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arxiv: 2605.16619 · v1 · pith:M5BWF7HMnew · submitted 2026-05-15 · ✦ hep-th

Evaporating Black Hole Interior and Complexity Evolution

Nicol\`o Bragagnolo , S. Prem Kumar This is my paper

Pith reviewed 2026-05-20 15:57 UTC · model grok-4.3

classification ✦ hep-th
keywords evaporating black holesJT gravityend-of-the-world braneblack hole complexityPage timereplica wormholessubsystem complexityquenched averaging
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The pith

In evaporating black holes the interior complexity grows linearly to a peak at Page time then decays exponentially.

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

The paper studies the interior of an evaporating black hole in a JT gravity model with an end-of-the-world brane, where evaporation is implemented by entangling the brane states with an auxiliary radiation system. It extracts a geodesic length from the boundary-to-brane two-point function and treats its renormalised value as a measure of the black hole subsystem's complexity. The computation includes non-perturbative contributions from spacetime wormholes and replica wormholes under quenched averaging. Unlike the linear growth followed by a late-time plateau in non-evaporating black holes, the evaporating case shows complexity that grows linearly early, reaches a maximum near the Page time of order the black hole entropy, and then falls exponentially, with the precise shape depending on the dimension of the radiation Hilbert space.

Core claim

The complexity of the black hole subsystem grows linearly at early times, reaches a maximum at the Page time of order the black hole entropy, and then decays exponentially; this evolution depends nontrivially on the dimension of the emitted radiation Hilbert space. The relative fluctuations of the interior length stay small before the Page time but become order one and eventually large later, signalling a loss of self-averaging in which the ensemble average is dominated by rare configurations.

What carries the argument

The renormalised geodesic length extracted from the boundary-to-brane two-point function, interpreted as subsystem complexity and computed with contributions from spacetime and replica wormholes under quenched disorder averaging.

If this is right

  • Complexity reaches its maximum around the Page time rather than continuing to grow or plateauing indefinitely.
  • Relative fluctuations of the interior length become large after the Page time, so that the ensemble-averaged complexity is dominated by rare realisations.
  • The detailed shape of the growth, peak, and decay depends on the dimension of the radiation Hilbert space.
  • Non-perturbative wormhole effects are essential for capturing the transition from self-averaging to non-self-averaging behaviour.

Where Pith is reading between the lines

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

  • If the length proxy holds, then measures of interior complexity may begin to decrease once information starts to be recoverable from the radiation after the Page time.
  • The loss of self-averaging after the Page time could be tested by comparing typical versus averaged realisations in toy models with explicit radiation degrees of freedom.
  • Similar decay behaviour might appear in other evaporating systems once the radiation Hilbert space dimension exceeds the black hole entropy.

Load-bearing premise

The renormalised geodesic length from the boundary-to-brane two-point function can be read as a faithful proxy for the complexity of the black hole subsystem.

What would settle it

A microscopic calculation of the same boundary-to-brane correlator in a non-gravitational random Hamiltonian model that produces a different late-time decay for the extracted length.

read the original abstract

We study the evolution of the interior of an evaporating black hole in a simple model of Jackiw-Teitelboim (JT) gravity with an end-of-the-world (EoW) brane, where evaporation is modeled by entangling the brane's internal states with an auxiliary radiation system. To probe the black hole interior, we consider a geodesic length extracted from a boundary-to-brane two-point function and interpret its renormalised value as a measure of subsystem complexity. Our computation, based on quenched disorder averaging, includes non-perturbative gravitational effects from both spacetime wormholes and replica wormholes, encoding ensemble averaging over the dual random Hamiltonian and brane-state couplings. Unlike non-evaporating black holes where complexity first grows linearly and then plateaus at late times $\sim{\cal O}(e^{S_{\rm BH}})$, we find that complexity evolution of the black hole subsystem in the evaporating case differs drastically, depending nontrivially on the dimension of the emitted radiation Hilbert space. It grows linearly at early times, reaches a maximum at Page time $\sim{\cal O}({S_{\rm BH}})$, and then decays exponentially. We further show that the relative fluctuations of the interior length remain small before the Page time but become of order one and eventually large at later times: this signals a loss of self-averaging, with the ensemble-averaged complexity dominated by rare configurations rather than by typical realisations.

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.

Referee Report

1 major / 1 minor

Summary. The paper examines the interior of an evaporating black hole in a JT gravity model with an end-of-the-world brane, where evaporation is modeled via entanglement with an auxiliary radiation system. It extracts a renormalized geodesic length from the boundary-to-brane two-point function and interprets this as a measure of black-hole-subsystem complexity. Using quenched averaging over random Hamiltonians and brane couplings (incorporating spacetime and replica wormholes), the computation shows linear growth at early times, a maximum near the Page time of order S_BH, and subsequent exponential decay—contrasting with the late-time plateau seen in non-evaporating cases. The abstract explicitly notes that relative fluctuations of the interior length remain small before the Page time but become O(1) and eventually large afterward, indicating loss of self-averaging.

Significance. If the central result holds, the work would be significant for clarifying how evaporation alters complexity evolution in holographic models, particularly the nontrivial dependence on radiation Hilbert-space dimension and the role of non-perturbative wormhole effects. The explicit inclusion of replica wormholes and the discussion of self-averaging loss constitute strengths that allow falsifiable statements about ensemble versus typical behavior.

major comments (1)
  1. Abstract and main text: The central claim that complexity 'decays exponentially' after the Page time is derived from the quenched ensemble average. The manuscript itself reports that relative fluctuations become O(1) and large at late times, so that the average is dominated by rare configurations rather than typical realizations. This directly affects whether the reported decay describes physical single-system evolution or is an averaging artifact; a quantitative comparison between the ensemble result and typical-sample behavior (or an explicit statement of the regime of validity) is required to support the claim of a 'drastic' difference from the non-evaporating plateau.
minor comments (1)
  1. Notation: the symbol for the renormalized geodesic length and its precise relation to the two-point function should be defined once in the main text with an equation reference rather than only in the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. The point raised regarding the interpretation of the ensemble-averaged complexity in the presence of large fluctuations is well-taken, and we have revised the manuscript to provide a clearer statement on the regime of validity of our results.

read point-by-point responses

  1. Referee: Abstract and main text: The central claim that complexity 'decays exponentially' after the Page time is derived from the quenched ensemble average. The manuscript itself reports that relative fluctuations become O(1) and large at late times, so that the average is dominated by rare configurations rather than typical realizations. This directly affects whether the reported decay describes physical single-system evolution or is an averaging artifact; a quantitative comparison between the ensemble result and typical-sample behavior (or an explicit statement of the regime of validity) is required to support the claim of a 'drastic' difference from the non-evaporating plateau.

    Authors: We agree that the exponential decay reported in our work is a property of the quenched ensemble average over random Hamiltonians and brane couplings. The manuscript already notes that relative fluctuations become O(1) at late times, indicating that the average is dominated by rare configurations. This loss of self-averaging is a key result, distinguishing the evaporating case from the non-evaporating one where a plateau is observed in the ensemble. To address the referee's concern, we will revise the abstract and relevant sections of the main text to explicitly state that our claims regarding the exponential decay pertain to the ensemble-averaged complexity, which may not correspond to the typical behavior of a single realization after the Page time. We have added a discussion clarifying the regime of validity: prior to the Page time, self-averaging holds and the ensemble result is representative, while afterward, the large fluctuations signal that individual systems may exhibit different evolution, potentially remaining closer to a plateau or showing different dynamics. Although a quantitative comparison to typical-sample behavior would be desirable, it would require computing the variance or higher moments of the complexity distribution or employing annealed averaging, which goes beyond the quenched averaging and replica wormhole techniques employed in this work. We believe the explicit statement of this regime, combined with the already-reported fluctuation analysis, sufficiently supports the claim of a drastic difference in the ensemble sense, which is the natural setting for our holographic model. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation is model computation with explicit interpretive assumption

full rationale

The paper computes the renormalized boundary-to-brane geodesic length via quenched averaging over random Hamiltonians and brane couplings in JT gravity with EoW brane, including replica wormholes. This length is then interpreted as subsystem complexity. The reported linear growth, peak at Page time, and subsequent exponential decay follow from the explicit gravitational path integral evaluation in the evaporating setup. No step reduces the final curve to a fitted parameter, self-definition, or self-citation chain; the loss of self-averaging after Page time is stated as an observed feature of the ensemble, not hidden in the derivation. The central claim therefore rests on independent holographic calculation rather than tautological input.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The model rests on standard assumptions of JT gravity and holographic duality plus the interpretive mapping of geodesic length to complexity; no explicit free parameters or new entities beyond the EoW brane are introduced in the abstract.

axioms (1)
  • domain assumption Jackiw-Teitelboim gravity with an end-of-the-world brane is a valid effective description for the evaporating black hole interior
    The entire setup is built on this model choice stated at the opening of the abstract.
invented entities (1)
  • End-of-the-world brane no independent evidence
    purpose: To model the black hole interior and its entanglement with radiation during evaporation
    The brane is introduced as part of the model; independent evidence is not provided in the abstract.

pith-pipeline@v0.9.0 · 5786 in / 1481 out tokens · 43746 ms · 2026-05-20T15:57:27.857043+00:00 · methodology

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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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We study the evolution of the interior of an evaporating black hole in a simple model of Jackiw-Teitelboim (JT) gravity with an end-of-the-world (EoW) brane... the renormalised value as a measure of subsystem complexity... replica wormholes... I(n) turns out to be a non-monotonic function of e^{S_BH}/k... E[X/(X+1)^2] with X~Poisson(e^{S_BH(t)-S_rad(t)})

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    the ensemble-averaged complexity of the evaporating black hole turns over near Page time and then decreases exponentially... relative fluctuations... become order one... loss of self-averaging

What do these tags mean?
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The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
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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

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