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arxiv: 2606.20314 · v1 · pith:VLU47Z6Dnew · submitted 2026-06-18 · ✦ hep-th · gr-qc

Macroscopic Black-Hole Remnants in a Nonlocal Field Theory: Towards Hawking Radiation in SFT

Feng-Yin Cheng , Pei-Ming Ho , Wei-Hsiang Shao This is my paper

Pith reviewed 2026-06-26 15:59 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords Hawking radiationblack hole remnantsstring field theorynonlocal field theoryinformation paradoxscrambling timetrans-Planckian modessmearing operator
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0 comments X

The pith

In a nonlocal theory based on string field theory, Hawking radiation from large black holes stops around the scrambling time due to suppression of trans-Planckian modes, resulting in a macroscopic remnant.

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

The paper calculates the time-dependent number of outgoing Hawking particles for a massless scalar field whose interaction with the dynamical black hole is modified by the smearing operator e^{ℓ²□}. It finds that while the standard Planck spectrum at the Hawking temperature appears at early retarded times, the particle number falls to zero shortly after the scrambling time u_scr = 2a log(a/ℓ). This early termination occurs because the nonlocal smearing makes the collapsing shell effectively invisible to high-momentum modes. A reader would care if this mechanism offers a concrete way for black hole evaporation to end without violating unitarity, by leaving behind a remnant whose size is set by the string length.

Core claim

Modifying the scalar field interaction via the smearing operator e^{ℓ²□} in the dynamical black hole background leads to the expectation value ⟨N̂(u)⟩ reproducing the Planck spectrum for u ≪ u_scr but approaching zero shortly thereafter, because the collapsing shell becomes invisible to trans-Planckian modes.

What carries the argument

The smearing operator e^{ℓ²□} that exponentially suppresses trans-Planckian interactions with the black hole.

If this is right

  • Hawking radiation terminates around the scrambling time u_scr ≡ 2a log(a/ℓ).
  • The black hole leaves a macroscopic remnant rather than evaporating completely.
  • The standard Planck spectrum is recovered at early times before the shutoff.
  • This provides an alternative resolution to the black hole information paradox via the remnant.
  • The collapsing shell is invisible to trans-Planckian modes after the scrambling time.

Where Pith is reading between the lines

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

  • The same nonlocal suppression may appear in other string-inspired models of black hole evaporation.
  • Information could remain stored in the macroscopic remnant rather than being carried away by late radiation.
  • Analog systems with modified dispersion relations might exhibit a similar early cutoff in emitted particles.

Load-bearing premise

The smearing operator e^{ℓ²□} applied to the scalar field interaction captures the essential nonlocal suppression of trans-Planckian modes in full string field theory.

What would settle it

A direct computation of the particle spectrum in the full string field theory around a black hole background that shows continued radiation past the scrambling time would falsify the claim.

read the original abstract

We demonstrate that, for a large black hole of radius $a$, Hawking radiation terminates around the scrambling time $u_{\text{scr}} \equiv 2a \log(a/\ell)$ due to the nonlocal, exponential suppression of trans-Planckian interactions inherent in string field theory (SFT). Modifying a massless scalar field's interaction with a dynamical black hole background via the smearing operator $e^{\ell^2\Box}$ (where $\ell$ denotes the string length scale), we calculate the time-dependent number expectation value $\langle \hat{N}(u) \rangle$ of outgoing Hawking particles at retarded time $u$. While the standard Planck spectrum at the Hawking temperature is reproduced at early times ($u \ll u_{\text{scr}}$), the particle number approaches zero shortly after the scrambling time. This early shutoff reflects the property that the collapsing shell becomes effectively invisible to trans-Planckian modes, offering an alternative resolution to the black hole information paradox via a macroscopic remnant.

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

2 major / 0 minor

Summary. The manuscript claims that modifying the interaction of a massless scalar field with a dynamical black-hole background via the smearing operator e^{ℓ²□} (with ℓ the string length) causes the time-dependent outgoing particle number expectation value ⟨N̂(u)⟩ to follow the standard Planck spectrum at the Hawking temperature for retarded times u ≪ u_scr, but to drop to zero shortly after the scrambling time u_scr ≡ 2a log(a/ℓ). This early termination is attributed to the nonlocal exponential suppression of trans-Planckian modes, rendering the collapsing shell effectively invisible to those modes and yielding macroscopic black-hole remnants as an alternative resolution of the information paradox.

Significance. If the chosen smearing operator faithfully represents the essential nonlocal structure of string field theory, the result would supply a concrete dynamical mechanism for the cessation of Hawking radiation around the scrambling time, thereby addressing the information paradox through macroscopic remnants rather than information loss or Planck-scale remnants. The approach is distinctive in its attempt to import SFT-inspired nonlocality into the calculation of time-dependent Hawking flux.

major comments (2)
  1. [Abstract] Abstract: the central claim that ⟨N̂(u)⟩ drops to zero after u_scr rests on the assumption that the operator e^{ℓ²□} applied to the scalar-shell interaction captures the essential nonlocal suppression of trans-Planckian modes present in full SFT. No derivation of this operator from the SFT action, nor comparison with alternative nonlocal regulators, is indicated; without such justification the reported shutoff may be an artifact of the specific regulator rather than a robust prediction.
  2. [Abstract] Abstract: the scrambling time is introduced as u_scr ≡ 2a log(a/ℓ), employing the identical length scale ℓ that parametrizes the smearing operator. It must be shown explicitly that this time scale arises dynamically from the mode-mixing calculation rather than being fixed by the choice of regulator; otherwise the early shutoff is not independently predicted.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and for identifying two key points that require clarification. We address each major comment below, indicating where revisions will strengthen the manuscript without altering its core claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that ⟨N̂(u)⟩ drops to zero after u_scr rests on the assumption that the operator e^{ℓ²□} applied to the scalar-shell interaction captures the essential nonlocal suppression of trans-Planckian modes present in full SFT. No derivation of this operator from the SFT action, nor comparison with alternative nonlocal regulators, is indicated; without such justification the reported shutoff may be an artifact of the specific regulator rather than a robust prediction.

    Authors: The referee is correct that the manuscript does not derive the smearing operator e^{ℓ²□} from the full SFT action; it is introduced as a standard phenomenological model for the nonlocal form factors that arise in SFT. We will revise the introduction and Section 2 to add an explicit justification, citing the SFT literature in which analogous exponential regulators are employed to capture high-mode suppression, and include a short comparison with other regulators (e.g., Gaussian) to argue that the qualitative shutoff is robust. This revision will make the modeling assumption transparent. revision: yes

  2. Referee: [Abstract] Abstract: the scrambling time is introduced as u_scr ≡ 2a log(a/ℓ), employing the identical length scale ℓ that parametrizes the smearing operator. It must be shown explicitly that this time scale arises dynamically from the mode-mixing calculation rather than being fixed by the choice of regulator; otherwise the early shutoff is not independently predicted.

    Authors: We agree that the dynamical origin of u_scr must be shown explicitly. In the Bogoliubov-coefficient integrals the smearing supplies an exponential cutoff whose effect, combined with the near-horizon redshift, produces the factor log(a/ℓ) at the moment when trans-Planckian contributions are suppressed. We will revise the relevant calculation section to display this step in detail, demonstrating that the logarithmic dependence emerges from the mode overlap rather than being inserted by hand. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces the smearing operator e^{ℓ²□} as an explicit modeling assumption for SFT nonlocality on the scalar-shell interaction. It then computes the time-dependent ⟨N̂(u)⟩ from this modified dynamics. The expression u_scr ≡ 2a log(a/ℓ) is presented as the characteristic scale at which the computed suppression becomes effective, with the early-time Planck spectrum and subsequent drop emerging as outputs of the calculation rather than inputs. No step reduces by definition to its own fitted parameters, no self-citation chain is load-bearing for the central result, and the reproduction of the Planck form at u ≪ u_scr followed by shutoff constitutes independent content from the nonlocal regulator. The provided abstract and context contain no quoted reduction of the form 'prediction = input by construction.'

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the modeling choice that the exponential smearing operator faithfully represents SFT nonlocality for trans-Planckian modes near a dynamical horizon; no independent evidence for this modeling choice is supplied in the abstract.

free parameters (1)
  • string length ℓ
    Sets both the strength of the smearing and the definition of the scrambling time; its value is not derived from the calculation.
axioms (2)
  • domain assumption The smearing operator e^{ℓ²□} applied to the interaction term correctly encodes the nonlocal suppression of trans-Planckian physics in SFT.
    Invoked when the authors state that they modify the scalar field interaction via this operator to capture SFT effects.
  • domain assumption The background is a collapsing shell whose geometry remains classical until the scrambling time.
    Required to define the dynamical black-hole background on which the smeared field propagates.

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Reference graph

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