Black holes and the butterfly effect

• The thermofield-double state realizes two entangled thermal systems holographically as the two boundaries of an eternal AdS black hole.Tracing out one side gives the other side a thermal density matrix, while the entanglement entropy equals the black-hole horizon area divided by 4G_N.Source: §1 · p. 3
• Thermal systems are conjectured to be fast scramblers, rendering small subsystems nearly thermal on a timescale t ∼ β log S.Source: §1 · p. 3
• The discussion connects disrupted behind-the-horizon correlations to firewall questions while emphasizing that the shock-wave geometry can defeat a naive firewall argument.The paper also notes that the smoothness requirement involves special entanglement across the horizon, which scrambling can disrupt.Source: §1 · pp. 3, 4, 6
• The paper uses mutual information I = S_A + S_B − S_A∪B and holographic extremal surfaces to track correlations between regions on opposite boundaries.Mutual information vanishes at leading order when connected and disconnected minimal surfaces exchange dominance.Source: §1 · p. 4
• A few quanta injected on the left at time −t_w are exponentially blueshifted relative to t = 0, so their backreaction creates a shock wave despite their initially small energy.In the BTZ setup, the perturbation is modeled by gluing solutions of masses M and M + E across a null surface, with E much smaller than M.Source: §1 · p. 5
• When E M e^(2πt_w/β) becomes order one, RT surfaces exchange dominance and mutual information drops to zero, beginning at t_* ∼ β/(2π) log(M/E).For a quantum with energy E ∼ T_H, this is the fast scrambling time; flat-space stringy effects do not change t_*, though stronger black-hole string effects are not excluded.Source: §1 · p. 5

2 A qubit model

• The numerical model evolves ten qubits in each of two systems using a nonintegrable Ising Hamiltonian with transverse and parallel fields.The coefficients are chosen following [31] to keep the Hamiltonian far from integrability.Source: §2 · p. 6
• The procedure prepares a thermofield-double state, applies σ_z to the fifth left qubit at time −t_w, and measures correlations between sites one and two and their thermofield doubles.Source: §2 · p. 6
• As t_w increases, mutual information falls sharply from its near-maximal unperturbed value and then levels off at a floor that decreases for larger total systems.Recent perturbations leave sites one and two unaffected because the disturbance has not yet propagated to them.Source: §2 · p. 7, Figure 1 · p. 7
• The spin-spin two-point function exhibits the same qualitative behavior, showing that sufficiently old small perturbations destroy special local thermofield-double correlations.Source: §2 · p. 7

3.1 Unperturbed BTZ

• The unperturbed thermofield double is dual to the maximally extended two-sided BTZ geometry, conveniently covered by Kruskal coordinates.The left and right asymptotic regions form the two boundaries of the eternal black hole.Source: §3.1 · p. 8, Figure 2 · p. 8

3.2 BTZ shock waves

• An early, small perturbation is modeled by gluing BTZ solutions across a null shell, producing a shock wave whose strength reflects the perturbation’s blueshift.The shell separates masses M and M + E, with E much smaller than M; in the relevant limit the geometry is described by a simple coordinate shift.Source: §3.2 · pp. 9–10
• The shock-wave construction uses a classical metric most directly justified for a large fixed number of quanta in the small G_N limit, although the authors expect the conclusions to extend to few quanta.The RT entropy calculation captures the contribution proportional to N^2 and may receive numerically large subleading or finite-λ corrections.Source: §3.2 · p. 10, §3.4 · p. 12

3.3 Geodesics

• The shock wave leaves large local curvature invariants absent but changes nonlocal geodesic distances, which encode the perturbation’s effect on probes crossing the interior.For equal-time points on opposite boundaries, the shock contribution increases the geodesic distance, especially when the perturbation was released far in the past.Source: §3.3 · pp. 11–12

3.4 Mutual information

• RT surfaces compute mutual information between equal-sized regions on opposite boundaries, with the connected and disconnected extremal surfaces determining whether I(A; B) is positive or zero.For sufficiently large regions, the connected surface initially dominates, but the mutual information decreases as the perturbation is moved earlier and can reach zero.Source: §3.4 · pp. 12–13
• For φ > π, mutual information approaches a finite positive floor, while regions with φ = π take the longest to relax.For smaller regions, the disconnected surface can dominate for all shock strengths, giving zero mutual information.Source: §3.4 · pp. 13–14

3.5 Correlation functions

• In the free-field approximation, cross-boundary correlators remain unchanged until t_w is of order t_* and then decay exponentially as the perturbation is moved earlier.Their onset occurs at the same scrambling scale found from RT mutual information, but their decay is smoother because the free-field approximation lacks the sharper inelastic turnoff.Source: §3.5 · pp. 14–15

Correlation-function probe

• The correlator is difficult to evaluate directly because it measures spacelike correlations rather than S-matrix scattering information.The authors instead estimate the relevant regime using flat-space string scattering and the expectation that weak scattering causes only weak changes in spatial correlations.Source: §4 · p. 16

String effects

• Flat-space stringy effects do not change the log S dependence of the time t∗ at which the correlator starts decreasing.String effects become order one somewhat before t∗ by an additive S-independent amount proportional to β log ℓ/l_s.Source: §4 · p. 16

Flat-space scattering

• High-energy string scattering produces transverse spreading, elastic phase-shift corrections, and suppressed inelastic effects at sufficiently small impact parameters.For b < bI, the transverse profile grows logarithmically with s, while the imaginary phase-shift contribution is suppressed by (l_s/b)^2.Source: §4 · p. 17

Black-hole scattering

• Applying flat-space scattering estimates to the black-hole background is reliable only when the horizon, curvature, and geodesic scales exceed the relevant string-scattering scales.The passage states that this condition fails near t∼t∗.Source: §4 · p. 17

Limitations

• The analysis cannot reliably exclude singularity effects or longitudinal string spreading that could make t∗ much shorter than β log S.The significance of longitudinal spreading appears gauge-dependent, and its relevance to this setup remains uncertain.Source: §4 · p. 17

String effects

• The Einstein-gravity contribution becomes order one at t∗ = β 2π log S, while the inelastic string phase shift is suppressed.The elastic effect is expected to be accurately captured by the homogeneous shock-wave geometry.Source: §4 · p. 17

5 Discussion

• Small early perturbations create highly blueshifted shock waves that disrupt correlations between the left and right field theories.The large boost produces the logarithmic time dependence and suggests a relation between sensitive dependence and scrambling.Source: §5 · p. 18
• The analytic shock solutions use carefully synchronized infall times, while realistic local perturbations spread infall times and can change the correlator’s decay from double-exponential to ordinary exponential.Hydrodynamical effects are absent from the shock solutions, motivating localized analyses that include spatial propagation and infall-time spread.Source: §5 · pp. 18–19
• The authors do not offer decisive insight into the firewall controversy and note that genuine AMPS firewalls before scrambling would invalidate the bulk calculation.They also cannot determine which assumptions in the stronger firewall counting argument should be relaxed.Source: §5 · pp. 19–21
• After a scrambling time, RT surfaces disconnect and correlators vanish, leaving no simple probes of the behind-the-horizon region.This is presented as an emerging pattern rather than a decisive characterization of the interior.Source: §5 · p. 19
• The shock-wave geometry defeats a naive firewall argument because the right boundary operator represents a mode far from the left black mode, rather than its interior partner.In the shock geometry, the black mode is naturally uncorrelated with the mode represented by ϕR [59].Source: §5 · pp. 19–20, Figure 4 · p. 20
• The analysis cannot reliably exclude stronger black-hole stringy effects that would produce a scrambling time smaller than the Einstein-gravity value.The CFT representation of the green mode is also mysterious, especially at higher energies where the shift description fails.Source: §5 · p. 21
• For fixed perturbation time, shock effects weaken as the infall time is delayed, whereas earlier perturbations can make the infaller experience an increasingly painful collision.The shock has several attributes of a firewall, but the discussion does not establish an actual AMPS firewall.Source: §5 · p. 21

A Haar scrambling

• The calculation uses Haar integrals to evaluate mutual information and correlation functions after applying a random unitary to one side.The approach follows a norm-based strategy related to Page’s analysis and would give identical results for a 2-design.Source: A Haar scrambling · p. 22
• For d1 = ∥ρAB − ρA ⊗ ρB∥1 with d1 ≤ 1/e, the distance lower-bounds mutual information via Pinsker and upper-bounds it via Fannes.Because of the square-root, d1 is difficult to average over U, motivating a further inequality derived from Cauchy–Schwarz applied to the eigenvalues.Source: A Haar scrambling · p. 22
• The derivation exploits that the reduced states of A and B remain maximally mixed for every unitary in the setup.This property simplifies the averaged norm calculation.Source: A Haar scrambling · pp. 22–23
• If subsystem A is smaller than half the left system, its one-norm distance is exponentially small in the extra Hilbert-space dimension of its complement.The resulting mutual-information bound is therefore exponentially small when A is significantly smaller than half the total system.Source: A Haar scrambling · p. 23
• The expected two-sided spin-spin correlator vanishes under unitary averaging, while its root-mean-square value is |L|^-1.This provides a direct correlation-function manifestation of scrambling in the qubit model.Source: A Haar scrambling · p. 23

B.1 Higher dimensions

• The analysis extends the BTZ shock construction to D-dimensional AdS black holes by estimating when the v-coordinate shift becomes order one.It does not compute geodesic distances or RT surfaces in higher dimensions.Source: §B.1 · p. 24
• A spherically symmetric null perturbation is matched across the horizon in Kruskal coordinates to determine the shift produced by a small-energy shell sent in at time tw.Near the horizon, the large-tw expansion makes the shift’s logarithmic dependence tractable.Source: §B.1 · p. 24
• The leading coefficient of the logarithm in t* is dimension-independent, although the appendix uses a proxy rather than a full higher-dimensional extremal-surface calculation.The coefficient follows from the near-horizon expansion and thermodynamic relations involving R, SBH, and β.Source: §B.1 · pp. 24–25
• t* ∼ β/(2π) log N^2 at fixed E, R, and T in any spacetime dimension.The estimate uses the condition that the Kruskal-coordinate shift α becomes one and the scaling GN ∝ N^-2.Source: §B.1 · p. 25

B.2 Solutions with localized sources

• A localized source on the horizon produces an angular shift h(Ω) governed by an Einstein-equation Green’s-function problem.The source is taken at the north pole of the transverse sphere and is modeled by localized null stress energy.Source: §B.2 · p. 25
• For a large AdS black hole, the shift is the Green’s function of a very massive field on the sphere and decays with angular distance.Its angular decay can be compared with the exponential growth generated by moving the perturbation earlier.Source: §B.2 · p. 25
• As tw increases, the level sets of the localized shift expand outward with a definite propagation speed.This compares spatial spreading along the angular directions with the perturbation’s exponential growth toward the past.Source: §B.2 · p. 25