The large D limit of dimensionally continued gravity
aa r X i v : . [ g r- q c ] M a r The large D limit of dimensionally continued gravity Gaston Giribet
Physics Department, University of Buenos Aires, and IFIBA-CONICET
Ciudad Universitaria, Pabell´on 1, 1428, Buenos Aires, Argentina.
Abstract
In a recent paper [1] Emparan, Suzuki, and Tanabe studied general relativity in the limitin which the number of spacetime dimensions D tends to infinity. They showed that, in suchlimit, the theory simplifies notably. It reduces to a theory whose fundamental objects, blackholes and black branes, behave as non-interacting particles. Here, we consider a differentway of extending gravity to D dimensions. We present a special limit of dimensionallycontinued gravity in which black holes retain their gravitational interaction at large D andstill have entropy proportional to the mass. The similarities and differences with the limitconsidered in [1] are discussed. Introduction
There exist several ways of extending general relativity (GR) to higher dimensions. The simplestone is retaining the form of Einstein-Hilbert Lagrangian density and then extend the action to D ≥ D > D = 4. In D >
4, the requirement of theequations of motion to be symmetric rank-two covariantly conserved equations of second orderdoes not select Einstein tensor uniquely. In addition, there exists the possibility to supplementEinstein-Hilbert action with dimensionally extended characters of the form χ n = Z ε a a ...a n ...a D R a a ∧ R a a ∧ ...R a n − a n ∧ e a n +1 ∧ e a n +2 ∧ ...e a D , (1)which, despite of being of order R n , yield second-order field equations. Then, it is natural toinquire about why not to include the whole hierarchy of characters χ n up to order ( D − / χ can be thought of as the dimensionalextension of Euler characteristic in D = 2 dimensions, in D > D = 4, for instance, the Gauss-Bonnet theorem implies that R terms of this sort do notmodify Einstein equations, as early noticed by Lanczos [2]; however, in D > χ n in higher dimensions. The theory of gravity in D dimensions whose action consists of all the dimensionally extended topological densities (1)up to n = ( D − / D dimensions.This digression about which is the natural extension of GR to D dimensions acquires par-ticular importance in relation to recent studies on the behavior of gravity in the large D limit[1]. This limit had already been considered in the literature, for instance in Refs. [4, 5, 6, 7],and it was recently revisited in [1] by Emparan et al., who observed that GR simplifies notablywhen D goes to infinity. In particular, they observed that in this limit the theory reduces toa theory of non-interacting particles. The fundamental objects of the theory, black holes andblack p -branes, exhibit vanishing cross-section and behave like dust matter.The idea of considering the large D limit of gravity theory can be motivated by the large N limit of gauge theories. The latter has shown to be a fruitful tool to investigate the structure Here we will work in the first order formalism; see Section 2 for conventions.
1f both Yang-Mills and Chern-Simons theories. Exceptis excipiendis, gravity theory can also beconsidered as a gauge theory for the local Lorentz group SO ( D − , /D expansion. Ofcourse, besides the mathematical analogy with the large N limit of gauge theories, the factthat D represents the dimensionality of the spacetime itself introduces additional conceptualdifficulties. Nevertheless, as explained in [1], this limit may still be considered and interestingphysical information can be extracted from studying it.Here, we will consider a different way of extending gravity to D dimensions and study thelimit of large D . More precisely, we will consider the gravity theory defined by the action thatincludes all terms (1) up to a given order R k , with k ≤ ( D − /
2. For this type of theories, thementioned analogy between the large N limit of gauge theories and the large D limit of gravityis even more direct since in the particular case 2 k + 1 = D the actions we will consider coincidewith Chern-Simons actions (CS) for the gauge group SO ( D − , < k + 1 < D , instead, one is in an intermediate situation, between GR and CS. Thiswill allow us to play between two extremes, between k = 1 and k = ( D − /
2. The fact ofhaving now two parameters, D and k , allows us to take the large D limit in different manners.For instance, we can take D going to infinity by keeping k fixed, but we also can take both D and k large in such a way that the quotient D/k remains fixed. In the latter case we will findthat, contrary to the limit considered in [1], the black holes happen to retain their gravitationalpotential in a finite region outside the horizon. At first, this might sound surprising since the R k terms of Lovelock theory are expected to introduce ultraviolet effects merely. In the wordsof [1], the fact that Riemann curvature tends to strongly localize close to the horizon indicatesthat the dust picture should still apply [in Lovelock theory] at least in some situations. We willsee that, although this is the case in certain situations, it is not true in general and Lovelockblack holes may actually retain the interactions at large D . As said, we will be concerned with Lovelock theory of gravity. The idea of considering Lovelocktheory in relation to the large D limit of gravity was already proposed in [1]. The action of the2heory can be written as follows S = κ − X D/ n =0 α n χ n (2)where the terms χ n are given by χ n = Z ε a a ...a n ...a D R a a ∧ R a a ∧ ...R a n − a n ∧ e a n +1 ∧ e a n +2 ∧ ...e a D (3)where R ab = R abµν dx µ ∧ dx ν is the curvature two-form, R ab = dω ab + ω ac ∧ ω cb , with ω ab = ω abµ dx µ being the spin connection one-form, and e a = e aµ dx µ is the vierbein one-form. Latin indices referto indices in the tangent bundle while Greek indices refer to indices in the spacetime. In (2) κ and α n are dimensionful constant that introduce new fundamental scales in the theory. We willdiscuss these scales below.The equations of motion are obtained by varying (2) with respect to the vierbein and thespin connection. Varying with respect to e a yields X D/ n =0 α n ( D − n ) ε aa a ...a D R a a ∧ ...R a n a n +1 ∧ e a n +2 ∧ ...e a D = 0 , (4)while varying with respect to ω ab yields X D/ n =0 α n n ( D − n ) ε aba a ...a D R a a ∧ ...R a n − a n ∧ T a n +1 ∧ e a n +1 ∧ ...e a D = 0 , (5)where T a = de a + ω ab ∧ e b is the torsion two-form. Equations (5) vanish if torsion is taken tobe zero. Notice this is sufficient but not necessary condition if D ≥
4. Here we will consider T a = 0. Then, the equations that remain to be solved are (4).In addition to considering (4) we will define our theory by specifying a criterion to choosespecial sets of coupling constants α n . We will follow the criterion of Ref. [8]. That is, we willdemand the theory to admit a unique maximally symmetric vacuum. This prevents the theoryfrom suffering from ghost instabilities [9] and other type of pathologies [10]. This requirementof a unique vacuum leads to the following choice of couplings constants [8] α n ≤ k = L n − k ) ( D − n ) Γ( k + 1)Γ( n + 1)Γ( k − n + 1) , (6)while α n>k = 0. Ipso facto, this introduces an additional parameter of the theory, k , whichrepresents the highest order R k in the action. This invites to define the critical dimension D c ≡ k + 1, which represents the minimum number of dimensions such that a term χ k in3he action would contribute non-trivially to the equations of motion. In other words, χ k is theChern-Weil topological invariant in D c − D = D c (i.e. D = 2 k + 1) the theory defined by (2)-(6) coincides with the Chern-Simons theory of gravity[11]. In the case D = D c + 1 the action admits to be written as a Pfaffian, and then it is oftenreferred to as the Born-Infeld action [12]. Hereafter, we will be viewing the gravity theory as abiparametric model, and consequently we will express all the formulae below as functions of D and D c .At first glance it might seem remarkable that demanding the theory to admit a uniquemaximally symmetric vacuum yields a relation between the coupling constants α n that makesall of them to be determined by a unique fundamental scale L . However, due to the plethora ofvacua in higher-curvature theory, such a requirement turns out to be actually very restrictiveand this is why, apart from Planck scale κ , L appears as the only relevant scale.About Planck scale, we find convenient to define Newton constant as follows κ = 2Γ( D − D − G D,D c (7)where G D,D c has dimensions of (length) D − D c +1 , such that the coefficient of the Einstein-Hilbertterm, α /κ , has dimensions of (length) − D as required. In (7),Ω D − = 2 π ( D − / Γ( D − ) (8)is the volume of the unit ( D − − ( D − D − L , (9)which is given by the coefficient α /κ in the action above. Classical black holes
Another interesting features of the set of theories defined by the choice (6) is the fact that theycan be solved analytically in a variety of examples. In particular, their spherically symmetric4olutions can be found explicitly for generic values of D and D c . These metrics take the form [8] ds = − f dt + f − dr + r d Ω D − (10)with f ( r ) = 1 + r L − (cid:16) r r (cid:17) D − D c ) / ( D c − . (11)In the particular case D c = 3 ( k = 1) this solution reduces to Schwarzschild-Tangherlinisolution of GR, as expected. In the cases D = D c , on the other hand, this solution coincideswith the Ba˜nados-Teitelboim-Zanelli solution for Chern-Simons gravity [13].The mass of solutions (10)-(11) can be computed by resorting to the Hamiltonian formalism[8]. The result is expressed in terms of the horizon radius r H as follows M = r D − D c H G D,D c (cid:18) r H L (cid:19) ( D c − / (12)up to an additive constant that can be set to zero for simplicity.At this stage we are ready to study the geometry of these black holes in the large D limit.In this limit the volume of the ( D − D − ∼ D − D/ , sothat it tends to zero. This means that the base manifold of the black hole shrinks in the large D limit. This was rephrased in [1] as the black holes having vanishing cross-section when D goesto infinity.Outside the horizon, the gravitational potential damps off faster as D increases. This impliesthat the gravitational interactions between Schwarzschild-Tangherlini black holes extinguishesin the large D limit. In the general case (11), the way the gravitational potential scales with D also depends on how D c scales. If D c remains finite in the large D limit, the behavior ofsolutions (10)-(11) would be qualitatively similar to that of [1]. However, if, instead, both D and D c are taken to infinity in a way that the quotient D/D c remains fixed, then the blackholes happen to retain their gravitational interaction outside the horizon. In this limit Ω D − still vanishes, but metric function (11) has a large D behavior f ( r ) ≃ r L − (cid:16) r r (cid:17) D/D c − , (13)and the gravitational potential remains finite. 5 uantum black holes Now, let us turn to discuss black holes in the quantum regime. The Hawking temperatureassociated to black holes (10)-(11) can easily be calculated to be T = ~ π ( D c − (cid:18) ( D − r H L + ( D − D c ) r H (cid:19) , (14)which reproduces the GR result for D c = 3. We observe that the theory for generic D and D c seems to exhibit Hawking-Page transition, provided L is finite. If D goes to infinity and D c remains fixed, temperature (14) diverges. Still, there is a point at which the specific heat changesits sign and the transition occurs. This happens at the scale r = L p ( D − D c ) / ( D − ≃ L. On the other hand, in contrast to what happens in GR, the presence of higher-curvature termspermits to take the large D limit in a way that T remains finite. This is achieved by taking D c to infinity as well by keeping D/D c fixed. For instance, if we define D c = D (1 − α ), then thescale at which the transition takes place is governed by α , obtaining r ≃ L √ α .Let us study the case of asymptotically flat solutions. This is obtained by taking the large L limit. In the theories defined by (2)-(6) this corresponds to having only the highest curvatureterm R k turned on. In this limit, we find T = ~ ( D − D c )2 π ( D c − r H . (15)The entropy, on the other hand, is S = πr D − D c +1 H ( D c − ~ G D,D c ( D − D c + 1) . (16)Because of the presence of higher-curvature terms in the action, these black holes happen notto obey the Bekenstein-Hawking area law. Instead, entropy is a different monotonic function ofthe horizon area A , namely S ∝ A D − Dc +1 D − . From (16) and (12) we also observe that even in theparticular limit in which the black holes retain their gravitational potential, the entropy and themass go S ∝ M when D is large. This implies that such a behavior is not necessarily associatedto the non-interacting picture, at least not in a simple way. Black p -branes The study of the thermodynamics of black holes (10)-(11) enables to study the thermodynamicalstability of other black objects of the theory. For instance, consider black p -branes. That is,6onsider solutions of the form Σ D − p × T p , with T p being a p -torus and Σ D − p being a black holeof the type discussed above. This type of solutions was considered in Refs. [14, 15], where itwas shown that metric ds = − f dt + f − dr + r d Ω D − − p + X pi =1 dz i with f ( r ) = 1 − (cid:16) r r (cid:17) D − p − D c ) / ( D c − are solutions of the theory (2)-(6) in the limit L → ∞ .One can analyze the thermodynamical instability of black p -branes by comparing the entropyof such a configuration with that of a black hole. This requires a careful analysis of the parame-ters involved in each configuration when comparing them in the microcanonical ensemble. Thethermodynamical stability analysis yields the following result for the quotient of entropies [14] S Black p -brane S Black hole = ( D − D c + 1)( D − D c − p ) (2 G D,D c ) λ M λ ( A D,D c ,p ) λ , (17)with A D,D c ,p = Γ( D − D c − p + 1)Γ( D − D − p − / D − D c + 1)Γ( D − − p )Γ(( D − / π p/ V ol ( T p ) (18)and with critical exponents λ = 1 D − D c − p − D − D c (19) λ = D − D c + 1 − pD − D c − p − D − D c + 1 D − D c (20) λ = 1 D − D c − p (21)From (17) we observe that the thermodynamical analysis of the black hole / black branetransition in this theory is qualitatively similar to that of GR: There always exists a criticalmass above which the black p -brane is the preferable configuration. The natural question arisesas to how this picture is modified in the large D limit. For instance, in the large D limitwith D c fixed, all the exponents λ , , tend to zero. This behavior is actually expected becausehere we are considering p fixed. A similar behavior is exhibited also in the limit in which thequotient D/D c remains fixed. An interesting limit is given by taking both parameters to infinityby keeping the difference D − D c finite. In this limit, exponents λ , , remain finite while A D,D c ,p ∼ D p/ /V ol ( T p ). It would be interesting to study the instability of p -brane solutionsof this theory in a similar way to what has been done in Refs. [16, 17] at large D . The analysisof mechanical stability, on the other hand, can hardly be accomplished for these theories. Thisis mainly because of two reasons: First, the higher-curvature terms in the action introducehigher powers of the derivatives that make the complexity of the equations to grow dramaticallyeven for large D . 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