cid:17) . (34)This theory was well studied in [31, 32, 33]. In this section, we adopt the notation and thelinearized formulae in [4]. The equations of motion are κ E µν = 0, where E µν = R µν − g µν R + Λ g µν + 2 αR ( R µν − Rg µν ) + (2 α + β )( g µν (cid:3) − ∇ µ ∇ ν ) R + β (cid:3) ( R µν − Rg µν ) + 2 β ( R µσνρ − g µν R σρ ) R σρ +2 γ (cid:0) RR µν − R µσνρ R σρ + R µσρλ R νσρλ − R µρ R νρ (cid:1) − γg µν (cid:0) R − R µν R µν + R µνρσ R µνρσ (cid:1) . (35)There are in general two distinct (A)dS vacua, whose the effective cosmological constantsare determined by the quadratic algebraic equation (3) with ∆ now replaced by ∆, givenby κ (cid:0) (Λ − Λ ) + ∆Λ (cid:1) = 0 , ∆ = ( Dα + β ) ( D − D − + γ ( D − D − D − D − . (36)The formulae (4) and (5) still hold but with ∆ being replaced by ∆ and the relateddiscussions are also valid here, except that now ∆ has the ( α, β ) dependence.10 .1 Conventional critical gravity The linearized equations of motion for the metric fluctuations around one of the two (A)dSvacua are [4]˜ κ eff G Lµν + κ (2 α + β ) (cid:16) ¯ g µν ¯ (cid:3) − ¯ ∇ µ ¯ ∇ ν + 2Λ D − g µν (cid:17) R L + κβ (cid:16) ¯ (cid:3) G Lµν − D − g µν R L (cid:17) = 0 , (37)where ˜ κ eff = κ (cid:0) (cid:1) , ˜∆ = DαD − + βD − + ( D − D − γ ( D − D − . (38)The trace equation turns out to be κ (cid:16) α ( D −
1) + Dβ (cid:17) ¯ (cid:3) R L − ( D − κ eff R L = 0 , (39)where κ eff has the same definition as (7), but now with ∆ being given by (36). (Note that∆ and ˜∆ become the same when α = 0 = β .) In [3, 4], it was proposed to consider4 α ( D −
1) + Dβ = 0 . (40)so that the scalar mode becomes non-dynamical. The equations of motion then impliesthat R L = 0 for a generic κ eff = 0. It follows from the gauge choice (12) that h = 0. Thelinearized equations of motion for the transverse and traceless modes now become − β (cid:16) ¯ (cid:3) − D − D − − M (cid:17)(cid:16) ¯ (cid:3) − D − D − (cid:17) h µν = 0 , (41)where M = − β − (cid:16) κ eff + 4 κ Λ β ( D − D − (cid:17) . (42)Hence, the theory contains in general one massless and one massive graviton, satisfyingrespectively (cid:16) ¯ (cid:3) − D − D − (cid:17) h ( m ) µν = 0 , (cid:16) ¯ (cid:3) − D − D − − M (cid:17) h ( M ) µν = 0 . (43)The absence of the tachyonic mode requires M ≥
0. When we further requires that M = 0,then the relation κ eff + 4 κ Λ β ( D − D −
2) = 0 , (44)defines the critical point at which the theory contains no massive graviton. As the equationon the fluctuation is a fourth order differential equation, there could be other kinds ofmodes, for example the logarithmic mode like the those in chiral gravity [5, 6] and massivegravity [35, 36] in three dimensions. 11 .2 New critical point We now consider a new critical condition. After imposing (40), instead of imposing (44),we impose the following condition κ eff = 0 , (45)where κ eff is define by (39). Explicitly, the constants α and β under this critical conditionbecome α = − D ( D − D − D − γ (cid:16) − Λ (cid:17) , β = 4( D − D − γ (cid:16) − Λ (cid:17) , (46)where ∆ is defined in (3). The new critical condition contains the one discussed in theEGB gravity, and it reduces to that when α = 0 = β . As in the case of the EGB gravity,this condition implies that the two (A)dS vacua coalesce into one, with the effective cosmo-logical constant Λ = 2Λ . The consequence of this is that the trace equation (39) becomesautomatically satisfied and the trace h of the metric fluctuations becomes non-dynamicalat the linear level. Given the conditions (40) and (45) on the parameters and the gaugechoice (12), we find that the linearized equations of motion are − κβ (cid:16) ¯ (cid:3) − D Λ( D − D − (cid:17)(cid:16) ¯ (cid:3) − D − D − (cid:17) ˜ h µν = 0 , (47)where ˜ h µν is transverse and traceless. Thus the theory contains one massless and onemassive graviton, satisfying respectively (cid:16) ¯ (cid:3) − D − D − (cid:17) ˜ h ( m ) µν = 0 , (cid:16) ¯ (cid:3) − D Λ( D − D − (cid:17) ˜ h ( M ) µν = 0 . (48)The mass square of the massive modes is given by M = 2Λ D − . (49)It was shown that the generalized Breitenlohner-Freedman bound for a spin- s field in anAdS background is given by [37, 38]( M BF s ) ≥ ( D − s + 5) Λ2( D − D − . (50)It is clear that M ≥ ( M BF2 ) for s = 2 and D ≥
3. It follows that the theory does notcontain tachyonic instability, although ghost modes are inevitable. When β = 0 and hence α = 0, it reduces to the critical point in the EGB theory, in which case even the gravitonbecome non-dynamical at the linearized level.12 .3 Exact solutions at the new critical point Even spherically-symmetric solutions are hard to find in quadratically-extended gravity forgeneric parameters. When γ = 0, the (A)dS Schwarzschild black hole with an appropriateeffective cosmological constant is a solution. It was recently demonstrated numerically infour dimensions that new black holes beyond the Schwarzschild one exist [39, 40]. Thisindicates that a variety of new black holes may exist in generally-extended gravities. Thesimplicity of the black hole solution at the critical point of the EGB theory suggests that ex-act solutions may be easier to construct in the new critical theory. Indeed for the parameters(46), we find a new solution under the spherical symmetric ansatz (20) with f = g r + k , h = ( p g r + k + µ ) . (51)where k characterize the topology of the solution, and g = 1 /ℓ is the inverse of the (A)dSradius, related to the effective cosmological constant byΛ = 2Λ = − ( D − D − g . (52)The Riemann curvature squared (30) is given by R = 9 g ( p g r + k + µ ) (cid:16) µ (11 g r + 3 µ + 11 k ) + (5 g r + 9 µ + 5 k ) p g r + k (cid:17) . (53)It is of interest to note that there is no curvature singularity at r = 0. Thus the solutiondescribes a smooth soliton for k = 1 and µ >
0, with the radial coordinate r runs from 0to asymptotic infinity. Using the formulae obtained in [18] for the Wald formalism, we findthat the mass of the soliton vanishes, or to be precise, δM = 0.We also obtain (A)dS planar black holes ( k = 0) for some specific γ : h = f = g r − µr D − , (54)The ( α, β, γ ) parameters, satisfying the critical condition (46), are given by { α, β, γ } = 1Λ {− D ( D − D − D − , ( D − D − D − , − D − D − D − } . (55)The solutions describe the black holes, but also with vanishing mass and entropy, accordingto the formulae in [18]. These solutions can be viewed as thermalized vacua, similar tothose found in the conformal gravity [34]. 13 .4 A further critical point When β = 0 and ˜ κ eff = 0, it follows from (37) that the linearized equations of motioninvolve only the trace scalar mode, with no kinetic term for any transverse mode. For non-vanishing α and γ , we find no exact black hole or soliton solutions. When α = 0, the theoryreduces to the EGB theory at the critical point. In D ≤ γ = 0, the theory reduces tothe well-known f ( R ) gravity with f = ( R − DD − Λ) , whose equations of motion reduce toa single scalar equation R = DD − Λ. It is of interest to note that the critical case where alllinear perturbations have no kinetic terms can only occur when α = 0 = β , in other words,in the EGB theory. In the previous sections, we have studied the new critical points of extended gravities wherethree cases emerge: (1) the whole kinetic terms of all h µν vanish; (2) that of h = h µµ vanishes;(3) that of the transverse h µν vanishes. We have obtained a large number of exact staticsolutions. In this section, we examine how these solutions change when we deviate fromthese critical points. This can help us to determine the physical meaning of the integrationconstants in the more general setting. First we consider the parametersΛ = − ( D − D − g + ( D − D − ( D − αg κ ( D − D + 9) , β = − D + 1) αD − D + 9 ,γ = − ( D − (3 D − D + 7) α D − D − D − D + 9) + κ D − D − g . (56)It reduces to the critical case when α = 0 and hence β = 0. In general dimensions, we findAdS planar black holes ( k = 0), with h = f = g r − µr D − , Λ = ( D − D − g . (57)Note that the form of the solution is identical to that in the EGB theory at the criticalpoint. We now show that the parameter µ is related to the massive spin-2 modes.Using the general formula worked out in [18], we find that the first law of black holethermodynamics dM = T dS holds, with M = ( D − ω πg (cid:16) κ − ( D − D − D − D − D + 9 αg (cid:17) µ , = ( D − g r π , S = ωr D − (cid:16) κ − ( D − D − D − D − D + 9 αg (cid:17) . (58)To understand the physical meaning of the µ parameter, let us consider the linearizedperturbation around the AdS vacuum, namely, h ( r ) = g r + k + h ( r ) , f ( r ) = g r + k + f ( r ) , (59)where h ( r ) and f ( r ) are small perturbation. For general parameters away from the criticalpoint, ( h , h ) are subject to fourth order differential equations, and the solutions are [18] h = − mr D − + ξ r D − − σ + ξ r D − σ ,f = − mr D − + ( D − − σ ) ξ D − r D − − σ + ( D − σ ) ξ D − r D − σ + η r D − − σ + η r D − σ , (60)where the parameter m is associated with the massless graviton mode, and σ = κβ − (cid:16) D ( D − α + ( D − D + 7) β + 8( D − D − γ − g − (cid:17) ,σ = κ D − α + Dβ (cid:16) D − g − − ( D − (cid:0) D − D − α − ( D − D + 32) β (cid:1) − D − D − D − γ (cid:17) . (61)It follows from the falloff behavior that ( ξ , ξ ) and ( η , η ) are associated with the massivespin-2 mode and massive scalar mode respectively. For the parameters (56), we find that σ = σ = 0. The linearized solutions now become h = − mr D − + ξ + ξ log rr D − , f = − mr D − + ρ + ρ log rr D − , (62)where the massive spin-2 mode and massive scalar mode coincide in the metric function f and we collectively denotes them by ( ρ , ρ ). Since for σ = 0 = σ , we have h = f ifthe solution involves the massive spin-2 but not the scalar modes. Thus the exact solutionwe obtained involves both the massive spin-2 and the scalar modes, but not the masslessgraviton mode.In the D = 5 dimension, the solution (57) is also valid for the spherical and hyperbolictopologies. The solution becomes ds = − f dt + dr f + r d Ω ,k , f = g r + k − µ ,β = − α , γ = − α + κ g , Λ = − g + 4 αg κ . (63)The mass, the temperature and the entropy can be calculated using the formulae in [18],and we find M = 3 ωkα π µ + 3 ω ( κ − αg )32 πg µ , = g r π , S = ωr (cid:16) κ − αg + 3 κkg r (cid:17) . (64)It is easy to verify that first law dM = T dS holds. It is of interest to note that for k = 0,the mass formula involves the linear as well as the quadratic term in µ . If we perform smallperturbation around the AdS vacuum, we find that at the linearized order ( k = 0) h = f = − µ − mr . (65)This implies that the scalar mode has no independent parameter in this case. In this subsection, we consider the deviation from the critical point discussed in section 3.We consider the parametersΛ = − ( D − D + ǫ − g , α = − ( D − ǫ + ( D − ǫ − D )8( ǫ − D − D + ǫ − ,β = ( D − ǫ − D + ǫ − ǫ − D − D + ǫ − ,γ = ( ǫ − ǫ − D − D + ǫ − D − D − D − D + ǫ − . (66)We obtain planar AdS black holes with f = g r − µr D + ǫ − . (67)When ǫ = 0, the parameters satisfy the critical conditions (40) and (45), and the cor-responding solution was obtained in section 3.3. Note that the solution with ǫ = 0 wasobtained in section 3.3. The solution with ǫ = − ( D −
3) was obtained in section 4.1.When ǫ = −
1, the solutions are also valid for spherical/hyperbolic topologies, namely ds = − f dt + dr f + r d Ω D − ,k , f = g r + k − µr D − , Λ = − ( D − D − g , α = 12( D − D − g , β = − α , γ = 0 , (68)For all these solutions, we find that the entropy and the mass vanish, with non-vanishingtemperature T = ( D + ǫ − g r π . (69) It is well known that for appropriate range of the coupling constants, the EGB gravityadmits two (A)dS vacua, and the linearized perturbations are the massless gravitons. We16nd that in one (A)dS vacuum the graviton has positive kinetic energy, whilst in the otherit has negative kinetic energy and hence is ghost-like. There exists a critical point of thecoupling constants for which the two (A)dS spacetimes coalesce and the linearized equationsof motion become automatically satisfied. The linear perturbation of the vacuum henceloose its interpretation as a graviton, and the EGB theory at the critical point describes agravity theory without graviton.We then derived the perturbative equations of motion at the quadratic order. We alsoconstructed the most general static solutions with spherical/toric/hyperbolic isometries. Us-ing the Wald formalism, we demonstrated that these solutions all had non-negative energies,indicating that the theory may not have nonlinear ghost excitations. One of the solutiondescribes a previously-known black hole with unusual asymptotic falloffs. We adopted Waldformalism to derive its the mass, entropy and temperature and hence the free energy. Wefound that the first law of black hole thermodynamics holds and furthermore there is alsoHawking-Page type of phase transition.We then considered more general theories involving up to quadratic curvature invariants.We found the critical points in the parameter space at which the linearized equation forthe scalar trace mode is automatically satisfied. Interestingly this allows us to find someexact black hole solutions. Alternatively, for some other choice of parameters, only thescalar trace mode has a kinetic term whilst the transverse modes do not. The case whereall linear perturbations have no kinetic terms occurs only in the EGB theory among thegeneral quadratically-extended gravities. We also considered the theories deviated fromthese critical points, which enables us to understand better the integration constants suchas the mass of the black hole at the critical point in a more general setting.We expect that these critical points commonly exist in Lovelock gravities or in generalhigher-derivative gravities. The lacking of any bilinear kinetic term or two-point functionis unusual from the point of view of both classical and quantum field theories and itsphysical implication is not clear at the moment. Although the critical point in EGB theorylies outside the causality regions [41, 42, 43, 44] of the parameter space, the interestingphenomenon warrants further investigations.
Acknowledgments
We are grateful to Zhao-Long Wang for useful discussions. Z.Y.F. and B.C. are supportedin part by NSFC Grants No. 11275010, No. 11335012 and No. 11325522. H.L. is supported17n part by NSFC grants NO. 11175269, NO. 11475024 and NO. 11235003.
References [1] K.S. Stelle,
Renormalization of higher derivative quantum gravity,
Phys. Rev. D ,953 (1977). doi:10.1103/PhysRevD.16.953[2] K.S. Stelle, Classical gravity with higher derivatives,
Gen. Rel. Grav. , 353 (1978).doi:10.1007/BF00760427[3] H. L¨u and C.N. Pope, Critical gravity in four dimensions,
Phys. Rev. Lett. ,181302 (2011) doi:10.1103/PhysRevLett.106.181302 [arXiv:1101.1971 [hep-th]].[4] S. Deser, H. Liu, H. L¨u, C.N. Pope, T.C. Sisman and B. Tekin,
Critical points of D -dimensional extended gravities, Phys. Rev. D , 061502 (2011) doi:10.1103/ Phys-RevD.83.061502 [arXiv:1101.4009 [hep-th]].[5] W. Li, W. Song and A. Strominger, Chiral gravity in three dimensions,
JHEP ,082 (2008) doi:10.1088/1126-6708/2008/04/082 [arXiv:0801.4566 [hep-th]].[6] A. Maloney, W. Song and A. Strominger, “Chiral Gravity, Log Gravity and Ex-tremal CFT,” Phys. Rev. D , 064007 (2010) doi:10.1103/PhysRevD.81.064007[arXiv:0903.4573 [hep-th]].[7] N. Johansson, A. Naseh and T. Zojer, Holographic two-point functions for 4d log-gravity,
JHEP , 114 (2012) doi:10.1007/JHEP09(2012)114 [arXiv:1205.5804[hep-th]].[8] D. Grumiller, W. Riedler, J. Rosseel and T. Zojer,
Holographic applications of log-arithmic conformal field theories,
J. Phys. A , 494002 (2013) doi:10.1088/1751-8113/46/49/494002 [arXiv:1302.0280 [hep-th]].[9] M. Hogervorst, M. Paulos and A. Vichi, The ABC (in any D) of logarithmic CFT, arXiv:1605.03959 [hep-th].[10] D. Lovelock,
The Einstein tensor and its generalizations,
J. Math. Phys. , 498(1971). doi:10.1063/1.1665613[11] R.M. Wald, Black hole entropy is the Noether charge,
Phys. Rev. D , 3427 (1993)doi:10.1103/PhysRevD.48.R3427 [gr-qc/9307038].1812] V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynam-ical black hole entropy,
Phys. Rev. D , 846 (1994) doi:10.1103/PhysRevD.50.846[gr-qc/9403028].[13] J. Crisostomo, R. Troncoso and J. Zanelli, Black hole scan,
Phys. Rev. D , 084013(2000) doi:10.1103/PhysRevD.62.084013 [hep-th/0003271].[14] F. Canfora, A. Giacomini and S.A. Pavluchenko, Dynamical compactification inEinstein-Gauss-Bonnet gravity from geometric frustration,
Phys. Rev. D , no. 6,064044 (2013) doi:10.1103/PhysRevD.88.064044 [arXiv:1308.1896 [gr-qc]].[15] D.G. Boulware and S. Deser, String generated gravity models,
Phys. Rev. Lett. ,2656 (1985). doi:10.1103/PhysRevLett.55.2656[16] R.G. Cai, Gauss-Bonnet black holes in AdS spaces,
Phys. Rev. D , 084014 (2002)doi:10.1103/PhysRevD.65.084014 [hep-th/0109133].[17] R. Aros, R. Troncoso and J. Zanelli, Black holes with topologically nontrivialAdS asymptotics,
Phys. Rev. D , 084015 (2001) doi:10.1103/PhysRevD.63.084015[hep-th/0011097].[18] Z.Y. Fan and H. L¨u, Thermodynamical first laws of black holes in quadratically-extended gravities,
Phys. Rev. D , no. 6, 064009 (2015) doi:10.1103/PhysRevD.91.064009 [arXiv:1501.00006 [hep-th]].[19] M. Cvetiˇc, G.W. Gibbons, D. Kubiznak and C.N. Pope, Black hole enthalpy and anentropy inequality for the thermodynamic volume,
Phys. Rev. D , 024037 (2011)[arXiv:1012.2888 [hep-th]].[20] D. Kastor, S. Ray and J. Traschen, Enthalpy and the mechanics of AdS black holes,
Class. Quant. Grav. , 195011 (2009) [arXiv:0904.2765 [hep-th]].[21] H.S. Liu, H. L¨u and C.N. Pope, Generalized Smarr formula and the viscositybound for Einstein-Maxwell-dilaton black holes,
Phys. Rev. D , 064014 (2015)doi:10.1103/PhysRevD.92.064014 [arXiv:1507.02294 [hep-th]].[22] S.W. Hawking and D.N. Page, Thermodynamics of black holes in anti-De Sitter space,
Commun. Math. Phys. , 577 (1983). doi:10.1007/BF012082661923] X.H. Feng, H.S. Liu, H. L¨u and C.N. Pope, Black hole entropy and viscositybound in horndeski gravity,
JHEP , 176 (2015) doi:10.1007/JHEP11(2015)176[arXiv:1509.07142 [hep-th]].[24] X.H. Feng, H.S. Liu, H. L¨u and C.N. Pope,
Thermodynamics of charged blackholes in Einstein-Horndeski-Maxwell theory,
Phys. Rev. D , no. 4, 044030 (2016)doi:10.1103/PhysRevD.93.044030 [arXiv:1512.02659 [hep-th]].[25] X.H. Feng and H. Lu, Higher-derivative gravity with non-minimally coupled Maxwellfield,
Eur. Phys. J. C , no. 4, 178 (2016) doi:10.1140/epjc/s10052-016-4007-y[arXiv:1512.09153 [hep-th]].[26] M. H. Dehghani and R. B. Mann, Lovelock-Lifshitz Black Holes,
JHEP , 019(2010) [arXiv:1004.4397 [hep-th]].[27] M. H. Dehghani and R. B. Mann,
Thermodynamics of Lovelock-Lifshitz Black Branes,
Phys. Rev. D , 064019 (2010) [arXiv:1006.3510 [hep-th]].[28] B. Chen, Z. Y. Fan and L. Y. Zhu, AdS and Lifshitz Scalar Hairy Black Holes inGauss-Bonnet Gravity, arXiv:1604.08282 [hep-th].[29] A. Anabalon, N. Deruelle, Y. Morisawa, J. Oliva, M. Sasaki, D. Tempo and R. Tron-coso,
Kerr-Schild ansatz in Einstein-Gauss-Bonnet gravity: an exact vacuum solu-tion in five dimensions,
Class. Quant. Grav. , 065002 (2009) doi:10.1088/0264-9381/26/6/065002 [arXiv:0812.3194 [hep-th]].[30] M. Banados, R. Olea and S. Theisen, Counterterms and dual holographic anoma-lies in CS gravity,
JHEP , 067 (2005) doi:10.1088/1126-6708/2005/10/067[hep-th/0509179].[31] S. Deser and B. Tekin,
Gravitational energy in quadratic curvature gravities,
Phys.Rev. Lett. , 101101 (2002) doi:10.1103/PhysRevLett.89.101101 [hep-th/0205318].[32] S. Deser and B. Tekin, Energy in generic higher curvature gravity theories,
Phys. Rev.D , 084009 (2003) doi:10.1103/PhysRevD.67.084009 [hep-th/0212292].[33] I. Gullu and B. Tekin, Massive higher derivative gravity in D -dimensional anti-deSitter spacetimes, Phys. Rev. D , 064033 (2009) doi:10.1103/PhysRevD.80.064033[arXiv:0906.0102 [hep-th]]. 2034] H. L¨u, Y. Pang, C.N. Pope and J.F. Vazquez-Poritz, AdS and Lifshitz blackholes in conformal and Einstein-Weyl gravities,
Phys. Rev. D , 044011 (2012)doi:10.1103/PhysRevD.86.044011 [arXiv:1204.1062 [hep-th]].[35] E.A. Bergshoeff, O. Hohm and P.K. Townsend, Massive gravity in three dimen-sions,
Phys. Rev. Lett. , 201301 (2009) doi:10.1103/PhysRevLett.102.201301[arXiv:0901.1766 [hep-th]].[36] E.A. Bergshoeff, O. Hohm and P.K. Townsend,
More on massive 3D gravity,
Phys.Rev. D , 124042 (2009) doi:10.1103/PhysRevD.79.124042 [arXiv:0905.1259 [hep-th]].[37] Y.X. Chen, H. L¨u and K.N. Shao, Linearized modes in extended and critical grav-ities,
Class. Quant. Grav. , 085017 (2012) doi:10.1088/0264-9381/29/8/085017[arXiv:1108.5184 [hep-th]].[38] H. L¨u and K. N. Shao, Solutions of free higher spins in AdS,
Phys. Lett. B , 106(2011) doi:10.1016/j.physletb.2011.10.072 [arXiv:1110.1138 [hep-th]].[39] H. L¨u, A. Perkins, C. N. Pope and K. S. Stelle,
Black holes in higher-derivative Grav-ity,
Phys. Rev. Lett. , no. 17, 171601 (2015) doi:10.1103/PhysRevLett.114.171601[arXiv:1502.01028 [hep-th]].[40] H. L¨u, A. Perkins, C.N. Pope and K.S. Stelle,
Spherically symmetric solutions inhigher-derivative gravity,
Phys. Rev. D , no. 12, 124019 (2015) doi:10.1103/ Phys-RevD.92.124019 [arXiv:1508.00010 [hep-th]].[41] D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge cor-relations,
JHEP , 012 (2008), arXiv:0803.1467 [hep-th].[42] J. de Boer, M. Kulaxizi and A. Parnachev,
AdS /CFT , Gauss-Bonnet Gravity, andViscosity Bound, JHEP , 087 (2010), arXiv:0910.5347 [hep-th].[43] X.O. Camanho and J.D. Edelstein,
Causality constraints in AdS/CFT from conformalcollider physics and Gauss-Bonnet gravity,
JHEP , 007 (2010), arXiv:0911.3160[hep-th].[44] A. Buchel, J. Escobedo, R.C. Myers, M.F. Paulos, A. Sinha and M. Smolkin,
Holo-graphic GB gravity in arbitrary dimensions,
JHEP1003