arXiv:0802.3318v2 [hep-th] 22 May 2008 o h ulo yeIBsrn hoyon theory string IIB type of dual the for bu 0 agrta 1 seas 1].Furthermore, is [17]). and also [16] (see order recently (1) leading the than lattice larger the 30% on about has calculated temperature been deconfinement also the above slightly QCD nteuiaiylmt(e ..[15]). e.g. gases atomic (see cold limit certain unitarity and 14] the at 13, created in 12, plasma 11, 10, quark-gluon [9, the RHIC include closest bound coming systems the liq- The to and bound. water bound the including satisfy lower substances helium known uid materi- universal all all far, a So for is als. bound] (1) (KSS) Kovtun-Starinets-Son that [the [8] in conjectured etr fteKSbudi hti is it that is bound KSS the of feature 24] 23, bound. 22, the 21, of [20, See discussions other 19]. [18, for bound the satisfy to found fclr and colors of mn hmi h nvraiyo h ai fteshear the of theories. ratio the gauge of universality coupled viscosity insights the strongly is striking them of yielded Among dynamics has 4] the 3, into 2, [1, correspondence oneeapet h S on,se[7.I particu- In [27]. see plausible bound, a KSS including the work, to of related counterexample closely For modification gravity [26]. holographic dual the the in terms derivative explored higher generic have we gravity. Einstein the to of possible corrections correctness on order constraint the higher important be- hand, an of other impose danger would the immediate bound On in gravity, is Ein- violated. of bound to ing theory the corrections quantum that any expect appears in it do occur we to of gravity that half stein Given bound the small time. violate generic the gravity order Einstein linearized to the corrections at Thus gravity. stein o l ag hoiswt nEnti rvt ulin dual gravity Einstein an with limit theories the gauge all for rmtepito iwo d/F,a interesting an AdS/CFT, of view of point the From h nid itr(d)cnomlfil hoy(CFT) theory field (AdS)/conformal Sitter anti-de The oiae ytevsns ftesrn adcp [25], landscape string the of vastness the by Motivated η N eateto hsc n srnm,Uiest fWaterlo of University Astronomy, and Physics of Department oteetoydensity entropy the to ∞ → oe,ta tuning that model, on nvsoiyaecreae,spotn h dao p a of idea the theories. which supporting consistent in correlated, all example are for concrete viscosity a on is This bound inconsistent. theory the rvt ul h ha icst oetoydniyratio, bound, density entropy viscosity to Starinets-Son viscosity shear the dual, gravity 1 λ nrcn okw hwdta,fracaso ofra edth field conformal of class a for that, showed we work recent In etrfrTertclPyis ascuet nttt o Institute Massachusetts Physics, Theoretical for Center ar Brigante, Mauro ste’ of opig twsfurther was It coupling. Hooft ’t the is α ′ and 2 eiee nttt o hoeia hsc,Wtro,On Waterloo, Physics, Theoretical for Institute Perimeter orcinto correction λ 3 η s eateto hsc,Safr nvriy tnod A9 CA Stanford, University, Stanford Physics, of Department ∞ → = 4 1 π η/s icst on n aslt Violation Causality and Bound Viscosity where , 1 η/s ogLiu, Hong s eo (16 below 5 ,7 8] 7, 6, [5, a encalculated been has η/s saturated N AdS o uegluon pure for η/s stenumber the is / η/s 25)(1 1 5 oetC Myers, C. Robert ≥ × Dtd eray 2008) February, (Dated: 1 yEin- by / u to due S / 4 5 4 π π nue ircuaiyvoaini h F,rendering CFT, the in violation microcausality induces ) and nti ae eage ntecneto h same the of context the in argue, we paper this In . (1) ontgaiyda,dsrbdb h lsia cinof = Λ action (below Gauss- classical [28] the with form by CFTs the described (3+1)-dimensional dual, of gravity class Bonnet a for lar, hspoie oceeeapei hc oe bound lower a which in on example concrete a provides This ut eie n[6,t hc erfrteraesfor readers [31] as the refer we which re- references. technical to and few [26], details a in on Our derived heavily end. sults rely the mysterious, will at is below possibilities obvious bound discussion two KSS discuss the we from and difference 36% The em[9 ssuppressed) is [29] term as-ontgaiy ossec fteter requires theory the (4+1)-dimensional of consistency of gravity, duals Gauss-Bonnet CFT Thus, (3+1)-dimensional inconsistent. is for and microcausality violates theory eepaieta hsrsl snnetraiein nonperturbative is result this that emphasize We ha icst ost eo[30]. zero to goes viscosity shear efudta [26] that found we where o utalnal orce au.Fo 3 h KSS the (3) From for value. violated corrected is linearly bound a just not h ttcbakbaeslto o 2 a ewritten be can (2) for solution brane black static The nti ae,w ilageta when that argue will we paper, this In f ds η/s 2 ( η/s I r 2 tpe Shenker, Stephen = ) = n h ossec fteter r correlated. are theory the of consistency the and ol ilt h ojcue Kovtun- conjectured the violate could , ,Wtro,OtroNL31 Canada 3G1, N2L Ontario Waterloo, o, ehooy abig,M 02139 MA Cambridge, Technology, f + = nossec fater n lower a and theory a of inconsistency − sil nvra oe on on bound lower universal ossible L r λ f 16 2 2 GB 2 ( ois(F)wt Gauss-Bonnet with (CFT) eories r 2 πG ai 2 Y,Canada, 2Y5, N2L tario ) λ 1 N L 1 GB N 2 ♯ 2 ( dt η s R Z   2 2 η s = 1 + d 35 USA 4305, − − λ ≥ 3 5 4 GB x 1 f 4 π s n h Yaida Sho and − 16 25 R ( √ 1 [1 r 1 L µν UIP0/2 MIT-CTP-3929 SU-ITP-08/02, − ) 6 >  2 − − dr g R n h Gibbons-Hawking the and 4 4 n as and 0 1 [ 4 2 µν π R λ λ +  GB GB − + . L r 2Λ R ] 2 2  . µνρσ

1 3 − X η/s i λ λ =1 3 GB GB R r r + dx 4 µνρσ 4  > → i 2   ! )] 100 9 , . 4 1 , λ the , the , GB (4) (3) (2) (5) (6) , 2 2 2

In (5), N♯ is an arbitrary constant which specifies the c2HzL UHyL speed of light of the boundary theory. We will take it to g 1 1 2 1 be N♯ 2 1+ √1 4λGB to set the boundary speed of light≡ to unity. The− horizon is located at r = r and + -1 -0.5 y
r+ 2 zturn 4 6 8 turn the Hawking temperature is T = N♯ πL2 . Such a solution z y 3,1 describes the boundary theory on R at a temperature 2 2 FIG. 1: Left: cg (z) as a function of z for λGB = 0.245. cg has T . 2 a maximum cg,max at zmax. As λGB is increased from λGB = 9 1 2 2 The shear viscosity can be computed by studying small 100 to λGB = 4 , cg,max increases from 1 to 3. cg (z) also 1 metric fluctuations φ = h 2 around the black brane back- serves as the classical potential for the 1D system (17). The ground (5). We will take φ to be independent of x1 and horizontal line indicates the trajectory of a classical particle. Right: U(y) [defined in (25)] as a function of y for λGB = x2 and write 0.245. dωdq − φ(t, ~x, r)= φ(r; ω, q) e iωt+iqx3 . (7) 2 2 K ′′ (2π) Here, Ω = z(1 λGB f˜ ) and Z f˜ − At quadratic level, the effective action for φ(r; ω, q) can N 2f˜ ˜′′ be found from (2) as (up to surface terms) 2 ♯ 1 λGBf cg(z)= 2 − ˜′ (14) z λGB f 1 z 1 dωdq 2 2 − S = C dz K ∂zφ K2 φ (8) −2 (2π)2 | | − | | can be interpreted as the local “speed of graviton” on a Z
constant r-hypersurface. As pointed out in [26], an im- where C is a constant, and portant feature of (14) is that cg can be greater than 1 9 2 ′ when λGB > (see Fig. 1). Note that cg has a maxi- K = z f˜(z λGBf˜ ), 100 2− mum cg,max > 1 somewhere outside the horizon and ap- ω˜ 2 ′′ K2 = K q˜ z 1 λGB f˜ . (9) proaches 1 near the boundary at infinity. We will restrict 2 ˜2 N♯ f − − 9   our attention to λGB > 100 for the rest of the paper. From standard geometrical optics arguments [32], in Primes above denote derivatives with respect to z and the large momentum limit, a localized wave packet of a we have introduced the following notation graviton should follow a null geodesic xµ(s) in the effec- r L2 L2 tive graviton geometry (13). More explicitly, write the iΘ(t,r,x3) z = , ω˜ = ω, q˜ = q, (10) wave function (7) in the form φ = e φen(t, r, x3) r+ r+ r+ where Θ is a rapidly varying phase and φen denotes a 2 2 ˜ L z 4λGB slowly varying envelope function. Inserting into (12), we f = 2 f = 1 1 4λGB + 4 (11). r+ 2λGB − r − z ! find at leading order µ ν From (8) one can derive the retarded two-point function dx dx eff gµν =0, (15) for the stress tensor component T12 in the boundary CFT ds ds and read off the shear viscosity (3) in the small q and µ µν µν with the identification dx g k g Θ. Given ω limit. In [26] it was also observed that for λ > ds eff ν eff ν GB translational symmetries in≡ the t and≡x directions,∇ we 9 and sufficiently large q, the equation of motion for φ 3 100 can interpret ω and q as conserved integrals of motion admits solutions which can be interpreted as metastable along the geodesic, quasiparticles of the boundary CFT. We will now argue that these quasiparticles can travel faster than the speed dt 2 dx3 2 1 of light and thus violate causality. We will first display ω = fN♯ , q = fN♯ 2 . (16) ds ds cg this phenomenon using graviton geodesics.     In a gravity theory with higher derivative terms, gravi- Assuming q = 0 and rescaling the affine parameter as 6 ton wave packets in general do not propagate on the light s˜ = qs/N♯, we get from (15) and (16) cone of a given background geometry. The equation of 2 motion following from (8) can be written as dr 2 2 ω = α cg, α . (17) µν ds˜ − ≡ q g˜ ˜ µ ˜ ν φ = 0 (12)   eff ∇ ∇ This describes a one-dimensional particle of energy α2 where ˜ is a covariant derivative with respect to the 2 µ moving in a potential given by cg. As is clear from Fig. 1, ∇ eff 2 eff effective geometryg ˜µν =Ω gµν given by geodesics starting from the boundary can bounce back to the boundary, with a turning point rturn(α) given by 1 1 geff dxµdxν = f(r)N 2 dt2 + dx2 + dr2 . (13) µν ♯ − c2 3 f(r) α2 = c2(r ) . (18)  g  g turn 3 400 000

9 300 000 In contrast, for λGB 100 , cg(z) is a monotonically in- ≤ VHzL creasing function of z and there is no bouncing geodesic. 200 000

For a null bouncing geodesic starting and ending at the 100 000 boundary, we then have 0 0 50 100 150 ∞ ∞ t˙ 2 α z ∆t(α)=2 dr = dr, FIG. 2: V (z) as a function of z for λGB = 0.2499 andq ˜ = 500. r˙ N 2 2 rturn(α) ♯ rturn(α) f α c Z Z − g q (19) Now consider the limitq ˜ . Since V1 is independent ∞ ∞ 2 → ∞ x˙3 2 cg ofq ˜, the dominant contribution to the potential is given ∆x3(α)=2 dr = dr, 2 2 > 1 r (α) r˙ N♯ r (α) 2 2 byq ˜ cg(z) except for a tiny region y q˜. Thus in Z turn Z turn f α cg ∼ − − this limit, we can simply replace V1(y) by V1(y)=0 for q (20) all y < 0 and V1(0) = + . Equation (21) can then be where dots indicate derivatives with respect tos ˜. written as ∞ In the boundary CFT we have local operators which 1 create bulk disturbances at infinity that propagate on ~2∂2ψ + U(y)ψ = α2ψ, ~ 0 (24) graviton geodesics sufficiently deep inside the bulk (r < − y ≡ q˜ → ω) [33]. In particular, we expect microcausality violation∼ in the boundary CFT if there exists a bouncing graviton where α was introduced in (17) and (see Fig. 1) geodesic with ∆x3(α) > 1 [34]. Now, as r r ∆t(α) turn max 2 → cg(y) y < 0 (α cg,max), a geodesic hovers near rmax for a long U(y)= . (25) → + y =0 time, propagating with a speed cg,max in x3-direction. ( ∞ Indeed, the integrals in (19) and (20) are dominated by In the ~ 0 limit, we can apply the WKB approxi- contributions near rmax. In such a limit, the ratio of the → mation. The leading WKB wave function eiΘ(t,r,x3) is integrand in ∆x3(α) to that in ∆t(α) near rmax is cg,max. ∆x3(α) just the rapidly varying phase of the geometric optics Thus, cg, > 1, violating causality. ∆t(α) → max approximation. The real part of α2 satisfies the Bohr- We will now show explicitly that the superluminal Sommerfeld quantization condition (with n some integer) graviton propagation described above corresponds to su- perluminal propagation of metastable quasiparticles [35] 0 1 ∆x3 q˜ dy α2 c2(y) = (n )π (26) in the boundary CFT with ∆t identified as the group ve- g y − − 4 locity of the quasiparticles. For this purpose, we rewrite Z turn q the full wave equation (12) in a Schr¨odinger form The above equation determines ω as a function of q for ∂2ψ + V (y)ψ =ω ˜2ψ (21) each given n. Taking the derivative with respect to q on − y both sides of (26), we find that the group velocity of the with ψ and y defined by quasiparticles is given by

dy 1 K dω ∆x3(α) = , ψ = Bφ, B = , (22) vg = = (27) ˜ ˜ dq ∆t(α) dz N♯f(z) s f and where ∆t(α) and ∆x3(α) are given by (19) and (20) re- spectively. Thus as argued in the paragraph below (20), 2 ˜2 ′ N♯ f ′′ f˜ ′ 2 2 vg approaches cg,max > 1 as α cg,max, violating causal- V (y)=˜q cg(z)+ V1, V1(y)= B + B . → B f˜ ! ity. In this limit the WKB wave function is strongly (23) peaked near rmax, reflecting the long time the geodesic In the above primes denote derivatives with respect to spends there. One can also estimate the imaginary part z. Note that y(z) is a monotonically increasing function of α2 (or ω), which has the form e−h(α)˜q with h(α) given of z with y 0 as z (boundary) and y by the standard WKB formula. Thus in theq ˜ → ∞ as z 1 (horizon).→ →c2(z ∞) is given by (14). →V −∞is a limit the quasiparticles become stable. Presumably local → g 1 monotonically increasing function of y (for λGB > 0) boundary operators that couple primarily to the long- −2 with V1(y = ) = 0 and V1 y as y 0. lived quasiparticles can be constructed by following [33]. 2 −∞ ∼ → Since cg is monotonically decreasing for r > rmax, To summarize, we have argued that signals in the for large enoughq ˜, V (y) develops a well and admits boundary theory propagate outside the light cone. In metastable states (see Fig. 2). The wave functions of such a boosted frame disturbances will propagate backward metastable states are normalizable at the AdS boundary in time. Since the boundary theory is nongravitational, and have an in-falling tail at the horizon, corresponding these are unambiguous signals of causality violation and to quasiparticles in the boundary CFT [35]. hence inconsistency. 4

Here we observe causality violation in the high momen- [6] P. Kovtun, D. T. Son and A. O. Starinets, JHEP 0310, tum limit. This is in agreement with the expectation that 064 (2003) [arXiv:hep-th/0309213]. causality should be tied to the local, short-distance be- [7] A. Buchel and J. T. Liu, Phys. Rev. Lett. 93, 090602 havior of the theory. Also, a sharp transition from causal (2004) [arXiv:hep-th/0311175]. [8] P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. to acausal behavior as a function of λGB is possible be- Lett. 94, 111601 (2005) [arXiv:hep-th/0405231]. cause of the limiting procedureq ˜ needed in our 68 → ∞ [9] D. Teaney, Phys. Rev. C , 034913 (2003) argument. A more rigorous derivation of these phenom- [arXiv:nucl-th/0301099]. ena using the full spectral function obtained from the [10] P. Romatschke and U. Romatschke, Phys. Rev. Lett. 99, Schr¨odinger operator would be desirable. 172301 (2007) [arXiv:0706.1522 [nucl-th]]. 658 We argued that, for a (4+1)-dimensional Gauss- [11] H. Song and U. W. Heinz, Phys. Lett. B , 279 (2008) 9 [arXiv:0709.0742 [nucl-th]]. Bonnet gravity, causality requires λGB . Thus, ≤ 100 [12] A. Adare et al. [PHENIX Collaboration], Phys. Rev. consistency of this theory requires, Lett. 98, 172301 (2007) [arXiv:nucl-ex/0611018]. [13] P. Romatschke, arXiv:0710.0016 [nucl-th]. η 16 1 . (28) [14] K. Dusling and D. Teaney, Phys. Rev. C 77, 034905 s ≥ 25 4π (2008) [arXiv:0710.5932 [nucl-th]].   [15] T. Schafer, Prog. Theor. Phys. Suppl. 168, 303 (2007) This still leaves rooms for a violation of the KSS bound. [arXiv:hep-ph/0703141]. We see two possibilities. [16] H. B. Meyer, Phys. Rev. D 76, 101701 (2007) First, it could be that Gauss-Bonnet theory with [arXiv:0704.1801 [hep-lat]]. 9 [17] S. Sakai and A. Nakamura, arXiv:0710.3625 [hep-lat]. λGB is consistent and appears as a classical limit of ≤ 100 [18] P. Benincasa and A. Buchel, JHEP 0601, 103 (2006) a consistent theory of quantum gravity, somewhere in the [arXiv:hep-th/0510041]. string landscape (see [27] for a plausible counterexample [19] A. Buchel, J. T. Liu and A. O. Starinets, Nucl. Phys. B to the KSS bound). Maybe this is how nature works and 707, 56 (2005) [arXiv:hep-th/0406264]. the KSS bound can be violated, at least by 36%. [20] T. D. Cohen, Phys. Rev. Lett. 99, 021602 (2007) Alternatively, it could be that there is a more subtle [arXiv:hep-th/0702136]. 0802 inconsistency in the theory within the window of 0 < [21] A. Cherman, T. D. Cohen and P. M. Hohler, JHEP , 9 026 (2008) [arXiv:0708.4201 [hep-th]]. λGB . These issues deserve further investigation. ≤ 100 [22] J. W. Chen, M. Huang, Y. H. Li, E. Nakano and We thank D. T. Son, A. Starinets, and L. Susskind D. L. Yang, arXiv:0709.3434 [hep-ph]. for discussions. MB and HL are partly supported by the [23] D. T. Son, Phys. Rev. Lett. 100, 029101 (2008) U.S. Department of Energy (D.O.E) under cooperative [arXiv:0709.4651 [hep-th]]. research agreement #DE-FG02-05ER41360. HL is also [24] I. Fouxon, G. Betschart and J. D. Bekenstein, Phys. Rev. supported in part by the A.P. Sloan Foundation and the D 77, 024016 (2008) [arXiv:0710.1429 [gr-qc]]. U.S. Department of Energy (DOE) OJI program. Re- [25] For a review, see M. R. Douglas and S. Kachru, Rev. 79 search at Perimeter Institute is supported by the Gov- Mod. Phys. , 733 (2007) [arXiv:hep-th/0610102]. [26] M. Brigante, H. Liu, R. C. Myers, S. Shenker and ernment of Canada through Industry Canada and by the S. Yaida, arXiv:0712.0805 [hep-th] [Phys. Rev. D (to be Province of Ontario through the Ministry of Research published)]. & Innovation. RCM also acknowledges support from an [27] Yevgeny Kats and Pavel Petrov, arXiv:0712.0743 [hep- NSERC Discovery grant and from the Canadian Insti- th]. tute for Advanced Research. SS is supported by NSF [28] B. Zwiebach, Phys. Lett. B 156, 315 (1985). [29] R. C. Myers, Phys. Rev. D 36, 392 (1987). grant 9870115 and the Stanford Institute for Theoretical 1 Physics. SY is supported by an Albion Walter Hewlett [30] As discussed in [26], λGB is bounded above by 4 for the theory to have a boundary CFT, and η/s never decreases Stanford Graduate Fellowship and the Stanford Institute below 0. for Theoretical Physics. [31] R. G. Cai, Phys. Rev. D 65, 084014 (2002) [arXiv:hep-th/0109133]. [32] C. W. Misner, K. S. Thorne, and J. A. Wheeler, San Francisco 1973, p. 1279. [33] J. Polchinski, arXiv:hep-th/9901076. [1] J. M. Maldacena, Adv. Theor. Math. Phys. 2, [34] To be precise this only indicates the presence of a pole 231 (1998) [Int. J. Theor. 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[36] D. Birmingham, I. Sachs and S. N. Solodukhin, Phys. [arXiv:hep-th/0205051]. Rev. Lett. 88, 151301 (2002) [arXiv:hep-th/0112055]; D. T. Son and A. O. Starinets, JHEP 0209, 042 (2002)