Black Holes in Quasi-topological Gravity
aa r X i v : . [ g r- q c ] J un Preprint typeset in JHEP style - PAPER VERSION arXiv:1003.5357 [gr-qc]
Black Holes in Quasi-topological Gravity
Robert C. Myers and Brandon Robinson , Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Department of Physics & Astronomy and Guelph-Waterloo Physics Institute,University of Waterloo, Waterloo, Ontario N2L 3G1, CanadaE-mail: [email protected], [email protected]
Abstract:
We construct a new gravitational action which includes cubic curvature in-teractions and which provides a useful toy model for the holographic study of a threeparameter family of four- and higher-dimensional CFT’s. We also investigate the blackhole solutions of this new gravity theory. Further we examine the equations of motion ofquasi-topological gravity. While the full equations in a general background are fourth-orderin derivatives, we show that the linearized equations describing gravitons propagating inthe AdS vacua match precisely the second-order equations of Einstein gravity. ontents
1. Introduction 12. Black Holes in Gauss-Bonnet gravity 33. New Curvature-Cubed Interaction 6 D ≥
4. Black Hole Solutions 10
5. Black Hole Thermodynamics 21
6. Equations of Motion 247. Discussion 27A. A New Topological Invariant? 29
1. Introduction
Recently, there has been some interest in gravitational actions with higher curvature actionsin the context of the AdS/CFT correspondence. For example, Einstein gravity in the AdSbulk defines a universality class of CFT’s in which the ratio of the shear viscosity to entropydensity is given by precisely η/s = 1 / (4 π ) [1, 2]. However, it is understood that addinghigher curvature interactions to the bulk gravity action leads to a broader class of CFT’sin which this ratio generally depends on the value of the additional gravitational couplings[3]. Further it is possible with some holographic constructions to violate the famous boundconjectured by Kovtun, Son and Starinets (KSS) [1] producing theories with η/s < / (4 π ).In certain string theory constructions, the appearance of curvature-squared interactionsproduces violations of the KSS bound [4] but these models only produce reliable results ina regime where the corresponding gravitational coupling is parametrically small. Hence atpresent, one can only deviate perturbatively away from the universality class defined byEinstein gravity in these string theory models.– 1 –owever, it is also of interest to explore situations where the gravitational couplings arefinite. For example, holography can yield new consistency conditions for the gravitationaltheories and their dual CFT’s. One theory which provides a useful toy model in thisregard is Gauss-Bonnet (GB) gravity. Even with a finite coupling for the curvature-squaredinteraction, this theory still provides some calculation control, which has been exploited inseveral recent holographic studies [5, 6, 7, 8, 9]. However, GB gravity only introduces asingle new coupling which limits the range of dual CFT’s which can be studied. A naturalgeneralization would be the further addition of interactions cubic in the curvature, as thisallows the investigation of the the full range of parameters in the three-point function ofthe stress tensor [10]. A straightforward extension of GB gravity would be to include thecubic interaction of Lovelock gravity [14]. However, because of the topological origin of theLovelock terms the cubic interaction only contributes to the equations of motion when thebulk dimension is seven or greater. In the context of the AdS/CFT correspondence, thismeans that such a term will be effective in expanding the class of dual CFT’s in six or moredimensions [8, 9]. Our key result in this paper is to construct a new gravitational actionwith cubic curvature interactions which provides a useful toy model to study a broaderclass of four (and higher) dimensional CFT’s, involving three independent parameters. Inthe following, we describe the construction for the new gravitational action and investigateblack hole solutions in this theory. We leave the detailed study of the properties of thedual class of CFT’s to a companion paper [10].An outline of the rest paper is as follows: We begin with a review of black hole solutionsand various aspects of these solutions in Gauss-Bonnet (GB) gravity coupled to a negativecosmological constant in section 2. Inspired by the GB equations of motion determiningblack hole solutions, we construct a new interaction that is cubic in curvatures and yieldssimilar simple solutions in section 3. Again we wish to emphasize that this interaction isnot the six-dimensional Euler density as appears in third-order Lovelock gravity. Further,we show in appendix A that the new interaction does not have a topological origin andhence we call the new theory: ‘quasi-topological gravity.’ We turn to a discussion of theasymptotically AdS black hole solutions in section 4. While the focus of this discussion isplanar black holes in five dimensions, we generalize the results to curved horizons and higherdimensions in sections 4.2 and 4.3. In section 5, we examine black hole thermodynamicsin the new theory, deriving some of the basic thermal properties of the black holes andthe corresponding plasmas in the dual CFT. We examine the equations of motion of quasi-topological gravity in section 6. While the full equations in a general background are fourth-order in derivatives, we show that the linearized equations describing gravitons propagatingin the AdS vacuum solutions are precisely the second-order equations of Einstein gravity.We conclude with a brief discussion of our results and future directions in section 7.While we were in the final stages of preparing this paper, two related preprints appearedin which exceptional new theories of curvature-cubed gravity were constructed. Ref. [11]constructs an interesting curvature-cubed theory in three dimensions. Up to a contri-bution proportional to the six-dimensional Euler density, the curvature-cubed interactionconstructed in five dimensions by [12] is identical to that studied here. Refs. [12, 13] arealso able to relate our interactions in D ≥
2. Black Holes in Gauss-Bonnet gravity
We begin here with a brief review of black holes in Gauss-Bonnet (GB) gravity. The lattercorresponds to a theory of gravity in which a curvature-squared interaction is added withthe form of the density for the Euler characteristic of four-dimensional manifolds, X = R abcd R abcd − R ab R ab + R . (2.1)Of course, this term will not affect the gravitational equations of motion if the dimensionof the spacetime is four (or lower), however, it makes interesting contributions for D ≥ D = 5 and witha negative cosmological constant: I = 116 πG Z d x √− g (cid:20) L + R + λL X (cid:21) . (2.2)We add some comments about higher dimensions at the end of this section.Let us present the ansatz for the metric of five-dimensional planar AdS black holes,which we will be using throughout the paper:d s = r L (cid:0) − N ( r ) f ( r ) d t + d x + d y + d z (cid:1) + L r f ( r ) d r . (2.3)Inserting this metric ansatz into the action (2.2) (and integrating by parts a number oftimes) yields I = 116 πG Z d x N ( r ) L (cid:2) r (1 − f + λf ) (cid:3) ′ (2.4)where the ‘prime’ indicates differentiation with respect to r . Schematically, the equationof motion coming from varying the lapse N takes the simple form [ r ( · · · )] ′ = 0 and so themetric function f is given by solving for the roots of a quadratic polynomial [18, 19]: λf ( r ) − f ( r ) + 1 − ω r = 0 . (2.5)The latter yields two solutions f ± ( r ) = 12 λ " ± s − λ (cid:18) − ω r (cid:19) (2.6)– 3 –ow varying δf yields a constraint which requires that N = constant , which we leaveunspecified for the moment. In the following, we will consider only the solutions with f − ,since the other branch with f + contains ghosts and is unstable [16] — as we will see inlater sections. With the choice f = f − , it is easy to verify that the horizon appears at r = r h = ω .In fixing the value of N , it is convenient to consider the solution with ω = 0,d s = r L (cid:0) − N ( r ) f ∞ d t + d x + d y + d z (cid:1) + L r f ∞ d r . (2.7)where we have adopted the notation f ∞ ≡ lim r →∞ f ( r ) = 12 λ h − √ − λ i . (2.8)We recognize eq. (2.7) as anti-de Sitter (AdS) space, presented in the Poincar´e coordinates.From g rr above, we also see that the AdS curvature scale is given by ˜ L = L/ √ f ∞ . Thismetric also makes apparent a convenient choice for the lapse, namely N = 1 /f ∞ , which weadopt in the following. In the AdS vacuum (2.7), this ensures that any motions in the branedirections are limited to lie within the standard light cone, i.e., − d t + d x + d y + d z .In the black brane solution (2.3), we still have lim r →∞ N f ( r ) = 1 and so this commentapplies in the asymptotic region. In the context of the AdS/CFT correspondence, thelatter means that the speed of light in the boundary CFT is simply c = 1.Examining the solutions in eq. (2.6) or f ∞ in eq. (2.8), we see that there is an upperbound at λ = 1 /
4. For larger values of λ , the gravitational theory does not have an anti-deSitter vacuum and the interpretation of the solutions (2.6) becomes problematic. In fact,using the AdS/CFT correspondence to demand consistency of the dual CFT, e.g., requiringthat the boundary theory is causal, imposes much more stringent constraints on the GBcoupling [5, 6] − ≤ λ ≤ . (2.9)We now turn to the thermodynamic properties of these GB black holes. The temper-ature of the black brane solutions is then given by the simple expression: T = 14 π r h f ′ | r h L N = ωπL N , (2.10)which when evaluated with N = 1 / √ f ∞ becomes: T = ωπL (cid:20)
12 (1 + √ − λ ) (cid:21) / . (2.11)The latter can be calculated by the standard technique of by analytically continuing themetric (2.3) to Euclidean time, τ = − i t , and choosing the periodicity of τ to ensure thegeometry is smooth at r h = ω . Next we evaluate the Euclidean action: I E [ T ] = 116 πG V ω NT L λ (cid:18) r ω (cid:16) λ − √ − λ (cid:17) − λ + 2 λ √ − λ (cid:19) , (2.12)– 4 –here V is the regulator volume obtained by integrating the ( x, y, z ) directions. Furtherwe have limited the radial integration from r = ω to r + to regulate the asymptotic or UVdivergence in I E . The divergent r contribution is removed with background subtractionusing the AdS vacuum (2.7), i.e., that is I E − I E remains finite in the limit r + → ∞ . Notethat in general, such a calculation would include a generalized Gibbons-Hawking term [20],as well as other boundary terms to regulate the divergences in the Euclidean action [21].However, these surface terms do not contribute to the final result for planar AdS blackholes when we use the background subtraction approach. Therefore we identify the freeenergy density as F = TV (cid:0) I E − I E (cid:1) = − ( π √ f ∞ L ) G T . (2.13)Then we may identify the energy and entropy densities as ρ = − T ddT ( F /T ) = 3( π √ f ∞ L ) T G , (2.14) s = − d F dT = (cid:0) π √ f ∞ LT (cid:1) G = 14 G ω L . One can confirm that the last result matches the entropy calculated using Wald’s techniques[22].These solutions are easily generalized from five to an arbitrary spacetime dimension, D . In this case, the action is conveniently parameterized as I = 116 πG D Z d D x √− g (cid:20) ( D − D − L + R + λ L ( D − D − X (cid:21) . (2.15)We also generalize the metric ansatz to include spherical and hyperbolic, as well as planar,horizons: ds = − ( k + r L f ( r )) N ( r ) dt + dr k + r L f ( r ) + r dℓ k (2.16)where dℓ k is given by k = +1 : d Ω D − (cid:0) metric on S D − (cid:1) ,k = 0 : 1 L D − X i =1 (cid:0) dx i (cid:1) , (2.17) k = − d Σ D − (cid:0) metric on H D − (cid:1) . Note that for k = ±
1, the above line element has unit curvature. With this general ansatzincorporating both curved horizons and D spacetime dimensions, f is determined by simplysolving for the roots of λf ( r ) − f ( r ) + 1 − ω D − r D − = 0 , (2.18)and the solutions take the form given in eq. (2.6) with the replacement ω /r → ω D − /r D − .The lapse is again a constant and as above we choose N = 1 / √ f ∞ . In general, the hori-zon is determined by f ( r h ) = − k L r h and so we only have r h = ω for the planar horizons,– 5 – .e., k = 0. In the case of the curved horizons, explicitly evaluating r h requires solving a( D − r h . Hence we only have a relatively simple solution in fivedimensions where: r h = 12 q − k L + 2 p k L − k λL + 4 ω . (2.19)
3. New Curvature-Cubed Interaction
As discussed in the introduction, we are motivated by considerations of the AdS/CFTcorrespondence to consider a curvature-cubed theory of gravity in five dimensions. GBgravity has a number of features which one might want to reproduce, such as providingsecond-order equations of motion and a family of exact black hole solutions. A naturalcandidate to extend these properties to a curvature-cubed theory would be the Lovelocktheory where the six-dimensional Euler density is added as a new gravitational interac-tion. However, this curvature-cubed interaction would only contribute to the equations ofmotion in seven and higher dimensions and hence will not contribute in the desired fivedimensions. While Lovelock’s work then indicates that it should not be possible to findan alternate action which yields second-order equations of motion, we begin by writingthe most general interaction including all possible curvature-cubed (or more precisely, six-derivative) interactions in five dimensions and attempt to tune the coefficients to producea simple equation for the black hole solutions, as discussed for GB gravity in the previoussection. We return to the equations of motion in section 6 and we demonstrate that thelinearized equations of motion in the AdS vacuum are indeed second-order.Let us begin by listing a basis of the possible six-derivative interactions:1. R c da b R e fc d R a be f R ab R bc R ca ∇ a R bc ∇ a R bc R cdab R efcd R abef R ab R ba R ∇ a R ab ∇ b R R abcd R abce R de R ∇ a R ∇ a R . R abcd R abcd R ∇ a R bcde ∇ a R bcde R abcd R ac R bd ∇ a ∇ c R abcd R bd In assembling this list, we have discarded any total derivatives, e.g., ∇ a ∇ a ∇ b ∇ c R bc and wehave simplified various expressions using the index symmetries of the Ricci and Riemanntensors. In particular, these symmetries allow us to reduce any other index contraction ofthree Riemann tensors to some combination of terms 1 and 2. Further, term 12 can bereduced to term 13 using ∇ a R ab = ∇ b R . Similarly, using the Bianchi identities, terms 9and 10 can be shown to be reducible to other terms and total derivatives as well. Hence, weare left with a list of 10 independent interactions which are cubic in curvatures. Combiningall of these interactions together in a single expression gives: √− g Z = √− g (cid:16) c R c da b R e fc d R a be f + c R cdab R efcd R abef + c R abcd R abce R de + c R abcd R abcd R + c R abcd R ac R bd + c R ba R cb R ac + c R ba R ab R + c R + c ∇ a R bc ∇ a R bc + c ∇ a R ∇ a R (cid:17) . (3.1)– 6 –t this point, we substitute the black brane metric ansatz (2.3) and evaluate eq. (3.1).The next step will be to see if the coefficients c i can be tuned to produce a result with thesame form as in eq. (2.6). In order to accomplish this task, we integrate by parts repeatedlyto put as many terms as possible in the form N ( r ) × ( · · · ) where the factor in brackets isindependent of the lapse function. The resulting expression then becomes √− g Z = − N L h(cid:16) r ( c + 2 c ) f f (6) + (cid:0) r (236 c + 496 c ) f + r (56 c + 28 c ) × f f ′ (cid:1)(cid:1) f (5) + (cid:0) r (24 c + 48 c + 20 c + 48 c + 12 c + 24 c + 12 c + 48 c + c ) f f ′′ + 8 r ( c + 2 c ) (cid:0) f ′ (cid:1) + r (5056 c + 384 c + 96 c + 2264 c +96 c + 96 c + 288 c + 960 c + 72 c ) f + r (72 c + 288 c + 120 c +1312 c + 192 c + 480 c + 192 c + 608 c + 84 c ) f f ′ (cid:1) f (4) + (cid:0) r (48 c +12 c + 48 c + 24 c + 16 c + 12 c + 8 c + 24 c + 48 c ) f (cid:0) f ′′′ (cid:1) + (cid:16) r (240 c + 96 c + 60 c + 100 c + 96 c + 224 c + 36 c + 144 c + 42 c ) (cid:0) f ′ (cid:1) + r (1494 c + 6069 c + 3054 c + 4380 c + 4012 c + 9776 c + 12000 c +1794 c + 2340 c + 18 c ) f f ′ + (cid:0) r (6 c + 24 c + 24 c + 12 c + 24 c + 12 c +6 c ) f ′ + r (1296 c + 920 c + 384 c + 708 c + 388 c + 2016 c + 1536 c +414 c + 888 c ) f ) f ′′ + r (1584 c + 8140 c + 36 c + 1440 c + 1212 c +4944 c + 1488 c + 5952 c + 17280 c + 19712 c ) f (cid:1) f ′′′ − r ( c + 4 c + c + 2 c + 4 c + 2 c + 4 c ) (cid:0) f ′′ (cid:1) + (cid:0) r (42 c + 84 c + 21 c + 84 c + 42 c +21 c + 84 c ) f ′ + r (10896 c + 4260 c + 18 c + 6384 c + 1308 c + 3696 c +1608 c + 2700 c + 4488 c + 1842 c ) f ) (cid:0) f ′′ (cid:1) + (cid:0) r (26544 c + 306 c +28092 c + 8790 c + 84000 c + 10740 c + 8916 c + 17472 c + 12600 c +35088 c ) f f ′ + r (324 c + 5436 c + 564 c + 6960 c + 82560 c + 8676 c +22608 c + 6069 c + 24384 c + 24192 c ) f + r (264 c + 252 c + 1056 c +720 c + 300 c + 624 c + 1920 c + 672 c + 408 c ) (cid:0) f ′ (cid:1) (cid:17) f ′′ − r (8 c +40 c + 400 c + 80 c + 16 c + 3 c + 16 c + 4 c ) f + r (434 c +1240 c + 301 c + 950 c + 592 c + 2800 c + 9 c + 361 c ) (cid:0) f ′ (cid:1) + r (10122 c +10236 c + 558 c + 10080 c + 2160 c + 29040 c + 12480 c + 8202 c +31296 c + 96000) (cid:0) f ′ (cid:1) + r ( − c + 5904 c + 24240 c + 504 c +23520 c + 7248 c + 5748 c + 91200 c + 5232 c )) f f ′ (cid:3) + · · · . (3.2)Note that not all terms can be put in the desired form with further integration by partsand so the ‘ · · · ’ indicates the presence of spurious terms containing factors like ( N ′′ ) /N ,for example. Focusing on the terms appearing explicitly in eq. (3.2), we find that choosingthe values of the c i ’s as1 . c = − c − c . c = − c − c – 7 – . c = 38 c + 32 c . c = 1556 c + 1114 c (3.3)3 . c = 157 c + 727 c . c = 04 . c = 187 c + 647 c . c = 0reduces this expression to the following simple form √− g Z = 127 N ( r ) L ( c + 2 c )( r f ) ′ . (3.4)At the same time, the spurious terms denoted by ‘ · · · ’ in eq. (3.2) also vanish with thischoice of coefficients. It is quite remarkable that there is enough freedom in the generalaction (3.1) to produce this simple result. In fact, we are still free to choose the (relative)values of c and c in constructing this interaction. Explicitly then, if we choose c =1 , c = 0, the new curvature-cubed interaction takes the form Z = R c da b R e fc d R a be f + 156 (cid:16) R abcd R abcd R − R abcd R abce R de +120 R abcd R ac R bd + 144 R ab R bc R ca − R ba R ab R + 15 R (cid:17) (3.5)or with c = 0 , c = 1, Z ′ = R abcd R cdef R ef ab + 114 (cid:16) R abcd R abcd R − R abcd R abce R de +144 R abcd R ac R bd + 128 R ab R bc R ca − R ba R ab R + 11 R (cid:17) . (3.6)The fact that we do not produce a unique interaction should not be surprising. Anycurvature-cubed interaction can be modified by the addition of the six-dimensional Eulerdensity X without affecting the equations of motion. In fact, we can infer the form of X by setting c = − c , in which case eq. (3.4) vanishes, as it must if evaluated for the six-dimensional Euler density. A standard normalization for the six-dimensional Euler densityis: X = 18 ε abcdef ε ghijkl R abgh R cdij R ef kl = 4 R cdab R efcd R abef − R c da b R e fc d R a be f − R abcd R abce R de + 3 R abcd R abcd R +24 R abcd R ac R bd + 16 R ba R cb R ac − R ba R ab R + R , (3.7)where in the first line, ε abcdef is the completely antisymmetric tensor in six dimensions andhence the corresponding expression only applies for D = 6. However, the first line alsomakes clear that this expression should vanish when evaluated in five (or lower) dimensions.This normalization corresponds to the choice c = 4 and c = −
8. That is, X = 4 Z ′ − Z . D ≥ c = 1 and c = 0. Comparing to eq. (3.7)– 8 –hows we are guaranteed that, if a nontrivial interaction exists, the result will be distinctfrom the six-dimensional Euler density. With this choice, we substitute the D -dimensionalextension of eq. (2.3) into the action (3.1). One then finds with a judicious choice of theremaining c i ’s, the result can be reduced to √− g Z D ∼ N ( r ) L D (cid:0) r D − f ( r ) (cid:1) ′ . (3.8)The required choice of coefficients (with c = 1 and c = 0) is:1 . c ( D ) = − D − D − D −
4) 5 . c ( D ) = − D − D − D − . c ( D ) = 3(3 D − D − D −
4) 6 . c ( D ) = 3 D D − D −
4) (3.9)3 . c ( D ) = 3 D (2 D − D −
4) 7 . c ( D ) = 04 . c ( D ) = 6( D − D − D −
4) 8 . c ( D ) = 0 . In practice, we determined the coefficients separately for D = 5 . . .
10 and found the generalexpressions above to fit the results in all of these cases. Given these expressions, the generalform of Z D becomes Z D = R c da b R e fc d R a be f + 1(2 D − D − (cid:18) D − R abcd R abcd R − D − R abcd R abce R de + 3 DR abcd R ac R bd (3.10)+ 6( D − R ab R bc R ca − D − R ba R ab R + 3 D R (cid:19) . One easily verifies that this result reduces to eq. (3.5) for D = 5.An important note is that, with the coefficients prescribed above for D = 6, theresulting action (3.8) is trivial, i.e., there is an overall factor of zero in this result. Hence Z does not actually produce a nontrivial interaction which is cubic in curvatures. Onemight be tempted to believe instead that Z yields another topological invariant in sixdimensions. Of course, there is no obvious known invariant to which Z might correspond[23]. In appendix A, we demonstrate that R d x √ g Z is not a topological invariant byexplicitly evaluating this expression for some nontrivial six-dimensional geometries. Hence,we refer to the theory of gravity extended with our new curvature-cubed interaction as‘quasi-topological gravity.’ At this point, we also note that our construction does not yielda nontrivial curvature-cubed interaction for D ≤ D > c arbitrary in our analysis above. Hence, we complete the discussion by gen-eralizing eq. (3.6) to higher dimensions using the formula: Z ′ D = 2 Z D + X . The final– 9 –xpression can be written as Z ′ D = R abcd R cdef R ef ab + 1(2 D − D − (cid:16) − (cid:0) D − D + 5 (cid:1) R abcd R abce R de + 32 (cid:0) D − D + 2 (cid:1) RR abcd R abcd + 12 ( D −
2) ( D − R abcd R ac R bd (3.11)+8 ( D −
1) ( D − R ab R bc R ca − D − RR ab R ba + 12 (cid:0) D − D + 6 (cid:1) R (cid:19) .
4. Black Hole Solutions
We have thus far constructed a new gravitational action that includes interactions up tocubic order in the curvature and which still yields particularly simple equations to findblack hole solutions. In this section, we complete the study of the black holes in this newtheory. While we have written an action for the theory in arbitrary number of spacetimedimensions, we will focus on the case D = 5 here. Further we begin by examining solutionsof the form given in eq. (2.3) and hence construct black holes with planar horizons [24, 18].The extension of this analysis to larger D and curved horizons (with spherical or hyperbolicgeometries) is straightforward and will be discussed briefly at the end of this section. We begin with the five-dimensional action: I = 116 πG Z d x √− g (cid:20) L + R + λL X + 7 µL Z (cid:21) (4.1)which extends the GB action with the addition of the curvature-cubed interaction Z .Next, as in section 2, we consider the following metric ansatz:d s = r L (cid:0) − N ( r ) f ( r ) d t + d x + d y + d z (cid:1) + L r f ( r ) d r . (4.2)Evaluating the action (4.1) with this metric then yields I = 116 πG Z d x N ( r ) L (cid:2) r (1 − f + λf + µf ) (cid:3) ′ (4.3)where the prime again denotes a derivative with respect to r . The variation δN now yields (cid:2) r (1 − f + λf + µf ) (cid:3) ′ = 0= ⇒ − f + λf + µf = ω r . (4.4)Similarly, satisfying the equation produced by taking the variation of f requires that either N ′ = 0 or − λf + 3 µf = 0. Since the latter is generally inconsistent with eq. (4.4), wearrive at N = constant . As in section 2, we choose N = 1 /f ∞ where f ∞ ≡ lim r →∞ f ( r ).This choice ensures that the speed of light in the boundary metric is just one.– 10 –e are now left with a cubic equation (4.4) to solve for f ( r ). To do so, we first makethe substitution f = x − λ µ , with which eq. (4.4) becomes: x − (cid:18) µ + λ µ (cid:19) x + 2 λ + 9 λµ + 27 µ (1 − ω r )54 µ ! = 0 . (4.5)This expression is further simplified by defining p = 3 µ + λ µ q = − λ + 9 µλ + 27 µ (1 − ω r )54 µ . (4.6)We then arrive at the depressed form of the equation: x − p x − q = 0 . (4.7)In the following discussion, note that the r -dependence is entirely contained in the coeffi-cient q . Before proceeding, we observe that there are three distinct cases depending on thesign of the discriminant, D = q − p , of eq. (4.7) with the following results:1. q − p > ⇒ q − p < ⇒ q − p = 0 ⇒ p = 0, we define α = (cid:16) q + p q − p (cid:17) , (4.8) β = (cid:16) q − p q − p (cid:17) , which allows the roots of eq. (4.7) to be written in the simple form using Cardano’s formula.Shifting these roots as above yields the following solutions: f = α + β − λ µ ,f = −
12 ( α + β ) + i √
32 ( α − β ) − λ µ , (4.9) f = −
12 ( α + β ) − i √
32 ( α − β ) − λ µ . If D = q − p > α and β can be taken as real and x corresponds to the single realroot. In this regime, the solution is then given by f = x − λ µ . If D = q − p < α and β are necessarily complex but implicitly eq. (4.9) still yields three unequal real roots. Inthis case, α and β can be chosen to have conjugate phases, i.e., α = √ p e iθ/ and β = √ p e − iθ/ (4.10)– 11 –here cos θ = q/p and sin θ = p p − q /p . Here, we are using the fact that p is alwayspositive in this domain. The solutions may then be cast in the explicitly real but implicitform: f = 2 √ p cos θ − λ µ ,f = −√ p (cid:18) cos θ √ θ (cid:19) − λ µ , (4.11) f = −√ p (cid:18) cos θ − √ θ (cid:19) − λ µ . While the precise form of f ( r ) is determined by eqs. (4.9) and (4.11), these results offerlittle insight into the physical properties of the corresponding solutions, e.g., which solutionsactually correspond to black holes. However, we will see below that much of the physicscan be inferred directly from the cubic equation (4.4).At this point, it is convenient to consider the AdS vacuum solutions. As discussed insection 2, the latter can be found by setting f ( r ) to be constant ( i.e., setting ω = 0) oralternatively taking the limit r → ∞ . Setting ω = 0 and f ( r ) = f ∞ in eq. (4.4) yields: h ( f ∞ ) ≡ − f ∞ + λf ∞ + µf ∞ = 0 . (4.12)With the choice N = 1 /f ∞ , the five-dimensional metric (2.3) becomes ds = r L (cid:0) − dt + dx + dy + dz (cid:1) + L f ∞ dr r , (4.13)which corresponds to the metric for AdS in Poincar´e coordinates. Further from the g rr component, we see that the radius of curvature of the AdS spacetime is˜ L = L /f ∞ . (4.14)Implicitly, we have assumed f ∞ >
0, which is not always the case — see discussion below.Now let us examine these solutions as functions of the couplings, i.e., in the µ − λ plane. In particular, if we insert the asymptotic limit r → ∞ , the discriminant of eq. (4.7)is useful in determining the number of vacuum solutions at different points in the parameterspace. As seen in eq. (4.6), p is unchanged in this limit since it is independent of r but q is slightly simplified: p = 3 µ + λ µ q ∞ = − λ + 9 µλ + 27 µ µ . (4.15)Given the previous discussion, we see that eq. (4.12) yields 3 real solutions for D ∞ = q ∞ − p ≤ D ∞ >
0. The vanishing of the discriminant, D ∞ = 0,reduces to a quadratic equation for µ with solutions µ = 227 − λ ±
227 (1 − λ ) / . (4.16)Eq. (4.16) generates the two (upper) curves in the µ - λ plane shown in figures 1 and 2. Inthe region bounded by these two curves, D ∞ < a) (g)(c)(d)(f)(h) (b) (e) - - - Λ Μ Figure 1:
The red and blue curves indicate the positive and negative branches of eq. (4.16),respectively, where D = 0. The region bounded by these two curves is where D <
0. The greencurve indicates p = 0. The letter labels refer to the various regions described in table 1. The blueshaded region indicates those couplings for which there exist asymptotically AdS black holes. while outside of these two curves, D ∞ > µ = 0. Ofcourse, µ = 0 is a special axis in the parameter space corresponding to the Gauss-Bonnettheory discussed in section 2. Note that the positive branch of eq. (4.16) crosses the λ -axisat λ = 1 /
4, which is precisely the critical coupling in the GB gravity. Recall that in GBgravity for λ < , there are two vacua while no vacuum solutions exist for λ > .Below we will find that another interesting boundary in the µ - λ plane is p = 0, whichcorresponds to the lowermost (green) curve in figures 1 and 2 – note that p = 0 is alwaysbelow both branches of D = 0, except at the points ( λ, µ ) = (0 ,
0) and ( , − ). Fromeq. (4.15), we see that p = 0 simply corresponds to µ = − λ . A distinguishing feature of p = 0 is that eq. (4.5) becomes a perfect cubic equation.As a point of clarification, we should note that when eq. (4.12) yields real roots, thevalue of f ∞ may be either positive (as assumed above) or negative. For example, it is easyto see that if µ > h ( f ∞ → −∞ ) ≃ µf ∞ < h ( f ∞ = 0) = 1 > N > f ∞ <
0. If instead we choose N = 1 / | f ∞ | , the metric (2.3)becomes ds = r L (cid:0) dt + dx + dy + dz (cid:1) − L | f ∞ | dr r , (4.17)which corresponds to a particular set of coordinates on five-dimensional de Sitter space,where we observe that r plays the role of time. The radius of curvature of this dS spacetimeis ˜ L = L / | f ∞ | . Even though with f ∞ <
0, eqs. (4.9) or (4.11) still yield nontrivialsolutions f ( r ), however, the corresponding metrics should be interpreted as (singular)– 13 – g) (c )(c )(c ) Λ - - - - Μ Figure 2:
A closer examination of the lower right quadrant of figure 1, covered by the regionsdenoted (c) and (g) in table 1. All three curves intersect at ( λ, µ ) = ( , − ), where the twobranches of eq. (4.16) end. Region (g) is bounded by these two branches. We divide (c) into: (c )where p <
0, (c ) and (c ) with p > cosmological solutions rather than black holes. Hence we will not consider solutions with f ∞ < i.e., whether or not the graviton isa ghost in a particular AdS vacuum. Recall that for GB gravity, we discarded one branchof the solutions in eq. (2.6) because the analysis of [16] showed that the graviton was aghost in these backgrounds. A similar analysis applies for quasi-topological gravity, as wewill see in section 6. The key factor distinguishing this feature of the various vacua is theslope of the cubic equation determining f ∞ : h ′ ( f ∞ ) ≡ − λf ∞ + 3 µf ∞ . (4.18)In section 6, we show that this expression appears as a pre-factor in the kinetic term forgravitons propagating in a given AdS vacuum. The kinetic term has the usual sign when h ′ ( f ∞ ) < h ′ ( f ∞ ) >
0. Hence the stable or ghost-free AdS vacuain table 1 are distinguished by having h ′ ( f ∞ ) >
0. Given that this factor is simply theslope of h ( f ∞ ), it is easy to see that since h ( f ∞ = 0) = 1 then if there is one AdS vacuum( i.e., one root with f ∞ > h ( f ) ≡ (cid:18) − ω r (cid:19) − f + λf + µf = 0 . (4.19)– 14 – ∞ µ λ Stable AdS Ghosty AdS dS BH solutiona + + + 0 0 1 -b + + – 0 0 1 -c + – + 1 0 0 f (in c ,c )d + – – 1 0 0 f e – + + 1 1 1 f f – + – 1 1 1 f g – – + 2 1 0 f h – – – 1 0 2 f Table 1:
Table of various vacua and black hole solutions. The column labeled ‘BH solution’indicates which root (4.9) yields the nonsingular black hole solution. In case (c), the black holesolution is only realized in the regions denoted (c ) and (c ) in figure 2. Now in the asymptotic limit r → ∞ , we simply recover eq. (4.12) and the roots matchthe vacuum solutions f ∞ . We can regard the effect of r decreasing through finite valuesas reducing the ‘constant’ term in the cubic polynomial of f and as a result, the roots of˜ h ( f ) shift away from f ∞ .To illustrate various possibilities, figure 3 plots an example of ˜ h ( f ) in case (g) withthree AdS vacua – see table 1. First we consider the smallest root f , which asymptoticallyreaches the AdS vacuum solution with the smallest value of f ∞ . As shown when r decreases,this root decreases moving monotonically to the left until it reaches f = 0 at r = ω . As r shrinks to even smaller values, f becomes negative and the solution becomes singularwith f → −∞ as r →
0. Of course, this behaviour is precisely that of a black hole with ahorizon at r = ω .Next consider the second root f in figure 3. In this case, the slope ˜ h ′ ( f ) is positiveand so the corresponding AdS vacuum contains ghosts. Note that as r decreases (or ω /r increases), the root now moves to the right, i.e., f grows as we move to the interior ofthe solution. This behaviour is problematic as it corresponds to a negative mass solutionand it seems to be connected to the ghost problems. As we discuss below, the solutionreaches a naked singularity at r = r (= 1 . ω , in this particular example) where thetwo roots, f and f , coalesce. One could overcome the problem with negative masses bysimply choosing the integration constant ω <
0. In this case, f moves to the left withdecreasing r but a naked singularity is still produced when the roots f and f coalesce.Finally we turn to f , the largest of the three roots in figure 3. Here again, the rootmoves to the left as r decreases so that f decreases as we move to the interior of thegeometry indicating a positive mass. However, as noted above, this root coalesces with f at r = r and becomes complex for smaller values of r . Defining f ( r ) = f and Taylorexpanding eq. (4.19) about this point, we find f ( r ) ≃ f + 2 γ ω r (cid:18) r − r r (cid:19) / where γ = −
12 ˜ h ′′ ( f ) = 32 | µ | f − λ . (4.20)– 15 – æææ æææ æææ ææææ f f f f - - h Ž Figure 3:
Graph of ˜ h ( f ∞ ) for µ = − .
005 and λ = 0 . r decreases.Here, the three curves correspond to ω /r = 0, 1 and 1.752 from the top to bottom. The slopeis negative for the roots, f and f , indicating that these are stable solutions while it is negativefor f indicating the graviton is a ghost in this background. At ω /r = 1 . f and f coalesce and a curvature singularity appears in both solutions. Further calculating the curvature using this result yields R abcd R abcd ∝ ω r L r − r ) (4.21)showing that the spacetime has a naked singularity at this point. One could again examinethese solutions with ω <
0. However, in this case, f moves to the right, indicating anegative mass, and a naked singularity arises with f → + ∞ as r → f correspondsto an asymptotically AdS black hole. The other roots, f and f , are both asymptoticallyAdS but produce spacetimes with naked singularities. Examining the other cases in thetable in a similar way, one finds that in each parameter regime with an AdS vacuum, thereis a single black hole solution corresponding to the smallest positive root of eq. (4.19). Theonly exception is case (c) where we must also be in the regions denoted (c ) or (c ). Theserestrictions are related to the possibility that a naked singularity will arise if the function˜ h ( f ) is not monotonic in the range f ∈ [0 , f ∞ ], as explained for the root f in the exampleabove – see also figure 4. First of all, in the region (c ), p is negative and ˜ h ( f ) has noextrema at all. In regions (c ) and (c ), p > f of˜ h ( f ). In region (c ) to the left of (g) where D ∞ <
0, both of the extrema f > f ∞ and so˜ h ( f ) is monotonic in the desired range. On the other hand, one finds 0 < f < f ∞ in region(c ) to the right of (g). Therefore in this parameter regime, the solution develops a nakedsingularity when r reaches the value where f = f . However, the solution corresponds toa black hole with smooth event horizon for parameters in the regions (c ) and (c ). All of– 16 – æææ - f - h Ž Figure 4:
The function ˜ h ( f ) plotted for µ = − .
045 and λ = 0 .
4, typical parameters in the region(c ). The single (real) root f corresponds to a ghost-free asymptotically AdS solution. However,when the radius reaches a value where ˜ h ′ ( f ) = 0 at the root, indicated with the black dot, thegeometry becomes singular. our results with regards to which root yields a black hole solution are summarized in table1. As in eq. (2.16), we can again generalize the metric ansatz to include spherical and hyper-bolic, as well as planar, horizons: ds = − (cid:18) k + r L f ( r ) (cid:19) N ( r ) dt + dr k + r L f ( r ) + r dℓ k (4.22)where dℓ k is given by k = +1 : d Ω ,k = 0 : 1 L (cid:0) dx + dy + dz (cid:1) , (4.23) k = − d Σ . As in eq. (2.17), we have above the metric on a unit three-sphere for k = +1 and on a three-dimensional hyperbolic plane with unit curvature for k = −
1. The analysis at the beginningof section 4.1 follows through unchanged. Implicitly, we will assume N ( r ) = 1 /f ∞ butmore importantly, the solutions are again determined by eq. (4.4):1 − f + λf + µf = ω r . (4.24)– 17 –ence one arrives at the same solutions for f as previously found. The difference betweenplanar and curved horizons is that the usual horizon equation g tt = 0 now becomes f = − k L r . We have not made a complete analysis of the structure of the new spacetimesthroughout µ - λ plane but let us make the following preliminary remarks.We can develop a qualitative picture of the solutions using the same graphical approachas in the previous section. However, as well as following the behaviour of the ˜ h ( f ), we mustnow also keep track of the critical value of f where a horizon can form when k = 0, i.e., f h ≡ − kL /r . While the mass parameter ω controls how quickly ˜ h ( f ) (and its roots)are shifting as r varies, the rate of change in f h is controlled by the cosmological constantscale L while the direction is controlled by k . Our first observation then is that if we havea black hole solution with k = 0, then for large masses ω ≫ L , we will always find thenew (curved) horizon equation with r h ∼ ω . That is, in this regime, the relevant root of˜ h ( f ) moves much more quickly as r decreases than f h . Hence the root reaches f = 0 at r = ω (as discussed for the planar horizons) while | f h | = L /ω ≪ µ - λ plane where they were found for planar horizons, in the previous section. However, inthese regions, one may find that there is a lower bound on the mass of these black holesolutions different from ω = 0.Let us consider small masses first for spherical horizons with k = +1 and f h = − L /r .As r decreases, the smallest positive root of eq. (4.19) moves to the left, approaching f = 0as r → ω . At the same time, f h starts at zero at r = ∞ and then moves to the leftto negative values as r decreases. Hence to form a horizon, the root must ‘catch up’ to f h . If we tune ω to smaller and smaller values, slowing down the rate at which the rootmoves, it becomes clear that it may never coincide with f h . For example, consider cases(d) with p >
0, (e), (f) and (h) in table 1. In each of these cases, there will be a value f < h ′ ( f ) = 0, where the spacetime develops a singularity as describedin section 4.1 – note that this previous discussion does not change for k = 0. Hence if f h ( ω ) = − L /ω ≤ f , then the root will not be able to reach f h before hitting f . Hencein this situation, the spacetime will contain a naked singularity. In cases (c), (d) with p < h ( f ) is monotonic for negative values of f . However, now for small r ( i.e., r ≪ ω, L ) the root behaves as f ≃ − [ ω / ( | µ | r )] / . Hence the root is growing much moreslowly than f h ∝ r − in this regime and again it becomes apparent that the root will nevercatch up to f h . Hence the solution will again have a naked singularity as r →
0. Thusour final conclusion is that for spherical horizons with k = +1 there will always be a lower(positive) bound on the mass parameter below which no black hole solutions exist. Thisis, of course, qualitatively, the same result as found for spherical black holes for Einsteingravity with a negative cosmological constant [25]. We have not calculated the exact valueof the lower bound but one must find ω ∼ L , with the precise proportionality constantdetermined by the gravitational couplings, λ and µ .We might add that since f h < k = +1, the discussion given in the previoussection remains unchanged for the solutions asymptotic to the ghosty vacua, the extrastable AdS vacuum in case (g) and the AdS vacuum in case (c ). That is, there will be no– 18 –pherical black hole solutions in any of these cases.Now let us turn to considering small masses for hyperbolic horizons with k = − f h = L /r . In this case, f h moves to the right to positive values as as r decreases and somore exotic possibilities arise. For example, even if ω = 0 and the root does not move, f h will increase and eventually coincide with the root. That is, a horizon will appear even ifthe mass is set to zero and by continuity, we must also find black holes with small negativemasses. Note that with ω , the root moves to the right to values larger than the initial f ∞ .Here, cases (e), (f) and (g), as well as the region denoted (c ), are distinguished because˜ h ( f ) is not monotonic for positive values of f > f ∞ . Hence the negative mass solutions willonly contain a horizon if f h catches up to the root before reaching the point where ˜ h ′ = 0.This will set a (negative) lower bound on the mass with ω ∼ − L where again the precisebound will be determined by λ and µ . Cases (c ), (d) and (h) with k = − h ( f ) is monotonic for f > f ∞ and so a singularity only developsas r → f → ∞ . However, in this regime, the root grows as f ≃ [ | ω | / ( | µ | r )] / while f h = L /r . Hence the solution will also contain a horizon with r h ≃ | µ | L / | ω | for arbitrary negative values of ω . Hence we have found that in any of the parameterregimes where planar black holes exist, there will be black holes with hyperbolic horizonswith negative masses. In cases (c ), (e), (f) and (g), there is a lower bound on how negativethe mass can become but in cases (c ), (d) and (h), the hyperbolic black holes can havean arbitrarily large negative mass. These results are not entirely surprising given thathyperbolic black holes with negative masses exist for for Einstein gravity with a negativecosmological constant [26], although there is a lower bound on the mass there.We might briefly also consider the effect of setting k = − i.e., , the solutions asymptotic to the ghosty vacua, theextra stable AdS vacuum in case (g) and the AdS vacuum in case (c ). It is clear that evenin these cases a smooth horizon forms with ω = 0 since f h = L /r moves to positive valuesas r decreases. Again by continuity, black hole solutions will also exist for small positiveand negative values of ω . As above, the largest root will grow as f ≃ [ ω / ( µ r )] / ina regime where r →
0. Since the critical value f h grows more quickly, we conclude thatthe solutions which asymptote to the ghosty vacua in cases (e) and (f) (with µ >
0) willform a horizon for arbitrarily large positive values of ω . Similarly, the solutions whichasymptote to the second stable AdS vacuum in case (g) or to the AdS vacuum in case(c ) will form hyperbolic horizons for arbitrarily large negative masses. Hence we find thatexotic hyperbolic black holes also exist in parameter regimes even where no planar blackholes formed. With the construction described in section 3.1, we extended quasi-topological gravity tohigher dimensions D ≥ D = 6. The general action for D ≥ D = 5) is: I = 116 πG D Z d D x √− g (cid:20) ( D − D − L + R + λL ( D − D − X – 19 – D − D − D − D − D + 16) µL Z D (cid:21) (4.25)where X and Z D are given in eqs. (2.1) and (3.10), respectively. We choose the metricansatz in eq. (2.16), which can describe D -dimensional black holes with planar, sphericalor hyperbolic horizons (for k = 0, +1 and –1, respectively). The coefficients of the action(4.25) are chosen so that substituting in this metric (2.16) yields I = 116 πG D Z d D x ( D − N ( r ) L D (cid:2) r D − (cid:0) − f + λf + µf (cid:1)(cid:3) ′ . (4.26)The lapse must again be constant and a convenient choice is N ( r ) = 1 /f ∞ as described insection 2 – with the implicit assumption that f ∞ >
0. If we wish to consider the vacuumsolutions, f ( r ) is also fixed to be constant with f ( r ) = f ∞ where these solutions are againdetermined by eq. (4.12). Hence the various vacuum solutions are again distributed asdescribed in table 1. Assuming f ∞ >
0, the D -dimensional metric becomes ds = − (cid:18) k + r L f ∞ (cid:19) dt f ∞ + dr k + r L f ∞ + r dℓ k (4.27)where dℓ k is given in eq. (2.17). These solutions correspond to a spherical ( k = +1), flat( k = 0) or hyperbolic ( k = −
1) foliation of
AdS D . From the g rr component, we see thatthe radius of curvature of the AdS D spacetime is˜ L = L /f ∞ . (4.28)In considering the black hole solutions, the only difference from the analysis for D = 5in the previous sections is that we replace: ω /r → ω D − /r D − . In particular, the lattersubstitution is made in q in eq. (4.6). Further, eq. (4.19) is replaced with˜ h ( f ) ≡ (cid:18) − ω D − r D − (cid:19) − f + λf + µf = 0 . (4.29)The remainder of the analysis and the results in section 4.1 carries over unaltered. Inparticular, table 1 correctly describes the planar black hole solutions for D ≥
7. Similarly,the discussion of black holes with curved horizons in section 4.2 remains largely unchanged.In certain cases, the discussion for small masses referred to the behaviour of the root,now, of eq. (4.29) as r approaches zero. In the present case, this behaviour changes to f ≃ [ ω D − / ( µ r D − )] / while the behaviour of the critical value remains f h = L /r .Hence the root is grows more slowly than f h for D = 5, at the same rate for D = 7 andmore quickly for D ≥
8. As a result, one finds, in cases (c), (d) with p < D = 5 and 7 but thelower bound is simply ω D − = 0 for D ≥
8. Further, the conclusion that hyperbolic blackholes exits in cases (c ), (d) and (h) with arbitrarily large negative masses only appliesfor D = 5. Similarly, some of the details about the formation of hyperbolic horizons, inregions of the coupling space where planar black holes do not exist, change depending thespacetime dimension D . – 20 –s noted above, this discussion applies for D ≥
7, but in those dimensions, one alreadyhas a cubic order theory in Lovelock gravity, which also reproduces eq. (4.29). Hence for D ≥
7, the black holes for quasi-topological gravity discussed here would be the same asthose in cubic Lovelock theory, as considered in [8, 27].
5. Black Hole Thermodynamics
We now turn to the thermodynamic properties of the black hole solutions of quasi-topologicalgravity. Our focus will be on the planar ( k = 0) black holes in five dimensions. The exten-sion of these results to curved horizons and higher dimensions is straightforward.First, we use the standard approach to calculate temperature: analytically continuethe metric to Euclidean signature with τ = − ıt and periodically identify τ to produce aneverywhere smooth Euclidean section. Interpreting the period of τ as the inverse temper-ature, we find T = 14 π r h f ′ | r h L √ f ∞ , (5.1)which assumes the lapse is chosen as N = 1 / √ f ∞ . For the planar black holes, r h = ω andfurther we will use f ( r = ω ) = 0. Then we can evaluate the f ′ | r h by differentiating theconstraint equation (4.4) and evaluating the result at r = ω . A simple calculation yields f ′ | ω = ω , giving: T = ωπL √ f ∞ . (5.2)In passing we note that extending this calculation to general dimensions and curvedhorizons yields T = 14 π √ f ∞ (cid:20) r h L f ′ | r h − kr h (cid:21) = D − π √ f ∞ " ω D − L r D − h r h r h + 2 λkL r h − µk L − kD − r h , (5.3)where the precise location of the horizon r h must still be determined for a spherical orhyperbolic horizon but, of course, r h = ω in the planar case ( k = 0).Next, we calculate the entropy and energy densities of the black holes following theEuclidean action approach, as already sketched for GB theory in section 2. That is, weidentify the Euclidean action for the black hole solution, as the leading contribution to thefree energy, i.e., I E ≃ F/T . Evaluating the Euclidean action yields I E [ T ] = − πG Z /T d τ Z R ω d r Z d x √ g E (cid:18) L + R + λL χ + 7 µ Z (cid:19) = − V πG T L √ f ∞ (cid:2) r (cid:0) − f ( r ) + 15 λf ( r ) − µf ( r ) (cid:1) (5.4)+ r (cid:0) − λf ( r ) − µf ( r ) (cid:1) f ′ ( r ) (cid:3) R ω . – 21 –ere V = R d x is the regulator volume for the (spatial) gauge theory directions and asusual, we have set N = 1 / √ f ∞ . The final result is simplified using the constraint (4.4) toproduce the following asymptotic expansion of f ( r ): f ≃ f ∞ − ω r − λf ∞ − µf ∞ ) + · · · . (5.5)Keeping only the divergent and finite terms in the limit R → ∞ , eq. (5.4) then reducesto: I E [ T ] = V πG ω T L √ f ∞ (cid:20) R ω f ∞ (cid:0) − λf ∞ + 9 µf ∞ (cid:1) − − λf ∞ + 3 µf ∞ − λf ∞ − µf ∞ (cid:21) . (5.6)We remove the divergence in this expression by subtracting the action for the AdS vacuum I E [ T ′ ] = V πG R T ′ L √ f ∞ f ∞ (cid:0) − λf ∞ + 9 µf ∞ (cid:1) , (5.7)where T ′ is chosen so that the periodicity of the AdS background matches that of the blackhole at the regulator surface r = R :1 T ′ = 1 T p f ( R ) √ f ∞ ≃ T (cid:18) − ω R f ∞ (1 − λf ∞ − µf ∞ ) (cid:19) . (5.8)Combining these expressions yields a simple expression for the free energy: F [ T ] = T (cid:0) I E [ T ] − I E [ T ] (cid:1) = − V ω πG L √ f ∞ = − (cid:0) π √ f ∞ L (cid:1) G T (5.9)where we have implicitly taken the limit R → ∞ above. As noted in section 2, for planarAdS black holes, we do not have to account for the generalized Gibbons-Hawking term[20] or the boundary counter-terms [21] in this calculation of the free energy. Finally, toeliminate the regulator volume, we work with the free energy density: F = F/V . Thenwe calculate the energy and entropy densities as ρ = − T ddT ( F /T ) = 3( π √ f ∞ L ) T G , (5.10) s = − d F dT = (cid:0) π √ f ∞ LT (cid:1) G = 14 G ω L . (5.11)Note that these expressions are ‘identical’ to those appearing in eq. (2.14) for GB gravity,however, f ∞ implicitly depends on the additional coupling µ . We also comment thatthese results obey the relation ρ = T s , as expected for a four-dimensional CFT at finitetemperature.
In this section, we verify that the entropy density in eq. (5.11) matches the Wald entropy[22]. Of course, we find agreement [28] but the results for the Wald entropy are also readily– 22 –xtended to higher dimensions and curved horizons. Wald’s prescription for the black holeentropy is S = − π I d D − x √ h Y abcd ˆ ε ab ˆ ε cd where Y abcd = ∂ L ∂R abcd (5.12) L is the Lagrangian and ˆ ε ab is the binormal to the horizon. For the static black holesconsidered here, Y = Y abcd ˆ ε ab ˆ ε cd is constant on the horizon and so the entropy is givensimply as S = − πY A, (5.13)where A = H d D − x √ h is the ‘area’ of the horizon. Let us divide up the terms in theaction (4.25) according to their powers of the curvature tensor: the Einstein term, the GBinteraction and the quasi-topological interaction. (Of course, one also has the cosmologicalconstant term but it does not contribute in Wald’s formula (5.12).) Following the aboveprescription, as usual, the Einstein term yields Y = − / (8 πG D ) and the resulting entropyis the expected Bekenstein-Hawking entropy S = A/ (4 G D ). Applying this formalism toGB terms, we find Y = − πG D λ L ( D − D − (cid:0) R − (cid:0) R tt + R rr (cid:1) + 2 R trtr (cid:1) , (5.14)where this expression applies for a general static black hole metric. One can think thatthe tensor components above are presented in an orthonormal frame or alternatively in acoordinate frame, as long as the indices are in precisely the raised and lowered positions asshown. Integrating this over the horizon gives the following contribution to the entropy as S = A G D λ L ( D − D − (cid:0) R − (cid:0) R tt + R rr (cid:1) + 2 R trtr (cid:1) = A G D D − D − − λ f ( r h )) . (5.15)where we are using the general metric (2.16) with N = 1 /f ∞ to evaluate the second line.Finally turning to the quasi-topological contribution in the action, we find Y = 14 πG D D − µ L ( D − D − D − D + 16) (cid:20) c (cid:0) R tmtn R rnrm − R tmrn R rmtn (cid:1) (5.16)+3 c R trmn R trmn + c (cid:18) R trtm R rm − R trrm R tm + 14 (cid:0) R mnpr R mnpr + R mnpt R mnpt (cid:1)(cid:19) + c (cid:18) R R trtr + 12 R mnpq R mnpq (cid:19) + c (cid:0) R tt R rr − R tr R rt + R rmrn R mn + R tmtn R mn (cid:1) + 34 c (cid:0) R rm R rm + R tm R tm (cid:1) + c (cid:0) R mn R mn + R (cid:0) R rr + R tt (cid:1)(cid:1) + 32 c R (cid:21) . We have left the coefficients arbitrary above but it is understood that they are to be fixed– 23 –s in eq. (3.9). Now integrating over the horizon yields S = − A G D D − µ L ( D − D − D − D + 16) (cid:20) c (cid:0) R tmtn R rnrm − R tmrn R rmtn (cid:1) +3 c R trmn R trmn + c (cid:18) R trtm R rm − R trrm R tm + 14 (cid:0) R mnpr R mnpr + R mnpt R mnpt (cid:1)(cid:19) + c (cid:18) R R trtr + 12 R mnpq R mnpq (cid:19) + c (cid:0) R tt R rr − R tr R rt + R rmrn R mn + R tmtn R mn (cid:1) + 34 c (cid:0) R rm R rm + R tm R tm (cid:1) + c (cid:0) R mn R mn + R (cid:0) R rr + R tt (cid:1)(cid:1) + 32 c R (cid:21) = A G D D − D − (cid:0) − µ f ( r h ) (cid:1) . (5.17)Combining all of these expressions, we arrive at the Wald entropy for quasi-topologicalgravity: S = A G D (cid:18) − D − D − λ f ( r h ) − D − D − µ f ( r h ) (cid:19) . (5.18)Evaluating this expression on a planar horizon yields the simple result, S = A/ (4 G D ), i.e., ,the higher curvature contributions vanish on planar horizons. If we divide by the regulatorvolume, this yields the entropy density: s = SV D − = ω D − G D L D − , (5.19)which, of course, agrees with the result in eq. (5.11) for D = 5. For the case of black holeswith curved horizons, as in section 4.3, we find that the entropy is given by S k = A G D (cid:18) D − D − λ k L r h − D − D − µ k L r h (cid:19) . (5.20)
6. Equations of Motion
We have found that the equations of motion for quasi-topological gravity take an incrediblysimple form with the ansatz (2.16) for a static AdS black hole – e.g., see eq. (4.26). In thissection, we would like to investigate the equations of motion in greater generality to gainsome further insight into this simplicity. In particular, we will see the linearized equationsof motion of graviton fluctuations also exhibit a certain simplicity. We have already arguedthat the new cubic interactions constructed in section 3 do not have a topological origin –see appendix A. Hence this cannot be the source of the simplicity noted above.Let us begin with the cubic-curvature interactions in eq. (3.1). First we set c = 0 = c and then find the general contribution these terms would make to the metric equations of– 24 –otion:1 √− g δIδg ab = c (cid:0) − R R ab + 3 ( R ) ; ab (cid:1) + c (cid:16) − R cd R dc R ab − RR ac R cb + ( R cd R dc ) ; ab − (cid:3) ( RR ab )+2( RR c ( a ) ; b ) c (cid:1) + c (cid:18) − R ac R cd R db − (cid:3) ( R ac R cb ) + 3( R cd R d ( b ) ; a ) c (cid:19) + c (cid:16) − R c ( a R b ) dce RR de + 2( R ( a | cde R ce ) | ; b ) d − (cid:3) ( R acbd R cd ) + ( R ( a c R b ) d ) ; cd − ( R cd R ab ) ; cd (cid:17) + c (cid:16) − RR acde R bcde − R ab R cdef R cdef + ( R cdef R cdef ) ; ab +4( RR acdb ) ; cd (cid:17) + c (cid:16) − R acde R bcdf R ef − R defc R def ( a R b ) c + ( R cdef R cde ( a ) ; b ) f + 12 (cid:3) ( R cdea R cdeb ) + 2( R edc ( a R b ) e ) ; cd + 2( R e ( ab ) c R ed ) ; cd (cid:19) + c (cid:16) R acde R defg R fgbc +6( R ( a cef R d b ) ef ) ; cd (cid:17) + c (cid:16) − R cdef R cgea R fgbd + 3( R edf ( b R a ) ecf ) ; cd − R ( a e b ) f R edf c ) ; dc (cid:17) + g ab (cid:18) c (cid:18) R − (cid:3) ( R ) (cid:19) + c (cid:18) RR cd R dc − ( RR cd ) ; cd − (cid:3) ( R cd R dc ) (cid:17) + c (cid:18) R cd R de R ec −
32 ( R ce R ed ) ; cd (cid:19) + c (cid:18) R cd R cedf R ef − ( R ecf d R ef ) ; cd (cid:17) + c (cid:18) RR cdef R cdef − (cid:3) ( R cdef R cdef ) (cid:19) + c (cid:18) R cdef R cdeg R fg −
12 ( R degc R degf ) ; ef (cid:19) + 12 c R cdef R ef gh R ghcd + 12 c R cdef R dgf h R gche (cid:19) . (6.1)Since we have eliminated the terms involving derivatives of the curvature from the action bysetting c = 0 = c , the above contributions contain at most terms with four derivativesof the metric, such as in (cid:3) ( RR ab ). This may be misleading, however, since we have madeno attempt to simplify this expression using, e.g., the Bianchi identities. A useful check isto choose the coefficients above so that eq. (3.1) corresponds to the six-dimensional Eulerdensity (3.7), i.e., choose c = − c , c = 4 and the remaining coefficients as in eq. (3.3).In this case, we were able to verify that any terms involving derivatives of curvatures canbe eliminated and the expected field equations involving only factors of the curvature werereproduced – e.g., see [27, 29]. Indeed Lovelock’s general discussion [14] dictates that X isthe only gravity Lagrangian which is cubic in curvatures for which the equations of motiondo not contain any derivatives of curvatures. Hence one must expect equations of motionthat include terms with derivatives of curvatures for any other choice of the coefficients c i and, in particular, for quasi-topological gravity with c i as in eq. (3.9). While we stillmade some effort to ‘tidy up’ the derivative terms in eq. (6.1), we did not produce anyparticularly illuminating results.As a next step, we examine the linearized equations of motion for a graviton pertur-bation in quasi-topological gravity. Hence we fixed the coefficients as in eq. (3.9) and thensubstitute into the above expression (6.1): g ab = g [0] ab + h ab where g [0] ab is a solution of thefull equations of motion. Again, the resulting expression of a generic fluctuation is rathercomplicated and so to proceed further, we restrict our attention to transverse tracelessgauge with ∇ a h ab = 0) and h aa = 0. This choice simplifies the result somewhat and the– 25 –our-derivative contribution is proportional to:( D − R cdef h de ; cf ( ab ) + R cd ( (cid:3) h cd ) ;( ab ) − R cd (cid:0) (cid:3) h c ( a (cid:1) ; b ) d + 2 D − R cd ( (cid:3) h ab ) ; cd +2 (cid:0) (cid:3) h c ( a (cid:1) ; de R | cde | b ) + g ab ( (cid:3) h cd ) ; ef R cedf + 2 R c ( a (cid:3) h c b ) − g ab R cd (cid:3) h cd + ( D − (cid:3) h cd R cadb − R ( D − (cid:3) h ab , (6.2)where we use the standard notation T ( ab ) = ( T ab + T ba ). Hence despite the reductionin the number of terms, we see that for general backgrounds, the linearized equations ofmotion for (physical) gravitons include four-derivative contributions.Of course, if we were considering gravitons propagating in flat space, these contribu-tions would all vanish because R abcd = 0 in the background spacetime. The same resultwould apply for any interactions which are cubic in curvatures for a flat background. How-ever, in the present context, it is natural to consider gravitons propagating in an AdS D background. In this case, of course, the background curvature is nonvanishing and soone might expect these terms (6.2) will still appear in the linearized equations of motion.However, further simplifications can be expected since AdS D is a maximally symmetricspacetime with R abcd = − L ( g ac g bd − g ad g bc ) . (6.3)Remarkably, one finds that upon substituting this background curvature into eq. (6.2), allof the remaining four-derivative terms cancel! Further, because the background curvature(6.3) is covariantly constant, there are no nontrivial terms with only three derivatives actingon the graviton. Therefore, in quasi-topological gravity, the linearized graviton equationin an AdS D background is only a second-order equation.With this result in hand, we next construct the linearized equation of motion for thegraviton in the AdS D vacuum solutions. Hence we must consider the full action (4.25) forquasi-topological gravity, including the cosmological constant, Einstein and Gauss-Bonnetterms as well. The equation of motion for the transverse traceless graviton can then bewritten as: − (cid:0) − λf ∞ − µf ∞ (cid:1) (cid:20) ∇ h ab + 2 f ∞ L h ab (cid:21) = 8 πG D ˆ T ab . (6.4)We have added a stress-tensor on the right-hand side, as might arise from minimally cou-pling the metric to additional matter fields or from quadratic or higher order contributionsin the graviton. The second bracketed factor on the right-hand side is the standard Ein-stein equation of motion for gravitons in an AdS background with curvature ˜ L = L/ √ f ∞ [30]. One can recognize the first bracketed factor as the slope of the cubic equation (4.12)determining f ∞ , i.e., see eq. (4.18). Hence this slope determines the sign of the gravitonpropagator or alternatively the sign of the coupling of the graviton to the stress tensor.We see that the appropriate sign for a well-behaved graviton is negative since this factor Note that here we are distinguishing ˜ L , the curvature scale of the AdS background, from L , the AdSlength scale appearing in the action. In particular, recall that in the vacuum solutions above, we found˜ L = L/ √ f ∞ . – 26 –educes to –1 when λ = 0 = µ . Hence as discussed in section 4.1, the AdS vacua are onlystable when the slope is negative while the graviton is a ghost in AdS backgrounds wherethe slope is positive.While restricting to transverse traceless gauge is a convenient simplification to deter-mine the linearized equations in an AdS background, we can do better. A straightforwardargument shows that the above gauge-fixed equations of motion (6.4) extend to the fulllinearized Einstein equations [30]: − (cid:0) − λf ∞ − µf ∞ (cid:1) (cid:20) ∇ h ab + ∇ a ∇ b h cc − ∇ a ∇ c h cb − ∇ b ∇ c h ca (6.5) − g [0] ab (cid:16) ∇ h cc − ∇ c ∇ d h cd (cid:17) + 2 f ∞ L h ab − ( D − f ∞ L g [0] ab h cc (cid:21) = 8 πG D ˆ T ab , where g [0] ab is the background AdS metric. We know that the full linearized equations comefrom a covariant expression and hence they must invariant under the ‘gauge’ transforma-tions: δh ab = ∇ a ε b + ∇ b ε a . Now the linearized Einstein equations (6.5) are certainlyinvariant under these transformations and reduce to eq. (6.4) upon fixing to transversetraceless gauge. However, there may be additional contributions which are both gaugeinvariant and completely vanish for transverse traceless modes. For example, the full equa-tions may include an additional contribution proportional to g [0] ab (cid:18) ∇ h cc − ∇ c ∇ d h cd + ( D − f ∞ L h cc (cid:19) , (6.6)which is gauge invariant but would not contribute in eq. (6.4). In this particular case, itis easy to argue that such a contribution could not arise from the variation of an action(quadratic in h ab ). However, given the action (4.25) for quasi-topological gravity, anotherapproach is to evaluate the quadratic action (using Mathematica) and subsequently exam-ining the equations of motion for some specific trial perturbations which are not transverseor traceless. In every case considered, the latter equations match precisely the resultsexpected from eq. (6.5). This clearly shows that there are no additional terms of theform given in eq. (6.6) but more importantly that there are no additional four-derivativecontributions in the full linearized equations without gauge-fixing. The fact that gravi-tons propagating in an AdS background simply obey the same equations of motion as inEinstein gravity plays an important role in understanding the holographic properties ofquasi-topological gravity [10].
7. Discussion
In section 3, we have constructed a new gravitational action which includes terms cubicin the curvature. Our construction was motivated by the simple equations (2.5) arisingto determine the black hole solutions in GB gravity. We were able to reproduce a similarstructure (4.4) for our new theory. We wish to emphasize how remarkable this result is. We would like to thank Miguel Paulos for discussions on this point and confirming that several trialperturbations satisfied eq. (6.5) with Mathematica. – 27 –n section 6, we showed that the full equations are fourth order in derivatives. However,in section 4, we found that once the geometry of the horizon is fixed, the static black holesolutions are fixed by a single integration constant! It seems that the symmetry imposed onthe background geometry must play an important role in producing the simplicity of thesesolutions. While our approach was to substitute the ansatz into the action, it would beinteresting to work directly with the equations of motion and formalize this result in termsof a ‘Birkhoff theorem.’ It would also be interesting to see to what extent this simplicityextends to spinning or electrically charged black holes in quasi-topological gravity.We also saw that the linearized equations of motion for gravitons propagating in theAdS backgrounds reduced to the same second order equations as for Einstein gravity insection 6. There we saw more or less directly that this simplification comes about due tothe maximal symmetry of the AdS spacetime. On the other hand, we should not expectthat the four-derivative contributions (6.2) to the equations of motion cancel in the blackhole backgrounds. Hence the quasinormal spectrum of the black hole solutions should bestudied in detail. In particular, this spectrum may reveal that these solutions are unstablefor certain values of the gravitational couplings.The simplicity of the graviton equations in an AdS background has the interestingconsequence that the standard holographic rules apply in matching the metric fluctuationsto the stress tensor of the CFT. In a general higher derivative theory, implicitly the gravi-ton would be matched with some higher dimension operator, as well as the stress tensor[6]. This interpretation arises because the higher derivative equations allow the metricfluctuations to have more that the standard asymptotic behaviour in the AdS geometry.In any event, this complication is evaded in quasi-topological gravity and so the effect ofthe higher derivative gravitational terms will only be felt in the higher n -point couplingsof the stress tensor.Having motivated the construction of quasi-topological gravity by considerations ofthe AdS/CFT correspondence, one might ask how the universality class of dual CFT’s hasbeen expanded. The gravitational theory is defined by three independent dimensionlessparameters: λ , µ and L D − /G D . In general, the three-point function of the stress tensor of aCFT in four or higher dimensions is also characterized by three independent (dimensionless)parameters [31]. This match in the counting of these parameters is not a coincidence, as thediscussion of [32] that holographically modelling the full range of these CFT parametersrequires the introduction of curvature-squared and curvature-cubed interactions in the bulkgravity theory. We make precise the mapping between the gravitational couplings and thedual CFT parameters, as well as exploring other holographic aspects of quasi-topologicalgravity in [10, 33].Another natural extension of this work is to consider analogous gravitational interac-tions with higher powers of the curvature. Of course, Lovelock gravity provides an infinitesequence of (curvature) n interactions which still allow for a certain calculational controlin the gravitational theory. However, because of their topological origin, these Lovelockterms only contribute to the equations of motion for D ≥ n + 1, i.e., in the context ofthe AdS/CFT correspondence, for dual CFT’s with d ≥ n . However, our expectation isthat our construction of five-dimensional quasi-topological gravity with curvature-cubed– 28 –nteractions can be extended general (curvature) n interactions and this has been verifiedby some preliminary calculations [9]. In fact, our conjecture is that in D ≥ n + 1, an in-dependent interaction can be constructed for each independent scalar contraction of (Weyltensor) n . For D < n + 1, Schouten identities will reduce the number of independent inter-actions, as seen in the present analysis of the curvature-cubed interactions. However, ourunderstanding of this issue remains incomplete, as we are still uncertain as to why therewas no effective curvature-cubed interaction in six dimensions. In any event, better under-standing the number of independent quasi-topological gravity terms with higher powers ofthe curvature will be an interesting direction of study. Important new insights into thisquestion were given in refs. [12, 13]. There it was shown that the curvature-cubed interac-tions constructed here can be simply expressed in terms of scalar contractions of the Weyltensor combined with the six-dimensional Euler density. Their construction immediatelygeneralizes to an infinite family of higher curvature interactions with similar properties. Acknowledgments
We thank Ted Jacobson, Barak Kol, Robb Mann, Miguel Paulos and Aninda Sinha forhelpful discussions and useful comments. We also thank Jorge Escobedo for his help inpreparing the figures and for proofreading the paper. RCM would also like to thankthe KITP and the Weizmann Institute for hospitality at various stages of this project.Research at the KITP is supported by the National Science Foundation under Grant No.PHY05-51164. Research at Perimeter Institute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontario through the Ministry of Research& Innovation. We also acknowledge support from an NSERC Discovery grant and fundingfrom the Canadian Institute for Advanced Research.
A. A New Topological Invariant?
In section 3.1, our construction produced an nontrivial interaction (3.10) for any numberof dimensions D ≥
5. However, we noted Z , the six-dimensional expression, did not con-tribute to the equations of motion for the black hole metric (2.3) (extended to D = 6). Inparticular, this also means that the the AdS vacua are unaffected by the addition of Z to the gravitational action. This behaviour is reminiscent of Lovelock gravity where, forexample, the Euler density X provides a nontrivial gravitational interaction for D ≥ D ≤ Z yields another topological invariant in sixdimensions. However, in the following, we demonstrate that R d x √ g Z is not a topolog-ical invariant by explicitly evaluating this expression for certain specific six-dimensionalgeometries.As our first test, we evaluate this expression on a deformed six-sphere with metric: ds = R (cid:2) dθ + sin θ (cid:0) a sin θ (cid:1) n d Ω (cid:3) (A.1)– 29 –here n (implicitly an integer) and a are constants defining the deformation away from theround six-sphere. We then find: Z S √ g Z = 5443 π , Z S √ g X = 768 π . (A.2)where we have normalized X as in eq. (3.7). Hence we see that both of these results areindependent of the deformation parameters. Of course, for X , this occurs because theintegrated expression is a topological invariant. While again suggestive for Z , this resultis by no means conclusive and hence we consider a further test.Next we consider the following metric in which the spheres in the direct product S × S are deformed: ds = R h dθ + sin θ (cid:0) a sin θ (cid:1) dφ i (A.3)+ L (cid:20) d ˜ θ + sin ˜ θ (cid:16) b sin ˜ θ (cid:17) d Ω (cid:21) where the deformation is characterized by the constants a and b . In this case, we find: Z S × S √ g Z = F ( a, b, R/L ) , Z S × S √ g X = 1536 π . (A.4)where F ( a, b, R/L ) is a complicated (and not particularly illuminating) function of bothdeformation parameters and the relative radius of curvature of the two spheres. Hence thisresult makes clear that Z does not yield a topological invariant. References [1] P. Kovtun, D. T. Son and A. O. Starinets, “Viscosity in strongly interacting quantum fieldtheories from black hole physics,” Phys. Rev. Lett. , 111601 (2005) [arXiv:hep-th/0405231];P. Kovtun, D. T. Son and A. O. Starinets, “Holography and hydrodynamics: Diffusion onstretched horizons,” JHEP , 064 (2003) [arXiv:hep-th/0309213].[2] A. Buchel and J. T. Liu, “Universality of the shear viscosity in supergravity,” Phys. Rev.Lett. , 090602 (2004) [arXiv:hep-th/0311175];A. Buchel, “On universality of stress-energy tensor correlation functions in supergravity,”Phys. Lett. B , 392 (2005) [arXiv:hep-th/0408095];P. Benincasa, A. Buchel and R. Naryshkin, “The shear viscosity of gauge theory plasma withchemical potentials,” Phys. Lett. B , 309 (2007) [arXiv:hep-th/0610145];D. Mateos, R. C. Myers and R. M. Thomson, “Holographic viscosity of fundamental matter,”Phys. Rev. Lett. , 101601 (2007) [arXiv:hep-th/0610184];K. Landsteiner and J. Mas, “The shear viscosity of the non-commutative plasma,” JHEP , 088 (2007) [arXiv:0706.0411 [hep-th]];N. Iqbal and H. Liu, “Universality of the hydrodynamic limit in AdS/CFT and themembrane paradigm,” Phys. Rev. D , 025023 (2009) [arXiv:0809.3808 [hep-th]].[3] For example, see:A. Buchel, J. T. Liu and A. O. Starinets, “Coupling constant dependence of the shearviscosity in N=4 supersymmetric Yang-Mills theory,” Nucl. Phys. B , 56 (2005)[arXiv:hep-th/0406264]; – 30 – . Benincasa and A. Buchel, “Transport properties of N = 4 supersymmetric Yang-Millstheory at finite coupling,” JHEP , 103 (2006) [arXiv:hep-th/0510041];A. Buchel, “Shear viscosity of boost invariant plasma at finite coupling,” Nucl. Phys. B ,281 (2008) [arXiv:0801.4421 [hep-th]]; “Resolving disagreement for η/s in a CFT plasma atfinite coupling,” Nucl. Phys. B , 166 (2008) [arXiv:0805.2683 [hep-th]];R. C. Myers, M. F. Paulos and A. Sinha, “Quantum corrections to η/s ,” Phys. Rev. D ,041901 (2009) [arXiv:0806.2156 [hep-th]].[4] Y. Kats and P. Petrov, “Effect of curvature squared corrections in AdS on the viscosity of thedual gauge theory,” JHEP , 044 (2009) [arXiv:0712.0743 [hep-th]];A. Buchel, R. C. Myers and A. Sinha, “Beyond η/s = 1 / π ,” JHEP , 084 (2009)[arXiv:0812.2521 [hep-th]];R. C. Myers, M. F. Paulos and A. Sinha, “Holographic Hydrodynamics with a ChemicalPotential,” JHEP , 006 (2009) [arXiv:0903.2834 [hep-th]].[5] M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, “Viscosity Bound Violation inHigher Derivative Gravity,” Phys. Rev. D (2008) 126006 [arXiv:htp-th/0712.0805]; “TheViscosity Bound and Causality Violation,” Phys. Rev. Lett. , 191601 (2008)[arXiv:0802.3318 [hep-th]];A. Buchel and R. C. Myers, “Causality of Holographic Hydrodynamics,” JHEP , 016(2009) [arXiv:0906.2922 [hep-th]].[6] D. M. Hofman, “Higher Derivative Gravity, Causality and Positivity of Energy in a UVcomplete QFT,” Nucl. Phys. B , 174 (2009) [arXiv:0907.1625 [hep-th]].[7] X. H. Ge and S. J. Sin, “Shear viscosity, instability and the upper bound of theGauss-Bonnet coupling constant,” JHEP , 051 (2009) [arXiv:0903.2527 [hep-th]];R. G. Cai, Z. Y. Nie and Y. W. Sun, “Shear Viscosity from Effective Couplings ofGravitons,” Phys. Rev. D , 126007 (2008) [arXiv:0811.1665 [hep-th]];R. G. Cai, Z. Y. Nie, N. Ohta and Y. W. Sun, “Shear Viscosity from Gauss-Bonnet Gravitywith a Dilaton Coupling,” Phys. Rev. D , 066004 (2009) [arXiv:0901.1421 [hep-th]];J. de Boer, M. Kulaxizi and A. Parnachev, “ AdS /CF T , Gauss-Bonnet Gravity, andViscosity Bound,” arXiv:0910.5347 [hep-th];X. O. Camanho and J. D. Edelstein, “Causality constraints in AdS/CFT from conformalcollider physics and Gauss-Bonnet gravity,” arXiv:0911.3160 [hep-th];A. Buchel, J. Escobedo, R. C. Myers, M. F. Paulos, A. Sinha and M. Smolkin, “HolographicGB gravity in arbitrary dimensions,” arXiv:0911.4257 [hep-th].[8] J. de Boer, M. Kulaxizi and A. Parnachev, “Holographic Lovelock Gravities and BlackHoles,” arXiv:0912.1877 [hep-th];X. O. Camanho and J. D. Edelstein, “Causality in AdS/CFT and Lovelock theory,”arXiv:0912.1944 [hep-th].[9] Miguel Paulos, unpublished.[10] R. C. Myers, M. F. Paulos and A. Sinha, “Holographic studies of quasi-topological gravity,”arXiv:1004.2055 [hep-th].[11] A. Sinha, “On the new massive gravity and AdS/CFT,” arXiv:1003.0683 [hep-th].[12] J. Oliva and S. Ray, “A new cubic theory of gravity in five dimensions: Black hole, Birkhoff’stheorem and C-function,” arXiv:1003.4773 [gr-qc]. – 31 –
13] J. Oliva and S. Ray, “A Classification of Six Derivative Lagrangians of Gravity and StaticSpherically Symmetric Solutions,” arXiv:1004.0737 [gr-qc].[14] D. Lovelock, “The Einstein tensor and its generalizations,” J. Math. Phys. , 498 (1971);Aequationes Math. , 127 (1970).[15] B. Zwiebach, “Curvature Squared Terms And String Theories,” Phys. Lett. B (1985) 315.[16] D. G. Boulware and S. Deser, “String Generated Gravity Models,” Phys. Rev. Lett. (1985) 2656.[17] J. T. Wheeler, “Symmetric Solutions To The Gauss-Bonnet Extended Einstein Equations,”Nucl. Phys. B (1986) 737;J. T. Wheeler, “Symmetric Solutions To The Maximally Gauss-Bonnet Extended EinsteinEquations,” Nucl. Phys. B (1986) 732;R. C. Myers and J. Z. Simon, “Black Hole Thermodynamics in Lovelock Gravity,” Phys. Rev.D (1988) 2434;R. C. Myers and J. Z. Simon, “Black Hole Evaporation and Higher Derivative Gravity,” Gen.Rel. Grav. , 761 (1989).[18] R. G. Cai, “Gauss-Bonnet black holes in AdS spaces,” Phys. Rev. D (2002) 084014[arXiv:hep-th/0109133].[19] S. Nojiri and S. D. Odintsov, “Anti-de Sitter black hole thermodynamics in higher derivativegravity and new confining-deconfining phases in dual CFT,” Phys. Lett. B (2001) 87[Erratum-ibid. B (2002) 301] [arXiv:hep-th/0109122];Y. M. Cho and I. P. Neupane, “Anti-de Sitter black holes, thermal phase transition andholography in higher curvature gravity,” Phys. Rev. D (2002) 024044[arXiv:hep-th/0202140];I. P. Neupane, “Black hole entropy in string-generated gravity models,” Phys. Rev. D (2003) 061501 [arXiv:hep-th/0212092];I. P. Neupane, “Thermodynamic and gravitational instability on hyperbolic spaces,” Phys.Rev. D (2004) 084011 [arXiv:hep-th/0302132].[20] R. C. Myers, “Higher Derivative Gravity, Surface Terms and String Theory,” Phys. Rev. D , 392 (1987).[21] V. Balasubramanian and P. Kraus, “A stress tensor for anti-de Sitter gravity,” Commun.Math. Phys. , 413 (1999) [arXiv:hep-th/9902121];R. Emparan, C. V. Johnson and R. C. Myers, “Surface terms as counterterms in theAdS/CFT correspondence,” Phys. Rev. D , 104001 (1999) [arXiv:hep-th/9903238];R. B. Mann, “Misner string entropy,” Phys. Rev. D , 104047 (1999)[arXiv:hep-th/9903229].[22] R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D , 3427 (1993)[arXiv:gr-qc/9307038];V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynamicalblack hole entropy,” Phys. Rev. D , 846 (1994) [arXiv:gr-qc/9403028];T. Jacobson, G. Kang and R. C. Myers, “On Black Hole Entropy,” Phys. Rev. D , 6587(1994) [arXiv:gr-qc/9312023].[23] T. Eguchi, P. B. Gilkey and A. J. Hanson, “Gravitation, Gauge Theories And DifferentialGeometry,” Phys. Rept. , 213 (1980). – 32 –
24] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, “Large N field theories,string theory and gravity,” Phys. Rept. , 183 (2000) [arXiv:hep-th/9905111].[25] S. W. Hawking and D. N. Page, “Thermodynamics Of Black Holes In Anti-De Sitter Space,”Commun. Math. Phys. , 577 (1983);E. Witten, “Anti-de Sitter space, thermal phase transition, and confinement in gaugetheories,” Adv. Theor. Math. Phys. , 505 (1998) [arXiv:hep-th/9803131].[26] R. Emparan, “AdS/CFT duals of topological black holes and the entropy of zero-energystates,” JHEP , 036 (1999) [arXiv:hep-th/9906040].[27] M. H. Dehghani and R. B. Mann, “Thermodynamics of rotating charged black branes inthird order Lovelock gravity and the counterterm method,” Phys. Rev. D , 104003 (2006)[arXiv:hep-th/0602243];M. H. Dehghani and R. B. Mann, “Thermodynamics of rotating charged black branes inthird order Lovelock gravity and the counterterm method,” Phys. Rev. D , 104003 (2006)[arXiv:hep-th/0602243];M. H. Dehghani and R. Pourhasan, “Thermodynamic instability of black holes of third orderLovelock gravity,” Phys. Rev. D , 064015 (2009) [arXiv:0903.4260 [gr-qc]];M. H. Dehghani and M. Shamirzaie, “Thermodynamics of asymptotic flat charged black holesin third order Lovelock gravity,” Phys. Rev. D , 124015 (2005) [arXiv:hep-th/0506227].S. H. Hendi and M. H. Dehghani, “Taub-NUT Black Holes in Third order Lovelock Gravity,”Phys. Lett. B , 116 (2008) [arXiv:0802.1813 [hep-th]].[28] V. Iyer and R. M. Wald, “A Comparison of Noether charge and Euclidean methods forcomputing the entropy of stationary black holes,” Phys. Rev. D , 4430 (1995)[arXiv:gr-qc/9503052].[29] Y. D´ecanini and A. Folacci “Irreducible Forms for the Metric Variations of the Action Termsof Sixth-Order Gravity and Approximated Stress-Energy Tensor,” Class. Quant. Grav. (2007) 4777 [arXiv:hep-th/0706.0691][30] For example, see:H. Liu and A. A. Tseytlin, “D = 4 super Yang-Mills, D = 5 gauged supergravity, and D = 4conformal supergravity,” Nucl. Phys. B , 88 (1998) [arXiv:hep-th/9804083];G. Arutyunov and S. Frolov, “Three-point Green function of the stress-energy tensor in theAdS/CFT correspondence,” Phys. Rev. D , 026004 (1999) [arXiv:hep-th/9901121].[31] H. Osborn and A. C. Petkou, “Implications of Conformal Invariance in Field Theories forGeneral Dimensions,” Annals Phys. , 311 (1994) [arXiv:hep-th/9307010];J. Erdmenger and H. Osborn, “Conserved currents and the energy-momentum tensor inconformally invariant theories for general dimensions,” Nucl. Phys. B , 431 (1997)[arXiv:hep-th/9605009].[32] D. M. Hofman and J. Maldacena, “Conformal collider physics: Energy and chargecorrelations,” JHEP , 012 (2008) [arXiv:0803.1467 [hep-th]].[33] R. C. Myers and A. Sinha, “Seeing a c-theorem with holography,” arXiv:1006.1263 [hep-th]., 012 (2008) [arXiv:0803.1467 [hep-th]].[33] R. C. Myers and A. Sinha, “Seeing a c-theorem with holography,” arXiv:1006.1263 [hep-th].