Localized shocks

The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.

Citation Roberts, Daniel A., Douglas Stanford, and Leonard Susskind. “Localized Shocks.” J. High Energ. Phys. 2015, no. 3 (March 2015).

As Published http://dx.doi.org/10.1007/jhep03(2015)051

Publisher Springer-Verlag

Version Final published version

Citable link http://hdl.handle.net/1721.1/97174

Terms of Use Creative Commons Attribution

Detailed Terms http://creativecommons.org/licenses/by/4.0/ JHEP03(2015)051 Springer March 10, 2015 : February 19, 2015 December 17, 2014 : : Published 10.1007/JHEP03(2015)051 b Accepted Received doi: . Using chaotic spin-chain numer- Published for SISSA by iHt e x [email protected] , W and Leonard Susskind iHt − e b,c ) = t ( x W [email protected] , . 3 Douglas Stanford a . In a lattice system, products of such operators can be represented ), where t 1409.8180 1 t The Authors. ( 1 We study products of precursors of spatially local operators, c x Gauge-gravity correspondence, AdS-CFT Correspondence, Black Holes

, [email protected] ...W ) n t ( n x Massachusetts Institute of Technology, Cambridge,Stanford MA, Institute U.S.A. for Theoretical PhysicsStanford and University, Department Stanford, of CA, Physics, U.S.A. School of Natural Sciences, InstitutePrinceton, NJ, for U.S.A. Advanced Study, E-mail: Center for Theoretical Physics and Department of Physics, b c a Open Access Article funded by SCOAP ArXiv ePrint: using tensor networks. In gauge/gravitysupported duality, they by are localized related to shock Einstein-Rosentwo waves. bridges descriptions, We generalizing find earlier a work geometrical in correspondence the between spatially these homogeneousKeywords: case. Abstract: W ics and gauge/gravity duality,grows we linearly show in that a single precursor fills a spatial region that Daniel A. Roberts, Localized shocks JHEP03(2015)051 ]. 17 – 15 ]. These 14 – ] studied the 10 5 ], and the pullback- 7 – 4 22 ], which have related conclusions.) We 19 , 18 14 6 21 – 1 – 19 ]. What types of gauge theory variables can describe they do not appear to be helpful. One way of stating the 20 1 ], mutual information and entropy [ 10 3 – 1 10 3 4 12 4 1 ] modification of the standard smearing function procedure [ 18 9 , 8 Two possible answers have been suggested. Hartman and Maldacena [ Arguments have been made that black holes in a typical state do not have an interior geometry [ 3.2 Precursor growth 3.3 ERB dual to multiple localized precursors 2.1 Precursor growth 2.2 TN for multiple localized precursors 3.1 Localized shock waves 1.1 Some terminology 1 time evolution of thethe thermofield growth of double the state interior and and the pointed growth of out a the tensor network(We relationship (TN) are description between grateful of to thewill an state. not anonymous address referee typical states for in pointing this out paper. [ problem is as follows: thea gauge short theory time, scrambles during and whichOn settles two down the sided to other correlations static hand, decay equilibrium and anto in mutual appropriately grow information defined for saturates. interior a geometrythis much of growth? longer the black time hole [ continues gauge theory? A number ofsided probes correlation of functions the [ interiorpushforward have [ been proposed. Theseprobes include work two- well for shortfor timescales, black but holes for in a timescales typical longer state, than a scrambling time, or 1 Introduction How is the region behind the horizon of a large AdS black hole described in the dual B Localized shocks in Gauss-Bonnet C Maximal volume surface and decoupled surface 4 Discussion A Localized shocks at finite time 2 Qubit systems 3 Holographic systems Contents 1 Introduction JHEP03(2015)051 – 24 (1.1) ], two 24 . In [ S e ∼ ] suggested that the interior 20 . w ], these features are also necessary ) of the Einstein-Rosen bridge con- 5 ) (1.2) iHt 1 e t x AdS ( ]. ` 1 x W ] that the length of the black hole interior 25 w ], Maldacena [ 23 , iHt ...W 21 − 22 ) – 2 – e n t , an operator local on the thermal scale. But as ( x n ) = x , we will extend this analysis to the case of spatially W w t W ( 3 x W ], is that the geometry of the minimal tensor network describing is increases, the stress energy is boosted and a gravitational shock ]. In section 20 w 25 t ), and find that it has the same “glued cone” geometry we inferred for w . t ( 2 x W = 0 this precursor is simply w t Spatially homogeneous precursors were analyzed using gauge/gravity duality in [ In this paper, we will continue exploring the relationship between tensor network ge- There are strong reasons to think that the tensor network and quantum circuit de- A second suggestion focused on the evolution of the quantum state as modeled by a advances, it becomes increasingly nonlocal. In a lattice system, such an operator can ]. Their action on a thermal state corresponds to adding a small amount of energy to an w wave is produced.behind Products the of horizon precursors [ createlocalized an precursor intersecting operators. networkhole We of dual will shock to study waves thethe spatial TN. geometry More generally, of we will the see two-sided that black the ERB geometry dual to multiple perturbations, can also be representedterize in in terms section of tensor networks, with geometries that27 we will charac- AdS black hole. As be represented in terms ofcharacteristic tensor TN networks, geometry and of we aalong will single their precursor argue slanted consists on faces. of general two grounds General solid that products cones, the of glued together precursor operators of localized precursor operators, each of the form For t ometry and Einstein geometry.hypothesis, We will following work [ inthe the setting entangled state of two-sided is blacknecting a holes. the coarse-graining two (on Our sides. scale We will consider TN and Einstein geometry associated to products scriptions are essentially thespecial same features, thing. such asnot time-translation A shared invariance QC by and is the unitarityfor most a of a general the special TN TN. gates, case to But which beTN of as are able and noted a to QC in TN. describe representations [ are the It black the has hole same. some interior. Thus we will assume that that the at a given timetime. is proportional The computational to complexity thegenerate is computational the the complexity state, size of and of it theof is the state us expected minimial at checked to quantum a the increase circuit refinementspherical same linearly that of shock for can this a wave geometries relationship long constructed time for in a [ family of states, corresponding to the could be understood asgauge a theory. refined According type toto of this size tensor picture, of network the the describing minimal overall the tensor length state network of that of the can thequantum interior represent circuit dual is the (QC). proportional state. It was conjectured [ Building on this work and ideas of Swingle [ JHEP03(2015)051 . is is w t C B (1.3) (1.4) v )] as the w t ( x ] and provides W such that [ 24 r y ). . For simplicity, let o w ] y t y ( . A localized precursor . However, for a suitably . x y w x ,W ]. W ) W iHt w 20 t might be nonzero in a large region, ( W e x C . In the spin chain system, we w W w [ . at location t † ) iHt ] ∗ y y − . This generalizes [ t e W perturbation has not had an order one − is the rank of the dual gauge theory. ,W AdS ) ` w t w N W t ( ( B x v holographic system, we will use the geometry W ≈ – 3 – is given by )[ N β )] is a local operator at point , where ( 2 w W y ρ t ( N n W x W log ) is associated to a region of influence, i.e. the region in [ w r π In the examples that we consider, this region consists of β t 1) dimensional volume of the region in 2 ) = tr ( | x 2 y − . We will define the radius of the operator , indicating that the W x d − ∗ x | , < t ) of an operator is the scrambling time, and the “butterfly effect speed” is the spacetime dimension of the boundary theory. This expression w w )]. This is the sense in which the operator is space-filling. ; this should be understood as the spread of the butterfly effect. w t t 2 w t ( d ( B t ( v N C 3 x W )] as the ( W [ log w s t π ( β ) is the precursor, and 2 x ) is a precursor of an approximately local operator ], where w t w W ( t [ 26 = ( [ x s is the scrambling time x refers to the space-time dimension of the boundary theory. ∗ W ∗ 1) t A precursor W A localized precursor which local operators have an order-one commutator with d t ) is sum of complicated products, each summand including nontrivial operators at most sites within − d We will see that the radius increases linearly with A central object in our analysis will be the size and shape of a precursor operator. w d t It would be more precise toIt optimize is over important all to operators distinguish growth from movement: the growing operator is not a superposition of • • • • ( 3 2 2( x chaotic system, this step is not necessary: theoperators butterfly at effect different locations. suggests that Ifbut any this it operator were would the will be case, do. W numerically the commutator small. The factthe that region the of commutator size is order one indicates that the operator For the convenience of the reader we will list some terminology used in this paper. is negative for times effect on any localoutwards at subsystem. speed However, after the scrambling1.1 time, the Some precursor terminology grows of the localized shock wave to determine Here q the size greater than or equala to ball one. centered atradius location of this ball. will check this numerically. In the large This behavior can be diagnosed using the thermal trace of the square of thewhere commutator, us consider unitary operators, so that the maximum of this quantity is two. We will define specifically, it agrees on scalesa large wide range compared of to examples relating TN and EinsteinIn geometry the [ spin chain andcovering a holographic region systems that that increases we outwards study, ballistically precursors with become respect space-filling, to the time variable local at different times and positions, agrees with the expected structure of the TN. More JHEP03(2015)051 = 1, (2.2) (2.1) . We 1 g − , we will T defines the 2.1 B v . In our numerics, , the coordinate size T ), where ∗ t , . . . , n )]. In a qubit model, the is similarly defined, but it 2 − w , t ( w is the rank of the dual gauge dec t x , : ( i 1 = 1 N W B [ Z i v . s hZ w = 0 by the action of a single qubit ≈ + t r i iHt where e 1 2 gX th site, )]. Size and radius are related: size is the Z i N w + w t ( ], and one for which it is integrable ( x -volume that passes through the ERB, from +1 iHt i d − 28 W Z – 4 – e = 0 on the right. Σ [ i We will see r t Z 5) [ r. . . ) = i 1 w matrix system through fast-scrambling dynamics, and X t − ( = 0 − T 1 N Z h = × N H 05, . 1 have entropy of order , we will use this pattern of growth to qualitatively analyze the − = AdS 2.2 ` g in the past. w t are the Pauli operators on the 1)-volume of this region defines the size i − ,Z = 0 on the left boundary, to is the spatial slice of maximal i d t 1)-volume of a ball of radius . In section = 8. We will consider two choices for the couplings: one for which the system a time ,Y w i − t x n max X d Σ time maximizes a functional obtained from the volume by dropping transverse gradients. size indicates the numberW of qubits affected at The radius of the affected( region is speed at which the precursor grows. The ( We will study the size of the precursor associated to In this section, we will study precursors in a simple qubit system. In section A perturbation applied to a degree of freedom on a specific site will evolve in two ways: • • • = 0). we use is strongly chaotic ( h The spin Hamiltonian we will use is an Ising system defined on a one-dimensional chain where of time geometry of the minimal tensor network for a2.1 product of precursor operators. Precursor growth on operator growth inabout the fast-scrambling the dynamics complementary of mechanismscan a of see single the spatial cell. important growth Thisnamely phenomena on paper qubits. by scales is studying mostly larger spatial lattices than of low dimensional objects;numerically simulate the system, and observe linear growth of the precursor as a function cells, and so in orderof to a represent cell the should thermal be state no at bigger temperature than it will spread out through the it will spread out through the lattice. In previous studies of unit black holes, the focus was 2 Qubit systems Black holes of radius group. This is the numbertheory. of degrees The of gauge freedom theory of dual a single to cutoff larger cell black of holes the can regulated gauge be represented as a lattice of such JHEP03(2015)051 . g 05, . ... and , the 1 (2.4) (2.5) (2.3) 9 k . The k Y − a Z Z 8 1 Z = − 7 k g Y , or 6 Y X . The reason this , 5 ...X .... 3 2 X X 4 X I = 1 3 ]]] + 1 Y X A 2 , | Z y 1 H,Z − [ X x | 2 c H, [ − w t H, 1 [ c 4 e 3 w , we define either as the number of k 3! it w y = 0. This system is integrable for all t h W ]). This nonlocal mapping relates ]] + 1 k k 32 x W – 5 – H,Z k [ = 1 and 0 c g H, . The size, according to the commutator definition, [ , to spin operators of the form ]. This bound states that , such strings are generated by the Baker-Campbell- k 1 1 k ≤ 2 w ] 2! . 31 t Z y – w t − ), with ) 29 is greater than or equal to one, with ,W ] 2 ) 1 2 ) in a basis of Pauli strings, e.g. X,Z,XX,XZ,YY,ZX,ZZ,IX,XXX,XXZ,XYY,XZZ,YXY,YYZ,ZXZ,XXXZ. 2.1 ] w /c i t w 1 t ( c ( x ( ,A H,Z 1 [ ) W Z ]]]]] = < [ w w t k ( )] it 1 w t H,Z Z − [ ( 1 . The fact that the free fermions propagate linearly in time corresponds, W k Z H, are constants that depend on the Hamiltonian. The norm is the operator [ [ Y r 2 1 − ) = H, , c [ k 1 w t ( , c H, 1 0 [ c 5) of the same spin chain alongside the integrable model, and plot the size of the such that tr[ Z ...X . i 2 H, The commutator is a useful measure of the size of the operator, but having an explicit This integrable system is clearly very special: even typically diffusive quantities such as It is not hard to see that some systems can saturate this linear behavior. A rather triv- [ This is an interesting case where growth and movement are related, despite footnote X 4 = 0 1 H, [ is possible is that the change of variables is nonlocal. Hausdorff formula For example, suppressing coefficients and site indices, one finds the sixth order term in the integrable system it begins to shrink. numerical representation allows us tothink understand the about growth expanding in otherStarting ways. from It the is simple helpful to operator To show that thish is not the case,precursor in we the numerically right analyze panelcorresponds a of to chaotic figure the version staircase plots. ( (solid blue) The and rate integrable of (dashed growth blue)the curves. is operator clearly The grows linear only to significant for the difference both occurs size the once of chaotic the entire chain — in the chaotic system it saturates and X in the spin variables, to a linear growth ofenergy the density operator. move ballistically. One mightis naively also guess exceptional, that and the that linear a chaotic precursor system growth will exhibit slower (perhaps diffusive) growth. ial example is the spinand chain ( can be solved bylocal mapping Jordan-Wigner to transformation a (see systemfermion e.g. of annihilation free [ operator spinless at Majorana site fermions via the non- where (infinity) norm, and the boundfaster) with is distance. valid as This bound longthan implies as linearly, that the the interactions radius of decay the exponentially operator (or can grow no faster and radius are interchangeable.sites As arate function at of which the operatorcommutator grows of can local be operators controlled [ using the Lieb-Robinson bound for the In this setting, where the operator begins at the endpoint of a one dimensional chain, size JHEP03(2015)051 5 . = 8 (2.6) = 0 n h 8 05, . That is, ) is the sum . t 1 7 B ( v k − for a quantum c , with different 2 = : 6 g mn ] )]. The smooth black Left , in the left panel of 5 w t w iHt ( t − 1 is three. Given a Pauli w e 4 Z t [ 3 . It agrees fairly well with ). Notice that the integrable s Y w 2 1 3 t I ( 1 1 Z X 2 . ) in w t k 1 ( k c k k as the sum of the squares of coefficients 0 , as a function of 8 7 6 5 4 3 2 1 0

k c

k } P sites k c k ), evolved with the chaotic . This is illustrated in figure c P { w = 0 Hamiltonian (dotted). t ( iHt h – 6 – 1 8 before the strings can grow in length. This can − Z e )] = 1 w Y 7 t = 1, ( g 1 Z 6 [ ). 2 w s t ( . We plot the 5 k k k c 4 w k t P k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 3 ∝ )] 2 w . We will use this pattern of growth to characterize the tensor networks t ( w 1 t Weight of strings of length k length of strings of Weight operator of Size 1 Z B [ needs to be converted to v 2 : for both types of evolution, the size grows linearly until it approaches the size of the s 1

0

1 0

Z w k ) (t c . Ballistic growth of the operator )] = . Again, the chaotic and integrable systems track each other well until the operator Right w increases, high order terms in the BCH expansion become important, corresponding t 1 ( To quantify this growth, we will group together Pauli strings according to their length. w x t W [ In this section,r we will assumeassociated that to a precursor product operatorsassociated of to grow localized the time at precursors. evolution operator system a To begin, of rate let eleven us sites. review the Each TN diagram geometry represents a formula for [ the staircase definition using thefact commutator. that The initial delay inbe the thought growth of is as due the to the time to “scramble”2.2 a single site. TN for multiple localized precursors becomes roughly as largesmoother as notion the of size entire by system. taking Using these coefficients, we can define a This is plotted as the black curves in the right panel of figure We define the lengthnon-identity of Pauli operator. a Paulistring For string representation example, of as the an the length operator,of highest of we all define site Pauli strings indexfigure of that length is associated to a curves show As to longer and more complicated Pauli strings. Hamiltonian (solid) and theof integrable the squares ofand the chaotic coefficients behavior of is Paulispins). rather strings similar of until length thesystem. strings After grow this to point,operator reach the the begins chaotically-evolving operator end to saturates, shrink. of while the The the chain blue integrably-evolving ( “staircase” curves show the size Figure 1 JHEP03(2015)051 . n i m with . . In 11 R . The TFD n | w iHt 11 − e iHt m iHt and e − e L . The grids of . . . δ 4 i 2 3 n i , and 2 2 i x m 1 i δ t W 1 n , 1 w m can be understood as the δ system, and the right ends iHt on one side of a maximally − = mn L e A A mn I operator, the cancellation would x . Concatenation of tensor networks W . 3 i n i| m | mn – 7 – A mn , and the line endpoints on the right represent the P 11 = i system. Contracting the tensors gives the wave function. i i| i R | , . . . , m i 1 operator is the identity on all sites except for the central one, 5 . m P . If we had not inserted the 3 W A w = ] pointed out the relationship between this linear growth and the geom- iHt . Line segments correspond to Kronecker delta contractions. Thus the Let us now understand the tensor network for a single precursor op- i 5 e . 11 A t | ], we can simplify the network by considering the partial cancellation be- and 24 ). The structure is illustrated in figure w w , . . . , n t . The line endpoints on the left represent the tensor decomposition of the 1 ( ], where we evolve backwards in time, insert the operator, then evolve forwards t iHt x n . The tensor network description of the identity operator (left) and operators − 33 W e , 9 Thought about either way, the basic feature of this network is that it grows linearly with We have presented the TN as giving a formula for an operator, but we can also think Intersections of lines represent a tensor with rank equal to the number of lines. In the D.S. is grateful to Don Marolf for pointing out an error in a previous version of this argument. 5 again. As in [ tween be complete. This meansby that the we linearly can growing remove precursor. thethe tensors After right in doing panel the so, of region we figure that obtain is the not simpler affected network shown in represents multiplication of operators,operator, and in in which the we lefttensor simply panel, network concatenate for we the the haverepresented networks a by for naive the TN dot. forfold This the [ naive tensor network is an explicit representation of a time- time. The work of [ etry of the Einstein-Rosen bridge of the time-evolved thermofieldOne double precursor. state erator this interpretation, the left endscorrespond correspond to to indices indices in inThis the the is an example ofwave a function general correspondence: for a an operator stateentangled given state: by acting with the operator intersecting lines represent a particularis contraction known of that a timenetworks, large evolution using number by a of a technique these local known tensors. Hamiltonian as can Trotterization. It be representedabout in it terms as of a such formula for an entangled pure state of two quantum systems index into eleven indices index as figure at the far left is a formula forfigure, the we identity have matrix: only four-fold intersections, corresponding to tensors Figure 2 successively larger values of JHEP03(2015)051 , at can (2.7) , and B 6 v 3 W . The red w iHt space, as shown W e . w 1 x iHt iHt − e e 1 x W 1 iHt space gives the minimalized TN. . − x 3 localized precursor operators . . . e n n iHt e n x W n – 8 – iHt − e ) = 1 t ( are not necessarily in time order. In fact, the most interesting case is 1 x i t Now, consider a product of ...W ) n t ( n x insertion. In the unshaded region, the forwards and backwards evolutions cancel. W W : the naive tensor network describing a precursor operator from the insertion of the operator, we only need to include a time fold of length ; the rest of the fold cancels. The minimalized TN geometry in figure . This procedure extends to higher dimensions, where one finds a geometry | Left B x . Fibering the position-dependent time fold over the . | 4 /v | x : the network after removing tensors that cancel. The dotted lines indicate contractions; their This “minimalized” tensor network can be understood in terms of a position-dependent We emphasize that the times − | 6 w Multiple precursors. the one in which the differences between adjacent times alternate in sign. distance t be constructed as thein fibration of figure this position-dependentconsisting fold of over two the solid cones, glued together along their slanted faces. Figure 4 The geometry at right is equivalent to the right panel of figure time-fold. Because the region of influence of the operator grows outwards at a rate network represents backwards timethe evolution, green the network black represents dot forwardslinearly represents time growing the evolution. local Shading indicates operator Right the region affected byendpoints the should be identified. Figure 3 JHEP03(2015)051 = . , as O n 5 increases, , . . . , t | 1 x t | . 1 ]). As < t 3 24 < t 2 t − < ; that is, the time location of the fold B /v | ]. Depending on the configuration of times, x | 24 , with a sign that depends on the direction of increases, the folds are pulled inwards. Eventually | B – 9 – x | /v | x = 0. We define the position-dependent time fold as . As ) through-going insertions (see [ x ± | 1 k j t − < t n 3 < t 2 t becomes − -dependent time-fold for the product of three local precursor operators x j < folds, and ( n ) inserted at the same spatial point and with 0 ≤ 1 operators on the r.h.s. . To form the minimalized TN, we cancel adjacent t = 0, we simply have the folded time axis defined by the times k ( = 0 with 0 x W iHt ) x 2 ± . The minimal tensor network describing a product of three precursor operators . Slices of the t e ( W ) In order to compare with holography, it will be useful to emphasize the representation 3 and t ( this will have each fold gets pulledassociated inwards, to by operator an amount the fold. This reflects thecancellation fact that between as forwards we move and farther backwards from the time insertion, evolution. there is At additional certain special values of Even with three operators, the geometry can be ratherof nontrivial, the as minimalized shown in TNwhere figure using all operators position-dependent are time-folds.follows. localized at Let At us beginin the with spherically the symmetric case case studied in [ As before, we canW form a naive TNregions by simply of concatenating forward the andprocedure networks is backward for explicit, evolution each but of outside ittheir the the insertion becomes positions complicated influence are as of varied, the and any for number systems of insertion. with operators spatial increases, This dimension as larger than one. Figure 6 inserted at the second and third folds merge, leaving a single fold. Figure 5 W JHEP03(2015)051 i x (3.1) (3.2) (3.3) , and TFD | ). This AdS 6 ` . We then , we begin } x j t { ]. By analyzing , the procedure } in order; second, i j 26 i i x τ { dx i dx 2 r , t + 2 (1) 0 by f dr t ) − r , we will be able to estimate the e ( 1 − . Fibering this geometry over the and − x 3.2 = f r + 2 dt ) . The minimalized TN is the fibration of this r ( , B , u/v – 10 – d f ) r /v − ( | 2 − ∗ j r r h x , and we minimize the length of the folded time axis − (1) j 0 − 2 τ 2 AdS f ` r x e − = ) = 2 = | ≤ | r Planar AdS black holes have only one scale, namely j ( ds . t 7 f uv ]. − 2.2 j 34 τ [ 1) transverse directions. It will convenient to use the smooth Kruskal | space. In general, it consists of flat regions glued together at the loci , which are defined in terms of − R v x d by a new variable and and j , we will study the geometry dual to a product of local precursor operators. L t u 3.3 must satisfy runs over ( } i i τ In the case where the operators are localized at general positions { We should clarify the role of the thermofield double state and the background black hole geometry. 7 , pairs of folds will merge and annihilate, leaving behind two fewer folds (figure | x Why couldn’t we study theelements same problem of with the vacuum AdSvacuum, precursor, as the the and dynamics background? is its The integrable.small commutators answer The expectation is with relevant that value commutators other matrix in mightblack the operators, be hole vacuum large depend geometry state as allows (this on operators, us can but the to be they study energy. seen have nontrivial explicitly matrix Near in elements. the the spin model numerics). The where coordinates, we can write the metric in terms of dimensionless coordinates as characterized in section 3.1 Localized shock waves Let us start byof reviewing two the CFTs AdS black hole dual to the thermofield double state a gravitational shock wave that distortsthe the transverse Einstein-Rosen bridge, profile as of incommutator [ this with other shock operators, wave, andIn in thus section the section size ofWe the will precursor as find a a function of detailed time. match with the geometry of the corresponding tensor networks, 3 Holographic systems In this section, wewill will be use highly holography geometric: to the study action of the a dynamics precursor on of the precursors. thermofield double The state analysis generates replace each subject to two constraints:the first, the time fold musttime pass fold through over each the of the folds. procedure defines a foldedspace time gives axis the as geometry a of function the of minimalized tensor network. is slightly more complicated. Towith define the the position-independent position-dependent time fold fold, at passing location through insertions at times | JHEP03(2015)051 . 8 π/d . (3.7) (3.8) (3.4) (3.5) (3.6) whose A W = 4 ] for AdS β = 0 frame depends on t is a function 40 0 , = 0 horizon, . a 0 i 26 a u 2 ). By evaluating du ) 7 x ( 10 h direction and stretched i ) i , ]. This connection was made u u i ( dx 38 δ i ) 9 dx uv ]. The connection to scrambling was TFD ) | , ( . ) 36 2 w A t x uv r ( ( L suppression from the gravitational + 0 B ). In terms of these Kruskal coordi- iH i a 0 ) e ) = N r + u ( x dx G ( i 1 direction (see figure uv δ ( W − v dx = 0. The inverse temperature is w f B /β ) 0 t dudv w L ) dr uv uv πt iH ( 2 r uv ]. ∞ e − – 11 – ( 2 B R e ) 35 A r +1 + ( E (1) ) in the = d AdS 0 − f ` x i ) = f = 1, or ( h r h ( r = 4 ∗ dudv uv ) r 2 AdS of the perturbation earlier, its energy in the TFD uu − ` | T ) uv w . However an operator corresponding to even a single quantum will ( t = , the geometry will be affected. w t N 2 A ) = 2 ( x 1, with integral of order one. The precise form of N − ds uv ( W h . . When this exceeds the log A | /β x π β | 2 AdS w 2 ` πt CFT: ], we will act on the thermofield double state with a precursor of an 2 = = e ∗ L 2 26 is not large, then the geometry will not be substantially affected by the t operator is an approximately local, thermal scale operator acting near loca- , the metric will not be exactly of this type. Corrections are analyzed in appendix ∝ ds w ∼ x w t t ]. w is small, then the boost must be large in order to overcome the suppression. The ] for the case of a flat space Schwarzschild black hole, and by [ W ], and the connection to gauge/gravity duality was made in [ t is the dimensionless asymptotic energy of the perturbation, and 26 direction. We can replace this by a stress tensor localized on the 39 37 N E v on the left boundary. We would like to understand the geometry dual to this state. G x The backreaction of this matter distribution is extremely simple. It was worked out If Following [ This potential important of this time scale was first noticed by [ For finite In order to have a well-defined notion of geometry, we should consider a coherent operator 9 8 = 0 with a shift of magnitude 10 energy is fixedlead as to a similar function effectssingle-particle on of case the as commutator giving that the we eikonal estimate phase below; [ themade metric in should [ be understoodprecise in in [ the This metric can beu understood as two halves of the AdS black hole, glued together along details of the perturbation, as well as the propagationfirst to in the [ horizon. black holes. The idea is to consider a shock wave ansatz of the form where concentrated within gets boosted coupling, associated stress energy distribution isin highly the compressed in the where the tion If the time perturbation. However, Schwarzschild time evolutionas acts near we the make horizon the as Killing a boost, time and The horizon of the black hole is at operator in the nates, the black hole metric can be written where the tortoise coordinate is JHEP03(2015)051 (ii) (3.9) , and (3.10) (3.11) | x | direction. chosen to v 2, and , reflecting c w t ≤ C ) in the ≤ by a delta function , = 0 is discontinuous. x ( 0 v o h a ] y ) is equal to one. | = 0) and plugging into , y ) 2 ,W x ) ) − ( u 0 w ( x t a | δ ( , 2 w x t w u π t β W 2 ), the surface ( [ has been defined with , † ] C | ). 2 3.8 y Ee x w | N 1 t µ ) and ( − N u ,W − x d AdS 2 ) ( ) ` log ∗ 2 − δ t w W d πG t π | − β − ( 2 (0) x w x 16 | t 1, the approximation of such that A ( ≈ W | π β . ) = 2 y 1 – 12 – )[ , we find the solution | u e − N β ( ) = x d AdS 0 − E ( | G x c` ρ ( x uδ | ) = n h x
log ( 2 h are both unitary operators, so that 0 π µ β 2 1, the solution to this equation will depend only on the y = 0. In the coordinates ( ) = tr + | = u i W y ∂ ∗ i |  t − ∂ x | x and − | , x slice through the one-shock geometry. The geometry consists of two halves w t x W ( . For C 1) (i) − . In the next section, we will use the interplay of these exponentials to 2 ), which we can replace with a delta function. The differential operator in d x ( x d ( 0 as the maximum distance a ) is the thermal density matrix. To simplify the analysis, we will consider the = w β . A constant 2 t ( ) can then be inverted in terms of Bessel functions. Expanding for large µ ρ 3.9 The strength of the shock wave is exponentially growing as a function of where setting in which they correspond to different bulkat fields. time As before, we will define the radius of the operator 3.2 Precursor growth To measure thecommutator-squared size of the precusor, let us consider the expectation value of the is incorrect; the power-law singularity in the denominator shouldthe be growing smoothed boost out. ofa the function initial of perturbation.determine However, the it growth is of exponentially the precursor suppressed operator as where the scrambling time absorb certain order-one constants. For where integral of eq. ( assuming a thermal-scale initial energy the curvature of thisEinstein’s metric equations, (setting we find a solution if Figure 7 of the eternal AdS blackThe hole, shock glued lies together along with the a surface null shift of magnitude JHEP03(2015)051 , ] ]. i 25 ψ | 42 , the , (3.15) (3.16) (3.12) (3.13) w t 41 , and the inner y operator is a null i W ) equal to one. We x 1 the shift becomes W ∼ 3.10 TFD ) | ] y ) (3.14) y ] for holographic systems − w 7 t – x ,W ( 5 ) , . It is helpful to think about h w ) i (log t w ( t O x direction, what is its overlap with . perturbation is highly boosted, and v − W TFD 1) [ | x 2 † ) (log ] − is not too large, so the relative boost is w d y O N W ]. The inner product of the states t operator creates a mild shock wave, and d ( , the rate at which entanglement spreads. w − x t x 25 E ,W ) 2( , log ) ) v i W ) in the ∗ W 0 E )[ w 1 µ t y s y t ψ x | ( W − − − = x ψ − – 13 – h w x w operator before or after the W = π t y ( t [ ( ( | 2 βµ y h i Re π h 0 B 2 2 βµ W v ψ = | , indicating that the commutator is small everywhere. − ∗ TFD = B t h was recently computed by [ , the region of influence spreads ballistically, with the v )] = w and t = 2 E w t v ) = i ( | ] x y 26 = 0 frame, in which the W − [ t TFD r | x | y , and [ W w ) 2 t ( The speed w N t C ( x 11 log direction of magnitude W π β v 2 . This is the speed at which precursors grow, or equivalently, the speed at = perturbation and shift it by B = i y v is the space time dimension of the boundary theory. The radius is negative for operator creates a field theory disturbance that propagates on this background. perturbation is not. Let us suppose that ∗ ψ | W t y d y then reduces to the following question: if we take the field theory state corresponding It is interesting to compare this speed with If the strength of the shock is sufficiently weak, the shift is small and the overlap is To calculate the above expectation value we will follow the procedure used by ref. [ W W We are grateful to Sean Hartnoll for emphasizing this point to us. less than the scrambling time i 0 11 w ψ which the butterfly effect propagates. On rather general grounds, entanglementlocal should operators. spread no faster than the commutator of Here, t However, for largervelocity values of where therefore determine the size offind the (for operator perturbations by setting with the order formula one ( energy close to one; we recover aof small order commutator. the However, typical once wavelengthproduct in the begins field to theory decrease. excitationoverlap created will Because by be the quite strength small of within the a shock few is thermal exponential times. in At order one precision, we can The difference between applying the shift in the | to the the original unshifted state? where this inner product in the the a small fraction of thethe Planck scale. Then the in the spherical shockUsing the case. purification by Similar the thermofield calculations state, were we also can write previously done by [ JHEP03(2015)051 n 14 . . . h ) and (3.17) (3.18) (3.19) (3.20) 1 x ( h = 4) for n ] for further d 25 .) . . . h ) GB x ( λ 1 h 1 13 d − ) do not have any symmetry, d . . i r 3.20 B v . , even as the shocks make the total spatial GB 1 d , we work out the shock solutions in 1 d  λ − TFD 4 − | AdS B 1 2 1 ) ... ` 1 − 2) = 1 + 1. t + ] shows that maintaining causality requires 1)] ( 1 d 1 − 45 x − p 2 d GB 4 is that the velocity is corrected to d ). This is a manifestation of chaotic dynamics in the ( λ d – 14 – 8 ≥ 1 + [2( − √ ...W d ) , and it exceeds the speed of light (in 1 q n = 1 2  t GB ( ]. In appendix E n λ = v ] was that the TN geometry reflects a coarse-graining of the ERB x . For spatially localized perturbations, there are two 43 ) = n 20 W GB λ . . . t , with equality at ( 1 E ] showed how to construct states of this type in the spatially t B v v The result for 25 , agrees with the geometry of the corresponding TN on scales large ≥ ) slightly enlarges the horizon, “capturing” the preceding shock and causing it 12 ∗ B t max v will be corrected in bulk theories that differ from Einstein gravity. . In general, the geometries dual to ( ], ref. [ B ]. (In fact the recent work of [ v AdS 44 47 ` , 4. In fact, Gauss-Bonnet gravity is known to violate boundary causality for 46 . At this level, the maximal spatial surface represents the entire ERB. 36 [ / . 3 0 AdS − ` − < < The point we will emphasize is that the intrinsic geometry of the maximal spatial slice The speed The timelike interval inside the ERB remains of order We are grateful toA Steve Shenker feature for of suggesting the this. multiple shock construction is that the application of an additional shock (with time 14 12 13 GB GB separation greater than to run from singularityCFT: to the application singularity of (seediscussion an figure and additional detail. precursor disrupts the fine-tuning ofvolume the very large. state. The Seeon conjecture [ scale of [ through the ERB, Σ compared to and finding the maximal surface exactly would require the solution of a nonlinear PDE. determined by thedifferences: times first, thesecond, the null geometry shifts aftersubstantially depend shocks affect on collide our analysis, is the for not transverse reasons generally explained position known. below. These regions will not Following [ homogeneous case, building theperturbations geometry are up small onepatches and shock of the AdS-Schwarzschild, at glued together times a along are time. their horizons, large, with If null the shifts the geometry masses consists of of the a number of an infinite number of massive higher spin fields3.3 for any nonzero ERB value of dualIn to this multiple section, localized we precursors willoperators, characterize the geometry dual to a product of localized precursor This speed increases forλ negative λ Stringy effects are consideredGauss-Bonnet in gravity. [ One can check that dual to Einstein gravity, with the result JHEP03(2015)051 that . The C = 0 slice . Across AdS all shocks ` x 8 ∼ (iii) 1. The , which maximizes a  dec ∗ all odd numbered times t . This was first pointed 2 m r insertions. We will calculate (ii) | − j = t W are negative, and r − = 0, , we use the following fact: in the j +1 ,... j x 4 max t | , t 2 t coarse-graining, becuase both are attracted to – 15 – ], and we will refer to the attractor surface as the AdS 5 ` . Each HM surface is intrinsically flat, so the surface passes through a shock connecting adjacent patches, it AdS ` max , obtained from the true volume by dropping terms involving ). dec in this transition region. However, the transition takes place over We will start by demonstrating agreement with the TN geometry V all shocks are centered near 3.20 max (i) agree at the level of an max are positive and all even nubered times . A slice through a three-shock geometry. The white pre-collision regions are cut and consists of approximately flat regions, joined by curved regions of size and Σ ,... Before we begin, let us make one more technical comment. In the analysis below, we Within each patch of a multi-shock wormhole, the maximal surface hugs the HM 3 gradients. Finding this surface is technically simpler, but we argue in appendix , t dec max 1 A simple example. in a case int which are strong, i.e. adjacentthrough time the differences are geometry large, dual to a configuration of this type is shown in figure volume-like functional, x Σ the same HM surfaces. flat regions grow largetheir in proportion size to and the shape,the time in TN between the associated limit to ( that they become large, andwill match focus to on the the geometry geometry of of the “decoupled maximal surface” Σ surface of that patch.transitions from As one HM Σ surfaceable to to the determine next. Σ an Due to intrinsic the distance lack of ofΣ symmetry, order we will not be exact AdS-Schwarzschild geometry, maximalspatial surfaces slice within the defined ERB byout are a (for attracted co-dimension constant two to Schwarzschild surfaces)Hartman-Maldacena a radius in (HM) [ surface. the separate patches. The darkregions. blue curve As is the the maximal shocksmore surface. grow of stronger, Dark each grey the indicates HM dark post-collision surface. grey regions shrink and the maximal surface tracks In order to understand the large-scale features of Σ Figure 8 pasted from the displayed white regions of AdS black holes. The pale blue curves are HM surfaces of JHEP03(2015)051 dec iHt e (3.23) (3.21) (3.22) ), and 1. The and . 3.21 ) iHt x 1. At smaller ( ). When this − e x ( +1 = 0. j |  h +1 -th segment is x x ) j j | x h ( ) ). This problem was j x x h ( ( j i h h . ) dependence of the position- | gradients, the surface can be increases, the only difference shifts associated to the even- = x | | | x indicates a slight modification ]. But now, keeping the same x i u x . | | | shocks, the intersection of Σ ∗ h 24 t x | . 2 is a fibration of the collection of n (log | over which the length of the folds x O | . As µ x | − x dec − j − ) t 2 ∗ B 2 t − − d − | /v j | t x +1 | x ± | j ( t 2 does not contain π β 2 – 16 – − represents further cancellation of behavior of the length of the e ∝ | shifts associated to odd-numbered perturbations dec ∗ t ∗ j t t v h 2 -independent shifts ) = ]. The logarithm in x , by solving a maximal surface problem in a spatially +1 x + 1 segments. Two of these connect to the asymptotic x j ( . This takes place when the length of a given segment | − 24 j j h n t 6 h . The . The essential point is that two of the shocks become very , the geometry of Σ − log x 9 2.2 +1 ]. For a configuration with j 5 t | is varied, all but an order one contribution to the length will come x 1 of them pass between shocks. All but an order-one contribution − and the logarithm, this agrees with the n ∗ t ] one finds that the large ], following [ . , the singularity in the denominator should be smeared out over the thermal 24 24 | x | AdS (counting from the right), we have a null shift of magnitude ` = 0 is a curve made up of j ∼ x There is an interesting point regarding nonlinearities during this transition. Classical This analysis is enough to cover the regions of So far, this is identical to the homogeneous case in [ Because the defining functional for Σ vanishes. In terms oftransition the is shock sketched in wave figure profiles,weak, this so corresponds the to resulting geometry effectively has two fewerGR shocks, nonlinearities in in keeping with the the collision TN. of shocks are proportional to from the region nearapproximately the flat. flat HM Varying segments, surfaces, that thus is, all a but fibration of an the order-one folded pieces timechange, of axis. but Σ their will number beannihilation remains of constant. folds, We as also in need figure to understand the merger and Apart from the dependent folds from section during single-site fast scrambling [ of linear growth, but it ispoint subleading is in that, the even limit as that the segments are large. An important configuration of shocks, we consider awill slice at be nonzero that thethe shocks segments will are be weaker, correspondingly according shorter, to the transverse profile in eq. ( analysis in [ These segments can be identified with pieces of the folded time axis at homogeneous shock geometry with studied in [ with boundaries, and to the length of these segments comes from regions near the HM surface. Following the numbered perturbations (left-moving shocks). Thisvalues form is of accurate for scale constructed independently at each The upper sign is(right-moving appropriate shocks), and for the the lower sign is appropriate for shock JHEP03(2015)051 3 h 2 h ) are (3.27) (3.25) and 3.20 2 h 1 , the strength h x such that either B k . , the shocks become j /v | | 0 x k | j > x ) rapidly becomes much B − region in Σ. x . This corresponds to the ( ∗ /v ) is of order one, nonlinear x | x k ) (3.26) . At left, both +1 ( AdS j ) (3.24) 1 j > t ` 1 6 x − h | − | | +1 k − ) j 1 j j k − x t j − x h ( t k ) x j j − − t x h − ( k x j − | j k − t j h t k B j t | − | | /v j | sgn( t | > − +1 k . The reason is that as we vary j B 1; nonlinear effects are small and the geometry has x ) = sgn( – 17 – ) = /v k | j  k AdS − t j ` 3 be the least index greater than t +1 x k h − j 2 − such that x +1 h − | +1 k , j k +1 ∗ | j − j k t x t j | 2 x t ). We do not know the geometry in this region, and we are | − sgn( | − | k sgn( 9 ]. There, as here, one can ignore through-going operators. In j 1 t a switchback: − 24 k j k − becomes order one (middle) and the surface passes through a nonlinear Given a general time-configuration of homogeneous shocks, the t j through-going: 3 location of interest is very weak. k +1 − h j k 2 j x t h +1 | k j be the first index t | 1 j . The ERB implementation of the transition in figure = 0. As a recursive step, let Let us therefore begin by determining which operators in a general product ( t or the insertion makes at the insertion makes the localized case, shocks cantheir also profile be at ignorable the because they are sufficiently farrelevant. away Let that first shock with appreciable strength in the frame of the maximal volume surface anchored smaller than one. The general case. only subtlety in constructing“switchback” Σ operators comes [ from the distinction between “through-going” and far from the post-collision regions.effects are However, when still important,post-collision but region the (figure shiftsunable are to small characterize the enough shapea that small of Σ piece the of can maximal Σ, passof surface. of the characteristic through Fortunately, size this shocks the corresponds changes exponentially, to and the product weaker. Eventually post-collision region. At yeteffectively larger one shock (right). The transition occurs over an order product is large, the effects are strong, but the null shifts ensure that the surface Σ passes Figure 9 are large, so the shaded post-collision regions are far from Σ. Increasing JHEP03(2015)051 ]. 23 , , and (3.28) (3.29) k j 22 , 20 space, and x ). In this case, x ]. The first line ( k 24 j h . . j B ]. But entanglement may /v | 52 j ), will be at least order one. – x x ]. ( 48 − 24 . +1 | x k j j x h − − | This implies that both the TN and ) 2 x | x 2 B − ) is stronger than ( µ d 15 x k | − . j /v ( ) j | ∗ h x t +1 +1 k − 2.2 j j − j t x h x ± | ( − – 18 – π β 2 x e − | ∗ ) = t , because the single-site “scrambling time” is order one for a qubit x 2 ( j 2.2 h | − j + 1) of the folded time axis is generalized to t j − system with a single-site perturbation [ ], spherically symmetric precursors were studied. Because of the . After extracting this subset, we further discard all through-going +1 x j N 24 t | ) in a large t − , as described at the end of section ( was not present in section becomes ignorable for reasons similar to those explained in [ x U ∗ k t j States obtained through the action of precursor operators provide a setting to test Each shock in this set now corresponds to a switchback of the time fold, so the geometry The ) and t 15 ( precursors studied in thislocal paper geometric provide properties an of opportunity ERBs and to TNs. relate a muchsystem. wider Its range appearance of inU this equation is consistent with the interpretation of extra cancellation between conjectured that these quantities are related to the geometry behind the horizonthese [ conjectures. In [ symmetry, it was only possiblea to single relate information-theoretic a quantity single — geometric complexity. quantity By — contrast, ERB-volume the — to spatially localized connection between spatial connectivity andnot entanglement be [ enough; the growth ofcontinues entanglement saturates to after a very evolve short time, forsubtle while than geometry a entanglement very entropy(in long to contrast time. encode to this classicallong evolution. Evidently systems) time, there as computational In does is complexity lattice the evolves need quantum form for systems for of an the quantities exponentially minimal more tensor network describing a state. It has been therefore that they agree. 4 Discussion Connections have been found between quantum mechanics and geometry, most notable the Although we will notsolution give to a the formal minimizationlocation problem proof, that one defines can the checkthe cross that ERB section this geometry of are the set fibrations minimal of of TN intervals the at is same the collection of intervals over the and the length of segment ( this transverse position shocks. To simplify notation, we re-index the remaining shocksis by nearly identical toprofiles are the replaced simple by case considered above. The only difference is that the the second condition ensures thatshock the shock of the second conditionthat ensures the that product the ofOperators the fold strengths failing switches of both back, the conditions and shocks, correspond the to second weak line shocks ensures that only mildly affect Σ at The first line of the first condition ensures that the time fold is through-going at JHEP03(2015)051 ]. ). u 55 ( δ to infinity and the mass w t The idea is one of universality: is finite, there are corrections to 16 w t – 19 – ], was that the structure of the minimal tensor held fixed. If 20 uu T ]. 43 ] for a description of continuous-time complexity. 54 , 53 generates the same state as one on the right. For a general product of precursor 3 It would be interesting to understand how wide this collection is. Strong coupling More precisely, we found a match between the TN associated to a product of precursors Our basic hypothesis, following [ But see [ 16 This solution is appropriate inof a particular the limit, perturbation where we tothis take zero, solution. with For example,source the point. shock Exact must solutions be with confined this within property the in future planar BTZ lightcone can of be the obtained from [ necessarily reflect the views of the John Templeton Foundation. A Localized shocks atIn finite the time main text of the paper, we presented a localized shock solution proportional to grateful for the hospitalitypart of by the National Aspen Science Center FoundationDepartment grants for of 0756174 Physics. Energy and This PHYS-1066293, underwas work and grant made was by Contract possible the supported Number in U.S. in part DE-SC00012567.dation. through the This The support publication opinions of a expressed grant in from this the John publication Templeton are Foun- those of the authors and do not Acknowledgments We are grateful tothe Don DOD Marolf through and the Steveis NDSEG Shenker grateful Program, for by for discussions. the the Fannie D.R. hospitality and is of John supported Hertz the Foundation, by Stanford and Institute for Theoretical Physics. D.S. is is to understand correctionsduality, to the the effects pattern of ofbulk. growth finite at The gauge role finite theory of coupling. inelastic couplingis and In the translate stringy gauge/gravity subject physics to in of stringy the [ context corrections of in precursors the and shock waves a continuum quantum field theory usingthe pattern tensor of networks. growth ofby precursor operators a leads wide to collection a of geometric quantum description systems. that is shared plays an important role, since precursors do not grow in free theories. A natural problem operators, we found ageometry match implied by between general the relativity. “minimalized” TN andin the a large-scale strongly spatial interacting latticecharacterizing an system, analogous and product the large-scalethat in spin features the systems of holographic are dual the theory. to ERB We black holes, do geometry or not that mean we know to precisely how imply to describe states of network, encoding the instantaneousreflects the state Einstein of geometry of theone the that holographic ERB. defines We boundary complexity emphasize — theory, thatmany that it directly TNs seems is to that the be can minimal pickedfigure TN describe out — by a the general given relativity. state. There are For example the naive TN on the left side of JHEP03(2015)051 . w t B , = 0. (A.7) (A.5) (A.6) (A.1) (A.4) A v , ) i : , x h 0 along u, . ( du uv uu A = 0. h t = 0, traveling from 0 in the past, with . v B du T w ν vB 0 t u dx − 0 coordinate should remain µ Z uu u h  N ) (A.2) dx 2 i i ) (A.3) ii , this is ∂ i µν . , A h πG h ) ) dependence. h i i B u v, x 1 . 1 2 ∂ γ v, x + of the N component of Einstein’s equations. , x − 1 i v, x A B 0 − 0 G − ) = 8 u 1 i (0) - u  dx − u, i u uu is small, the perturbation can be very A ( u , x γu, γ h ∂ 0 ( dx
uu γu, γ N , ) ( u . h 0 − uu γu, γ G ( ui u uv uu u∂ H ( ii ui ( = 0 horizon. Physically, it is clear that the du 2 A T ii (1) refers to h 0 − γ γH H B h i N – 20 – u . If u O v u t ∂ 0 + components of the metric become larger, but the ∂ u B Z 3 πG v∂ ) = ) = ) = ∂ direction of a null curve near i i i a 2 ui Λ. To linear order in  v h − ) = ) dudv 2 + i uu i ) B g x x < v A 2 ( − u, v, x u, v, x u, v, x , x and uv ( ( ( + 0 ( δv + ii , 0 ui uu A R 0 h uu h h B u AB h − uu ( (1), where the g 2 is the backreaction of a perturbation at time ui O = 1 2 h − 2 N 2 i ). To do so, we consider the H − ∂ , the G x ds D 2 ( γ a uu h ∼ = R ) )+ i , and . We would like to show that this quantity is approximately equal to the i uu 0 = w x t , x E u ( 0 π β 2 uu , , this is the shift in the δv = 0 grows large. That is, the effect on a late infaller of an early perturbation (on e ) E h u 2 i u γ ( = ∂ 1 µν γ a h We will take this component of Einstein’s equations, and integrate it Let us consider the quantity As we increase We begin with a planar black hole metric with an arbitrary perturbation Without constructing the exact solution explicitly, we will show in this appendix that − 0 = 0 to a ( Einstein’s equations set this equal to 8 Integrating by parts, we find where the first term was simplified using the background equations of motion for For small u shock wave profile We define finite as the same side) does notform grow with the time separation. We will use this fact below, in the where entire profile is compresseddisturbance to towards the the geometry at an order one value large. This solution can be constructed by applying a boost to a reference solution the shock profile issmall accurate in this for region, but it will be large compared to We are interested in the response to an order one source at time fully tuned high energyshould insertion be at low the energy, boundary. andprinciple, the In the solution our exact will setting, linearized not theUnfortunately, backreaction be these boundary can precisely functions operator be localized are calculated on not using the known retarded light exactly cone. propagators. for the In black hole background. However, those solutions are precisely localized on the light cone, corresponding to a care- JHEP03(2015)051 (1) 0 f (B.3) (B.4) (B.5) (B.6) (B.1) (B.2) (A.8) . That N , and  G ) β 2 R ] is + 59 . – µν in the metric. This (1) 57 , R ] i O i µν N N dx R G i , 4 4, the Gauss-Bonnet term # ) ) x dx d − ( ) + 2 − i 1 r d < r ) will therefore agree with the − i , x d + , as in Einstein gravity. x − ), but the transverse dependence µνρσ 0 δ ( d 2 R u, (1 ∝ GB δv . For , dr ( λ ) 1 (1). The strength of the perturbation α 0 x uu 3) . GB − µνρσ ( ) f (1) =  λ 0 ] h r R − 4 ( ( f
2 AdS d du T GB f 2 α ` evaluated at the horizon. Since the metric − λ 0 17 µ ) rescales the source, effectively changing the π/N 0 . 1 u 4 + + 2)( 0 B GB + 2 was modified. The error did not propagate elsewhere in GB q Z − v − i λ 1) = 4 – 21 – ] λ dt w ∂ 1 d N − t 2 i ] N β ( − ∂ 1 . The solution 2 AdS p , but because the relationship between N d A, B, B ` " πG γ ) , we therefore have − = ( w t r γ ) = d ( π α ) β 1 + 2 GB 2 f r  + GB λ e ) = 8 GB 2 1 2 λ − i R ( (1) in λ x h ( B  = O v ] with a negative cosmological constant, the action is ) = 2 g ] in the region that this profile is large compared to δv 2 AdS r ) | 56 ( ` − N 2 i x (1 + 2 , we pass to Kruskal coordinates and assume a shock wave ansatz . ). Plugging into the GB equation of motion, with a stress tensor | f ∂ (1) at large √ . The factor (1+2 are related to µ = w 1 t x O − i 1) 2 a 3.8 , and Gauss-Bonnet coefficient 3.1 B a w − 2 t +1 v d = 1, and one can check that ds − d π ( β are 2 d 0 d AdS r = ), we find the condition ` e a 0 x ( w Z = u N ( t 1 2 G π N relationship between boosts and − 2 µ βµ d w δ 1 t a dimensionless parameter. The planar black hole solution [ πG ) < π β 2 u | 16 ( e x δ GB | ) is unchanged. The important difference is the presence of λ = x Following section ∝ ( D.S. is grateful to Juan Maldacena for pointing out a mistake in v1 of this appendix, which stated S h 17 uu , up to a correction that is still grows with thehas boost been factor rescaled, one finds that the the paper. where again scrambling time by a smallof amount of order log(1 +changes 2 the temperature such that now The horizon is at as the same formT as ( with with with AdS radius is topological. It will be convenient to rewrite this coefficient as shock profile is, for B Localized shocks inIn Gauss-Bonnet Gauss-Bonnet gravity [ components at This is the same equationγ satisfied by the shock profile. We see that it is accurate at large where the constants JHEP03(2015)051 (C.1) (C.2) , i is much easier i dx dec i . The left panel is . = 0. 2 dx ) u 10 ) v i ∂ uv direction. ( coordinate. Here, we will ( 2 x B x − d and the relationship between + 4 B i 2 dec A − ,Σ v : the portion of the surface to the left of v dudx 2 at the horizon i u / max ∂ ∂ ) ) 1 x ). The pullback metric is then i − ( Right d uv h ( A u, x = ( AB v v − − – 22 – 2 gradients are small. However, Σ r x v du xdu is negative, so the first term is positive. The decoupled u 1 ∂ v − ) u d slice through the shock wave geometry, with a slice of the ∂ d uv slice through the shock wave geometry, with the corresponding x ( Z x A − d AdS h ` obtained by dropping all gradients in the 2 AdS = 2 ` dec V = V : a constant b dy is given by dropping the second term inside the square root. a = 0 at the left boundary, and Left is the maximal volume codimension one surface crossing the wormhole, anchored t is the “decoupled maximal surface.” Specifically, it maximizes a modified volume . dy = 0 on the two asymptotic boundaries. dec t ab V max dec G Σ at Σ functional, We can parametrize the surface using We can use the symmetry of the shock wave geometry to reduce the problem to one • • It is useful to keep involume mind that conditions and the volume of both pieces is in the unperturbed blacka hole representation geometry. of We a illustrate constant maximal this volume in surface figure shownside in of blue. the shock Onregion (the the is other right, a is maximal we volume related display surface by in only a the the unperturbed symmetry). region black The hole on geometry, portion one with of boundary the surface in this The coarse-grained features of these surfacesthe are volume very comes similar from to a eachto other, region work because where with, most because of thestudy defining the equation relationship is between decoupled the in surfaces the in the example setting of a single shock wave. C Maximal volume surfaceIn and this decoupled appendix, surface we willthem. give First, some let details us about recap Σ the definitions: Figure 10 slice of the maximalthe volume shock surface is shown a in maximalconditions blue. volume shown. surface The in full the surface unperturbed is black obtained hole by geometry, with gluing the two boundary such pieces together. JHEP03(2015)051 , ) is . In u is an | (C.4) (C.3) ( const ) v r m ( = r , we plot f -derivative | w = vu 11 t will be very p r 1 − dec d r ), the surface tends x and Σ . In figure

(

h

- V - V dec dec dec ) (Σ ) (Σ will be proportional to dec V max dec , 0.4 0.2 0 1 0.8 0.6 . ) ) x ( , the decoupled surface 6 = 1 + 1 theory on a spatial circle. , the size of the precursor saturates, dec h x π d (Σ ∼ log ]. For large 5 ) maximizes 5 is maximal. The second follows from x ∗ dec t 1 V − − dec d w ≤ 4 max d t . The surfaces Σ ) * 1 h , most of the surface is near this radius, and ≥ −t ) h w max x 3 t ( – 23 – h , that maximizes the function (Σ Z m V r 2 ∝ ≤ ) will differ, but the regularized volume in this region is ) dec gradients are small. To put it differently, 1 ), for the BTZ setting of . For large dec ∗ dec t x (Σ (Σ max − dec 0 V -derivative of the volume. 0

] using techniques from [ w , we can find upper and lower bounds on the volume of the V 40 20 80 60

t w

120 100

V and Σ dec dec t ) (Σ 24 dec and Σ for any surface, and that Σ ). The gap is quite small, and is roughly proportional to the , the regularized decoupled volume of Σ max . x dec h . Numerically, the gap between the bounds is quite small, and appears dec C.3 ∗ V t . ≤ x − w V coordinates, this special radius corresponds to a surface given by t . The blue curve (left axis) shows the decoupled volume of the decoupled surface, ) and the gap in the bounds for a shock in the BTZ geometry, as a function of u, v ), as a function of ( , the surface Σ dec dec m Integrating over In the spatially homogeneous case, the contribution to the volume coming from the We can explain this as follows. At a fixed value of Using the surface Σ r (Σ (Σ = dec dec r subleading at large region near this surfacesimilar is in proportional to this log region,attractor because for both Σ the surfaces will therefore agree at a coarse-grained level. Away from the special surface given by finding aproblem maximal was studied surface in into [ a hug sptially a homogeneous fixedterms shock of radius background. in theindependent This interior, of the fact that V the strength to be proportional to the maximal surface, The first inequality follows from the fact that Σ V Initially, the volume grows quadratically, but afterand a ( the volume growslower linearly. bounds in The eq. red ( of curve the (right volume axis) itself. shows the gap between the upper and Figure 11 JHEP03(2015)051 , ] , ]. Phys. (C.5) ]. , D 67 SPIRE Phys. Rev. SPIRE , IN (2004) 126007 IN ][ ][ D 70 ]. hep-th/0506118 Phys. Rev. [ , SPIRE ]. Holographic probes of IN and other spaces x. ][ ]. 1 Phys. Rev. 5 ]. ]. ]. AdS dynamics from conformal ]. − , hep-th/9808017 arXiv:1211.2887 d [ SPIRE d [ (2006) 086003 AdS IN 1 Local bulk operators in AdS/CFT: a , and in fact proportional to the SPIRE ≥ SPIRE ][ SPIRE SPIRE ) h SPIRE The black hole singularity in ]. IN x IN IN IN ( Bulk and transhorizon measurements in IN ][ h Black holes: complementarity or ][ ][ D 73 ][ ][ Z ]. (2013) 081 SPIRE ∝ ) IN (1999) 104021 arXiv:1102.2910 07 ][ [ ]. SPIRE max – 24 – IN Constructing local bulk observables in interacting Phys. Rev. (Σ ]. D 59 ][ , Inside the horizon with AdS/CFT JHEP V arXiv:1305.7244 SPIRE On geodesic propagators and black hole holography , [ − Mutual information between thermo-field doubles and IN arXiv:1201.3664 hep-th/0306170 ) arXiv:1311.1200 arXiv:1207.3123 SPIRE [ [ arXiv:1303.1080 ][ Time evolution of entanglement entropy from black hole [ [ [ IN dec [ A framework for the landscape ), which permits any use, distribution and reproduction in (2011) 106009 ]. (Σ Phys. Rev. hep-th/0002111 , [ dec V D 83 (2012) 165 (2004) 014 hep-th/9905186 (2014) 011601 Entanglement tsunami: universal scaling in holographic thermalization Entanglement growth during thermalization in holographic systems SPIRE (2013) 062 (2013) 014 [ IN 02 10 CC-BY 4.0 (2014) 066012 02 ][ 05 112 hep-th/0212277 This article is distributed under the terms of the Creative Commons [ hep-th/9808016 (2000) 044041 D 89 Phys. Rev. JHEP JHEP , JHEP , , , JHEP , ]. On the construction of local fields in the bulk of , (2000) 066007 D 62 SPIRE IN hep-th/0408133 [ AdS/CFT firewalls? Anti-de Sitter space-times D 62 boundary view of horizons and locality [ AdS/CFT field theory Phys. Rev. Lett. Phys. Rev. AdS/CFT disconnected holographic boundaries interiors Rev. (2003) 124022 I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, H. Liu and S.J. Suh, H. Liu and S.J. Suh, B. Freivogel and L. Susskind, I.A. Morrison and M.M. Roberts, T. Hartman and J. Maldacena, J. Louko, D. Marolf and S.F. Ross, P. Kraus, H. Ooguri and S. Shenker, L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, D. Kabat, G. Lifschytz and D.A. Lowe, A. Almheiri, D. Marolf, J. Polchinski and J. Sully, V. Balasubramanian, P. Kraus, A.E. Lawrence and S.P. Trivedi, I. Bena, A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, -derivative of the volume. [9] [6] [7] [8] [4] [5] [1] [2] [3] w [14] [15] [11] [12] [13] [10] Open Access. Attribution License ( any medium, provided the original author(s) and source areReferences credited. It follows thatt the difference is subleading at large while the difference will be proportional to JHEP03(2015)051 ]. , , ] ]. SPIRE ]. ]. ] IN [ Commun. , Phys. Rev. D 90 Commun. , , Phys. Rev. SPIRE SPIRE , (2012) 065007 (2014) 067 IN IN , [ ][ 03 ]. (2014) 046009 D 86 arXiv:1312.3296 Phys. Rev. arXiv:1007.3957 [ , (2013) 034 [ SPIRE D 90 JHEP arXiv:0808.2096 IN , [ (2003) 021 [ 10 arXiv:1402.5674 An apologia for firewalls (1988) 1615 . , ]. 04 Phys. Rev. KITP “Fuzz or Fire” conference A 3 , (2014) 046 ]. JHEP arXiv:1210.6996 SPIRE , Phys. Rev. [ (2008) 065 JHEP 12 , IN (2011) 050405 ]. , ][ 10 SPIRE ]. IN 106 Quantum black hole evaporation JHEP ][ SPIRE arXiv:1311.3335 ]. , IN arXiv:1008.5137 (2013) 012 , Unitarity and fuzzball complementarity: ‘Alice JHEP , talk given at SPIRE , ]. ][ , Classical and quantum gravity effects from IN 09 – 25 – ][ SPIRE Int. J. Mod. Phys. , Cambridge University Press, Cambridge U.K. (2011). , Strong and weak thermalization of infinite IN SPIRE ]. ][ IN JHEP arXiv:0708.4025 , ][ [ Phys. Rev. Lett. The finite group velocity of quantum spin systems Multiple shocks Black holes and the butterfly effect , SPIRE Spectral gap and exponential decay of correlations Complexity and shock wave geometries math-ph/0507008 Gauge/gravity duality and the black hole interior ]. ]. ]. ]. IN [ Fast scramblers ]. ]. Black holes as mirrors: quantum information in random [ arXiv:1307.4706 [ (2007) 120 SPIRE SPIRE SPIRE SPIRE arXiv:1304.6483 SPIRE SPIRE [ IN IN IN IN IN IN hep-th/9304128 09 [ ][ ][ ][ ][ Cool horizons lead to information loss [ (2006) 781 ][ Eternal black holes in Anti-de Sitter (1972) 251 Disrupting entanglement of black holes Locality in quantum systems arXiv:1406.2678 Spacetime from Entanglement [ New concepts for old black holes Addendum to computational complexity and black hole horizons Computational complexity and black hole horizons 28 265 JHEP Quantum phase transitions Entanglement renormalization and Holography , (2013) 018 (2013) 171301 ]. ]. 09 (1993) 2670 111 SPIRE SPIRE IN hep-th/0106112 arXiv:1405.7365 IN arXiv:1306.0622 arXiv:1307.5915 arXiv:0905.1317 D 48 subsystems [ [ planckian energy superstring collisions Math. Phys. [ nonintegrable quantum systems Math. Phys. [ [ arXiv:1403.5695 (2014) 126007 [ August 19–30, University of California, U.S.A. (2013). [ JHEP Lett. fuzzes but may not even know it!’ P. Hayden and J. Preskill, Y. Sekino and L. Susskind, J.M. Maldacena, D. Amati, M. Ciafaloni and G. Veneziano, K. Schoutens, H.L. Verlinde and E.P. Verlinde, M.B. Hastings and T. Koma, M.B. Hastings, S. Sachdev, L. Susskind, M. Ba˜nuls,J. Cirac and M. Hastings, E.H. Lieb and D.W. Robinson, S.H. Shenker and D. Stanford, S.H. Shenker and D. Stanford, S. Leichenauer, L. Susskind, L. Susskind, D. Stanford and L. Susskind, B.D. Chowdhury, J. Maldacena, B. Swingle, D. Marolf and J. Polchinski, S.G. Avery, B.D. Chowdhury and A. Puhm, A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, [37] [38] [34] [35] [36] [30] [31] [32] [33] [28] [29] [25] [26] [27] [22] [23] [24] [19] [20] [21] [17] [18] [16] JHEP03(2015)051 . , , B 42 ]. ]. 61 73 Phys. , (2006) (1995) (1985) SPIRE ]. SPIRE 08 IN [ (1990) 138 55 Nucl. Phys. IN (1985) 315 , ][ B 436 ]. SPIRE quant-ph/0701004 JHEP Gen. Rel. Grav. IN , B 335 , Fortsch. Phys. , B 156 ][ , (2002) 084014 SPIRE ]. IN Prog. Theor. Phys. [ arXiv:1408.2823 ]. , Causality constraints on , D 65 Nucl. Phys. ]. , Phys. Rev. Lett. SPIRE arXiv:1412.6087 , IN Nucl. Phys. Viscosity bound violation in , SPIRE Phys. Lett. arXiv:0712.0805 , ][ , IN [ SPIRE ][ IN Phys. Rev. ]. ][ arXiv:1005.3035 ]. , ]. A covariant holographic entanglement arXiv:1407.5597 , SPIRE SPIRE SPIRE IN IN (2008) 126006 Black hole horizons and complementarity [ IN ][ – 26 – ][ ]. arXiv:0705.0016 [ hep-th/9901012 (2010) 2429] [ [ D 77 hep-th/0603001 [ SPIRE Conformal collider physics: energy and charge correlations The geometry of quantum computation D 19 (1985) 613 IN ]. ]. Cool horizons for entangled black holes Stringy effects in scrambling Black holes, shock waves and causality in the AdS/CFT ][ String generated gravity models 99 (2007) 062 Holographic derivation of entanglement entropy from AdS/CFT Aspects of holographic entanglement entropy ]. (1999) 010 The effect of spherical shells of matter on the Schwarzschild black The gravitational shock wave of a massless particle Switchbacks and the bridge to nowhere SPIRE SPIRE Phys. Rev. ]. 07 hep-th/9502074 , IN IN 02 [ arXiv:0803.1467 ][ ][ SPIRE Building up spacetime with quantum entanglement [ (2006) 181602 IN SPIRE JHEP ][ 96 IN Blue-sheet instability of Schwarzschild wormholes , JHEP [ , ]. Curvature squared terms and string theories The black hole interpretation of string theory Int. J. Mod. Phys. arXiv:1306.0533 [ . (1995) 7053 On gravitational shock waves in curved space-times Gauss-Bonnet black holes in AdS spaces [ (2008) 012 ]. ]. ]. SPIRE 05 IN Commun. Math. Phys. D 52 hep-th/0605073 hep-th/9408169 (1985) 173 [ , [ [ SPIRE SPIRE SPIRE hep-th/0109133 IN IN IN [ higher derivative gravity correspondence [ 2656 [ entropy proposal (2010) 2323 (2013) 781 (1985) 1401 Phys. Rev. Lett. 045 corrections to the graviton three-point coupling hole Rev. JHEP 253 721 [ R.-G. Cai, M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, B. Zwiebach, D.G. Boulware and S. Deser, L. Susskind and Y. Zhao, M.R. Dowling and M.A. Nielsen, G.T. Horowitz and N. Itzhaki, V.E. Hubeny, M. Rangamani and T. Takayanagi, M. Van Raamsdonk, J. Maldacena and L. Susskind, S. Ryu and T. Takayanagi, S. Ryu and T. Takayanagi, X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, T. Dray and G. ’t Hooft, I.H. Redmount, Y. Kiem, H.L. Verlinde and E.P. Verlinde, S.H. Shenker and D. Stanford, D.M. Hofman and J. Maldacena, K. Sfetsos, G. ’t Hooft, T. Dray and G. ’t Hooft, [58] [59] [56] [57] [53] [54] [55] [50] [51] [52] [48] [49] [45] [46] [47] [42] [43] [44] [40] [41] [39]