PaperlayerView exact revision on arXiv

Source-linked AI summary

Multiple Shocks

Stephen H. Shenker, Douglas Stanford

arXiv:1312.3296v2hep-th

TL;DR

The paper asks how two highly entangled CFTs can have weak local two-sided correlations. It perturbs the thermofield double with thermal-scale local operators and constructs the resulting shock-wave geometries. The resulting long wormholes and correlation suppression are governed qualitatively by chaotic dynamics and the fast scrambling time.

  • Problem

    The paper addresses how states can retain large two-sided entanglement while having very weak local correlations.

    Source: §3 · p. 15, §4 · p. 18

  • Method

    The authors perturb the thermofield double with local W operators at different times and construct the dual geometries from their shock-wave backreaction.

    Source: §1 · p. 4

  • Results

    The perturbations produce long wormholes with two-sided correlators small at all times, while the scrambling time controls the resulting geometry.

    Source: §1 · p. 4, §4 · p. 18

  • Takeaways & Limitations

    Einstein gravity can accommodate highly entangled states with weak local two-sided correlation through boost-enhanced shock-wave backreaction.

    Source: §4 · p. 18

    These states are atypical because the W operators increase energy without increasing two-sided entanglement, and the correlator estimates omit field backreaction and nonlinear effects.

    Source: §4 · p. 19

Abstract

from arXiv · show

Using gauge/gravity duality, we explore a class of states of two CFTs with a large degree of entanglement, but with very weak local two-sided correlation. These states are constructed by perturbing the thermofield double state with thermal-scale operators that are local at different times. Acting on the dual black hole geometry, these perturbations create an intersecting network of shock waves, supporting a very long wormhole. Chaotic CFT dynamics and the associated fast scrambling time play an essential role in determining the qualitative features of the resulting geometries.

1 Introduction

The paper studies how operations that delocalize the thermofield double’s local left-right entanglement alter its geometric dual. Widely separated perturbations generate intersecting shocks and very long wormholes, while scrambling controls when local correlations disappear.

1 Introduction

  • The construction probes the tension between smooth black-hole geometry, which relies on special local left-right entanglement, and the weak correlations of typical two-sided states.The thermofield double has highly atypical local subsystems of the two CFTs entangled across the sides.Source: §1 · pp. 2–4
  • For a single early perturbation, mutual information between smaller-than-half local subsystems drops to zero when the waiting time is of order t∗.The same perturbation disrupts the Ryu-Takayanagi surface through the wormhole.Source: §1 · p. 3
  • Equal-time two-sided correlators can become small when the observation time differs from the perturbation time by the scrambling scale, but there remains a time near the perturbation when the correlator is order one.The cited geodesic estimate is subject to potentially important nonlinear corrections at large relative boosts.Source: §1 · p. 3
  • Widely separated perturbations generate an intersecting network of shock waves, and sufficiently many shocks or large time separations produce a wormhole that is long in every boost frame.This indicates weak local correlation between the two boundaries at all times.Source: §1 · p. 4
  • The scrambling time t∗ governs how successive perturbations affect the geometry: a second operation after t∗ erases the first disturbance and pushes its shock toward the singularity.Thus, chaotic CFT dynamics determine the qualitative shock-wave structure.Source: §1 · p. 4
  • These weakly correlated geometrical states are not typical Hilbert-space states: the perturbations alter energy statistics, leave a distinguished detectable time, and require an atypical left reduced density matrix.Their boost behavior can also produce a frame in which an infalling observer encounters a high-energy shock near the horizon.Source: §1 · pp. 4–5

2 Wormholes built from shock waves

The paper constructs black-hole geometries by iteratively adding thermal-scale, boundary-localized perturbations to the thermofield double, producing intersecting shock waves and increasingly long wormholes. The fast scrambling time controls when shocks collide, disappear from the boundary, and organize into kinked or smooth multi-shock geometries.

2.1 One shock

  • The one-shock state can be interpreted with either an explicitly timed source or a time-independent Hamiltonian, with operator ordering fixed for observables before t1.The construction treats the perturbation as a thin null shell, justified by its highly boosted kinematics.Source: §2.1 · pp. 5–6
  • For one shock, a boundary-localized operator of energy E much smaller than M produces negligible backreaction until the perturbation is boosted at times t1 ∼ t∗.The resulting geometry is formed by gluing BTZ regions across a null surface with a shift in the Kruskal coordinate.Source: §2.1 · pp. 5–6

2.2 Two shocks

  • The two-shock geometry is built recursively by selecting a Cauchy surface, adding W(t) near the left boundary, and evolving the data forwards and backwards.In the folded-bulk description, successive perturbations occupy different sheets and are matched across colliding shells using continuity and the DTR regularity condition.Source: §2.2 · pp. 7–8
  • When a second perturbation is added sufficiently early, the increased horizon radius captures the first shock, moving it from the boundary onto the singularity.The shock coordinate becomes positive when (t2 − t1) ≈ t∗, while the final exponentially growing collision term dominates around (t2 − t1) ≈ 2t∗.Source: §2.2 · p. 9, Figure 3 · p. 9
  • The capture of the first perturbation is interpreted as scrambling: acting with W(t2) about a scrambling time before t1 disrupts the tuning that made a perturbation visible at t1.The corresponding operator commutator becomes approximately two when |t1−t2| ∼ t∗, diagnosing chaotic dynamics.Source: §2.2 · p. 10

2.3 Many shocks

  • Many-shock geometries arise by alternating shocks from opposite corners, whose null shifts extend the wormhole and permit a recursive construction for arbitrary multi-W states.For alternating times ±tw, the interior develops a discrete translation symmetry, allowing the geometry to be analyzed through a unit cell.Source: §2.3 · pp. 10–11
  • The wormhole length grows with the number and strength of shocks, reaching values above S and implying very weak local correlation ∼e−(const.)L between the boundaries.The estimate is an upper bound when energies become exponentially large or when the geodesic calculation leaves the controlled regime.Source: §2.3 · p. 12
  • Large α produces kinked geometries with small post-collision regions near singularities, while small α can smooth a dense shock network into a continuous geometry.For intermediate α, central white regions are collision-insensitive, but shaded regions can depend on string- and Planck-scale physics; the small-α metric agrees numerically with the recursive construction.Source: §2.3 · pp. 12–14, Figure 7 · p. 14

3 Ensembles

The paper defines ensembles of typical states and uses time-translation invariance to constrain whether such states can possess smooth, long-wormhole geometries. Boosting arguments imply that smooth duals of typical states face severe geometric restrictions, while the constructed long-wormhole states remain atypical.

Defining typical states

  • Time evolution preserves eigenstate coefficient magnitudes while changing phases, motivating time-translation-invariant ensembles whose energy-dependent distributions define typical states.The ensemble generalizes Haar-random states by allowing a smooth function of energy subject to normalization.Source: §3 · pp. 15–16

Generating typical states

  • A weak left-right wire can thermalize the combined system, and sufficiently long evolution produces states that are difficult to distinguish from typical ensemble states using linear measurements.The construction allows overlap within π/4 of any typical state, corresponding to indistinguishability better than roughly 80%.Source: §3 · p. 16
  • Typicality may require times ranging from at least S to recurrence scales near ∼e^eS, far longer than the times for which the paper’s geometric constructions remain reliable.The intermediate scale ∼e^S is also identified as potentially relevant for representing states as superpositions of earlier-time states.Source: §3 · p. 17

Constraints on smooth typical geometries

  • Time-translation invariance makes every boosted evolution of a presumed smooth typical state typical, so an infalling observer would encounter either mild or violent collisions depending on the boost.At t ∼ t∗, the transformed geometry gives Bob a violent collision even when the original geometry gives a mild one.Source: §3 · p. 17, Figure 8 · p. 17
  • Avoiding boost-dependent collisions would require regions outside the wormhole to resemble the empty eternal black hole, joined to the long wormhole along an unusual infinite-timelike-length locus.This geometry differs from the intuitive long thin wormhole and may suggest additional quantum states at the UV boundary.Source: §3 · pp. 17–18, Figure 9 · p. 18
  • Another possibility is that truly typical states lack smooth geometries outside region II, so an infalling observer encounters a firewall.This is presented as an alternative to constructing boost-invariant smooth geometries for typical states.Source: §3 · p. 18

4 Discussion

The discussion identifies long-wormhole geometries with highly entangled states whose two-sided correlators remain small, and explains how scrambling, atypicality, and neglected nonlinear effects qualify this construction.

Discussion

  • The construction yields highly entangled two-CFT states with two-sided correlators small at all times, using boost-enhanced backreaction to lengthen the wormhole.When perturbations are sufficiently separated, their shock-wave backreaction must be included.Source: §4 · p. 18
  • The scrambling time t∗ controls the geometry: separations Δt ∼ 2t∗ produce high-energy shocks, whereas many more closely spaced perturbations yield smoother wormholes.For separations exceeding t∗, only the outermost perturbation remains locally detectable in the CFT description.Source: §4 · pp. 18–19
  • Although these states reproduce very small typical-state-like correlations, they are atypical because they have a distinguished detectable time and inject energy without increasing two-sided entanglement.Boosting them also produces a high-energy horizon shock, unlike the expected invariance of a truly typical ensemble.Source: §4 · p. 19
  • Superpositions of geometries might represent typical states, but large off-diagonal contributions can invalidate semiclassical reasoning and make this possibility difficult to assess.The paper therefore does not exclude geometric descriptions arising from superpositions of multi-W states.Source: §4 · p. 19
  • The geodesic estimate neglects field backreaction, providing an upper bound on the correlation while leaving open whether nonlinear effects could realize e−S correlations with shorter wormholes.Such alternatives could involve high-energy shocks running between the singularities.Source: §4 · p. 19
  • The methods can construct states containing a few particles behind the horizon, but constructing field operators in that region remains open.The result establishes state construction without supplying actual interior field operators.Source: §4 · p. 19

A Recursion relations for many shock waves

The appendix formulates recursion relations for a translationally invariant network of intersecting shocks and numerically verifies that the resulting geometry agrees with the smooth wormhole metric in the α → 0 limit.

Recursion setup

  • Discrete translational invariance reduces the shock-wave network to recursions for collision radii r_n and BTZ parameters R_n, with Fig. 10 labeling these geometric quantities.The geometry is represented by the radii of collision vertices and the BTZ regions between collisions.Source: A Recursion relations for many shock… · p. 20, Figure 10 · p. 20

Recursion relations

  • One recursion follows directly from the DTR relation at each vertex, while the second is obtained by propagating Kruskal coordinates across a plaquette from known vertex radii.The top-vertex radius is determined after solving for the required change in the Kruskal coordinate u.Source: A Recursion relations for many shock… · p. 20

Initial conditions and geometry

  • For R = 1, initial data R_1 = r_1 = 1 together with R_2 and r_2 from the two-shock solution completely determine the geometry through the second-order recursions.The construction also computes the timelike distance between successive plaquette vertices for comparison with the smooth solution.Source: A Recursion relations for many shock… · p. 21

Numerical check

  • At α = 0.01, numerical solutions for r_n agree excellently with g(τ) when plotted against total geodesic distance from the initial slice.This provides the appendix’s numerical check of the recursive construction against the smooth wormhole metric.Source: A Recursion relations for many shock… · p. 21

B Vaidya matching conditions

The Vaidya matching construction fixes the null-coordinate profile by imposing continuity of the metric and sphere size across a matching surface. For the chosen g(τ), the geometry is fixed except for the overall length of the central wormhole region.

Matching conditions

  • The matching calculation is performed in a top-left Vaidya region by requiring the metric to be C1 across the null slice.Continuity of the S1 gives r = g(τ), while matching null-vector normalizations relates the two coordinate systems.Source: B Vaidya matching conditions · p. 21
  • The matching equations relate g′(τ), V, and ρ(V), first determining V(τ), then inverting to τ(V), and finally fixing ρ(V).The C1 condition also prevents a δ-function stress tensor from traveling along the null surface.Source: B Vaidya matching conditions · pp. 21–22

Scope of the construction

  • Although ρ(V) cannot be computed exactly for the specific g(τ), the matching conditions completely determine the geometry up to the undetermined overall length of the central wormhole.The residual freedom is global rather than a failure to determine the local matching geometry.Source: B Vaidya matching conditions · p. 22
Loading 1312.3296v2…