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Black Hole Information Recovery in JT Gravity

Zsolt Gyongyosi, Timothy J. Hollowood, S. Prem Kumar, Andrea Legramandi, Neil Talwar

arXiv:2209.11774v2hep-th

TL;DR

The paper studies how information carried by an infalling object is recovered from black-hole radiation, using generalized entropy in JT gravity where backreaction is exactly solvable. It analyzes Hayden–Preskill recovery, redundant encoding, python’s-lunch extremal surfaces, and infalling-observer experience. It finds that decoding complexity depends exponentially on the black-hole entropy shift and diary entropy, while horizon crossing remains smooth until the observer is scrambled out.

  • Problem

    The paper asks how an energy- and entropy-carrying diary affects black-hole information recovery, radiation encoding, decoding, and the experience of an infalling observer.

    Source: §1 · p. 3, §1.1 · p. 8

  • Method

    The analysis uses the generalized entropy and QES formalism in JT gravity across quasi-adiabatic shockwaves, logarithmic corrections, and finite-width infalling pulses.

    Source: §1 · pp. 3–4, §1.1 · p. 8

  • Results

    The information is recoverable from multiple radiation subsets, python’s-lunch decoding obeys log C „ (∆SBH + S(D))/2 ≥ S(D), and an infalling observer has a smooth horizon experience before being radiated out.

    Source: §6 · pp. 38–39

  • Takeaways & Limitations

    The results support Hayden–Preskill recovery with redundant radiation encoding and connect the recovery surface’s python’s-lunch structure to decoding difficulty.

    Source: §1.1 · p. 8, §6 · pp. 38–39

    The analysis assumes a greybody factor controlled by a single overall coefficient ξ, an assumption stated to hold for Schwarzschild or JT gravity with a semi-reflective barrier.

    Source: §2.1 · p. 12

Abstract

from arXiv · show

We consider the issue of information recovery for an object carrying energy and entropy into a black hole using the generalized entropy formalism, in the context of JT gravity where the backreaction problem can be solved exactly. We verify the main aspects of the Hayden-Preskill scenario but with some refinements. We show that the information is encoded in the Hawking radiation in a redundant way, as expected for a quantum error correcting code. We show how quantum extremal surfaces associated to information recovery have the form of a python's lunch and thereby show that the complexity of decoding is exponential in a combination of the entropy shift of the black hole and the entropy of the object. We also show that an infalling observer must have a smooth experience at the horizon and we calculate their endurance proper time inside the black hole before they are radiated out.

1 Introduction

The introduction frames information recovery for an infalling diary in JT gravity using generalized entropy and quantum extremal surfaces, whose backreaction is exactly solvable. It connects the Hayden–Preskill scenario to islands, redundant radiation encoding, decoding complexity, and the experience of an infalling observer.

1 Introduction

  • JT gravity makes the backreaction of an infalling energy- and entropy-carrying system exactly solvable, enabling a semiclassical analysis of Hayden–Preskill information recovery.The paper expects many results to apply more universally beyond this model.Source: §1 · p. 3
  • The QES formalism computes microscopic radiation entropy by extremizing over islands, even though the effective semiclassical and fundamental radiation states occupy different Hilbert spaces.The holographic map embeds the effective state into the microscopic theory, while the QES formula remains usable without knowing the microscopic theory.Source: §1 · pp. 3–4
  • After the Page time, an island saddle makes Hawking partners part of the radiation description, so their entanglement no longer contributes and the entropy is set approximately by the QES area, SBH.Before the Page time, the radiation instead has Hawking’s thermal entropy and no island dominates.Source: §1 · pp. 5–7, Figure 1 · p. 6

1.1 The Hayden-Preskill scenario

  • Recovery is characterized by maximal mutual information between D and R and zero mutual information between D and the remaining black hole B, yielding a factorized ρ_DB.This corresponds to the entropy relation S(R) = S(D) + S(B).Source: §1.1 · p. 7
  • Information can be recovered redundantly from different tensor factors of the radiation, a feature associated with quantum error-correcting codes.The redundancy is presented as part of the robustness of these codes.Source: §1.1 · pp. 7–8
  • The analysis uses three levels of QES calculations: quasi-adiabatic shockwave treatment, logarithmic scrambling-time corrections, and finite-width pulses probing the python’s lunch.The final level connects the new generalized-entropy extremum to decoding difficulty after sufficient information has evaporated into radiation.Source: §1.1 · p. 8
  • The paper asks when diary information returns in radiation, whether it can be recovered from late or disconnected subsets, and how backreaction affects smooth horizon crossing and interior experience.It also asks whether arbitrary radiation processing can affect the diary inside the black hole.Source: §1.1 · p. 8
  • The framework follows Hayden and Preskill [38] while incorporating diary backreaction, late-radiation recovery, logarithmic entropy corrections, python’s-lunch decoding structure, and a smooth but finite infalling-observer experience.These topics organize the paper’s subsequent sections and motivate its detailed JT-gravity treatment.Source: §1.1 · p. 9

2 Islands-in-the-stream with infalling objects

This section examines how infalling objects modify island saddles through energy and entropy backreaction in JT gravity and how their information is recovered from Hawking radiation. It shows that the resulting recovery has quantum-error-correcting-code structure, including complementary-pair recovery and scrambling-time corrections controlled by logarithmic entropy terms.

2 Islands-in-the-stream with infalling objects

  • The calculations use JT gravity but are expressed to allow analogy with higher-dimensional black holes in the s-wave approximation, with greybody effects represented by ξ in 1 < ξ ≤ 2.The greybody-factor treatment assumes frequency and temperature dependence appropriate to Schwarzschild or an imposed semi-reflective JT barrier.Source: §2 · p. 9, §2.1 · p. 12

2.1 Review of islands-in-the-stream without infalling objects

  • Without infalling objects, island QESs lie close to and inside the horizon, and the island contribution reduces the matter entropy to radiation entropy on the symmetric difference R a ˜I.The corresponding generalized entropy combines the black-hole entropy at the QES with Srad(R a ˜I), under the adiabatic approximation.Source: §2.1 · pp. 10–11

2.2 Adding infalling objects

  • Infalling object D is modeled as a narrow energy-and-entropy pulse whose energy backreaction jumps the black-hole entropy and enlarges possible QES endpoints to include t0.Its entropy contributes with opposite signs depending on whether D lies inside the island, while the analysis assumes a small entropy jump that exceeds thermal-time radiation entropy.Source: §2.2 · pp. 12–13

2.3 Information recovery in the early radiation

  • For early infall, recovery begins just before the Page time and completes just after it, with mutual information increasing continuously between the relevant saddle transitions.The processing interval includes both black-hole scrambling and energy- and entropy-backreaction effects; subsequent radiation emits entropy comparable to Sp(D) in the reversible limit.Source: §2.3 · pp. 15–16

2.4 Recovery in the late radiation

  • Late-radiation recovery is possible using a sufficiently large interval ˜R and a sufficiently small preceding interval A, with smaller A lowering the recovery time.The condition Srad(A) < Srad(R6) implies A ⊂ R6, because R6 is uncorrelated with D and need not be included.Source: §2.4 · p. 17

2.5 Scrambling time

  • Logarithmic corrections replace SBH(u) by SBH(u) − c/24 log(SBH(u)/c), interpreting these terms as the black-hole scrambling time.This correction does not apply to the QES immediately before D falls in; that case is deferred to section 3.Source: §2.5 · p. 17

2.6 A quantum error correcting code

  • Information about D is recoverable from both early and suitably chosen late radiation, demonstrating redundant encoding characteristic of a quantum error-correcting code.The shared radiation R7 is necessary overlap, yet D’s information cannot be recovered from R7 alone.Source: §2.6 · p. 18
  • D’s information can be recovered from the complement of R7 even though R7 alone is insufficient, demonstrating pairwise rather than single-subset recovery.Among R1 ∪ R6, R7, and the future radiation B, any pair recovers D while no single subset does.Source: §2.6 · p. 18
  • The radiation structure is exactly analogous to the three-qutrit QECC, with Q1 representing D, Q2 the black hole, and Q3 previously emitted radiation [53].Q2 is entangled with Q3, while Q1 never interacts with the already dispersed Q3.Source: §2.6 · p. 18
  • For the qutrit model, no single qutrit contains information about Q1, whereas any pair contains mutual information 2S(Q1).The corresponding relations are I(Qj, Q1) = 0 and I(Qj ∪ Qk, Q1) = 2S(Q1).Source: §2.6 · p. 18

3 Solving the backreaction in JT gravity

This section develops an analytically tractable JT-gravity treatment of backreaction for localized infalling matter in an evaporating black hole, determining the geometry, entropy evolution, and island saddles. It then analyzes the QES behavior and the timing and extent of information recovery, including the processing delay and recovered mutual information.

3 Solving the backreaction in JT gravity

  • JT gravity analytically solves the backreaction of arbitrary infalling matter, using a localized shockwave D and a large-c CFT bath to collect Hawking radiation.The geometry is an AdS2 patch matched to a half-Minkowski auxiliary region, with semiclassical control supplied by large c.Source: §3 · p. 19

3.1 Some aspects of black holes in JT gravity

  • The dilaton and boundary map V(t) determine the evaporating JT geometry, with exact evolution for arbitrary infalling matter and an adiabatic temperature T(t) = T0e^-kt/2.The analysis neglects extremal entropy and uses the adiabatic regime, where the black-hole entropy is linear in temperature.Source: §3.1 · pp. 19, 20, 22

3.2 Including an infalling system

  • When D falls in, the black-hole energy jumps by ED, the event horizon shifts to ˜U = 0, and the entropy shift determines the parameter λ.For diary energy much smaller than the initial black-hole energy, λ ≈ ∆SBH/(2SBH(t0)) ≪ 1.Source: §3.2 · pp. 23–24

3.3 Islands

  • Island generalized entropy includes the diary entropy according to whether D overlaps the island, while the dominant QES lies just inside the horizon during slow evaporation.For R, the diary term is S(D) when the shockwave crosses the island and zero otherwise; for R Y ¯D, the assignment reverses.Source: §3.3 · p. 26
  • The QES freezes in front of D at v → t0 − [2πT(t0)]^-1 log(ΔSBH/c), with U → ΔSBH/[6SBH(t0)].This reproduces the result of [7, eq. (92)].Source: §3.3 · p. 27
  • Freezing before the shockwave requires inequality (2.12); otherwise, for smaller diaries, the shockwave backreaction is negligible in this approximation.At later times, a new saddle with v > t0 exists only after u > t0 + [2πT(t0)]^-1 log[SBH(t0)/c], and D lies in its island.Source: §3.3 · pp. 27–28

3.4 Information recovery

  • Before the island I2 is favored for S(R ∪ D), the mutual information remains zero, with R6 defining the upper bound of this vanishing-correlation interval.The refined delay includes the logarithmic corrections anticipated by Hayden and Preskill [38].Source: §3.4 · p. 28
  • The information-processing delay includes logarithmic black-hole-entropy corrections and backreaction, after which the mutual information rises to I(R,D) = 2S(D).The delay is associated with R6 = [t0,u], while the final rise occurs when the radiation region becomes R2 = R6 ∪ R7 and the radiation saddle transitions I1 → I2.Source: §3.4 · p. 28

4 Python’s lunch

The paper identifies information-recovery saddles as python’s lunches: two minimal QES surround a maximal QES, whose entropy height controls decoding complexity. Exact and numerical JT analyses show that the maximum forms in the diary interval, while an island-in-the-stream sweep gives a complementary picture.

4 Python’s lunch

  • The python’s lunch consists of two minimal QES with a maximal QES between them, and its height is conjectured to quantify decoding complexity.The obstruction to decoding is associated with postselection in the tensor-network picture.Source: §4 · p. 29

4.1 Finding the maximum QES

  • The exchange of minimal-QES dominance signals that D’s recovery information has evaporated, requiring a maximum QES in or at the boundary of the D interval.This behavior appears for both the square energy pulse and operator-quench models, indicating universal behavior in these examples.Source: §4.1 · pp. 31–32, Figure 3 · p. 32, Figure 4 · p. 32
  • For R, the maximum lies at v “ t0 ` ϵ, whereas for R Y D it lies inside the D interval because competing entropy terms have different monotonicities.The black-hole entropy increases with v, while the D-dependent entropy increases for R and decreases for R Y D.Source: §4.1 · pp. 32–33
  • The maximum-QES entropy is SmaxpRq « SBHpt0q ` ∆SBH ` SpDq ` SradpR2q, while the relevant minimum is Smin “ SI1pRq.The entropy difference therefore combines the black-hole entropy shift with the diary entropy in the decoding-complexity estimate.Source: §4.1 · p. 33
  • In the reversible case, decoding complexity is dominated by C „ exppSpDqq, while irreversible absorption increases the computational cost.The result agrees with the predictions in [47] and [59].Source: §4.1 · p. 33

4.2 Island in the stream and python’s lunch

  • The island-in-the-stream sweep reproduces the python’s lunch: after the diary falls in, a second minimum and an intervening maximum appear, with the maximum corresponding to an island just after the diary.The diary-only local maximum is termed a python’s snack when previously emitted radiation has already been decoded.Source: §4 · p. 29, §4.2 · pp. 33–34, Figure 5 · p. 34

5 Effect of the infalling system

The infalling system clarifies how radiation operations, Hawking-partner encoding, and an observer’s interior experience remain consistent with information recovery and a smooth horizon.

5.1 Measurements outside affecting the inside

  • After the Page time, no operation on radiation R can affect an object D that falls in later, because their reduced state factorizes and I(R,D)=0.This is consistent with the island region having been encoded in the radiation rather than remaining meaningfully behind the horizon.Source: §5.1 · p. 35

5.2 Encoding of Hawking partners

  • The smooth-horizon result resolves the apparent monogamy conflict: partner modes can be encoded in early radiation while backreaction changes which island saddle dominates.Disrupting the horizon entanglement would instead produce a divergent energy density, or firewall.Source: §5.2 · pp. 36–37
  • The purifier of Hawking modes R2 is redundantly encoded in both early radiation R1 and the remaining black hole, realizing the quantum-error-correcting structure.For small R2, the early-radiation correlation is quantified by I(R1,R2)=ξ+1.Source: §5.2 · p. 36
  • Backreaction makes sufficiently large R2 uncorrelated with early radiation, allowing D to experience the entangled inertial vacuum across the true horizon without extreme non-locality.The transition occurs when the island I2 dominates, at the minimum radiation size needed to recover D.Source: §5.2 · pp. 36–37
  • The observer’s accessible interior is limited by the island crossing, which occurs at proper time τ=8λτ_s/3 for the stated QES coordinate.The trajectory uses proper time τ, with horizon crossing at τ=0 and singularity at τ=τ_s.Source: §5.2 · pp. 37–38

6 Discussion

The paper applies generalized entropy in JT gravity to recover infalling information through multiple radiation subsets, refine scrambling-time estimates, and relate decoding complexity to backreaction and diary entropy.

6 Discussion

  • Information can be recovered from different Hawking-radiation subsets, including radiation emitted after the diary falls in, through entanglement-wedge reconstruction and quantum error correction.The JT setup makes the infalling object’s backreaction exactly solvable.Source: §6 · p. 38
  • The scrambling delay contains a usual c log(S_BH/ΔS_BH) term and an additional ΔS_BH−S(D) term caused by irreversible black-hole absorption.When the first law is not saturated, this thermodynamic contribution delays information return and prevents the black hole from behaving as a mirror.Source: §6 · p. 38
  • log C ≳ (ΔS_BH+S(D))/2 ≥ S(D), linking python’s-lunch decoding complexity to the entropy shift and diary entropy.The result is obtained using the conjectured relation between complexity and python’s-lunch size.Source: §6 · pp. 38–39
  • Backreaction allows an infalling observer to encounter a smooth horizon while their information later becomes accessible outside when they cross the island.The corresponding ultra-relativistic endurance proper time depends on ΔS_BH.Source: §6 · p. 39

A Multipartite generalized entropy in the shockwave geometry

The appendix extends the generalized-entropy variational problem to multipartite radiation and an infalling object in the adiabatic shockwave geometry, deriving approximate QES conditions and competing saddles.

A Multipartite generalized entropy in the shockwave geometry

  • For arbitrary radiation intervals, the matter theory is taken to be c massless Dirac fermions, whose multiple-interval entropy formula makes the variational problem tractable.The analysis assumes each relevant interval is large compared with the thermal scale.Source: A Multipartite generalized entropy in… · pp. 40–41
  • In the adiabatic approximation, QES contributions simplify when surfaces lie close to the would-be horizon U=0 or true horizon ũ=0.The working assumption is UaVa≪1 for Va<1 and ũaṽa≪1 for Va>1.Source: A Multipartite generalized entropy in… · p. 40
  • Extremization yields VaUa≈c/(48S_BH(v_a)) and ũVaũa≈c/(48S_BH(v_a)), consistent with the assumed near-horizon regime when S_BH≫c.These approximate solutions justify the initial QES assumption in the adiabatic approximation.Source: A Multipartite generalized entropy in… · p. 41
  • A hierarchy among endpoint coordinates places each QES near an interval endpoint, while configurations with an odd number of smaller coordinates are maxima rather than relevant entropy saddles.The hierarchy follows because radiation intervals are large compared with the thermal scale.Source: A Multipartite generalized entropy in… · pp. 41–42
  • The diary modifies QES solutions near its shockwave: for one branch Ua=1, while another branch has Ua=λ/3 and va shifted logarithmically beyond t0.These competing saddles produce logarithmic entropy corrections, including a correction when the QES is stuck before the diary.Source: A Multipartite generalized entropy in… · pp. 42–43
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