Holographic micro thermofield geometries of BTZ black holes

Journal of High Energy Physics, Jun 2017

We find general deformations of BTZ spacetime and identify the corresponding thermofield initial states of the dual CFT. We deform the geometry by introducing bulk fields dual to primary operators and find the back-reacted gravity solutions to the quadratic order of the deformation parameter. The dual thermofield initial states can be deformed by inserting arbitrary linear combination of operators at the mid-point of the Euclidean time evolution that appears in the construction of the thermofield initial states. The deformed geometries are dual to thermofield states without deforming the boundary Hamiltonians in the CFT side. We explicitly demonstrate that the AdS/CFT correspondence is not a linear correspondence in the sense that the linear structure of Hilbert space of the underlying CFT is realized nonlinearly in the gravity side. We also find that their Penrose diagrams are no longer a square but elongated horizontally due to deformation. These geometries describe a relaxation of generic initial perturbation of thermal system while fixing the total energy of the system. The coarse-grained entropy grows and the relaxation time scale is of order β/2π. We clarify that the gravity description involves coarse-graining inevitably missing some information of nonperturbative degrees.

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Holographic micro thermofield geometries of BTZ black holes

Received: April Published for SISSA by Springer Open Access 5 6 7 c The Authors. 5 6 7 0 Department of Chemistry, Brown University 1 Center for Theoretical Physics of the Universe 2 Department of Physics, Ewha Womans University 3 Department of Physics, Sejong University 4 Physics Department, University of Seoul 5 nd that their Penrose diagrams are no 6 Institute for Basic Science , Seoul 08826 , Korea 7 Providence , Rhode Island 02912 , U.S.A We nd general deformations of BTZ spacetime and identify the corresponding thermo eld initial states of the dual CFT. We deform the geometry by introducing bulk elds dual to primary operators and nd the back-reacted gravity solutions to the quadratic order of the deformation parameter. The dual thermo eld initial states can be deformed by inserting arbitrary linear combination of operators at the mid-point of the Euclidean time evolution that appears in the construction of the thermo eld initial states. The deformed geometries are dual to thermo eld states without deforming the boundary Hamiltonians in the CFT side. We explicitly demonstrate that the AdS/CFT correspondence is not a linear correspondence in the sense that the linear structure of Hilbert space of the underlying CFT is realized nonlinearly in the gravity side. We also longer a square but elongated horizontally due to deformation. These geometries describe a relaxation of generic initial perturbation of thermal system while xing the total energy of the system. The coarse-grained entropy grows and the relaxation time scale is of order =2 . We clarify that the gravity description involves coarse-graining inevitably missing some information of nonperturbative degrees. AdS-CFT Correspondence; Black Holes; Conformal Field Theory Holographic micro thermo eld geometries of BTZ black holes Contents 1 Introduction 2 3 4 Einstein scalar system Linearized perturbation Linearized solution including back reaction Boundary stress tensor and horizon area Convenient form of coordinates Field theory construction Other examples of micro-geometries Bulk dynamics Conclusions B Other perturbation with m2 6= 0 Introduction one may study various aspects of gravity and eld theories in a rather precisely de ned horizon and gravitational singularities. is the inverse of PI CI OI e 4 H0 with H0 denoting generic perturbation of thermal system can be achieved. Namely such states are still which we take as . present work. Penrose diagram and horizon area. In section 4, we present the eld theory description various scalar perturbations. partially overlap with ours in this paper. We begin with the three dimensional Einstein scalar system S = There are also in general interactions between these bulk elds, which we shall ignore in this note. The dimension of the corresponding dual operator is related to the mass by d) = `2m2 and recover it whenever it is necessary. The Einstein equation reads and the scalar equation of motion is given by m2 2 gab = @a @b = 0 known AdS3 M4 background where M4 may be either T 4 or K3 [10]. Thus our The BTZ black hole in three dimensions can be written as ds2 = where the coordinate ' is circle compacti ed with ' ' + 2 . Of course here we turn is ensured if the Euclidean time coordinate tE has a period Gibbons-Hawking temperature is then 2 = 2 `R . The corresponding The mass of the black hole can be identi ed as The boundary system is de ned on a cylinder T = M = ds2B = dt2 + `2d'2 theory is related to the Newton constant by Thus the entropy of the system becomes while the energy of the system can be expressed as in terms of the quantities of CFT. Linearized perturbation Introducing new coordinates ( ; ; x) de ned by c = 2 R 4G 3` 2G S = M = x = tanh `2 = should be xed further. the BTZ black hole metric (2.5) can be rewritten as ds2 = d 2 + d 2 + cos2 dx2 A( ; ; x) B( ; ; x) = ( ; ; x) equations of motion (2.3) and (2.4) reduce to (@~A)2 + (@xA)2 A @~2A = 2A A2 (@~ )2 + AB (@x ) 3(@~B)2 2B @~2B + @~B @~ @xA@xB + 2B@x2A = 0; As a power series in , the scalar eld may be expanded as ( ; ; ') = in the leading order becomes tan @ h + tan @ h + @~2h R2 cos2 @'2 h = 0 . Here for simplicity, we shall consider h = cos2 sin a = b = 0( ) + 1( ) cos 2 0( ) + 1( ) cos 2 tan @ h + tan @ h + @~2h h = 0 corresponding to integral dimensions. only. Let us organize the series expansions of the metric variables by A = A0 1 + B = B0 1 + The leading order equations for the metric part then become A0 = cos2 ; B0 = of motion up to some extra homogeneous solutions. Using the remaining components = 2 with the simplest solution of (3.7). Linearized solution including back reaction (1 + 6 cos 2 + 5 cos 4 ) + c1 tan + (1 + tan ) (5 + cos 4 + 6 (2 + cos 2 ) tan ) + c3 cos2 + c4(2 + cos 2 ) tan (13 + 16c3) cos2 cos sin in (3.8) can be written in more convenient forms (1 + 6 cos 2 + 5 cos 4 ) + (1 + tan ) (5 + cos 4 + 6 (2 + cos 2 ) tan ) + c3 cos2 (13 + 16c3) cos2 cos sin + 0 = 1 = 0 = c2 1 = 0 = 1 = 0 = c2 1 = where we introduce One then nds ( 0 + 1 cos 2 ) = ( 0 + 1 cos 2 ) = ( ; ) = 1 ( ; ) = 1 0 = 1 = 0 = c2 1 = (5 + cos 4 (1 + 6 cos 2 + 5 cos 4 ) cos 4 + ( 0 + 1 cos 2 ) + O( 4) ( 0 + 1 cos 2 ) + O( 4) (1 + 2 cos2 ) cos 2 6(2 + cos 2 )) + c3 cos2 (25 + 16c3) cos2 We now require that A and B have expansions ( 0 + 1 cos 2 ) + O( 4) = ( x c2 = 2116 and c3 = one has 9 . This choice xes the freedom of coordinate scaling. Therefore 0)2, one may 0 = 1 = 0 = 1 = 0( ) = 10 cos2 ) + (1 + tan ) (1 + 2 cos2 )(1 + tan ) (21 + 12 cos2 )(1 + tan ) (1 + tan ) One further nds 0 = 1 = 0 = 1 = In this coordinate system, the (orbifold) singularity is still located at . Hence the directional coordinate is ranged over [ other hand, the spatial in nity is at 0( ) so that the coordinate is ranged over nds the Penrose diagram is elongated horizontally in a -dependent manner. We nd that Boundary stress tensor and horizon area may be identi ed as hO(t; ')i = 8 G`3 cosh2 t`R2 tanh `2 = expanded as 0)4. Let us de ne ( ) by ( ) = 0( ). The functions A and B can be hO(0; ')i = 0 @t hO(t; ')ijt=0 = td = A = B = q cos 2 + where q = 3 transformation, the metric becomes denotes higher order terms in By the coordinate q cos 2~ sin2 ~ + q sin 2~ sin 2~ + d~2 + d~2 + cos2 ~ dx2 = ~ + = ~ + which are time independent. ( ) = 27 9 cos2 +22 cos4 24 cos6 changed. The mass and pressure are then given by M = 2 ` p = The horizon area (length) becomes A( ) = 2 R 1 In the region near of A( ), from which one nds A( 2 , our small approximation breaks down since the coe cient of 2 , our small approximation breaks down since the coe cient of 2 term becomes too large. A(0) is given by A(0) = 2 R 1 Convenient form of coordinates we make the following coordinate transformation cos 2 cos 2 + Then the metric turns into the form a = a = bx = sin 2 + cos2 cos2 (11 + 4 cos2 ) cos 2 10 cos2 ) + 4 cos2 ) cos 2 3 + 4 cos2 (2 coordinate ranges ; =2; =2]. Here we choose tE ranged over [ de ned by g (tE; ') = g( tE; ') Oy(t; ') = O(t; ') We associate the interval [ 2 1; 1). This { 11 { initial state is given by j (0; 0)i = p hnjU jmi jmiL U is in general given by The Lorentzian time evolution is given by the Hamiltonian where the left-right Hamiltonians are identi ed with U = T exp dtEH(tE) HLT (tL) 1 dtL + 1 HR(tR) dtR HL(tL) = H HR(tR) = H itR is relevant to the initial ket state. left in nities. Then with 0( ), one has the relations tR = sin tL = tER = sinh E tLE = which identi es the boundary times tR and tL. Since is ranged over [ 2 ; 2 ], one sees that tR and tL are ranged over ( above becomes 1; 1) as expected. Now by analytic continuation, the where, from the Euclidean geometry, one nds that E is ranged over ( nds that tER can be chosen to be ranged over ( 4 4 ; ) whereas tLE to be ranged over Note that the points tE = 4 is not associated with the right nor the left boundaries of the 1; 1). One without deforming the Hamiltonian. of g( E ; '). For the Janus deformation in [8], one nds the analytic continuation indeed works. We leave further clari cation of this issue to future works. { 12 { Thus the time evolution is given by j (tL; tR)i = T exp i dt0LHLT (t0L) 1 T exp HR(t0R) j (0; 0)i by the von Neumann de nition R(tR) = trLj (tL; tR)ih (tL; tR)j SR(tR) = trR R(tR) log R(tR) tary operator U = T exp h i R0tR dt0RHR(t0R)i. For the undeformed case with Hamiltonian tion is fully preserved, which implies that the corresponding ne-grained (von Neumann) bative objects in string theory. In quantum eld theory on R S1, one may prove that boundary is given by hO(t; ')i = h (0; 0)j1 O(t; ') j (0; 0)i can be evaluated perturbatively as hO(t; ')i = ` trO(t; ')O( i(s H0 + O( 3) (4.15) { 13 { given by trO(t; ')O(t0; '0)e m= 1 approximation. See [8, 13] for the normalization factor. of ' with hO(t; ')i = dx g0(s) cosh 2 (t + is) + cosh x t0) + cosh 2 ` (' '0 + 2 m) + i g0(s) can be determined by demanding The function g0(z) is identi ed as [ cosh(v + iu) + cosh x]2 = cosh2 v g0(z) = where O njc (j 0) and O njs (j 1) are de ned by operator precisely at tE = . (There is, however, an example where the Lorentzian simpli es as U in (4.6) is modi ed to O njc = O njs = d' cos j' d' sin j' O (t; ')jt=0 O (t; ')jt=0 4 creates a deformation of state without combination of operators V = corresponding to such deformation of states described in the above. Other examples of micro-geometries The corresponding expectation value can be given by gn(s) = hO(t; ')in = hO(t; ')i0 + O( 3) larly by hn( ; ) = h1( ; ) = hO(t; ')i1 = form is di erent from that of the previous solution. h( ; ) = 0 h0( ; ) + 1 h1( ; ) solves the linearized scalar eld equation in (3.7) where 1 are real. From this problem from the beginning. One nds nonvanishing n2 implies f n1n2 ( ; ) = ( ) where is ranged over [ ( ); 0R( )]. One nds that R=L( ) = 2 ( ; =2) = 1 + 2GR=L( ) + O( 4) a( ; ) = b( ; ) = 12 a210( ; ) + 0 1 a201( ; ) + sin 3 18 cos 2 + cos 4 ) relation GL( 0; 1) = GR( 0; function g(s) by illustrated for various 1 with The corresponding scalar eld reads g0(s) = h0 = 1 + sin and the vev becomes hO(t; ')i = to present. The choice will also give the scalar solution given by g(s; ') = gn(s) = hn( ; ) = Finally we consider the case of massive scalar whose dual operator O has a general the case with `2m2 = 3 ( given by h = cos 1(sin ) + 2Q 1(sin )) Note that this reduces to (3.12) or (5.12) for massless case ( h = cos3 construct a rather general state by the insertion { 17 { with V = P O200, one can choose the linear combination which leads to g(s; ') = 0 g0(s) + 0 g0(s) V = ( 0 + i 0)O200 = C200 O200 h( ; ) = h = string theory. Bulk dynamics better to use the coordinates ( ; ) 2 [ as introduced in section 3.3. Namely h( ; ) = he( ; ) + ho( ; ) he( ; ) = n hn( ; ) ; ho( ; ) = discuss properties of this solution. First of all, there is a symmetry ) = ) = hn( ; ) { 18 { ) = ) = he( ; ) h(0; ) = q1( ) @ h( ; )j =0 = q2( ) condition is relaxed. Now we shall give an initial condition at = 0 by perturbations can be independent from each other. Note that the set which leads to the symmetry of the solution j n = 0; 1; 2; the Dirichlet boundary condition. The cos factor here follows from the fact that we are now the basis is given by j n = 0; 1; 2; argued below, our gravity description fails near the singularities are spacelike. Away from particular nothing special happens near horizon regions. { 19 { gather lies in the right side of horizons that are colored in red. description by the boundary eld theory. In particular R(tR) contains all those microis available obBut he cannot cross the past horizon of the right side, which is the 45 red line in gure 6. basically from the causality imposed by the horizon. Note however that the e ect is of order 2 since missing information is mainly due to the horizon change that is of order 2 . The higher order contributions including gravthe information regarding the perturbative gravity uctuation may be restored. { 20 { metric description of micro-geometries. missing information is stored in such nonperturbative degrees. Conclusions vacuum, which is the thermalization of any initial perturbation. We will report the related study elsewhere. Acknowledgments in 2017. { 21 { For n = 1 case, one has h1 = cos2 sin (1 A = cos2 B = a = b = a = ( b = the form 0 = 1 = 2 = 0 = 1 = 2 = = 1 = 1 37 cos2 + 126 cos4 120 cos6 ) + (9 + 15 cos2 + 44 cos4 (1 + 2 cos 2) tan (3 + 23 cos2 + 150 cos4 144 cos6 ) + (3 + 24 cos 2 72 cos4 ) tan (111 + 23 cos 2 + 94 cos4 (27 + 3 cos 2 + 122 cos4 60 cos6 ) + 120 cos6 ) (37 + 20 cos 2 4 cos4 ) tan 8 cos4 ) tan 13 cos 2 + 234 cos4 240 cos6 ) + 4 cos 2) tan 18(1 + 2 cos2 ) cos 2 + (1 + 8 cos2 24 cos4 ) cos 4 ] + O( 4) 2[74 + 40 cos2 + 2( 9 + 8 cos4 ) cos 2 4 cos2 ) cos 4 ] + O( 4) { 22 { cos2 (37 126 cos2 + 120 cos4 ) 44 cos2 + 36 cos4 ) 222 cos2 + 144 cos4 ) cos2 (37 106 cos2 + 60 cos4 ) 146 cos2 + 120 cos4 ) 234 cos2 + 240 cos4 ) Along the future horizon the horizon length is a monotonically increasing function sin2 ~(9 cos 2~ 2 cos 4~) + (9 sin 2~ sin 4~ cos 2~) sin 2~ + 0 = 1 = 2 = 0 = 1 = 2 = = ~ + = ~ + 0( ) = 2 ( =2; ) 18 cos 2 + cos 4 ) + O( 4) Penrose diagram in gure 7. As in section 3.2, the metric can be transformed to the standard BTZ metric (3.29) by the coordinate transformation, 279 + 93 cos2 782 cos4 + 3064 cos6 4272 cos8 + 1920 cos10 A(0) = 2 R 1 A coordinate transformation [36 sin 2 (cos 2 + 2 sin 2 ) + sin 4 (cos 4 + 4 sin 4 )] (74 + 18 cos 2 cos 2 + cos 4 cos 4 ) gives the metric of the form (3.35) with a = 74 cos2 + 252 cos4 252 cos2 + 240 cos4 + 6 cos 2 ( 18 + 3 cos2 44 cos4 + 36 cos6 ) cos 4 ( 6 + cos2 174 cos4 + 144 cos6 )] + 6 cos 2 (33 + 44 cos2 + cos 4 ( 47 + 18 cos2 ( 7 + 8 cos2 )] 97 cos2 + 60 cos4 ) a = bx = and (A.3) with 0 = 1 = 2 = explicit form of the solution is given by h = cos3 cos 2 (330 16 cos2 (15 64 cos2 + 60 cos4 )) cos 4 ( 15 + 76 cos2 960 cos4 + 960 cos6 )] 45 cos2 18 cos4 + 40 cos6 ) + (165 + 275 cos2 132 cos4 + 108 cos6 ) (1 + 2 cos 2) tan (15 + 115 cos2 402 cos4 144 cos6 ) 24 cos4 ) tan { 24 { (27 + 20 cos2 + 4 cos4 ) tan 115 cos2 + 118 cos4 + 120 cos6 ) + 8 cos4 ) tan 65 cos 2 + 18 cos4 240 cos6 ) 4 cos 2) tan 22(1 + 2 cos2 ) cos 2 (1 + 8 cos2 24 cos4 ) cos 4 ] + O( 4) 2[54 + 40 cos2 + 8 cos4 + 16 cos4 ) cos 2 4 cos2 ) cos 4 ] + O( 4) 0 = 1 = 2 = 0 = 1 = 2 = cos2 (45 + 18 cos2 cos2 (55 + 132 cos2 108 cos4 ) cos2 (5 + 42 cos2 + 144 cos4 ) cos2 (135 + 154 cos2 cos2 (55 + 2 cos2 120 cos4 ) 18 cos2 + 240 cos4 ) 1 = 2 = = 1 = 1 0( ) = 2 ( =2; ) 22 cos 2 cos 4 ) + O( 4) = ~ + = ~ + sin2 ~(11 cos 2~ + 2 cos 4~) + (11 sin 2~ + sin 4~ cos 2~) sin 2~ + { 25 { A( ) = 2 R 1 + 375 + 125 cos2 + 50 cos4 264 cos6 + 848 cos8 640 cos10 A(0) = 2 R 1 A coordinate transformation gives the metric of the form (3.35) with a = a = bx = [44 sin 2 (cos 2 + 2 sin 2 ) sin 4 (cos 4 + 4 sin 4 )] (54 + 22 cos 2 cos 2 cos 4 cos 4 ) 45 cos2 18 cos4 + cos 2 ( 660 + 110 cos2 + 264 cos4 216 cos6 ) + cos 4 ( 30 + 5 cos2 + 282 cos4 + 144 cos6 )] cos2 [6(45 + 18 cos2 + 2 cos 2 (605 + cos 4 (235 132 cos2 + 108 cos4 ) 18 cos2 (29 + 8 cos2 )] [3885 + 120 cos2 872 cos4 + 480 cos6 2 cos 2 (975 + 8 cos2 ( 125 + 44 cos2 + 60 cos4 )) cos 4 (75 380 cos2 + 192 cos4 960 cos6 )] Open Access. any medium, provided the original author(s) and source are credited. 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Dongsu Bak, Chanju Kim, Kyung Kiu Kim, Jeong-Pil Song. Holographic micro thermofield geometries of BTZ black holes, Journal of High Energy Physics, 2017, 1-28, DOI: 10.1007/JHEP06(2017)079