Extremal black holes in the Hořava-Lifshitz gravity
aa r X i v : . [ h e p - t h ] M a y Extremal black holes in the Hoˇrava-Lifshitz gravity
Hyung Won Lee, Yong-Wan Kim, and Yun Soo Myung Institute of Basic Science and School of Computer Aided ScienceInje University, Gimhae 621-749, Korea
AbstractWe study the near-horizon geometry of extremal black holes in the z = 3 Hoˇrava-Lifshitzgravity with a flow parameter λ . For λ > /
2, near-horizon geometry of extremal black holesare AdS × S with different radii, depending on the (modified) Hoˇrava-Lifshitz gravity. For1 / ≤ λ ≤ /
2, the radius v of S is negative, which means that the near-horizon geometryis ill-defined and the corresponding Bekenstein-Hawking entropy is zero. We show explicitlythat the entropy function approach does not work for obtaining the Bekenstein-Hawkingentropy of extremal black holes. e-mail address: [email protected] Introduction
Recently Hoˇrava has proposed a renormalizable theory of gravity at a Lifshitz point [1],which may be regarded as a UV complete candidate for general relativity. At short distancesthe theory of Hoˇrava-Lifshitz (HL) gravity describes interacting nonrelativistic gravitonsand is supposed to be power counting renormalizable in (1+3) dimensions. Recently, theHL gravity theory has been intensively investigated in [2], its cosmological applications in[3, 4], and its black hole solutions in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 17, 19, 20,21, 22, 23, 24, 25, 26, 27, 28, 29].There are two classes of Hoˇrava-Lifshitz gravity in the literature: with/without theprojectability condition where the former (latter) implies that the lapse function dependson time (time and space).Concerning the spherically symmetric solutions without the projectability condition,L¨u-Mei-Pope (LMP) have obtained the black hole solution with a flow parameter λ [5] andtopological black holes were found in [6]. Its thermodynamics were studied in [9, 11] butthere remain unclear issues in defining the ADM mass and entropy because its asymptotesare Lifshitz for 1 / ≤ λ < < z ≤ λ = 1 black hole solution in asymptotically flat spacetimes using the modified HLgravity without the projectability condition [10]. Its thermodynamics was defined in [13]but recently, the entropy was argued to take the Bekenstein-Hawking form [30, 31]. Parkhas obtained a λ = 1 black hole solution with two parameter ω and Λ W [18]. Within theprojectable theories, their black hole solutions are less interesting [32].Before proceeding, we wish to point out that the Hoˇrava black holes (LMP and KSblack holes) are completely different from the IR black holes (Schwarzschild-AdS andSchwarzschild black holes). The Hoˇrava black holes have their extremal black holes, whereasthe IR black holes do not have extremal black holes. In the Hoˇrava black holes, the charge-like quantities (pseudo charge) are related to the cosmological constant − W in the LMPblack holes and parameter ω in the KS black hole. This feature is very special, in comparedto the Reissner-Norstr¨om-AdS and Reissner-Norstr¨om black holes with the electric (mag-netic) charge “ Q ” which were obtained from the relativistic theories. We wish to mentionthat in the non-relativistic theories, the notion of horizon, temperature, and entropy wasnot well-defined [33]. Although most solutions have horizons, a part of solutions appears tobe horizonless for particles with ultra-luminal dispersion relations.In order that the thermodynamics of an extremal black hole is explored completely, thewhole spacetimes should be known, including the near-horizon geometry and asymptotic2tructures. However, for 1 / ≤ λ <
3, the LMP black holes have asymptotically Lifshitzwhich is not yet understood fully. In this sense, we call these Lifshitz black holes. Thus,it is still lack for understanding thermodynamics of the Lifshitz black holes because theconserved quantities are not defined unambiguously.In this work, we investigate the near-horizon geometry (AdS × S ) of the extremal blackholes to explore the unknown solution in asymptotically flat spacetimes and to understandthe Lifshitz black holes. For this purpose, we obtain AdS × S by exploring the KSsolution and the LMP solution. Then, we compare these with curvature radius v of AdS and curvature radius v of S obtained by solving full Einstein equations on the AdS × S background directly. For λ > / × S with different curvatures, depending on the (modified) Hoˇrava-Lifshitz gravity.For 1 / ≤ λ < / v of S is always negative and thus the correspondingnear-horizon geometry is ill-defined. Hence, this case will be ruled out as candidate for theextremal black holes in the Hoˇrava-Lifshitz gravity. Introducing the ADM formalism where the metric is parameterized [34] ds ADM = − N dt + g ij (cid:16) dx i − N i dt (cid:17)(cid:16) dx j − N j dt (cid:17) , (1)the Einstein-Hilbert action can be expressed as S EH = 116 πG Z d x √ gN h K ij K ij − K + R − i , (2)where G is Newton’s constant and extrinsic curvature K ij takes the form K ij = 12 N (cid:16) ˙ g ij − ∇ i N j − ∇ j N i (cid:17) . (3)Here, a dot denotes a derivative with respect to t . An action of the non-relativistic renor-malizable gravitational theory is given by [10] S HL = Z dtd x h L K + L V i , (4)where the kinetic term is given by L K = 2 κ √ gN K ij G ijkl K kl = 2 κ √ gN (cid:16) K ij K ij − λK (cid:17) , (5)with the DeWitt metric G ijkl = 12 (cid:16) g ik g jl − g il g jk (cid:17) − λg ij g kl (6)3nd its inverse metric G ijkl = 12 (cid:16) g ik g jl − g il g jk (cid:17) − λ λ − g ij g kl . (7)The potential term is determined by the detailed balance condition (DBC) as L V = − κ √ gN E ij G ijkl E kl = √ gN ( κ µ − λ ) (cid:16) − λ R + Λ W R − W (cid:17) − κ η C ij − µη R ij ! C ij − µη R ij ! ) . (8)Here the E tensor is defined by E ij = 1 η C ij − µ (cid:16) R ij − R g ij + Λ W g ij (cid:17) (9)with the Cotton tensor C ij C ij = ǫ ikℓ √ g ∇ k (cid:18) R jℓ − Rδ jℓ (cid:19) . (10)Explicitly, E ij could be derived from the Euclidean topologically massive gravity E ij = 1 √ g δW T MG δg ij (11)with W T MG = 1 η Z d xǫ ijk (cid:16) Γ mil ∂ j Γ lkm + 23 Γ nil Γ ljm Γ mkn (cid:17) − µ Z d x √ g ( R − W ) , (12)where ǫ ikl is a tensor density with ǫ = 1. In the IR limit, comparing Eq.(4) with Eq.(2)of general relativity, the speed of light, Newton’s constant and the cosmological constantare given by c = κ µ s Λ W − λ , G = κ π c , Λ = Λ W . (13)The equations of motion were derived in [3] and [5]. We would like to mention that the IRvacuum of this theory is anti de Sitter (AdS ) spacetimes. Hence, it is interesting to take alimit of the theory, which may lead to a Minkowski vacuum in the IR sector. To this end,one may deform the theory by introducing a modified term of “ µ R ” ( ˜ L V = L V + √ gN µ R )and then, take the Λ W → c = κ µ , G = κ π c , λ = 1 . (14)4aking N i = 0 and K ij = 0, a spherically symmetric solution could be obtained withthe metric ansatz ds SS = − N ( ρ ) dτ + dρ f ( ρ ) + ρ ( dθ + sin θdφ ) . (15)Substituting the metric ansatz (15) into ˜ L V with R = − ρ ( ρf ′ + f − L SSV = κ µ N − λ ) √ f " λ − f ′ − λ ( f − ρ f ′ (16)+ (2 λ − f − ρ − w − Λ W )(1 − f − ρf ′ ) − W ρ . Here the parameter ω controls the UV effects w = 8 µ (3 λ − κ , (17)which is positive for λ > /
3. This is possible because the metric ansatz shows all the allowedsinglets which are compatible with the SO (3) action on the S . In other words, the ADMdecomposition of HL gravity implies naturally the presence of a spherically symmetric staticsolution, in addition to the time-evolving solution of hypersurfaces for foliation preservingdiffeomorphisms [1]. For λ = 1 ( w = 16 µ /κ ≡ / Q ) and Λ W = 0, we have Kehagias-Sfetsos (KS) black holesolution where f and N are determined to be N = f = 2( ρ − M ρ + Q ) ρ + 2 Q + p ρ + 8 Q M ρ . (18)Here M may be related to the ADM mass. It seems that the metric function f looks likethat of Reissner-Nordstr¨om (RN) black hole. The outer (event) and inner (Cauchy) horizonsare given by ρ ± = M ± q M − Q (19)which is the same form as in the RN black hole. The extremal black hole is located at ρ ± = ρ e = M = Q. (20)This is why we take ω to be the charge-like quantity (pseudo charge) Q . In the limit of Q → ω → ∞ ), we recover the Schwarzschild black hole as the IR black hole.5owever, for large ρ , we approximate the metric function as f → − Mρ + 4 M Q ρ + · · · , (21)which is different from the RN-metric function f RN = 1 − Mρ + Q ρ . (22)At this stage, we define the Bekenstein-Hawking entropy for the extremal KS black hole as S BH = A πQ . (23)In order to explore the near-horizon geometry AdS × S of extremal black hole, we introducenew coordinates t and r using relations t = ǫ Q τ, r = ρ − Qǫ , (24)where ǫ is an arbitrary constant. Then, the extremal solution can be expressed by newcoordinates as ds = − r ( Q + ǫr ) + 2 Q + p ( Q + ǫr ) + 8 Q ( Q + ǫr ) 9 Q dt + ( Q + ǫr ) + 2 Q + p ( Q + ǫr ) + 8 Q ( Q + ǫr )2 r dr + ( Q + ǫr ) d Ω . (25)Taking the ‘near-horizon’ limit of ǫ → ds = v − r dt + dr r ! + v d Ω (26)with v = 3 Q , v = Q . (27)This shows clearly AdS × S in the Poincare coordinates. Its curvature is given by (4) R KS = R AdS + R S = − Q + 2 Q . (28) At present, we may have two kinds of entropy: one is the logarithmic entropy of S = A + πω ln h A i [20, 35]and the other is the Bekenstein-Hawking entropy S BH = A [30, 31]. The former can be obtained from thefirst law of thermodynamics dM = T H dS provided M and T H are known. The temperature is defined fromthe surface gravity at the horizon and thus, it is independent of the asymptotic structure. On the otherhand, the ADM mass M depends on the asymptotic structure because it belongs to a conserved quantitydefined at infinity. Hence, the entropy also depends on the mass M . However, it is hard to accept thelogarithmic entropy without considering quantum or thermal corrections. Therefore, it would be better touse the area-law of the Bekenstein-Hawking entropy to derive the ADM mass using the first law. In thiswork, we choose the Bekenstein-Hawking entropy as the entropy of the Hoˇrava black holes.
6n this case, the KS black hole solution interpolates between
AdS × S in the near-horizongeometry and Minkowski spacetimes at asymptotic infinity.However, the RN black hole has the Bertotti-Robinson metric as its near-horizon geom-etry ds = Q − r dt + dr r ! + Q d Ω . (29)Its curvature is given by (4) R RN = − Q + 2 Q . (30)Finally, we mention that the general solution with λ is not found in asymptotically flatspacetimes because third-order derivatives make it difficult to solve the Einstein equation. In this case, we take ω = 0 in Eq. (16) by dropping the modified term of µ R . TheL¨u-Mei-Pope (LMP) solutions for z = 3 Hoˇrava-Lifshitz gravity are given by f ( x ) = 1 + x − αx p ± ( λ ) , N ( x ) = x q ± ( λ ) q f ( x ) , (31)where x = p − Λ W r, p ± ( λ ) = 2 λ ± √ λ − λ − , q ± ( λ ) = − λ ± √ λ − λ − . (32)In this work, we choose p − ( λ ) = p ( λ ) and q − ( λ ) = q ( λ ) only. Its extremal black hole with f ( x e ) = 0 and f ′ ( x e ) = 0 are located as x e = 0 , for 13 ≤ λ ≤
12 ; x e = s p ( λ )2 − p ( λ ) = s λ − √ λ − − √ λ − , for λ > . (33)The Bekenstein-Hawking entropy for the extremal LMP black holes is defined by S BH = πx e h − Λ W i , (34)where − W is interpreted as a pseudo charge. Hence the non-zero entropy is available for λ > / f ( x ) can be expanded around extremal point to find itsnear-horizon geometry AdS × S as f ( x ) ≈ f ′′ ( x e )2 ( ǫρ ) + O ( ǫ ) , (35)where x − x e = ǫρ and ˜ t = ǫAt with A = f ′′ ( x e ) x q ( λ ) e p − Λ W . (36)7he line element is given by [19] ds = 2 f ′′ ( x e ) h − Λ W i − ρ d ˜ t + dρ ρ ! + h x e − Λ W i d Ω (37)with f ′′ ( x e ) = 4 − p ( λ ) . (38)Comparing the above expression with AdS × S , we obtain v = 12 − p ( λ ) h − Λ W i = v p ( λ ) , (39) v = p ( λ )2 − p ( λ ) h − Λ W i . (40)We express the Bekenstein-Hawking entropy as S BH = πv . For λ = 1, we obtain the LMPcase x e = 13 , v = 23 h − Λ W i = 2 v , v = 13 h − Λ W i . (41)Importantly, from Fig. 1, it is problematic to define the near-horizon geometry of extremalblack hole for 1 / ≤ λ ≤ / v is negative for 1 / ≤ λ < / p ( λ ) <
0) and v iszero for λ = 1 / p ( λ ) = 0). That is, its near-horizon geometry is ill-defined. In addition,the Bekenstein-Hawking entropy is zero.It seems that the Bertotti-Robinson geometry (RN black hole) of v = v = − / Λ W isrecovered from λ = 3 LMP black hole with x e = 1 f ( x ) = 1 + x − αx, N ( x ) = p f ( x ) x . (42)However, its asymptotic spacetimes are completely different from asymptotically flat space-times of the RN black hole.For 1 / < λ <
3, the LMP black hole solutions interpolate between AdS × S with v > v in the near-horizon geometry of extremal black hole and Lifshitz at asymptoticinfinity, while for λ >
3, these interpolate AdS × S with v < v in the near-horizon andMinkowski spacetimes at asymptotic infinity. × S In this section, we wish to find v -and v -solutions by solving full Einstein equations onthe AdS × S background with L K = 0. This study will be very useful for a further workon Hoˇrava black holes because it may provide a hint to explore the unknown black holesolutions. 8 Λ -0.5-0.250.250.50.7511.251.5v Figure 1: v ( λ ) (dotted curves) and v ( λ ) (solid curves) graphs with Λ W = −
1. We find v (1 /
3) = 1 / , v (1 /
2) = 1 / , v (1) = 2 /
3, and v (3) = 1, while we observe v (1 /
3) = − / , v (1 /
2) = 0 , v (1) = 1 /
3, and v (3) = 1.A variation to ˜ L V with respect to N is modified as δ ˜ L V δN = 0 : √ g " κ µ (Λ W − ω )8(1 − λ ) R − κ µ Λ W − λ ) (43)+ κ µ (1 − λ )32(1 − λ ) R − κ η (cid:16) C ij − µη R ij (cid:17) = 0 . A variation to with respect to the shift function N j is trivial for finding a black hole solutionas δ ˜ L V δN i = 0 → trivial . (44)A variation to L K + ˜ L V with respect to g ij is changed to δ ˜ L V δg ij = 0 : E ij ≡ κ µ (Λ W − ω )8(1 − λ ) E (3) ij + κ µ (1 − λ )32(1 − λ ) E (4) ij − κ µ η E (5) ij − κ η E (6) ij = 0 , (45)where E (3) ij takes the modified form E (3) ij = N (cid:16) R ij − Rg ij + 32 Λ W Λ W − ω g ij (cid:17) − (cid:16) ∇ i ∇ j − g ij ∇ (cid:17) N (46)and all remaining terms are the same as in [5].Considering AdS × S in Eq. (26), we have N = √ v r, g ij = diag h v r , v , v sin θ i ,R ij = diag h , , sin θ i , C ij = diag h , , i , K ij = diag h , , i (47) R = 2 v , K = 0 . N -variation leads to the equation obtained when varying with respect to v as κ µ √ v sin θ λ − v r " W v + 2( ω − Λ W ) v − λ + 1 = 0 . (48)The g ij -variation leads to three component equations E rr = κ µ v / λ − v r " W v + 2( ω − Λ W ) v − λ + 1 = 0 , (49) E θθ = κ µ r λ − v / v " v (cid:16) W v + 2 λ − (cid:17) − (cid:16) ( ω − Λ W ) v + λ (cid:17) v = 0 , (50) E φφ = sin θ E θθ = 0 . (51)Eqs. (48) and (49) give the same equation for v as3Λ W v + 2( ω − Λ W ) v − λ + 1 = 0 . (52)For Λ W = 0, this equation is solved to giveΛ W v = 13 (cid:18) − ω Λ W (cid:19) ± vuut λ − (cid:16) − ω Λ W (cid:17) . (53)For Λ W <
0, the solution is v ( λ, ω, Λ W ) = 13 (cid:18) − ω Λ W (cid:19) vuut λ − (cid:16) − ω Λ W (cid:17) − h − Λ W i . (54)From Eq. (50), we find for Λ W = 0 v ( λ, ω, Λ W ) = 6 λ − (cid:16) − ω Λ W (cid:17) − s λ − (cid:16) − ω Λ W (cid:17) λ − (cid:16) − ω Λ W (cid:17) − s λ − (cid:16) − ω Λ W (cid:17) v ( λ, ω, Λ W ) . (55)Eqs. (54) and (55) are our main results. × S -solution For Λ W = 0, Eq. (52) becomes 2 ωv − λ + 1 = 0 , (56)which gives the solution v = 2 λ − ω . (57)10rom Eq. (50) with Eq. (57), one has the relation between v and v v = 4 λ − λ − v . (58)The Bekenstein-Hawking entropy for extremal black holes with λ may take S BH = πv = π h λ − ω i . (59)For λ = 1, we recover the KS AdS × S exactly as v = 3 v , v = 12 ω = Q (60)whose entropy leads to Eq. (23). Although the black hole solution with arbitrary λ is notyet known in asymptotically flat spacetimes, its near-horizon geometry of extremal blackhole with λ could be found from Eqs. (57) and (58). For λ > /
2, we can define extremalblack holes because both v and v are positive. × S solution Here we comment on the ω = 0 case. Eq. (54) provides the solution v v = √ λ − − h − Λ W i = h p ( λ )2 − p ( λ ) ih − Λ W i . (61)From Eq. (55) with ω = 0, we obtain v = λ − λ − √ λ − v = v p ( λ ) . (62)This is the same result as was found from the previous section. × S solution We consider the general case of ω = 0 and Λ W = 0. As is shown in Fig. 2, it is observedthat v > λ > /
2, irrespective of any values of ω and Λ W <
0. It could beconfirmed from the fact that v = 0 implies λ = 1 /
2. We note that the λ = 1 solutiondescribes the near-horizon geometry of Park’s solution [18]. Also, from Fig. 3, we confirmthat v is always positive.Finally, the entropy of extremal black hole is determined to be S BH = πv (63)where v is given by Eq. (54). 11 Λ -0.5-0.250.250.50.751v H Λ , Ω , L W L Figure 2: v ( λ, ω, Λ W ) graphs with Λ W = −
1. We find that v > , = 0 , < λ > , = , < , respectively. Four graphs show for ω = 0 , . , , and 10 from top to bottom. Λ H Λ , Ω , L W L Figure 3: v ( λ, ω, Λ W ) graphs with Λ W = −
1. It shows that v > λ > . Four graphsare for ω = 0 , . , , and 10 from top to bottom (along v -axis). Plugging Eq. (47) into ˜ L V leads to the Lagrangian on AdS × S ˜ L AdS × S V = κ µ v sin θ λ − v (cid:16) W v + 2( ω − Λ W ) v − λ + 1 (cid:17) . (64)which is nothing but N δ ˜ L V δN (65)12n Eq.(48). After integration over S , it takes the form of entropy function [36]˜ L AdS V = πκ µ v λ − v " W v + 2( ω − Λ W ) v − λ + 1 . (66)A variation of ˜ L AdS V with respect to v leads to the known equation (52) πκ µ λ − v h W v + 2( ω − Λ W ) v − λ + 1 i = 0 , (67)while a variation with respect to v takes a different form3Λ W v + (2 λ −
1) = 0 . (68)The entropy function ˜ L AdS V is zero if Eq. (67) is used. Since this function contains v as aglobal factor, we cannot determine v by varying the entropy function with respect to v .It seems that this is a handicap in the entropy function approach to the Hoˇrava black holesbecause the HL gravity is a non-relativistic theory. However, we have determined v in Eq.(54) when using full Einstein equations on the AdS × S background. Now we wish toclarify why the reduced Lagrangian (66) is not useful to determine v . The entropy functionapproach is equivalent to taking into account (48) and (49). On the other hand, the fullEinstein equation on the AdS × S provides two more equations (50) and (51) than theentropy function approach. These determine a relation between v and v , as is shown inEq.(55). In this work, we have studied near-horizon geometry AdS × S of HL black holes. Animportant relation is v = v p and v = p − p [ − Λ W ] for the LMP black holes [5]. Also wehave obtained v = λ − λ − v and v = λ − ω for generalized KS black holes. The λ = 1 casecorresponds to the KS solution: v = 3 v , v = ω [10]. We regard ( − W , ω ) as pseudocharges as magnetic charge “ Q ” in RN and RN-AdS black holes.Since v is zero for λ = 1 /
2, we have to classify whole extremal black holes according tothe λ -value. For λ > / × S with different curvatures, depending on the (modified) Hoˇrava-Lifshitz gravity. For 1 / ≤ λ < / v of S is negative and thus the corresponding near-horizon geometryis ill-defined. Hence this case should be discriminated from the extremal black holes in theHoˇrava-Lifshitz gravity. This is clear because as is shown in Eq.(33), the Lifshitz blackholes with 1 / ≤ λ ≤ / ≤ z ≤
2) have degenerate horizons located at the origin likemassless BTZ black hole in three dimensions.13e hope that the black hole with arbitrary λ could be found soon in asymptoticallyflat spacetimes because its near-horizon geometry of extremal black hole are known as Eqs.(57) and (58). Also, we expect to obtain a general black hole whose near-horizon geometryis given by Eqs. (54) and (55) soon.Finally, the entropy function approach does not work for obtaining the Bekenstein-Hawking entropy of extremal Hoˇrava black holes because the Lorentz-symmetry was broken. Noted added –After the appearance of this work in arxiv, considerable research has in-dicated that for 1 / ≤ λ ≤
1, the projectable version of HL gravity has a serious problemcalled as strong coupled problem. In the projectable theories, the authors [37, 38] havefirst argued that ψ is propagating around the Minkowski space but it has a negative kineticterm, showing a ghost instability. In this case, the Hoˇrava scalar becomes ghost if the soundspeed square ( c ψ ) is positive. In order to make this scalar healthy, the sound speed squaremust be negative, but it is inevitably unstable. Thus, one way to avoid this is to choose thecase that the sound speed square is close to zero, which implies the limit of λ →
1. However,in this limit, the cubic interactions are important at very low energies which indicates thestrong coupled problem [39]. This invalidates any linearized analysis and any predictabilityof quantum gravity is lost due to unsuppressed loop corrections. The authors [40] triedto extend the theory to make a healthy HL gravity, but there has been some debate as towhether this theory is really healthy [41, 42, 43].On the other hand, in the Hamiltonian approach to the nonprojectable HL gravity,the authors [44] did not consider the Hamiltonian constraint as a second class constraint,which leads to a strange result that there are no degrees of freedom left when imposing theconstraints of the theory. Moreover, the authors [45] have claimed that there are no solutionof the lapse function which satisfies the constraints. Unfortunately, it implies a surprisingconclusion that there is no evolution at all for any observable. However, more recently, itwas shown that the IR version of HL gravity ( λR -model) is completely equivalent to thegeneral relativity for any λ when employing a consistent Hamiltonian formalism based onDirac algorithm [46].The projectability condition from condensed matter physics may not be appropriatefor describing the (quantum) gravity. Instead, if one does not impose the projectabilitycondition, the HL gravity leads to general relativity without the strong coupling problem inthe IR limit. We note that this work was carried out without the projectability conditionand thus, was nothing to do with the strong coupling problem for 1 / ≤ λ ≤ cknowledgement Y. Myung thanks Li-Ming Cao, Sinji Mukohyama, and Mu-In Park for helpful discussions.H. Lee was supported by the National Research Foundation of Korea (NRF) grant fundedby the Korea government (MEST) (No. 2009-0062869). Y. Kim was supported by theKorea Research Foundation Grant funded by Korea Government (MOEHRD): KRF-2007-359-C00007. Y. Myung was supported by Basic Science Research Program through theNational Research Foundation (NRF) of Korea funded by the Ministry of Education, Scienceand Technology (2009-0086861).
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