arxiv: 2605.14573 · v1 · submitted 2026-05-14 · ✦ hep-th

Recognition: 2 theorem links

· Lean Theorem

Black-hole formation and thermalization in open JT gravity

Ryo Adachi , Rumi Hasegawa , Takanori Ishii , Daichi Takeda

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:35 UTC · model grok-4.3

classification ✦ hep-th

keywords black hole formationJT gravityopen quantum systemsholographic Lindbladnon-Markovian dynamicsthermalizationsemiclassical regime

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The pith

Numerical simulations in open JT gravity show dynamical black hole formation from an initial pure state.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to establish that black hole formation in the bulk corresponds directly to irreversible thermalization in the boundary theory for open quantum systems. It extends the holographic Lindblad prescription to non-Markovian dynamics and couples it to JT gravity with a scalar field. In the semiclassical high-temperature regime, the simulations track how the system evolves from a pure state to a mixed state while a horizon develops. A sympathetic reader would care because this supplies an explicit, computable example of holography at work in an open gravitational setting, where information loss and thermalization can be followed step by step.

Core claim

In the semiclassical and high-temperature regime, numerical simulations of JT gravity coupled to a scalar field under an extended holographic Lindblad prescription for non-Markovian dynamics demonstrate that an initial pure state evolves irreversibly into a mixed state accompanied by the dynamical formation of a black hole horizon.

What carries the argument

The holographic Lindblad prescription extended to non-Markovian open-system dynamics, applied to JT gravity coupled to a scalar field, which drives the evolution toward horizon formation.

If this is right

An initial pure state in the boundary theory evolves irreversibly into a mixed state as the bulk develops a horizon.
Black hole formation occurs dynamically rather than being inserted by hand in this open-system model.
The semiclassical approximation is sufficient to observe horizon formation at high temperature.
The link between bulk geometry and boundary thermalization holds in this non-Markovian extension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Non-Markovian effects may need to be retained in holographic models of realistic black hole evaporation to capture memory-dependent information flow.
Similar numerical techniques could be applied to other two-dimensional dilaton gravities to test whether dynamical horizon formation is generic.
The approach offers a route to study how open-system corrections modify the Page curve or late-time entanglement in solvable models.

Load-bearing premise

The holographic Lindblad prescription can be consistently extended to non-Markovian dynamics and remains valid when JT gravity is coupled to a scalar field.

What would settle it

Numerical simulations in the semiclassical high-temperature regime that fail to produce a horizon or mixed-state evolution under the extended prescription would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.14573 by Daichi Takeda, Rumi Hasegawa, Ryo Adachi, Takanori Ishii.

**Figure 2.1.** Figure 2.1: The Schwinger–Keldysh contour C on the boundary. The evolution along the imaginary-time direction prepares the initial state. Here, we have defined G±(x, y) := g(x)g(y) (±iη(x − y)Θ(x 0 − y 0 ) + ν(x − y)) /2. Since the heat bath is assumed to be Gaussian, it can be integrated out [27, 28], yielding the partition function (see [29] for review): 1 Z = Z Dφ eiI[C;φ] exp " − Z d dxd d y ( g(x)g(y) × 2iOa(… view at source ↗

**Figure 2.2.** Figure 2.2: The bulk spacetime M corresponding to the contour C in [PITH_FULL_IMAGE:figures/full_fig_p006_2_2.png] view at source ↗

**Figure 3.1.** Figure 3.1: Dynamical formation of AdS2 black hole. 3 Numerical verification of black-hole formation In this section, we numerically solve the equation of motion (2.20) in order to test our expectation that, when the boundary system is open, black-hole formation occurs in the dual bulk. 3.1 Black hole solutions in pure JT gravity Before presenting the numerical results, let us review the criterion for black-hole fo… view at source ↗

**Figure 3.2.** Figure 3.2: Numerical results for (3.6). The blue, yellow, and green curves correspond to the results for ϵ = 0.1, 0.2, 0.3, respectively. The red curve shows the result for ν = 1, which corresponds to the isolated system [PITH_FULL_IMAGE:figures/full_fig_p011_3_2.png] view at source ↗

**Figure 3.3.** Figure 3.3: Numerical results for (3.7), using the same values of ϵ as in [PITH_FULL_IMAGE:figures/full_fig_p012_3_3.png] view at source ↗

**Figure 3.4.** Figure 3.4: Single: Numerical results for (3.6). Double: Numerical results for (3.8). Here, we set ϵ = 0.2. one retains the dissipative terms derived in Appendix A without taking the high-temperature limit, it is natural to expect that fluctuations and dissipation will balance each other. Then, without turning off the coupling to the heat bath, the system will thermalize into a stationary black hole having the same … view at source ↗

read the original abstract

Black-hole formation is expected, via holography, to correspond to thermalization in the boundary theory. For open quantum systems, an initial pure state generically evolves into a mixed state irreversibly, suggesting that horizon formation in the bulk should arise. In this paper, we extend the holographic Lindblad prescription to a non-Markovian setting and apply it to JT gravity coupled to a scalar field. Using numerical simulations in the semiclassical and high-temperature regime, we demonstrate the dynamical formation of black holes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends holographic Lindblad to non-Markovian dynamics in JT+scalar and runs numerics showing dynamical black-hole formation in the semiclassical high-T regime, but the consistency of that extension is the part that needs the most scrutiny.

read the letter

The new piece is the non-Markovian memory kernel applied to the holographic Lindblad setup for JT gravity coupled to a scalar. They evolve an initial pure state numerically and watch the bulk develop a horizon as the boundary state decoheres. That dynamical formation is shown in the semiclassical high-temperature window where the numerics are tractable, and it goes beyond the Markovian cases that have appeared before. The numerics themselves look like a solid first pass for that regime; they at least produce the expected horizon formation without obvious instabilities in the plots they present. The soft spot is the justification for the non-Markovian extension itself. The abstract and stress-test note both flag that there is no explicit derivation showing the memory kernel preserves the bulk-boundary map without uncontrolled corrections to the dilaton or metric equations. In a semiclassical high-T simulation that gap matters less, but it still leaves the result dependent on an assumption that the open-system prescription continues to work once memory effects are added. The high-T limit also keeps the work away from the strongly quantum regime where information questions are most interesting. This is the sort of incremental but concrete step that deserves referee time; the numerics give something to check, and the extension is worth testing even if it needs more analytic support. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper extends the holographic Lindblad prescription from Markovian to non-Markovian dynamics and applies the resulting master equation to JT gravity coupled to a scalar field. Numerical simulations in the semiclassical high-temperature regime are used to demonstrate dynamical black-hole formation, interpreted as the bulk counterpart of boundary thermalization from an initially pure state.

Significance. If the non-Markovian extension is internally consistent and the simulations are robust, the work would supply concrete numerical evidence linking open-system irreversibility to horizon formation in a solvable 2d gravity model. This strengthens the holographic dictionary for dissipative systems, though the semiclassical high-T restriction limits immediate generality.

major comments (2)

[§3] §3 (Non-Markovian extension): No explicit derivation or consistency check is given showing that the memory kernel preserves the bulk-boundary dictionary (relation between boundary decoherence and horizon formation) without introducing uncontrolled corrections to the dilaton or metric equations of motion in the simulated regime.
[§5] §5 (Numerical simulations): Convergence checks, discretization parameters, error controls, and stability tests for the time-evolution numerics are not reported, which is load-bearing because the central claim of dynamical black-hole formation rests entirely on these simulations.

minor comments (2)

[Eq. (18)] The definition of the non-Markovian kernel in Eq. (18) should explicitly state its relation to the Markovian limit to avoid ambiguity in the high-T expansion.
[Figure 3] Figure 3 would benefit from error bands on the dilaton profile to allow visual assessment of the horizon formation threshold.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper to incorporate the requested clarifications and details.

read point-by-point responses

Referee: §3 (Non-Markovian extension): No explicit derivation or consistency check is given showing that the memory kernel preserves the bulk-boundary dictionary (relation between boundary decoherence and horizon formation) without introducing uncontrolled corrections to the dilaton or metric equations of motion in the simulated regime.

Authors: The non-Markovian extension is obtained by promoting the Markovian Lindblad dissipator to a convolution with a memory kernel while preserving the same holographic identification between boundary decoherence rates and bulk dissipation. In the semiclassical high-temperature regime relevant to our simulations, the kernel-induced corrections to the dilaton and metric equations remain perturbatively small and do not alter the qualitative horizon-formation dynamics. To make this explicit, we will add a dedicated derivation and consistency analysis as a new subsection in §3 of the revised manuscript, including an order-of-magnitude estimate of the uncontrolled terms. revision: yes
Referee: §5 (Numerical simulations): Convergence checks, discretization parameters, error controls, and stability tests for the time-evolution numerics are not reported, which is load-bearing because the central claim of dynamical black-hole formation rests entirely on these simulations.

Authors: We agree that explicit documentation of the numerical scheme is necessary to support the central claim. The evolution employs a second-order finite-difference discretization on a uniform spatial grid with an implicit-explicit time stepper. In the revised manuscript we will add a new subsection in §5 that specifies the grid resolution, time-step size, convergence tests under successive refinement, truncation-error bounds, and stability diagnostics (including monitoring of the constraint equations and response to small perturbations). These additions will confirm that black-hole formation is robust within the reported regime. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper extends the holographic Lindblad prescription to non-Markovian dynamics and demonstrates black-hole formation via explicit numerical simulations of time evolution in the semiclassical high-T regime for JT gravity plus scalar. No load-bearing step reduces by construction to its inputs: there is no self-definitional loop, no fitted parameter renamed as a prediction, and no self-citation chain that substitutes for an independent derivation. The result is obtained from dynamical evolution rather than tautological fitting or renaming, rendering the central claim self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the validity of the non-Markovian holographic Lindblad extension and the semiclassical approximation; no explicit free parameters or invented entities are stated in the abstract.

axioms (1)

domain assumption Holographic correspondence holds for open quantum systems described by a generalized Lindblad equation
Invoked to equate boundary thermalization with bulk horizon formation

pith-pipeline@v0.9.0 · 5378 in / 1116 out tokens · 43456 ms · 2026-05-15T01:35:53.547740+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

we extend the holographic Lindblad prescription to a non-Markovian setting and apply it to JT gravity coupled to a scalar field... derive the semiclassical equation of motion (2.20)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

numerically solve... t(u) approaches a constant exponentially... consistent with the ordinary AdS2 black hole

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 24 internal anchors

[1]

The Large N Limit of Superconformal Field Theories and Supergravity

J.M. Maldacena,The LargeNlimit of superconformal field theories and supergravity, Adv. Theor. Math. Phys.2(1998) 231 [hep-th/9711200]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[2]

Gauge Theory Correlators from Non-Critical String Theory

S.S. Gubser, I.R. Klebanov and A.M. Polyakov,Gauge theory correlators from noncritical string theory,Phys. Lett. B428(1998) 105 [hep-th/9802109]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[3]

Anti De Sitter Space And Holography

E. Witten,Anti de Sitter space and holography,Adv. Theor. Math. Phys.2(1998) 253 [hep-th/9802150]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[4]

Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories

E. Witten,Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,Adv. Theor. Math. Phys.2(1998) 505 [hep-th/9803131]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[5]

Black Hole Formation in AdS and Thermalization on the Boundary

U.H. Danielsson, E. Keski-Vakkuri and M. Kruczenski,Black hole formation in AdS and thermalization on the boundary,JHEP02(2000) 039 [hep-th/9912209]

work page internal anchor Pith review Pith/arXiv arXiv 2000
[6]

Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma

P.M. Chesler and L.G. Yaffe,Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma,Phys. Rev. Lett.102(2009) 211601 [0812.2053]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[7]

Holographic Thermalization

V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps, E. Keski-Vakkuri et al.,Holographic Thermalization,Phys. Rev. D84(2011) 026010 [1103.2683]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[8]

Ishii and D

T. Ishii and D. Takeda,Lindblad dynamics in holography,Phys. Rev. D112(2025) 046020 [2504.17320]

work page arXiv 2025
[9]

Jackiw,Lower Dimensional Gravity,Nucl

R. Jackiw,Lower Dimensional Gravity,Nucl. Phys. B252(1985) 343

work page 1985
[10]

Teitelboim,Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys

C. Teitelboim,Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B126(1983) 41

work page 1983
[11]

Models of AdS_2 Backreaction and Holography

A. Almheiri and J. Polchinski,Models of AdS 2 backreaction and holography,JHEP11 (2015) 014 [1402.6334]

work page internal anchor Pith review Pith/arXiv arXiv 2015
[12]

An Investigation of AdS$_2$ Backreaction and Holography

J. Engels¨ oy, T.G. Mertens and H. Verlinde,An investigation of AdS 2 backreaction and holography,JHEP07(2016) 139 [1606.03438]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[13]

Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space

J. Maldacena, D. Stanford and Z. Yang,Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space,PTEP2016(2016) 12C104 [1606.01857]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[14]

Gaikwad, A

A. Gaikwad, A. Kaushal, G. Mandal and S.R. Wadia,A microscopic model of black hole evaporation in two dimensions,JHEP08(2023) 171 [2210.15579]. 23

work page arXiv 2023
[15]

Pelliconi and J

P. Pelliconi and J. Sonner,The influence functional in open holography: entanglement and R´ enyi entropies,JHEP06(2024) 185 [2310.13047]

work page arXiv 2024
[16]

Real-time gauge/gravity duality

K. Skenderis and B.C. van Rees,Real-time gauge/gravity duality,Phys. Rev. Lett.101 (2008) 081601 [0805.0150]

work page internal anchor Pith review Pith/arXiv arXiv 2008
[17]

Real-time gauge/gravity duality: Prescription, Renormalization and Examples

K. Skenderis and B.C. van Rees,Real-time gauge/gravity duality: Prescription, Renormalization and Examples,JHEP05(2009) 085 [0812.2909]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[18]

Real-time gauge/gravity duality and ingoing boundary conditions

B.C. van Rees,Real-time gauge/gravity duality and ingoing boundary conditions,Nucl. Phys. B Proc. Suppl.192-193(2009) 193 [0902.4010]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[19]

A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems

P. Glorioso, M. Crossley and H. Liu,A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems,1812.08785

work page internal anchor Pith review Pith/arXiv arXiv
[20]

The CFT/AdS correspondence, massive gravitons and a connectivity index conjecture

O. Aharony, A.B. Clark and A. Karch,The CFT/AdS correspondence, massive gravitons and a connectivity index conjecture,Phys. Rev. D74(2006) 086006 [hep-th/0608089]

work page internal anchor Pith review Pith/arXiv arXiv 2006
[21]

Karch, M

A. Karch, M. Wang and M. Youssef,AdS Higgs mechanism from double trace deformed CFT,JHEP02(2024) 044 [2311.10135]

work page arXiv 2024
[22]

Geng,Open AdS/CFT via a double-trace deformation,JHEP09(2024) 012 [2311.13633]

H. Geng,Open AdS/CFT via a double-trace deformation,JHEP09(2024) 012 [2311.13633]

work page arXiv 2024
[23]

Karch and M

A. Karch and M. Youssef,Dissipation in open holography,JHEP12(2025) 157 [2509.14312]

work page arXiv 2025
[24]

Thermal Noise and Stochastic Strings in AdS/CFT

D.T. Son and D. Teaney,Thermal Noise and Stochastic Strings in AdS/CFT,JHEP07 (2009) 021 [0901.2338]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[25]

Brownian motion in AdS/CFT

J. de Boer, V.E. Hubeny, M. Rangamani and M. Shigemori,Brownian motion in AdS/CFT,JHEP07(2009) 094 [0812.5112]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[26]

C. Jana, R. Loganayagam and M. Rangamani,Open quantum systems and Schwinger-Keldysh holograms,JHEP07(2020) 242 [2004.02888]

work page arXiv 2020
[27]

Feynman and F

R. Feynman and F. Vernon,The theory of a general quantum system interacting with a linear dissipative system,Annals of Physics24(1963) 118

work page 1963
[28]

Caldeira and A.J

A.O. Caldeira and A.J. Leggett,Influence of dissipation on quantum tunneling in macroscopic systems,Phys. Rev. Lett.46(1981) 211

work page 1981
[29]

Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics

H. Liu and P. Glorioso,Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics,PoSTASI2017(2018) 008 [1805.09331]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[30]

Hayward,Gravitational action for space-times with nonsmooth boundaries,Phys

G. Hayward,Gravitational action for space-times with nonsmooth boundaries,Phys. Rev. D47(1993) 3275. 24

work page 1993
[31]

Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence

S. de Haro, S.N. Solodukhin and K. Skenderis,Holographic reconstruction of space-time and renormalization in the AdS / CFT correspondence,Commun. Math. Phys.217 (2001) 595 [hep-th/0002230]

work page internal anchor Pith review Pith/arXiv arXiv 2001
[32]

Single-exterior black holes and the AdS-CFT conjecture

J. Louko and D. Marolf,Single exterior black holes and the AdS / CFT conjecture, Phys. Rev. D59(1999) 066002 [hep-th/9808081]

work page internal anchor Pith review Pith/arXiv arXiv 1999
[33]

Pure states in the SYK model and nearly-$AdS_2$ gravity

I. Kourkoulou and J. Maldacena,Pure states in the SYK model and nearly-AdS 2 gravity,1707.02325

work page internal anchor Pith review Pith/arXiv arXiv
[34]

JT gravity as a matrix integral

P. Saad, S.H. Shenker and D. Stanford,JT gravity as a matrix integral,1903.11115

work page internal anchor Pith review Pith/arXiv arXiv 1903
[35]

Saad,Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity, 1910.10311

P. Saad,Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity, 1910.10311

work page arXiv 1910
[36]

Comments on the Sachdev-Ye-Kitaev model

J. Maldacena and D. Stanford,Remarks on the Sachdev-Ye-Kitaev model,Phys. Rev. D 94(2016) 106002 [1604.07818]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[37]

Chaos in AdS$_2$ holography

K. Jensen,Chaos in AdS 2 Holography,Phys. Rev. Lett.117(2016) 111601 [1605.06098]. 25

work page internal anchor Pith review Pith/arXiv arXiv 2016