aa r X i v : . [ h e p - t h ] N ov CQUeST-2009-0265
A Note on Black Holes in AsymptoticallyLifshitz Spacetime
Da-Wei Pang † † Center for Quantum Spacetime, Sogang UniversitySeoul 121-742, Korea pangdw‘at’sogang.ac.kr
Abstract
We investigate several aspects of exact black hole solutions in asymptotically Lif-shitz spacetime, which were proposed in 0812.0530. Firstly, we calculate the tidalforces and find that in the near horizon region of such black hole backgrounds, thetidal forces diverge in the near extremal limit. Secondly, we evaluate the Wilsonloops in both extremal and finite temperature cases. Finally, we obtain the cor-responding shear viscosity and square of the sound speed and find that the ratioof shear viscosity to entropy density takes the universal value 1 / π in arbitrarydimensions while the square of the speed of sound saturates the conjectured bound1 / ontents By now, the AdS/CFT correspondence [1, 2, 3] is the unique approach which relatesstrongly coupled field theories to weakly coupled gravity theory. It has been extensivelyinvestigated in the past decade and its validity has been widely recognized in the the-oretical high-energy physics community. Recently there has been enormous progress onthe application of AdS/CFT correspondence, or even the more general gauge/gravity cor-respondence to physical systems in the real world, such as AdS/QCD and holographicmethods for condensed matter physics. Two nice reviews are given by [4, 5].It is well known that certain questions which are difficult to deal with in the field theoryside, become more transparent and more tractable in the gravity side via the AdS/CFTcorrespondence. In condensed matter physics there are many strongly coupled systems,so it is widely hoped that the AdS/CFT correspondence can provide some useful tools forstudying condensed matter physics. Recently interesting gravity models dual to variouscondensed matter systems have been proposed [6]-[11].Special attention has been paid to gravity duals of Lifshitz-like fixed points, which is1nitially proposed in [12]. One can find critical phenomena with unconventional scalingbehavior in many condensed matter systems t → λ z t, x → λ x , (1.1)where z = 1. A toy model realizing this scaling behavior with z = 2 is the so-calledLifshitz field theory, L = Z d xdt (( ∂ t φ ) − κ ( ∇ φ ) ) . (1.2)The corresponding gravity dual takes the following form [12] ds = L ( − r z dt + dr r + r d x ) , (1.3)where d x = dx + · · · dx d . This metric exhibits the following scale invariance t → λ z t, r → rλ , x → λ x . (1.4)Note that when z = 1, it turns out to be the usual AdS d +2 spacetime. In four-dimensionalspacetime, the corresponding action is a gravity theory with negative cosmological con-stant, coupled with abelian gauge fields A (1) , B (2) S = Z d x √− g ( R − − Z ( ∗ F (2) ∧ F (2) + ∗ H (3) ∧ H (3) ) − c Z B (2) ∧ F (2) , (1.5)where F (2) = dA (1) , H (3) = dB (2) and the cosmological constant Λ = − /L .There are also several generalizations along a similar way. Various anisotropic gravitysolutions in general spacetime dimensions with different scaling behavior were discussedin [13]. The aspects of holography in general anisotropic, non-relativistic backgroundswere investigated extensively in [14]. The geometry of Lifshitz spacetime was studiedin [15]. Furthermore, the embedding of such anisotropic gravity background with some-what different scaling behavior into string theory was realized quite recently in [16], wherethe corresponding black brane configurations were also obtained.However, although we can study the zero-temperature Lifshitz spacetime, it is difficult toobtain exact black hole solutions in the context of the original action (1.5). The black holein asymptotically Lifshitz spacetime was constructed in [17] using numerical methods,while Lifshitz topological black holes were obtained in [18]. It should be pointed outthat an exact solution of topological Lifshitz black holes was obtained in [18] for certainparticular case. 2hen discussing the aspects of holography in more general anisotropic, non-relativisticbackgrounds in [14], an exact solution with finite temperature was obtained by making useof a different action. It can be seen that by performing some coordinate transformation,the black hole solution is asymptotically Lifshitz-like. In this note we discuss severalaspects of this exact solution. In section 2 we rewrite the black hole solution in a moretransparent way. Then in section 3 we calculate the tidal forces and it turns out that suchtidal forces become divergent in the near horizon region, while the horizon area remainslarge. In this sense, this type of Lifshitz black hole is “naked” [19]. In the next sectionwe evaluate the Wilson loops in this asymptotically Lifshitz black hole background, theresults agree with previous examples in the extremal limit and the finite temperaturecases are calculated numerically. We discuss the hydrodynamic properties in section 5.It can be shown that the ratio of shear viscosity to entropy density is 1 / π in arbitrarydimensions and the square of the speed of sound is 1 / In this section we review the asymptotic Lifshitz solutions proposed in [14], includingthe extremal solution and black hole solution. We will rewrite the solutions in a moretransparent way.Consider the following action in ( d + 2)-dimensional spacetime S = 116 πG d +2 Z d d +2 x √− g [ R − − ∂ µ φ∂ µ φ − e λφ F µν F µν ] , (2.1)where Λ is the cosmological constant and the matter fields are a massless scalar and anabelian gauge field. The equations of motion can be written as follows: ∂ µ ( √− ge λφ F µν ) = 0 , (2.2) ∂ µ ( √− g∂ µ φ ) − λ √− ge λφ F µν F µν = 0 , (2.3) R µν = 2 d Λ g µν + 12 ∂ µ φ∂ ν φ + 12 e λφ F µρ F νρ − d g µν e λφ F µν F µν . (2.4)3e make the following ansatz for the metric ds = L [ − r z f ( r ) dt + dr r f ( r ) + r d X i =1 dx i ] , (2.5)where z ≥ F rt .We can obtain the following expression for F rt by solving (2.2) F rt = qe − λφ r z − d − , (2.6)where q is a constant which can be related to the charge of the black hole. Furthermore,solving the tt and rr components of (2.4)we can arrive at ∂ r φ∂ r φ = 2( z − dr . (2.7)When z = 1, it can be easily seen that the solution is φ = φ = const. The full solutioncan be obtained by solving the remaining equations of motion ds = L [ − r dt + dr r + r d X i =1 dx i ] ,φ = const , F rt = 0 , Λ = − d ( d + 1)2 L . (2.8)It is simply the AdS solution in Poincar´e coordinates. It also admits black hole solutionwith f ( r ) = 1 − r d +1+ /r d +1 and other fields remaining the same as the AdS solution.When z = 1, from (2.7) we can obtain φ = ± p z − d log r, (2.9)where we have taken the integration constant to be zero without loss of generality. Simi-larly, we can summarize the extremal solution as follows ds = L ( − r z dt + dr r + r d X i =1 dx i ) ,F rt = qe − λφ r z − d − , e λφ = r λ √ z − d ,λ = 2 dz − , q = 2 L ( z − z + d ) , Λ = − ( z + d − z + d )2 L . (2.10)4t is just the Lifshitz spacetime with non-trivial dilaton and gauge fields. It should bepointed out that the finite temperature generalization ds = L ( − r z f ( r ) dt + dr r f ( r ) + r d X i =1 dx i ) , f ( r ) = 1 − r z + d + r z + d , (2.11)is also a solution to the equations of motion with the same field configuration. Thus thefinite temperature solution is an asymptotically Lifshitz black hole.Now let us focus on the asymptotically Lifshitz black hole solution. The temperature is T H = ( z + d ) r z + π , (2.12)and the black hole entropy is S BH = V d G d +2 L d r d + , (2.13)where V d denotes the volume of the d dimensional spatial coordinates. One can rewritethe entropy as a function of temperature S BH = V d L d G d +2 ( 4 πz + d ) dz T dz , (2.14)which exhibits the expected behavior of an anisotropic scale invariant theory.The thermodynamic quantities can be obtained via the Euclidean path integral method,which were calculated explicitly in [14]. Here we shall not dwell on the details but onlylist some useful results. Consider the following Euclidean action I E = − πG d +2 Z d d +2 x √− g [ R − − ∂ µ φ∂ µ φ − e λφ F µν F µν ] − πG d +2 Z d d +1 x √ hK, (2.15)where the second term is the Gibbons-Hawking boundary term. After substituting thebackground configuration, the Euclidean action turns out to be I E = − r z + d + L d V d β H πG d +2 , (2.16)where β H = 1 /T H . We can calculate the other thermodynamic quantities in a standardway as soon as we obtain the Euclidean action. For example, the mass of the black holeis M = r z + d + dL d V d πG d +2 , (2.17)5nd the charge of the black hole is given by Q = 132 πG d +2 Z e λφ ( ∗ F ) = qL d V d πG d +2 . (2.18)From (2.16) we can see that there is no interesting phase structure for this asymptoticallyLifshitz black hole, as the Euclidean action is always negative. This can also be seen fromthe heat capacity C = dMdT = ∂M/∂r + ∂T /∂r + . (2.19)Using (2.12) and (2.17), we can obtain C = dV d r d + zG d +2 , (2.20)which shows that the black hole is always thermodynamically stable. In this section we will calculate the tidal forces of the Lifshitz black hole, following [19].It has been shown that there exist a class of black holes whose horizon area is large andall curvature invariants are small near the horizon, while any object falling in experienceslarge tidal forces outside the horizon. As the region of large tidal forces is visible to distantobservers, such black holes are called “naked”.Recall the metric ds = L ( − r z f ( r ) dt + dr r f ( r ) + r d X i =1 dx i ) , f ( r ) = 1 − r z + d + r z + d , (3.1)and the vielbein in the static frame is given as( e ) µ = − Lr z f ( r ) / ∂ µ t, ( e ) µ = Lr − f ( r ) − / ∂ µ r, ( e i ) µ = Lr∂ µ x i . (3.2)Consider timelike geodesics in the above background, with proper time τ and tangentvector u µ = dx µ /dτ . The constants of motion can be written as follows E = L r z f ( r ) ˙ t, p i = L r ˙ x i , (3.3)6here an overdot denotes d/dτ . For simplicity, we just consider radial geodesics, i.e. p i = 0. We can arrive at the following expression due to the normalization condition u µ u µ = − r = E L r z − − r L f ( r ) . (3.4)The parallel-propagated orthonormal frame ( e ′ ) µ = u µ can be obtained by a boost of theoriginal static frame ( e ′ ) µ = u µ = − E∂ µ t + ˙ rL r f ( r ) ∂ µ r ≡ cosh α ( e ) µ + sinh α ( e ) µ ( e ′ ) µ = sinh α ( e ) µ + cosh α ( e ) µ , (3.5)where cosh α = E [ L r z f ( r )] − / and the other components remain invariant. It can beseen that the boost parameter α diverges at the horizon.The components of the Riemann curvature in the boosted frame can be calculated byworking out the components in the static frame first and then performing some transfor-mations. However, there is another simple route to calculate such quantities which hasa more direct physical meaning [19]. We will calculate R ′ k ′ k which correspond to tidalforces in the transverse directions. Consider a class of radial infalling geodesics whosetangent vector is u µ and the deviation vectors are η i = ∂/∂x i , we have u ν ∇ ν η σ = u ν Γ σνρ η ρ = ˙ HH η σ , (3.6)where H = Lr . Thus the geodesic deviation equation gives R µνρσ u µ η ν u ρ = − u µ ∇ µ ( u ν ∇ ν η σ ) = − ¨ HH η σ . (3.7)Therefore R ′ i ′ i = R µνρσ u µ ( e i ) ν u ρ ( e i ) σ = − ¨ HH = ( z − E L r z + 1 L [1 + ( z + d − r z + d + r z + d ] . (3.8)The enhancement of the curvature in the geodesic frame leads to the term proportionalto E in the above expression. It can be seen that if we take the conserved quantity E to be very large, the tidal force can be made arbitrarily large. Conversely, we can also7ake the tidal force very small. Thus in order to avoid such ambiguities, we assumethat the conserved quantity E is chosen to be order one. It is sufficient to keep the termproportional to E only, as such term represents the difference between the static frameand the boosted frame. Then the tidal force in the near horizon region is given by R ′ i ′ i = ( z − E L r z + . (3.9)It can be easily seen that the tidal force vanishes in z = 1 case then we will consider twodifferent near-extremal limits with the assumption z > • r + << M fixed → L ∼ r − ( z + d ) /d . (3.10)Then the tidal force becomes R ′ i ′ i = ( z − E r z/d − z + . (3.11)The horizon area satisfies A ∝ L d r d + ∼ r − z + . (3.12)So r + << d = 2, the tidal force is( z − E r , that is, the tidal force also turns out to be very small. In this limitthe black holes are not “naked”. When d >
2, the requirement that the tidal forceis large gives 4 + 4 zd − z < → z > d d − d − . (3.13)It can be seen that the z = 2 case can never lead to “naked” black holes. • r + << A ∝ L d r d + >> → Lr + >> . (3.14)The tidal force turns out to be R ′ i ′ i ∝ Lr + ) r z − . (3.15)Thus it is possible to have “naked” black holes only in the z > Wilson loops
In this section we study Wilson loops for asymptotically Lifshitz black holes. The Wilsonloops describe the behavior of quarks by hanging strings from the boundary where thequarks locate at the ends of the strings. Although it is quite difficult to embed Lifshitzspacetime into string theory, the calculations presented here can provide some qualitativeinformation. Consider rectangular Wilson loops in Euclidean spacetime, the dynamics isdescribed by the Nambu-Goto action S = − Z dσ p det h ab , h ab = g µν ∂ a X µ ( τ, σ ) ∂ b X ν ( τ, σ ) , (4.1)where X µ ( τ, σ ) denote the string coordinates and τ, σ parametrize the string worldsheet.In the following we will focus on five-dimensional asymptotic Lifshitz black holes, whosemetric is ds = L [ − r z f ( r ) dt + dr r f ( r ) + r ( dx + dx + dx )] .f ( r ) = 1 − r z +3+ r z +3 . (4.2)Then we can obtain the equations of motion( r z +2 f ( r ) x ′ p r z − r ′ + r z +2 f ( r ) x ′ ) ′ = 0 , ( r z − r ′ p r z − r ′ + r z +2 f ( r ) x ′ ) ′ = 12 (2 z − r z − r ′ + ∂ r ( r z +2 f ( r )) x ′ p r z − r ′ + r z +2 f ( r ) x ′ , (4.3)where the prime stands for derivative with respect to σ .One possible static configuration is a pair of straight macroscopic strings which arestretched between r = ∞ and r = r + . The corresponding total energy is E = 2 L Z ∞ r + r z − dr. (4.4)The other possible configuration is a macroscopic U-shape string whose each end is con-nected to the quark and anti-quark at the boundary. In the static gauge σ = x , theequations of motion turn out to be( r z +2 f ( r ) p r z − r ′ + r z +2 f ( r ) ) ′ = 0 , ( r z − r ′ p r z − r ′ + r z +2 f ( r ) ) ′ = 12 (2 z − r z − r ′ + ∂ r ( r z +2 f ( r )) p r z − r ′ + r z +2 f ( r ) . (4.5)9e can arrive at the following result by extremizing the action f ( r ) r z +2 p r z − r ′ + f ( r ) r z +2 = const = f / r z +1min , (4.6)where r min is r coordinate of the string tip which is the closest to the horizon and f min ≡ f ( r ) | r = r min . Note that ∂r/∂x = 0 at r = r min . From the above expression we can rewrite x as a function of r x = Z rr min dr r f ( r ) / q ( rr min ) z +2 ( ff min ) − . (4.7)Thus the boundary distance between the endpoints of the string is given by ℓ = 2 Z ∞ r min dr r f ( r ) / q ( rr min ) z +2 ( ff min ) − . (4.8)The total energy of the U-shape string with inter-quark separation ℓ is E = L Z ℓ − ℓ dx p r z − r ′ + r z +2 f ( r )= 2 L Z dr r f ( r ) / p r f ( r ) − r f min . (4.9)Finally, the heavy quark potential is given by V = E − E , (4.10)where we have subtracted the contribution of two straight strings.In the extremal background, i.e. r + = 0, it can be seen that these results agree with thosegiven in [17] and these expressions reduce to those of [20] and [21] when z = 1. In the finitetemperature case [22], the integration can be worked out analytically by making use ofthe elliptic integral. Unfortunately, here we cannot obtain analytical results thus we haveto evaluate the integrals numerically. For simplicity we just consider the five-dimensionalcase with z = 2, that is, f = 1 − r /r . The expressions for the boundary distance ℓ andthe potential energy can be rewritten as follows ℓ = 2 r + √ a − a Z ∞ a √ y p ( y − y − a ) − ( y − a )] dy, (4.11)10 Figure 4.1: The boundary distance between the endpoints of a string ℓ as a function of a , with r + = 1.and V = 2 r L [ Z ∞ a dy ( y / p y − p ( y − a ) − ( y − a ) − y ) −
12 ( a − , (4.12)where we have introduced y ≡ r/r + and a ≡ r min /r + . Note that the parameter a shouldbe larger than 1, that is, the string always stays outside the horizon.The boundary distance between the endpoints of a string ℓ as a function of a is shownin Fig. 4.1, while the potential energy as a function of the boundary distance ℓ is shownin Fig. 4.2. Compared to the results in [22], it can be seen that these functions exhibitsimilar behavior. The boundary distance between the endpoints of a string has a maximumvalue ℓ max . For a fixed ℓ < ℓ max , there are two possible U-shape string configurations attwo different values of a . The energy of the U-shape string is plotted in Fig. 4.2. Theconfiguration with smaller a has a nearly zero potential energy and the configuration withlarger a has lower energy. The potential crosses zero at ℓ = ℓ ∗ . The pair of straight stringshas lower energy than the U-shape string configuration once ℓ > ℓ ∗ . In this section we discuss the hydrodynamic properties of such asymptotically Lifshitzblack holes, including the shear viscosity and the speed of sound. We will see that the11 .3 0.4 0.5 0.6 l - - - - - (cid:144) L Figure 4.2: The potential energy as a function of the boundary distance ℓ , with r + = 1.ratio of shear viscosity to entropy density is 1 / π in arbitrary dimensions, which saturatesthe well known KSS bound [24], while the square of the speed of sound is 1 /d . It shouldbe pointed out that in five-dimensional case, the square of the speed of sound is 1 / η . We will apply the Kubo formula η = − lim ω → ω Im G R ( ω, ~k = 0) , (5.1)where G R is the retarded two-point function of the scalar mode of the stress tensor G R ( ω, ~k = 0) = − i Z d d xdte iωt θ ( t ) < [ T xy ( t, ~x ) , T xy (0 , > . (5.2)Following the prescription proposed in [27], the linearized Einstein equation for φ ≡ h xy ( r ) e − iωt is the scalar wave function in the same background, due to the SO (2) symmetryof rotations in the xy − plane.Recalling the black hole solution (2.11), we make coordinate transformation u = r z + d + /r z + d for convenience. Then the black hole metric turns out to be ds = L [ − ( r z + d + u ) zz + d f ( u ) dt + 4( z + d ) u f ( u ) du + ( r z + d + u ) z + d d X i =1 dx i ] , f ( u ) = 1 − u . (5.3)Assuming Φ( ω, u ) = φ ( u ) e − iωt , the scalar wave equation ✷ Φ = 1 √− g ∂ µ ( √− gg uu ∂ u φ ) + g tt ∂ t ∂ t φ (5.4)12ives u f ( u ) ∂ u ( f ( u ) u ∂ u φ ) + u zz + d β ω φ = 0 , (5.5)where β − ≡ r z + ( z + d ). Following the standard procedure, we set φ k = (1 − u ) α andrequire that the most singular terms at u = 1 cancel as well as the incoming wave boundarycondition. These requirements finally fix α = − i βω. (5.6)Next, we choose φ k ( u ) = (1 − u ) − i βω (1 + i βωF ( u )), then substitute this expression backto the scalar wave function (5.5). Note that in order to calculate the shear viscosity, wejust need the perturbation up to the first order of ω . Furthermore, F ( u ) should be zeroat u = 1. It can be easily obtained that φ k ( u ) = (1 − u ) − i βω (1 − i βω ln 1 + u . (5.7)Finally combining the flux factor F = K √− gg uu φ − k ( u ) ∂ u φ k ( u ) , (5.8)where K = − πG d +2 is the coupling constant and the retarded green function G R ( ω, ~k = 0) = − F | u → , (5.9)as proposed in [28], we can obtain η = L d r d + πG d +2 . (5.10)Therefore we can arrive at the famous KSS bound ηs = 14 π . (5.11)For the speed of sound, we first note that the thermodynamic quantities of the black holeshould be identified with the quantities in the field theory side as { I E , M, S BH , T H } ↔ { Ω /T, E, S, T } (5.12)where Ω denotes the thermodynamic potential. Recall the results given in Section 2, I E = − r z + d + L d V d β H πG d +2 , T H = ( z + d ) r z + π , M = r z + d + dL d V d πG d +2 . (5.13)13hen by using the fact that the thermodynamic potential Ω isΩ = − P V d , (5.14)where P denotes the pressure, we can obtain P = 1 d EV d = 1 d ǫ, (5.15)where ǫ is the energy density. Thus the speed of sound is given by c s = ∂P∂ǫ = 1 d . (5.16)Note that in five dimensional spacetime, i.e. d = 3, we have c s = 1 /
3, which saturatesthe bound conjectured in [25], [26].
There has been enormous progress on applying the AdS/CFT correspondence, or themore general gauge/gravity correspondence to systems in condensed matter physics. Inthis note we discuss several aspects of the exact black hole solutions in asymptoticallyLifshitz spacetime. We firstly rewrite the solution proposed in [14] in a more convenientway. Then we show that the tidal forces in the near horizon region tend to be infinity inthe near-extremal limit, in which sense the black hole is “naked”. We also evaluate theWilson loops both analytically and numerically in the extremal and finite temperaturecases. Finally, we investigate the hydrodynamic properties of the black holes and findthat the shear viscosity and the speed of sound both saturate the conjectured bounds.There are several directions which worth further studying. Firstly, the embedding ofthe original Lifshitz background (1.1)into string theory is still unknown. However, someLifshitz backgrounds with different scaling behavior have been realized in string theoryin [16], where the configurations were comprised by D3-D7 and D4-D6 branes. It may beexpected that we can embed (1.1) into string theory by superposing more different typesof D-branes.Secondly, it has been observed in [29] that the Lifshitz fixed point has ultralocal correlatorsat finite temperature. Thus it would be interesting to calculation the correlation functionsin the black hole background, following the prescription in [28], and compare the results14ith those obtained in the field theory side. Furthermore, it is necessary to build upa systematic holographic renormalization method [30] in such asymptotically Lifshitzbackground.Finally, an interesting model of quantum gravity was proposed by Horava quite re-cently [31]. In 3 + 1 dimensions, this theory has a z = 3 fixed point in the UV and flowsto a z = 1 fixed point in the IR, which is just the classical Einstein-Hilbert gravity theory.Furthermore, it has been found that there exist black hole solutions in Horava-Lifshitzgravity [32]. Thus it is interesting to study the relations between the asymototicallyLifshitz black hole and black holes in Horava-Lifshitz gravity. Acknowledgements
We would like to thank Rong-Gen Cai, Li-Ming Cao, Qing-Guo Huang, Yunseok Seoand Wei-shui Xu for useful discussions and kind help. This work was supported by theKorea Science and Engineering Foundation(KOSEF) grant funded by the Korea govern-ment(MEST) through the Center for Quantum Spacetime(CQUeST) of Sogang Universitywith grant number R11-2005-021.
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