CORE JHEP12(2014)046 brought to you by by you to brought Springer - Publisher Connector - Publisher Springer by provided Springer October 10, 2014 December 4, 2014 : November 23, 2014 : : 10.1007/JHEP12(2014)046 Received Published Accepted doi: Published for SISSA by [email protected] , . 3 1312.3296 The Authors. Black Holes in String Theory, AdS-CFT Correspondence, Holography and c

Using gauge/gravity duality, we explore a class of states of two CFTs with a ,

[email protected]

Department of Physics, StanfordStanford, University, CA 94305, U.S.A. E-mail: Stanford Institute for Theoretical Physics, Open Access Article funded by SCOAP ArXiv ePrint: determining the qualitative features of the resulting geometries. Keywords: condensed matter physics (AdS/CMT) large degree of entanglement, butare with constructed very weak by local perturbing two-sidedthat correlation. the are thermofield These local states double atbations state different create with times. an thermal-scale intersecting operators Chaotic Acting network CFT of on dynamics shock the and waves, dual supporting the black a associated hole very fast geometry, long scrambling these wormhole. time pertur- play an essential role in Abstract: Stephen H. Shenker and Douglas Stanford Multiple shocks View metadata, citation and similar papers at core.ac.uk at papers similar and citation metadata, View JHEP12(2014)046 ]. L 10 (1.2) (1.1) , is of w t ]. 5 ]. We showed that 6 . ] context, as Van Raamsdonk R i 3 , n 2 | L i n | 2 / 15 n S βE − log e π β 3 2 n – 1 – X . This structure is closely related to the smooth = 2 R / ∗ 1 1 t Z 1 17 ] = 9 – i 7 TFD | (right), in the thermofield double state ] pointed out that a random unitary transformation applied to R 4 8 5 entanglement in this state is highly atypical, as local subsystems of 3 1 14 11 LR ] controversy has highlighted the conflict between the special local entangle- 1 (left) and L We examined this situation in detail in our study of scrambling [ Van Raamsdonk [ The importance of this time scale in black hole physics was pointed out in earlier work, including [ 2.2 Two shocks 2.3 Many shocks 2.1 One shock 1 special local entanglement. Thisorder happens the fast when scrambling the time time [ since the perturbation, hence the geometry behindwhich the can horizon. create a Certain pulse unitaries of correspond radiation propagating to justa local behind local operators, the operator horizon on [ of the energy, left if hand applied boundary early that enough, only scrambles injects the one left thermal hand quantum Hilbert worth space and disrupts the are entangled with localgeometry subsystems of of thegeometry eternal can black respond hole. to operations that The delocalize primary the goalthe entanglement. left of handed this CFTunchanged, paper leaves but the is density will to matrix change explore describing the how right relation handed between CFT degrees observables of freedom on both sides and of a CFT, The particular 1 Introduction The firewall [ ments required for smooth geometrytension and become especially the clear randomness in of thehas typical two sided emphasized. states. black hole The Aspects [ two of sided this eternal AdS Schwarzschild black hole is dual to two copies A Recursion relations for manyB shock waves Vaidya matching conditions 3 Ensembles 4 Discussion Contents 1 Introduction 2 Wormholes built from shock waves JHEP12(2014)046 , ]. 2 S/ 14 (1.3) , − e . This 13 1 ∼ ) is a conse- as a family of w drops to zero t 2 I ) that diagnoses − t A, B ( of the two CFTs. For I R ], using the length of the 6 is order one. ⊂ ] could be consistent with a i ) B ) is highly boosted relative to t 21 ( w , t is fixed. Using the Eigenvector R L are disturbed. At late times, two- . ϕ i i ) R ⊂ t ( , and is never larger than H , with operators at equal Killing time A L S i TFD ) + ϕ − TFD | ] passing through the wormhole [ t | h e ) ( L 1 ]. Using (2+1) Einstein gravity and ignor- R t 12 H ( ∼ , ϕ 20 1 ) ] suggests a different potential interpretation. – t 11 ( 24 – 2 – 15 L . is the inverse temperature. From the bulk point ϕ R ...W h β ) H is large, the relative boost between the geodesic and n | t + ( w L t ], they showed that the two point correlator between n H − W t 23 | , ] analyzed the behavior of truly typical two sided states where 22 21 is of order the scrambling time. In the bulk it is related to at which the correlator | ], when w . The fact that this expression depends only on ( w t 6 t
∼ /β − ) t ∗ t t | |− w t . − ∗ t | t ( π a class of geometries obtained by perturbing the left side of the thermofield is the black hole entropy and ∼ 2 e 2 w S t = 0 frame, creating a shock wave, as illustrated in the right panel of figure In fact, very little is known about more general states. To this end, we explore in The work of Maldacena and Susskind [ Marolf and Polchinski [ As pointed out in [ The two point correlation function t 1 + Here, we mean time evolution with 2 , there is a time / w section double state with a string of unitary local operators with order-one energy, states in which the localsided entanglements present correlations in become small becausethe of wormhole. the increasing This lengthsmooth suggests of but that the very geodesic the long threading behavior wormhole linking found the in two [ sides. for any choice ofcorrelators in times the for shock wave thetheir geometry two discussed result above. as operators. Marolf evidence and for This Polchinski a interpreted is “non geometrical” in connection contrastThese between authors with the considered the two the sides. time behavior evolution of of the thermofield double state the strength of these effects. the average energy ofThermalization the Hypothesis total [ Hamiltonian local operators on the two sides is typically the shock wave isto very the large. correlation This function makesbut result in likely are this the paper important. possibility weguide that We will are ignore to nonlinear them. currently the corrections exploring We important hope these phenomena. the effects Einstein In gravity results any will event be they a useful should serve as a lower bound to ing nonlinear effects, the correlationgeodesic function connecting was the computed in correlated1 [ points. Roughly, thequence of result the decreases boost liket symmetry a of power the of eternal black hole. It is clear that, for any choice of subsystems smaller than half,when one finds that the leading contribution to on opposite sides,become also small diagnoses if thegeodesics and relation hence probes between the degrees geometry [ of freedom and should of view, the perturbationthe sourced at an earlyshock time disrupts (large the Ryu Takayanagi surfaceThe area [ of this surfacethe is special used to entanglement calculate between the local mutual subsystems information where JHEP12(2014)046 . ] ]. E 27 , we 26 (2.1) , 3 . 25 L ρ ] to amplify the 6 ]. The paper [ operator creates a 14 W , i shock waves. We will outline an iterative n TFD | operators inject some energy into one of the ) 1 – 3 – t W ( W . Certain technical details of the shock wave construction ) are geometrical, but they are not drama-free. In particu- 4 1.3 acts unitarily on the left CFT and raises the energy by an amount at which a local perturbation is detectable in the left CFT. In order plays a central role in the construction, indicating that the geometry n ) and their bulk duals provide examples of how Einstein gravity can ∗ W t t 1.3 , creates a second disturbance and erases the first. This manifestation of ∗ t is assumed to be of order the temperature of the black hole, much smaller than E ]. We consider a CFT state of the form 6 ], it was noted that adding matter at the boundaries of the eternal black hole would We will conclude in section AdS/CFT applications of wide wormholes have previously been discussed in [ In general, the duals to ( The states ( The timescale 24 double [ where the operator The scale contains further discussion of the connection between chaos and2 geometry described here. Wormholes built from2.1 shock waves One shock Let us begin by reviewing the geometrical dual to a single perturbation of the thermofield In [ make a widesimilar, wormhole but describing we less addperturbation, than a and maximal small leaving the amount entanglement. total entanglement ofwormhole near matter, maximal. Our is relying The related examples length on to of are the the the resulting effect absence of of [ local two-sided entanglement [ will emphasize that the classThis of constrains truly the typical possible states form should of be a invariant under smooth such geometricalare boosts. dual recorded to in a two appendices. typical state. distinguished time to make states with weak two-sided correlation, we paylar, the by price boosting of the an geometryinfalling atypical one observer way collides or with another, a one high can always energy find shock a very frame near in the which horizon. an In section AdS boundary and onto the singularity. accommodate weak two-sided correlations,This but is they for are not multipleCFTs, typical reasons. making in the First, the energy Hilbert the statistics space. not precisely thermal. Second, the operators leave a correlation between the two boundaries at all times. is sensitive to chaoticshort-distance dynamics disturbance in in the the CFT.greater CFT. than The The application applicationscrambling of of is a a represented in second, the at bulk time by separation the second shock wave pushing the first off the that these states areprocedure dual that to builds geometries the with geometryexplore one a shock small wave part atseparations of a and/or the time. the diverse Using number class this of ofthe method, shocks metrics two is we dual asymptotic will large, to regions one states becomes finds of that very this the long form. wormhole in If connecting all the boost time frames, indicating weak local If the time separations are sufficiently large, the boosting effect described above means JHEP12(2014)046 1 in acts (2.2) ∼ . Since W is built E 4 1 and large W E , backreaction must be ∗ t Details of the shock wave operator immediately after 5 ∼ gauge theory. ) consists of a perturbation 1 t W N 2.1 = 0 surface increases as t ]. For the remainder of this section, , is sufficiently early, the boost relative to 31 /β – 1 1 t in the large- 28 πt 2 2 ) in different ways. One option would be N Ee , the perturbation is boosted relative to the 2.1 1 ∼ t – 4 – =0) ) consists of a perturbation that emerges from the past t ( p 2.1 E increases downwards on the left boundary. t , which is proportional to N ], following earlier work by [ /G 6 1 operator would then be time-ordered relative to other operators in an ) we will stick to the second, ordering the 1 ∼ t W M To keep the bulk solutions as simple as possible, we will assume that 3 . ; the M 1 . The geometry dual to eq. ( is the inverse temperature of the black hole. Once t β = 0 slice generates backreaction in that frame (right). Note that the horizons no longer meet. With this understanding, the bulk dual to the state ( One can think about the expression ( t For a large AdS black hole dual toIn a our state conventions, with the temperatureNotice Killing of that time order we the have AdS represented scale, the we matter have as a thin-wall null shell. Physical perturbations will have 3 4 5 , a good approximation to this metric consists of two pieces of the same BTZ geometry, , and then falls through the future horizon, as shown in the left panel of figure 1 1 AdS units, while some spatial width, andkinematics they that we might will follow consider massive in trajectories. this paper, However, it because will of be the permissible to highly treat boosted all matter in this way. included. The resulting geometry ismetric sketched are in given the in right [ panel. we will work in theconvenience; (2+1) the dimensional essential setting features of generalizet the to BTZ higher black dimensions. hole. For This small is for technical However, if we increaseoriginal frame, the and Killing the time energy relative to the horizontal where the state vector. that emerges from thet past horizon ofthe the energy black scale hole, of the approaches perturbation the is boundary order at one, time backreaction on the metric is negligible. at time expectation value. Another option isa to strictly understand it time-independent as Hamiltonian. the We statethe will of first occasionally a interpretation, system but use evolving where language with it appropriateoperators makes to before a difference (i.e. for expectation values involving the mass in an approximately sphericallyfrom symmetric local manner. operators We in will such also a way assume that that itto acts understand near the it boundary as of a the thermofield bulk AdS double space. state that was actively perturbed by a source Figure 1 horizon and falls throughthe the future horizon (left). If JHEP12(2014)046 , ∞ (2.5) (2.6) (2.3) (2.4) to + . On the 2 and evolve t 2 −∞ t to 1 t coordinate by amount geometry. The left panel v state by adding a perturbation . W 2 ) on the single-shock geometry W . 2 dφ t 2 /β ( ) ) ∗ t W . In general, the prescription is as . uv − i i 1 operators immediately after the state t Φ − 2 ( | ) π 2 ), we add a perturbation at (1 W , e TFD 2 uv i | 2 ) R Φ ∼ 1 | t ) + ( t and select a bulk Cauchy surface that touches /β (1 + ( – 5 – 1 i W πt ) Φ W | 2 2 t e dudv and evolve backwards in time from ( 2 . On the second sheet (a portion of the left panel of ` 1 M 1 . We add the second perturbation and evolve forwards t E 4 W t 2 4 t − , and the dashed blue line is the Cauchy surface that i to = = α 2 state is constructed from the one- . We record the data on that surface, add the perturbation TFD = 0 surface, with a null shift in the and then evolving forwards and backwards. | t −∞ ds W ) u 2 1 t t ( W ) near the boundary, and evolve the new data forwards and backwards. t ( W . Our prescription to order the , we use the above procedure to build the two- ∞ 2 . The dual to a two- ), we add a perturbation at ]. The two-shock geometry corresponds to a folded bulk with three sheets. On the 2 We can understand this prescription in terms of the “folded” bulk geometries discussed In figure 32 figure final sheet (a portion offorwards the to right panel + of figure means that we focushowever on we the use final each fold of the of sheets the in bulk, our extending iterative it construction in procedure. time from represents the state touches the left boundary atand time backwards in time, producing the geometry shown onin the [ right. first sheet, we evolve from assuming that we alreadyfollows: know we the start with geometrythe the for geometry left for boundary atcorresponding to time To construct the bulk dual, weconstructed simply above. need to In act order with understand to do how to this, construct it is the helpful bulk to dual generalize to our a problem state slightly, and 2.2 Two shocks Next, we consider a state of the form Here, we are using Kruskal coordinates for each of the patches, with metric near the boundary at time glued together across the Figure 2 JHEP12(2014)046 . 6 ], ∗ . 6 t 2 2 /` (2.9) (2.7) (2.8) 2 t (2.12) (2.10) (2.11) ≈ b . ) R  1 2 − t e − 2 M` t is the mass of N is the factor in G M 8 f − 2 . coordinate of the left- 2 r ` , with is the AdS length. The ) 2 v  ` . 2 E ` ) ) + 1 M` t E N M 2 − ( G ` . N 2 + 2 2 2 t G ( , . b = 8 in the two coordinate systems, we ) M 2 2 , the radius of the collision. In turn, Rt/` of the Schwarzschild solution in the ( 2 R r = 8 r 1 e ( /` /` t 2 b N 2 R ) ) r S − 2 2 R f , t t G M ) 8 − − = r 1 1 coordinates are conserved, respectively, by sinh uv uv ( t t l ( ( − v b b f E − 2 ] R R r 2 1 1 + e e  + 35 E – 6 – and ) = . Explicitly, for the 2+1 dimensional BTZ case, , v – − = 2 r = 2 u M ( 1 + 1 33 Ω  coordinate of the right-moving shock is b r R , the evolution is nearly linear and this equation 2 ). To find the precise relation, it is simplest to use d f + r ` 1 2 ) Rt/` as = u ) using the time coordinate at which the trajectory hits t ), we find r r − E b ( v e E v t r − + , is determined by R 2.8 f + 2 + = 2 is the mass of the black hole and or R t and u M dr u M , the size of the sphere, to be continuous at the join. Second, 1 M u ( r = − N t f , while the G 2 into eq. ( M + 8 /` 2 value of their collision as r 1 − t , where b r 2 2 the AdS length. The R fdt r e ` −  − M` 2 ` = N t refer to the top, bottom, left and right quadrants, and 2 G M 8 is determined by ds N − r G 2 8 r t, b, l, r is the BTZ radius in the lower quadrant, defined by − b It is clear from the figure that the two shells collide on the final sheet. Our assump- 2 R ) = r 6 r is set by the time difference (  ( The final, exponentially growing term begins to dominate the first term when ( This determines the Plugging this value of we can determine the value of the left boundary: In particular, in themoving Kruskal system shock of is the bottom quadrant, the where the radius of thethe horizon, black hole and right-moving and left-moving radial null trajectories. Using the Kruskal conventions in [ post-collision region in terms ofr the other masses and Kruskal coordinates. Byfind matching that the size of the If the collisionimplements takes conservation place of energy at ofthe large the equation shells. plays a However, even similar beyond role, the fixing linear the regime, mass where the metric f DTR condition becomes tions of spherical symmetry andby thin pasting walls together make it AdS-Schwarzschildconditions: possible geometries first, to with we construct different require the masses.we full have geometry the There DTR are regularity two condition [ JHEP12(2014)046 . . i v − . In 2 t (2.13) (2.14) = TFD | 2 ) 1 state, the t ( coordinate, Rt/` 1 e and at W v W 1 ) t 2 , one might have t coordinate in the t ( 2 u is carefully tuned to W i . i ) sufficiently early, then 2 and t ( . In this one- TFD i | TFD 2 ) shell in the Kruskal system ) W | ]. ), and the 1 1 t t )] ) becomes approximately equal ( ( 36 2 TFD . t , but at much earlier times it is | 2 ( . If we additionally perturb this 2.11 1 ) W W 2 1 t , this delicate tuning is upset, and 2 /` 2.14 . t t 1 2 ∗ t ( t 2 ,W Rt ) e 1 ≈ W . t ) ) ( 1 M E 1 1 3 t 4 t ( 1 W [ − + † W 2 2 )] t – 7 – 2 /` t 1 ( . We find 2 Rt 2 coordinate in eq. ( e suggests that we interpret the “capture” of the first , indicating that ( /` r ∗ 2 ,W ∗ t t ) Rt 1 ≈ − t − ( v e 1 ≈ W [ | 2 ) a scrambling time before /` 2 is negative, then the shell hits the boundary at time 2 t t . This large commutator is a sharp diagnostic of chaos: perturbing l ( v ∗ TFD fails to materialize. R t h ) operator creates a perturbation in the UV at time coordinate of the trajectory of the − W 1 t t e ( v state to have perturbations near the boundary both at ), plugging in the | ∼ 2 W t shifts earlier, the time at which the original shock reaches the boundary shifts W 2.9 2 − t 1 t | . As ) perturbation approaches the boundary at time We can also think about this effect in terms of the square of the commutator The presence of the timescale To analyze this effect in detail, it is again helpful to use Kruskal coordinates. The key Given that a 1 is positive, then the shell runs from singularity to singularity. To find the t ( v minus two terms involvingAccording to the the overlap bulk solution of the just time described, separation the is overlap greater ofto these than two states once should beone small if quantum perturbs all quanta a scrambling time later [ the perturbation at Expanding this out, we find two terms that each give a numerical contribution of one, boundary and onto the singularity, when ( perturbation in terms ofproduce scrambling. an Indeed, atypical the perturbationstate state by in acting the with UV at time we can use eq. ( left Kruskal system The coordinate becomes positive, indicating that the shock wave has moved off the left first shock, as shown in the right panel ofis figure to determine the of the left quadrant. If If fact, if the time differencecan is understand greater this than by scrambling,W going this back is to not the thevery case. left close panel to In the of the horizon. bulk, figure the we outward If jump we of add the the horizon second due perturbation to the increase in mass will be enough to capture the Figure 3 later, eventually moving onto the singularity (right). expected a two- JHEP12(2014)046 (2.16) (2.15) making this time 7 , , is very small com- w E . t w ± t − states are drawn. In each case, the . i W TFD /β | w ) limit in the gauge theory allows us to take 1 πt t 2 ( e perturbation, allow us to construct the dual N 1 M E – 8 – W 4 ...W . = ) n α t ], these states can be combined as different sheets of an ( n → ∞ 32 . The large- w W t M , one finds a very wide array of possible metrics. We will n directions, have the effect of extending the wormhole to the left, , . . . , t . , with 1 v t 4 , and all odd-numbered times are equal to states described in this paper, we are not sure how or whether they w t and → ∞ W u w t . The thermofield double and the first six multi- 0 and → Because of the null shifts, all but one of the shock waves run from singularity to We will also assume that the asymptotic energy of each shock, Here, we are backing off the limit 7 that all shocks runAt from the singularity level to ofunlike singularity, the leaving the bulk no multi- theory, locally therecan distinguished is be time. constructed nothing in wrong the with CFT. these geometries. However, alternate in the as illustrated in figure singularity. Still, the leftmostlocally one distinguished touches in the the boundary CFT. at One time can also consider bulk solutions with the property E/M held fixed. In this limit,straightforward: the we iterative alternately construction process addcorner, described shocks above and traveling becomes backwards forwards rather in in time from time the from top the left bottom left. The associated null shifts, which By varying the times focus on a particulartimes slice are through equal the to space of these states,pared in to which the unperturbed all mass even-numbered of two-shock collisions. This meanswith that the the matching recursive conditions procedure discussedto for above, arbitrary adding together states a of the form not. Using the“accordion” geometry. time-folded bulk of [ 2.3 Many shocks A general geometry built from spherical shock waves can be analyzed in terms of a sequence Figure 4 next geometry is obtainedleft from the corner. previous The by adding gray a regions shock are either sensitive from to the the top left details or of bottom a collision, but the white regions are JHEP12(2014)046 L ) . (2.17) (const − , and the e S and taking e ∼ α ∼ geometry shown . Let us begin by n . We can make this W α is large compared to large. Such wormhole α by fixing n 8 four- . k S α but not -independent deficit, this is . = 0). The length of such a α for large ∼ n
0 α and/or L , v n 2 α log O α/ n and take the time differences to be + = n  u 2 and 2 2 α α ) is a geodesic in the BTZ geometry passing , by making 5 in the iterative procedure, the geometry to the 1 + S operators source shocks of varying strength. – 9 – 2) to ( k  1 W α/ − for small = are equal in magnitude and alternating in sign means nα cosh , v } n i t { = = 0 ` L 2). Thus the regularized length across the entire wormhole is u / . First, let us consider the case in which . Instead, we could fix 2 α α M , this symmetry would be broken by a smoothly varying mass profile in the E ∼ (1 + 1 − SE ∼ cosh ` . In this case, the energies of the shocks are extremely high . times the length across the central layer of a unit cell. The portion of the geodesic S . A geodesic passes across a portion of the wormhole. It intersects the null boundaries of δM 6 n , then the mass of the left black hole will be larger than that of the right by an S Having computed the length, we would like to understand the qualitative shape of the Using this translation invariance, we can understand the full geometry of the wormhole Our assumption that the Notice that at finite 8 ∼ regions are pushed nearthe the details singularities, of and the almost collisions.in none This figure of should be the clear geometry from is the affected large- by wormhole, increasing from right to left.would also If we be relax broken the by assumption of the equal fact times, that this translation different invariance geometrical computation of thegeodesic correlator estimate is as completely an out upper of bound on control. the Weunit true interpret cell correlator. the as a functionone. of The construction of the geometry is very simple in this limit, because the post-collision geometries therefore describebetween CFT the states two with sides. veryn weak Note, local however, correlation amount that if weof make order geodesic is This function interpolates between length large, and in particular greater than computing the length ofthat the wormhole, passes i.e. from the the regularizedsimply left length boundary of the tothat shortest the passes geodesic right. through this Up unitfrom to cell Kruskal (see an coordinates figure ( can understand this as follows:left after of step all shocks willthat be region during unperturbed subsequent AdS-Schwarzschild. steps The is geometry therefore that independent gets of by built studying in a “unit cell,” for which the geometry depends on Figure 5 the central regions halfway across their width. that the interior region of the resulting wormhole has a discrete translation symmetry. We JHEP12(2014)046 , . s ]. – 2 α − 6 39 37 D – [ to be /` (2.18) (2.19) 2 37 [ 2 s − ` 2 α x,x D T M ∼ to be pure /` 2 2 s α ` 2 N and φ,φ . By requiring G T g M 2 τ,τ T α N is small enough that G α 2 2 dφ 2 ) dφ 2 τ ) ( . In order for . g τ s 2 τ ( g /` + . 3 . The mild boost means that doubling the 2 ` 2 s τ + /` 2 τ 2 ` τdx dx 2 cos 2 ) 1 + sin τ cos , and the shaded regions will be sensitive to ( 2 2 h ` – 10 – log of order M + + 2 τ /β 2 = 0, we find that the metric is uniquely determined 2 α w 1, we have geometries similar to those in figure τ dτ πt sin dτ 2 2 2 e fixed) it is natural to guess that the large kinks of size . ` ` . After a collision, these can be deflected by an angle − allowing an analysis in terms of an averaged stress tensor. 1 − − α S 9 Thus, we look for a solution to Einstein’s equations with αn = , is small. Roughly, we support the wormhole with a large number of = 1 = geometry is shown. Notice that the post-collision regions are small 2 2 ) /α 10 τ ds ) must be proportional to cos W R ds ( (with , which we now restore) as: τ g 1, inelastic stringy effects, proportional to ( R α h four-  collisions before hitting the singularity, but if the initial inhomogeneity is small, α α and /α 1 ` ∼ small enough that the probability of oscillator excitation per collision, α 11 will be smoothed out, . The large- 6 Specifically, we make an ansatz For small values of For intermediate values of We need In a realistic setting, the shocks won’t be exactly spherically symmetric. Suppose we build each shell We are grateful to Raphael Bousso for making this suggestion. 9 ], will be important in determining the form of this stress energy. As an example, though, 11 10 deflections will tend to cancel, and the total effect will remain small. times the number ofrelatively collisions, soft 1 quanta, withmass boost of factor the left black hole only leadsas to a a sum wormhole of ofEach particles length localized experiences on the (up to the scales and compute the stresscosmological tensor constant, implied Einstein’s equations.pure cosmological In constant order plus for that traceless the matter, solution we be find differentiable an at equation for we can ignore theseradial effects. null matter moving in bothsymmetry. directions, and with translation symmetry plus spherical string and Planck scale physics. in figure For most values of 39 we will work out the geometry appropriate for the case in which of the geometry. The central white regioninvariant is in unaffected each by collision details is of of the order collisions, but the Mandelstam Figure 6 and isolated near the singularities. The Kruskal diagram at the bottom emphasizes the kinkiness JHEP12(2014)046 , S limit of the dense ). α τ ( g , the value in a typical state found by S − e – 11 – states aside and address the question of whether . 01, solving the recursion relations numerically, and . W B = 0 α as a function of proper time in the direction orthogonal to 1 S ]. However, as we will emphasize in the discussion section, the 21 ) gives us the translationally invariant part in the interior of the 0 limit (bottom). . By taking 2.19 → A α . The wormhole created from a large number of weak shocks (top) becomes a smooth , and notice that the intersecting network of shocks in the interior of the wormhole 7 In this section, we will put the The metric ( standard phase space measure onfor an energy finding shell a in phase phase spacemeasure. space region. determines For the Typical an probability regions ergodicof are system time those time such with a evolution typical probability system reproducesstates in spends can this this also in probability. be a The defined region fraction as is ones equal that to occur the typically measure in of the the time region. evolution of So the typical system. consistent with a twoMarolf point and correlator of Polchinski [ order states constructed in this manner are not typical intruly the typical two-CFT states Hilbert could space. state” be more described carefully. This by concept smooth is straightforward geometries. in classical First statistical mechanics. let The us define “typical 3 Ensembles In the previous sectiondescribing we weak have correlation discussed betweena a large the family number left of of and shocks geometries right or with CFTs. large long time wormholes, In separations, particular, the by wormhole taking length can exceed figure is matched to the emptyonly exteriors one across direction. a These regionmass regions in profile are which therefore determined the a in shock piece appendix waves of are the moving BTZ-Vaidya in spacetime, with computing the size ofthe the symmetry axis, we find excellent agreement with thewormhole. function To completewith the the geometry, BTZ we need exteriors. to Here, understand how we to go patch back it to together the shock wave construction sketched in geometry in the In order to checknetwork that of this shock metric waves, we actuallyetry write corresponds in down to recursion appendix relations the for small the patched-together geom- Figure 7 JHEP12(2014)046 ) 12 D (3.4) (3.5) (3.6) (3.7) (3.1) (3.2) (3.3)

t r . iE (this time will f − s e 0 r c ∗ r c = 1 is satisfied (up to r X i

ψ then normalizes the probability | i ψ t s ! h N ) provides a natural, but not for nearly all ) max E s 2 | | !  0 s s 2 3.3 E  ) c c | ( . s f || s ) to this case is s i 2 2 E s c s ( f | / c 2 ) | P 2 2 E 3.2 f s | | / 2 t s E 2 = / = s ( c | ) | 2 t s i 2 iE f s is sufficiently chaotic, and that the initial | c 2 0 s | ψ − ). The factor =1 D | c / iE | e s X 2 H S | s − + i| =1 D s c e 2 s − X c | ψ 0 s s | c

c s −| − | – 12 – ∗ s e X ( c

s iHt − c . − = exp e 2 s e 0 s c i d | = 1. The ensemble ( c exp ) ∼ . 2 χ t 2 ) 4 π ( Z d ) |h ∼ i s s ψ ) c ) | ψ s E i | 2 , there is unique distribution that is invariant under U( t ( . This measure gives a natural notion of a typical state. In ( such that ) = d E ψ f s ( | D t  P 1 ( 2 2 ) max E /D i s ( s P 2 χ πf Y | P 2 f ( = 1 Z 2 π  P 2 ) 2 f i 1 s s ψ N X X | ( = = ≈ P i χ | d , the probability is proportional to i D ψ is chosen so that the state normalization condition | is smooth over the spread in energies of the system being sampled, and satisfies the d f f Z We now turn to the question of how time evolution can approximate this ensemble. Quantum mechanics is different. If a state Random matrix techniques show that the eigenstates of a random Hamiltonian are distributed in the 12 surate, so we can findtypically a be time double-exponential in the entropy distribution. In the final equality, we used the normalizationsame condition way for as states obtained by acting a random unitary on a reference basis. In the second equality, we have used the assumption that all energy levels are incommen- we compute which just changes the phases of the Assuming that the Hamiltonian ofstate the is system typical with respectstate to to this distribution, within then a time distance evolution eventually of brings order this one of nearly all states in the ensemble. To see this, where normalization condition unique, notion of a typical state. Note that this ensemble is invariant under time evolution, where small fluctuations), 2 a less completely randomon situation the we energy expect of the states. probabilities A in natural an generalization ensemble of to ( depend transformations. This is given byFor acting large on a reference state with a Haar random unitary. Time evolution does not changephase. the magnitude But of there the are coefficient naturala of notions Hilbert an of space eigenvector, a of only distribution dimension its for the magnitudes. For example, in JHEP12(2014)046 . . i i D S e ψ | e = 0, ) t TFD ∼ B | t − wormhole is i ψ | )? We expect the 3.3 that are typical of the | This overlap is enough to s black hole at time c | i 13 ψ ] and adding a weak “wire” | , after which point states can S 21 e 14 ∼ . II . Now thermalize by evolving is described by a smooth geometry with , we form an ensemble of states that is R i t ψ H | + L = 0 his experience will be the same as falling ). Therefore, after some time the state comes to have coefficients – 13 – s H B E t ) would have to be a surface containing timelike ( = H 9 f 0 , so that all quanta can equilibrate across the wire. An on the figure to be essentially the same as the empty H is also typical, and hence by assumption also described S , Bob will experience a violent collision. ∗ i t ψ | ∼ ) t t ] might be candidates for these. This possibility arose in a discussion with − 40 I,II,III 4, we could take our initial state and evolve it with two different chaotic . If t 4 of any typical state in that ensemble. + Ω where π/ − 0 in eigenstates of π/ = H i B ). By choosing a random time = t t ) for an appropriate TFD | 3.3 H at time i ψ | It typical states are dual to smooth geometries, avoiding this boosting effect would Having defined these ensembles, we will now use their time-translation invariance to The ensemble generated by the wire raises a question of time scales: how much evolu- In our specific situation we will imagine following [ To improve upon the The “mirror operators” of [ 13 14 Juan Maldacena and Edward Witten. the standard ones at the UV boundary of quadrant Hamiltonians (“wires”) for variousdifferent lengths time of evolution intervals. time in various orders. To be safe one should use order eternal black hole.empty This regions is would a have powerfullocus to constraint on be on the joined the Penrose incurves form of diagram some of infinite (figure way such length, onto geometries. quiteIf a different we These long from imagine the this wormhole. intuitive curvethe notion to The dual of be joining of a boost a long cut invariant, the thin off configuration wormhole. CFT. in This quadrant suggests IV that resembles there are other quantum states present than he might experience a mildIf collision. Bob But falls consider the intointo geometry this associated geometry with U( at time require all three regions derive a constraint. Suppose that aa typical long state wormhole. Thenby U( a smooth geometry withboost. a long This wormhole. is Roughly, dangerous: thea two imagine light geometries that ray are part behind related of the by a the horizon. matter If supporting Bob the starts falling into the Another potentially interesting timebe scale written is as the a time rence superposition timescales, if of relevant, naively wouldconstructions orthogonal be of vastly states the longer at previous than earlier sections those are over times. which reliable. the These geometrical recur- operator, with probability better than roughly 80%. tion is required to produce areasonable state to that allow we at may least treat(extreme?) a as time upper typical? bound As is a provided lower by bound, the it quantum seems recurrence time, schematically forward with U( invariant under time translation.expansion How of similar isdistribution this ( ensemble towithin ( an overlap of ensure that the states cannot be distinguished, with an optimal measurement of a linear between the leftimagine and the right wire allowing sides theleft that exchange and of right lets one sides the quantum everyΩ with large system which is thermal number as a of energy smeared thermal between aHamiltonian product times. the whole of Denote local thermalize. this operators wire in by the We an left can and operator right systems and the total JHEP12(2014)046 W (4.1) shock. , create n ∗ t , he would 2 W ∗ are greater t | ∼ i t t ) disturbs the ∼ − n − t ( B t n ; in bulk language, +1 1 i W t − | n t ]. If the time between per- 6 ) has a locally detectable dis- 4.1 , i TFD | ) 1 t ( = 0 and experiences a mild interaction with the 1 B . Roughly, the action of t n – 14 – t ...W ) n t ( different times, if the separations n n W emerges as an important dynamical timescale in the construc- ∗ t . A candidate for the geometrical dual to a typical state? ]. An observer falling through the horizon immediately encounters a 4 [ II Figure 9 shock is captured by a tiny increase in size of the horizon due to the ]. . Bob falls in from the boundary at 1 1 , our bulk analysis indicated that the CFT state ( − ∗ n t The scrambling time Of course another possibility is that typical states do not have smooth geometries W -suppressed backreaction associated to each perturbation [ N state includes operators localthan at turbance only at thedelicate tuning “outermost” required time for athe local perturbation to appear at time turbations is sufficiently large, their shockthe wave backreaction wormhole. must be included, lengthening tion of the metrics. Forkinked example, geometries perturbations with at high widely energy separated shocks,time times, while separation ∆ large lead numbers to of smoother perturbations wormholes. at As smaller a second example, even though a multi- and they provide constructiblelators examples that of are highly small entangledG at states all with times. two-sided The corre- key geometrical effect is boost enhancement of the 4 Discussion In the contexttwo-sided of AdS 2+1 black dimensional holeperturbations geometries Einstein of with gravity, the long we thermofield wormholes. have double state These identified of geometries a two are CFTs, large dual class to of outside of region firewall [ Figure 8 stress energy supporting theexperience solution. a If dramatic he interaction. jumped in at a much earlier time JHEP12(2014)046 . , 0 ]. R 2 n n t R 41 → − α 2 r ) = r ( f ). τ , and the BTZ ( } g n ). We would like to r ). { as 10 τ , we will then be able to 2.19 τ . One of these equations is operators increase the energy n R W (see figure and } n n ) depends on r R n ) at a given vertex, with r { 2.7 = 1. In order to do so, we will write recursion – 15 – R = ` (determined by 1 local correlation between the two sides. S S − e . 3 ∼ , and then compute the geodesic distance “straight up” from the ) in the case n ’th. Identifying this with the interval in τ R n ( g and n r We need two recursion relations, one each for Exploiting the discrete translational invariance of the arrangement of shock waves, Using the methods discussed in this paper it is straightforward to construct states By estimating correlators using geodesic distance, we have ignored the backreaction of One could attempt to build a typical state out of a basis consisting of the multi Although these states display the very small correlation between L and R characteristic states, each described by a geometry. It might seem unlikely that a superposition of relations for first collision to the confirm that the radius of the given simply by applying the DTR relation eq. ( limit, one finds agreement with the smooth metric givenwe in can eq. represent ( theparameters metric of in the terms geometries ofcheck between the the collisions, function radii of the collisions, A Recursion relations for manyIn shock this waves appendix,network we of will intersecting shock write waves. the By recursion solving these relations relations for numerically the in the translationally-invariant Acknowledgments We are grateful toJuan Raphael Maldacena, Bousso, Persi Don Diaconis, Marolf,discussions. Patrick This Joe Hayden, work Polchinski, Stefan is Lenny Leichenauer, supported Susskind, in part and by Edward NSF Witten Grant 0756174. for to represent states with containing a few particlesregion behind is the an horizon. open Constructing and interesting actual problem. field operators in this semiclassical reasoning invalid. the field sourced by the correlatedon operators. the correlation, Although an this should interestingfor provide possibility an is relatively upper that short bound nonlinear wormholes effects might with make high it possible energy shocks running between the singularities the horizon. If typicalkind states discussed are in dual section to smooth geometries, they would haveW to be of the distinct geometries could again be representedin as a expectation geometry, but values, this is the difficult to large exclude: number of off diagonal terms will dominate, rendering of typical states, they areat atypical which a in shock important wave ways. approacheswithout the They boundary. increasing have Also, the a the two-sided distinguished entanglement.entanglement time, is In very sharply a peaked, typical andAnother ensemble, deficits feature are the highly of distribution suppressed these of in the states measure is [ that boosting them gives a high energy shock wave on JHEP12(2014)046 2 ). R τ ( g (A.2) (A.3) (A.4) (A.5) (A.6) (A.1) as a function n r . parameter of the BTZ 1 1 at the bottom vertex. − − n n R b r r u + − = . n n . v 2 that is necessary to reach the n 2 R R r of the side and bottom vertices. = 1, the initial conditions are α 1 u = s . 1 , and the − 1 1 2 n n R . u − − 2 n r
r n − 2 2 n r 2 n 1 + 4 r and ∆ derived above, along with the r , r − . n − n p r b 2 b 2 b 2 n 1 r − 2 2 tan u u u + ∆) + ∆) − R = 2 n b b 1 2 n − − − 2 R u u − R 2 n ( 1 + 1 n n n r r r r − + = . + − + − – 16 – 1 1 + ( n 2 n 1 2 n 2 n r ,R R n R R − = R 2 2 R n 1 1 R = n α α r − − r n +1 n n 2 − +1 r r n R 2 n r 1 1 + = + − R = n n at the vertices is labeled +1 2 R R ), together with these initial conditions, completely determine 1 n r . The radius of the top vertex is then determined by r S n s ’th plaquette is A.4 r 1 n − ), one can check that the timelike distance from the bottom vertex fixed, we solve for ∆, the change in 2.4 b ) we must have 2 tan ) and ( u 2.9 = A.1 and ∆, we find the recursion relation v 01, numerically solving the recursion relations, and plotting b , and assume that we know the radii . n u R . The size of the = 0 = 1. Since the recursion relations are second order, we also need to determine α 1 . These can be found using the two-shock solution: r 2 r For a wormhole that connects BTZ regions with = 1 to the top vertex of the Taking of the total geodesic distance from the initial slice, one finds excellent agreement with The equations ( the geometry. In orderthe to geodesic compare distance with “straightKruskal the upwards.” metric smooth eq. Using wormhole, ( we also need to compute R and radius of the side vertex, Eliminating Let us choose aThen Kruskal using frame eq. for ( this patch in which Now, holding This gives To get the other equation, weparameter proceed as follows. We focus on a given plaquette, with BTZ Figure 10 geometry forming each plaquette is labeled JHEP12(2014)046 ≈ ) ) as V (B.3) (B.1) (B.2) ( V ]. across ρ ( ρ 1 ] C SPIRE IN [ ) along the join. By τ ( However, it is clear that (1976) 107 . arXiv:1307.1796 g 2 [ (2003) 021 16 , from which one finds = 2 dφ ]. 2 04 r A 57 r Rτ . 2 + − )) property of the metric relates the SPIRE R . V (2014) 038 JHEP 1 ) exactly. ) ( IN , ≈ τ τ C V ][ ( ( ) 11 ( Black Holes: Complementarity or dτ 0 0 `drdV τ ρ g g ( Phys. Lett. g τ , + + 2 implies that 2 Z 2 1 JHEP ` )) S , 2 dV V ) using − – 17 – (
) is determined by matching onto the metric in V 2 τ ( -function stress tensor traveling along the null surface. ( r V δ ρ g ( ) = ρ − τ . Rearranging these equations, we determine = ( 2 arXiv:1207.3123 2 ) 2 [ r V ) V ( ), which permits any use, distribution and reproduction in − V ρ ( ) ]. ρ along the surface. The ), and fix V ( = V 2 ( 2 on the horizon, and it increases in the inward null direction (i.e. ρ Continuity of the dV (2013) 062 τ SPIRE Evaporating Firewalls ds )) ) along the matching surface via = IN . This is a portion of the geometry τ 15 02 V 2 ( CC-BY 4.0 −∞ ][ Eternal black holes in anti-de Sitter ( ) 7 2 V ), we were not able to compute τ ρ ( This article is distributed under the terms of the Creative Commons τ 0 ( g − g JHEP 2 Thermo field dynamics of black holes , r ]. . = ( ) across a null slice. In particular, we require that the metric should be coordinate is RV/` SPIRE dτ . Requiring the inner product of these vectors to be continuous across the matching 2 ) 1 IN hep-th/0106112 τ V [ Firewalls? [ 2.19 S ( Re M. Van Raamsdonk, A. Almheiri, D. Marolf, J. Polchinski and J. Sully, J.M. Maldacena, W. Israel, Requiring continuity alone would allowA a surprisingly good approximation to the metric is 0 + 15 16 [4] [1] [2] [3] `g R Open Access. Attribution License ( any medium, provided the original author(s) and source areReferences credited. For our specific these conditions completely fix thecentral geometry, region up of to the the wormhole. undetermined overall length of the Next, invert this to find 2 normalization of the outward-pointing null vectors,the by matching the derivative of thesurface, size of we find follows. First, find The up and to theeq. right). ( Thethe function matching surface. taking the derivative along thethe patching inward-pointing null surface, vectors we in can the relate two the coordinate normalization systems. between In this way, one finds that We will work outlower the panel of matching figure condition in detail for the top left Vaidya region in the B Vaidya matching conditions JHEP12(2014)046 , , ]. Phys. 61 , ] (2000) D 67 , (1991) Phys. Rev. SPIRE Phys. Rev. , , IN D 61 (2014) 067 ][ A 43 03 ]. Phys. Rev. ]. Fortsch. Phys. , , ]. (1994) 888 ]. Phys. Rev. arXiv:0808.2096 JHEP SPIRE Rindler Quantum Gravity , [ , Phys. Rev. SPIRE IN , IN [ ][ ]. ]. SPIRE E 50 SPIRE ]. ]. IN IN arXiv:1211.2887 ][ [ ][ SPIRE SPIRE ]. (2008) 065 IN IN SPIRE SPIRE The black hole singularity in ]. ]. ][ IN IN ][ 10 ][ Phys. Rev. ][ A covariant holographic entanglement SPIRE , Quantum black hole evaporation (2013) 081 IN SPIRE SPIRE ]. hep-th/0506202 IN IN ][ JHEP [ arXiv:1101.6048 07 , ][ ][ , ]. – 18 – ]. SPIRE arXiv:0705.0016 arXiv:1206.1323 [ IN [ Inside the horizon with AdS/CFT JHEP Three dimensional Janus and time-dependent black Holographic particle detection ][ SPIRE On geodesic propagators and black hole holography , hep-th/0603001 ]. arXiv:0708.4025 Mutual information between thermo-field doubles and (2006) 044 IN [ SPIRE hep-th/0306170 [ arXiv:1303.1080 [ ][ IN Time Evolution of Entanglement Entropy from Black Hole Cool horizons for entangled black holes Black holes and the butterfly effect [ 04 ][ Gauge/Gravity Duality and the Black Hole Interior SPIRE ]. hep-th/0701108 (2007) 062 Fast Scramblers Holographic derivation of entanglement entropy from AdS/CFT [ IN Black holes as mirrors: Quantum information in random hep-th/0002111 ]. Excursions beyond the horizon: Black hole singularities in arXiv:1307.4706 [ ][ 07 (2012) 235025 [ JHEP (2007) 120 SPIRE , (2004) 014 29 IN (2013) 014 (2006) 181602 SPIRE ]. hep-th/9304128 09 ][ 02 IN [ JHEP [ 05 96 (2007) 068 , hep-th/0212277 Quantum statistical mechanics in a closed system Chaos and quantum thermalization [ 02 Addendum to Fast Scramblers JHEP arXiv:1306.0533 (2000) 044041 JHEP Extracting data from behind horizons with the AdS/CFT correspondence , [ JHEP (2013) 171301 , ]. , hep-th/9906226 [ JHEP (1993) 2670 111 D 62 , . SPIRE cond-mat/9403051 arXiv:1306.0622 IN [ (2013) 781 holes Lett. 2046 AdS/CFT hep-th/0402066 Yang-Mills theories. I. Rev. (2003) 124022 disconnected holographic boundaries Interiors 044007 Phys. Rev. Lett. entropy proposal subsystems D 48 [ [ Class. Quant. Grav. J. Maldacena and L. Susskind, D. Bak, M. Gutperle and S. Hirano, D. Marolf and J. Polchinski, J.M. Deutsch, M. Srednicki, J. Kaplan, G. Festuccia and H. Liu, J. Louko, D. Marolf and S.F. Ross, P. Kraus, H. Ooguri and S. Shenker, L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, T. Hartman and J. Maldacena, V. Balasubramanian and S.F. Ross, S. Ryu and T. Takayanagi, V.E. Hubeny, M. Rangamani and T. Takayanagi, I.A. Morrison and M.M. Roberts, K. Schoutens, H.L. Verlinde and E.P. Verlinde, L. Susskind, P. Hayden and J. Preskill, S.H. Shenker and D. Stanford, Y. Sekino and L. Susskind, B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, [8] [9] [6] [7] [5] [24] [25] [21] [22] [23] [19] [20] [16] [17] [18] [14] [15] [11] [12] [13] [10] JHEP12(2014)046 , , , , 73 , ]. (1995) Nucl. Phys. SPIRE ]. , IN ]. [ (1990) 1796 B 436 Commun. Math. SPIRE , SPIRE IN D 41 IN ][ [ Prog. Theor. Phys. , (1988) 1615 ]. An Apologia for Firewalls Nucl. Phys. , ]. A 3 Phys. Rev. ]. SPIRE , IN SPIRE ][ IN SPIRE Bulk and Transhorizon Measurements in ][ arXiv:1310.6335 IN ]. ]. arXiv:1311.7379 [ [ ]. , ]. SPIRE SPIRE Superstring Collisions at Planckian Energies Classical and Quantum Gravity Effects from SPIRE IN IN Int. J. Mod. Phys. IN ]. – 19 – Aspects of generic entanglement ][ ][ , gr-qc/9905011 ][ (1985) 613 [ ]. SPIRE Time dependent black holes and thermal equilibration (2014) 086010 IN 99 [ arXiv:1201.3664 [ ]. SPIRE State-Dependent Bulk-Boundary Maps and Black Hole D 89 IN Null particle solutions in three-dimensional (anti-)de Sitter [ ]. Shock wave geometry with nonvanishing cosmological constant (1999) 3465 The Effect of Spherical Shells of Matter on the Schwarzschild The Gravitational Shock Wave of a Massless Particle Internal structure of black holes SPIRE (1993) 307 IN 40 hep-th/0410166 arXiv:1304.6483 arXiv:0708.3691 (2012) 165 quant-ph/0407049 ][ SPIRE [ [ [ 10 [ IN (1987) 81 Phys. Rev. 10 [ , Blue-Sheet Instability of Schwarzschild Wormholes String-theoretic unitary S-matrix at the threshold of black-hole production Butterflies on the Stretched Horizon Commun. Math. Phys. . B 197 On gravitational shock waves in curved space-times JHEP (2006) 95 , (2004) 001 (2013) 018 (2007) 034 , ]. J. Math. Phys. (1985) 173 11 09 12 265 , hep-th/9408169 [ SPIRE IN JHEP Complementarity Phys. Phys. Lett. Planckian Energy Superstring Collisions (1985) 1401 [ JHEP AdS/CFT Black Hole Class. Quant. Grav. 721 spaces JHEP B 253 K. Papadodimas and S. Raju, P. Hayden, D.W. Leung and A. Winter, D. Amati, M. Ciafaloni and G. Veneziano, D. Amati, M. Ciafaloni and G. Veneziano, G. Veneziano, E. Poisson and W. Israel, A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, T. Dray and G. ’t Hooft, I.H. Redmount, K. Sfetsos, R.-G. Cai and J.B. Griffiths, L. Susskind, T. Dray and G. ’t Hooft, M. Hotta and M. Tanaka, D. Bak, M. Gutperle and A. Karch, [40] [41] [37] [38] [39] [35] [36] [32] [33] [34] [30] [31] [27] [28] [29] [26]