JHEP07(2016)112 Springer July 21, 2016 May 13, 2016 June 21, 2016 : : : Received Published Accepted 10.1007/JHEP07(2016)112 doi: Published for SISSA by . 3 1603.08004 The Authors. Gauge-gravity correspondence, Holography and condensed matter physics c

Exact solutions to Einstein’s equations for holographic models are presented , [email protected] Racah Institute of Physics,Jerusalem The 91904, Hebrew University Israel of Jerusalem, E-mail: Open Access Article funded by SCOAP Keywords: (AdS/CMT), Spacetime Singularities ArXiv ePrint: loops are discussed.and The the boundary Laplacian condition in for thethere IR the is is current-current examined. correlation a There function geometry is normalizable is no solution obtained infalling wave after in in a the the dimensional IR, reduction. IR. but instead, In a special case, a hyperscaling-violating and studied. Thethe IR less geometry generic has case ais of asymptotically timelike the AdS. cousin BKL This ofsolution’s (Belinski-Khalatnikov-Lifshitz) solution the appearance describes singularity, Kasner is and a singularity, an holographic the whichsoliton. interpolation RG UV is flow between The between the causality constraint them. planar is AdS The always black satisfied. hole The and entanglement entropy the and AdS Wilson Abstract: Jie Ren Asymptotically AdS spacetimes with asingularity timelike Kasner JHEP07(2016)112 18 (1.3) (1.1) (1.2) ]. The . 1 0 r = r , still an exact solution to  2 , .  dy 2 γ = 1 f 3 dy 2 + γ + 12 2 2 + dx 2 dx β β = 0, and the horizon is at f + 7 + r 2 + 2 f 2 dr f dr + – 1 – 2 + 5 , α 2 10 fdt dt 16 α − 1 = 1 f S  γ 2 − 2 + r L  2 β 2 = r L + 2 17 = α ds 2 ds . The AdS boundary is at 3 ) : 0 are constants satisfying the Kasner conditions 5 1 γ r/r 14 ( − , and β = 1 , α f black hole in Poincar´ecoordinates is The following “deformation” of the planar AdS black hole is 4 3.1 Geodesics 3.2 Entanglement entropy and Wilson loops where where Einstein’s equations gravity in the bulk.Wick Another rotation basic of solution is thedeconfinement the AdS phase AdS black transition soliton, hole. between which theAdS appears At AdS as a soliton a and critical double the temperature, AdS there black is hole a [ confinement- 1 Introduction The planar AdSa black strongly hole coupled plays quantum a field basic theory role in in the the AdS AdS/CFT boundary correspondence, is mapped where to a classical B Notes for an integral C Kaluza-Klein reduction on 5 Relation to hyperscaling-violating geometries 6 Discussion A Relation to other coordinates 3 Geodesics and extremal surfaces 4 Two-point correlation functions Contents 1 Introduction 2 Anisotropic solution and boundary stress tensor JHEP07(2016)112 , ) 1.2 0 or (1.4) ] and → 16 – T 14 ] for some related earlier . 7 2 , timelike Kasner singularity 6 ) alone demands that it must dy γ 2 1.2 r ]. Justifications for the timelike 9 + ¯ is completely different from a finite ]. The timelike BKL singularity has 2 0 4 ]. A key physical quantity to calculate r , dx 3 β 18 2 = . At first sight, this solution interpolates r r A , the IR geometry is + ¯ r 2 is the (finite temperature) horizon of the AdS is regular, and the solution is the AdS soliton. ¯ r − – 2 – 0 0 d 0 r r ], including the causality constraint [ r 5 + = = √ 2 r r ]), anisotropic geometries with a regular horizon do dt = α r 2 13 ¯ r − ]. The IR geometries are hyperscaling-violating geometries. ] in Ricci-flat spacetime after a double Wick rotation. At 8 2 ]. We will show that these two conditions are always satisfied = 2 17 ] as an IR fixed in AdS/CFT; see [ 5 ds ) is asymptotically AdS, and thus describes an RG flow from the ]. 6= 1, the IR geometry at 1.2 ] (and also [ . From the phenomenological point of view, there are anisotropic ma- 11 α , = 0. In this case, = 1. In this case, 12 1 10 γ γ = ), there is a naked singularity, and we call it 0 β r = 0, ]. When the Gubser criterion is satisfied, the geometry can be obtained as an β 9 = = = 1, r ; a property of the latter case is that the spectral functions in these geometries can α α black hole. To my taste, the remarkable simplicity of the metric ( According to [ Naked singularities are not uncommon in AdS/CFT. We have plenty of examples However, when • • → ∞ = 0 ( ¯ be interesting physically. We willand explore its various holographic higher aspects dimensionalthe of generalizations, entanglement the entropy, including solution and ( one-point Wilsonillustrated loops. and in figure two-point In functions, theterials bulk, in the condensed geometry matter physics, is for anisotropic, example, as [ BKL singularity have beenthe studied accessibility to in the [ IRfor [ the anisotropic solution.boundary conditions Beside, at werelatively the well will IR. behaved. examine another All constraint these related suggest to that the the timelike BKL singularity is have a hard gap [ not exist inGubser pure criterion gravity, without which matter impliesnot fields. necessary that condition the However, to the BKL justify Gubser singularity a criteron does naked is not singularity a [ satisfy sufficient but the superstring or 11D M-theory [ The parameters of thecriterion [ hyperscaling-violating geometries areextremal limit constrained of by a the finiteT temperature Gubser black hole. The extremal limit is at either been recently studied inworks. [ The solution ( AdS boundary as the UV, to the timelike Kasnerof singularity supergravity as systems the IR. with naked singularities from consistent truncations of the 10D This is the Kasnerr solution [ as distinguished to theis spacelike Kasner the singularity less in generic cosmology.generic case BKL The singularity of Kasner has singularity the oscillatory BKL behaviors [ (Belinski-Khalatnikov-Lifshitz) singularity, while the temperature horizon. By defining ¯ holographic RG flow, as explainedbetween in the appendix AdS black hole and the AdS soliton: This solution is obtained by mapping the time evolution of a Bianchi type I Universe to a JHEP07(2016)112 (2.3) (2.1) (2.2) ,  , 2 i UV IR  dx i p f 2 + 1) L n n =1 , i ( X n +1 + n + 2  f

0

R r dr r · · ·   + g – 3 – 2 − − dt √ t x p = 1 f +2 f n − d , we show that the causality constraint is always satisfied  2 3 2 Z r L = = S 2 action r ds +2 , we obtain a hyperscaling geometry after a KK reduction for a special n 5 , we examine the boundary conditions in the IR for solving correlation 4 is the AdS radius. The Einstein’s equations have the following solution: . Schematic plot of the geometry, i.e., the RG flow from UV to IR. The UV is asymptoti- L This paper is organized as follows. In the next section, we present the anisotropic We can obtained a hyperscaling-violating geometry after a Kaluza-Klein (KK) reduc- where We consider the AdS where functions. In section case of the anisotropic solution.discuss some Finally, further we summarize issues. the properties of this solution2 and Anisotropic solution and boundary stress tensor solution in arbitrary dimensions, obtain thethe stress energy tensor conditions. of the In boundary section CFTby and analyzing examine the nullloops. geodesics. In We section also examine the entanglement entropy and Wilson in the IR forcorrect boundary perturbation condition equations. in the For IR the is timelike normalizability. Kasnertion singularity, of however, a the violating special geometry case cannot of becriterion the is connected anisotropic marginally to violated, solution. a while the finite The null temperature energy lower-dimensional condition black hyperscaling- is hole; marginally the satisfied. Gubser Figure 1 cally AdS. As thebe IR expanding. is approached, The some Kasner directions singularity will in be the contracting, IR while is someis line-like directions or the will hyperplane-like. conductivity. For a black hole, we impose the infalling wave boundary condition JHEP07(2016)112 (2.6) (2.8) (2.9) (2.4) (2.5) (2.7) = 1) (2.10) (2.11) 0 r is ( . ) z +2 n and z ( r O = 1 throughout this 0 = 0 directions. + r i , . 2 p ν z  , 2 i dx ] ) , . µ log k = 0. 2 i
( dx ν g h dx ) [ 0 directions will be contracting, +1 i µν dx ν = 1 and . n p i dx n µν 2 p z X > µ µ L 2 f . 1. i X n ¯ r = 1 p 2 ) is dx +1) ) is c 2 +1 n ) + =1 2 i n n n i =1 ( X | ≤ p , z + i X 2.2 i , z h  µ µν ν p + µ | x n + =0 + m x ( 2 i +1 X ), the relation between 2 ( µ + 1, and it reflects the conformal anomalies +1) n ¯ r n µν z m 2 ( µν d g +1) 2.7 2 h dt g g b n t ( n + – 4 – p , . We will set + g 2 + 1 + 2 L 0 k < n N ν f 2 r l  +1 dt µ z t n l = 1 − dz πG ) and ( = p 2 z + 1) i 2 2 a = with r p ¯ r 2 16 2 n ) in Fefferman-Graham coordinates: + − 2.2 ) r z dz ( L − k n ( =0  i + ( 2.2 X g ··· = = = 2 2 0 behavior of 2 ¯ r i z 2 L 2 + d , the IR geometry of ( → µν r ds ds = = (2) T z ]. In our case, we will see that h 2 g 2 − 2 0 ds z 19 ds r When the IR is approached, the + √ 1 = ]. We will briefly review the procedure, and apply it to the anisotropic (0) ] for studies about this case; the geometry has oscillatory behaviors by swapping g ) describes the less generic case of the timelile BKL singularity. In the generic case, the r 7 = 0. , ) satisfies the Kasner conditions 20 0 directions will be expanding. We may also have ] is a function of ) toward the singularity. t 5 , 2.5 ) r p k ) = < ( 2.5 19 g i ≡ , z [ p µ 0 µν x p ( ( X g i p By comparison of the metric ( We can obtain the stress-energy tensor of the boundary CFT by the holographic renor- By defining ¯ The metric ( 1 (3+1)-dimensional ansatz is Please refer toexponents [ of ( Thus, we obtain the geometry ( where of the boundary CFT [ The stress-energy tensor is solution. Write the metric in Fefferman-Graham coordinates where the near boundary implies that the cosmologicalsingularity constant at is ¯ irrelevant inwhile the the IR. There is a timelike Kasner malization [ This is the Kasner solution in Ricci-flat spacetime after a double Wick rotation, which The IR limit ofpaper. the geometry Obviously, the is Kasner at conditions require and JHEP07(2016)112 +1) n ). By (2.17) (2.18) (2.12) (2.13) (2.15) (2.16) (2.14) ( n / 2 ≤ i ≥ vanishes at i be the affine ii ≤ . ) (no summa- p g λ 0 + t 2.9 ≤ p γ without loss of gen- . . ). Furthermore, when 0 ≤ γ 3 4 +1 ≤ / n ≥ ]. We will study geodesics direction (1 2.15  γ 3) β i 21 +1 ≤ √ , , n 3 ) i 1 − 0 / p 1 − (1 2 t − z/z p ) implies ( + 1) +1 +1 n n 0 0 n i . z z ( , 1 + ( 2.16 + 1) . ,  3 0 gives − 3 n / ( 1 / = 1 ≥ ) needs an extra condition 1 2 + 1 i ) gives  for all N N 0, and that the product of all ≤ i n n n ≤ β L L – 5 – i → πG πG β ≥ p 4 4 ≤ t tt , h ≤ for all p g ≥ 0 = = | 0 is the pressure along the i t i i +1 +1 ¯ p p 0 for all i n n ii p 00 ) ) T ≥ 0 0 T ≥ | ), the stress-energy tensor is traceless. , ) combined with ( ) as an example, and assume i ,  1 p ≡ h 1 z/z z/z ≡ h 2.4 2.4 1.2 i ( ≤ + ¯ ¯  ¯ p ≤  − α α 1 1 + ( , the weak and the strong energy conditions are also satisfied, but i ≤ ≤ = 3 3 / f / solution ( 2 3) . First, the energy density ¯ 4 i √ p (1 + and ): t ii p is the energy density, and ¯ T . is satisfied for all i  i p The null energy condition ensures Take the AdS The energy conditions for the boundary CFT give nontrivial constraints for the pa- Note that these energy conditions are for classical field theories. For a quantum field theory, the energy 2 ≥ t Geodesics and extremal surfaces are bulkfirst. probes The in null AdS/CFT [ geodesics are closely related to thedensity causality can constraint. be negative Let dueAdS to soliton the is Casimir negative. effect. Thus, For the example, constraints the discussed energy below density are of not the strictly CFT necessary. dual to the the IR, although there are contracting and expanding spatial3 directions. Geodesics and extremal3.1 surfaces Geodesics The dominant energy condition is more stringent, and implies the dominant energy condition (¯ for all erality. The null, weakimplies or strong energy condition combined with the Kasner conditions The null energy condition (¯ The first Kasner condition in ( p rameters where ¯ the first Kasner condition in ( The boundary stress-energy tensor cantion be for straightforwardly obtained by ( where JHEP07(2016)112 and 1 for (3.4) (3.5) (3.1) (3.2) (3.3) is the E − for null t i P . Therefore, = i . p E ), as suggested  = +1, 0, = , ) that a geodesic 2 i t r κ 2.2  is P p 2 i i 2 r ˙ p x r i f p = 0 and f n =1 = 0 if 0, which implies that the κ i X , where n =1 κ i X eff + → , V 2 = + 1 eff E ν 2 . f ˙ ˙ t V r x ≥ 2 i p µ ˙ r i t x f ˙ ; and + x p p i . The spacelike geodesics can reach i . In the above inequality, ∆ 2 i p 1). i 2 ˙ − µν r p t f /f /f t g 2 2 dt . κ < i p < p equals the time along a null geodesic in /f > p | = f P E t 2 i i , which implies that its tip is repulsive. − t direction. Since i p x · bry 2 p − p i P |  i Z i ) = 0. In the IR limit, there are three cases: , which implies that the horizon is attractive. P  f ] to justify the solution ( r E 2 P ≥ 2 2 ( ≥ 0 if 1 r r E t 14 → eff – 6 – = p ], and do not require this constraint. ≥ dt V i t eff dλ ˙ t , P p /f p 21 ) = V eff t 2 → − r 2 p V bulk Z ( /f 1 and all r E i f . Form this action, we can see that the energy ; Z eff i eff E/f λ = P V < = ν | = 1). ˙ t = x > p i E p µ 0. The deepest point in the bulk (largest | t p t x ˙ ) can be obtained by x p direction, and consider a small interval of the affine parameter = 1). ,V λ ∆ ∆ ≤ i ( µν t r p eff 0 if V dλ g ) = 0 ≤ r 0, which gives ( Z eff ≤ eff V = V 0 eff S + V ), we can see that 2 ˙ r are conserved quantities (no summation) 3.2 i 0, it requires P ≥ 2 r singularity is neither attractive nor repulsive. The AdS black hole ( The AdS soliton (one of The Kasner singularity ( ]. For two spatial points on the boundary, the causality constraint requires that the We will use the causality constraint [ Assume a photon is propagating along the 5 • • • . From ( λ where we used time along the null geodesic in the bulk, and ∆ If this is satisfied, spacelike separated pointsthroughout along the the bulk. boundary are Assume also spacelikeon a separated the photon AdS is boundary propagating in the along∆ a null geodesic with two ends by [ fastest null geodesic connecting these two points always lies in the boundary, namely geodesics, we have the null geodesics candeeper get in access the to bulk the than bulk null if geodesics [ Since ˙ can reach is given by the smallest root of The geodesic equation for spacelike, null, and timelike geodesics, respectively. The equation for where the dots are withmomenta respect to parameter along the geodesics, and the geodesics are extrema of the action JHEP07(2016)112 and , by (3.7) (3.8) (3.9) (3.6) i , the r (3.11) (3.10) r = > p f ∂ , we have 1 t √ ) case, this p σ m r 4 r = = is determined by n r dr , m r β dr , . = 2 (AdS − ), we obtain 4 ]. Consider a strip with α )  n − 2 i m 1 3.8 23 , f , dx ) , i α r/r 2 2 22 p dr . 0 ( / − f f ) x 4 1 α ) ) β − n m 2 − n f r m m =2 α / i X + 1) ) (1 m − , and is always satisfied. f + β f 1 ), the following quantity is conserved: i n + r/r . The value of ) f/f 1 r √ (  ( 2 ( 2( − (1+ m 2 = x > p − f − dr = − β ( t
f α 0 − f/f 2 p 2 r − 1 q ( ) 1 rf r n/ − n ) 1 ( ) r s f 0 0 m = m 2 1 dimensions. In the – 7 – and is independent of the anisotropy. x x / + for a connected surface. At β Z ) ν 0 − 2 ), and plugging it into ( β f ˆ r 1 n 0 f/f r/r r n ( + ( µ − x n r + , we can demonstrate that the causality constraint is α ≤ n 3.9 ( ∇ p λ 1 p r A − − 2 µν m f / f r ≤ = h ) 0 m α , and infinite width in other directions. The metric of the n ). With the asymptotically AdS solution as a full RG flow, h Z r from ( =  ): 1 − r  0 2 − x 1 (1 Z = r 2.5 x K 3.9 √ f 1 l ≡ 2 x − is the condition that the null geodesics can probe into the bulk, = n n x i 2 d A . The entanglement entropy is calculated from codimension-2 ex- ds Z = > p ], another condition for a physical geometry is that extremal surfaces t = S p ). Solving S 17 m , r 5 solved from ( ( is the area of the other 0 be the maximal value of f x 1 . Since the Lagrangian does not contain = m − ], it was argued that the causality constraint is satisfied as long as r n in the direction 5 m A . f l y → ∞ Let In [ ) = r ( 2 0 which has been regulatedintegrating by a UV cutoff at where is equivalent to aσ holographic Wilson loop whose worldsheet coordinates are x The area of the bulk surface is where width codimension-2 surface in the bulk can be written as which is always positive for all 0 Entanglement entropy tremal surfaces in the bulk by the Ryu-Takayanagi formula [ 3.2 Entanglement entropy andAccording Wilson to loops [ anchored at the AdS boundary shouldthe not extrinsic have any curvature barrier toextrinsic to access curvature the be is IR, which positive. requires For the radial normal vector ˆ always satisfied. analyzing the IR geometry ( we conclude that while the causality constraint does not require the boundary. By integrating along JHEP07(2016)112 . B , the (3.12) (3.13) (3.14) max , similar max . We choose l l ): < l < l , the connected ) in appendix x  3.9 ) below. crit α l max B.1 n − 2 . The metric for the 1 ) 1 ) 3.19 p m m . . l > l ≡ !  r/r 2 2 ( f/f β ]. When ( dr dr . 
24 − 2 [ 2 2 ) / ) 1 and r ) r ( β 0 i (  t crit 0 for a given quark-antiquark separation n x + β l p ¯ q i 0 2 x r direction, separated by α p ( β A 1+ l/ f 1 ≡ − f f ]. Consider a string connecting a quark x f n =1 α + i X 1 , and the profile of the string is given by 25 and 0. Therefore, there is always a maximal ≡ 1  r ) = + = 1 − x A Z 0, so there is a maximal length r m 1 f ( r r 1 = − . This plot can also be thought of as a qualitative r − f . Let α > eff 1 C max – 8 – n l  + σ α > 2 − A 2 + − 2 A dt = and α dt t ,V f 2 0 t p f l/ A − = q − −  0 2

B ) = 0 1 (disconn) r 2 σ r 1 r S ( = eff = 2 , we can write the above integral in terms of ( V 2 , the connected surface exists. When m ds ds + 2 max r/r  3 C = dr dx ), as illustrated in figure u l < l  r ( 0, there are two connected solutions x . We want to calculate the quark-antiquark potential to see whether the for the connected surface, according to the analysis of ( β > . Schematic plot of a string connecting a quark and an antiquark on the AdS boundary. max + . There is also a disconnected solution l ]. When α We can write an effective potential for a zero-energy particle from ( By defining max 24 If the string ends in an arbitrary direction, the metric for the string worldwheet is 3 and an antiquark on thethe AdS worldsheet boundary coordinate in the as the function string worldsheet is For the anisotropic solution,length we have 1 Wilson loops anisotropic solution describes a confined orcan deconfined state. be calculated The quark-antiquark by potential a holographic Wilson loop [ to [ surface does not exist.disconnected There surface may will be dominate: a critical length When l < l representation of extremal surfaces. For the anisotropic solution, we have 1 Figure 2 JHEP07(2016)112 . , , . m B r → ∞ (3.19) (3.16) (3.15) → ∞ (3.17) (3.18) ]. For ≤ ) ) 28 ]. Equa- r r ∗ m ( – 0 r 15 ( x ≤ l 26 . ) in appendix is determined by  β B.1 m 4 + , we have r ), we obtain ) α . We have 0 when 0 m ) dr , dr , m r ∗ m ) m 4 r 2 3.15 ≤ , the disconnected string is the “time” [ ) 0 = r/r x m ( x r , f/f β eff max ( 2 f dr . V , there is no connected string β / r/r ) 4 ( − + . + β ) α 2 m 1 1 max + ) − r m /T α 2 −  ( m β / m f β ) denoted by < l < l f + ( r/r β NG . The value of α ( l > l α 1+  S m ) f/f = f f r ( (1+ crit − m l = β − q β anisotropic geometry is in a non-confining ) = − f 2 r l + 1 ) = 1 2 f/f r ( . The zero-energy particle will turn around 4 α r ) − ( ¯ q ) ( q 3 f m 0 . When Z m V eff s ), the following quantity is conserved: – 9 – x + r 2 ), and plugging it into ( T 1, we always have ( / r/r . When 2 max 0 ( f/f x l 1) = ( 2 x < − crit r 3.16 for a connected string. At h l α p p ,V ( r m − 2 f m / r √ ) 0 m β ): σ 2 r < r Z from ( 2 + ], although the connected string solution exists. r  ) = 0 0 d α r Z ( = ( 28 x 3.16 f – T Z l 2 eff , if no other constraints are imposed. , we can write the above integral in terms of ( i V 26 = = m + , which is always positive as long as the geometry is not the AdS for the connected string. 2 NG γ NG r/r have a maximum,  S S l − = at a finite “time”. Equivalently, there is a maximal quark-antiquark max ). Solving dr dx l u 0, there will be a maximal value of 0, 0. For any 0 m m  r solved from ( r 0 for some = 1 ( 0 f β < x < β be the maximal value of β > = β > i = + ) can be rewritten as m is the time duration of the Wilson loop. + p r + α r + m α α + f T α as illustrated in theat left plot of figure separation 3.16 t , p A qualitative way to analyze the behavior of the string is by an effective potential, For There may be a critical length By defining Let 4 • in which a zero-energytion particle ( is moving, and the coordinate which implies that we alwaysto have a the connected anisotropic string geometry solution.i.e., is In in this a case, confined the phase. CFT dual This iscontribution possible dominates in [ higher dimensions, When solution. This isAdS similar to the Debyesoliton. screening Therefore, in the the CFT blackphase. dual hole to geometry the [ AdS The quark-antiquark potential is given by which has been regulatedintegrating by a UV cutoff at where where Since the Lagrangian does not contain The Nambu-Goto action for the string is JHEP07(2016)112 x 0, a , we (4.3) (4.1) (4.2) iωt β > (3.20) ∗ m − e + α r < r = ) = 0: t m ∗ m r δA ). The right plot ( < r 0 eff max V l . ∗ m r = 0 is r . = 0 , x 1 +1 a n α ···  2 ) , the zero-energy particle spends 1+ 4 + ω b . direction. For simplicity, we denote ∗ m f is 1 r is determined by at a finite “time” ( ). The left plot is for the case + 1 − x (0) x x + n a m a m , such that for any 0 = eff r r 0 x r 0 V 3.19 ≡ 1) m a 1) iωa m r = − 4 x − r  n 1) r n + 1)( ( x 2 ( x a a n − r − ( . n + 2 n – 10 – ( − F 1 4 4 − ··· r near the AdS boundary . When by the null energy condition of the boundary CFT. 0 −  ) = f + x m f β ω r a ) = , in which the zero-energy particle spends an infinitely long ( β (0) x ≥ σ ∗ m ∗ m ≤ 1 a r is the expectation value, and there are logarithmic terms r . − α r = = ∗ m 1) r (1 ] for applications of AdS/CFT to condensed matter physics. x ≤ m − ), and the equation for  a r = n ( x . The maximal value of 30 r + a , 2.2 3 00 x 29 m a r 0 and ) with a Maxwell term . We assume 0 when 0 β < 1 p 2.1 ≤ 0. There is a maximal value of + ≡ α . To calculate the conductivity, we turn on a perturbation eff β V is the source, β < eff . Schematic plot of the effective potential in ( 0 V is odd. The DC conductivity is x (0) + and a n right plot of figure α have an infinitely long “time”, when it approaches the turning point, as illustrated in the t p The action is ( • 4 ≡ when This equation is solvedbelow. with The appropriate asymptotic boundary behavior of conditions in the IR to bewhere discussed α Conductivity around the background ( We will investigate thefunctions. boundary We begin condition with thethe in current-current conductivity. the correlation See function, IR which [ We for is perturb responsible the solving system for by two-point an correlation electric field in the 4 Two-point correlation functions is for the case “time”, when it approaches Figure 3 in which a zero-energy particle will turn around at JHEP07(2016)112 in ] for (4.8) (4.9) (4.4) (4.5) (4.6) (4.7) 11 in both → ∞ ,  ˜ V 00 → ∞ ff . There is an = 0. ˜ V . x ν + 4 2 . There is also an 0 . If 2+ / r f 1 ], for example. ξ → −∞ 1) 31 2 , → ∞ − C ˜ x V a . The boundary condition β ˜ + V 1 2 . − ν β = 1. − α 4 , − 2 − 2 ν +2 α → ∞ 1 / α . α ( 1 )  x ξ ˜ r − β V 1 − 1 ˜ a f − 2 1 − C 2 α ) is not normalizable at ω 2 − f x n (1 = ∼ ( | = α 1) − ) ν 2 | r x 1+ Y − , we will have infalling wave by a Hankel = 0. The leading order of the Schr¨odinger a ωξ − )˜ = ( β ξ ξ ν x ( , ν = 0. The conductivity is gapped if ˜ a ˜ ξJ + 2 V . And + 1) – 11 – 4 → ∞ 1 | . The general solution is a linear combination of / ν α | p n + C ξ ( 1 n 2 ( Y 0 in either the UV or the IR, then the Schr¨odinger 2 x ξ 1 5 2 C , α − 16 ˜ a ≤ 3). 2 2 2 2 and dξ − / − d ν ˜ ) | V ) + ≥ ] 1 ν α 0 | 1 − = n J − ωξ 1 is always at rf ( (1+ ˜ ) V ν = f β → − ξ = r − ξJ 3). For the AdS soliton, we have p dr dξ ≥ 2, both solutions are normalizable, and 1 2(1 1, both solutions are normalizable, and / 2 n C 1 − r < 4 = < | nf x [ ν 6 | | ˜ a α ν in the UV (when 1, only one solution is normalizable, and | f < 2) 1, the IR limit < 2 | ≥ / − ν → ∞ | n is an integer, we need to use ( ˜ α < If in the IR isthe uniquely UV specified (when by ambiguity to specify a boundary condition inIf the 1 IR. The conductivityambiguity is to gapless. specify aV boundary condition in the IR. The conductivity is gapped if If 0 ν In the IR limit, the Schr¨odingercoordinate is After a change of variables by = The way to obtainIf a Schr¨odingerequation can be found in appendix C of [ • • • 5 6 ˜ V As a comparison, iffunction. the In IR limit ourmore is case, examples about at however, this the type boundary of boundary condition conditions. is There normalizability. are three See cases: [ which is independent ofBessel the functions dimension Since potential is where the derivatives are taken withthe respect UV to the and original the coordinate gap), IR, so then does the theequation conductivity. Schr¨odingerequation has has If a a continuous discrete spectrum, spectrum and (and the conductivity a is hard gapless. The potential Schr¨odinger is we can obtain a Schr¨odingerequation JHEP07(2016)112 7 is d = 0. (5.1) ω (4.12) (4.13) (4.10) (4.11) ], and two 33 ), we always 1.2 ), the IR geometry 2.2 , supergravity in [  5 , 2 d Φ . dx 1 = 2) solution ( 4 − + n α . We can take the source of the ( − 1) = 0, there is a radially conserved 4 ··· f Φ = 0 − 2 n 2 n α + ω ( x ˜ r 2 − a 2 1 . 1+ r ω , f 0 x ∼ dx 4 a / + 2 2 β + ]. At ˜ Φ = 1 0 ξ − − 2 1 n Φ − 32 ˜ r r f d is the hyperscaling violation exponent, and  – 12 – = r n θ ]. For the AdS , = 0) is the Drude weight of the conductivity. This + ˜ V 2 2 z − ω Π = / ). The leading order of the potential Schr¨odinger in 11 2 ) , 0 ˜ r dt α f f 1 4.5 10 ), we obtain that Π is a constant in the IR. Thus, Π is − = 0 has a delta function, despite the system is neutral. ].  ), Π( (1+ 1 [  f 11 ω 4.9 + θ d 4.3 2 00 = r | ≥ Φ ν | = ˜ dr dξ 2 ds = 0: 0 supergravity in [ = 1, and by ( 4 (0) x a 1 under the null energy condition of the boundary CFT. For higher dimensions, . We can study other two-point correlation functions in the same manner. Con- Φ) = 0 is is the Lifshitz scaling exponent, > | z iωt ν 1 will give nontrivial constraints to the parameters in general. | − Some remarks about the DC conductivity are as follows. The DC conductivity can If the boundary condition in the IR is uniquely defined regardless of how to resolve e It is less understood that the conductivity of a zero density system can have a delta function at ( 7 = 0 case of above, and thus one solution will contain a log term. Therefore, the boundary | ≥ 2 ν where the number of spatial dimensions in the AdS boundary. This can only happenexamples at from zero AdS temperature. There is an example from AdS We will show that afteris a dimensional equivalent to reduction a of a hyperscaling-violating special geometry case of ( which is independent of eitherν the dimension or other parameters.condition in This corresponds the to IR the given by the normalizability is unambiguous5 and well-defined. Relation to hyperscaling-violating geometries we obtain a Schr¨odingerequation ( the IR is Similarly, after a change of variables by means that the conductivity at Laplacian sider tensor perturbations,∇ which satisfy the Laplace equation. The Laplace equation quantity satisfying Π Substituting here the solutionthe ( same constant atperturbation the AdS boundary, and Π the singularity, we require have | be calculated by the membrane paradigm [ JHEP07(2016)112 ) r 5.7 (5.6) (5.7) (5.8) (5.9) (5.2) (5.3) (5.4) (5.5) , and (5.11) (5.10) 1 1, and S → on r y ,  . . 2 ,    , ) dy 2 ) ) d 2 γ 2 d  f 2 i dx dx ∂φ . ( dx + + 1 2 ) + 2 d d =1 = 0 − i ··· X ··· dx γ . , + αφ + + 2 + 2¯ d β + 1 2 2 1 2 e dy 2 f + − ··· dx dx dr ¯ . βφ ( ( γ 1 dβ θ 1 2 β θ + β + 2) − e 2 − 2 f d + 2 1 = r d 2 r + d . 2 θ/d + , and the geometry is dx L + ¯ r 2 ( + + 2 +2 2 β 2 f 2 d ¯ r z = ˜ /r + 1)( f dr ¯ s d γ 0. If we compactify the coordinate ]. For more details about the KK reduction, see is that the hyperscaling-violating geometry ( ): β d d – 13 – , θ = f + ( fdt , α + d + + 34 is always satisfied β α θ αφ [ 2 2.2 1) 2 2 = + 2¯ θ + γ < f − θ γ/d e z ¯ − − α − z dt dr dt = 1 ¯ r ¯ ( d ¯ R d βφ α d 1 1 α = = 2 2 γ 2 f +  − e ¯ r r 2 and θ ¯ g = 2 , we obtain an IR geometry − + = r z − − − ds z dt  ¯  β α √  dβ − /d f 2 1 1 r +2 + γ/d γ/d d 2 − √ 2+2 f α d r ) by  = r 2 = 2 1 = ¯ Z r 5.1 = r 2 ds 0, which requires + 1) and 2 = = ¯ s d ds in ( ¯ 2 ( d S d r ds 2 p / α > β > . The action after the dimensional reduction is 0 is the IR limit. This is a hyperscaling-violating geometry. The coordinate ¯ C = 1 → r α A consequence of the relation The dilaton is determined by We will examine a special case of ( cannot be regarded as an extremalviolating limit of geometry a is finite temperature continuouslyGubser geometry. connected criterion If a is to hyperscaling- satisfied, a the finite extremal geometry temperature is geometry, given i.e., by the The following relation between By comparing the two metrics, we obtain define a new variable ¯ where ¯ is related to the ˜ The boundary is conformal to Minkowski space. After we take the IR limit where ¯ appendix We assume do a KK reduction,violating we geometry in obtain the a IR. spatially The isotropic reduction geometry, ansatz which is written has as a hyperscaling- where JHEP07(2016)112 (5.12) (5.13) (5.14) . This i ), but the . The near- , but not for 4 ] for all → ∞ , we will obtain 0 35 d i ii → ∞ p g T and higher dimen- . , h 0 r 4 ≥  2 i t ≥ p ) dy i θ , there is always only one γ will have an infinitely large 4 f − 0 d z 0 d =1 T i X + d + . 2 i 0. In our case, the Gubser criterion ) 1)( θ -dimensional torus dx > − 0 θ i z − d − ) β + d z θ d f ( ( d − d =1  z i X h r r + + , d . The extremal limit is at  0 – 14 – 2 h )( f − will have an infinitely large radius. We can analyze θ ≥ dr 1
− + S θ d = 1 2 − f dt < r < r α 1) f vanishes. − − ii z g  ( 2 d 1 , and 0 r 0), and some spatial directions are expanding ( h ) r θ = 0. After a KK reduction on a → − 2 = 0 in the IR limit. As the IR is approached, some spatial directions are ii > d g r i ds ( β → tt g 0 and α > In the IR, we docondition not is have infalling imposed wave by boundaryto normalizability. conditions, the In instead, AdS the this soliton boundary aspect,normalizable than the solution to solution in the is AdS the moresolution black similar will IR, hole. give but nontrivial For for constraints AdS to higher the dimensions, parameters. a unique normalizable The causality constraintsurfaces is to always probe satisfied, the IR. and thereThe is boundary CFT no is barrier inhigher a for dimensions non-confining in extremal phase general. for the anisotropic AdS The null energy conditionensures for the boundarycontracting ( CFT requires product of all spatial The null energy condition for the hyperscaling-violating geometry is [ In the IR limit, the compactified +2)-dimensional geometry. Similarly, the compactified • • • • d We have studied anisotropic solutionssions. to They Einstein’s describe equationstimelike in a Kasner AdS holographic singularity, RG which flowconclusions is from are a an as less follows. asymptotically generic AdS case boundary of to the a BKL singularity. The key where a ( size in the IR limit. 6 Discussion a more general dimensional reduction, for example These inequalities are always satisfied in our case; the second one takes the equal sign. The horizon is at extremal solution exists only whenis ( marginally violated, and the near-extremal solution does not exist. where JHEP07(2016)112 and the 8 ]; we did not 39 , 38 ]. 11 ]. anisotropic solution in Fefferman-Graham 5 41 , cases? Other questions are as follows. – 15 – 5 40 ], the system admits an anisotropic solution with a , there is no infalling wave in the IR, but we still have 12 4 ] to study anisotropic plazmas at strong coupling; the and AdS 4 36 ] conclude that there exists infalling wave at the IR. According 37 ) is obtained by the Bianchi Type I Universe, but are there coun- 1.2 . A special case of the AdS ] and a later work [ 36 cosmological models to avoid the Bigwe Bang resolve singularity, such the as singularity bouncing in models. the Can IR analogously? The IR geometry in thisHow work to is construct the less an generic RGcase case flow of of from the the the timelike timelike asymptotically BKL BKL AdS singularity. singularity? boundary toThe the time generic evolution of the Universe is mapped to a holographic RG flow. There are It will be interesting ifcan more general we analytic generalize solutions the can solution be with obtained. an For electromagnetic example, The field? solution ( terparts of other Bianchi types of the Universe? The geometry is not connected to a finite temperature black hole. In a spacial case, we obtain a hyperscaling-violating geometry after a KK reduction. There is a large liturature about anisotropic plasmas, notably [ Further investigations of the anisotropic solution are necessary. Are there further A possible scenario to remedy the singularity is as follows. Starting from a gravity We do have some supergravity systems in which the gauge field vanishes as the extremal limit is taken, • • • • • 8 work is partially supportedIsrael by Science the Foundation American-Israeli Center Bi-Nationaland of Budgeting Science Excellence Committee Foundation, and and the The the Israel I-Core Science Program Foundation “The of Quantum thefor Universe”. example, Planning the 1-charge and 2-charge black holes in the appendix C of [ solutions in other contexts, for example, [ Acknowledgments I thank Steve Gubser, Chris Herzog, Elias Kiritsis, and Li Li for helpful discussions. This Both [ to our detailed analysis innormalizability as section a well-defined boundary condition. address these topics. Matter fields are involved in these geometries. There are anisotropic Some remarks coordinates has appearedsolution in looks [ complicated and does not have an explicit relation to the Kasner geometry. irrelevant in the IR, we can get a Kasnerevidence singularity to in justify the the IR. nakedsolution, singularity? especially What for is the the AdS field theory dual to the anisotropic system coupled toregular matter horizon. fields [ Thengeometry as is the described extremal by the limit anisotropic is solution. taken, In the a weaker matter case, fields if the vanish, matter fields are JHEP07(2016)112 ) r ( G (A.8) (A.4) (A.5) (A.6) (A.7) (A.1) (A.2) (A.3) .  r | ] Λ | . . 3 2 i 44 , 2 i , p p ) p , , 2 42 1 2 2 i 2 2 i =0 3 =1 i X dy i  X dx γ dx ) ) = h r = t ( ( i i p , U + U p tanh ] for a Bianchi type I Universe: 3 , 2 3) 3) | ¯ 2 r 2 i 2 / / =0 3 | 3 1 Λ =1 1 i X i | dx X Λ − − dx β | i i i p p r h p 2 ( = 2( 2( t 2 e = e
r i = 1), we obtain [ = 1 = ] r  3 = 1 = U =1 3 | i 2 + ¯ =1 2 X , 2 i X 2 Λ 1 =0 43 2 G . We can use the following solutions for | i 3  , G + X 3: ] for details. ¯ r 2 3 f / ). If the geometry has a cosmological constant, / 3 2 d 2 – 16 – 42 / 44 ) p 2 /L t , e dt 2.4 ) + ( + 3Λ = 1
+ 3Λ r − 2 2 r G ( 1 and 0 2 | m ˙ dt = G − + G Λ 1 | 2 2 − + 2Λ = 6 3 = α ds 2 dt 0 − 1 + cosh p  − fh dr − = = ; please read [ /G , G /G , 2 2 = G sinh | 2 ds ds = 1 = 1 Λ | ds 0 1 ˙ 3 U U p ) has three commuting Killing vectors, which implies that it is related = 1.2 G radius are related by satisfy the Kasner conditions ( 4 ): i r p ( U We consider a negative cosmological constant here. The cosmological constant and In the Ricci-flat case, we have the Kasner solution [ we obtain the following metric with After the coordinate transformation There are other cases for the AdS and where where After a double Wick rotation ( the above solution can be generalized to [ where The solution ( to a Bianchitransformations. type I We Universe willstraightforwardly after consider generalized a (3+1)-dimensional to arbitrary double spacetimes, dimensions. and Wick the rotation result and can appropriate be coordinate A Relation to other coordinates JHEP07(2016)112 3.2 (B.1) (B.2) (B.3) (A.9) . (A.10) (A.11) m ) can be r m r ( l du . 2 / du . 1 2 − ) has an explicit relation b a/ . ) − 2 u b 1.2 ) = 0, the solution is the AdS 2. dy d du , − / 2 α u 1 4 r (1 u − − b 1 + ¯ at the maximal value of ) ≈ − d 2 (1 u , a u k b > Z  dx 2 d u m 2 ) d r ¯ r 1 L ∼ r → ∞ u d 0 m (¯ r d ) 1 is 1 is − m Z f + r du − . m 2 1 − (1 r b + → → ) 1 m ¯ r a/ ( ) k r 2 l 4 2 )
d  u (¯ ¯ 2 m m r ) ), which are much more cumbersome than the r u u ¯ r r r f (¯ − r – 17 – m d f − r − 1 A.4 cannot reach 1, because what is inside the square 0 1 , there are three cases as follows: + − ) = ) = ) as ) as (1 Z = 0) = 0. We want to know whether √ r r m k 2 m m m (¯ (¯ 0. r m (1 r u r r h f m dt r d ( ( 2, and finite when ) or ( 1 2 r l l / 0 ( ¯ r l 1 a > Z − A.2 ) = 2 = ) = 2 m ≤ − ). r 2 m ) = 2 b ( r l ( , since ds 1.2 m l r m ( r l and the freedom of rescaling the AdS radius and the horizon size, we = ¯ r r 1. Obviously, / ≤ = 1 m r r 0. The maximal value of 0. The leading order of ≤ = 1, the solution is the AdS black hole, and when = 0. The leading order of α It is zero at a < root cannot be negative. We will have It is divergent when a > a The anisotropic solution in Poincar´ecoordinates has a remarkably simple form with a • • • divergent. At the maximal value of where 0 B Notes for an integral The calculation of theinvolves holographic the entanglement following entropy integral: and Wilson loops in section cosmological constant, unlike ( case without a cosmological constant.between the Moreover, the UV solution and ( the IR. soliton By defining can obtain the solution ( When where JHEP07(2016)112 , # 2  (C.6) (C.7) (C.8) (C.9) (C.1) (C.2) (C.3) (C.4) (C.5) (C.11) (C.10) dt ) 2 c γ γ f f , + + . α  2  2 . f s 2 α F −  ): x ( 2 f d . αφ ]. After eliminating β cs F − 5.2 f 30 +1)¯ αφ [ + = 0 d + 1)¯ α . x 2( tx 2 − dy a 2)¯ − h f  D e dr  ) , − 2( 1 4 2 2 2 iωt | 0 t c − ) + − tt γ . D e − e g 2 A ( | f 1 4 2 ZA . cy , − ) dt + + = 2 2 2Λ) − . + − c 2 c = ∂φ 2 ) 2 γ s tx dz γ − ( ) c st ( ¯ γ rr α β f f 1 2 γ δg g f 2 f γ R | ¯ βφ ∂φ f ( + 2 → + ω + 2 ( tt − − + r 2 α g e 2 g 1 2 + -dimensional action is | s α s f and , − 2 αφ α + + ( α f  D − 2¯ s x f √ 2 f e α − 2) 2 D a x + x ( ¯ − s f − αφ – 18 – d 2¯ d − 0 x iωt cs +1 1 − β e a = − + 2) − 0 sy , y D αφ f D S  ) e 2¯ d = d 2Λ ¯ 2 + e αφ + rr rr t /d d 1)( = 1 g L 1 g 2 − Z 2 A = ct ) t f − − ¯ xx 2 xx dr R = e c + 1)( → , the metric becomes +1 γ  δA D d g g S θ g g + ¯ t /d g 2 D f ( 2( − 2 − − ds + + √ 2+2 dt p √ √ 2 r cosh 2 ¯ Z x s R Z c is ( = α γ D  ≡ x f ) as the starting point of the model, and calculate the conductivity. d + 1)-dimensional action ¯ ¯ f g α c + a γ − − + + 00 x D ( Z C.7 -dimensional metric. The α 2 a √ s f and + 1)-dimensional metric as = = dimensions. The Lagrangian after the dimensional reduction is D α +2 2 ¯ θ d d f S D ¯ . The dilaton is s d − d dA Z has sinh − is the 2 " = 2 = ≡ 2 ¯ x s 1 ¯ r F d s d S We can treat ( We can also obtain an Einstein-Maxwell-dilaton (EMD) system from the dimensional = , the equation for 2 tx ds We turn on theh perturbations The gauge field is where The metric is where reduction. After applying the following Lorentz boost to the metric ( where In the above expressions, where We start from the ( We write the ( C Kaluza-Klein reduction on JHEP07(2016)112 , 1, , > | (C.12) (C.13) , ν | , , , x a (2015) 168 1 2 , ) 09 2 c γ , f , ]. Effective holographic + ]. ]. JHEP (2010) 151 2 , s 0. The leading order of the 11 α SPIRE , f IN → ) SPIRE SPIRE = 3), we always have − γ ][ ξ α IN IN ( ]. 1 D ][ ][ − − JHEP − γ . , β + 4 . 2 β 2(1 Oscillatory approach to a singular point SPIRE = 1 ( ]. +2 IN = α d − ][ f SPIRE (1970) 525 2 +1 (1978) 466 d IN , ν − – 19 – 19 ][ hep-th/0002160 hep-th/9803131 arXiv:1601.02599 r 4 On the motion of particles in the field of a naked [ / Investigations in relativistic cosmology = 1 A 66 ). The IR limit is at 2 x ξ − ˜ a 4.5 2 ν Timelike BKL singularities and chaos in AdS/CFT hep-th/9903214 ]. ), which permits any use, distribution and reproduction in [ Adv. Phys. ]. . = (1998) 505 [ (2000) 679 [ , , . 1 2 ˜ 4 2 Study of anisotropic black branes in asymptotically anti-de Sitter . V Phys. Lett.  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